ETT-311 SIGEx
Lab Manual
Signals & Systems Experiments with Emona SIGEx
Volume 1
ETT-311 SIGEx
Lab Manual
Signals & Systems Experiments with Emona SIGEx
Volume 1
Signals & Systems Experiments with Emona SIGEx
Volume S1 - Fundamentals of Signals & Systems
Authors: Robert Radzyner PhD
Carlo Manfredini B.E., B.F.A.
Editor:
Carlo Manfredini
Issue Number: 1.2
Published by:
Emona Instruments Pty Ltd,
78 Parramatta Road
Camperdown NSW 2050
AUSTRALIA.
web: www.emona-tims.com
telephone: +61-2-9519-3933
fax: +61-2-9550-1378
Copyright © 2011 Emona TIMS Pty Ltd and its related entities.
All rights reserved. No part of this publication may be
reproduced or distributed in any form or by any means, including
any network or Web distribution or broadcast for distance
learning, or stored in any database or in any network retrieval
system, without the prior written consent of Emona Instruments
Pty Ltd.
For licensing information, please contact Emona TIMS Pty Ltd.
NI, NI ELVIS II/+, NI LabVIEW are registered trademarks of
National instruments Corp.
The TIMS logo
Emona TIMS Pty Ltd
Printed in Australia
is a registered trademark of
EMONA SIGEx Lab Manual Volume 1
Contents
Introduction
(i)
How to handle, install and power up SIGEx™:
ESD Warning, Unpacking, Handling, Set-up and ESD Preventative Measures
An introduction to the NI ELVIS II/+ test equipment
S1-01
An introduction to the SIGEx experimental add-in board
S1-02
SIGEx board circuit modules
NI ELVIS functions
SIGEx Soft Front Panel descriptions
Special signals – characteristics and applications
S1-03
Pulse sequence speed throttled by inertia
Isolated step response of a system
Isolated pulse response of a system
Sinewave input
Clipping
Systems: Linear and non-linear
S1-04
Conditions for linearity
The VCO as a system
A feedback system
Testing for additivity
Frequency response
Unraveling convolution
S1-05
Unit pulse response
The superposition sum
A rectified sinewave at input
A sinewave input
Mystery applications
Integration, correlation & matched filters
Auto-correlation function of PRBS sequences
Cross-correlation function of PRBS sequences
ACF & matched filtering
Determining impulse response using input/output correlation
Matched filtering using “integrate & dump” circuitry
S1-06
Exploring complex numbers and exponentials
S1-07
Complex numbers and complex functions
Exponential functions
Build a Fourier series analyzer
S1-08
Constructing waveforms from sine & cosine
Computing Fourier coefficient
Build a manually swept spectrum analyzer
Analyzing a square wave
Spectrum analysis of various signal types
S1-09
Spectrum of impulse trains
Spectrum of filtered impulse trains
Duty cycle & sampling
Sync pulse train
Spectrum of PN sequences
Analog noise generation (AWGN)
Non-linear processes
Time domain analysis of an RC circuit
S1-10
Step response of the RC
Impulse response of the RC
Exponential pulse response of the RC
Synthesising transfer functions
Poles and zeros in the Laplace domain
S1-11
System with feedback only – allpole
Impulse response of LTIC systems
Feedback and feedforward – poles & zeros
Allpass circuit
Critical damping & maximal flatness
Sampling and Aliasing
S1-12
Through the time domain – PAM, Sample & Hold
Through the frequency domain
Aliasing and the Nyquist rate
Multi-frequency impulse spectrum
Uses of undersampling in Software Defined Radio
Getting started with analog-digital conversion
PCM encoding & quantization
PCM decoding & reconstruction
Frame synchronisation & quantization noise
S1-13
Discrete-time filters with FIR systems
S1-14
Graphical plotting of response from poles & zeros
Notch filter creation using two-delay FIR
Poles and zeros in the z plane with IIR systems
S1-15
Relating roots to coefficients in the quadratic polynomial
IIR without feedforward – a second order resonator
IIR with feedforward – second order filters
Viewing spectrum with broadband noise & FFT
Dynamically varying the poles & zeros
Using the “Digital Filter Design” toolkit
Discrete-time filters – practical applications
S1-16
Dynamic range at internal nodes
Transposed Direct form 2 IIR
Implementations with high sampling rates
Appendix A: SIGEx Lab to Textbook chapter table
S1-A
Appendix B: Quickstart guide to using SIGEx
S1-B
References
Electrostatic Discharge Sensitive Product
CAUTION: The inputs/outputs of this product can be damaged if subjected to
Electrostatic Discharge (ESD).
To prevent damage, industry-standard ESD prevention measures must be
employed during installation, maintenance, and operation.
Refer to page v for detailed information.
Introduction
This experiment manual is designed to provide a practical “hands-on”, experiential, lab-based
component to the theoretical work presented in lectures on the topics typically covered in
introductory “Signals and Systems” courses for engineering students.
Whilst it is predominantly focused on all electrical engineering students, this material is not
exclusively for electrical engineers. With an understanding of differential equations, algebra of
complex numbers and basic circuit theory, engineering students in general can reinforce their
understanding of these important foundational principles through practical laboratory course work
where they see the “math come alive” in real circuit based signals. This provides a foundation for
further study of communications, control, and systems engineering in general.
Students take responsibility for the construction of the experiments and in so doing learn from their
mistakes and consolidate their knowledge of the underpinning theory, which at times is particularly
abstract and hard to grasp for these early engineering students. They are not constrained by the
software and need to be systematic in debugging their own systems when results do not meet their
expectations.
The common reaction of early students when confronted with “complex analysis” is one of confusion
and regression to “rote-learning” in order to survive the examination process. This manual has as its
predominant aim to create real, “hands-on” implementation of the theory, in such a way that the
student can directly articulate and connect the mathematical abstractions with real world
implementations. It is a journey of personal discovery where the motto is “why is it so ?”
The use of “modeling” is the fundamental tool in this, and other NI ELVIS-based EMONA boards (such
as DATEx, FOTEx and HELEx), and it has been shown that experimenting with scaled models of real
world systems allows students early-on to get a tangible “feel” for principles that they may later
utilize in real world commercial workplace environments. As well, students tend to “believe” results
from “real hardware” rather than from software simulations, and this supports their “learning by
doing”.
The authors sincerely hope that students using this equipment and guided by this manual will complete
with a sense that complex numbers and systems analysis “makes sense” and is somehow more “real” and
applicable to real world problems. In this way they may successfully use these principles in solutions to
future problems they will encounter.
Carlo Manfredini
Sydney, November 2013
How to handle, install and power up SIGEx™ with NI ELVIS II/+
ESD Warning – ESD Sensitive Product
Caution Although this product has been designed to be as robust as possible, ESD
(Electrostatic Discharge) can damage or upset this product. This product must be
protected at all times from ESD. Static charges may easily produce potentials of several
kilovolts on the human body or equipment, which can discharge without detection.
Industry-standard ESD precautions must be employed at all times.
The Emona ETT-311 board is designed and intended for use as an experimental platform
for hardware or software in an educational/professional laboratory environment. To
facilitate usage, the board is manufactured with its components and connecting traces
openly exposed to the operator and the environment. As a result, ESD sensitive (ESDS)
components on the board, such as the semiconductor integrated circuits, can be damaged
when exposed to an ESD event. To indicate the ESD sensitivity of the Emona ETT-311
board, it carries the symbol shown at left.
Unpacking, Transporting, and Storage
When unpacking the Emona ETT-311 board from its shipping carton, do not remove the board from the
antistatic packaging material until you are ready to complete the installation. Before unwrapping the
antistatic packaging, discharge yourself by touching a grounded bare metal surface, touching an
approved anti-static mat, or wearing an ESD strap. When transporting or storing the Emona ETT-311
board, first place it in an antistatic container or packaging.
Handling and Setup
Handling the Emona ETT-311 board can damage the board components if ESD prevention measures are
not applied. Before handling or setup, equalize your potential with the board by touching one of the
integrated ESD discharge pads. During all handling and setup, ESD prevention measures must be
applied. In addition, the Emona ETT-311 board should be handled by the edges. Touching exposed
circuits, components or connectors could result in an ESD event. When setting up Emona ETT-311
board, observe the following guidelines to minimize the potential impact of ESD:
• Patch the desired experiment blocks using 2mm patch cords.
• Set switches and other controls to initial settings.
• Ensure the NI ELVIS is powered correctly by plugging the NI ELVIS AC/DC power supply brick into
an appropriate AC outlet.
• Move the board power switch ON.
Operation
When operating the Emona ETT-311 board, ESD can cause upset as well as damage to the
board components. Therefore, apply ESD prevention measures whenever operating the Emona
ETT-311 board. In addition, observe the following guidelines:
• Do not touch exposed traces or components on the board while the board is powered on.
• Exercise caution when manipulating switches, buttons, knobs, and other controls while the
board is powered on.
ESD Prevention Measures
ESD prevention measures focus on reducing or eliminating the build-up of static charge that
may result in an ESD event that could damage or upset sensitive electronics. To minimize the
potential for an ESD event, implement the following measures:
• Perform all work at an approved work station.
• Use an approved antistatic mat to cover your work surface.
• Wear a conductive wrist strap attached to the antistatic mat and a good earth ground.
• Before handling or beginning work, equalize your potential with the board by touching one of
the “GND” pads.
Handling SIGEx
When holding SIGEx, always
hold the circuit board by the
edges, as illustrated.
Hold SIGEx by the edges
How not to hold SIGEx
Ensure NI ELVIS II/+ Prototype Board Power OFF
Before installing the SIGEx add-in module in the
NI ELVIS II/+ PROTOTYPE PCI SLOT, always
check the PROTOTYPING BOARD POWER switch is
in the
OFF position.
Ensure Prototype Board Power is OFF
Installing SIGEx on
NI ELVIS II/+
When installing the SIGEx
add-in module in the NI
ELVIS II/+ PROTOTYPE
PCI SLOT, always
carefully check the
alignment is correct
before pushing SIGEx into
position.
Carefully align SIGEx with the
NI ELVIS socket
Carefully push SIGEx into position
Check for correct bracket alignment
Power up SIGEx
After SIGEx is correctly positioned, turn the NI
ELVIS II/+ Prototyping Board Power switch ON.
Turn NI ELVIS Prototype Board Power ON
Before removing SIGEx™ from the NI ELVIS II/+
Ensure NI ELVIS II/+ Prototype Board Power is OFF
Before removing the SIGEx add-in module from the
NI ELVIS II/+ PROTOTYPE PCI SLOT, always turn
the PROTOTYPING BOARD POWER switch OFF.
ALWAYS STORE THE SIGEx BOARD IN A ESDSAFE PLASTIC BAG – AS SUPPLIED WITH
ORIGINAL PACKAGING.
Ensure Prototype Board Power is OFF
before removing SIGEx
SIGEx ™ QUICK START - INSTALLATION AND OPERATION
The following guide outlines the steps required to install and use the NI ELVIS platform and the
Emona SIGEx add-in module.
1. Installation of NI components
• Install LabVIEW on your PC – follow NI LabVIEW Installation Guide
• Install NI ELVIS software – follow the NI ELVIS installation Guide
2. Installation of Emona SIGEx components
• Install Emona SIGEx software using the SIGEx CD-ROM supplied. Refer to page 21 of this User
Manual for detailed installation instructions. The installation will include the SIGEx SFP, the
User Manual and Lab Manual PDF files.
3. Setting-up the NI hardware components
• Make all power and interface connections between the PC, NI DAQ and NI ELVIS
• Power-up NI DAQ and NI ELVIS; Launch the NI ELVIS LAUNCHER software. This will most
likely occur automatically when NI ELVIS is detected.
4. Set-up of the Emona SIGEx add-in module
• Confirm the NI ELVIS’ “PROTOTYPING BOARD POWER” switch is OFF.
• Holding the SIGEx board by its outer edges, position it on the top of the NI ELVIS platform.
Carefully align the SIGEx circuit board with the NI ELVIS PCI socket and slide into position.
5. Operation
• Power-up the SIGEX add-in module by switching the NI ELVIS’ “PROTOTYPING BOARD
POWER” switch to ON. Confirm the the SIGEx board’s 3 red power LEDs at the left hand side
are lit.
• Patch together experiments following the instructions of the SIGEx Lab Manual. The SIGEx kit
includes all equipment required: patching wires and oscillscope leads.
• The NI ELVIS platform powers the SIGEx add-in module, provides the test instruments and
interface between NI LabVIEW and the SIGEx add-in module.
6. SIGEx front panel conventions
• The SIGEx add-in module follows strict labelling conventions to assist the student in building
and exploring experiments. For details see page "iii" of this User Manual. In brief:
Terminals with a CIRCLE label only take analog signals. (typically + 2V)
Terminals with a SQUARE label only take digital type signals. ( +5V TTL-level)
INPUT terminals are always on the LEFT of each functional block.
OUTPUT terminals are always on the RIGHT of each functional block.
7. Power down and unplug SIGEx
• Always switch the NI ELVIS’ “PROTOTYPING BOARD POWER” switch to OFF before removing
the SIGEx add-in module.
• Please handle the SIGEx add-in module by the outer edges of the circuit board and store in the
anti-static bag provided.
THIS PAGE IS INTENTIONALLY BLANK
Experiment 1 – An introduction to the NI ELVIS II/+ laboratory
equipment
Preliminary discussion
The digital multimeter and oscilloscope are probably the two most used pieces of test equipment
in the electronics industry. The bulk of measurements needed to test and/or repair electronics
systems can be performed with just these two devices.
At the same time, there would be very few electronics
laboratories or workshops that don’t also have a DC Power
Supply and Function Generator. As well as generating DC
test voltages, the power supply can be used to power the
equipment under test. The function generator is used to
provide a variety of AC test signals.
Importantly, NI ELVIS II has these four essential pieces of
laboratory equipment in one unit (and others). However,
instead of each having its own digital readout or display (like
the equipment pictured), NI ELVIS II sends the information
via USB to a personal computer where the measurements are displayed on one screen.
On the computer, the NI ELVIS II devices are called “virtual instruments”. However, don’t let
the term mislead you. The digital multimeter and scope are real measuring devices, not software
simulations. Similarly, the DC power supply and function generator output real voltages.
As well as the instruments mentioned above, the NI ELVIS II has available eight analogue
inputs and two analog outputs which can be controlled and written to by our LabVIEW program
and the input readings processed and displayed on screen. This allows for the creation of many
more custom "virtual instruments" which may be required in a particular experimental setup.
The experiments in this manual make use of several of the available analogue inputs as well as
several digital inputs and outputs which, in conjunction with the SIGEx board, are able to
implement two groups of programmable gain amplifiers for use throughout this manual.
Rather than utilising several independent instruments from the NI ELVIS as does the other
EMONA plug-in accessory boards (such as EMONA DATEx Telecoms-Trainer and the EMONA
FOTEx Fiber optics trainer), these instruments are all merged into one full-screen virtual
instrument for the SIGEx board known as the SIGEx Main soft front panel (SFP). With an easyto-use tabbed layout, each experiment has its requisite instrumentation grouped within tabs by
experiment.
1-2
© 2011 Emona Instruments
Experiment 1 – An introduction to the NI ELVIS II/+ test equipment V1.2
When an NI ELVIS1 unit is connected to a PC it will automatically run the Instrument Launcher
panel as shown below:
Figure 1: NI ELVIS II/+ Instrument Launcher panel
This panel gives the user access to each individual instrument. Several of these independent
instruments are used by SIGEx experiments. These are the FUNCTIONS GENERATOR (FGEN),
the DYNAMIC SIGNAL ANALYSER (DSA) and at times the SCOPE (Scope).
When using NI ELVIS with the EMONA SIGEx board to conduct signals and systems
experiments the user will run the SIGEx Main SFP VI shown below:
Figure 2: SIGEx Main SFP
1
Throughout this manual, NI ELVIS II & II+ are referred to, however the SIGEx board
and software work equally well on the NI ELVIS I platform, with the NI ELVIS I
FUNCTION GENERATOR used in manual mode ONLY, and NI ELVIS in BYPASS mode.
Experiment 1 – An introduction to the NI ELVIS II/+ test equipment
© 2011 Emona Instruments
1-3
There are 19 TABS for use with the experiments in this Volume of the manual.
These instruments take their signals directly from the SIGEx board via the EMONA
ETT-040 Universal Base Board, into the ELVISmx circuitry, and after processing by LabVIEW
are displayed on screen as required.
The combination of the LabVIEW programmability of the NI ELVIS unit as well as the numerous
analog and digital inputs and outputs available make it convenient to create customised
instrumentation for use in real world hands-on experimentation. The EMONA SIGEx board is a
good example of this integration of available hardware and the software control.
1-4
© 2011 Emona Instruments
Experiment 1 – An introduction to the NI ELVIS II/+ test equipment V1.2
Experiment 2 – An introduction to the EMONA SIGEx
experimental add-in board for NI ELVIS
Preliminary discussion
The experiments possible with the EMONA SIGEx board bring together worlds of
mathematical theory and practical implementation. We are able to explore, in a hands-on
manner, the representation of physical processes by mathematical models and test and measure
the benefits and limitations of such models. We explore the complementarity of the time and
frequency domains and practice thinking and theorizing in both. Through measurements,
calculations and observations we are able to consolidate our understanding of these domains.
The SIGEx board customizes the instrumentation available on the NI ELVIS to create
experiment-specific instruments which can be used to create many different circuit structures.
As well, the ability to programmatically control, measure and automate our measurements using
LabVIEW bring us closer to real-world practices of system control and monitoring.
Although the principles of being studied date back several centuries their application in real
world devices is continually being explored and implemented. The instrumentation used has
changed substantially however the rigorous nature of the mathematical process remains the
same and is a skill which is best learned in a hands-on manner.
By implementing the many mathematical model and theorems in real hands-on circuit based
experiments, the student reinforces and actualizes their understanding of these principles to
create a solid foundation for future learning.
An important skill for the engineer and scientist is the ability to take rigorous and precise
measurements, often repetitively, in order to study the phenomena at hand. The EMONA SIGEx
Signals & Systems Experimenter (ETT-311) provides an abundance of opportunities to learn and
practice experimental methodology in a variety of related topics which are common ground for
engineering students of several disciplines.
2-2
© 2011 Emona Instruments
Experiment 2 – An introduction to the SIGEx experimental add-in board V1.2
The experiment
For this experiment you will familiarize yourself with the various instruments available on the
SIGEx board and how they are used.
It should take you about 10 minutes to read this experiment and explore these functions.
Pre-requisites:
You should have completed the introductory chapter 1 so that you’re familiar with the equipment
setup and capabilities.
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Experiment 2 – An introduction to the SIGEx experimental add-in board
© 2011 Emona Instruments
2-3
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should observe
the USB light turn on (top right corner of ELVIS unit).The PC may make a sound to indicate that
the ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that
all three power LEDs are on. If not call the instructor for assistance.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with
the NI ELVIS that you’re using.
9.
You’re now ready to work with the NI ELVIS/SIGEx bundle.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the
STOP button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the
top of the window. This will allow the program to conduct an orderly shutdown and close the
various DAQmx channels it has opened.
Ask the instructor to check
your work before continuing.
2-4
© 2011 Emona Instruments
Experiment 2 – An introduction to the SIGEx experimental add-in board V1.2
EMONA SIGEx board overview
The SIGEx board is a collection of independent circuit blocks which each implement a single
simple function. No one block is a complete experiment, however several blocks together can
implement a wide variety of different experiments. The block inputs and outputs are patched
together with 2 mm patching leads according to the block diagram as documented in this Lab
Manual or from the many texts available on this topic.
EMONA SIGEx board layout
This chapter discusses the functionality of each module briefly and further details such as
specifications are contained in the EMONA SIGEx User Manual.
NI ELVIS II/ SIGEx bundle
Experiment 2 – An introduction to the SIGEx experimental add-in board
© 2011 Emona Instruments
2-5
SIGEX board circuit modules
Sequence Generator
The SEQUENCE GENERATOR provides a source of periodic data
streams which are output as 5V logic and bipolar level signals.
DIP switches allow the selection of 4 different streams.
A periodic SYNC pulse is output once per frame.
The module is clocked by a single input logic level clock. This will
typically come from the PULSE GENERATOR or FUNCTION
GENERATOR/SYNC outputs.
The state of the DIP switches at any time is displayed on the SIGEx
SFP along with a description.
Limiter
The LIMITER amplifies an incoming signal with DIP switch selectable
gain levels and to a fixed level, creating an amplitude limited output
signal.
It is typically used with bipolar analog sinusoidal signals or bipolar line
coded data streams.
RC Network
The RC NETWORK provides R and C elements which can be arranged as
either an RC circuit which acts as a LPF, or as a HPF.
The elements are floating and one end needs to be connected to GND.
Rectifier
The Rectifier provides half wave rectification of an incoming signal with
a non ideal diode component which has a forward voltage drop.
This is typically used with sinusoidal signals.
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© 2011 Emona Instruments
Experiment 2 – An introduction to the SIGEx experimental add-in board V1.2
Multiplier
The Multiplier provides four quadrant multiplication of two analog input
signals. Its overall gain is approximately unity and it is used to model
any multiplication process that may occur in a block diagram.
Integrate & Dump/Hold
Both Integrate and Dump as well as Integrate and Hold is available in
this circuit block. Usually clocked by the bit clock of an incoming
sequence, it is used to integrate over a single period of a waveform in
correlation and filtering functions.
Baseband Low Pass Filter
This LPF has a 4th order Butterworth response and serves both as a
“system under investigation” and for general filtering functions.
PCM Encoder
This module implements PCM encoding of a single analog signal. It
outputs an 8 bit frame along with a periodic Frame Sync pulse.
It can be used with both DC signals as well as sinusoids and serves to
allow specific investigation of the encoding process.
It has a maximum sampling rate of 2.5ksps ( 20kbps PCM data stream),
and so can be used with signal frequencies below the Nyquist limit of
1.25kHz.
PCM Decoder
This module implements PCM decoding of an 8 bit PCM digital data
stream from the PCM Encoder.
The Frame Sync is necessary to achieve synchronization and there is no
reconstruction filter on the output to allow investigation of quantization
issues.
Experiment 2 – An introduction to the SIGEx experimental add-in board
© 2011 Emona Instruments
2-7
Tuneable Low Pass Filter
This module is an adjustable LPF. It implements a 8th order Elliptic
filter with an adjustable corner frequency. The output signal level is
also adjustable, and it can accept analog and TTL level digital signals.
There is no anti-aliasing filter on the input so users need to be aware of
the bandwidth of their incoming signal.
Integrators
These 3 independent circuits are simple integrator circuits with a common DIP-switchselectable integration rate. They are used for continuous time integration ( unlike the
Integrate & Dump/Hold unit which operates over a single period only.)
They are used in Laplace domain experiments.
The DIP switch settings is displayed in the SIGEx SFP along with the approximate integration
rate.
Unit delays with Sample & Hold
The Sample & Hold is an analog sampler circuit which holds the sampled value for a single period
of the incoming TTL level clock signal. The unit delays are similar in that they hold the incoming
analog value at their input for a single clock period.
All 4 units share a common clock signal.
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© 2011 Emona Instruments
Experiment 2 – An introduction to the SIGEx experimental add-in board V1.2
Triple and dual input adders
There are 3 adder sections. Two identical triple input adder sections and a dual input adder.
The triple input adders, a & b, have adjustable gains. These gains are adjusted via the SIGEx
SFP and are typically used to implement the taps in feedback and feedforward systems.
The dual input adder has unity gain and is used for general purpose addition.
The GAIN ADJUST knob is read by the SIGEx SFP software and can be used to manually
adjust adder gains.
Experiment 2 – An introduction to the SIGEx experimental add-in board
© 2011 Emona Instruments
2-9
NI ELVIS functions blocks available on the SIGEx board
Pulse generator / Digital out
This module makes available the built in Pulse Generator from NI ELVIS
which has a very broad range of frequency and duty cycle control. This
is controlled from the SIGEx SFP and is usually used to provide digital
clock signals to experiments.
D-OUT-0 is a single digital output line which is available but currently
unused in experiments.
Function generator
This module makes available the built in Function Generator from NI
ELVIS which is a multifunction generator, with variable signal types,
variable amplitude and variable frequency. It is controlled via its own
instrument panel which available from the NI ELVIS Instrument
Launcher panel.
Analog out
This module makes available the built in dual analog outputs from the
DACs.
These outputs are controlled from various SIGEx experiment TABs and
can be modified to create any periodic waveforms required.
2-10
© 2011 Emona Instruments
Experiment 2 – An introduction to the SIGEx experimental add-in board V1.2
EMONA SIGEx Soft Front Panel (SFP) descriptions
The EMONA SIGEx Soft Front Panel serves both to control elements of the SIGEx hardware,
as well as provide experiment specific measuring instrumentation in a handy, experiment-perTAB based layout.
The layout is arranged so as to fit on screen easily with all parameters in view.
The source code VI’s are provided on the SIGEx CD so that users can modify and customize the
SFP arrangement and functionality if required.
SIGEx is designed for university and college users and access to the LabVIEW “Digital Filter
Design” toolkit is expected for full functionality to be available.
EMONA SIGEx Soft Front Panel
ADDER gain entry panel
The triple input adders have variable gains which are set from the entry controls on the SFP.
These gains can also be set programmatically as is done in several experiment TABS. The
onscreen gains are transferred to the hardware automatically and continuously.
Coefficient selector panel
The position of the onboard GAIN ADJUST knob can be interpreted as a range of values set to
a particular adder gain control. The radio button panel is used to select a particular gain control,
or none. The center value and step size of each increment from the GAIN ADJUST knob must
Experiment 2 – An introduction to the SIGEx experimental add-in board
© 2011 Emona Instruments
2-11
also be set. This allows either a broad range of values or a narrow focused range of values to be
adjustable via the knob.
Pulse Generator panel
In the panel the frequency and duty cycle of the PULSE GENERATOR block can be set. As well
the spare D-OUT-0 line can be toggled.
SG Sequence type and Integrator Gain readouts
These readouts mimic the selection of the onboard DIP switches and the text briefly describes
the signal type selected for convenience. Details of signals in the SIGEx User Manual.
Analog OUT viewer
This graph indicator displays the actual signal currently being output from the ANALOG OUT
terminals from the DACs. These vary depending on the experiment selected, and this readout is
convenient when SCOPE channels are being used for other signals.
SCOPE Trig level, trig slope, triggered LED, trig select, timebase etc
These controls are for the SFP scopes embedded in various experiment TABs.
Trig level sets the voltage level the trigger looks for. Usually set to 0 or 1 V
Trig slope allows triggering on either the positive or negative edge of a signal.
Triggered LED is ON (green) when a trigger point , as defined above, is detected.
Trig select determines which channel acts as the trigger.
Timebase varies the amount of real signal time to be captured and displayed. Total time
displayed is selectable.
RUN/STOP enables halting of the scope display for close inspection.
Y autoscale ON: enables toggling of the Y axis autoscale function for stable signal viewing with
varying amplitude signals.
This built in scope is a convenient, customized signal display for use in specific experiments.
Spectrum display is also available in certain TABs when required.
Note for ELVIS 2+ users
Due to the independent scope instrumentation available in the ELVIS 2+ it is possible to
simultaneously use the independent scope from the Instrument Launcher panel as well as viewing
the signals in the TAB based scope display.
The Dynamic Signal Analyser (DSA), a spectrum analyser, can also be used, but not at the same
time as the independent scope.
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© 2011 Emona Instruments
Experiment 2 – An introduction to the SIGEx experimental add-in board V1.2
Laboratory Experiment ‘X’ TABS
Each experiment in the SIGEX Lab Manual has its own SFP TAB if required.
Select the TAB as required and the appropriate instrumentation will be displayed. Labs 3 to 18
have TABs available.
Some graphs also have cursors enabled. These are very useful for taking accurate & quick
measurements.
HINT: Right-clicking on a graph will display extra available options you can use. Different
options are available when you right-click while the SFP is not running eg: setting a graph from
linear to log display is done while SFP is not running.
Digital Filter Design TAB
This TAB makes available several of the digital filter design features from the toolkit in one
handy display. The user should select a filter type from which the transfer function will be
calculated. The coefficients from the transfer function are extracted and setup on the SIGEx
hardware as the triple ADDER gains when required by the user. This can be seen on the SFP.
The calculated responses are displayed onscreen.
To view the actual signals and responses from the hardware, switch to a TAB which contains a
scope and FFT, for example the ZOOM FFT TAB, whilst inputting an appropriate source signal.
Note that SIGEx is limited to implementing only up to 2nd order filters. A red “error” LED will
highlight when orders >2 are selected.
Experiment 2 – An introduction to the SIGEx experimental add-in board
© 2011 Emona Instruments
2-13
ZOOM FFT TAB
This TAB contains a scope display, a spectrum display, and a zoomable view of the FFT display.
This TAB is a general purpose display TAB and is not associated with any particular experiment.
The FFT display is a 1000 point display, and the “# samples” control allows the user to select a
zoom window from 0 to 1000 points to display alongside. The “zoom region” slider enables the
zoom region to be selected from the overall 1000 point FFT display.
PZ PLOT TAB
This TAB uses components from the Digital Filter Design toolkit to calculate and plot the poles
and zeros on the unit circle from the coefficients of the transfer function as it is set up on the
SIGEx board.
The coefficient values from the triple ADDER gain controls are read by this TAB and plotted as
the equivalent poles and zeros in real time.
This is especially interesting when the coefficients are being varied manually by the onboard
GAIN ADJUST knob, in that the user can see the poles and zeros moving about the unit circle
in real time alongside the hardware.
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© 2011 Emona Instruments
Experiment 2 – An introduction to the SIGEx experimental add-in board V1.2
Experiment 3 – Special signals – characteristics and applications
Achievements in this experiment
Time domain responses are discovered: step and impulse responses as paradigms for the
characterization of system inertia; sinewaves were used as probe signals; clipping was applied to
the recovery of a digital signal.
Preliminary discussion
Bandwidth is a term that has been in the engineering vocabulary for many decades. Its usage
has extended over time, especially in the context of digital systems. It has become
commonplace now to mean information transfer rate, and all Internet users know that
broadband stands for fast, and better. There are highly competitive markets demanding top
performance – ever higher speed whilst maintaining a low probability of corruption. However, as
speed is increased, obstacles emerge in the form of noise, interference and signal distortion. At
the destination these limitations become digital errors, resulting in pixellated images, and audio
breaking up.
Engineers involved in the design of these systems must assess the suitability of numerous
components and sub-units e.g. adequate speed of response ?, too noisy, distorted? They will
need to benchmark the behaviour of subsystem. The procedures that are used for modelling and
testing must be universally accepted.
The most important consideration affecting the speed of a digital signal is the switching
process to produce a change of state. The switching time can never be instantaneous in a
physical system because of energy storage in electronic circuitry, cabling and connecting
hardware. This energy lingers in stray capacitance and inductance that cannot be completely
eliminated in wiring and in electronic components. The effect is just like inertia in a mechanical
system.
A universal procedure is needed to characterize, measure and specify ‘inertia’. Various
paradigms have become established over many years of application. One of these is the step
response. For this reason, the step function has become one of the special signals in systems
engineering.
There are other signal types of importance. The sinusoid or sinewave heads the list of the
range of applications. There are many others, including the impulse function, ramps,
pseudonoise waveforms and pseudorandom sequences, chirp signals.
This Lab has its focus on signals that are most needed for basic operations. Other signals will
be introduced progressively in succeeding labs.
Figure 1: step, impulse and sinusoid signals
3-2
© 2011 Emona Instruments
Experiment 3 - Special signals – characteristics and applications V1.2
The experiment
In Part 1 we investigate how signals are distorted when a system's response is affected by
inertia, and discover signals that are useful for probing a system's behaviour.
In Part 2 we introduce the sinewave, and observe how the systems investigated in Part 1
respond to inputs of this kind.
Signals that have been subjected to amplitude limiting, also known as clipping, are commonly
encountered when excessive amplification is used, such as in audio systems, resulting in overload
distortion. In Part 3 we generate clipped signals and examine a useful application of clipping.
As this experiment is a process of discovery, we will name the blocks which represent the
channel “ System Under Investigation” until we have familiarized ourselves with their actual
characteristics.
It should take you about 45 minutes to complete this experiment.
Pre-requisites:
Familiarization with the SIGEx conventions and general module usage. A brief review of the
operation of the SEQUENCE GENERATOR module. No theory required.
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Figure: TAB 3 of SIGEx SFP
Experiment 3 - Special signals – characteristics and applications
© 2011 Emona Instruments
3-3
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should observe
the USB light turn on (top right corner of ELVIS unit).The PC may make a sound to indicate that
the ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that
all three power LEDs are on. If not call the instructor for assistance.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with
the NI ELVIS that you’re using.
9.
You’re now ready to work with the NI ELVIS/SIGEx bundle.
10.
Select the EXPT 3 tab on the SIGEx SFP.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the
STOP button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the
top of the window. This will allow the program to conduct an orderly shutdown and close the
various DAQmx channels it has opened.
Ask the instructor to check
your work before continuing.
3-4
© 2011 Emona Instruments
Experiment 3 - Special signals – characteristics and applications V1.2
Part 1a – Pulse sequence speed throttled by inertia
In this set of exercises we continue the digital theme introduced above and explore the
behaviour of signals in transit through a channel that has a limited speed of switching.
SEQUENCE
SOURCE
S.U.I.
Figure 1a: block diagram of the setup for observing the effect of
a system (SUI) on a digital pulse sequence.
Figure 1b: SIGEx model for Figure 1a.
11.
Patch up the model in Figure 1b. The settings required are as follows:
PULSE GENERATOR: FREQUENCY=1000; DUTY CYCLE=0.50 (50%)
SEQUENCE GENERATOR: DIP switch to UP:UP for a short sequence.
SCOPE: Timebase 10ms; Rising edge trigger on CH0; Trigger level=1V
Set up the CH0 scope lead to display the LINE CODE output of the SEQUENCE GENERATOR
12.
Measure the smallest interval between consecutive transitions . Compare this with the
duration of one period of the clock by moving the scope lead to view the SEQUENCE
GENERATOR CLK input from the PULSE GENERATOR.
Question 1
What is the minimum interval of the SEQUENCE GENERATOR data ?
We could think of these sequences as streams of logic levels in a digital machine, possibly
representing digitized speech or video. The information elements in this stream are the unit
Experiment 3 - Special signals – characteristics and applications
© 2011 Emona Instruments
3-5
pulses. They are sometimes called symbols. Verify that there is one symbol per clock period.
Since the clock frequency is 1000 Hz, the symbol rate is 1000 per second. The symbols in this
sequence have only two possible values, so they are called binary symbols, and the transmission
rate is commonly expressed as bits/sec.
13.
Connect the CH0 lead to the output of
the BASEBAND LPF module (BLPF) and
connect the CH1 lead to the output of the
TUNEABLE LPF module (TLPF).
Set the TLPF FREQ so the output appears
similar to that from the BASEBAND LPF, as
shown in Figure 1c.
TLPF GAIN: set knob to 12 o’clock
Figure 1c: example signals
Note the presence of oscillations on both signals and the differences between them. Where
possible you should venture comments. You are not expected to have any prior knowledge of
these waveforms.
Question 2
Describe the signal transitions for both outputs:
14.
With the clock remaining unchanged on 1000 Hz measure the time for each signal to
change state. Is it the same for low to high (amplitude) as for high to low ? Specify the
reference points you are using on the amplitude range, eg 1% to 99%, 10% to 90%. Note these
values in the table below. “Freeze” the signals using the “RUN/STOP” SFP switch in order to
take your measurements, and use the TRIGGER SLOPE control to select between rising and
falling edge capture.
NOTE: Disconnect the RC NETWORK when measuring the other systems as it loads the output
LINE CODE signal slightly and affects the measurements.
TIP: Calculate the levels you wish to measure and use the X & Y cursors as guidelines.
3-6
© 2011 Emona Instruments
Experiment 3 - Special signals – characteristics and applications V1.2
Table 1: transition times for sequence data
Range
(%)
BLPF@1kHz
(us)
TLPF@1kHz
(us)
BLPF@1.5kHz
(us)
TLPF@1.5kHz
(us)
10-90 rising
10-90 falling
1-99 rising
1-99 falling
15.
Next, increase the clock frequency to around 1.5 kHz. Repeat the measurements in Task
14 above, and compare the two sets of results.
16.
Progressively increase the clock frequency, and carefully observe the effect on the
output waveform. Note that something significant occurs above 2 kHz. Confirm that below
2 kHz the original transitions can be unambiguously discerned at the channel output, even
though they are not sharp. Describe your observations as the clock is taken to 3 kHz and above.
Are you able to correctly identify the symbols of the original sequence from the distorted
output waveform? Estimate the highest clock frequency for which this is possible. Venture an
explanation for the disappearance of transitions in this channel.
Question 3
Describe the signal transitions for both outputs:
In the next segment we will closely examine the shape of the transition corresponding to an
isolated step excitation.
Part 1b – isolated step excitation of a system
STEP
SOURCE
S.U.I.
Figure 1c: block diagram of step excitation arrangement
Experiment 3 - Special signals – characteristics and applications
© 2011 Emona Instruments
3-7
Figure 1d: SIGEx model Figure 1c.
17.
Connect signals as shown in Figure 1d above. Connect CH0 to the BLPF output and CH1 to
the TLPF output, and view both signal on the scope. Settings are as follows:
18. PULSE GENERATOR: FREQUENCY=250;
DUTY CYCLE=0.50 (50%)
SCOPE: Timebase 2ms; Rising edge trigger on
CH0; Trigger level=1V
Confirm that the scope time base is set to
display not more than two transitions. Use
RUN/STOP to freeze scope display.
Figure 1e: example signals:50% figure
Observe the channel's response to a single transition (you can use scope trigger and other time
base controls to display a LO to HI transition or a HI to LO transition). Confirm that the shape
of the output transition is similar to the shapes you observed in Task 13 above.
When the response to a step excitation is isolated in this way, so that there is no overlap
with the responses of neighbouring transitions, it is known as the step response.
Note the presence of oscillations and the relatively long settling time to the final value
(sometimes known as ringing -- a term that goes back to the days of manual telegraphy and
Morse code). Compare with the waveform in Task 13 .
Note that some of the transitions observed in Task 13 occur before the previous
transition response has completely settled.
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© 2011 Emona Instruments
Experiment 3 - Special signals – characteristics and applications V1.2
The risetime of the step response is an indicator of the time taken to traverse the transition
range. Various definitions can be found according to the application context. The frequently
used 90% criterion is suggested as a convenient choice for this lab.
19.
Measure and compare the risetime of the three step responses. Use this to estimate
the maximum number of transitions per second that could be accommodated in each case (ignore
the effect of the oscillations). Compare this with the results in Task 0..
Table 2: transition times for step input
Range
(%)
BLPF
(us)
TLPF
(us)
RCLP
(us)
10-90 rising
10-90 falling
Graph 1: step response waveforms
Experiment 3 - Special signals – characteristics and applications
© 2011 Emona Instruments
3-9
Part 1c – isolated pulse response of a system
An isolated pulse can also be used as an alternative to the use of an isolated step as the
excitation to “probe” the behaviour of the system. The variable duty cycle of the PULSE
GENERATOR serves as source of this signal.
PULSE
SOURCE
S.U.I.
Figure 1e: block diagram of pulse response investigation
Figure 1f: model for pulse response investigation
20.
Leave the patching as per the previous section, with the PULSE GENERATOR output
connected to both S.U.I. With the frequency of the PULSE GENERATOR still set to 250 Hz,
progressively reduce the DUTY CYCLE in steps as follows: 0.4, 0.3, 0.2, 0.1, 0.05 (5%).
When you reach 0.1, move in steps of 0.01 eg. 0.09, 0.08, 0.07,... and observe the effect on the
pulse width and pulse interval. Note that the transitions are not affected. As you continue to
reduce the duty cycle, and thus reduce the input impulse width, the flat top between
transitions gets shorter, and ultimately disappears. Since the rising transition is not able to
reach its final value, it is not surprising that the amplitude of the pulse gets smaller.
Question 4
Describe what happens when you reach 10% and 5% duty cycle ?
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© 2011 Emona Instruments
Experiment 3 - Special signals – characteristics and applications V1.2
21.
Are you able to determine the ‘demarcation’ pulse width -- i.e. after which the response
shape remains unchanging? Record the duty cycle value at which this occurs for all SUI’s in the
table below.
Table 3: pulse response readings
BLPF
TLPF
RCLPF
Duty cycle
“demarcation” value
Calculated pulse width (us)
% of step response
Period of oscillations (us)
22.
Using the known PULSE GENERATOR frequency and the measured duty cycle, calculate
and tabulate the input pulse width.
23.
Express this as a percentage of the step response risetime, using the values from the
previous section on step response, and note these values in the table above.
Reflect on this for a moment, i.e. the response shape remaining apparently independent of
the input pulse width -- this is an interesting discovery.
24.
Move the scope leads so as to view the input pulse as CH0 and one of the SUI outputs on
CH1.Note that for the both there are oscillations. The presence of these oscillations provides
an opportunity for additional observations of shape changes as the width of the input pulse is
reduced. There are many ways of testing this, eg. the number of sidelobes, their relative
amplitudes, the intervals between zero crossings.
25.
For each SUI, set the pulse width to the “demarcation” value and measure the period of
the oscillations following the pulse. Note these in the table above.
You have demonstrated that, provided the time span of the excitation signal is sufficiently
concentrated, the shape of the response pulse is entirely determined by the characteristics of
the system. We could think of this as the striking of a bell, or tuning fork, or of the steel
wheel of a train to detect a crack. The system is hit with a short sharp burst of energy.
INSIGHT: The response shape is not affected by the input signal.
The energy burst used as input is called an impulse. The resulting response is called the impulse
response. An impulse function is a mathematical construct derived from a physical pulse. The
idea is straightforward. The pulse width is reduced to an infinitesimal value while maintaining
Experiment 3 - Special signals – characteristics and applications
© 2011 Emona Instruments
3-11
the energy constant. Naturally this implies a very large amplitude. The impulse function plays a
central role as one of the fundamental signals in systems theory, with numerous ramifications.
In the above exploration we discovered practical conditions that make it possible to generate a
system's natural response or characteristic, i.e. a response that is not affected by the exact
shape of the input excitation. Concurrently we have discovered a path to the definition of the
impulse function and a vital bridge to link this mathematical abstraction to the world of physical
signals.
26.
With the setup unchanged, measure the delay at the peak of the output pulse and
compare this with the delay of the step response measured earlier.
27.
Return to your records of the step responses obtained in Steps 17 & 18. For each case,
carry out a graphical differentiation with respect to time (approximate sketches are sufficient,
however take care to achieve a good time alignment to identify key features). Compare these
results with the records obtained in Task 23. As a useful adjunct exercise, consider a slightly
modified step function in which the transition is a ramp with a finite gradient, though still quite
steep. Carry out the differentiation with respect to time on this function, and compare with the
above. Record your conclusion.
Graph 2: differentiations of step response waveforms
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© 2011 Emona Instruments
Experiment 3 - Special signals – characteristics and applications V1.2
You have demonstrated that, provided the time span of the excitation signal is sufficiently
concentrated, the shape of the response pulse is entirely determined by the characteristics of
the system. We could think of this as the striking of a bell, or tuning fork, or of the steel
wheel of a train to detect a crack. The system is hit with a short sharp burst of energy. The
response shape is not affected by the input signal.
The energy burst used as input is called an
impulse. The resulting response is called the
impulse response. An impulse function is a
mathematical construct derived from a
physical pulse. The idea is straightforward.
The pulse width is reduced to an infinitesimal
value while maintaining the energy constant.
Naturally this implies a very large amplitude.
The impulse function plays a central role as
one of the fundamental signals in systems
theory, with numerous ramifications.
In the above exploration we discovered practical conditions that make it possible to generate a
system's natural response or characteristic, i.e. a response that is not affected by the exact
shape of the input excitation. Concurrently we have discovered a path to the definition of the
impulse function and a vital bridge to link this mathematical abstraction to the world of physical
signals.
Experiment 3 - Special signals – characteristics and applications
© 2011 Emona Instruments
3-13
Part 2 – Sinewave input
As mentioned in the introduction, sinewaves are encountered in a large number of applications.
The special role of the sinusoidal waveshape for system characterization is explored in
Experiment 2, and further developed in Experiment 4. In this segment we just get our toes
wet. We carry out some basic observations and compare the sinewave response of the various
S.U.I’s with the impulse response obtained above.
S.U.I.
Figure 2a: block diagram of setup for sinewave investigation
Figure 2b: patching model for Figure 2a.
28.
Connect the FUNC OUT output from the FUNCTION GENERATOR to the inputs of both
S.U.I. Launch the NI ELVIS Intrument Launcher and select the FUNCTION GENERATOR. Set
up the FUNCTION GENERATOR as follows:
Select: SINE wave
Voltage range: 4V pp
Frequency: 100 Hz
Press RUN when ready.
Connect CH0 of the scope to the output of the FUNCTION GENERATOR, and CH1 to output of
S.U.I.
Progressively increase the frequency from 100 Hz to 10 kHz and observe the effect on the
amplitude of the output signal. Make a record of your findings in the form of a table of
3-14
© 2011 Emona Instruments
Experiment 3 - Special signals – characteristics and applications V1.2
amplitude vs frequency. Enter your results into the table on the TAB3 SFP, which will plot
those results. Consider the possible advantage of using log scales.
To enable a “log” Y axis, stop the SIGEX SFP program, right click the plot graph, select Y scale >
Mapping > Log. To return to Linear, repeat this process and select “Linear”.
Table 4: amplitude vs frequency readings
Frequency (Hz)
BLPF (Vpp)
TLPF(Vpp)
RCLPF(vpp)
29.
Refer to the results you obtained and sketched of the step response in Question 19.
Notice the similarity of the step response shape to a half cycle of a sinewave. Estimate the
frequency of the matching sinewave. Examine the graph obtained in the above task and see
whether any feature worth noting appears near this frequency.
Question 5
What frequency would a matching sinewave have ?
Question 6
Describe what happens to the frequency response plotted on the SFP at this frequency ?
Experiment 3 - Special signals – characteristics and applications
© 2011 Emona Instruments
3-15
30.
Return to the observations you recorded in Task 19. A physical mechanism was proposed
there to explain the reduction in pulse response amplitude as the width of the input pulse was
progressively made smaller. Consider whether the reduction in output amplitude of the sinewave
with increasing frequency could be explained through a parallel argument.
Question 7
What was the mechanism described earlier ?
Part 3: clipping
A common example of voltage clipping or limiting occurs in amplifiers when the signal amplitude
is too high for the available DC supply voltage headroom. In audio systems clipping is
undesirable as it causes distortion of the sound. However, in other applications, a clipped signal
can be useful.
We examine the operation of the voltage LIMITER and try out an application. First we find out
how it can be used to convert a sinewave to a square wave.
Figure 3a: block diagram for clipping a sinewave
Figure 3b: wiring model for Figure 3a
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© 2011 Emona Instruments
Experiment 3 - Special signals – characteristics and applications V1.2
31.
Patch up the system in Figure 3b. As we will be using the MEDIUM mode of the
LIMITER unit, the on-board switches must be set accordingly (swA= OFF, swB= OFF). Tune the
FUNCTION GENERATOR to 1200Hz and select SINUSOIDAL output with 4 V pp.
Set scope as follows:
SCOPE: Timebase 2ms; Rising edge trigger on CH0; Trigger level=0V
Display the output and input of the LIMITER, and observe the effect of changing the amplitude
at the AMPLITUDE control of the FUNCTION GENERATOR. Make it larger and smaller.
Record your findings in the form of a graph showing p-p output voltage vs p-p input voltage. You
can plot your readings on the graph below.
Graph 3: CLIPPER input and output readings
Next we use the CLIPPER as a primitive digital detector.
32.
Patch up the SIGEx model in Figure 3d (note that it is an extension of the model in
Figure 1b). The LIMITER should be in the same setting as before (OFF:OFF). Display the
outputs of the LIMITER and of the BLPF. Begin with the clock rate near 1.5 kHz. As before,
the timebase should be adjusted to provide a useful balance between detail and range of
observation. Examine the two signals and consider the possible interpretation of the output as a
restored or regenerated form of the original digital sequence.
Experiment 3 - Special signals – characteristics and applications
© 2011 Emona Instruments
3-17
SEQUENCE
SOURCE
S.U.I.
Figure 3c: block diagram for clipping a digital pulse sequence
Figure 3d: model for block diagram of Figure 3c
33.
As you gradually increase the clock frequency (as in Task 16), carefully watch for the
disappearance of transitions or pulses in the CLIPPER output. When this happens, wind the
frequency back slightly and determine the highest frequency that allows detection without
visible errors. Compare the result with your previous findings in Task 16, i.e. without using the
LIMITER.
Figure 4: example of signals in & out of LIMITER
34.
Compare with the results obtained in Part 1 and record your conclusions, i.e., about the
practicality and usefulness of the clipper as an "interpreter" to recover the data in the
distorted signal .
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© 2011 Emona Instruments
Experiment 3 - Special signals – characteristics and applications V1.2
Question 8
How does this setup compare to the previous findings without a LIMITER ?
In the above we have used only continuous-time waveforms. Discrete-time signals and systems
are introduced in Lab 4.
Experiment 3 - Special signals – characteristics and applications
© 2011 Emona Instruments
3-19
Tutorial questions
Q1 The impulse function was described in Part 1. Explain why the step function is a
better alternative in a practical context. Show how the impulse response can
be obtained from the step response. Is this indirect procedure for measuring
the impulse response theoretically equivalent, or does it involve an
approximation?
Q2 Consider a system with step response rise-time of 4 µs. What information does this
provide about the impulse response?
Q3 a. Consider the waveform at the yellow X output of the SEQUENCE GENERATOR (as
in Part 1). Suppose the p-p voltage is 3.9 Volt and the clock is 2 kHz. What is
the average power into a 1 Ohm load?
b. Suppose the waveform is passed through BASEBAND LOW PASS FILTER and
the p-p output amplitude is also 3.9 Volt. Is the power greater or less than
at the channel input? State the reasoning (hint: consider the waveform
shape required to have the average power exceed that of the waveform at
the channel input).
c. Consider two different sequences as above. One has N transitions per period,
the other has N + 4. Explain why the number of transitions does not affect
the average power for the signal format at the channel input. Is the
answer the same at the output? If no, in which case will the average power
be greater? Indicate why. Hint: math not required, just consider how the
average is worked out.
Q4 A 60 kHz sinewave is applied at one input of a MULTIPLIER, and a 59 kHz sinewave
at the other input. The amplitudes are both 2 Volt p-p. Use a suitable
formula to show that the MULTIPLIER output is the sum of two sinewaves.
Calculate their respective frequencies. The MULTIPLIER output is fed to a
system similar to BASEBAND LOWPASS FILTERS, with step response risetime 300 µs. Describe the signal at the output of this system, if any.
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© 2011 Emona Instruments
Experiment 3 - Special signals – characteristics and applications V1.2
Experiment 4 – Systems – Linear & Nonlinear
Achievements in this experiment
Understanding of the special role of sinusoids in linear systems. Knowledge and experience of
testing for linearity.
Preliminary discussion
Lab 1 was about special signals. The aspect that was special was their use for getting
characteristic information about systems. However, although the focus of that lab was on
signals, a system was nevertheless needed as a vehicle for the action. In this lab the focus will
be on the system, and as might be expected, it is signals that will be doing the talking.
Moreover, it will emerge that the story in Lab 1 was incomplete. We will discover a signal set
that is perfectly matched to a special class of systems.
We get started with explorations of various linear and nonlinear systems, and look into memory
effects.
Next, tests for linearity are implemented.
The lab concludes with a ‘lateral’ investigation of frequency response.
Pre-requisite work
Question 1
Write down a formula to express the square of a sinusoid in terms of a double angle argument.
Question 2
What is the meaning of differential linearity ?
Consider the two conditions for linearity for a system S defined by y = S(x)
Scaling or homogeneity:
a.y = S(a.x)
Additivity:
y1 + y2 = S(x1 + x2)
Question 3
How would you apply these formulas in testing systems for linearity in this Lab ? How many
replicas of the system are needed for the additivity test ?
4-2
© 2011 Emona Instruments
Experiment 4 - Systems – Linear & Nonlinear V1.2
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should observe
the USB light turn on (top right corner of ELVIS unit).The PC may make a sound to indicate that
the ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that
all three power LEDs are on. If not call the instructor for assistance.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with
the NI ELVIS that you’re using.
9.
You’re now ready to work with the NI ELVIS/SIGEx bundle.
10.
Select the EXPT 4 tab on the SIGEx SFP.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the
STOP button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the
top of the window. This will allow the program to conduct an orderly shutdown and close the
various DAQmx channels it has opened.
Experiment 4 - Systems – Linear & Nonlinear
© 2011 Emona Instruments
4-3
Experiment
Part 1: Getting started
In this Part we have a quick look at some simple systems and check for linearity. First the
LIMITER, next the RECTIFIER.
S.U.I.
Figure 1: block diagram for sinusoidal investigation
Figure
2:
patching diagram for a “clipping” system
11.
Connect the FUNC OUT terminal of the FUNCTION GENERATOR to the LIMITER
input. Connect CH0 to this input. Connect CH1 to the LIMITER output.
The settings are as follows:
FUNCTION GENERATOR: FREQUENCY=1000; AMPLITUDE: 1 V pp; SINE waveform selected
LIMITER: DIP switch to OFF:OFF for “medium” clipping. Refer to SIGEx User Manual for
details.
SCOPE: Timebase 4ms; Rising edge trigger on CH0; Trigger level=0V
12.
Adjust the amplitude of the SINE waveform from 1 v pp to around 10 V pp. Take various
readings throughout this range and document in the table below.
HINT: Use the cursors on the graph of TAB 4 to quickly take peak readings, then double.
4-4
© 2011 Emona Instruments
Experiment 4 - Systems – Linear & Nonlinear V1.2
Table 1
Input amplitude
(Vpp)
LIMITER amplitude
(Vpp)
RECTIFIER amplitude
(Vpp)
Question 4
Does this system (CLIPPER) satisfy the scaling test for linearity? Show your reasoning.
Figure
3:
patching diagram for a “rectifying” system
Experiment 4 - Systems – Linear & Nonlinear
© 2011 Emona Instruments
4-5
13.
Connect the FUNC OUT terminal of the FUNCTION GENERATOR to the RECTIFIER
input. Connect CH0 to this input. Connect CH1 to the RECTIFIER output, as per Figure 3.
The settings required are as follows:
FUNCTION GENERATOR: FREQUENCY=1000; AMPLITUDE: 1 V pp; SINE waveform selected
SCOPE: Timebase 4ms; Rising edge trigger on CH0; Trigger level=0V
14.
Adjust the amplitude of the SINE waveform from 1 Vpp to around 6 Vpp. Take various
readings throughout this range and document in the table 1 above. As the output signal appears
only to be positive going, measure its positive amplitude (which is also its total peak to peak
value). Show that this system satisfies the homogeneity test over part of the input range (i.e.
where the range excludes the zero input point).
Question 5
Does this system (RECTIFIER) satisfy the scaling test for linearity? Show your reasoning.
In the above, consider the meaning and applications of the term ‘differential linearity’ (also
known as 'incremental linearity').
For comparison, you will now repeat the above with an analog MULTIPLIER. Observe the effect
of changing the AMPLITUDE. Record your findings concerning linearity.
15.
Connect the FUNC OUT terminal of the FUNCTION GENERATOR to the MULTIPLIER
inputs. Connect both inputs in parallel. Connect CH0 to this input. Connect CH1 to the
MULTIPLIER output, as per Figure 4.
The settings required are as follows:
FUNCTION GENERATOR: FREQUENCY=1000; AMPLITUDE: 1 V pp; SINE waveform selected
SCOPE: Timebase 4ms; Rising edge trigger on CH0; Trigger level=0V
Figure
4-6
© 2011 Emona Instruments
4:
patching diagram for a “multiplying” system
Experiment 4 - Systems – Linear & Nonlinear V1.2
Table 2
Input amplitude
(Vpp)
MULTIPLER amplitude
(Vpp)
Question 6
Does this system (MULTIPLIER) satisfy the scaling test for linearity? Show your reasoning.
Note that the application of the additivity test is not needed when the homogeneity test has
failed. A complication with the additivity test is the need for replicas of the system under test,
hence the homogeneity test is normally applied first.
Unlike the BLPF channels, investigated in Lab 3, the responses of the above systems appear as
effectively instantaneous, i.e. the systems are ‘memoryless’ over the time scales of interest
above. However, closer examination with the scope time base in the nanosecond ranges reveals
that the responses are not as ‘instantaneous’ as might seem. Thus the idea of a ‘memoryless’
system is only relative in a practical context.
Part 2 – the VCO as a system
In this part we explore the output-input characteristics of the VCO (voltage controlled
oscillator) and apply linearity tests.
Experiment 4 - Systems – Linear & Nonlinear
© 2011 Emona Instruments
4-7
Figure 5: model for investigation of VCO as a system
16.
Connect as shown in Figure 5.
Settings are as follows:
FUNCTION GENERATOR: FREQUENCY=2000 Hz; AMPLITUDE: 4 V pp; SINE waveform
selected
MODULATION TYPE: FM
SCOPE: Timebase 4ms; Rising edge trigger on sinewave output; Trigger level=0V
17.
Display both the input DC voltage and the output sinusoid. Ensure that the scope is
triggering on the sinusoid signal for a stable display.
18.
Observe the effect of varying the DC input voltage. Document output frequency vs
input DC voltage in the table below. (The scope can be used to measure the DC voltage and
frequency).
Table 3
Input DC voltage
(V)
4-8
© 2011 Emona Instruments
VCO output frequency
(Hz)
Experiment 4 - Systems – Linear & Nonlinear V1.2
19.
Consider whether the VCO is a linear system, ie in terms of output frequency vs input
DC voltage. For this purpose should you focus on incremental frequency rather than absolute
frequency ? (cf differential linearity).
Question 7
Is the VCO a linear system ? Explain your reasoning.
Figure 6: model for investigation of VCO as a system with rapidly “varying” input level
20.
Replace the DC signal with the signal output from the MULTIPLIER, where the input to
the MULTIPLIER is both the DC voltage at DAC-1 and the fixed sinewave at DAC-0. This
effectively gives you amplitude control over the sinewave by using the DC voltage as the
multiplication factor. Observe the effect of varying the DC input voltage on the MULTIPLIER
output, as well as the VCO output.
21.
Observe the VCO output. Consider possible applications (hint: synonym for ‘varying’ is
‘modulating’). View the input and output of the VCO with both scope leads. Trigger the scope on
the input signal for a stable display.
A note about scope triggering:
As a general rule, it is best to trigger the scope on the lowest frequency signal being displayed.
This causes the scope to display the slower signal with stability, and have the faster signal
moving relative to that. In this case, although the two signals displayed are not synchronised, it
is easy to see the influence of the lower frequency signal on the VCO output. Try both ways and
see if you agree.
Question 8
What applications could the VCO with varying output frequency be used for ?
Experiment 4 - Systems – Linear & Nonlinear
© 2011 Emona Instruments
4-9
Part 3 – a feedback system
We will apply both linearity tests to a simple feedback system implemented with the LAPLACE
module. First we carry out some basic exploration of the operation of the INTEGRATOR.
22.
Patch up the model in Figure below.
Settings are as follows:
FUNCTION GENERATOR: FREQUENCY=1000 Hz; AMPLITUDE: 4 V pp; SINE waveform
selected
MODULATION TYPE: NONE
LIMITER DIP switches: DOWN:DOWN
INTEGRATOR RATE DIP switches: UP:UP
SCOPE: Timebase 2ms; Rising edge trigger on CH0; Trigger level=0V
Figure 7: patching model for sawtooth generation
Figure 8: scope display for sawtooth generation
23.
Measure the p-p amplitude of the output sawtooth waveform. Calculate the slope (slew
rate) of the sawtooth and use this to write a formula for the INTEGRATOR output in terms of
the input.
4-10
© 2011 Emona Instruments
Experiment 4 - Systems – Linear & Nonlinear V1.2
Question 9
What is the formula for the INTEGRATOR output ?
24.
Repeat this for the other three available INTEGRATOR rate settings. If a particular
INTEGRATOR rate is too high, then note this.
Question 10
What are the formulae for the other INTEGRATOR rate settings ?
Next we apply negative feedback to the “open loop” integrator above.
25.
Patch up the modules according to the figures below to add a feedback path to the
system.
Settings are as follows:
FUNCTION GENERATOR: FREQUENCY=500 Hz; AMPLITUDE: 2 V pp; SQUARE waveform
selected
MODULATION TYPE: NONE
INTEGRATOR RATE DIP switches: UP:UP
SCOPE: Timebase 4ms; Rising edge trigger on CH0; Trigger level=0V
ADDER GAINS: b1=1.0; b2= - 1.0
Figure 9: block diagram for a feedback system
Experiment 4 - Systems – Linear & Nonlinear
© 2011 Emona Instruments
4-11
Figure 10: patching diagram for a feedback system
Ensure that you have used and set the
ADDERS as per the figure and settings above.
We are interested in negative feedback, hence
the negative gain setting for b2.
View the output and input of the
INTEGRATOR as CH0 and CH1 respectively.
Your display should look like Figure 11.
Figure
11:
example signals
Observe the displayed waveforms of Fig 11 and satisfy yourself that they represent step
responses (briefly display the input waveform as a reminder). Note that these responses are
made up of exponential segments.
Examine the waveform at the ADDER output (input to the INTEGRATOR), and confirm that it
looks like the impulse response corresponding to the step response at the INTEGRATOR output.
26.
On the scope, measure the time taken for the exponential curve to decay by 50% and
use this to calculate the time constant.
27.
Refer to your preparation work on tutorial Q1c and compare the theoretical step
responses with those observed above.
Question 11
Use the value of the b2 gain, and INTEGRATOR constant you measured above to determine the
time constant of the exponential responses. Compare this with the value obtained from your
measurement.
4-12
© 2011 Emona Instruments
Experiment 4 - Systems – Linear & Nonlinear V1.2
28.
Observe the effect of varying the b2 gain, and check that this is in general agreement
with the predictions of the theoretical formulas. In particular, take a close look at b2 values of
0, -0.1, -0.2, ... and confirm that you can see effect addition (or subtraction) of the output from
the input, as you subtly introduce feedback.
29.
Explain why this feedback system is not memoryless in the time scale of our
observations. Determine the location of the system memory (see Tutorial Q1d).
30.
Change the input amplitude and confirm that the system satisfies the homogeneity
condition for linearity. Switch the FUNCTION GENERATOR back to SINEWAVE output to
demonstrate that the result is the same with a sinusoidal input.
Question 12
Write a differential equation for this first-order feedback system. Assume initial conditions
are zero. Show that with a sinusoidal function of time as input, the output is also sinusoidal.
Show that this also happens when the input is a complex exponential. Which special property of
complex exponential functions provides the key?
Part 4: testing for additivity
Next we test for the additivity condition.
The block diagram below depicts a model for this, that requires three identical copies of the
system under test ie: s(a), s(b) and s(c).
Test signal x1 is a sinewave from ANALOG OUT DAC-0; x2 is a squarewave of opposite phase
from ANALOG OUT DAC-1.
31.
Select the appropriate signals as outputs from the ANALOG OUTPUT block by switching
ON the “Part 4 Signal Select” button on TAB “Lab 4”.
x
1
S(α)
x
2
x
1
x
2
S( x1 + x 2 )
sum at input
S(β)
y
S( γ)
y
1
Sx
1
+ S x2
2
Figure 12: test method using matched copies of the system
Experiment 4 - Systems – Linear & Nonlinear
© 2011 Emona Instruments
4-13
Figure 13: patching model for generating the S(x1 + x2) & Sx1 + Sx2
arrangements simultaneously
We have available three INTEGRATORS which will act as our systems S(x).
32.
As a first step, confirm that each S(x) is identical, by inputing the same signal into each
and comparing their outputs. You can use the square signal from DAC-1 as an interesting and
familiar source. Setup the INTEGRATOR RATE DIP switches as UP:UP
33.
Patch up the 2 models as per the figure above.
Settings are as follows:
INTEGRATOR RATE: UP:UP
ADDER GAINS: a0=a1=b0=b1= +1.0; a2=b2=0;
SCOPE: Timebase = 4ms, Trig level = 0V
34.
Focus firstly on system S(x1 + x2). View all points of the system and confirm that the
signals are as expected.
Figure 14: example of (x1 + x2) before and after processing by system S
35.
Now view the signals of system Sx1 + Sx2 and confirm that all signals are as expected.
Produce an accurate record each of the systems output waveform on the graph below.
4-14
© 2011 Emona Instruments
Experiment 4 - Systems – Linear & Nonlinear V1.2
Graph 1: additivity signals
Question 13
Does the outcome indicate that the linearity conditions have been met for these two test
inputs?
36.
Vary the gains of one system and observe the effect. Remember to ensure that gains
a1=b1 and a0=b0 to maintain a consistent comparison.
Question 14
Does the outcome during variation indicate that the linearity conditions are still maintained for
these two test inputs?
Experiment 4 - Systems – Linear & Nonlinear
© 2011 Emona Instruments
4-15
Part 5 – frequency response
In Lab 3 (Part 2) we found that when a sinewave of constant amplitude was applied to the input
of a BBLPF channel, the output amplitude was dependent on the frequency. A sinewave was
chosen for that test because we were making a comparison with the shape of the impulse
response of the BBLPF, which was very similar to a sinusoidal segment.
The idea of frequency response using sinewaves is well known in the audio culture, for example,
as used in the testing of amplifiers, microphones, etc. . Here we ask: why have sinewaves been
singled out for this -- could another kind of waveshape be used to measure frequency response,
for example, a squarewave? There is a quick way to find out.
S.U.I.
S.U.I.
Figure 15:block diagram for investigation
with a squarewave and a sinewave
37.
Measure the frequency response of BBLPF with a square wave. Use the set up and
method in Lab 3. Use the square wave available at the FUNCTION GENERATOR block, FUNC
OUT terminal, by selecting SQUAREWAVE output (this signal is a bipolar squarewave, as
opposed to the TTL level squarewave available at the SYNC output). Explain your criterion for
the amplitude measurement at the output (specifically, to deal with the shape changes and
ringing oscillations). Enter your readings into the table below.
Input frequency
(Hz)
4-16
© 2011 Emona Instruments
Square wave
output (V)
Sine wave
output (V)
Experiment 4 - Systems – Linear & Nonlinear V1.2
Question 15
How are you able to use the square wave for this test ?
38.
Next, try with sinewaves. The measurement is very straightforward. Unlike
squarewaves, the sinewaves emerge at the output with no apparent change of shape. The issue
of amplitude measurement criterion does not arise. Enter your readings into the same table.
Consider the possibility of other waveforms that may also have this property. Relate this to the
math in Tutorial Q 1.
And now, a role for sinusoids in quantifying nonlinearity.
In the world of the discerning audiophile, speed of response is of paramount importance, and we
have seen that this can be quantified in terms of a time domain response, and also through the
medium of the sinewave frequency response. An equally essential facet of sound quality is
linearity. The presence of nonlinearities in amplifiers, microphones, etc generates distortion and
diminishes the listening experience. Hence it is important to have paradigms for quantifying
deviation from linearity.
It is interesting that the sinusoidal waveform is also a basis for nonlinearity modelling. Think
about how this might be implemented. However, you will find it more rewarding to reconsider
this after discovering the fertile valleys of the frequency domain, in Labs 8 & 9.
The systems used so far have not included a discrete-time example. This is done in later
experiments.
Tutorial questions
Q1 a. Write a differential equation for the first-order feedback system in Part 4.
Assume initial conditions are zero. Show that with a sinusoidal function of
time as input, the output is also sinusoidal. Show that this also happens when
the input is a complex exponential. Which special property of complex
exponential functions provides the key?
b. For each of the two possible outputs, write expressions for the ratio of output to
input when the input is exp(jωt). Show how this result can be used to obtain
a formula for the amplitude response when the input is the real signal
sin(ωt).
c. Modify the differential equation in (a) to include nonzero initial conditions. Make
the input zero and solve for the natural response (i.e., the homogeneous
solution). Use this to obtain the step responses at each output. Apply the
findings in Lab 1 to convert the step responses to impulse responses.
d. Find out how to make an integrator with an operational amplifier.
Experiment 4 - Systems – Linear & Nonlinear
© 2011 Emona Instruments
4-17
e. Suppose that in (b) jω is replaced by the general complex variable s = σ + jω.
Suggest an interpretation for s in a practical context.
f. Consult your favourite reference to see how the Laplace transform is applied to
solve linear differential equations with constant coefficients.
Q2 What is the difference between a halfwave rectifier and a fullwave rectifier ? Show
how to make a full-wave rectifier with two half-wave rectifiers.
Q3
Consider a multiplier with 1.5V DC to input y. Is this system linear wrt input x?
Repeat this with a 2 Volt p-p sinewave at input y. Consider this test with
other waveforms at input y (for example a squarewave, a rectified sinewave).
Next, apply a 2 kHz sinewave to the two inputs connected in parallel. Write an
equation for the output in terms of the input. Is the scaling (homogeneity)
test for linearity met? Show that the output waveform is sinusoidal. What
is its frequency? Suggest an application for this system ?
Q4 From the Lab, should we conclude that nonlinear systems are generally memoryless?
Suggest examples to the contrary. Hint: a speed control governor in a steam
engine, analogous method for amplitude compression control in audio systems.
Q5
Consider a cascade of two of the first-order systems in Q 1 . The INTEGRATOR
constants are τ1 and τ2 (in seconds),
the feedback gains are α1 and α2,
respectively. The input is sin(ωt). Write down the output/input amplitude
ratio of the overall system. Plot this vs frequency (use convenient numerical
values for the constants).
Q6
Consider again the two first-order systems in Q 5, however this time connect them
in parallel, with output 2 subtracted from output 1. Repeat Q 5 for this case.
Compare the two configurations.
Repeat with the outputs added and compare with the above.
Q7
4-18
Consider the sawtooth waveform generated with the integrator. Draw a block
diagram showing how this waveform could be used with a VCO and a scope in
x-y mode to produce a rudimentary frequency response display.
© 2011 Emona Instruments
Experiment 4 - Systems – Linear & Nonlinear V1.2
Experiment 5 – Unraveling Convolution
Achievements in this experiment
Carry out a step-by-step dissection of the convolution process in a discrete-time system. Use
this to discover the convolution formula. Demonstrate that convolution can be visualized as a
running average of successive values of the input signal. Observe a special property applying to
sinewaves. Demonstrate the operation of a filter in the time domain.
Preliminary discussion
For many students, the first encounter with convolution is an abstract mathematical formula in
a textbook. This lab offers a more evocative experience. By tracing the passage of some basic
signals through a simple linear system, you will be able to observe the underlying process in
action, and, with a little arithmetic, discover a formula as it emerges from the hardware.
Refresh your basic trigonometry: you will need sin(ω.t) + A.cos(ω.t + φ) expressed as a single
sinusoid.
If any of the modules is unfamiliar, spend a little time with the SIGEx User Manual. This will
give you a headstart in setting up the lab.
We patch up a delay line with two unit delays and three taps with independently adjustable gains
as in Figure 1. In the first exercise you will set these gains to given values, and observe the
output when the input is a single pulse (more exactly, a periodic sequence of single pulses). This
is an important preliminary as it introduces the unit pulse response.
+
OUTPUT
INPUT
b2
b1
b0
UNIT
DELAY
UNIT
DELAY
Figure 1a: general block diagram of the system
Next, you will observe the output when a pair of adjacent pulses is used as input. This near
trivial example provides us with a springboard to the general case. It will demonstrate how
convolution operates as an overlapping and superposition of unit pulse responses. A second more
general input sequence is then used to reveal a deeper insight and to provide a vehicle for
setting up the key formulas.
In the remainder of the lab we use sinewaves as inputs. This takes us to the rediscovery of the
special role of sinusoids in linear time-invariant systems.
In the final exercise we carry out and unravel a disappearing act.
It should take you about 40 minutes to complete this experiment.
5-2
© 2011 Emona Instruments
Experiment 5 – Unraveling Convolution V1.2
Pre-requisites:
Familiarization with the SIGEx conventions and general module usage. A brief review of the
operation of the S/H & UNIT DELAY module from SIGEx User Manual. No theory required.
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should observe
the USB light turn on (top right corner of ELVIS unit).The PC may make a sound to indicate that
the ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that
all three power LEDs are on. If not call the instructor for assistance.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with
the NI ELVIS that you’re using.
9.
You’re now ready to work with the NI ELVIS/SIGEx bundle.
10.
Select the EXPT 5 tab on the SIGEx SFP.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the
STOP button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the
top of the window. This will allow the program to conduct an orderly shutdown and close the
various DAQmx channels it has opened.
Experiment 5 – Unraveling Convolution
© 2011 Emona Instruments
5-3
Experiment
Part 1 – Setting up
11.
Patch up the model as shown in Figure 2.
Settings are as follows:
PULSE GENERATOR: FREQUENCY=1000Hz; DUTY CYCLE: 0.5 (50%)
SEQUENCE GENERATOR: DIPS UP/UP
SCOPE: Timebase 10ms; Rising edge trigger on CH0; Trigger level=1V
+
OUTPUT
INPUT
b2
b1
b0
UNIT
DELAY
UNIT
DELAY
Figure 1b: general block diagram of the system
Figure 2: patching diagram of the system in Figure 1
12.
The required signal appears at the SEQUENCE GENERATOR SYNC output as a 5V signal
and needs to be reduced in amplitude using the available a0 GAIN function . Using the scope,
check that you have a periodic sequence of a single 1V pulse in a frame of 31 pulse periods.
Confirm the pulse width is 1ms. Adjust the a0 gain value to have a pulse amplitude of 1V
precisely. Adjust the SCOPE trig level to suit.
5-4
© 2011 Emona Instruments
Experiment 5 – Unraveling Convolution V1.2
Note that as the incoming pulse is being clocked by the same clock as the unit delay blocks, the
pulse is already aligned with the unit delay clock and is thus already a discrete signal. For this
reason it can be input directly into the unit delay without use of the S/H block.
Part 2 – unit pulse response
Before proceeding with the examination of the system response, the delay line “tap” gains must
be set. For the first case we shall use b0 = 0.3, b1 = 0.5 and b2 = − 0.2 (see Figure 1). These
settings have been chosen arbitrarily as interesting and varied values for this exercise. Adjust
each gain in turn on the SFP and use the scope to confirm your settings.
Question 1
Describe a procedure for confirming the GAIN at each tap ?
Question 2
Display the delay line input signal (i.e. at the first z-1 block input) and the ADDER output signal.
Measure and record the amplitude of each pulse in the output sequence.
Note that the system output is a sequence of three contiguous pulses with amplitudes in the
same ratio as the adder input gains. Could this have been predicted from Figure 1?
Indeed, since the single pulse at the input is generating delayed and scaled replicas as it travels
down the delay line. These are then summed in the adder.
Thus we have the system’s response to an isolated pulse. From this we can define the unit pulse
response h(n) as the response when the amplitude of the input pulse is unity.
From your measurements, show that the unit pulse h(0) = b0 , h(1) = b1 , h(2) = b2 in this
example.
The presence of delayed energy is normally expected in real-life systems, whether electrical or
mechanical. For example, due to inertia in mechanical systems. Similarly in electric circuits, we
have energy storage effects in reactive components such as capacitors and inductors.
Experiment 5 – Unraveling Convolution
© 2011 Emona Instruments
5-5
Part 3 – The superposition sum
13.
Adjust the SEQUENCE GENERATOR DIP switches to position UP:DOWN to select the
sequence of two contiguous pulses. Using the same gain settings as in Part 2, observe the output
signal. Note that it consists of four nonzero pulses per frame. Measure and record the
amplitude of each pulse.
14.
Verify that the output sequence is simply the sum of two offset unit pulse responses.
Use the graph below to show your computation.
Graph 1: unit pulse pair summation
Question 3
What is meant by “superposition”. Discuss how this exercise above relates to superposition and
the “additivity” principle.
5-6
© 2011 Emona Instruments
Experiment 5 – Unraveling Convolution V1.2
Question 4
What do you expect to see if this exercise were expanded to two or more contiguous pulses ?
Explain.
Part 4: rectified sinewave at input
In this exercise the input is a little more interesting than in Part 3: a sequence of three or
four pulses of different amplitude. We obtain the source of this signal from the ANALOG OUT
DAC-0. We then pass this analog signal to the SAMPLE/HOLD block to be sampled and this
becomes our discrete sequence of pulses. Note that the PULSE GENERATOR and DAC signal
generator share the same internal clock and hence no slippage occurs in the scope displays.
Note that although the signal is sampled and becomes “discrete” it has not become a “digital”
signal. This is an important distinction. Rather it now exists as sequential discrete samples of
the original signal. More about sampling and its implications in several later experiments.
Settings are as follows:
PULSE GENERATOR: FREQUENCY=800Hz; DUTY CYCLE: 0.5 (50%)
SCOPE: Timebase 10ms; Rising edge trigger on CH0; Trigger level=1V
Confirm that the sinewave from the DAC-0 is 100Hz, 2V peak, before entering the RECTIFIER.
We will treat the sinewave as a continuous signal and ignore the very small steps present as
these have no consequence to our procedure.
Question 5
Note the amplitude of the half wave rectified sine and explain why its amplitude is reduced
relative to the input ?
Experiment 5 – Unraveling Convolution
© 2011 Emona Instruments
5-7
Figure 3: Block diagram of system with series of pulses as input.
Figure 4: SIGEx patching model for rectified sinewave input
5-8
© 2011 Emona Instruments
Experiment 5 – Unraveling Convolution V1.2
15.
Maintaining the same values for the b gains as in Part 1 (i.e. b0 = 0.3, b1 = 0.5 and b2 = 0.2), display the input (i.e., SAMPLE-HOLD output as shown in Figure 5a) and output signals.
Sketch the original half-rectified sinusoid and discrete output from the SAMPLE/HOLD in the
graph below.
Graph 2: inputs and sampled outputs
Confirm that there are 8 samples of each half wave sine input. This is expected as the sampling
clock is 800Hz and the input sinewave is 100 Hz.
Note there are four nonzero pulses in the delay line input sequence and six in the summer
output. The reason for this will emerge as we proceed.
Again, we will carry out a deconstruction of the output signal in terms of the time offset
contributions, i.e. we will trace the output pulse amplitudes in terms of the
input values. We could use the same method as in Part 3, however, there is an
interesting alternative. We will separately observe and compare the three individual
contributions into the output ADDER, i.e. the signals added through gains b0, b1 and
b2 in turn. Examples of these signals are shown in Figure 5b below.
Experiment 5 – Unraveling Convolution
© 2011 Emona Instruments
5-9
Figure 5a: example input signal and 4-level sampled signal
Figure 5b: example individual bo, b1 & b2 inputs to adding junction
16.
We begin with the contribution through b2. Temporarily disconnect the leads
corresponding to inputs b0 and b1. Observe and record the input and output signals in the graph
above. Confirm for yourself that this result is as expected. View the sampled input on one scope
channel and the individual output on the other scope channel.
17.
Now repeat for the b1 and b0 contributions. Only the outputs need be recorded since
the same input is used. Again, verify that the results are as expected.
When completed, you will have three scaled replicas of the input with time offsets.
18.
For each time slot, sum the contributions of the three output records and plot the
result. Verify that this agrees with the output signal produced when the three leads are
reconnected to the adder inputs.
Question 6
How does this process relate to the principle of “superposition” ?
In summary, we have just passed a sequence of discrete values (our sampled input) through a
system with a particular response (a series of unit delays with multiplying taps) to produce an
accumulation ( the adder) of discrete product terms which are then output.
Each one of these individual contributions is the scaled and shifted version of the unit pulse
response of this system. Another way of thinking about this system as as it being a series of
weighting coefficients.
Now we need to think about representing this process mathematically.
An obvious way to start is to write a separate formula for the six nonzero output pulse
amplitudes . Let us name them y(1), y(2), …, y(6).
5-10
© 2011 Emona Instruments
Experiment 5 – Unraveling Convolution V1.2
19.
Label them on your sketch above. Pay attention to the orientation you use.
Each consists of a sum of three products, i.e., of a tap gain b0, b1, or b2 and an input pulse
amplitude.
The input sequence has eight elements (as previously observed, four of these are zero). Label
these x(1), x(2), …, x(8) (you should find it convenient to choose x(1) = x(2) = x(7) = x(8) = 0).
20.
Label these on your sketch above.
Numeric indexing is useful as an aid in looking for the general pattern.
The next step is to use symbolic indexing so that the set of formulas can be condensed into one.
Here are some of the formulas as they appear with numeric indexing:
y(3) = b0 .x(3) + b1 .x(2) + b2 .x(1)
y(4) = b0 .x(4) + b1 .x(3) + b2 .x(2)
y(5) = b0 .x(5) + b1 .x(4) + b2 .x(3)
y(6) = b0 .x(6) + b1 .x(5) + b2 .x(4)
Question 7
Write down the formula for y(2) and y(1) ? Discuss any unexpected differences .
The general pattern is readily apparent. With symbolic indexing, we can replace these with the
single formula
y(n) = b0 .x(n) + b1 .x(n -1) + b2 .x(n - 2)
where n represents the position in the sequence as a discrete time index.
In Part 2, we found that the unit pulse response h(k) for this system is
h(0) = b0 , h(1) = b1 , h(2) = b2 .
Hence we can express the formula for y(n) as
y(n) = h(0) .x(n) + h(1).x(n - 1) + h(2).x(n - 2)
i.e.,
y(n) = sum over range of k {h(k).x(n - k)}
This simple formula is known as convolution.
Experiment 5 – Unraveling Convolution
© 2011 Emona Instruments
5-11
For the discrete signal case we have implemented it is also known as the convolution sum or the
superposition sum. It can also be expressed in this more compact form:
y(n) =
y(n) = x(n) * h(n) where * represents the “convolution” operator.
It provides a neat way of packaging the procedures you carried out in your investigation, and it
allows the range of k to be extended to any desired value. It is the mathematical representation
of the interaction between the input sequence and the unit pulse response of the system.
Note in particular the term x(n-k). Since we are computing in terms of k, this is a time reversed
version of x(k). This issue is put into context further in a later experiment introducing
correlation, an operation which has a similar looking equation. This subject is covered in detail in
the text by Oppenheimer p90 (see Reference section for details)
Question 8
Explain why this term is reversed and what does this mean ?
For the continuous signal case, the summation is achieved using integration and is known as the
convolution integral or the superposition integral. This topic is covered in detail in many texts
including Lathi and Oppenheimer. (see Reference section for details)
In brief, the convolution integral for continuous signals comes about by increasing the number
of discrete samples whilst reducing their width to a limit of 0, such that the sum of many
products can be represented by the integration of the product.
Its equation is
y(t) =
In the above exercise, we obtained the result by means of a superposition of dissected parts of
the unit pulse response. In Part 3 we also used superposition; however, it did not involve
breaking up the unit pulse response. This method was easier to apply for that case because the
input sequence had only two pulses.
Compare the two methods when used with the input in Part 4 (do this with the help of sketch
graphs as before). Consider possible advantages of one over the other as a vehicle for deriving
the convolution formula.
21.
Repeat this process with a new system response. Let each tap be equal to 0.333 ie:
b0=b1=b2=0.333.
5-12
© 2011 Emona Instruments
Experiment 5 – Unraveling Convolution V1.2
Question 9
What is a common label for this response ?
Part 5: sinewave input
Up to this point we have used relatively short input sequences. This made it easy to trace the
passage of the pulses through the system since output segments remained distinct and could be
readily referenced to the corresponding input segment. However, we need to consider whether
the formula that we obtained on this basis is also valid in the more general and usual situation
when the input signal is an ongoing stream without breaks.
For this purpose we will go over the steps in Part 4 using a sinewave as input. This means there
is no "natural" reference marker, hence you will need to designate one.
Nevertheless, keeping track of time points will be straightforward since the signal is periodic
and the number of samples per period is relatively small.
22.
We use the setup in Part 4, and re-use “full sinewave” output at DAC-0 by bypassing the
RECTIFIER.
As before, since the sinewave frequency is a submultiple of the PULSE GENERATOR clock, no
slippage occurs in the scope displays.
23.
Proceed as in Part 4, connecting only one of the three adder inputs. As in Part 4, observe
and record each of the separate output sequences for each input in turn.
As you would expect, these are scaled and delayed replicas of the input, i.e. they are sampled
sinewaves. Carry out spot checks to verify that amplitude and phase relative to the input are
correct. (Reminder, since there are eight samples per period, the phase shift corresponding to
a unit delay is 45 degrees.)
Experiment 5 – Unraveling Convolution
© 2011 Emona Instruments
5-13
24.
For each time slot, add up the contributions of the three output records obtained above
and plot the result. Verify that this agrees with the output signal produced when all the input
leads are reconnected to the adder.
Graph 3: sinewave input signals
Now we are ready to revisit the formula obtained in Part 4. Proceed as before and show that
the formula remains valid for this case.
Question 10
Show that the formula remains valid.
This completes the basics of the convolution formula.
5-14
© 2011 Emona Instruments
Experiment 5 – Unraveling Convolution V1.2
In the next task we take a closer look at the output signal obtained in Part 5, and show that it is
a sampled sinewave. This is a discrete-time example of the investigation in Lab 4, where we
discovered that when the input of a linear system is sinusoidal, the output will also be sinusoidal.
25.
Confirm that the eight pulses making up one period of the output sequence represent
samples of a sinewave. A straightforward method is to exploit the sum of squares identity.
Since there are eight samples per period, you can match pairs of samples that are 90 degrees
apart (how many pairs can you find?). Note that knowledge of the peak amplitude is not essential
for this (all that is needed is to show that the sum of the squares is the same for each pair).
Question 11
Show your working for the sum of squares analysis ?
26.
Use the FUNCTION GENERATOR tuned to 100 Hz to provide the alternative
unsynchronized sinewave input. Set it to an amplitude of 2Vpeak (4V pp). The resulting slippage
effectively produces an interpolation of the samples -- a useful exploitation of something that
is usually unwanted!
Now that we have a simple way of displaying the input and output sinewaves, an interesting
additional item to examine is the theoretical verification of the measured output/input
amplitude ratio and phase shift. The peak values are clearly apparent, so the amplitude ratio
measurement is straightforward. Similarly, the now discernible zero crossings can be used for
the phase shift measurement.
The math for the theoretical verification is done by means of the application of the formula for
reducing the sum of sinusoids .
In the next Part we investigate an example of effects that can be obtained by means of suitably
chosen values for the tap gains.
Experiment 5 – Unraveling Convolution
© 2011 Emona Instruments
5-15
Part 6: mystery application
Here we explore a special application of convolution. The setup is the same as for Part 5,
except for new tap gains in the delay line. Two sets will be tested.
The first is b0 = 0.3 , b1 = 0.424 , b2 = 0.3
The second set is b0 = -0.3 , b1 = 0.424 , b2 = -0.3
27.
When the tap gains have been adjusted to the first set of values,( b0 = 0.3 , b1 = 0.424 ,
b2 = 0.3), display the output, and measure the amplitude and phase relative to the input as in
Part 5. Confirm that this is in agreement with theory.
28.
Alter the tap values to the second set of values (b0 = -0.3 , b1 = 0.424 , b2 = -0.3) and
repeat the measurements. Is the outcome predicted by theory ?
29.
Vary the frequency of the sinewave over a suitable range to demonstrate that sinewave
inputs with frequencies near 100 Hz are heavily attenuated.
Question 12
Why is the outcome obtained above described as filtering?
In other labs you will discover how to choose the tap gains to design different kinds of
responses versus frequency.
5-16
© 2011 Emona Instruments
Experiment 5 – Unraveling Convolution V1.2
Tutorial questions
Q1 Look up your textbook to find out the name given to systems that have the structure
in Figure 1, i.e. without feedback. Consider an alternative discrete-time
system that has a single delay element with feedback (refer to your textbook
if needed). Show how to apply the convolution formula in this case.
Q2 Consider the condition(s) required to avoid infinite output values in applications
where the unit pulse response is not time limited. Show to avoid an unstable
output in the feedback system in Q1.
Q3 Consider a process that consists of taking a running average of a data sequence, such
as atmospheric pressure. Suppose we do this by taking the sum of 50% of the
middle value, and 25% of the preceding and following values. Can this process
be described as convolution? If so, write down the unit pulse response.
Q4 Using integration instead of discrete summation, write down a continuous-time
version of the convolution formula in terms of the system impulse response.
Experiment 5 – Unraveling Convolution
© 2011 Emona Instruments
5-17
5-18
© 2011 Emona Instruments
Experiment 5 – Unraveling Convolution V1.2
Experiment 6 – Integration, convolution, correlation
and matched filters
Achievements in this experiment
You will be able to intuitively visualize the processes of integration, convolution, correlation and
“matched filters”. You will appreciate the similarities and differences between these processes.
You will see them in action with real signals in the time domain and understand their use in
practical applications. You will learn another Greek word.
Preliminary discussion
Any two signals may differ in a number of ways. They may have different amplitude
distributions, mean values, autocorrelation functions, variances etc. During transmission and may
have undergone amplitude variation, bandwidth limiting and phase delay effects.
At the receiver is useful to apply and processes to recover the transmitted information from
the received signal and to maximise the SNR of the received signal.
Some of the tools we have at our disposal are processes such as integration, correlation,
decision point timing and thresholding.
All of these processes have the equivalent processes in the frequency domain, however in this
experiment we will concentrate on time domain effects.
Figure 1: SIGEx SFP screenshot with TAB 6 selected
6-2
© 2011 Emona Instruments
Experiment 6- Integration, convolution, correlation V1.2
Pre-requisite work
Question 1
For both a maximal length PRBS, of 31 and 63 bit length, calculate the ACF function values for
all possible positions.
Question 2
Calculate the sequence from a 5 bit LFSR using feedback taps 5 & 3.
Question 3
(a) For the set up in Fig11, write down an expression for x(t) in terms of the input y(t) and the
S.U.I. impulse response h(t).
(b) Write down an expression for the CCF of x and y, and substitute the expression for x from
(a).
(c) Demonstrate that the result in (b) can be reduced to the convolution of h(t) and the ACF of
the input.
(d) Show that if the ACF of y is an impulse function, the output of the cross-correlator gives
h(t) (with a scaling factor).
(e) Demonstrate that if the input is white noise the ACF is an impulse
Question 4
(a) In the term "matched filter" what are the items that are matched?
(b) What is the role of the MF in a digital communication receiver?
(c) describe the operation of the "integrate & dump" process in a digital communication receiver
(d) explain why the I&D receiver is effectively a filter with a square pulse as its impulse
Experiment 6- Integration, convolution, correlation
© 2011 Emona Instruments
6-3
response
(e) extend (d) to explain why the I&D receiver is the matched filter for square pulseform data
sequences in additive white noise
6-4
© 2011 Emona Instruments
Experiment 6- Integration, convolution, correlation V1.2
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should observe
the USB light turn on (top right corner of ELVIS unit).The PC may make a sound to indicate that
the ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that
all three power LEDs are on. If not call the instructor for assistance.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with
the NI ELVIS that you’re using.
9.
You’re now ready to work with the NI ELVIS/SIGEx bundle.
10.
Select the EXPT 6 tab on the SIGEx SFP.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the
STOP button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the
top of the window. This will allow the program to conduct an orderly shutdown and close the
various DAQmx channels it has opened.
Ask the instructor to check
your work before continuing.
Experiment 6- Integration, convolution, correlation
© 2011 Emona Instruments
6-5
Experiment
This experiment consists of 5 parts. These parts can be presented separately or in
combination depending on the lab time available or curriculum preferences.
Part 1: Auto-correlation & Cross-correlation of PRBS sequences
A PN or PRBS sequence, known as pseudo-noise sequence or Pseudo Random Bit Sequence, is a
periodic digital sequence with well-known and defined sequence length. Maximal length
sequences are simple to generate and three of such sequences are available at the SEQUENCE
GENERATOR module. Refer to the User Manual for specific details.
We will be mostly using a bipolar version of the PRBS. Rather than being a 5V TTL level data
stream, it will be a +/- 2V data stream. This lends itself better to analog processing and is the
same spectrally except for the lack of a DC component.
In this part of the experiment we wish to compare a sequence with itself, and with delayed
versions of itself. This process is known as an auto-correlation, meaning a correlation or
comparison, with itself (from the Greek word “autos” meaning “self”).
As shown in Figure 3, to effect the comparison in an analog manner, we will input the same
signal into both inputs of the “correlator”. We will then multiply the incoming signals by each
other and then integrate over the sequence period. If any part of a signal is the same the other,
then the product will be maximum. Any differences between the incoming signals will result in a
smaller signal. By detecting the maximum product point we can determine the instant at which
the sequences are most alike. The signal output by the correlator is in this case known as the
auto-correlation function (ACF). It is a function of the delay index, or “lag” between inputs.
Another important point is that the comparison is carried out over the entire period of the
signal, which in this case is the entire period of the sequence. Remember that these PRBS
sequences are repetitive.
Figure 2: correlator operation diagram
6-6
© 2011 Emona Instruments
Experiment 6- Integration, convolution, correlation V1.2
In Figure 2 is sketched the operation of the implemented “correlator” for two cases. On the
left is the condition where the two inputs are aligned. On the right is the case where the two
inputs are not aligned, even though it is the same bit sequence. It is easy to see how the
combination of the MULTIPLER and INTEGRATE & DUMP modules are easily able to discern
between these two conditions. The final “ACF output value” is an indication of the level of
similarity between the two inputs.
11.
To implement this suggested methodology, patch together the modules as shown in
Figure .
Figure 3: block diagram for ACF of SG PRBS sequence. DELAY = n/fclk; n=0,1,2 or 3.
Figure 4: patching diagram for ACF of PRBS sequence
Settings are as follows:
PULSE GENERATOR: Frequency=3000 Hz; DUTY CYCLE=0.5 (50%)
SCOPE: Timebase 40ms; Rising edge trigger on CH0; Trigger level=0V
SEQUENCE GENERATOR: DIPS set to UP:UP (short sequence)
12.
Confirm for yourself that this signal at points B,C & D are delayed versions of the
original signal from point A using the scope.
Experiment 6- Integration, convolution, correlation
© 2011 Emona Instruments
6-7
13.
The reference input is connected to be MULTIPLIER input X DC. The input to be
compared is input to the MULTIPLIER input Y DC. Confirm that the lead from Y DC is currently
connected to the reference input X DC at terminal A. We will now compare the original sequence
with itself.
14.
View the output of the MULTIPLIER, kXY, and confirm that it is a constant DC level.
Question 5
What voltage is the output of the MULTIPLIER ? Explain why this is so
15.
View the INTEGRATE & DUMP output and confirm that it is ramping up until reaching
saturation as expected. The INTEGRATE & HOLD output will hold the maximum voltage reached
which is the voltage at saturation.
Note that the INTEGRATE & DUMP module is a clocked module and resets its output ie: dumps
its output at the end of every clock period. In this setup we are clocking this module with the
SYNC pulse from the SEQUENCE GENERATOR which occurs at the beginning of the sequence,
and hence has a period equal to the length of the sequence.
16.
Estimate the actual voltage that the ramp would have reached if the module had not
saturate. This is easily done by calculating the rate of the ramp in V/sec and multiplying by the
period of the integration.
Question 6
What voltage would the ramp have reached if it had not saturated ?
This voltage represents the degree of similarity between the two inputs to the MULTIPLIER
module. As you would expect it is at a maximum as the inputs are identical and aligned.
17.
Now to compare with a non-identical sequence we delay the original sequence by several
bit periods and repeat the correlation. Move the lead connected to Y DC to terminal B.
NOTE: although we are comparing the SAME sequence against itself, it is no longer aligned (ie:
bit0 versus bit0) and so at various bit positions it will differ.
6-8
© 2011 Emona Instruments
Experiment 6- Integration, convolution, correlation V1.2
Figure 5: examples of I&D, I&H, and CLK signals during autocorrelation,
at various instants
18.
View the output I&D as well as input IN. Confirm that the output I & D is no longer
ramping to saturation but is varying in both positive and negative direction. This is expected as
the signal from the MULTIPLIER is no longer a DC voltage but is a varying bipolar signal.
19.
Measure the voltage at output I & H. This is the held voltage following integration at the
end of the clock period. And it is the same for each period as the sequences are repetitive.
Question 7
What voltage is at the I & H output ?
20.
Move the flying lead connected to Y DC to terminals C & D, and note the held voltage
alongside the voltage when the lead was at terminal B.
It is safe to assume that the held voltage will be more or less the same for the remaining other
delayed bit positions. Refer to the “correlator operation” diagram above if this is not clear.
Experiment 6- Integration, convolution, correlation
© 2011 Emona Instruments
6-9
21.
Sketch these held voltages versus the delay index “n” with no delay being n=0 .
Remember that there are 31 possible positions . You will currently be plotting readings for only
4 positions.
Graph 1: Plot of I & H voltages vs. delay position n
The plot you have just sketched is a portion of what is known as the autocorrelation function
(ACF) of that particular sequence. It is normally calculated using the following equation:
rxx(߬) =
Question 8
How well do these results correspond with your theoretical expectations from the pre-lab
preparation work ?
The purpose of this exercise was to show you how these equations can be manifested as real
signals. An ACF will show the degree of uniqueness of a particular signal, and degree of
similarity, which is minimal when dealing with such MLS sequences used for coding and
encryption purposes.
6-10
© 2011 Emona Instruments
Experiment 6- Integration, convolution, correlation V1.2
22.
Repeat these steps and determine the ACF for the 63 bit sequence output from
SEQUENCE GENERATOR when the DIPs switches are set to UP:DOWN. This is a maximal
length sequence also.
23.
Sketch this ACF on the same graph as for the previous results.
Question 9
How well do these results correspond with your theoretical expectations ?
24.
Measure and plot the ACF for the sequence from the SEQUENCE GENERATOR for DIP
switch position DOWN:UP.
Question 10
Based on your measured ACF, what can you say about this sequence ?
25.
Set the SEQUENCE GENERATOR back to DIP switch position UP:UP and change the
patching according to figure .
Figure 6: patching for the ACF of PRBS sequence, with DAC-based PRBS
26.
On TAB 6, select radio button “SG PRBS”. An exact copy of the 31-bit PRBS output from
the SEQUENCE GENERATOR is available at DAC-0. This sequence is generated by LabVIEW
and so is the clock signal which is output from DAC-1. As well, from the TAB 6 of the SIGEx
Experiment 6- Integration, convolution, correlation
© 2011 Emona Instruments
6-11
SFP you can vary the delay of this output sequence from 0 to 30 bit periods. This will allow you
to take readings for the entire ACF.
27.
The clock signal provided from DAC-1 is 3.3kHz. This is similar to the previous clock
rate used. View the SYNC signal from SEQUENCE GENERATOR on CH1, and the INTEGRATE
AND DUMP output. Confirm that this signal as stable and that the product of the MULTIPLIER
module is being integrated. You may need to stop and restart the SFP to initialise this mode.
28.
Vary the delay index “n” on the SFP TAB 6, and observed the change in the integration.
As you vary this control you will observe in the ANALOG OUT viewer window, which displays the
output signals DAC-1 & DAC-0, that the sequence is shifted by the number of bits selected by
control “n”. In this exercise you are able to shift one of the sequences over the complete range
of possible positions, that is from 0 to 31 bits. Switch Y-AUTOSCALE off for a stable display.
29.
View both the INTEGRATE AND DUMP output as well as the INTEGRATE AND HOLD
output with the scope. As you vary the delay index “n” from 0 to 30, ie: over 31 bits, note the
output level from the INTEGRATE AND HOLD output and plot this value versus the delay index
on the graph used in the previous exercise.
Question 11
What observations can you make about this signal from its ACF ?
Part 2: Cross-correlation function (CCF) of PRBS sequences
Now that you are familiar with a methodology for correlating sequences, it will be interesting to
compare two different sequences. The same methodology is used, except given that we know the
sequences are different, we will expect there to be multiple smaller correlation points and no
one single major similarity. However, the cross-correlation process will give us this information
in an objective manner.
The sequence from the SEQUENCE GENERATOR when DIPs are in the UP:UP position is as we
know from above, a Maximal Length Sequence (MLS) of 31 bits in length, generated using a
Linear Feedback Shift Register (LFSR) with 5 registers, and using taps 5 & 3.
There are a few other tap settings which will create a MLS from a 5 register LFSR, and the
signal output from ANALOG OUTPUT: DAC-0 is generated using taps 5 & 2.
In this exercise you will compare our reference sequence from the SEQUENCE GENERATOR
and this new sequence and plot the cross-correlation function.
For two signals, x(t) and y(t), the cross-correlation function for a time limited pulse is :
rxy(߬) =
6-12
© 2011 Emona Instruments
Experiment 6- Integration, convolution, correlation V1.2
30.
Patch together the modules as per the figure below. Notice that whilst it is essentially
the same, the source of the bit clock is from the DAC-1 output. The ANALOG OUTPUT module
is providing the “master clock” for both itself and the SEQUENCE GENERATOR. In this way we
are guaranteed that both sequences bit output transitions are aligned. As well, the SYNC pulse
which occurs once every 31 bits is output from the SEQUENCE GENERATOR and provides the
integrate period clock for the INTEGRATE AND DUMP module.
Figure 7: block diagram for cross-correlation of differing PRBS
Figure 8: patching diagram, for cross-correlation of differing PRBS
Settings are as follows:
SCOPE: Timebase 40ms; Rising edge trigger on CH1; Trigger level=1V
SEQUENCE GENERATOR: DIPS set to UP:UP (short sequence)
31.
The clock signal provided from DAC-0 is 3.3kHz. This is similar to the previous clock
rate used. View the SYNC signal from SEQUENCE GENERATOR on CH1, and the INTEGRATE
AND DUMP Output. Confirm that this signal as stable and that the product of the MULTIPLIER
module is being integrated.
Experiment 6- Integration, convolution, correlation
© 2011 Emona Instruments
6-13
32.
Vary the numeric control “n” on the SFP TAB 6, and observe the change in the
integration. As you vary this control you will observe in the ANALOG OUT viewer window, which
displays the output signals DAC-1 & DAC-0, that the sequence is shifted by the number of bits
selected by control “n”. In this exercise you are able to shift one of the sequences over the
complete range of possible positions, that is from 0 to 31 bits.
33.
View both the INTEGRATE AND DUMP output as well as the INTEGRATE AND HOLD
output with the scope. As you vary the shift index “n” from 0 to 30, that is 31 bits, note the
output level from the INTEGRATE AND HOLD output and plot this value versus the shift index
on the graph used in the previous exercise.
Question 12
What observations can you make from this cross-correlation about the nature of the two
sequences ?
Question 13
Write down the 31-bit pattern for both PRBS sequences here. Note also the number of bit
pattern “runs” ? Why is the pattern “00000” not present ?
6-14
© 2011 Emona Instruments
Experiment 6- Integration, convolution, correlation V1.2
Part 3: Auto-correlation & matched filters
34.
Select the radio button “exponential pulse”, which will output an exponential pulse at
DAC-1 and a time-shiftable copy at DAC-0. The time-shift is controlled by the DELAY INDEX
control which varies from 0-100 for one complete period. Stop and restart the SFP to initialise
after selecting.
35.
View this pulse and sketch at on a graph below. Measure and document the time constant
and peak voltage on your sketch. NB: time constant is time to decay to 36% (1/e) of initial value.
Graph 2: using exponential pulses
36.
View both outputs from the ANALOG OUT block, DAC-0 and DAC-1, and confirm that
you are able to delay them relative to each other by varying the delay index “n” on the TAB 6 of
the SFP. You can also use the ANALOG VIEWER window to view this.
Experiment 6- Integration, convolution, correlation
© 2011 Emona Instruments
6-15
Figure 9 : Patching for ACF of exponential pulses
37.
Patch together the experiment as per Figure 9. Vary the delay index control “n” over its
full range (0-100) and plot the ACF on the graph above as previous sections. For convenience,
take a reading every 10 index points.
The output signal you have just plotted will be the auto-correlation function (ACF) for the
exponential pulse.
Question 14
How could you describe this function ?
38.
Refresh your memory about the impulse response of the RC NETWORK block and
sketch this impulse response in the graph above.
REMINDER: To measure the impulse response of the RC NETWORK simply input a narrow pulse
from the PULSE GENERATOR into the RC network and measure its output response. Set the
PULSE GENERATOR to FREQUENCY = 100 Hz, DUTY CYCLE = 0.02 (2%).
If the signal is applied to a system, and that signal is a time-reversed version of the impulse
response of a system, several interesting observations can be made.
Firstly, let the system be the RC NETWORK, which has an impulse response shaped like a
decaying exponential pulse. Let us name it, h(t).
Secondly, if the signal to be applied is a time-reversed version of that response, then it will look
like a rising exponential pulse, and let us name it x(t).
6-16
© 2011 Emona Instruments
Experiment 6- Integration, convolution, correlation V1.2
Then, the output of the system, y(t), will be the convolution of the input with the impulse
response.
Expressed mathematically:
y(t) = x(t)*h(t)
:convolution
y(߬) = ∫x(t).h(߬-t) dt
Since h(t) = x(-t)
y(߬) = ∫x(t).x(t + ߬) dt
y(t) is the same as the auto-correlation (ACF) of the input x(t).
As well, when a signal is input to a system whose impulse response is equal to a time-reversed
version of the input, then the system is called a “ matched filter”. Called this simply because it's
characteristic response is “matched” to the input waveform. A “matched filter” is the optimum
filter for the input waveform to which it is matched, in order to maximise the SNR at the
receiver. More about that later in this experiment.
What is of interest at this point is the observation that passing a signal through its “matched”
filter results in an output which is equal to the autocorrelation of the input signal. So we can see
that autocorrelation and convolution involving matched filters are related.
39.
Patch up the experiment in figure 10, and plot the output waveform on a graph above.
Note that in this step there is no shifting of static signal, but that this signal simply passes
through the system, RC NETWORK block, and is acted upon, or modified, by the system
according to its characteristic response. That interaction is what we call “convolution” .
(This example is covered in a pre-lab preparation question of Experiment 10.)
Figure 10 : Patching diagram for inputting “shaped” pulse into its “matched filter”
So far in this experiment you have encountered the implementation of two similar equations.
The one for convolution, with the time-reversed component, and one for correlation, with no
time-reversal. They have both involved integration over a period of time, T, and because we are
working with time-limited single pulses we can restrict the integration to a single complete
period of the pulse or sequence, rather than the general form of the equation where the
integration is carried out over the time period -∞ < t +∞ .
Experiment 6- Integration, convolution, correlation
© 2011 Emona Instruments
6-17
Part 4: Determining impulse response using input/output correlation
By now you should be starting to get a feeling for the relationship between input, output and the
system response. It is important to know the characteristic response of the system in order to
be able to effectively pass information through that system. In this part of the experiment we
will explore a system’s response by probing it with random noise and seeing what the correlation
between its input and output waveforms can tell us about this system.
Remember that when we probe a system using an impulse, we are in fact inputting many
simultaneous frequencies into that system. Consequently its output, its impulse response, are
those many frequencies either being increased, reduced, delayed or left intact by the system.
In order for this method to work correctly the bandwidth of the input should be an order of
magnitude greater than the bandwidth of the SUI.
Question 15
Is the bandwidth of the proposed input for this exercise adequate for this application ?
40.
Patch up the experiment in figure 11 . The RC NETWORK is to be implemented as the
System under Investigation (SUI). The clock signal at DAC-1 is 3.3kHz.
Figure 11: PRBS noise S.U.I , followed by correlator
6-18
© 2011 Emona Instruments
Experiment 6- Integration, convolution, correlation V1.2
Figure 12: Patching diagram for S.U.I investigation
41.
Select the radio button “SG-PRBS”. Once again to run the correlation you will need to
shift the reference SG-PRBS over its entire range of the delay positions, and plot the output of
the correlator (the INTEGRATE AND HOLD output value) versus the delay index value.
42.
Plot this waveform on the graph below with the RC NETWORK implemented as the
System under Investigation. Do a quick run through to determine the range of values to expect.
This will help for plotting. Then quickly plot every point.
Experiment 6- Integration, convolution, correlation
© 2011 Emona Instruments
6-19
Graph 3: in/out correlation plots
Question 16
Describe the output waveform from the correlator for the RC NETWORK SUI ?
Resembles the impulse response of an RC NETWORK.
Figure 13 : in/out waveforms for both SUI; RC NETWORK & TLPF
6-20
© 2011 Emona Instruments
Experiment 6- Integration, convolution, correlation V1.2
43.
Vary the SUI from the RC NETWORK block to the TUNEABLE LPF. Set the
FREQUENCY control knob to the 9 o'clock position. Sweep the correlator from 0 to 30 while
viewing the correlator output and set the GAIN level of the TUNEABLE LPF to avoid the
correlator output voltage being greater than 10V. (This is to avoid saturation of a signal).
44.
Plot the output of the correlator (the INTEGRATE AND HOLD output value) versus the
delay index value. Plot this waveform on the graph above.
Question 17
Describe the output waveform from the correlator for the TUNEABLE LPF ?
Even though the PRBS used as the noise source has only a short length this simple exercise has
been adequate to confirm in principle that correlation of input and output noise waveforms from
a system can be used to determine the impulse response of the system.
Part 5: Matched filtering using “integrate and dump” circuitry
The majority of transmitted data signals are in the form of square pulses. The optimum filter at
the receiver for maximising the signal to noise ratio for square pulses would have a square
impulse response. Since square pulses are symmetrical the time-reversal is not relevant.
By now you know that the output signal of such a filter is the convolution of the input signal and
the filter impulse response, which is the integration of the product of these two signals over
the bit period (for finite length signals). The product of two DC values (ie: square pulse & square
response) is a DC value. At the end of the bit period whatever the result of the integration of
this DC value is the filter output value for that bit alone.
Hence the combination of the multiply function and an “integrate and dump” function will
implement an optimum matched filter for square pulse data signals.
This is the same correlator structure we’ve been using throughout this experiment. In this case
we can skip the MULTIPLIER as the signals are both DC and it would only add a scaling factor.
45.
Patch together the experiment setup below in figure 15 .
Experiment 6- Integration, convolution, correlation
© 2011 Emona Instruments
6-21
Figure 14: block diagram for integrate & dump filtering
Figure 15: patching diagram for integrate & dump filtering
46.
Set the PULSE GENERATOR Frequency = 500 Hz, DUTY CYCLE = 0.5 (50%).
Select the radio button “ noise”, for an output of uniform white noise at DAC-1.
47.
At the SFP, setup the signal gain = 1.0 and the noise gain = 1.0. View the various signals:
noise free original data, data signal with noise, “integrate and dump” signal, and “integrate and
hold” signal. Notice the “filtering action” of the I & D on the wideband noise.
In communications jargon, the ratio of Signal to added Noise is known as the SIGNAL-NOISE
RATIO, SNR, and is usually expressed in decibels (dB), according to the following formula,
where V1 is the input signal level and V2 is the output signal level:
Gain (or attenuation) in dB = 20 log (V2/V1)
6-22
© 2011 Emona Instruments
Experiment 6- Integration, convolution, correlation V1.2
Figure 16 : examples of in/out waveforms for I & D filter
Question 18
How many errors do you estimate are occurring in the recovered data signal after the filter ?
48.
Reduce the signal gain to = 0.5 and leave the noise gain = 1.0. View the various signals:
noise free original data, data signal with noise, “integrate and dump” signal, and “integrate and
hold” signal. You are reducing the SNR.
49.
Reduce the signal gain and increase the noise gain to 2.0 (maximum) until you determine
that some errors may be occurring in the output signal.
Figure 17 :noise free data signal and I&D output after addition of noise,
with zero errors
Question 19
How do you determine when errors are occurring ? At what signal levels did this occur ?
Experiment 6- Integration, convolution, correlation
© 2011 Emona Instruments
6-23
50.
Plot some examples of the input and output signals to the INTEGRATE AND DUMP
filter for various levels of SNR.
Graph 4: integrate & dump filtering
You should be able to see in this exercise that despite the level of noise being very high relative
to the signal level integration process is a very powerful noise rejection capability. Put simply,
this is because the noise has an average value of zero whereas the signal as a distinctly positive
or negative average value. To see this even more clearly for yourself view the output of the
integrate and dump block for the signal alone and for the noise alone.
It can be shown theoretically that this integrate and dump filter is the optimum matched filter
for the square pulse data signals were used in this exercise. Hopefully in this experiment you
have been able to see this principle in action for yourself with real electrical signals.
51.
Reduce the signal level gain to 0 at the SFP and view the integration of the noise signal
alone. Have the noise signal gain set to 1.0
52.
Reduce the noise level gain to 0 at the SFP and view the integration of the data signal
alone. Have the data signal gain set to 1.0
6-24
© 2011 Emona Instruments
Experiment 6- Integration, convolution, correlation V1.2
Tutorial Questions
Q1. Determine the CCF between a sine wave and a cosine wave of the same frequency.
Q2.
(a) In working out the solution of the MF what are assumptions about the noise and the filter?
(b) What is the reason why gaussian noise is assumed in the theoretical derivation of the MF?
(c) What is a "power-limited system"? Give an example.
(d) The use of square pulseforms is not suitable in bandlimited applications. Explain.
Q3.
(a) What is the Fourier transform of the ACF?
(b) What happens to the correlation interval of the ACF when the bandwidth of the signal is
decreased (e.g. by means of a lowpass filter)?
(c) What is the ACF of white noise?
(d) The two-sided power spectrum of a given signal is a gaussian bell. What is the ACF shape? (
(e) What is the ACF of a sinewave?
Q4.
(a) Describe three methods for measuring the impulse response
(b) List disadvantages of using a narrow pulse input
(c) List advantages of the CCF method
(d) List advantages of the step function method
Experiment 6- Integration, convolution, correlation
© 2011 Emona Instruments
6-25
References
Lynn.P.A, “An introduction to the analysis and processing of signals”; Macmillan
Langton.C, ”Linear Time Invariant (LTI) Systems and Matched Filter”, www.complextoreal.com
G.R. Cooper and C.D. McGillem (Purdue), "Probabilistic methods of Signal and System Analysis",
Holt Rinehart and Winston 2nd Ed 1986
6-26
© 2011 Emona Instruments
Experiment 6- Integration, convolution, correlation V1.2
Experiment 8 – Exploring complex numbers and exponentials
Achievements in this experiment
You will be understand the fundamentals of complex numbers and their usage in signals analysis. You
understand the significance of the exponential and the use of the complex exponential ejθ. You
understand their relevance in terms of real signals.
Preliminary discussion
“That this subject [imaginary numbers] has hitherto been surrounded by
mysterious obscurity, is to be attributed largely to an ill-adapted notation.
If, for instance, +1, -1, √-1 had been called direct, inverse and lateral units,
instead of positive, negative and imaginary (or even impossible)
then such an obscurity would have been out of the question.
-Carl Friedrich Gauss (1777-1855)
This quote from Gauss is a good place to start in our exploration of what complex numbers mean and
how they are useful to mathematicians, scientists and engineers. From the very earliest use of these
concepts, confusion has abounded as to what and imaginary number actually is and how it is useful. By
relating as much as possible to real signals and their manipulation, this experiment hopes to clarify
and consolidate the understanding for the student about this issue.
Figure 1: portrait of Leonhard Euler 1707-1783
Leonhard Euler, one of the great mathematicians, was born in Basel,Switzerland. He achieved his
master’s degree at age 16. Private student of Johann Bernoulli, he then went on to author over 900
publications covering many diverse topics in mathematics and science. His memory was legendary and
he could recite the entire Aeneid word for word. A famous equation (worth reading about) is
Euler’s identity:
eiπ + 1 = 0
Although in general mathematics the imaginary part of a complex number is identified by the letter
“i”, in electrical engineering in order to avoid confusion with the symbol for current “i”, the symbol
for the imaginary part of a complex number uses the letter “j”. As there is much overlap between
mathematics and electrical engineering, you will at time find these symbols used interchangeably.
7-2
© 2011 Emona Instruments
Experiment 7 – Exploring complex numbers & exponents V1.2
Pre-lab preparation
Question 1
Confirm your understanding of the algebra associated with complex number by solving these
equations using the binomial method :
a) (3 + i2) + (5 – i6)
b) (3 + i2) x (5 – i6)
c) (3 + i2) - (5 – i6)
d) (a + ib) + (c + id)
e) (a + ib) x (c + id)
f) (a + ib) x (a – ib)
Question 2
Confirm your understanding of the algebra associated with complex number by solving these
equations using vectors. Sketch your working on the graph below:
a) (3 + i2) + (5 – i6)
b) (3 + i2) x (5 – i6)
c) (3 + i2) - (5 – i6)
d) (a + ib) + (c + id)
e) (a + ib) x (c + id)
f) (a + ib) x (a – ib)
Graph 1: Vector arithmetic
Experiment 7 – Exploring complex numbers & exponents
© 2011 Emona Instruments
7-3
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should observe the
USB light turn on (top right corner of ELVIS unit).The PC may make a sound to indicate that the
ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that all
three power LEDs are on. If not call the instructor for assistance.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with the
NI ELVIS that you’re using.
9.
You’re now ready to work with the NI ELVIS/SIGEx bundle.
10.
Select the Lab 7 tab on the SIGEx SFP.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the STOP
button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the top of the
window. This will allow the program to conduct an orderly shutdown and close the various DAQmx
channels it has opened.
Ask the instructor to check
your work before continuing.
7-4
© 2011 Emona Instruments
Experiment 7 – Exploring complex numbers & exponents V1.2
Experiment
Complex numbers
In the pre-lab preparation work you refreshed your understanding of the arithmetic to do with
complex numbers. Each complex number can be represented as a point at two-dimensional plane and
when joined to the origin of the plane with a line they are known as vectors. They can be expressed
in either rectangular or polar coordinates depending on convenience for the task at hand.
Figure 2: diagram of complex number
The relationship between the Cartesian and polar coordinates systems is as follows:
x = cos (θ); y = sin(θ);
also it is convenient to express the number using Eulers formula:
eiθ = cos(θ) + isin(θ)
The use of i for the vertical component denotes that it is the component in the vertical or imaginary
axis. The use of θ is fairly obvious as the angle the number makes with the horizontal or real axis.
However the introduction of ‘e’ is somewhat more perplexing, so we will briefly mention it here.
Overview of the derivation of eix using series
Knowing the following series:
ex = 1 + x + x2/2! + x3/3! + x4/4! + …
sin x = x – x3/3! + x5/5! - …
cos x = 1 – x2/2! + x4/4! - …
Where n! is factorial n
Now, substituting ix for x, we get
eix = 1 + ix + (ix)2/2! + (ix)3/3! + (ix)4/4! + …
And knowing that i2 = -1; i3 = -i; i4 =5 1; i = i and substituting in
eix = 1 + ix - x2/2! - ix3/3! + x4/4! + ix5/5! - …
If we separate the I terms, it gives
eix = (1 – x2/2! + x4/4! - …) + i(x –x3/3! + x5/5! - …)
From above we note that these terms are sin x and cos x, so
eix = cos x + i.sin x
Experiment 7 – Exploring complex numbers & exponents
© 2011 Emona Instruments
7-5
Complex functions
If the angle θ is replaced with a varying angle, varying in time, such as wt, then we have:
ejwt = cos(wt) + jsin(wt)
As the angle θ is now a function of time, then the vector is also moving in relation to time, and is in
fact rotating at the rate w, which corresponds to w/2π revolutions per second. We can call this
vector a “phasor”.
Let us take a look at what this means with real signals.
In this TAB, the SCOPE and XY GRAPH will BOTH display the actual signal is as connected to by the
scope leads. The phasor plot will display signals derived from the amplitude and phase settings on the
tab itself and output to the ANALOG OUT terminal , DAC-1 & DAC-0. These will be the same signals
if you make connections as outlined in experimental procedure.
Figure 3: patching for DAC1 to CH1, DAC0 to CH0, and both to ADDER
11.
Patch up a SIGEx model of the system in Figure.
Connect CH1 scope lead to DAC-1 output and CH0 lead to DAC-0 output.
Settings are as follows:
SCOPE: Timebase = 40ms, Trigger on CH1, Trigger level =0V
DAC1: Reference amplitude = 1; Phase = 0 degrees
DAC0: Reference amplitude = 1; Phase = -90 degrees
Question 3
Write the equation for signals at DAC-1 and DAC-0 as a function of time in the form: A.cos(wt + θ).
Think of the centre of the scope timeline as the instant t=0.
12.
Observe phasor plot and XY graph and confirm they are as expected.
Question 4
Explain why the XY graph displays a circle ?
7-6
© 2011 Emona Instruments
Experiment 7 – Exploring complex numbers & exponents V1.2
13.
Vary the phase settings for DAC-1 and DAC-0 and observe the phasor plot and XY graph.
Confirm your understanding of what they display.
14.
Disconnect the scope lead from DAC-0. The graphs will now only display signal from DAC-1.
Pay attention to the XY graph.
15.
Reconnect DAC-0, and disconnect the scope lead from DAC-1. The graphs will now only display
signal from DAC-0. Again pay attention to the XY graph.
Question 5
Explain the signal as viewed on the XY graph ?
16.
Vary the settings as follows :
Ref amplitude DAC1 = 1; Phase = -15 degrees
Ref amplitude DAC0 = 1.2; Phase = 75 degrees
Question 6
Write the equation for signals at DAC-1 and DAC-0 as a function of time in the form: A.cos(wt + θ).
17.
Vary the settings back as follows :
Ref amplitude DAC1 = 1; Phase = 0 degrees
Ref amplitude DAC0 = 1; Phase = 90 degrees
18.
Leave the CH1 scope lead connected to DAC1 output and move the CH0 scope lead to the
output of the adder “f+g”. You are now viewing the sum of the two sinusoids.
Question 7
Measure and document the equation for the sum of the two sinusoids. Compare this with the
expected resultant using the phasor method.
19.
Leave the CH1 scope lead connected to DAC1 output and leave the CH0 scope lead to the
output of the adder “f+g”. You are still viewing the sum of the two sinusoids. Set up phases to be 0
and 180 degrees. Try also 0 and -180 degrees.
Question 8
What is the output sum signal for these settings ? Is this expected ? Explain.
Experiment 7 – Exploring complex numbers & exponents
© 2011 Emona Instruments
7-7
Figure 4: SIGEx TAB 7 SFP detail displaying real signals via scope
On the phasor plot can be seen two phasors representing the outputs from DAC-1 and DAC-0, both
sinusoidal signals. The phasor plot shows the relative phase difference an amplitude of each signal
but not the rotation. Since are both rotating at the same frequency we do not need to display that
rotation. Consequently let us explore the addition of real signals and compare the results with those
expected from vector addition.
From above we have stated that
ejwt = coswt + j.sinwt
hence
v(t) = Acos(wt + θ)
= Re { Aej(wt+θ) }
= Re { Ae(jθ).ej(wt) }
Ae(jθ) sets up the initial conditions at t=0, ie amplitude & angle
As well, it can be shown that
v(t) = Acos(wt + θ) = A/2.ejθ.ejwt + A/2.e-jθ.e-jwt
Each term is a rotating phasor, with A/2. ejθ.ejwt rotating in a positive direction (counter clockwise)
and A/2e-jθ.e-jwt rotating in the negative direction.
Negative direction meaning negative frequency, which can be thought of in the same way as a motor
spinning in the reverse direction.
7-8
© 2011 Emona Instruments
Experiment 7 – Exploring complex numbers & exponents V1.2
These two terms show that sinusoid is the sum of two complex exponentials, which are the conjugate
of each other.
Let us display these conjugate signals with the SIGEx board.
20.
Settings as follows :
Ref amplitude DAC1 = 1; Phase = 20 degrees
Ref amplitude DAC0 = 1;
Set “Phase follow DAC-1” mode switch to ON
Setting “Phase follow” mode ON will automatically set the relative phase of the DAC-0 signal to be
the negative of the phase for the DAC-1 signal. That is, it will become the “conjugate” signal. Both
phasors will be symmetrical about the real axis.
21.
Vary the DAC-1 phase setting and observe the phasor plot showing the phasors moving
relative to each other. Bear in mind that the sum of these two vectors will fall on the real axis and
inscribe a real signal.
Next you will measure and plot the resultant real signal amplitude.
22.
Connect the scope lead CH-1 to DAC-1 and CH-0 to the adder output “f+g”.
23.
Vary the DAC-1 phase from 0-360 degrees in 15 degree steps and note these values in the
Table below. Take the peak amplitude value reading, not the peak to peak value. Watch the phasor
plot and document the amplitude as negative when the resultant signal is in the negative half-plane.
Experiment 7 – Exploring complex numbers & exponents
© 2011 Emona Instruments
7-9
Phase
(degrees)
Resultant amplitude
(Vpk)
Phase
(degrees)
Resultant amplitude
(Vpk)
0
2
210
-1.75
30
1.75
240
-1
60
1
270
0
90
0
300
1
120
-1
330
1.75
150
-1.75
360
2
180
-2
Table 1: resultant amplitude readings
24.
Plot these measured values on the graph below.
Graph 2:plot of resultant from measurements
7-10
© 2011 Emona Instruments
Experiment 7 – Exploring complex numbers & exponents V1.2
Question 9
What is the equation for this resultant signal ?
Exponential functions
When a quantity increases or decreases at a rate which is proportional to its value at that instant
then that process is known as an exponential process. Expressed as an equation, exponential decay is:
dN/dt =-kN where k is the decay constant
Solving this equation for N gives us:
N(t) = N0e-kt; where N0 is the initial value at t=0.
However, the general form for this equation is
N(t) = N0.b-kt ; where 0 < 1/b < 1
For every time period where t = 1/k, then N(t) will reduce by the factor of 1/b.
In a case where b=2, the function will halve every time constant.
In the case where b=e, then the function will reduce by a factor of 1/e (0.36) every time constant.
In this section we will use a simple loop to cause a pulse of known amplitude to decay at a variable
rate. We can control the rate of decay by adjusting the gain controls of the feedback path.
25.
Patch together the experiment as shown in figure .
Figure 5 : block diagram for exponentially decaying pulse generation
Experiment 7 – Exploring complex numbers & exponents
© 2011 Emona Instruments
7-11
Figure 6: patching diagram for decaying pulse generator
Settings are as follows:
ADDER gains: a0 = 1; a1 = 0.8; b=1; b1=0.36
SEQUENCE GENERATOR: DIPS: UP:UP
SCOPE: Timebase: 40ms; Trigger on CH0; Trig level = 1.0V
PULSE GENERATOR: Frequency = 1000Hz, Duty cycle=0.5 (50%)
26.
Confirm that the gain settings are as indicated and view the output of both loops
simultaneously with both scope leads.
27.
Confirm for yourself that the signal decaying rapidly is decaying at the rate 1/e = 0.36. You
would expect this as each sample out of the loop will be approximately 0.36 times less than the
previous input. Once the initial pulse is input to the loop, the input settles to zero and only feedback
signals remain in the loop.
28.
Try varying the gain a1 to various values between zero and one. In particular, set gain a1 to
0.5 and view this signal.
Question 10
What is the equation for this resultant signal for a1=0.5 ?
Question 11
What is another term for the time constant when a1=0.5 ?
What is most interesting of all these exponential waveforms is that when the decay rate equals ‘e’,
then the rate of change equals the value of the function itself at that instant. This only occurs when
the decay rate equals e. This form is known as natural decay, and occurs in many natural processes,
and is the basis for natural logarithms. Whilst it is outside the scope of this lab manual, it is very
interesting to read about the number e, epsilon, and is well covered in one of the references.
7-12
© 2011 Emona Instruments
Experiment 7 – Exploring complex numbers & exponents V1.2
Experiment 8 – A Fourier Series analyser
Achievements in this experiment
Compose arbitrary periodic signals from a series of sine and cosine waves. Confirm the Fourier
Series equation. Compute fourier coefficients of a waveform. Build and use a Fourier Series
analyser. Demonstrate that periodic waveforms can be decomposed as sums of sinusoids.
Introduce complex notation.
Preliminary discussion
In Lab 3 we discovered that sinewaves are special in the context of linear systems (time
invariance assumed). Unlike other waveforms, a sinewave input emerges at the output as a
sinewave. We saw in Lab 4 that this makes it possible to completely characterize the behaviour
of a system in this class by simply measuring the output/input amplitude ratio and the phase
shift of the sinewave as a function of frequency.
Now, this is fine if we only need to process sinewaves -- is it feasible to make use of this when
dealing with other kinds of inputs ? For example, when the input is a waveform like the
sequence of digital symbols we investigated in Lab 4.
One possibility is to see whether a non-sinusoidal waveform could be expressed as a sum of
sinewaves, over a suitable frequency range, either exactly, or even approximately. If this could
be done, the system output can then be obtained by exploiting the additivity property of linear
systems, i.e., first obtain the output corresponding to each sinusoidal component of the input
signal, then take the sum of the outputs. This was covered in Lab 4.
In this Lab we explore this idea by means of an ancient technique based on the generation of
beat frequencies -- somewhat like when a musician uses a tuning fork. We will begin by adding
together many beat frequencies and view the resulting waveform. From there we will look at the
equations and use what we know from trigonometry to decompose waveforms into their
constituent components.
Pre-requisites:
Familiarization with the SIGEx conventions and general module usage. A brief review of the
trigonometry required will be covered as needed.
8-2
© 2011 Emona Instruments
Experiment 8 – A Fourier Series analyser V1.2
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should observe
the USB light turn on (top right corner of ELVIS unit).The PC may make a sound to indicate that
the ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that
all three power LEDs are on. If not call the instructor for assistance.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with
the NI ELVIS that you’re using.
9.
You’re now ready to work with the NI ELVIS/SIGEx bundle.
10.
Select the Lab 8 tab on the SIGEx SFP.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the
STOP button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the
top of the window. This will allow the program to conduct an orderly shutdown and close the
various DAQmx channels it has opened.
Experiment 8 – A Fourier Series analyser
© 2011 Emona Instruments
8-3
The experiment
Part 1 – Constructing waveforms from sine & cosines
A simple sinusoid with zero phase can be represented by the equation
(Eqn 1)
And a cosinewave with a frequency of n times can be simply represented by the equation
(Eqn 2)
Any wave which is an integer multiple of another frequency is known as a harmonic of the
frequency. So equation 2 represents the harmonics of the signal in equation for n >1.
We can use the “harmonic summer” simulation in Experiment TAB 8 to view and sum multiple
signals which are harmonics of the fundamental signal ie: the signal for n=1. Harmonics for which
n is even are known as even harmonics, and when N is odd are called odd harmonics. This
simulation allows us to view and sum any sinusoid for which 1 <= n <= 10, that is, the fundamental
and nine harmonics. The numeric entry boxes allow you to enter the amplitude of each sinusoid.
11.
Experiment a little by entering various amplitudes into the numeric entry boxes for the
cosine row. Confirm for yourself visually that what is displayed is as you would expect.
12.
Set all amplitudes to zero, except for the “1st” component, the fundamental, which you
can set to 1. Moving from the second to the ninth sequentially, set each amplitude to the equal
to 1. As you do this notice how the combined signal is changing. Do this for the cosine row ONLY
at this point.
Question 1
How would you expect the summation of to look if you could add up many more harmonics ?
Question 2
What is its peak amplitude and is this as expected ?
Question 3
Is the fundamental an odd or even function ? Is the summation odd or even ?
8-4
© 2011 Emona Instruments
Experiment 8 – A Fourier Series analyser V1.2
Question 4
Write the equation for the summation of the 10 signals ? Is it symmetrical about the X axis?
Question 5
Vary the amplitudes and notice how the signal changes . You may set the amplitude of certain
components to 0 as you see fit. Can you create a wave form which starts at a zero value ?
Write the equation for your new varied amplitude signal ? Does it start at a zero value ?
13.
Sketch your arbitrary wave form, for which you have just written the equation, below.
Graph 1: sine wave summation
Experiment 8 – A Fourier Series analyser
© 2011 Emona Instruments
8-5
The general form of the equation for the summation of cosine harmonics is as follows:
where
are the amplitude values of each cosine wave.
14.
Set all cosinewave amplitudes equal to 0 and now set the sine waveform amplitudes equal
to 1, starting from the first harmonic and moving sequentially until the 10th harmonic. Notice
how the summation changes as you add harmonics.
Question 6
How would you expect the sine summation of to look if you could add up many more harmonics ?
Question 7
What is its peak amplitude and is this as expected ? Is this an odd or even function ?
Question 8
Vary the amplitudes and notice how the signal changes . You may set the amplitude of certain
components to 0 as you see fit. Can you create a wave form which starts at a non-zero value ?
Write the equation for your new varied-amplitude signal ? Does it start at a non-zero value ? Is
it symmetrical about X axis.?
The general form of the equation for the summation of sine wave harmonics is as follows:
where
are the amplitude values of each sine wave.
The general form for the summation of both sine and cosine harmonics is:
8-6
© 2011 Emona Instruments
Experiment 8 – A Fourier Series analyser V1.2
Even functions are symmetrical about the Y axis.
Odd functions are not symmetrical about the Y axis, but appear to be inverted about the X axis
on the negative side of the Y axis.
Sine and cosine waves, as well as their sums of sine and cosines, are always symmetrical about
the X axis. Being symmetrical means that they cannot represent a DC offset. In order to have a
DC offset we must add a constant to the equation.
15.
Add a DC component to the signal by inputting the value into the DC numeric entry box ?
Let us add a value a0 to represent that constant and the general equation for our arbitrary
wave form becomes
where N is max number of harmonics present. In our experiment above, N=10.
This equation describes any arbitrary wave form with the proviso that it is periodic. Any
periodic waveform, no matter how complicated, can be represented by the summation of many
simple sine and cosine waveforms. This equation is known as the Fourier series equation,
naturally enough named after Jean Baptiste Joseph, Baron de Fourier, 1768 – 1830.
Figure 1: Portrait of Joseph Fourier
Let us have a further look at this important equation.
When N equals one, this frequency known as the fundamental frequency, is also called the
resolution frequency. Being the smallest frequency in this series it defines the minimum
separation between components in a particular waveform.
In the Fourier series equation above we have grouped waves in terms of whether they are sine
or cosine waveforms. Let us now modify this equation and group individual components by their
harmonic number. This is easy enough and is as follows
Experiment 8 – A Fourier Series analyser
© 2011 Emona Instruments
8-7
Grouping in this way allows us to think of each frequency as having both a sine and cosine
component.
Let us have a quick revision of what adding a sine and a cosine of the same frequency together
may result in.
16.
Using the “harmonics summer” simulation in the Experiment 8 TAB of the SIGEx SFP,
set all amplitudes including that of the DC equal to 0. Set the amplitude of the sine and cosine
first harmonic equal to 1. View the resulting summation.
Question 9
Write the equation for the summation of these 2 waves ? Write the equation for the summation
in terms of the sine wave with a non zero phase shift.
17.
Vary the amplitudes from 0 to 1 and notice how the signal changes .
Question 10
Describe how the summation changes as you vary the respective amplitudes?
Question 11
For a particular pair of amplitudes you have set, write the equation for the summation in terms
of sine and cosine as well as its equivalent polar representation ?
8-8
© 2011 Emona Instruments
Experiment 8 – A Fourier Series analyser V1.2
18.
Sketch a vector or for phasor representation of these two signals and their resulting
summation, known as the resultant.
Graph 2: components & resultant
For each harmonic, nwt, a signal may have a sine and cosine component, implemented by their
own respective amplitude, an and bn. It is helpful to think about these component pairs as a
single entity at a particular frequency. The only difference between them being their respective
amplitude and the respective orientation or phasing. We know that a sine wave is 90° out of
phase with a cosine wave, a quality which makes those two components orthogonal to each other.
By orthogonal we mean that they operate independently of each other. And so we need a
notation and document these two-dimensional pair of components and this notation is provided
to us by Euler with his famous equation:
Where j the notes the component at 90° out of phase with the other component, hence the sine
and cosine pair.
Experiment 8 – A Fourier Series analyser
© 2011 Emona Instruments
8-9
Hence we can replace all sine/cosine pairs at a particular harmonic n with the complex
exponential function ejnwt, and in this way the Fourier series can be rewritten as
where Cn represents the resultant from the pair of respective amplitudes for the sine and
cosine components. ie: Cn2 = bn2 + an2, and is known as the “complex fourier series”.
Euler's formula allows us to process the sine/cosine component pair , simultaneously, rather
than individually. It is a form of two-dimensional notation where the sine and cosine components
for a frequency are instead treated together as a resultant with particular phase ie polar
notation. This notation is known as “complex” notation and was introduced in the previous
experiment, Experiment 7.
Remember that the practical part of this experiment just completed above shows us how a
resultant waveform is represented by two orthogonal components.
Part 2 – Computing Fourier coefficients
In the previous part of this experiment we discovered that we can construct any arbitrary
periodic waveform by the summation of a number of sine and cosine harmonics. This knowledge
is the basis of signal synthesis and in this part of the experiment we will do the reverse
process, that is, the analysis of an arbitrary wave form to discover the presence and amplitude
of its constituent harmonics. We call these amplitudes the Fourier coefficients of the waveform
as they are the coefficients within the Fourier series equation we have just derived.
Armed with only some basic trigonometry we will now explore some more qualities of sinusoids.
Before we create our arbitrary wave form to be analysed let us determine some rules with
which we can build our analyser.
We know that the area under a sine wave or a cosine wave over a timeframe of one period will
always be equal to 0. Consequently we know that the area under a sine wave or cosine wave for a
timeframe of any number of periods will also be equal to 0. To determine the area
mathematically we would integrate over the timeframe of interest. This process would give us
the average value of the signal during that time. This average value is the DC component of the
signal.
Figure 2: model for integrating one period of a sinusoid
8-10
© 2011 Emona Instruments
Experiment 8 – A Fourier Series analyser V1.2
The INTEGRATE & DUMP module can be used to integrate over a single period as denoted by
the input clock. This is well suited to integrating periodic waveforms such as sinusoids and
products thereof.
In this next exercise we will integrate a sinusoid over a single period to prove the assertion that
the area under a sine wave or a cosine wave over a timeframe of one period will always be equal
to 0.
19.
Connect the model as per Figure 2 above. The clock connection is necessary for this
module, unlike for the continuous time INTEGRATORS on the SIGEx board.
Settings are as follow:
SCOPE: Timebase: 4ms; Trigger on Ch0; Connect Ch0 to CLK input; Level=1V
FUNCTION GENERATOR: Select SINE output, AMPLITUDE = 4vpp, FREQUENCY=1kHz
Refer to the SIGEx User Manual for instructions on how the I&D and I&H functions work if
this is not obvious. NOTE: The I & D period is from positive clock edge to positive clock edge.
Question 12
What is the output value at the end of the integration period ? HINT: the I&H function will
hold the final value.
20.
While viewing the I&D output, to broaden your understanding of the integration process,
change the incoming signal from sine to triangle to bipolar squarewave at the FUNCTION
GENERATOR VI and confirm that the integrated output is as you would expect over one period.
You should be able to confirm that integration is the accumulation of the total signal charge,
with positive signals adding to the total, and negative signals subtracting from the total.
DC extraction using LPF
Another way of extracting the DC component of a signal is by using a filter which is tuned to a
sufficiently low-frequency to exclude all harmonics except for the DC component. The
integration process you have just explored is also a form of filtering, (as discussed in
Experiment 6 on matched filtering.) We will use the filtering method next.
Note that in this next part of the experiment the signals we are using are generated from data
arrays in LabVIEW and output in a synchronized manner from the ANALOG OUT terminals
DAC-0 and DAC-1.These signals are thus synchronized to each other, just as they appear on
screen and in the textbook.
21.
Using the left hand side of Experiment 8 TAB of the SIGEx SFP, select a sine wave , 1st
harmonic. Connect DAC-1 to the input of the TUNEABLE LPF and view both input and output
with the scope.
Settings are as follow:
SCOPE: Timebase 4ms; Rising edge trigger on CH0; Trigger level=0V
TUNEABLE LPF: GAIN=mid position
Experiment 8 – A Fourier Series analyser
© 2011 Emona Instruments
8-11
Adjust the frequency control, fc, of the TLPF from fully clockwise moving counterclockwise
until the output signal is a DC value of approximately 0 V.
Try this test for several harmonics. Change the “sine harmonic” value to 2,3 or 5 etc to try this.
INSIGHT 1: average value of an integral number of periods of sine or cosine waves equals zero.
Note that as far as the filter is concerned a single cosine wave is the same as a sine wave, so
Insight 1 holds true for both. You can vary the phase of the sinewave out of DAC-1 using the
“sine phase” control. The signal out of DAC-0 remains constant as a cosine wave.
22.
Set the “sine harmonic “ value back to 1. Set “sine phase” to 0. View DAC-1 with CH1
and DAC-0 with CH0.
Settings as follows:
SCOPE: Timebase 4ms; Rising edge trigger on CH0; Trigger level=0V
DAC-1 will be a sine wave and DAC-0 a cosine wave, relative to each other.
Connect as shown in the figure below. View the output of the MULTIPLIER as well as the output
of the TUNEABLE LOW PASS FILTER.
Figure 3: comparing sinusoids
You will now multiply a sine wave by a cosine wave and determine its average value.
Repeat this with the second and third harmonic of the sinusoid.
Question 13
What is the average value of these three products ?
INSIGHT 2: average value of an integral number of products of any sine or cosine harmonics
equals zero.
8-12
© 2011 Emona Instruments
Experiment 8 – A Fourier Series analyser V1.2
23.
Set the “sine harmonic “ value back to 1. Set “sine phase” to 90. View DAC-1 with CH1
and DAC-0 with CH0.
Settings as follows:
SCOPE: Timebase 20ms; Rising edge trigger on CH0; Trigger level=1V
DAC-1 will now be a cosine wave and so will be DAC-0.
NB: This is relative to the start of the signal data array. See the ANALOG OUT VIEWER
window for confirmation.
Connect as shown in the figure above.
View the output of the MULTIPLIER as well as the output of the TUNEABLE LOW PASS
FILTER with the scope.
You will now multiply a cosine wave by itself and determine its average value.
Question 14
What is the average value of the product of a cosine by itself ?
INSIGHT 3: average value of an integral number of products of any sine by its harmonic equals
zero.
INSIGHT 4: average value of an integral number of products of any sine by itself equals a non
zero value.The same holds for cosine.
Question 15
Write the complete formula for the product of a cosine, Acoswt, by itself? What do the terms
represent?
At this point we can see that the only product to yield a non-zero average value is that of a sine
wave by the same sine wave. The same is true for cosine. So if we multiply an arbitrary
waveform by a particular “probing” sine or cosine harmonic and extract the average value of this
product there will only be non-zero when that arbitrary wave form contains that particular
“probing” harmonic as one of its constituent components. This is a very powerful insight and
allows us to have a simple tool with which to analyse arbitrary wave forms.
Let us now give this a try.
24.
Construct the following arbitrary wave form using the HARMONIC SUMMER on the
right hand side of the SIGEx SFP, Experiment 8 TAB.
Settings are as follows :
Cosine amplitudes: 1, 0, 0.5,0,0,1,0,0,0,0
Sine amplitudes: 0,0.3,1,0,0,0,2,0,0,0
DC level: 0.5
Experiment 8 – A Fourier Series analyser
© 2011 Emona Instruments
8-13
Switch “to DAC-0” ON: This will output the summation to DAC-0. See the ANALOG VIEWER
window.
Set “sine harmonic” =1; Set “sine phase = 0. This signal is output to DAC-1.
SCOPE: Timebase 4ms; Rising edge trigger on CH0; Trigger level=0V
Figure 4: comparing sinusoids
25.
You will need to now ensure that the overall gain of the MULTIPLIER and TLPF is unity.
We know from the User Manual specifications that the MULTIPLIER gain is approximately
unity, whereas the TLPF GAIN is variable. Connect both MULTIPLIER inputs to the same 1st
harmonic sinewave and view the output. While viewing the TLPF output, turn the fc control
clockwise to pass the entire signal. Adjust the TLPF gain precisely for a 4Vpp output, so as to
have unity gain throughout the “analyser” path.
26.
Return to your previous setup but do not touch the TLPF GAIN setting for the rest of
the experiment. At this point we are multiplying the arbitrary wave form which investigated by
a first harmonic sine wave.
Figure 5: example of SIGEx SFP Tab “Lab 8” including
harmonic generator bottom left, and summer on the right
8-14
© 2011 Emona Instruments
Experiment 8 – A Fourier Series analyser V1.2
27.
View the product at the output of the multiplier on CH0 and the TLPF output. Adjust
the TLPF frequency control to isolate the DC component. Note the value of the DC level as the
amplitude of the first harmonic sinusoid in the table below. Increase harmonics one by one,
adjust theTLPF frequency setting if necessary and note the amplitude for the remaining 10 sine
harmonics in table.
Now to analyze for the cosine components.
28.
Set “sine harmonic” back to 1. Change “sine phase” to 90. This will convert the sine wave
into a cosine wave , as sin(wt + 90) = cos(wt). Repeat the above steps to all 10 cosines and enter
your measurements of the TLPF DC output level into the table .
Harmonic
number
sine
(V)
cosine
(V)
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
10th
DC (V) =
Table of measured coefficients
29.
Connect the input of the TLPF directly to the arbitrary wave form at DAC-0 to measure
the DC value of that waveform alone, and enter into the table at “DC (V) = “.
Question 16
How do your readings compare with expectations ? . Explain any discrepancies .
Experiment 8 – A Fourier Series analyser
© 2011 Emona Instruments
8-15
30.
While you have the experiment setup, vary the harmonic summer settings and see the
resulting output from the TLPF. Experiment a little with values while you have the system setup
and ready.
8-16
© 2011 Emona Instruments
Experiment 8 – A Fourier Series analyser V1.2
Part 3 – Build a manually swept spectrum analyzer
In the previous part of this experiment we use synchronised signals which were generated in
LabVIEW and output via the DACs of the ANALOG OUT module. In this part of the experiment
we will analyse the constructed arbitrary signal using an independent sinusoid generator, that is,
the FUNCTION GENERATOR.
Figure 6:building a “wave analyzer”
31.
Once again construct the following arbitrary waveform using the HARMONIC SUMMER
of the SFP, Experiment 8 TAB. Launch the FUNCTION GENERATOR as well.
Settings are same as above, and are as follows :
Cosine amplitudes: 1, 0, 0.5,0,0,1,0,0,0,0
Sine amplitudes: 0,0.3,1,0,0,0,2,0,0,0
DC level: 0.5
Switch “to DAC-0” ON
SCOPE: Timebase 20ms; Rising edge trigger on CH0; Trigger level=0V
FUNCTION GENERATOR: Frequency=1000Hz, Amplitude=4Vpp, Sine wave selected
32.
Move the input to the multiplier which was connected to DAC-1 to the FUNC OUT
terminal of the FUNCTION GENERATOR . This signal will be a 4Vpp sine wave at 1000 Hz. View
both inputs to the multiplier on the scope. One is the arbitrary waveform, the other is the 1000
Hz sine wave.
Question 17
What do you notice about their phase relationship ? Is this to be expected ? Explain.
If the inputs are drifting relative to each other then you would expect the product of the
inputs to also be slowly varying.
33.
View the sinusoid and the product, output of the multiplier, at the same time and
confirm that it is slowly varying. Trigger the SCOPE on the sinusoid to get a more stable display.
As it varies it will pass through the point of interest, the interface point that we wish to
measure. This point occurs twice per cycle lasts creating both are maxima and minima. If we
detect either the maximum of the minima the absolute value of these will give us the average
DC value which we require.
Experiment 8 – A Fourier Series analyser
© 2011 Emona Instruments
8-17
You may wish to review your understanding of why these maximas and minimas occur when
signals drift relative to each other. You can use both trigonometry and/or the DAC outputs and
the MULTIPLIER module to revisit this topic.
34.
Now view both the output from the FUNCTION GENERATOR and the output of the
TLPF. Adjust the frequency of the TLPF to isolate only the DC component and confirm that it
has a maxima and a minima. HINT: if the DC value is varying too slowly, or you are impatient, or
just inquisitive (…a good thing), increase the frequency at the function generator by 1 Hz. This
will make your DC value vary at a rate of approximately 1 Hz (…much easier to view).
Trigger on the sinewave from the FUNCTION GENERATOR for a stable display.
HINT: use Y-AUTOSCALE ON/OFF to center the display
35.
Vary the frequency of the function generator in steps of 1000 Hz starting from 1000
Hz up to 7000 Hz. Note the maximum value of the DC output in the table below. It will be more
accurate to measure peak to peak and then halve it.These will be the measured amplitudes of
the constituent harmonics making out the arbitrary wave form. How do they compare with the
actual amplitude value entered into the SFP itself.?
HINT: to vary the FUNCTION GENERATOR Frequency, place the cursor alongside the digit to
be varied and then use the UP and DOWN keyboard arrows to vary the digit value.
Input
frequency (Hz)
TLPF output
pp swing (V)
Half of
pp (V)
Entered
values
Calculated
resultant (V)
1000
2000
3000
4000
5000
6000
7000
DC (V) =
Table of measured coefficients
8-18
© 2011 Emona Instruments
Experiment 8 – A Fourier Series analyser V1.2
Question 18
Can you explain if your readings differ in some places from the actual value ?
HINT: you have one value per harmonic instead of two. Consider the previous discussion above
about resultants in your answer. And allow for MULTIPLIER and TLPF gains (although you should
have set the overall gain to unity in an earlier part of the experiment).
Whilst the theoretical modeling with the synchronized sine and cosine waves from the previous
part is able to yield the constituent sine and cosine components, a real world signal without the
benefit of synchronization can only measure the resultant amplitude. This is shown in the
exercise just completed.
36.
This section of the experiment was about building a manually swept spectrum analyser.
The function generator instrument has the ability to be automatically swept. As an exercise in
automation enter an appropriate start, stop frequency, say 1001 & 10001 Hz, as well as step size
of 1000 Hz and a step interval of 2 seconds and allow the function generator itself to sweep the
analysing frequency while you view the amplitude of the DC value from the analyser which
corresponds to the amplitude of each harmonic present. Display the FUNCTION GENERATOR
sinewave also, and trigger the scope on it for a stable display of 4Vpp. Set the AUTOSCALE to
OFF.
Congratulations, you have just constructed a swept spectrum analyzer using basic mathematical
blocks.
Experiment 8 – A Fourier Series analyser
© 2011 Emona Instruments
8-19
Part 4 – analyzing a square wave
We will now use our manually swept spectrum analyser to investigate which harmonics at present
in a square wave. The experiment set up is as for the previous section except that now the input
is taken from the PULSE GENERATOR module output.
Figure 7: working with a square wave
37.
Vary the patching as shown in figure 7 above
.
Settings are as follows:
FUNCTION GENERATOR: Frequency=1000Hz, Amplitude=2Vpp, Sine wave selected
PULSE GENERATOR: FREQUENCY=1000Hz, DUTY CYCLE=0.5 (50%)
SCOPE: Timebase 4ms; Rising edge trigger on CH1; Trigger level=0V
38.
Vary the frequency of the function generator in steps of 1000 Hz starting from 1000
Hz up to 7000 Hz. (Remember, you can vary the input frequency by 1 or 2 Hz to speed up the
resultant DC oscillation display).Note also the maximum value of the DC output in the table
below. These will be the amplitudes of the constituent harmonics making up the squarewave. You
may know from theory that a square wave of 50% duty cycle contains only odd harmonics. This
can help you focus your investigation around the appropriate frequencies.
8-20
© 2011 Emona Instruments
Experiment 8 – A Fourier Series analyser V1.2
Input
frequency (Hz)
TLPF output
amplitude (V)
Scaled measured values
(V)
Calculated
resultant (V)
1000
2000
3000
4000
5000
6000
7000
DC (V) =
Table of measured coefficients for squarewave
Question 19
Why are some of the harmonics hard to detect ?
Varying the duty cycle of the PULSE GENERATOR from the SFP control “DUTY CYCLE”, for
example to 0.2 (20%) will introduce even harmonics.
Question 20
Can you now detect even harmonics in the squarewave of 20% duty cycle ?
View the output from the MULTIPLIER to see what signal you are measuring and extracting the
DC value from.
Experiment 8 – A Fourier Series analyser
© 2011 Emona Instruments
8-21
Comparing measured Fourier series coefficients to theory
From the theory the relative value of the Fourier series coefficients for a 50:50 square wave
is a series like so:
1, 1/3, 1/5, 1/7….1/n where n is the odd harmonic present in the squarewave.
Question 21
Compare your measured coefficients for the first 4 odd harmonics as ratios to that expected
by theory ? Remember to normalize the measurements for the comparison.
References
Langton.C.,”Fourier analysis made easy ”, www.complextoreal.com
8-22
© 2011 Emona Instruments
Experiment 8 – A Fourier Series analyser V1.2
Experiment 9 – Spectrum analysis of various signal types
Achievements in this experiment
You will use a spectrum analyser to observe real signals in the frequency domain. You will
discover important relationships between time and frequency domain characteristics of various
classes of signals.
Preliminary discussion
We are well familiar with the virtues of the scope as our eyes for observing signals and
waveforms in the time domain. In this Lab we discover the reality of the frequency domain
through the eyes of the spectrum analyser.
Various experiments introduced in earlier labs are extended through the use of the spectrum
analyser. An important theme is the observation of special properties, such as those relating to
periodic waveforms, and to discrete-time signals.
By understanding the frequency domain characteristics of various signal types, we can further
develop our ability to think in both the time AND frequency domain when considering signals.
The two go hand in hand and it is essential for the engineer to be well versed in both.
Having hands-on experience with signals and their spectrums will consolidate the theory,
particularly of broad spectrum signals, in a way that supports the learning of
telecommunications principles in other coursework.
Signals and their spectrum are the foundation for several other fields of study.
9-2
© 2011 Emona Instruments
Experiment 9 – Spectrum analysis V1.2
Pre-requisite work
Question 1
What is the conversion equation for a linear voltage scale to a logarithmic scale ?
Question 2
What linear ratio does a -6dB gain equal ?
Question 3
List some of the more important characteristics of PN sequences:
Question 4
Multiply a sinewave with a squarewave so as to create a halfwave rectified sinewave and
calculate its spectrum:
Experiment 9 – Spectrum analysis
© 2011 Emona Instruments
9-3
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should observe
the USB light turn on (top right corner of ELVIS unit).The PC may make a sound to indicate that
the ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that
all three power LEDs are on. If not call the instructor for assistance.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with
the NI ELVIS that you’re using.
9.
You’re now ready to work with the NI ELVIS/SIGEx bundle.
10.
Select the Lab 9 tab on the SIGEx SFP.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the
STOP button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the
top of the window. This will allow the program to conduct an orderly shutdown and close the
various DAQmx channels it has opened.
Ask the instructor to check
your work before continuing.
9-4
© 2011 Emona Instruments
Experiment 9 – Spectrum analysis V1.2
Experiment
Spectrum of the impulse train
We will generate a train of impulse responses, and pass these through a BASEBAND LPF channel
and display the spectrum both before and after filtering.
Figure 1: pulse source to S.U.I.
11.
Patch together the experiment as shown in Figure above.
Settings are as follows:
PULSE/CLK GENERATOR: 500 Hz; DUTY CYCLE=0.1 (10%)
SCOPE: Timebase 40 ms; Rising edge trigger on CH1; Trigger level=1V
Connect CH1 to input, CH0 to output of BASEBAND LPF. Ensure FFT: Y-scale set to Linear
mapping.
12.
Observe the pulse train on the scope. Measure the width and repetition interval, and
confirm that these are as you would expect for the settings on the PULSE GENERATOR you
have selected.
NB: Its always a good idea to measure and confirm that signals set up on equipment are as
expected, to avoid errors and wasted time.
13.
Observe the spectrum of this pulse train. The spectrum analyser should be set for a
linear frequency axis and linear scale for the vertical axis. Vary the timebase, which in return
varies the frequency scale, to provide a convenient balance between range and resolution in the
frequency display.
You can also switch between this experiment’s tab and the ZOOM FFT TAB for more control
over the FFT display if you prefer without disrupting any settings.
14.
Note that the spectrum consists of discrete components. Confirm that the interval
between components is determined by the pulse frequency.
Experiment 9 – Spectrum analysis
© 2011 Emona Instruments
9-5
Figure 2 (a): Spectrum of single width impulse with linear scaling
(b) Spectrum of double width impulse with linear scaling
Question 5
At what frequencies do the nulls occur at ?
15.
Increase the width of the pulse, to double, but not the frequency, by varying the duty
cycle of the pulse train. Set the PULSE GENERATOR DUTY CYCLE to 0.2 (20%)
16.
Confirm that the frequency interval between nulls is determined by the pulse width.
Question 6
What is the mathematical relationship between null spacing and the pulse width ?
17.
Confirm that the spacing of the individual components of the spectrum, ie: its
harmonics, have not changed with the double width pulse.
18.
Make suitable measurements of the spectrum amplitude to show that the shape of the
envelope is of the form sin(x)/x for both impulse widths. Confirming the ratio of envelope peaks
would be one important characteristic to check.
19.
Verify that the general form of the spectrum is not affected by the clock frequency ie:
that it still maintains the sin(x)/x shape.
Question 7
What are the characteristics of the sin(x)/x form that you are looking for ?
9-6
© 2011 Emona Instruments
Experiment 9 – Spectrum analysis V1.2
20.
Now, reduce the width of the pulse, but not the frequency, by again varying the duty
cycle of the pulse train. Set the PULSE GENERATOR DUTY CYCLE to 0.02 (2%)
21.
Confirm that the frequency interval between nulls is determined by the pulse width ?
You may need to vary the scope timebase to alter the FFT frequency display range.
22.
Try a final reading with a duty cycle of 1 %. Set the PULSE GENERATOR DUTY CYCLE
to 0.01 (1%)
Question 8
What is the general trend that you are observing as the duty cycle tends towards 0 ?
Question 9
Using the various findings so far, what shape you expect the spectrum of a single pulse, that is,
a pulse train with very large separation between pulses, to have ?
Spectrum of the filtered impulse train
23.
Return to the original single width pulse settings.
Settings are as follows:
PULSE/CLK GENERATOR: 500 Hz; DUTY CYCLE=0.1 (10%)
SCOPE: Timebase 40 ms; Rising edge trigger on CH1; Trigger level=1V
Connect CH1 to input, CH0 to output
24.
Observe the scope display of the pulse response at the BASEBAND LPF channel output.
Figure 3(a) impulse after filtering (b)post–filter spectrum
25.
Ensure that the pulse width is narrow enough such that the filtered pulse output is the
impulse response of the channel.
Experiment 9 – Spectrum analysis
© 2011 Emona Instruments
9-7
26.
Observe the frequency response of this output and compare with the response of the
unfiltered input train. This is most easily done by viewing input and output simultaneously on
both channels of the scope. Sketch these responses below:
Graph 1: impulse train responses
27.
Determine and plot the response characteristic of the BASEBAND LPF. This will be the
ratio of the output to the input. If you have viewed the frequency responses using a log/dB
scale on the vertical axis, then this process is a straightforward subtraction of input from the
output. Modify the FFT display Y scale Mapping settings to be set to Logarithmic for these
measurements.
Duty cycle and sampling
In the section above you have determined the frequency response of a periodic pulse train with
variable duty cycle. This is the typical signal used as a sampling pulse train. Sampling will be
covered in a later experiment (X12 – Sampling) in this manual.
It suffices at this point to emphasize that the spectrum of the sampling signal will be
interacting with the message to be sampled, hence it is imperative to be familiar with it and its
characteristics.
Start with a square wave from the PULSE GENERATOR, frequency = 500Hz, set to duty cycle =
50%. Slowly reduce the duty cycle from 50, to 40, then 30 and finally 20%, and observe the
9-8
© 2011 Emona Instruments
Experiment 9 – Spectrum analysis V1.2
occurrence of even harmonics. You should realise that at 20% you have the same spectrum
studied above and you should have an insight into the origin of the harmonics present.
“Sinc pulse” trains
In previous experiments we have encountered spectrums of rectangular pulse trains which are
shaped by the sin(x)/x form. This sin(x)/x characteristic is known as the “sinc function”.
A rectangular shaped signal in the time domain results in “sinc function” shaped frequency
response spectrum. In this part of the experiment we investigate the spectrum of pulse with
sin(x)/x shaping in the time domain ie a sinc function in time. For convenience we will use a
repetitive train of sinc pulses.
28.
A sinc pulse train is available at the ANALOG OUT : DAC-1 module. View this signal with
a SCOPE channel in both the time and frequency domain.
Settings are as follows:
SCOPE: Timebase = 10ms, Trig level = 0.5V
With this signal we have a unique situation where the spectrum is finite. In order for this to
occur this sinc pulse must itself be infinite in the time domain, and in fact a true single sinc
pulse is only asymptotically limited in time. That is, it tends to zero in both positive and negative
time directions. In this experiment we have made a approximation by repeating the pulse in a
train and terminating the oscillations after about 20 cycles.
Figure 4: sinc pulse and its bandlimited spectrum; exponential pulse for comparison
Question 10
At what time instants does the sync pulse have a zero crossings ?
29.
Sketch both the sinc pulse and its spectrum in a graph below, and annotate the diagrams
of all the relevant measurements.
Experiment 9 – Spectrum analysis
© 2011 Emona Instruments
9-9
Graph 2: sync pulse train time and frequency responses
30.
As per the steps taken above in the impulse section, apply this sinc pulse train to the
BASEBAND LPF module and observe its input and output in both the time and frequency domain.
Plot the output frequency response on the graph above.
Figure 5: TUNEABLE LPF time & frequency response to the sinc pulse train
31.
It is also interesting to see the difference in response between the BASEBAND LPF and
the TUNEABLE LPF. Pass the sync pulse train through the TUNEABLE LPF and slowly reduce
the -3dB cutoff frequency to the same as that of the BASEBAND LPF as found above. Notice
the difference in the roll off rate.
9-10
© 2011 Emona Instruments
Experiment 9 – Spectrum analysis V1.2
Applying an impulse train and sync pulse to a system is equivalent to applying many simultaneous
sinusoids.
Spectrum of pseudorandom sequence
In this set of exercises we examine and compare the spectra of various forms of pseudorandom
sequences. Usually known as PRBS (Pseudo random bit sequences).We will begin with short
clocked Maximal Length sequences, i.e. the kind of sequence available at the SEQUENCE
GENERATOR module. Next we will examine longer length clocked Maximal Length sequences.
Finally we will look at a non-Maximal length sequence.
It is interesting to consider how pseudorandom sequences are generated. Firstly, it has to be
said that they are named “pseudo” random as they are not truely random sequences. They are
actually periodic and completely deterministic. However they have numerous qualities which
make them useful as signals.
PN sequences are created by structures such as shown in Figure 6. These structures can either
be implemented in hardware via shift registers, or in software.They are usually called LFSR
(Linear feedback shift registers).
.
Figure 6: block diagram for the of SIGEx SEQUENCE GENERATOR sequence (DIPS
UP:UP)
In this example, 5 registers are used, resulting in a sequence of period 25 – 1. There is one less
than the maximum number possible as state [00000] is considered illegal.
Question 11
Why do you suppose state [00000] is illegal ?
32.
Patch together the experiment in Figure 7 and view both the output X as well as the
SYNC output from the SEQUENCE GENERATOR.
Experiment 9 – Spectrum analysis
© 2011 Emona Instruments
9-11
Figure 7: patching diagram for PN sequence from SEQUENCE GENERATOR
Settings are as follows:
PULSE GENERATOR: Frequency = 2000 Hz, Duty cycle = 0.5 (50%)
SEQUENCE GENERATOR: DIPS set to UP:UP
SCOPE: Timebase = 100ms, TRIG set to CH1, connected to SG SYNC; level = 1V
CH0 connected to LINE CODE output, CH1 connected to SYNC
The LINE CODE output from the SEQUENCE GENERATOR is a zero DC version of the X
output, and is more convenient for viewing due to no DC component.
33.
View the spectrum of this signal. Can you see where the nulls occur. Vary the scope
timebase so that you can see the separation between the individual harmonics of the PN
sequence. Notice the sin(x)/x form of the spectrum. Use the ZOOM FFT TAB for more
resolution.
Question 12
Where do the nulls occur ? What is the separation between harmonics ? What do these values
relate to ?
to be approx 66Hz.Theory states 2000/31 = 64.5 Hz
34.
Switch to a 6-bit generated sequence with length of 26 – 1 = 63 bits. This is with
SEQUENCE GENERATOR DIPs set to UP:DOWN. Set the SCOPE timebase to 100 ms.
35.
Confirm that the nulls and harmonic separation are in accordance with the relationships
we uncovered in the previous part of the experiment.
When considering a PN sequence such as this example, it is easy to see the repetition of such
short sequences, and hence to understand that it is not truely random. You can even count the
one and zeros, and runs of ones and zeros and prove to yourself some of the characteristics of
these maximal length sequences. Try this for yourself.
You can also see that if such a sequence is correlated against itself,ie: autocorrelated, then it
will only have one position in which there is strong correlation. Hence these sequences are useful
for coding in spread spectrum type systems.
But what about a non-maximal length sequence. It follows by considering the description, that a
non-maximal length sequence repeats in less than the maximum possible periods.
9-12
© 2011 Emona Instruments
Experiment 9 – Spectrum analysis V1.2
36.
Switch to another 6-bit generated sequence. This is with SEQUENCE GENERATOR
DIPs set to DOWN:UP. This sequence however is non-maximal length, and you can see that it
repeats 3 times within the period time of the previous sequence. As well, its spectrum is quite
different than for a maximal length sequence. It has an irregular form. Vary the timebase and
look at the separation between harmonics.
Figure 8: Non-maximal length sequence spectrum, clocked at 2kHz.
37.
Finally, switch to the longest sequence on the SIGEx board, a 14-bit generated maximal
length sequence, with length of 214 – 1 = 16k bits. This is with SEQUENCE GENERATOR DIPs
set to DOWN:DOWN. You may need to vary the clock rate and scope timebase to effectively
measure this sequence.
Question 13
Where do the nulls occur ? What is the separation between harmonics ? What do these values
relate to ?
Analog noise generation (AWGN)
As we have just seen with the PN sequences, the spectrum of composed of many evenly spaced
harmonics with a rolloff to repetitive nulls. If we wish to create a signal composed of many
evenly spaced harmonics of equal amplitude, we can isolate a small region of the PN sequence
spectrum and use that as our signal. Using a low pass filter we can attenuate all harmonics above
say 10% PN CLOCK frequency and then we are left with a signal with White Gaussian Noise
characteristics which is useful in various experiments as a noise source. AWGN stands for
“additive white Gaussian noise” meaning the type of noise which is imposed on a signal travelling
through a particular noisy channel.
Experiment 9 – Spectrum analysis
© 2011 Emona Instruments
9-13
38.
Patch together Figure 9 and view the output of the TUNEABLE LPF. Set the TUNEABLE
LPF control both to fully clockwise at this stage. View the input and output signals to the filter
in the time domain and the frequency domain.
Figure 9: block diagram for PRBS generated analog noise
Figure 10: patching diagram for PRBS analog noise
Settings are as follows:
PULSE /CLK GENERATOR: Frequency= 2000, Duty cycle = 0.5 (50%)
SEQUENCE GENERATOR: DIPS set to UP:UP (short sequence)
SCOPE: Timebase = 100ms.
39.
With the cutoff frequency of the TUNEABLE LPF above 12kHz, you can view the
unfiltered PN sequence as a bipolar sequence. We are using the bipolar output from the
SEQUENCE GENERATOR as we want a bipolar analog output with no DC component.
40.
Slowly reduce the cutoff frequency until you have reached approximately 200-300 Hz.
You can see this from the spectrum display. Consider the output of the filter in the time
domain. Is it a satisfactory noise signal ?
Question 14
How many harmonics are visible in the filter output ?
41.
Switch the SEQUENCE GENERATOR DIPS to DOWN:DOWN. This will select the
longest sequence. Now take a look at the output from the filter. Would you describe this as a
satisfactory noise signal ? It certainly looks like “noise” with no visible repetition.
9-14
© 2011 Emona Instruments
Experiment 9 – Spectrum analysis V1.2
Question 15
How many harmonics are visible in the filter output ? Calculate this.
Question 16
Is this analog noise signal periodic ? What is its period ? Calculate this
When analysing the frequency response of systems, rather than sweeping a single frequency
across the band, it is convenient to input a multi-frequency signal such as this PN noise, and
then view the system response spectrum all in one display.
This approach is used is several later experiments in this SIGEx Lab Manual and is particularly
useful when used with an averaging display.
Non-linear processes: clipping, rectification and harmonic multiplication
Clipping or amplitude limiting a sinusoid can often occur in a system due to overload conditions.
It may even be done on purpose in cases where the amplitude does not carry any information and
class C amplifiers are to be used. Class C amplifiers are not linear and the maintenance of
amplitude of the signal is not critical. These are typically used with FM signals, which are signals
which do not carry any information in their amplitude but depend solely on the frequency and
phase of the signal.
Although slight clipping of a sinusoid does not appear to be drastic when viewed in the time
domain it will be interesting to see what effect it has on the frequency domain.
42.
Patch together Figure 11 and view the output of the LIMITER block. View the input and
output signals to the LIMITER in the time domain and the frequency domain.
Figure 11: clipping a sinusoid signal
Experiment 9 – Spectrum analysis
© 2011 Emona Instruments
9-15
Settings are as follows:
FUNCTION GENERATOR: Select SINE output, frequency = 1k, amplitude= 4Vpp.
LIMITER DIPS set to UP:UP
SCOPE: Timebase = 10 ms.
43.
Patch together the figure above and view both the input and output of the LIMITER
block. Vary the LIMITER DIP switch settings to all for possible positions and sketch
observations on a graph below.
Graph 3: clipped signals and spectra
You can see that different levels of clipping are applied to the input sinusoid depending on the
position of the DIP switch. Label your sketch with these levels, you may wish to use terms such
as low, medium, high.
Question 17
What effect does a higher level of clipping have on the spectrum of the clipped signal ?
9-16
© 2011 Emona Instruments
Experiment 9 – Spectrum analysis V1.2
44.
Vary the frequency of the input sinusoid and observe the change in the harmonics of the
clipped output.
Question 18
What is the relationship between the input frequency and the output harmonic frequencies ?
45.
Switch from the LIMITER block to the RECTIFIER block, and observe the spectrum of
the rectified sine wave. Plot the responses on the previous graph.
Question 19
What can you say about the spectrum of the rectified sine wave ? Is this what you would have
expected ? Refer back to your pre-lab preparation questions.
Question 20
Is the clipping process a linear or non-linear process ? Explain.
Experiment 9 – Spectrum analysis
© 2011 Emona Instruments
9-17
Tutorial questions
Q1
What are some of the characterictics of non-linear processes ?
9-18
© 2011 Emona Instruments
Experiment 9 – Spectrum analysis V1.2
Experiment 10 – Time domain analysis of an RC circuit
Achievements in this experiment
You will analyse a simple network using tools such as steps, impulses, exponential pulses and
sinusoids to compare theory and practical results. You will then synthesize and test an
equivalent network using a 1st order feedback structure.
Preliminary discussion
The RC circuit is a simple form of network which involves electrical charge storage elements. It
is these “storage elements” such as capacitors and inductances which create the delays in
signals often referred to as “leads” and “lags” in signal phases relative to each other.
Figure 1: RC network; circuit and block diagram
From the experiment on Exponentials, remember that RC is the time taken to for an unit
exponential to decay to e-1 = 1/e = 0.37 of its initial value
10-2
© 2011 Emona Instruments
Experiment 10 – Time domain analysis of an RC circuit V1.2
Pre-requisite work
Question 1: the step response
This question follows the integration method in Section 4 of S.K.Tewksbury's notes
http://stewks.ece.stevens-tech.edu/E245L-F07/coursenotes.dir/firstorder/cap-difeq.pdf
(a) Apply elementary circuit theory to show that the voltage equation for the RC circuit in
Figxxx is
V_in (t) = i (t). R + V_cap (t) = i (t). R + Q(t)/C
Eq prep1.1
Where Q(t) is the charge in capacitor C.
(b) Show that this can be expressed as
(d/dt)(V_in) = R.di/dt + i/C
Consider the case where V_in(t) is a step function of amplitude V_o and the capacitor charge
Q(t) = 0 at t = 0. Show that for t > 0 (d/dt)(V_in) = 0 and the DE reduces to
di/dt = - a. i
[a = (1/RC)]
Use (d/dt)log_e(i) = 1/i to show that the solution of the DE is
i(t) = i_o exp( - a.t)
(t > 0)
[i_o = V_o/R]
(c) Use Eq 1.1 to show that
V_out(t) = V_cap(t) = V_in(t) - R.i_o exp( - a.t)
Hence the step response V_out/V_in = (1 - exp(- a.t))
(d) Plot the result in (c) for a = 1000
(e) What is the asymptotic value of the step response as t increases indefinitely? Show that
the step response rises to (1 -1/e) of its final value at t = 1/a.
Question 2: the impulse response
(a) Describe the main properties of the theoretical impulse function.
Show that differentiation of the unit step function wrt t produces a unit impulse at t = 0. Apply
this to the step response result in Question P1(c) to show that the impulse response h(t) of the
RC circuit is
a. exp(- a.t)
Experiment 10 – Time domain analysis of an RC circuit
© 2011 Emona Instruments
10-3
(b) Explain why the impulse function can only be approximated in practice.
Sketch an impulse approximation realized as a finite width pulse. Explain why an excessively
narrow pulse is undesirable in practical applications. Estimate a pulse width that would be
suitable for use with the case in Question P1. Indicate your reasoning.
(c) Using the property in (a) we could generate the impulse response by first recording the step
response, then differentiating. Compare this alternative with the use of a finite width pulse
input. Include discussion of signal peak limitations and output amplitude considerations.
(d) Show that the impulse response falls to 1/e of its initial value at
t = 1/a
Question 3: convolution and response to an exponential input
This question introduces convolution and its application in the analysis of systems like the RC
circuit in Q. P1.
(a) The convolution of the time functions x_1 and x_2 can be expressed as
[for t > 0]
x_1 * x_2 =
Note that the convolution is a function of t and that tau is a dummy variable that has no further
role after integration.
Show that changing the order (x_2 * x_1) does not change the result.
Show that if x_1 is a unit impulse the convolution x_1 * x_2 = x_2(t).
Suppose we approximate a continuous time signal x_1(t) as a sum of very narrow contiguous
pulses, each of which can be thought of as representing an impulse function (each with its
individual amplitude). Suppose next that this pulse train representation of x_1(t) is then applied
as input to the system introduced in Q. P1. Each of the pulses in the train will produce an
individual output that will be a close (weighted) approximation to the system's impulse response.
The overall output will be the sum of these (overlapping) weighted impulse response
approximations.
Demonstrate that this sum is effectively the convolution of x_1 and the system's impulse
response h(t). (Invoke the usual limit methods to morph the discrete sum into a continuous time
integral.)
(b) Show that for t > 0, the convolution for the case
x_1(t) = exp(- a1.t) and x_2(t) = exp(- a2.t)
[a1 N.E. a2]
is (1/(a2 - a1)) . (exp(- a1.t) - exp(- a2.t)
(c) Sketch the graph of the result in (b) versus t over the range t > 0. Show that for positive
values of a1 and a2 the function is positive for t > 0, and that it is zero at t = 0 and t-> infinity.
Find the peak and the corresponding value of t for a1 = 0.5 and a2 = 1.1 .
(d) Use the results in (a) and (b) and in Q. P2(a) to obtain the response of the RC circuit in Q.P1
when the input is
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© 2011 Emona Instruments
Experiment 10 – Time domain analysis of an RC circuit V1.2
x_1(t) = exp(- a1.t)
(e) Repeat the tasks in (b) and (c) for the case a1 = a2 = a.
NB: a useful reference for this question is Schuam Laplace Transforms (1965); p45 (convolution
of two exponentials)
Question 4: response to a sinusoidal input
In Q. P1 we sought the output of the RC circuit in Fig xxx for the case in which the input is a
step function. This result was extended in Q. P2 and P3 for an impulse function input and for an
exponential input. Now we examine the solution when the input is sinusoidal. This case is of
special importance in this work as it opens the way to powerful tools for the solution of systems
of much greater complexity than the introductory example under investigation here.
(a) Use the result in Q.P1(a) to show that
V_in (t) = RC.(dV_out/dt) + V_out (t)
Eqn. P4.1
To simplify the analysis we will use the complex exponential A_in.exp(jwt) to represent the
input sinusoid [recall that exp(jwt) = cos(wt) + j.sin(wt)].
In Q. P1(b) we obtained a solution of the DE by direct integration. However, sometimes it turns
out that invoking a "feeling lucky" approach can provide the desired result:
a solution of the form
V_out = A_out . exp(j.phi_out) . exp(jw.t)
is substituted into the RHS in the above DE.
Show that this is a solution for a suitable value of A_out . exp(phi_out). (The suitable value is
the one that makes the RHS = LHS). With A_in = 1, show that the sought value is
A_out . exp(phi_out) = 1/(1 + jwRC)
Hence show that V_out = V_in . 1/(1 + jwRC) = V_in . (1/RC)/ (jw + (1/RC))
Note that this result has a very interesting feature:
the output has the same form as the input.
[To discover the importance of this property it is worthwhile to think about the use of other
waveforms to express the output in terms of the input. For example, a squarewave, a periodic
ramp, a sawtooth (an optional lab exercise). ]
(b) Use the result in (a) to obtain a formula for the ratio of output amplitude to input amplitude
as a function of w for 1/RC = 1000 (rad/sec). Sketch the result, and find the value of w for
which the ratio is 3dB.
Question 5: solution using the Laplace Transform
(a) Look up the definition Y(s) of the Laplace transform of the
Experiment 10 – Time domain analysis of an RC circuit
© 2011 Emona Instruments
10-5
function y(t). Show that the Laplace transform of (d/dt)y(t) is sY(s).
Solve Eqn P4.1 as a function of s by applying the Laplace transform to both sides (note that no
restriction is imposed on the form of the input) .
Compare this result with the solution obtained with input
V_in (t) = A_in.exp(jwt)
Comment on similarities and differences.
(b) The transfer function is defined as V_out(s)/V_in(s). Use the result in (a) to write down the
transfer function of the RC circuit.
(c) Find the Laplace transform of y(t) = exp( - a.t). Compare this with the transfer function in
(b).
(d) What is the relationship between the transfer function and the impulse response that is
apparent from (c)?
(e) On the basis of (d), what is the operation in the s domain that corresponds to convolution in
the time domain? Confirm your answer by looking up the convolution theorem.
Question 6: synthesized model of RC circuit
(a) Consider Eqn P4.1 in the Laplace domain, i.e.,
s.V_out = a . V_in + (- a) . V_out
Use the block diagram in Task 25 as a guide to model this equation using an integrator (1/s).
Note that s.V_out(s) appears at the integrator input.
(b) In practical applications the use of a scaled integrator (k/s) may be necessary. Adjust the
system equation so that the LHS is (s/k).V_out, and modify the model accordingly.
(c) Suppose k = 200 and a = 1000. Determine the corresponding value of a1 in the block diagram
in Task 25.
10-6
© 2011 Emona Instruments
Experiment 10 – Time domain analysis of an RC circuit V1.2
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should observe
the USB light turn on (top right corner of ELVIS unit).The PC may make a sound to indicate that
the ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that
all three power LEDs are on. If not call the instructor for assistance.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with
the NI ELVIS that you’re using.
9.
You’re now ready to work with the NI ELVIS/SIGEx bundle.
10.
Select the Lab 10 tab on the SIGEx SFP.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the
STOP button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the
top of the window. This will allow the program to conduct an orderly shutdown and close the
various DAQmx channels it has opened.
Ask the instructor to check
your work before continuing.
Experiment 10 – Time domain analysis of an RC circuit
© 2011 Emona Instruments
10-7
Experiment
Part 1: Step response of the RC network
In previous experiments you have been introduced to the step response as a useful signal with
which to investigate a system.
In this part of the experiment we will measure the step response of an RC network and compare
it to our theoretical expectations.
Figure 2: block diagram of RC network; wiring diagram of experiment
11.
Wire together the RC NETWORK block with the PULSE GENERATOR block as the
source of input signal.
Settings are as follows:
PULSE/CLK GENERATOR: 50 Hz; DUTY CYCLE=0.5 (50%)
SCOPE: Timebase 20ms; Rising edge trigger on CH1; Trigger level=1V
Connect CH1 to input, CH0 to output
From the preliminary discussion at the beginning of this lab, the unit step response of an RC
network is:
h(t) = [1 - e-t/RC].u(t)
12.
Drawing upon your pre-lab preparation work , use the actual values of the SIGEx RC
network to calculate the expected step response signal.
Use the values: R = 10,000 ohm, C = 100nF (100 x 10-9 F); hence RC = 1 x 10-3, and 1/RC = 103 =
1000. The time constant for this circuit is 1 ms.
Measure the input step size in volts.
Question 7
How long will it take this RC NETWORK to rise to a level 37% below its final level ?
10-8
© 2011 Emona Instruments
Experiment 10 – Time domain analysis of an RC circuit V1.2
Question 8
Calculate the expected real circuit step response of the RC NETWORK using the real circuit
values and real circuit input values. These values are available in the User Manual. For your
convenience they are R=10kohm, and C=100nF
13.
Confirm that the measured step response corresponds with your theoretical
expectations. Characteristics of the exponential waveform were discussed in Experiment 7 on
exponentials.
Figure 3: exponential waveform Ae(-t).u(t); where u(t) is Heaviside function
Experiment 10 – Time domain analysis of an RC circuit
© 2011 Emona Instruments
10-9
14.
Sketch the step response on the graph below. Show all relevant time constants and
voltage levels.
Graph 1: step and impulse responses
Impulse response of the RC network
In earlier experiments where we introduced the impulse function, it was noted that a true
impulse, with infinitesimally small width, and infinite amplitude, cannot be actually physically
generated. It can however be approximated, by a pulse of finite height and nonzero duration.
Our criteria for whether the approximation is adequate is that the impulse is sufficiently brief
that there is no discernible change in the “system under investigation’s” output shape as the
pulse width is reduced. This is the criteria we will apply here in creating an impulse.
15.
Maintain the same wiring for this next step. You now need to input an impulse signal to
the RC network under investigation. This is easily achieved by varying the duty cycle of the
PULSE GENERATOR block as follows. Leave the FREQUENCY = 50 Hz, and set DUTY CYCLE to
0.05 (5%).
16.
Notice how the output amplitude is diminished, and the rise time is still finite and easily
visible. The input pulse is not a close enough approximation to a true impulse for our
measurements. What is the pulse width currently ?
10-10
© 2011 Emona Instruments
Experiment 10 – Time domain analysis of an RC circuit V1.2
17.
Further reduce the DUTY CYCLE to 0.01 (1%) and increase the FREQUENCY to 100 Hz
for viewing convenience, and to narrow the pulse further.
Question 9
What is the width if the impulse. What is its maximum amplitude ?
18.
Remove the scope lead to the input signal and view only the output impulse
response.Trigger the scope and set the trig level to suit the signal size ie: it is now much less
than 1 V.
19.
Sketch the impulse response on the graph above. Use a new voltage scale for
convenience.
From the preliminary discussion at the beginning of this lab, the unit impulse response of an RC
network is:
h(t) = (1/RC).e-t/RC.u(t) = 1000. e-1000t.u(t)
Question 10
What is the equation for the measured impulse response using actual circuit values ? How does
this compare with theory ?
Question 11
Explain why the impulse response reaches the peak value that it does.
HINT: superposition of 2 step responses is involved.
Experiment 10 – Time domain analysis of an RC circuit
© 2011 Emona Instruments
10-11
Response of the RC network to an exponential pulse
In this segment we investigate the response to a more general input. As an example we consider
an exponential pulse with a different time constant. This provides an opportunity to revisit
convolution as per Prep Question P3.
20.
Switch the input signal for this next step to the ANALOG OUTPUT DAC-0. There is
present an exponential pulse signal with the equation:
x(t) =1.e-500t.u(t)
Figure 4: Wiring for RC network with exponential pulse input
21.
View both input and output signals. Trigger the scope and set the trig level to suit the
signal size ie: it is now much less than 1 V.
22.
Notice how the rise time of the output is easily visible with a particular time constant,
and well as the decay time having its own time constant.
Question 12
What is the equation for the output signal and how does it compare with the theoretical output
expected from this network ? Refer to your work in preparation question 3.
10-12
© 2011 Emona Instruments
Experiment 10 – Time domain analysis of an RC circuit V1.2
23.
Sketch the exponential pulse response, along with input, on the graph below.
Graph 2: exponential pulse response
Experiment 10 – Time domain analysis of an RC circuit
© 2011 Emona Instruments
10-13
Synthesising an RC NETWORK transfer function
24.
In the preparation exercises it was shown that the impulse response of this RC
NETWORK is given by:
h(t) = (1/RC).e-t/RC.u(t) = 1000. e-1000t.u(t)
and the corresponding system response in the Laplace domain by
H(s) = (1/RC)/(s + (1/RC)) = 1000/(s + 1000)
It was also shown that the Laplace domain equation can be modelled with the block diagram
below,
where k/s represents integration with scaling factor k.
Note that the value of k is set by the INTEGRATION RATE DIP switches.
Figure 5: block diagram of synthesised “RC network”
Question 13
Show that RC = | 1/(k.a1)|; where |k.a1| = 1000
Figure 6: patching diagram of synthesised “RC network”
10-14
© 2011 Emona Instruments
Experiment 10 – Time domain analysis of an RC circuit V1.2
25.
Patch together this system and investigate its performance using step and impulse
responses as per settings below. Sketch the responses below.
Settings as as follows for step response:
FUNCTION GENERATOR: Select Squarewave output, 1Vpp, 0.5V offset, 100Hz frequency
SCOPE: Timebase 10ms
ADDER GAINS: a0 = 1.0; a1 = -0.1; a2 = 0
INTEGERATION RATE: DIPS set to UP:UP
For impulse response use the PULSE GENERATOR as a source with frequency = 100Hz, Duty
cycle = 0.02 (2%)
Graph 3: step and impulse response of synthesized system
26.
You would expect that a system like this will need a “small” amount of feedback as the
change to the original step is not great. However the initial value suggested (-0.1) is not
accurate enough. It is a good starting point as the time constant is similar. However the signal
amplitude is too large.
27.
Change the values of a0 and a1 until you get a perfect match with the actual RC
NETWORK. To do this, view both outputs of the RC NETWORK and the synthesised “RC
NETWORK” at point Y together and then make adjustments accordingly.
Experiment 10 – Time domain analysis of an RC circuit
© 2011 Emona Instruments
10-15
HINT: for convenient manual adjustment, you can set the GAIN ADJUST knob on the SIGEx
board to vary the a1 gain coefficient, and then hand adjust the level.
Figure 7: Step responses: real RC network & synthesised “RC network”. Not fully aligned.
Question 14
What values of a0 and a1 have you found give your synthesised system a perfect match to the
actual RC NETWORK ?
28.
In order to calculate the actual transfer function of this synthesised system you will
need to also know the INTEGRATION RATE of the integrator used.
In a previous experiment this was measured and you should use the same procedure to
remeasure this value. In brief, input a bipolar squarewave to the integrator, with frequency
around 300Hz, and amplitude less than 2Vpp from the FUNCTION GENERATOR. The
INTEGRATION RATE is equal to ramp voltage spread /ramp time/ input voltage.
This value is the k value from Figure 5 above.
Question 15
What is the signal at the input to the integrator ? Is this expected ? Explain:
Yes,
Question 16
Using the measured values above, what is the actual transfer function for your synthesised
network which matches the actual RC network ? Show your working.
10-16
© 2011 Emona Instruments
Experiment 10 – Time domain analysis of an RC circuit V1.2
Question 17
Explain any discrepancies you find between expected theory and measurements. What sources
of error are responsible for these ?
29.
Use a varying sinusoid signal to plot the frequency response of the synthesised system
and the actual RC network simultaneously. They should track each other.
Settings are as follows:
FUNCTION GENERATOR: Select SINEWAVE output, 8Vpp, 0V offset, 50Hz frequency
SCOPE: Timebase 20ms. Set Y AUTOSCALE to OFF
Start from 50 Hz, and take 10 measurements up to 2 kHz.
Graph 4: frequency response and bode plot of synthesized system
Question 18
Express the 3dB frequency in radians/sec and compare with your answer in Question 11(2)
Experiment 10 – Time domain analysis of an RC circuit
© 2011 Emona Instruments
10-17
30.
Sketch the bode plot for the system on the same graph as the frequency response from
the previous step. Show your working.
Tutorial questions
Q1
(convolution by graphical method)
In this exercise we use a graphical method to evaluate the convolution of two exponential
functions, i.e., the impulse response of the RC circuit and the input signal in Task 21. The
convolution formula is given in prep question P3 (a).
Note that you will need to carry out the multiplication of the two functions in the integrand for
up to ten values of t. This multiplication and plotting of the result as a function of the
integration variable tau could be done with the aid of a computer to save time. It will be
evident that the the area under the product will be small when t is large and also when near
zero.
Compare the outcome with the theoretical result using the method in prep question P3 (a), and
with your experimental records.
Q2
re convolution theorem
(a) Use the results in Q. P5 to obtain the Laplace transforms of the input and the impulse
response.
(b) Express the product of the transforms in (a) as a partial fraction sum.
(c) Obtain the inverse transform of the result in (b) and compare the outcome with the graph of
the output response in T24.
N.B. There is no need to invoke the inversion formula - each of the two components in the
partial fraction sum in (b) is an elementary transform that inverts by inspection).
(c) State the convolution theorem and the class of systems for which it applies. Does your
result support the convolution theorem?
Q3
(re exponential input)
This question considers the possibility of using a calibrated exponential input to measure the
time constant of the RC circuit. Suppose 1/alfa1 is the unknown time constant of the system
under test and 1/alfa2 the time constant of the input exponential. Investigate the behaviour of
the output as alfa2 is varied, focussing on the magnitude and position of the peak.
10-18
© 2011 Emona Instruments
Experiment 10 – Time domain analysis of an RC circuit V1.2
Experiment 11 – Poles and zeros in the Laplace domain
Achievements in this experiment
You will discover how poles and zeros can be used to visualize frequency responses graphically at
a glance, in a minimal-math zone. You will be able to use this knowledge to intuitively design low
order continuous-time responses. You will be ready to extend this concept to higher order
systems, and discrete-time applications
Preliminary discussion
In the early days of linear system design, long before the advent of programmable computing,
numerous graphical techniques were invented to get around the arithmetic tedium required to
calculate system responses. One of these is the pole-zero technique. A century on, this scheme
has remained in the system engineer's tool kit because it provides a medium for the intuitive
visualization of system behaviour, especially in regard to stability and sensitivity. And, with
some circuit structures, there is a bonus: by observing the movement of the poles and zeros,
you can intuitively predict system response variations as components are tuned.
Part 1: We examine the behaviour of lowpass and bandpass responses generated with the
second order feedback system in Fig. 1 in the context of poles and zeros.
Part 2: The findings in Part 1 are extended to create special filters realized with the feedbackfeedforward system in Fig. 2. The pole-zero paradigm is applied to discover intuitive
interpretations of the associated equations.
Part 3: We discover how maximally flat and critically damped responses are generated.
It should take you about 60 minutes to complete this experiment, not including the preparation
to be done before the hands-on lab work.
11-2
© 2011 Emona Instruments
Experiment 11 – Poles & zeros in the Laplace domain V1.2
Pre-requisite work
Preparation will be required in order for the hands-on lab work to make sense. This guided
preparation is a revision of theory you will have covered in lectures and is presented below as a
number of computation exercises. This work should be completed before attempting the lab.
Question 1
For the system in Figure 1, obtain a differential equation relating the output x0(t) and the input
u(t). Show by substitution that x0 = ejωt is a solution and determine the corresponding input u(t)
that produces this output.
If you are uncomfortable with a complex-valued function to represent the behaviour of a
system that is supposed to operate with real-valued signals, x0 = cos(ωt) or sin(ωt) could be
used. However, you will quickly discover that the exponential function has a very useful
property that simplifies the math considerably. Remembering that cos(ωt) is Re{ ejωt}, you can
carry out the analysis with ejωt then simply take the real part of the result. Practitioners
generally don't bother with the formality of taking the real part. Moreover, complex valued
signals are easily realized in digitally implemented systems, and indeed, frequently used, for
example in modulators and demodulators of dial-up modems.
INPUT
x2
x1
x0
u
-a1
-a0
Figure 1: schematic of 2nd order integrator feedback structure without feedforward.
Question 2
From the above, with x0 = ejωt, obtain an expression for the ratio x0/u as a function of jω (not
just "ω"; the reason for this will emerge shortly). Note that this ratio is complex valued. Then,
obtain its magnitude and phase shift as functions of ω (not jω).
Experiment 11 – Poles & zeros in the Laplace domain
© 2011 Emona Instruments
11-3
Question 3
From the results in Question 1 above, plot the magnitude |x0/u| versus ω (radians/sec) for the
case a0 = 0.81, a1 = 0.64 . Note that there is a progressive fall off as ω increases. Hence, we
can think of this system as realizing a lowpass filter.
Graph 1:response plot
Question 4
We now consider an alternative way of getting the response. With a little algebra we create a
graphical medium that will provide an intuitive environment for visualizing and generating both
magnitude and phase responses.
First, return to the expression for x0/u obtained in (a) and replace "jω" by the symbol "s". Look
upon s merely as a convenient stand in for jω. It is not necessary to ascribe any deeper
significance to this substitution for the purposes of this lab. The result is the (complex-valued)
rational function
x0/u = H(s) = 1/(s2 + a1.s + a0)
(Eqn 1).
For the case a0 = 0.81, a1 = 0.64 (as in (b), express the denominator quadratic in the factored
form (s - p1)(s - p2), where p1 and p2 are the roots. Show that these are given by
p1 = 0.9(cos(110.8°) + j.sin(110.8°)) = 0.9expj0.616π
p2 = 0.9(cos(110.8°) - j.sin(110.8°)) = 0.9exp-j0.616π
11-4
© 2011 Emona Instruments
(Eqn 2).
Experiment 11 – Poles & zeros in the Laplace domain V1.2
Question 5
Express the complex points p1 and p2 from equation 2 above as the non-exponential complex
form of a + ib, that is, with a real and imaginary part.
Question 6
Next, we look at a graphical approach for evaluating the factors (s - p1) and (s - p2). Place
crosses ("x") on a complex plane at the locations corresponding to p1 and p2, as obtained in (c)
above. Place a dot at the point 1.2 on the j axis, i.e., the complex value jω = j1.2 . Join this point
and the crosses at p1 and p2 with straight lines. Satisfy yourself that the lengths of these
joining lines are |jω - p1| and |jω - p2|. Noting that 1/|H(jω)| is the product of these two
magnitudes, estimate |H(j1.2)|.
Graph 2: vector subtraction
Experiment 11 – Poles & zeros in the Laplace domain
© 2011 Emona Instruments
11-5
Question 7
Use the idea above to obtain estimates of |H| at other frequencies and thus produce a sketch
graph of |H| over the range 0 to 5 radian/s. (ie: ω will range from 0 to 5). Notice that the
presence of a peak in the response is obvious from the behaviour of the vector from p1 as the
dot on the j axis is moved near p1. Note that this vector has much greater influence than the
other vector, especially near the peak. Compare this estimate with the computed result you
obtained in (b). Plot at least 4 points over this range, choosing your points to reflect the
important characteristics of this response.
Explain why the vector from p1 has a greater influence on the peak of the response.
Graph 3:plot the calculated magnitude response |H(jw)|
Question 8
The roots p1 and p2 of the denominator polynomial of H(s), marked as crosses on a plane of the
complex variable s are known as poles of H(s). Note that in the example case, p2 is the complex
conjugate of p1. Why is this so?
11-6
© 2011 Emona Instruments
Experiment 11 – Poles & zeros in the Laplace domain V1.2
Question 9
Derive Eqn1 from the schematic (block) diagram, Figure 1, without using the differential
equation step. That is, treat the integrator as a "gain" of value 1/s and process the equations as
algebra.
Question 10
Next we proceed to the system in Fig 2. Note that this is a simple extension of the feedback
only system in Fig. 1. Use Eqn 1 to obtain the output/input equation y/u ,
y/u = H_y (s) = (b2.s2 + b1.s + b0)/(s2 + a1.s + a0)
Eqn3
(i) Consider the case b0 = 2.0, b1 = 0, b2 = 1.0 . Show that the roots of the numerator for these
coefficients are z1 = 0 + j1.414, z2 = 0 - j1.414 . Place an "o" on these points on the same s plane
diagram you used to mark the poles, Graph 2. The roots of the numerator are known as "zeros".
b2
x2
b1
x1
b
2
x0
b0
y
u
-a1
-a
0
Figure 2: schematic of 2nd order integrator feedback structure with feedforward
combiner.
Experiment 11 – Poles & zeros in the Laplace domain
© 2011 Emona Instruments
11-7
Question 11
Using the zeros with the method from Question 6, carry out the graphical estimation of the
numerator of Eqn 3 at s = j1.2. Note that this is a special case, with the zeros located on the j
axis (since b1 = 0). Hence, the lines joining the point jw and the zeros will lie on the j axis.
Combine the numerator and denominator estimates to obtain |H_y(j2)|. Extend to other values
of w, and sketch the magnitude response |H_y(jω)|. Comment on the presence of a null at ω =
1.414.
Question 12
(optional) Compute |H_y(jω)| from Eqn 3 and assess the quality of the estimate based on poles
and zeros.
Question 13
With a0 and a1 as in Question 3, apply the pole-zero method to obtain approximate graphs of the
magnitude response for the following cases:
b1 = 1, b0 = b2 = 0
b2 = 1, b0 = b1 = 0
b2 = 1, b1 = −a1, b0 = a0
b2 = 1, b1 = 0, b0 = a0
State the name of the response type corresponding to each case (e.g., bandstop, allpass, etc).
For the allpass case, plot the phase and/or group delay response (group delay = − d(phase)/d ω).
Find out and note here an application for the allpass response.
11-8
© 2011 Emona Instruments
Experiment 11 – Poles & zeros in the Laplace domain V1.2
Graph 4: various responses
Question 14
The integrators in Figs 1 and 2 were depicted as having unity gain. A practical realization
normally has an associated gain constant. The corresponding integrator equations have the form
x0 = k . ∫(x1) dt
x1 = k . ∫(x2) dt
Note that k is not dimensionless. Its unit is sec-1. The SIGEx INTEGRATOR modules provide a
choice of four values of k, selectable by means of on-board switches. The switches are labelled
“INTEGRATION RATE” and the selection and associated value is displayed on the SIGEx SFP.
Suppose k = 12,500 sec-1 is selected. Modify the frequency scale for the response in (b) above
to reflect this choice of k. Explain your reasoning here.
Experiment 11 – Poles & zeros in the Laplace domain
© 2011 Emona Instruments
11-9
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should observe
the USB light turn on (top right corner of ELVIS unit).The PC may make a sound to indicate that
the ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that
all three power LEDs are on. If not call the instructor for assistance.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with
the NI ELVIS that you’re using.
9.
You’re now ready to work with the NI ELVIS/SIGEx bundle.
10.
Select the EXPT 11 tab on the SIGEx SFP.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the
STOP button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the
top of the window. This will allow the program to conduct an orderly shutdown and close the
various DAQmx channels it has opened.
Ask the instructor to check
your work before continuing.
11-10
© 2011 Emona Instruments
Experiment 11 – Poles & zeros in the Laplace domain V1.2
The experiment
Part 1 – System with feedback only
We will use the SIGEx model in Fig.3 . This is an implementation of the feedback system in Fig.2
Fig 3: SIGEx model of second order feedback system
11.
Patch up the model as shown in Figure 3.
Settings are as follows:
INTEGRATION RATE: Switches to ON/ON ( k = 12,500)
ADDER GAINS: a0=-0.81; a1=-0.64; a2= +1.0
FUNCTION GENERATOR: Sinewave selected, FREQUENCY=1k; Amplitude= 2V pp
SCOPE: Timebase 10ms; Rising edge trigger on CH0; Trigger level=0V
Special notice re nomenclature: The feedback and feedforward gains are implemented with the
SFP gain controls of the dual triple-input ADDERs on the SIGEx board. We will always refer to
the symbols a0, a1, a2, b0, b1, b2, as depicted in Figures 1 and 2. These symbols have been used in
almost all popular textbooks over many decades.
NOTE: When first wiring up this system, and after setting up the gains correctly, the output
may appear “stuck” at the power supply limits eg: 10-11 V. In this case the integrator is
“saturated” and needs to be discharged. To do this, briefly connect a lead from the saturated
output you are viewing, to a GND terminal on the SIGEx board. This will discharge the output to
0V, and the integrator will then perform normally.
Question 15
Measure and plot the gain frequency response at the output of the second integrator (x0) onto
Graph 5. Confirm that this is a lowpass response similar to the theoretical predictions you
obtained in prep item (Q3) (the rescaling of the frequency axis will be calculated next). Note
the -3dB cut-off frequency and the frequency at which the response drops to -30dB. Measure
the overshoot (if any) and note the frequency of the peak.
Experiment 11 – Poles & zeros in the Laplace domain
© 2011 Emona Instruments
11-11
In the theoretical preparation we had an integrator gain of unity and rad/sec as the frequency
unit. Hence, to carry out the rescaling you will need to convert from rad/sec to Hz, in addition
to multiplying by the integrator gain (prep item (Q14)).
To convert from angular frequency, ω in rad/sec to frequency, you multiply by 2π. (Remember, 1
complete revolution, 1 cycle, 360°, is 2π radians).
Since the theory had a theoretical integration rate of 1, and our real circuit integrators has an
integration rate of approximately 12,500, we must multiply the frequency value by the
integration rate of 12,500. This gives the scaled, non-normalised, real circuit frequency value.
Although you may have already measured the actual integration rates on the SIGEx board in a
previous experiment, it is worthwhile to repeat this measurement again to ensure your values
are as accurate as possible.
The integrator gains are readily measured by means of a balanced squarewave as input to the
open loop integrator. This produces a sawtooth waveform at the integrator output (there may
be a DC offset in the output; it is of no consequence in this measurement). A balanced
squarewave can easily be obtained from the FUNCTION GENERATOR.
Settings are as follows:
FREQUENCY = 1kHz
AMPLITUDE= 2Vpp
DC OFFSET = 0V
Select SQUARE wave output
Fig 4: SIGEx model for measuring Integration Rate
Fig 5: typical scope display during measurement
11-12
© 2011 Emona Instruments
Experiment 11 – Poles & zeros in the Laplace domain V1.2
12.
Patch together as mentioned above and view a single segment of waveform. Measure the
rate of this segment, ie: rise/run as the units V/s and note here. Ensure that you use a segment
of waveform which allows you to take an accurate measurement. Avoid signal segments which
have reached the supply rails as these will be more difficult to measure. As well, note the exact
voltage level of the input signal for the duration of the ramping signal. (In this setup it will be
1V, hence you will be dividing by 1.)
Question 16
Calculate the integration rate as (rise(V)/run(s)) / input voltage (V). The units for integration
rate are sec-1. Repeat your measurement for a falling ramp and confirm that the magnitudes are
equal. Compute rates for all 4 switch positions in case you need this information later on.
13.
Next, plot the poles of this system: apply the method in Questions 3 and 4 (you will find
it convenient to calculate the poles on the basis the normalized transfer function first, and then
denormalize the frequency scale in a separate step). Show the values of the real and imaginary
parts, and the radius (√a0) relative to the origin of the s plane. Use the method in prep item
Question 4 to estimate the magnitude response and verify that the key features match the
location of the poles (eg. overshoot, bandwidth).
Graph 5: poles and magnitude response
Experiment 11 – Poles & zeros in the Laplace domain
© 2011 Emona Instruments
11-13
14.
To contrast with the lowpass response at node x0, we consider and measure the gain
frequency response at node x1, the output of the first integrator. You will discover that this is
a bandpass response.
Question 17
Measure the frequency of the response peak, the 3dB frequencies, and hence, the 3dB
bandwidth.
15.
Next, we examine this response in the context of the pole-zero diagram. Briefly return
to prep item (c) to obtain the transfer function for output at x1. In setting up the differential
equation you will already have shown that x1 is the time derivative of x0, hence x1 = jω.x0, or s.x0.
Hence the transfer function for x1 has the same denominator and poles as x0. The only
difference is in the numerator, which changes to s, introducing a zero at s = 0. On the basis of
the pole-zero diagram, demonstrate that the frequency response for output at x1 must be
bandpass.
Question 18
Calculate the geometric and arithmetic means of the 3dB frequencies. Compare this with the
peak frequency. Consider which of these gives the closer agreement. This is not easy to
resolve as the peak is quite flat, and pinpointing it can be challenging. It turns out that for this
type of second-order system the peak is at the geometric mean of the 3dB frequencies (see Tut
Q.2). Since these can be measured more accurately, this provides a better alternative for
measuring the resonance frequency. From Tut Q.2 it is readily shown that this formula is not
restricted to a 3dB bandwidth criterion. You may like to put this to the test, e.g. for the 6dB
frequencies.
Question 19
In Tut Q.2 it is shown that the bandpass response peak is at (√a0) rad/sec. Using this formula
and measurement results obtain an alternative estimate of the scaling factor, and compare this
with the results of the integrator gain measurements in T1.3. Consider which of these is the
more reliable .
Record these results for use in Tut Q.2.
11-14
© 2011 Emona Instruments
Experiment 11 – Poles & zeros in the Laplace domain V1.2
16.
With the setup as in section 15, keeping a1 undisturbed, vary a0, around 10 percent, and
observe the effect on the resonance frequency and bandwidth. You may wish to use the manual
GAIN ADJUST knob on the SIGEx board to vary each parameter in turn. Remember to setup
its range to suit your parameter. (Use of the GAIN ADJUST feature is covered in the SIGEx
User Manual as well as Experiment 2).
Repeat this with a1, keeping a0 contant near its original setting. Satisfy yourself that a0
controls the peak frequency, without affecting the bandwidth. Likewise, observe that a1
controls the bandwidth without affecting the peak frequency.
Question 20
Consider practical uses of these properties and record your comments.
17.
Plot the locus of the poles corresponding to the above tests. Interpret your findings
graphically in terms of the pole positions, e.g. the real part is a1/2. These considerations are
explored theoretically in Tut. Q2.
Graph 6: locus of poles
Experiment 11 – Poles & zeros in the Laplace domain
© 2011 Emona Instruments
11-15
Examining the impulse response of the LTIC system
Fig 6: SIGEx model for impulse response of system
Settings are as follows:
PULSE GENERATOR: 400 Hz with DUTY CYCLE=0.05 (5%)
INTEGRATION RATE: Switches to ON/ON ( k = 12,500)
ADDER GAINS: a0=-0.81; a1=-0.64; a2= +1.0
SCOPE: Timebase 4ms; Rising edge trigger on CH0; Trigger level=1V
18.
Display the output of the PULSE GENERATOR on SCOPE CH0 and x0 on SCOPE CH1. You
should be able to view 1 instance of the impulse clearly. Reduce a1 to decrease the damping of
the response, so that several cycles of decaying oscillation are displayed, as per Figure 7.
You may wish to use the manual GAIN ADJUST knob on the SIGEx board to vary this
parameter. Remember to setup its range to suit your parameter.
Question 21
Record your value of a1 here as you will need it later.
Measure the interval between zero crossings (or peaks) to obtain an estimate of the frequency.
Record this value for comparison with a frequency domain measurement after completion of
T1.8.
Progressively reduce a1 further, until it reaches its minimum value of 0, and observe the effect
on the decay rate of the oscillatory ("ringing") response (a0 is to remain undisturbed).
Question 22
Record your observations.
11-16
© 2011 Emona Instruments
Experiment 11 – Poles & zeros in the Laplace domain V1.2
Fig 7: impulse response of system for reduced a1 coefficient
If slowly decaying ringing is still present with the a1 level set to 0, continue to slowly increase
its value in a positive direction ie: positive gain.
At some point the decaying oscillations will be replaced by sustained self oscillations. Confirm
that these oscillations remain when the input is disconnected and measure the frequency.
Question 23
Record your findings.
19.
Measure the peak-to-peak amplitude (adjust the gain control of the scope as needed).
Plot an approximate locus of the poles. Label the points at which any observations of interest
occur in the shape of the response. Comment on the behaviour of the impulse response when
the poles have positive real part.
You may find that the onset of instability does not occur exactly at the value of a1 that you
would expect from theory. Consider practical issues that may cause this.
Experiment 11 – Poles & zeros in the Laplace domain
© 2011 Emona Instruments
11-17
Graph 7: pole locus with varying a1 coefficient
Question 24
Return to the setup in Fig. 3 and with a0 back to the same position as in step 18, recorded in
Q21, measure the resonance frequency at point x1 (the bandpass filter output). Compare this
result with the time domain frequency measurements of the impulse response oscillations.
11-18
© 2011 Emona Instruments
Experiment 11 – Poles & zeros in the Laplace domain V1.2
Part 2 – Feedback and feedforward
In this Part we experiment with ideas introduced in prep items Q10-13.
20.
Using the model of Fig. 3 that you patched up in Part 1, use the “b” ADDER on the SIGEx
board to implement the feedforward gains b2, b1, b0 as follows:
[b2, b1, b0] = [1, 0, 2]
[a2, a1, a0 = [1, -0.64, -0.81]
INTEGRATION RATE: DIPs set to UP:UP
b2
x2
b1
x1
b
2
x0
b0
y
u
-a1
-a
0
Figure 8: schematic of 2nd order integrator feedback structure with feedforward combiner (from
preparation section of experiment)
Fig 9: SIGEx model of second order feedback system
Experiment 11 – Poles & zeros in the Laplace domain
© 2011 Emona Instruments
11-19
21.
Measure the gain response at y and compare with the results in prep Questions 10-12.
Note that this response is lowpass. The zero at j1.41 produces a null at this frequency. Compare
this response with the allpole response of x0 obtained in Part 1. In particular, examine the
trade-off between depth of attenuation in the stopband and cut-off rate in the transition band.
Confirm the frequency of the null. Plot the response on Graph 8.
Graph 8: response
Question 25
Decrease b0 progressively and observe that this causes a reduction of the gain at low
frequencies. Continue until the gains at low and high frequencies are close to equal.
You may wish to use the manual GAIN ADJUST knob on the SIGEx board to vary this
parameter. Remember to setup its range to suit your parameter.
Check that the null is still present. This realizes a bandstop filter, also known as a "notch" filter.
Measure b0 (and a0 in case it was altered). Verify that b2 is still set to unity.
11-20
© 2011 Emona Instruments
Experiment 11 – Poles & zeros in the Laplace domain V1.2
Question 26
Show that this response is obtained when b0 = a0 (with b2 = 1). This can be done quickly using Eqn
3 in prep Question 9: at low frequency, substitute s = 0; at high frequency use 1/s = 0.
Question 27
From prep Question 11 we expect the deepest notch when b1 is zero. Examine whether this is
the case in your implementation. Vary b1 above and below zero and find the value that gives the
deepest notch. Suggest why there may be a discrepancy between theory and practice.
Check the integrator gain by comparing theoretical and measured values of the null frequency.
Consider possible practical causes for any discrepancies.
Question 28
Select a different integrator constant: suggested dip switch position DOWN UP (i.e k around
29,000/s) and measure the new null frequency.
Experiment 11 – Poles & zeros in the Laplace domain
© 2011 Emona Instruments
11-21
The ALLPASS circuit
22.
Using the same model of Fig.8 that you patched up in Part 2, restore the ADDER gains
as follows:
[b2, b1, b0] = [1, 0, 2]
[a2, a1, a0 = [1, -0.64, -0.81]
INTEGRATION RATE: DIPs set to UP:UP
Gradually vary the b1 value from 0 in a negative direction and observe the depth of the notch
decrease. Continue until there is no dip. Carefully trim b1 for a response that is flat across the
entire frequency range. You may wish to use the manual GAIN ADJUST knob on the SIGEx
board to vary this parameter. Theoretically, the condition for this response ("allpass") is b1 = a1
(with b0 = a0 and b2 = 1). Suggest why the theoretical condition may not be satisfied in the
hardware if there are discrepancies.
Plot the poles and zeros and show graphically that the magnitude response is allpass (refer to
prep item Q13), on Graph 9.
Question 29
Measure and plot the phase shift vs frequency and, again compare with your expectations from
the pole-zero plot, on Graph 9.
Graph 8:allpass responses
11-22
© 2011 Emona Instruments
Experiment 11 – Poles & zeros in the Laplace domain V1.2
Viewing frequency responses using FFT
Having measured several responses by taking individual single frequency measurements, you can
understand that it is worthwhile implementing a more automated methodology for viewing
system responses across a spectral range. By using a multi-frequency input signal one can view
the output of a system with an FFT display and see the complete frequency response of the
system. A suitable input signal is a pseudo-noise signal with a flat spectrum at the frequencies
of interest. This signal was investigated in a previous experiment (Experiment 9) and so will not
be discussed in detail here.
Figure 10: example output spectrums for feedback and feedforward system
with multi-frequency input signal
23.
Patch together the input noise signal as shown in Figure 11 and view the output of the
system with the scope, by using the scopes FFT display. Settings are as follows:
PULSE GENERATOR: Frequency = 30kHz, Duty cycle=0.5 (50%)
TUNEABLE LPF: Corner frequency set to approx. 3kHz
SCOPE: Timebase : 100ms
Figure 11: SIGEx model for a multi-frequency, flat spectrum signal source. TUNEABLE LPF Fc
generally set to 10-15% of CLK frequency for flat response
24.
Repeat several of the previous exercises with this method and compare your
observations with your previous findings. You will find the exercises involving the use of the
Experiment 11 – Poles & zeros in the Laplace domain
© 2011 Emona Instruments
11-23
GAIN ADJUST knob to be particularly interesting and dynamic, as you can explore the
interaction of parameters in real time.
Switch to TAB “ZOOM FFT” for a more detailed display of the time & frequency domains.
Part 3 – Critical damping & maximal flatness
You may have come across the terms "maximally flat" and "critically damped". To explore these
ideas we return to the allpoles lowpass filter used in Part 1. Although the feedforward
subsystem is not required, there is no need to alter the setup you have been using in Part 2 as
we will be ignoring the feedforward output. Notice that the feedforward section has no impact
on the performance of the feedback section.
25.
Set the “a” ADDER gains as for Part 1 of this experiment, as follows: [a2, a1, a0] = [1, 0.64, -0.81]. Display the lowpass output x0, and carry out a quick scan of the gain response with
the FUNCTION GENERATOR set to a sinewave output. Note the significant overshoot at the
high end of the passband. Gradually vary the a1 gain control and watch the reduction in
overshoot. Continue until the overshoot is barely observable. Don't forget to check for any
variation of input amplitude that may affect your observations.
Question 30
Record the values of a0 and a1 that realize this outcome. This response is known as maximally
flat. In Tut Q.8 you are invited to show that the formula for a maximally flat second order
allpole is a1 = √(2.a0).
26.
Critical damping is a condition that pertains to the step response. To generate a step
response we use a low frequency squarewave as input from the PULSE GENERATOR.
Settings are as follows:
PULSE GENERATOR: 400 Hz with DUTY CYCLE=0.5 (50%)
INTEGRATION RATE: Switches to ON/ON ( k = 12,500)
ADDER GAINS: a0=-0.81; a1=-0.64; a2= +1.0
SCOPE: Timebase 400us/div; Rising edge trigger on CH0; Trigger level=1V
Display x0 and the input squarewave and observe the overshoot in x0. Progressively increase a1
until the overshoot is barely observable. You may wish to use the manual GAIN ADJUST knob
on the SIGEx board to vary this parameter. Remember to setup its range to suit your
parameter.
11-24
© 2011 Emona Instruments
Experiment 11 – Poles & zeros in the Laplace domain V1.2
Figure 12: example underdamped and critically damped step responses
Question 31
Record the values of a0 and a1 that realize this outcome. This response is known as critically
damped. It is of interest in control systems as it realizes the most rapid risetime without
overshoot. This idea also finds application in the context of Gaussian filters. Further
exploration of critical damping is provided in Tut. Q.9.
Experiment 11 – Poles & zeros in the Laplace domain
© 2011 Emona Instruments
11-25
Tutorial questions
Q1 From prep item (c): determine the position of the pole p1 in terms of the coefficients a0
and a1, i.e., show that with p1 expressed in polar form p1 = ρ.exp(j.θ), we have
ρ = sqrt(a0)
and cos(θ) = a1/2 ρ
Q2 Allpole bandpass filter:
a.
Derive the transfer function and squared magnitude response formula for the second
order resonator in Fig. 1. Obtain the conditions for the poles to be complex valued; show this
graphically, i.e., on a plane with axes a0 and a1.
b.
Show that there is a magnitude response peak at w0=sqrt(a0) (hint: save time and effort:
use the reciprocal of the response formula; manipulate this to reveal a simple and quick method
without differentiation!).
c. In the Lab you discovered that a0 tunes the resonance frequency, and a1 controls the
bandwidth with almost no interaction. Confirm this serendipitous property analytically (hint: the
result in part a of this question).
d. Derive the 3dB points and show that the bandwidth is given by a1(subject to conditions for
complex poles being satisfied). Show that the product of the 3dB frequencies is equal to the
square of the resonance frequency w0 (i.e.,equal to a0).
Hint: inspect the simple formula that revealed the peak. Look for terms of the form (w – 1/w)2
e. Apply these results to the measurements in T1.4.
Q3 State the coefficient conditions to realize an allpass. It would be useful to have an allpass
realization in which the values of a0 and b0, and of a1 and b1 are implemented by single controls,
respectively. Show how Fig 2 can be modified to achieve this. Indicate a possible practical
difficulty with this idea (Hint: refer to your findings in T2.7).
Q4 Obtain the theoretical phase and group delay responses for the allpass in T2.7.
Q5 As mentioned in T2.6, switched-capacitance filters (SCF) are used in the SIGEx TUNEABLE
LOWPASS FILTER. Find out how this works and explain how the tuning is implemented in this
module.
Q6 What is an elliptic filter? Explain how to use the structure in Fig 2 to implement elliptic
filters. What is the required value of b1?
Q7 Show how the system in Fig. 2 can be used to implement a tuneable notch with a single
tuning control. Indicate applications of notch filters
Q8 Derive the conditions for a maximally flat allpole 2nd order LPF (i.e., transfer function, and
poles).
Q9 Derive conditions for critical damping; does critical damping of the impulse response occur
at the same pole values as for step response?
Q10 (optional) In this exercise we consider the application of poles and zeros in the design of a
fourth order BPF as a cascade of two biquad sections (Fig. 2). Suppose a centre frequency of
0.9 rad/s is specified. The design is to provide two zeros on the j axis, one above the passband,
one below.
11-26
© 2011 Emona Instruments
Experiment 11 – Poles & zeros in the Laplace domain V1.2
a. Plot a suitable pole zero constellation. Estimate the pole positions so that the bandwidth is
around 0.1 rad/s. To achieve a flat passband the poles should be slightly offset ("staggered ").
Note that if they are too far apart, an excessively deep trough will result. To estimate the
overall magnitude response, plot estimates of the individual biquad responses on a dB gain scale.
Use the results obtained in T2.2 as a guide. Note that one section will be lowpass and the other
highpass. Students with access to four LAPLACE modules are encouraged to try this at the lab
bench. You will find it very helpful to exploit the orthogonality of a0 and a1 (see Q2) as you trim
your design. Adjust the placement of the zeros to achieve a practical balance between width of
transition bands and depth of attenuation in the stopbands.
b. Note that there are two pole zero pairings available, low pole with high zero, and viceversa.
The usual criterion for the choice is dynamic range, ie. avoiding large peaks. Which is the best
pairing?
c. Suppose the specification had called for a bandwidth of 0.02 rad/s. Consider the practical
challenges faced by the designer. One of these is dynamic range at internal nodes. Estimate the
selectivity that could be realistically achieved with SIGEx LAPLACE modules using the
structure in Fig 2. Indicate your reasons. Other biquad structures are available for
consideration. These can be found in standard textbooks.
d. Show a fourth-order system realized as a single stage of the form of Fig. 2. List practical
issues that should be considered in comparing an implementation of the above filter as a single
stage versus cascaded biquads.
Experiment 11 – Poles & zeros in the Laplace domain
© 2011 Emona Instruments
11-27
11-28
© 2011 Emona Instruments
Experiment 11 – Poles & zeros in the Laplace domain V1.2
Experiment 12 – Sampling and Aliasing
Achievements in this experiment
You will be able to intuitively visualize the spectrum of a sampled signal, and aliasing. You will be
able to use this to gain an intuitive understanding of sampling theorems for minimum sampling
rates.
Preliminary discussion
The conversion of analog signal to digital format involves two stages:
- first, the capture of "frozen" sample values
- then, the digitization of these frozen analog values
Further processing may be applied to improve the storage efficiency, ie, to reduce the memory
needed to a minimum.
In this lab we are concerned only with the sampling process. It is evident that the choice of
sampling rate is the paramount issue: Too slow means that some details are lost with samples
too far apart. If the sample spacing is too fine, resources are wasted, i.e. storage and
processing time. A suitable balance between these considerations is needed.
You will start with the sampling of some typical signals, then observe the recovery of the
continuous-time signals from sample sequences at various rates. From this you will be able to
discover the link between the minimum sampling rate and bandwidth.
This lab opens the door to gaining an intuitive understanding of the theory and practical issues
underlying sampling.
In Part 1 you set up sampling operations of selected test signals and carry out observations in
the time domain. Next, you investigate the reverse process, recovering the analog signal, and
examine the effect of various sampling rates.
In Part 2 you retrace the time domain investigations of Part 1 with observations in the
frequency domain. This provides a systematic structure for the processes involved and makes
possible intuitive mathematical interpretation. Equipped with this insight, you will be able to
easily formulate criteria for choosing efficient sampling rates.
12-2
© 2011 Emona Instruments
Experiment 12 – Sampling and Aliasing V1.2
Pre-requisite work
Question 1
Look up or derive the trigonometric identity for the product of two sines expressed as a sum.
Confirm that the frequencies in this sum are (f1 + f2) and |f1 - f2|, where f1 and f2 are the
input frequencies. Confirm that the output components are of equal magnitudes.
Question 2
Look up or derive the Fourier series of a squarewave of duty ratio other than 50% (25% and 1%
say). Note the sinx/x shaped spectrum envelope. Locate the frequency of the first null of the
envelope for each case and note the relationship with the pulse width.
Now consider the 50% duty ratio case. Comment on the disappearance of the even harmonics.
Question 3
Derive the spectrum of the product of a sinewave and a 1% duty ratio squarewave. You can do
this easily by using superposition with the results in Question 1 and Question 2. For convenience,
make the frequency of the squarewave around five times the sinewave frequency. Plot the
resulting spectrum.
Experiment 12 – Sampling and Aliasing
© 2011 Emona Instruments
12-3
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should observe
the USB light turn on (top right corner of ELVIS unit).The PC may make a sound to indicate that
the ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that
all three power LEDs are on. If not call the instructor for assistance.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with
the NI ELVIS that you’re using.
9.
You’re now ready to work with the NI ELVIS/SIGEx bundle.
10.
Select the EXPT 12 tab on the SIGEx SFP.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the
STOP button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the
top of the window. This will allow the program to conduct an orderly shutdown and close the
various DAQmx channels it has opened.
Ask the instructor to check
your work before continuing.
12-4
© 2011 Emona Instruments
Experiment 12 – Sampling and Aliasing V1.2
Experiment
Part 1: through the time domain
11.
In this exercise we observe the sampling of a sinewave. Patch up the model in Fig 2.
Fig 1: block diagram for sampling with narrow pulses
Fig 2: SIGEx model for sampling with narrow pulses
The sampling signal is obtained from the FUNCTION GENERATOR square wave output which is
set up specifically to drive the MULTIPLIER block like a switch. When the sampling signal is
non-zero, ie at 1V, the input sinewave is passed. However when the sampling signal is zero volts,
then the input signal is not passed, and 0V is output. This emulates an open/close switch.
Settings are as follows:
FUNCTION GENERATOR: Squarewave selected; 10kHz; 1Vpp, 0.50V Offset, with DUTY
CYCLE=50%
SCOPE: Timebase 2ms; Rising edge trigger on CH0; Trigger level=0V
ANALOG OUTPUT (DAC-1): 4Vpp sinewave at 1kHz
12.
View the sinewave input to the MULTIPLIER on CH0, and the sampling signal at the
MULTPLIER on CH1. Confirm that they are as expected.
13.
Display the analog input and the output sampled at 10,000 samples/sec. Try one or two
other sampling rates and various DUTY CYCLES settings. These parameters are controlled from
Experiment 12 – Sampling and Aliasing
© 2011 Emona Instruments
12-5
the FUNCTION GENERATOR instrument panel. Slippage between signals will occur due to lack
of synchronisation between signals.
This demonstration of the sampling process is evocative because it makes it possible to view the
samples as we imagine them, as individual narrow pulses with amplitude proportional to the
sample value, over the entire width of the pulse. However in most applications further
processing will be involved, such as encoding this sampled level as a digital representation (i.e.,
analog-digital conversion). Hence the sampled value must be held while this is carried out. A
sample-and-hold device is provided in the DISCRETE-TIME section of the SIGEx board.
14.
Repeat step 13 using sample and hold. Refer to Fig 4 for the SIGEx model. Compare the
outcome with step 13 .
Fig 3: block diagram for sampling with full width pulses (using Sample/Hold)
Fig 4: SIGEx model for sampling with full width pulses (using S/Hold)
Fig 5: example signals from SIGEx model of Fig 2 & 4
12-6
© 2011 Emona Instruments
Experiment 12 – Sampling and Aliasing V1.2
Next we consider the recovery of the original analog waveform from the sample train. We will
use a lowpass filter to smooth out the jagged corners of the stepped signal generated with the
Sample/Hold. This has a good chance of succeeding when variations between samples are
relatively small.
Fig 6: block diagram for recovering the analog signal
15.
Attempt the recovery of the analog signal from the stepped sample train from the
SAMPLE/HOLD by means of the TUNEABLE LOWPASS FILTER (Fig 7). Set the Fc tuning knob
to full clockwise. Set the GAIN to give a gain of 1 (knob at mid-range). Start with a high
sampling rate, 10k samples/sec, say. Display the filter output and observe the effect of
reducing the filter bandwidth. Compare this output with the original unsampled signal.
16.
Since the sample rate is set by the clock signal, we will interchangeably refer to the
sampling rate as a clock rate, and use the unit “Hz” rather than “samples/sec”.
Question 4
Repeat this for a few other sampling rates, from 10000Hz, down to 2000Hz, say. Document your
readings in Table 1 below. From these observations, what is the minimum sampling rate you
consider adequate to allow recovery of the analog signal without too much distortion, on the
basis of this sampling format (i.e. using the SAMPLE/HOLD function).
Table 1: sample rate readings for recovery from S/H
Sample rate (Hz)
Experiment 12 – Sampling and Aliasing
TLPF setting
(approx.position)
Recovered
amplitude (V)
© 2011 Emona Instruments
12-7
Question 5
Repeat the procedures in step 15 for recovery using the TUNEABLE LPF using the sample train
generated with the system in Fig 2, i.e. with narrow pulses. Document your readings in Table 1
below. Compare the outcome with those obtained with the S/Hold method. Do you expect one
of these sample formats to be better for interpolation to analog form? Is this borne out by
your results?
Table 2: sample rate readings for sampled pulse train recovery
Sample rate (Hz)
17.
TLPF setting
(approx.position)
Recovered
amplitude (V)
Leave the TLPF Fc control set as per the last few results, for the next few questions.
Question 6
Examine the step and impulse responses of the filter at the settings that give you the best
outcomes. Measure risetime and related properties and compare with the sample interval. 1 Use
the PULSE GENERATOR module set to 10Hz, and various DUTY CYCLES settings to achieve this
easily.
Question 7
For the same settings as in step 17, carry out a quick examination of the frequency response of
the filter. Obtain and record the 3dB cut-off frequency, and the attenuation of the stop-band.
1
You may wish o refer back to your notes from “Experiment 3: Special signals”, where step and
impulse responses were covered.
12-8
© 2011 Emona Instruments
Experiment 12 – Sampling and Aliasing V1.2
As we have already seen, the sinusoid has a special role in linear systems. It turns out that the
sampling properties of sinewaves make it possible to establish precise limits for sampling rates.
This is developed thoroughly in Part 2.
18.
Carefully observe the result when the sampling rate is less than two samples per period
(e.g. less than 2kHz). The frequency of the sinewave recovered at the filter output will have
changed (use the filter output as trigger source for the scope). The recovered signal is not the
sinewave at the input of the sampler.
This is easily achieved by slowly decreasing the FREQUENCY setting of the FUNCTION
GENERATOR. Using the “down arrow” to slowly decrement frequency works well.
Check that this outcome occurs with both sampling formats.
One way to see how this comes about is to plot the sample points on graph paper and draw a
smooth curve through these points by eye. The new sinewave is called an alias of the original.
The effect is known as aliasing. Try this below for a sampling rate much less than 2kHz, say
1500Hz. Draw 4 cycles of the 1kHz sinewave as a reference on Graph 1 below.
Graph 1: alias waveforms
Confirm that the sum of the original and alias frequencies = sampling frequency.
Insight into these outcomes is best achieved from a frequency domain perspective.
Experiment 12 – Sampling and Aliasing
© 2011 Emona Instruments
12-9
Part 2: through the frequency domain
In Part 1 we examined sampling and reconstitution through observations in the time domain.
However, the mathematical structure underlying these processes is more readily revealed in the
frequency domain.
In the next task we examine the spectrum of the product of two sinewaves. This will be needed
later as a tool for analyzing the spectrum of sampled signals. Patch together some blocks as
shown in Figure 7.
Figure 7:block diagram for product of two sinewaves
19.
Set the FUNCTION GENERATOR frequency to 5 kHz. Display the MULTIPLIER output
as well as the lower frequency input sinewave from ANALOG OUTPUT: DAC-1.
Settings are as follows:
FUNCTION GENERATOR: Sine wave output, 2Vpp; 0V offset.
SCOPE: Timebase: 4ms; Ch0: DAC-1 ; Ch1: MULTIPLIER output
The two lines straddling the FUNCTION GENERATOR frequency are clearly displayed. Confirm
that the outcome agrees with the theoretical predictions in Pre-lab preparation Q1. This is an
important fundamental result which you must be familiar and comfortable with.
Next, the spectrum of the sampled sinewave.
20.
Keep this patching but change the signal type output from the FUNC OUT terminal of
the FUNCTION GENERATOR to recreate the patching of the sampled sinewave from Figure 2.
Settings are again as follows:
FUNCTION GENERATOR:Squarewave output; 5kHz; 1Vpp, 0.50V Offset, with DUTY
CYCLE=50%
SCOPE: Timebase 4ms; Rising edge trigger on CH0; Trigger level=0V
ANALOG OUTPUT (DAC-1): 4Vpp sinewave at 1kHz
21.
Disconnect the scope lead from the MULTIPLIER output and use it to view the sampling
squarewave only. Confirm you understand why its spectrum is a series of odd numbered
harmonics, including a DC component. This was covered in Pre-lab preparation Q2.
22.
Reconnect the scope lead to the MULTIPLIER output and view the sampled sinewave
along with the sampling squarewave. Observe the spectral lines straddling each of the
squarewaves harmonics. Each one of these harmonics is straddled by a sum and difference
signal. Satisfy yourself that each of these sinewave components can be considered as being
separately multiplied by the sampler input, i.e. the 1kHz sinewave. Thus, by focusing on just one
Fourier component of the squarewave pulse train in turn, we are able to build the array of line
12-10
© 2011 Emona Instruments
Experiment 12 – Sampling and Aliasing V1.2
pairs of the form observed earlier, each pair centered at the respective harmonics of the
FUNCTION GENERATOR squarewave frequency. Confirm you understand completely why the
spectrum looks like this.
Question 8
Explain why the sampled signal spectrum looks the way it does and specifically relate this to
your understanding of pre-lab preparation item 1 & 2.
In Prep Q2 you showed that the squarewave signal can be expressed as a Fourier Series, i.e. as
the weighted sum of sinewaves (in this instance at 5kHz, 15kHz,...) for a 50% duty cycle signal.
Vary the duty cycle of the squarewave by changing the DUTY CYCLE value in the FUNCTION
GENERATOR control panel to 25%.
Satisfy yourself that you understand the source for the appearance now of even harmonics also.
Question 9
Note the frequency of the first and second nulls in the spectrum and explain why they are at
those frequencies.
Experiment with the duty cycle and view the effect of other duty cycles upon the spectrum. Try
using a 10% sampling duty cycle. Confirm with your theoretical understanding.
Aliasing and the Nyquist rate
In the next tasks we revisit the investigation in step 18, where aliasing was discovered at
sampling rates below a critical limit. The view through the frequency domain reveals a
straightforward mathematical interpretation. Once again we will use the FUNCTION
GENERATOR to generate the sampling clock. Whilst we are mainly interested in the frequency
display, in the time domain scope display some clock slippage will be visible but is not of concern.
With the same patching as before in the previous step (Figure 7), select a 25% duty cycle so as
to have both odd and even harmonics present:
Settings are again as follows:
FUNCTION GENERATOR:Squarewave output; 5000 Hz; 1Vpp, 0.50V Offset,
with DUTY CYCLE=25%
SCOPE: Timebase 20ms; Rising edge trigger on CH0; Trigger level=0V
ANALOG OUTPUT (DAC-1): 4Vpp sinewave at 1000Hz. View on CH1
Selecting a time base of 20ms allows us to have higher resolution in the frequency domain and
see the harmonics more closely, although the individual samples in the time domain display are
more difficult to discern.
23.
Begin with a sampling rate of 5kHz, which is 5 samples per period of the 1kHz sinewave.
Display the spectrum of the sampled output (on CH1) as well as the spectrum of the sampling
Experiment 12 – Sampling and Aliasing
© 2011 Emona Instruments
12-11
pulse train (on CH0). Now progressively reduce the sampling rate in 500Hz steps ie 4500Hz,
4000Hz, 3500Hz, and observe the effect on the spectrum. Carefully keep track of the
positions of the spectrum images about each harmonic of the sampling clock. Note that the only
component that is not shifting is the one at 1000 Hz, corresponding to the input sinewave.
Use RUN /STOP to hold the time domain display if slipping.
24.
When you have reduced the sampling rate to 3000Hz, locate the component adjacent to
the one at 1000Hz. It should be at 2000Hz. We denote this component as the lower sideband
of the first spectrum image. (The upper sideband of the first spectrum image is at 4000Hz).
Continue reducing the sample rate and carefully follow the further movement of this component.
Note the sample rate when the lower sideband is at the same frequency as the input sinewave.
Question 10
At what sampling rate does the lower sideband of the first spectrum image become located at
the same frequency as the input sinewave ?
25.
Proceed further, to 1500Hz say, and note that the first lower sideband is now at a
frequency below that of the input sinewave. Examine the overall spectrum and check that you
are able to identify the pairing of the sidebands about their respective mirror frequencies.
Note that the image patterns/pairs are now overlapping.
26.
Proceed to 1000Hz . Now note that the output also contains a DC component. If you
were sampling at the exact rate of the input signal, and were exactly in phase as well (something
we are not able to due to slippage between the sampling pulses and the input) then you would
expect to sample the sinewave at the exact same point in each period, resulting in a DC output.
Consider this especially in relation to the frequency domain display you have.
Now that we have seen the effect of the sample rate on the spectrum of the sampled sinewave,
we are ready to focus on the recovery of the analog input with a lowpass filter as examined in
step 3 and 4. We will watch the spectrum of the filter input and output as the cut-off
frequency is tuned to suppress the unwanted components, and discover the challenges that arise
as we strive for the lowest achievable sampling rate.
27.
Add the TUNEABLE LPF to the experiment setup as shown in Figure 6. Display the
spectrum at the TUNEABLE LOWPASS FILTER output. Set the sampling frequency back up to
5kHz and the TUNEABLE LOWPASS FILTER cut-off to the highest available (Fc knob fully
clockwise). Set the TUNEABLE LPF GAIN to approximately 1 (mid position). Progressively
reduce the TUNEABLE LOWPASS FILTER cut-off so that all components of the sampled
spectrum are suppressed except one. Note its frequency and compare this with the spectrum
of the sampled signal. Satisfy yourself that this is as expected.
Repeat this with the FUNCTION GENERATOR at 1500Hz. Again, compare the result with the
spectrum of the sampled signal and verify the validity of the outcome. Why is the original
analog signal not recovered in this case? Why is the term alias is used to describe this non valid
output?
Question 11
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© 2011 Emona Instruments
Experiment 12 – Sampling and Aliasing V1.2
You should be able to recover a clean sinewave. What is its frequency ? Where does it come
from?
In the next segment we look for the lowest sampling rate that allows proper recovery of the
input sinewave without generating alias components.
28.
Return to the setup at the start of step 13. Decrease the sampling rate to 2000Hz and
tune the TUNEABLE LOWPASS FILTER as before. Continue reducing the sampling rate and
retuning the TUNEABLE LOWPASS FILTER until the filter can cleanly resolve the desired
input component at 1000Hz, but not reduce its amplitude. We are trying to isolate the wanted
component alone and separate it from the unwanted “images”. Note the sampling rate
corresponding to this situation, and the frequency of the first unwanted sideband. Satisfy
yourself that this frequency corresponds to the lower edge of the stopband of the filter and
that the edge of the passband is at 1000 Hz.
Consider how the sampling rate could be decreased further if a filter with tighter transition
band was available. In theory, an ideal “brickwall” filter is needed to achieve ideal results. We
will have to make do with the capabilities of the TUNEABLE LPF. Check the SIGEx User Manual
to inform yourself about its performance characteristics.
Question 12
Why is it not possible to recover the analog input when the number of samples per cycle of the
input sinewave is less than two?
Question 13
What is the minimum sampling rate that allows a filter to be able to recover the original
sinewave signal without any other unwanted components ?
Multi-frequency input spectrum
Up until now we have simply worked with a single frequency input sinewave for simplicity. The
sampling theorem also works with multiple component input signals such as voice or noise. You
have discovered that the minimum sampling rate that can be used to recover a message
correctly is twice the bandwidth of the signal to be sampled. This rate is commonly known as the
“Nyquist rate”.
In this section we will repeat the previous steps quickly, but with a multi-frequency input signal.
This will allow us to enhance our visualisation of the sampled spectrum and the occurrence of
“aliasing”. There is a two component signal available at ANALOG OUTPUT DAC-0. It has an easy
to recognise “triangular” envelope. This signal is arbitrary and has been created to assist in
visualisation of the sampling process.
Experiment 12 – Sampling and Aliasing
© 2011 Emona Instruments
12-13
Figure 8: example of a 2-component signal at DAC-0: time and frequency representation
Figure 9: example of a sampled multi-frequency signal, in time and frequency domains
29.
Repeat the steps 23,24,& 25 in the section above: “Aliasing and the Nyquist rate” whilst
using this multi-frequency signal source. Start with a sampling rate of 10kHz to more clearly see
the images.
Notice how the triangular spectrum is reflected about each harmonic of the sampling signal,
forming a series of spectral pairs.
OPTIONAL: Uses of undersampling in Software Defined Radio
This section is optional and builds upon your understanding of the sampling process.
As has been seen so far, the sampling process can also be thought of as a simultaneous, multifrequency multiplying process, where the sampling waveform is a series of harmonics with
decreasing amplitudes. We also know precisely what the outcome of each multiplication will be.
With this in mind, we can take a closer look at the sampling theorem for lowpass signals. Rather
than stating that a signal must be sampled at least at twice the rate of the highest component
in that signal, it states that the signal must be sampled at twice the bandwidth of the signal in
question regardless of its position in the passband. This means that signals in the passband need
only be sampled a minimum of twice the signal’s bandpass bandwidth, and NOT twice the
absolute frequency of the passband signal.
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© 2011 Emona Instruments
Experiment 12 – Sampling and Aliasing V1.2
For example, a DSBSC signal of a message with 2Khz bandwidth, at a bandpass frequency
centered at 100kHz (such as is used in EMONA DATEx DSB-SC experiments) need only be
sampled at 2 * 2kHz, and not 2 * 102kHz.
The reason for this is important and has hopefully become evident in previous sections. The
sampling process creates upper and lower sideband pairs about each harmonic of the sampler.
The “Nyquist rate” limitation exists to avoid the created “pairs” overlapping each other and
creating new and unknown components inside the message bandwidth.
This means we can sample at frequencies lower than the passband which is less demanding of
the sampling circuitry and has been utilised in contemporary wireless front-end convertors
which sample directly at RF frequencies, an already demanding task.
In this part of the experiment, we will firstly create a passband DSB-SC signal with a two-tone
“message” signal at a “carrier” frequency of 10kHz. We will then sample this signal at greater
than the 4000Hz (twice the message bandwidth of 2000Hz), but much lower than the actual
maximum frequency of the passband signal itself, 11kHz multiplied by 2 ie: 22kHz.
Remembering that the sampling signal can have harmonics, Fs, 2Fs, 3Fs,… should alert you to the
fact that we are again simply doing a simultaneous, multi-frequency multiplication in which the
smaller 10kHz component (2nd harmonic of 5000Hz) will multiply our message band down to
baseband, where we can directly receive it. This is the essence of a direct down-conversion
receiver in telecommunications.
In effect we are carefully exploiting the “aliasing” effect which in the previous section we were
careful to avoid. Notice that we need to select a sampling frequency which has an integer
harmonic multiple equal to the center of the passband signal.
30.
Patch together the experiment setup in Figure 10. The ANALOG OUTPUT: DAC-0 will
supply the passband DSB-SC signal at 10kHz. Select the “PASSBAND” option on the SFP TAB 12
with the toggle switch.
Figure 10: direct down-conversion receiver using undersampling
31.
View both inputs to the MULTIPLIER. One is the passband signal, with 2 upper sideband
components, and 2 lower sideband components with a total bandwidth of 2kHz. The other is the
sampling signal with harmonics at DC, 5kHz, 10kHz, 15kHz visible.
Experiment 12 – Sampling and Aliasing
© 2011 Emona Instruments
12-15
Figure 11:examples of inputs and outputs to down-convertor in time and frequency domains
Settings are again as follows:
FUNCTION GENERATOR:Squarewave output; 5kHz; 1Vpp, 0.50V Offset, with DUTY
CYCLE=25%
SCOPE: Timebase 10ms; Rising edge trigger on CH0; Trigger level=0V
32.
Remove the scope lead from the sampling signal and use it to view the output of the
MULTIPLIER. Confirm that there is a small image of the “triangular” message from 0 to 1kHz ie
at baseband.
33.
View the input and output to the TUNEABLE LPF and gradually tune out all frequencies
except for the baseband message. The characteristic “triangular” envelope of the message in
the frequency domain lets us see that the signal is in fact available at baseband. There will be
some modulation of the signal levels due to non-synchronisation of the sampling signal.
Synchronisation is a major issue in this type of down conversion.
It is also labelled “undersampling” due to the use of low sampling rates, and these techniques
feature in modern Software Defined Radio and wireless systems.
34.
Experiment with different sampling rates and confirm that the signals you see are in
accordance with theory. Be aware of the phenonema of negative frequencies “folding” about the
0Hz point and being reflected into the positive frequency domain. This is often the source of
unaccounted-for components.
12-16
© 2011 Emona Instruments
Experiment 12 – Sampling and Aliasing V1.2
Tutorial questions
Q1 re need for anti-aliasing filter:
Explain the role of the anti-aliasing filter in the sampling process. Show that
aliasing is caused by the sampling process, hence the anti-aliasing filter must
be ahead of the sampler.
Consider a signal of bandwidth 3.5kHz in the presence of wideband noise. The
noise spectrum is uniform and has a bandwidth of 500kHz. The SNR is 30dB.
The signal is sampled at 8k/s. What is the signal-to-noise-ratio (SNR) of the
sampled signal if a suitable anti-aliasing filter is used. Compare this with the
SNR that would be obtained if an anti-aliasing filter were not used.
Q2 re trade-off: excessively sharp cut-off lowpass filters.
Describe why the use of sharp cut-off anti-aliasing and interpolation lowpass
filters helps achieve sampling rates close to the theoretical limit. Indicate
the disadvantages of sharp cut-off lowpass filters, and the criteria for
obtaining a practical compromise.
Q3 re step 2
Suppose in the model of Fig 1 we were able to increase the duty cycle to the
point of taking up the entire interval between samples, would this be
equivalent to S/Hold ? Describe how you would implement a S/Hold device
(clue: a capacitor is needed to hold the sampled analog voltage).
Q4 Why is S/Hold used in practical DACs, in preference to narrow pulses?
Q5 x/sinx correction
Data sheets of commercial digital to analog converters mention "x/sinx"
correction. Why is this needed?
Experiment 12 – Sampling and Aliasing
© 2011 Emona Instruments
12-17
12-18
© 2011 Emona Instruments
Experiment 12 – Sampling and Aliasing V1.2
Experiment 13 – Getting started with analog-digital conversion
Achievements in this experiment
Practical knowledge of coding of an analog signal into a train of digital codewords in binary
format using pulse code modulation (PCM), i.e., analog to digital conversion. Understanding of the
decoding process, quantization issues and reconstruction of the output signal. Appreciation of
the need for both bit clock and frame synchronisation .
Preliminary discussion
This lab introduces the basics of digital-analog signal conversion with the PCM ENCODER and
PCM DECODER modules. The formatting of a PCM signal, will be examined in the time domain.
Part 1 deals with encoding. The recovery of the analog signal is covered in Part 2.
How did the idea of PCM Encoding come about ?
Imagine you wanted to send an analog signal such as voice across a noisy channel. Naturally the
received signal would be subject to a degree of additive noise, as well as other forms of signal
distortion due to bandwidth limitations. It may become unrecognisable and hence useless.
By converting that analog signal into packets of digital data, we are able to benefit from the
high level of noise immunity inherent in digital signals. Although digital data transmission has its
own issues, mainly, bandwidth limitations, we can be certain of the ability to reconstruct the
message if the digital error rate is kept within limits.
It should take you about 45 minutes to complete this experiment, not including the preparation
to be done before the hands-on lab work.
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© 2011 Emona Instruments
Experiment 13 – Getting started with analog-digital conversion V1.2
Pre-requisite work
PCM ENCODER basics
The PCM ENCODER module converts an analog input into a stream of digital codewords. As
investigated in Lab 12, the maximum allowable input bandwidth will depend on the sampling rate.
The amplitude range must be matched to the input range of the encoder. If excessive,
overloading will occur. If too small, noise and interference will degrade signal quality. The
appropriate range is within the ± 2.5 volts range. This is specified in the SIGEx User Manual.
A step-by-step description of the operation of the module follows:
i.
the module is driven by an external TTL clock.
ii.
the input analog signal is sampled periodically. The sample rate is determined by the
external clock.
iii.
the sampling process is carried out as a sample-and-hold operation, within the module.
The held output ( i.e., the amplitude of the analog signal at the sampling instant) is not available
on the front panel of the module 1.
iv.
the PCM encoding is carried out by comparing each sample amplitude with a finite set of
amplitude levels. These are distributed (uniformly, for linear sampling) within the range
± 2.5 volts. These are the system quantizing levels.
v.
each quantizing level is assigned a number, starting from zero for the lowest (most
negative) level, with the highest number being (L-1), where L is the available number of levels.
vi.
each sample is assigned a digital (binary) code word representing the number associated
with the quantizing level which is closest to the sample amplitude. The number of bits ‘n’ in the
digital code word will depend upon the number of quantizing levels.
Question 1
Show that n = log2(L):
vii.
the code word is assembled into a time frame together with any other bits as may be
required (described below). In the PCM ENCODER (and many commercial systems) a single
extra bit is added, in the least significant bit position. This is alternately a one or a zero.
These bits are used for frame synchronization in the decoder.
viii.
the frames are transmitted serially. They are transmitted at the same rate as the
samples are taken. The serial bit stream is accessible on the front panel of the module.
1 Sampling with a sample and hold operation is introduced in Lab12 where you are able
to display the "held" output.The PCM ENCODER effectively has a “sample and hold”
unit within its front end.
Experiment 13 – Getting started with analog-digital conversion
© 2011 Emona Instruments
13-3
ix.
also available from the module is a synchronizing signal FS (‘frame synch’). This signals
the end of each data frame.
Before proceeding with the experiment tasks, we briefly review the essential features of the
ENCODER module.
Figure 1:Layout of the PCM ENCODER
The SIGEx board layout of the module is shown in Figure 1. Technical details are described in
the SIGEx User Manual.
Familiarize yourself with the purpose of each of the input and output connections, and the
three-position toggle switch. Counting from the top, these are:
• FS: frame synchronization, a signal which indicates the end of each data frame.
• Vin:: the analog signal to be encoded.
• PCM DATA: the output data stream, the examination of which forms the major part of this
experiment.
• CLK: this is a TTL (red) input, and serves as the MASTER CLOCK for the module. Clock rate
must be 20 kHz or less. For this experiment you will use a 10kHz TTL signal from the PULSE
GENERATOR module.
PCM time frame
Each binary word is located in a time frame. The time frame contains eight slots, with one
clock period per slot. These slots hold the bits of a binary word, numbered 7 through 0, from
first to last. The least significant bit (LSB) is contained in slot 0.
13-4
© 2011 Emona Instruments
Experiment 13 – Getting started with analog-digital conversion V1.2
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should observe
the USB light turn on (top right corner of ELVIS unit).The PC may make a sound to indicate that
the ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that
all three power LEDs are on. If not call the instructor for assistance.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with
the NI ELVIS that you’re using.
9.
You’re now ready to work with the NI ELVIS/SIGEx bundle.
10.
Select the EXPT 13 tab on the SIGEx SFP.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the
STOP button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the
top of the window. This will allow the program to conduct an orderly shutdown and close the
various DAQmx channels it has opened.
Experiment 13 – Getting started with analog-digital conversion
© 2011 Emona Instruments
13-5
The experiment
We are now ready to proceed. Note that the PCM ENCODER is the only module used in
Part 1.
It is not necessary to become involved with how the PCM ENCODER module achieves its purpose.
What will be discovered is what it does under various conditions of operation.
PCM encoding and quantisation
Examination with DC input
We begin with some basic aspects of the analog to digital conversion process. For this purpose
we will use a constant (DC) analog input. This ensures completely stable oscilloscope displays,
and enables easy identification of the quantizing levels.
11.
Patch the PCM ENCODER as shown in Figure 2.
Figure 2:patching diagram for PCM ENCODER
Settings are as follows:
PULSE GENERATOR: 10,000 Hz with DUTY CYCLE=0.5 (50%)
SCOPE: Timebase 2ms; Rising edge trigger on CH0; Trigger level=1V
Zero input to PCM ENCODER
12.
connect the PCM INPUT terminal to a GND terminal on the SIGEX board.
13.
display FS on CH0. Adjust the timebase to show three frame markers. These mark the
end of each frame, that is, the last bit of the frame.
14.
display the CLK signal on CH1.
Question 2
Record the number of clock periods per frame.
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© 2011 Emona Instruments
Experiment 13 – Getting started with analog-digital conversion V1.2
Question 3
Currently the analog input signal is zero volts (since INPUT is grounded). Before checking with
the scope, consider what the PCM encoded output might look like. Can you assume that it will be
00000000?. What else might it be, bearing in mind that this PCM ENCODER outputs offset
binary format.
Question 4
On CH1 display the signal at PCM DATA output.The display should be similar to that in Figure 3
(possibly with fewer frames). Is it in agreement with your expectations?
The same codeword appears in each frame because the analog input is constant. This codeword
represents the 8-bit binary output number, corresponding to the zero volt analog input.
Figure 3: PCM Encoder input and output signals
Confirm the following three points:
1. the number of slots per frame is 8
2. the location of the least significant bit is coincident with the end of the
frame
3. the binary word length is 8 bits
Variable DC input to PCM ENCODER
15.
remove the ground connection, and connect the INPUT to output of the ANALOG
OUTPUT: DAC 1. This programmable output is set to be a VARIABLE DC voltage, which is
controlled from the control on the Lab13 TAB on the SIGEX SFP. Sweep the DC voltage slowly
Experiment 13 – Getting started with analog-digital conversion
© 2011 Emona Instruments
13-7
backwards and forwards over its complete range, and note how the data pattern within each
frame changes in discrete jumps.
Question 5
Adjust VARIABLE DC to its maximum negative value. Record the DC voltage and the pattern of
the 8-bit binary number.
Question 6
Slowly increase the amplitude of the DC input signal until there is a sudden change to the PCM
output signal format. Record the format of the new digital word, and the input DC voltage at
which the change occurred. Use the INCREMENT arrows on the digital value entry box for a
steady stable increase in DC value.
16.
continue this process over the full range of the DC supply. Note a selection of DC
voltages and corresponding binary output words in the table below.
Table 1: DC VOLTAGE input vs. PCM codewords
DC VOLTAGE (V)
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© 2011 Emona Instruments
8 bit PCM codeword
Experiment 13 – Getting started with analog-digital conversion V1.2
17.
draw a diagram showing the quantizing levels and their associated binary numbers in the
graph below.
Graph 1:DC to binary word plot
Review of the 8-bit data format
Question 7
On the basis of your observations so far, provide answers to the following:
*
*
*
*
*
*
*
what is the sampling rate ?
what is the frame width ?
what is the width of a data bit ?
what is the width of a data word ?
how many quantizing levels are there ?
are the quantizing levels uniformly (linearly) spaced ?
what is the the minimum quantized level spacing ? How does this compare to theory ?
Experiment 13 – Getting started with analog-digital conversion
© 2011 Emona Instruments
13-9
Question 8
The relationship between the sampled input voltage and the output codeword has been
described above. Suggest some variations of this relationship that could be useful ?
Working with periodic signals
In this section we consider the challenge of generating a stable display of the encoded output
with a conventional scope when a periodic input signal is used. Carry out the three tasks below
to see the problem.
This difficulty does not arise if you have access to an instrument with signal capture capability,
such as a digital scope. You can use the RUN/STOP button on the SFP to hold a single screen
display of the data when necessary.
In Part 2 this issue is further examined when we display the reconstructed signal at the
decoder output.
18.
Take a periodic signal from the ANALOG OUTPUT: DAC0 terminal.
Question 9
Adjust the scope to display this waveform. Record its shape and frequency. Check whether
this conforms with the Nyquist criterion. Show your reasoning.
19.
Display the PCM DATA output on CH1. Again, synchronize the scope to the frame signal
FS. Display two or three frames on CH1. Notice how they differ as expected.
20.
Move the scope lead from the FS and view the INPUT signal on CH0. You are now viewing
the input sinusoid as well as the output codeword as in Figure 4. Freeze the display occasionally
(using RUN/STOP) and look at the relationship between input and output. See if you can see the
pattern.
13-10 © 2011 Emona Instruments
Experiment 13 – Getting started with analog-digital conversion V1.2
Figure 4: input and output from PCM ENCODER
PCM decoding and reconstruction
In this part we proceed with the decoding2 of the output of the PCM ENCODER to reconstitute
the analog input. Before proceeding with the experiment tasks, we briefly review the
OPERATION of the DECODER.
PCM DECODER
The operation of the PCM DECODER module is as follows:
•
a frame synchronization signal FS is received from the transmitter.
•
the binary number representing the coded (and quantized) amplitude of the sample from
which it was derived, is extracted from the frame.
•
the quantization level represented by the input codeword is determined.
•
a voltage proportional to this amplitude level is generated.
•
this voltage appears at the output terminal OUTPUT for the duration of the respective
frame.
•
signal reconstruction is completed by lowpass filtering, as seen in Lab 12. A built-in
reconstruction filter is not provided in the module so that you can see the quantization
steps in the output. The recovered signal will not be identical to the original due to
quantizing. Other imperfections may be due to aliasing distortion, as already considered in
Lab 12.
The DECODER is driven by an external clock which must be synchronized to that of the
ENCODER. It is possible to “embed” the frame sync signal into the data, but this feature is
outside the scope of this lab and is not included in the functionality of the SIGEx board3.
An alternative to automatic frame synchronization is to steal the synchronization signal, FS,
from the PCM ENCODER module. Use of this signal would assume that the clock signal to the
2 it is common practice to refer to demodulation from analog signals, and decoding from digital signals.
3 This feature is available on the EMONA DATEx Telecommunications Trainer board for NI ELVIS
Experiment 13 – Getting started with analog-digital conversion
© 2011 Emona Instruments
13-11
PCM DECODER is of the correct phase. This is assured in this experiment, but would need
adjustment if the PCM signal is transmitted via a bandlimited channel. Hence the use of
embedded frame synchronization information in some systems.
The front panel of the DECODER module is shown in Figure 3. Technical details are described in
the SIGEx User Manual.
Figure 5: Layout of the PCM DECODER
The following is a summary of the input and output connections :
•
FS: connect an external frame sync. signal here
• PCM DATA: the PCM signal to be decoded is connected here.
• OUTPUT: the decoded signal.
• CLK: this is a TTL (red) input, and serves as the CLOCK for the module. Clock rate must be
20 kHz or less. For this experiment you will use the same clock as used for the PCM
ENCODER.
Figure 6: PCM Encoding and decoding patching diagram: input from DAC 0.
TLPF serves as recontruction filter
13-12 © 2011 Emona Instruments
Experiment 13 – Getting started with analog-digital conversion V1.2
Decoder Experiments
21.
The PCM ENCODER provides the digital input for the DECODER. Patch together as
shown in Figure 7.
Settings are as follows:
PULSE GENERATOR: 10,000 Hz with DUTY CYCLE=0.5 (50%)
SCOPE: Timebase 20ms; Rising edge trigger on CH0; Trigger level=1V
TUNEABLE LPF: Fc set to max, GAIN set to max.
22.
Connect CH1 of the SCOPE SELECTOR to the OUTPUT of the PCM DECODER, and CH0
to the INPUT signal at the ENCODER. Trigger the SCOPE on the INPUT signal. Your display
should resemble Fig. 7.
Figure 7: input and output signals
The recovered analog signal & reconstruction
In this section we use a periodic signal to observe the quantized appearance of the decoder
sample-hold output.The internal DAC output implements a “zero-order hold” operation, in that it
holds the value for a complete clock period. This has an inherent filtering effect, prior to any
reconstruction filtering.
The OUTPUT signal is distinctly different from the input signal. The output is only updated once
every frame, hence there is a large step change between output samples. This difference is
known as “sampling distortion”.
As well the “granularity” of the output only having 256 possible levels will contribute some
error, known as “quantizing distortion”, however, it is not easily observable with this input signal.
In the previous section you will have calculated the step size due to quantization by the encoder.
You can see, qualitatively, that the stepped output signal bears a resemblance to the input. We
saw in Lab 12 that the use of a lowpass filter made it possible to smooth out the steps. The
need for pulseshape equalization in the form of so called “x/sinx” correction was also mentioned
there.
Experiment 13 – Getting started with analog-digital conversion
© 2011 Emona Instruments
13-13
Question 10
Momentarily, vary the clock rate from 10,000 to 20,000 Hz. How does this affect the “sampling
distortion” viewable in the output signal ?
An interpolation LPF is provided in the TUNEABLE LPF block.
23.
Adjust the TUNEABLE LPF in the output path of the PCM DECODER to vary the filters
corner frequency, and observe the reconstructed signal. Adjust the TUNEABLE LPF GAIN
control to set the output signal to the same amplitude as the input. This makes the overall
conversion gain equal to unity. The PCM DECODER does not have the same conversion gain as the
ENCODER.
Question 11
View the input to the TUNEABLE LPF, ie the output of the PCM DECODER and compare with the
INPUT sinusoid. What is the gain of the PCM DECODER itself.
Again viewing the output from the TUNEABLE LPF, which is acting as the reconstruction filter,
slowly reduce the Fc until the steps are eliminated, and the original signal is recovered.
Question 12
Can you explain the source of the delay between input and output signals ? Both with and
without the TUNEABLE LPF ?
Question 13
Momentarily, vary the clock rate from 10,000 to 20,000 Hz. How does this affect the required
Fc needed to recover the signal without distortion ?
13-14 © 2011 Emona Instruments
Experiment 13 – Getting started with analog-digital conversion V1.2
Tutorial Questions
Q1 from your knowledge of the PCM ENCODER module, obtained during preparation for the
experiment, calculate the sampling rate of the analog input signal. What can you say
about the bandwidth of an input analog signal to be encoded ?
Q2 define what is meant by the data ‘frame’ in this experiment. Draw a diagram showing the
composition of a frame for the 8-bit coding scheme
Q2 quantizing distortion decreases as the number of quantizing levels is increased. Explain why
an excessive increase of the number of quantizing levels may incur a bandwidth
penalty: describe how to manage this trade-off. Look up Shannon's formula and
show how it relates to this trade-off. Explain whether this issue could be observed
in this lab ?
Q3 explain why a DC input provides a stable display of the PCM DATA output on a conventional
scope. Why is the display ‘unstable’ when the input is a sine wave (for example) ?
Q4 look up “two’s complement” encoding and find out a field of application where it is used.
What is the advantage of “offset binary” encoding used in this lab? Devise or look
up a method to convert from offset binary to two’s complement.
Q5 carry out a brief internet search of the principles and applications of companding. Compile a
summary of its advantages. Compare this with published information about the ‘A’
and ‘µ’ companding laws used respectively in Europe and the USA.
Q6 two sources of distortion in the reconstructed message have been identified; they were
called sampling distortion and quantizing distortion.
a) assuming a sample-and-hold type sampler, what can be done about minimizing
sampling distortion ?
b) what can be done about minimizing quantizing distortion ?
Why is “x/sinx” correction required with sample-hold interpolation? (Refer to Lab
12)
Q7 describe how to determine the specification for the reconstruction filter used in the
decoder ?
Experiment 13 – Getting started with analog-digital conversion
© 2011 Emona Instruments
13-15
13-16 © 2011 Emona Instruments
Experiment 13 – Getting started with analog-digital conversion V1.2
Experiment 14 – Discrete-time structures:
Finite Impulse Response (FIR) filters
Achievements in this experiment
You will be able to relate the response of a discrete-time (DT) FIR filter to its transfer
function. You will use the zeros of the transfer function to visualize frequency responses
graphically at a glance, without math. You will be able to use this knowledge to intuitively
design low order discrete-time responses. You will be ready to extend this concept to
recursive DT filters and to higher order applications.
Preliminary discussion
In Lab 11 we used poles and zeros as an intuitive aid in working with continuous-time filters. In
this Lab we apply the same idea for discrete-time applications. DT filters can be implemented
with or without the use of feedback. The latter filters are generally known as nonrecursive
or Finite Impulse Response (FIR).
The transfer function of a nonrecursive filter can be expressed as a polynomial. Since there is
no denominator, there are no poles, only zeros, which makes it simpler for getting started. As
with the continuous-time case, you can intuitively predict and track system responses from the
zeros.
The absence of feedback is an important advantage in many applications as there is no risk of
unstable behaviour. This is very useful when working with adaptive filters.
Serious practical realizations of FIR filters generally require a large number of delay
elements. The most demanding are the bandpass. However, the notch filter works well as a
two-delay FIR example. We will use this example here to examine the relationship between the
frequency response and the coefficient values through the interpretation of zero positions in
the z plane.
14-2
© 2011 Emona Instruments
Experiment 14 – Discrete time Structures: FIR V1.2
Preparation
This preparation provides essential theory needed for the lab work to make sense.
Question 1
Consider the system in Figure 1, where nT are the discrete-time points, with T sec denoting
the unit time delay, i.e. the time between samples. Show that the difference equation relating
the output y(nT) and the input u(nT) is
Y(nT) = b0.u[nT] +
b1.u[(n - 1)T] + b2.u[(n - 2)T]
(Eqn 1).
Show by substitution that ejnTω is a solution, i.e. show that when the input is ejnTω , y(nT) is
ejnTω multiplied by a constant (complex-valued);
ω is the frequency of the input in
radians/sec.
+
OUTPUT
INPUT
b2
b1
b0
UNIT
DELAY
UNIT
DELAY
Figure 1: schematic of FIR filter with two unit delays
In Lab 11 we used a complex exponential input to represent the behaviour of a system that is
supposed to operate with real-valued signals. You could consider using u[nT] = cos(nTω) or
sin(nTω) instead. However, the use of the exponential function simplifies the math
considerably. We have already seen that cos(ωt) is Re{exp(jωt)}, so, you can carry out the
analysis with ejnTω , then simply take the real part of the result. After a while, working with
complex exponential functions to represent sinusoids becomes second nature and we don't even
bother thinking about taking the real part. Many practical systems implemented digitally
actually operate with complex-valued signals, for example modulators and demodulators
working with quadrature signals.
From the above, with input u(nT) = ejnTω , show that
H = y/u = b0 + b1. e-jTω + b2. e-j2Tω
(Eqn 2).
Note that H is not a function of n.
Question 2
Use this result to obtain a general expression for the magnitude of y/u as a function of ω. You
will need to first write down the real and imaginary parts.
Set T = 1 sec for the time being, and plot the result for the case b0 = 1, b1 = -1.3 ,
b2 = 0.9025 over the range ω = 0 to 2.π rad/sec. Label the frequency axis in Hz as well as
rad/sec. You should find there is a significant dip in the response near 0.13Hz.
Question 3
Experiment 14 – Discrete time Structures: FIR
© 2011 Emona Instruments
14-3
As in Lab 7, we consider an alternative way of getting frequency responses. We will create a
graphical medium to provide an intuitive environment for visualizing and generating both
magnitude and phase responses.
First, return to the expression for y/u obtained in (a) and replace "exp(jTω)" by the symbol
"z". Look upon z merely as a convenient macro for exp(jTω) . At this point there is no need to
ascribe any deeper significance to this substitution. The result is the (complex-valued)
polynomial
y/u = H(z) = b0 + b1.z-1 + b2.z-2 = z-2 . [b0. z2 + b1.z + b2 ]
(Eqn 3).
For the case b0 = 1, b1 = - 1.3 , b2 = 0.9025 (from (Q2)), express the quadratic in the brackets
in the factored form (z - z1)(z - z2), where z1 and z2 are the roots. Show that these are given
by
z1 = 0.95e j0.260π
z2 = 0.95e -j0.260π
(Eqn 4).
Satisfy yourself that the magnitude response of H can be expressed as
|H(ω)| = |(ejT ω - z1)|.|(ejT ω - z2)|
(Eqn 5).
Write down the corresponding expression for the phase of H.
Question 4: Graphical plotting of poles & zeros
We are now ready to proceed with a graphical approach for evaluating the factors (z - z1) and
(z - z2) in Eqn 3. Place an "o" on a complex plane (we will refer to this as the z plane) at the
locations corresponding to z1 and z2, as obtained in Eqn 4. With T = 1, we will get an estimate
of |H| at ω = π/5.
Place a dot at the point ejπ/6. Join this point and the point z1 with a straight line. The length of
this line is |(ejπ/5- z1)|.
Do the same with z2 to obtain |(ejπ/5 - z2)|. From Eqn 5, the desired estimate of |H(π/5)| is
simply the product of the lengths of these two lines.
Question 5
By repeating this for other values of ω we are able to get a quick estimate of the graph of |H|
versus ω. It's important to note that the locus of ejTω is a circle of unity radius centered at
the origin (known as the unit circle). Hence, the general shape of the frequency response is
easily estimated by simply running a point counter-clockwise along the circumference of the
unit circle, starting at (1, 0). Note that the idea is just a variant on the procedure introduced
in Lab 11, where we moved the frequency point along the j axis. Compare the outcome with the
result computed in (Q2).
Notice that the presence of the trough in the response can be seen at a glance from the
behaviour of the vector from the "zero" z1 as the dot on the unit circle is moved near z1. By
comparison, the rate of change of the other vector is small over that range.
Question 6
Modify Fig 1 by replacing the unit delays with a gain of 1/z and show that Eqn 3 follows by
inspection using simple algebra, without the need to work through the difference equation
14-4
© 2011 Emona Instruments
Experiment 14 – Discrete time Structures: FIR V1.2
step. While this is only a minor simplification in this example, it is very useful in more
complicated cases, especially where feedback loops are involved.
Although z was originally introduced in (Q3) as just a substitution for ejTω, our interpretation
appears to have been extended in (Q4) to include any complex number. Consider whether this
is the case, and why.
Question 7
In the above example, we had the sample interval T = 1. Suppose T = 125 microsec. Adjust the
frequency axis for this value of T. Extend this result for any value of T. How are the zeros of
H(z) affected by the value of T? Why is it appropriate to use T = 1 normally?
Show that |H(ω)| is periodic, and determine the period in Hz.
Experiment 14 – Discrete time Structures: FIR
© 2011 Emona Instruments
14-5
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should
observe the USB light turn on (top right corner of ELVIS unit).The PC may make a sound to
indicate that the ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that
all three power LEDs are on. If not call the instructor for assistance.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with
the NI ELVIS that you’re using.
9.
You’re now ready to work with the NI ELVIS/SIGEx bundle.
10.
Select the Lab 14 tab on the SIGEx SFP.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the
STOP button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the
top of the window. This will allow the program to conduct an orderly shutdown and close the
various DAQmx channels it has opened.
Ask the instructor to check
your work before continuing.
14-6
© 2011 Emona Instruments
Experiment 14 – Discrete time Structures: FIR V1.2
Experiment
Part 1 Notch filter using two-delay FIR:
We will be implementing the filter in Fig. 1. Figure 2 shows the SIGEx wiring diagram. The
gains b0, b1, b2 in the triple ADDER module realize the coefficients b0, b1, b2, respectively.
11.
Patch up the SIGEx model in Fig. 2.
Settings are as follows:
ADDER gains: b0 =1.0; b1 = -1.3; b2 = 0.902
PULSE/CLK GENERATOR: 10 kHz, Duty cycle = 0.5 (50%)
FUNCTION GENERATOR: 1000 Hz; 4Vpp, Sinewave output selected
SCOPE: Timebase 4ms; Rising edge trigger on CH0; Trigger level=0V
View the input signal on CH0, and output with CH1. Check that the time interval between
samples is consistent with the clock frequency.
Figure 2: SIGEx wiring model of FIR filter with two unit delays
12.
Measure and plot the magnitude response over the range 300 Hz to 3kHz using the
FUNCTION GENERATOR. Note that the amplitude measurement could be a little challenging
at the upper frequencies due to the increasing interval between samples relative to the signal
period. It is suggested you vary the timebase appropriately, and use the RUN/STOP and YAUTOSCALE buttons when necessary to achieve a stable display.
Confirm that the response has a deep notch.
Experiment 14 – Discrete time Structures: FIR
© 2011 Emona Instruments
14-7
Graph 1:response plot
Question 8
Measure the notch frequency and the depth relative to the response at DC. Also measure the
time delay as a function of frequency at several points of interest.
13.
Using the values of b0, b1, b2 entered, apply the method developed in the preparation
section to plot the zeros of this filter in the z-plane. Check that you have the correct sign of
the coefficients (the zeros should be in the right half-plane). Compare the measured
frequency response with the plot of the zeros. Verify that the measured notch frequency
agrees with the position of the zeros (remember, the frequency at the 180 degree point on the
unit circle corresponds to f_sample/2 = 1/2T Hz).
14.
Decrease the b1 gain slightly and observe the effect on the magnitude response.
Measure the new notch frequency. Use this to determine the new positions of the zeros and
the corresponding value of b1. Verify the agreement between measurement and theory.
Question 9
Determine and note the new notch frequency, for the b1 gain entered. Document the
relationship between b1 and notch frequency
14-8
© 2011 Emona Instruments
Experiment 14 – Discrete time Structures: FIR V1.2
15.
This time, reduce b2 gain very slightly (order of 2-3%, say) and again, observe the
effect on the magnitude response. The outcome should be a reduced notch depth. Try to
determine the approximate position of the zeros from the notch depth (notch depth relative
to DC gain, say).
Part 2 Using notch filter to eliminate interference
In the next exercise we use the notch filter to remove an interference tone. Consider this
practical situation: we are trying to receive a message at frequency f1 from a distant
transmitter, but in our neighbourhood there is another transmitter sending an unwanted signal
at frequency f2 affecting our reception of f1.
16.
Keep the PULSE/CLK GENERATOR frequency at 10,000Hz and maintain the
coefficients values as for Part 1 of this experiment.
17.
Use the ANALOG OUTPUT as a source of signals f1 ,and f2. Add these using the “f +
g” ADDER block to create f1 + f2 .
Confirm that DAC-0(f1 ) = 500Hz sinusoid, and DAC-1 ( f2 ) = 1300 Hz sinusoid.
18.
View each output individually as well as the sum of the outputs. View them both in the
time domain as well as in the frequency domain.Show that the f2 signal can be suppressed with
the notch filter. Remember that the notch is set to 1.3kHz.
Question 10
What is the level of attenuation of the f2 signal for the original zero positions.
19.
Vary the value of b1 slightly in each direction and notice the affect on the output.
Remember that b1 controls the frequency of the notch. This is a finding from a previous part
of this experiment.
NOTE: To vary the value of a control slightly, set the cursor next to the digit you wish to
vary and then use the UP and DOWN arrow buttons on the control.
As well, the GAIN ADJUST knob on the SIGEx board itself can be used to vary by hand
the coefficient value of the ADDER.
Question 11
From your previous findings in this experiment, what change is required to gain b1 to reduce
the notch frequency ?
Experiment 14 – Discrete time Structures: FIR
© 2011 Emona Instruments
14-9
20.
View both the input two-tone signal on CH1 and the output signal from the notch filter
on CH0. In particular, view them on the FFT display of Lab14 SFP.
21.
As well, switch between the Lab 14 TAB and the PZ PLOT TAB on the SFP while making
these adjustments. Notice how the zeroes move when b1 is adjusted. In particular, interpret
the notch frequency from the position of the zeroes. By entering the sample clock frequency
into the “clock freq” control on the PZ PLOT TAB, you can automate the calculation of the
notch frequency somewhat. Ensure you understand what is being calculated.
Figure: example of PZ PLOT TAB displaying a FIR structure. Zeroes only.
Remember that the POLE-ZERO PLOT is an essential diagram for interpreting frequency
response of a structure.
Question 12
What is the equation relating theta of the zero to the frequency of the zero, as implemented
in the PZ PLOT TAB ?
22.
Vary the value of b1 until you eliminate the lower frequency component of the two-tone
message signal. You are exploring using the selectivity of a filter through variation of its
transfer function. Use both the FFT display on TAB14 and the PZ PLOT TAB during this
process so you develop an appreciation for the relatedness of these two different display
representations.
1410
© 2011 Emona Instruments
Experiment 14 – Discrete time Structures: FIR V1.2
Figure : Notch filter set to attenuate the lower component of the message
Question 13
For what value of b1 did you achieve the maximum attenuation of the lower message component
RELATIVE to the higher component ? What levels did you measure ?
23.
Pass the notch filter output, which is sampled and discrete, through a reconstruction
filter such as the BASEBAND LPF to “clean up” the signal. Experiment again with the b1
coefficient whilst viewing the now, continuous, output signal as you select between each
message component. See how “clean” a sinusoid you can recover. Experiment with attenuating
both components.
HINT: if using the GAIN ADJUST knob to control b1, you can set the available output range of
the knob so as to be able to “fine tune” your coefficient value by hand.
Question 14
What components is the TUNEABLE LPF attenuationg in order to give a “clean” signal ?
Experiment 14 – Discrete time Structures: FIR
© 2011 Emona Instruments
14-11
Tutorial Questions
Q 1.
In Eqn 4 the zeros are complex conjugates. Examine other cases in this Lab
and consider whether this is a coincidence or an outcome of a general
property of FIR filters.
Q 2.
Lowpass FIR filters are sometimes called moving average (MA) filters.
Using an example, explain the reason behind this.
Q 3.
For the system in Prep (c) show that when b0 = b2 the zeros are on the unit
circle. Plot a graph of the frequency of the zeros versus b1/b0. Tip: to fast
track the math, factor out z.
Interpret the result in the context of step 9.
Q 4.
Consider a zero at the origin of the z plane. Show that its magnitude
response is constant and its phase response linear. Show that the factor z-2
in Equation (3) generates a pair of poles at the origin. What is the effect of
these poles on the overall response H_y?
Q5.
The z plane method generates frequency responses that are periodic. Is this
an artificial mathematical side effect resulting from the use of complex
variables, or does this reflect reality (refer to the Lab on sampling)?
Q. 6.
Find the zeros for the following 4-delay FIR filters (hint: exploit the
symmetry)
(i)
[1, 2sqrt2, 4, 2sqrt2, 1]
(ii) [1, -2sqrt2, 4, -2sqrt2, 1]
(iii) [1, 0, 0, 0, 1]
(iv) [1, 0, 1, 0, 1]
(v)
[1, 0, 2, 0, 1]
(vi) [1, 1, 1, 1, 1]
(hint: this is a geometric progression)
(vii) [1, sqrt3, 2, sqrt3, 1]
State the type of filter realized in each case (lowpass/ highpass/ other).
Compare cases (i) and (ii), and derive a rule of thumb for converting a
lowpass FIR filter to highpass (make reference to the coefficients and to
the zeros).
Which of these is relevant to the implementations in step 12 and 15?
Confirm that in each case the zeros are on the unit circle. Is it a
coincidence, or is there a systematic property that these examples share?
1412
© 2011 Emona Instruments
Experiment 14 – Discrete time Structures: FIR V1.2
Q. 7.
Show that the phase responses of the filters in Q.6 are linear.
Q. 8
In Prep (h) consider an alternative case: move the zeros at +/- 150deg to
180deg (the other zeros remaining fixed). What is the effect on bandwidth
and attenuation in the stop band? What is the result if all the zeros are
placed at 180deg?
Suppose the FIR has 48 zeros. Discuss alternatives for the design of a
lowpass filter: is it preferable to spread the zeros across the stopband or to
place all the zeros at 180deg?
Q 9.
Find and plot the zeros for the coefficients in step 13. Note that the
transfer function is a truncated GP. Apply this to simplify the factorization.
Q 10.
Write down the transfer function of a first-order continuous-time LPF. The
impulse response of this filter is a decaying exponential that can be matched
approximately to the unit pulse response of the FIR in step 13. Obtain the
time constant and pole of the CT LPF that matches the FIR. Estimate the
bandwidth of the CT filter. Compare this with the bandwidth of the FIR.
Q11.
Show how an analog FIR can be implemented with short equal lengths of
coaxial cable as delay elements. Satisfy yourself that such a filter can be
used with continuous-time inputs, i.e., without sampling. On this basis we
should expect a transfer function in terms of the Laplace variable s. Obtain
such a transfer function for the case in prep (b) and show that it is a
quadratic in exp(-sT), where T is the delay of a cable element. Show that in
order to apply the graphical method for estimating responses, a z plane
representation is needed, with z = exp(sT). Filters of this kind were used
with early TV systems in the 1940's. The inventor was Kalmann (not to be
confused with Kalman filters in optimization theory).
Q12.
Consider the periodicity issue in Q5 again, in the context of Q11. Satisfy
yourself that frequency responses of this continuous time system are also
periodic.
Experiment 14 – Discrete time Structures: FIR
© 2011 Emona Instruments
14-13
1414
© 2011 Emona Instruments
Experiment 14 – Discrete time Structures: FIR V1.2
Experiment 15 – Poles and zeros in the z plane: IIR systems
Achievements in this experiment
You will be able to interpret the poles and zeros of the transfer function of discrete-time filters to
visualize frequency responses graphically at a glance, without math. You will be able to use this
knowledge to intuitively design recursive/IIR discrete-time responses.
Preliminary discussion
In Lab 11 we discovered how poles and zeros can be used as an intuitive tool for analyzing and
designing continuous-time (CT) filters. Next, in Lab 14 we examined discrete-time (DT) FIR filters
and found the same ideas could be applied there. The complex "s" plane was replaced with the
complex "z" plane, and the unit circle used instead of the j axis for the representation of frequency.
Because zeros only are involved in FIR filter work, this provided a convenient gateway to getting
started with z-plane ideas.
In this Lab we will investigate more general DT filters that are characterized with both poles and
zeros. These filters are known as recursive since they use feedback, and also as Infinite Impulse
Response (IIR). With feedback we will be able to realize much higher selectivity than possible with a
comparable complexity FIR implementation. The most conspicuous example is the second-order
resonator, which will open the way to achieving realistic bandpass responses. As we proceed, we will
find many parallels with the CT(continuous time) filter experiments in Lab 11.
Part 1: we examine the behaviour of the basic second-order resonator implemented without zeros.
Part 2: zeros are introduced to generate lowpass, bandpass, highpass and allpass responses using the
Direct Form 2 structure.
15-2
© 2011 Emona Instruments
Experiment 15 – Poles and Zeros in the z plane V1.2
Pre-requisite work:
This preparation extends the theory covered in Lab 14 to include poles.
INPUT
x0
UNIT
DELAY T
u
x1
UNIT
DELAY T
x2
-a1
-a2
Figure 1: schematic of 2nd-order feedback structure without feedforward.
Question 1
Consider the feedback system in Figure 1.
Show that the difference equation relating the adder output x0(nT) and the input u(nT) is
x0(nT) = u[nT] - a1.x0[(n - 1)T] - a2.x0[(n - 2)T]
(Eqn 1),
where nT are the discrete time points, T sec denoting the unit delay, i.e. the time between samples.
Show by substitution that ejnTw is a solution, i.e. show that when x0(nT) is of the form ejnTw, the input
u(nT) is ejnTw, multiplied by a constant (complex-valued); w is the frequency of the input in
radians/sec; (the use of complex exponentials for the representation of sinusoidal signals is
discussed in Lab 8, 10 and 13.
From the above, with input u(nT) = ejnTw obtain
x0/u = 1/[ 1 + a1. e-jTw + a2. e-j2Tw]
(Eqn 2).
Note that x0/u is not a function of the time index n.
Question 2
Use this result to obtain a general expression for |x0/u| as a function of w.
Tip: to simplify the math, operate on u/x0 instead of x0/u, expressing the result in polar notation.
Experiment 15 – Poles and Zeros in the z plane
© 2011 Emona Instruments
15-3
Set T = 1 sec for the time being, and plot the result for the case a1 = -1.6 , a2 = 0.902 over the range
w = 0 to π rad/sec. Label the frequency axis in Hz and in rad/sec. You should find there is a peak in
the response near 0.09Hz.
Graph 1:response plot
Question 3
Replace "exp(jTw)" by the symbol "z" in Eqn 2. The result is
H_x0(z) = x0/u = 1/(1 + a1.z-1 + a2.z-2 ) = z2 / ( z2 + a1.z + a2 )
(Eqn 3).
The quadratic ( z2 + a1.z + a2 ) can be expressed in the factored form (z - p1)(z - p2).
15-4
© 2011 Emona Instruments
Experiment 15 – Poles and Zeros in the z plane V1.2
Figure 2: Notes on the graphical interpretation of pole-zero plots
Experiment 15 – Poles and Zeros in the z plane
© 2011 Emona Instruments
15-5
Reviewing the finding of roots of the quadratic polynomial
From equation 1 above
x0(nT) = u[nT] - a1.x1(nT) - a2.x2(nT) substituting x1=x0.z-1 and x2=x0.z-1.z-1 we arrive at
x0 = u –a1x0/z1 – a2x0/z2
Grouping x0 terms:
x0(1 + a1/z1 + a2/z2) = u
At this point we can see that although we started with negative gains in the circuit model, we now
have positive values as coefficients in the quadratic equation.
Further, we arrive at:
x0/u = z2/(z2 + a1z1 + a2) which we earlier named Eqn 3.
We now have the general quadratic form with positive coefficients.
INSIGHT: positive coefficients result in negative gains in the actual implementation
This quadratic (z2 + a1z + a2) can be expressed in factored form, as (z-p1)(z-p2)
Remember that z, p1 & p2 are complex numbers. You can think of these as vectors: from the origin of
the z plane to a 2 dimensional point on that plane.
Each factor ie: (z-p1) and (z-p2) is a difference vector between a general point z, who’s locus we
restrain to the unit circle, and the 2 specific roots p1 & p2. It will be a vector, having direction and
magnitude, and can be expressed in polar notation as r⁄θ, or rejθ, or in Cartesian notation as (a + ib).
Both these representation are complex numbers.
If we define p1 as (σ + iw) and its conjugate, p2 as (σ – iw) we can express the quadratic factors as:
(z- p1)(z- *p1) = z2 + p.*p – pz – *pz
Switching to polar notation for convenience, p.*p = rejθ. re-jθ = r2
So that leaves z2 + r2 –z.(p + *p), and if using Cartesian notation in this instance for convenience, ie. p
= σ + jw then p + *p = 2σ so
z2 +(-2 σ)z + r2 = z2 -2 σ z + r2 = z2 +a1z + a2
Relating coefficients gives a1 = -2σ and a2 = r2
For stability the poles must always be inside the unit circle, hence 0 < a2 < 1
Changes in a1 directly influence the real component of the pole position
a2 has a square law relationship with r of the pole.
Other relationships, such as θ, w, imag part, can be derived from these easily with trigonometry.
The general solution for the roots of the quadratic polynomial x2 + a1x + a2 is:
x = -a1/2 +/- i√[a2-(a1/2)2]
With these equations in mind consider how changes in the coefficients from the math will move the
poles or zeros about the unit circle, and influence the response of the system.
This lab aims to make you more familiar of the interrelationships between these parameters.
15-6
© 2011 Emona Instruments
Experiment 15 – Poles and Zeros in the z plane V1.2
Using the values of a1 and a2 given in Question 2 above, find the roots p1 and p2 (express the result in
polar notation). Mark the position of p1 and p2 on the complex z plane with an "x" to indicate that
they represent poles. The distance between these points and the unit circle is of key importance.
This is a parallel process to that in Lab 11 where we plotted zeros. A similar procedure was carried
out in Lab 11 for a CT transfer function in the complex variable s.
Write down a formula for p1 in terms of a1 and a2. Note that p1 may be real or complex depending on
a1 and a2. Determine the conditions for p1 to be complex valued. For this case, express p1 in polar
notation. Take note of the fact that |p1| does not depend on a1 (this will be useful later). Obtain p2
from p1.
Question 4
Satisfy yourself that the magnitude response of H_x0 is given by
|H_x0(w)| = 1/[|(ejTw - p1)|.|(expjTw - p2)|]
(Eqn 4).
This provides the key for the graphical method described in Lab 13 to obtain an estimate of the
magnitude response. Again, we will use T = 1 .
Plot the magnitude of the denominator for selected values of w over the range 0 to π. The quantity
|(ejTw - p1)| becomes quite small and changes rapidly as the point on the unit circle is moved near p1.
Plot additional points there as needed. Invert to get |H_x0(w)| and compare this with the result you
obtained in (b).
Experiment 15 – Poles and Zeros in the z plane
© 2011 Emona Instruments
15-7
Graph 2:response plot
b0
x0
b1
x1
x2
b2
1/Z
y
1/Z
u
-a
1
-a2
Figure 3: block diagram of 2nd-order Direct-form 2 structure with feedforward.
15-8
© 2011 Emona Instruments
Experiment 15 – Poles and Zeros in the z plane V1.2
Question 5
Modify Fig 1 by replacing the unit delays with a gain of 1/z and show that Eqn 3 follows by inspection
using simple algebra.
Question 6
Apply this idea to show that the transfer function for the system in Fig. 3 is
H_y (z) = y/u = (b0 + b1.z-1 + b2.z-2 ) /(1 + a1.z-1 + a2.z-2 )
(Eqn5)
Question 7
Use the graphical pole-zero method (covered in Experiment 14) to obtain estimates of the magnitude
responses for the following cases (0 to Nyquist freq):
(i) b0 =b2 = 1, b1 = 2, a1 and a2 as in Question 2.
(ii) b0 = b2 = 1, b1 = - 2, a1 and a2 as in Question 2
(iii) b0 = 1, b1 = 0, b2 = - 1, a1 and a2 as in Question 2
Which of these is lowpass, highpass, bandpass?
Experiment 15 – Poles and Zeros in the z plane
© 2011 Emona Instruments
15-9
Graph 3:pole-zero method
Question 8
Consider a DT system with sampling rate 20kHz. Obtain estimates of the poles and zeros that
realize a lowpass filter with cut-off near 3kHz. Obtain a highpass filter using the same poles.
Question 9
For the same sampling rate as in Question 8 obtain estimates of the poles and zeros that realize a
bandpass filter centered near 3.1kHz, with 3dB bandwidth 500Hz. HINT: review Question 7
15-10
© 2011 Emona Instruments
Experiment 15 – Poles and Zeros in the z plane V1.2
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should observe the
USB light turn on (top right corner of ELVIS unit).The PC may make a sound to indicate that the
ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that all
three power LEDs are on. If not call the instructor for assistance.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with the
NI ELVIS that you’re using.
9.
You’re now ready to work with the NI ELVIS/SIGEx bundle.
10.
Select the EXPT 15 tab on the SIGEx SFP.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the STOP
button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the top of the
window. This will allow the program to conduct an orderly shutdown and close the various DAQmx
channels it has opened.
Ask the instructor to check
your work before continuing.
Experiment 15 – Poles and Zeros in the z plane
© 2011 Emona Instruments
15-11
Experiment
Part 1: IIR without feedforward: a second-order resonator
In this part we implement and investigate the system in Fig 1.
Figure 4: patching diagram for feedback structure without feedforward from Figure 1
11.
Patch up a SIGEx model of the system in Fig 1.
Settings are as follows:
ADDER GAINS: a0=1; a1=+1.6; a2= -0.902
PULSE GENERATOR: 20kHz, DUTY CYCLE=0.5 (50%)
FUNCTION GENERATOR: Sinewave selected, FREQUENCY=1k; Amplitude= 2V pp
SCOPE: Timebase = 4ms, Trigger on input signal, Trigger level = 0V
Question 10
Calculate the poles corresponding to these values. Measure and plot the magnitude response at the
output of the feedback adder. Note and record the resonance frequency and the bandwidth. Use
the poles to graphically predict these parameters; compare with your measurements.
HINT: Do a quick sweep of frequency range, and turn AUTOSCALE off at maximum amplitude, then
sweep the range more slowly while taking measurements. This will enable you to see the variation in
gain of the output more easily than with AUTOSCALE on.
Question 11
Decrease |a1| by a small amount ( around 5-10%, say) and measure the effect on the resonance
frequency and bandwidth. Use this to estimate the migration of the poles. Does this agree with your
expectations?
Question 12
15-12
© 2011 Emona Instruments
Experiment 15 – Poles and Zeros in the z plane V1.2
Repeat step 3 for a 5% decrease of a2. Compare the effects of varying a1 and a2. Which of these
controls would you use to tune the resonance frequency? Use the formulas you obtained in the
preparation to explain this.
NOTE about TAB “ PZ plot” on the SIGEx SFP.
This panel calculates the poles and zeros relating to the currently set ADDER gains which relate to
the coefficients of the transfer function. Use this visualization tool to confirm your understanding .
You may wish to move back and forth between the experiment TAB and the “PZ plot” TAB as
required during the experiment.
Figure 5: “PZ plot” TAB from SIGEx SFP
You may wish to use the manual GAIN ADJUST knob on the SIGEx board to vary these parameters.
Remember to setup its range to suit your parameter.
Question 13
With a1 unchanged, gradually increase a2 and observe the narrowing of the resonance. Continue until
you see indications of unstable behaviour. At that point, remove the input signal and observe the
output (if needed, increase a2 a little more). Is it sinusoidal? Measure and record its frequency.
Measure a2. Calculate and plot the pole positions. Note especially whether they are inside or outside
the unit circle.
Experiment 15 – Poles and Zeros in the z plane
© 2011 Emona Instruments
15-13
Figure 6: patching diagram for step response
12.
We will now repeat step 11 in the time domain. Use the PULSE GENERATOR as a clock
source and SEQUENCE GENERATOR to set up a unit pulse input. The SYNC signal from the
SEQUENCE GENERATOR will act as a repetitive unit pulse source. What matters is that the unit
pulses are far enough apart that each pulse is a unique event to the system under investigation.
Setting are as follows:
PULSE GENERATOR: 20kHz, DUTY CYCLE=0.5
SEQUENCE GENERATOR: DIPS set to UP:UP (short sequence)
ADDER gains: a0=1.0; a1=1.6; a2=-0.902; b0=1.0
Question 14
Begin with a2 around -0.9. Describe the effect on the response as the magnitude of a2 reduces.
Measure the frequency of the oscillatory tail of the response and compare with your observations in
step 5.
Figure 7: typical pulse response
15-14
© 2011 Emona Instruments
Experiment 15 – Poles and Zeros in the z plane V1.2
Graph 4:pole only response plot
Figure 8: Setup with feedforward and feedback sections implemented, as per Figure 9 below
Experiment 15 – Poles and Zeros in the z plane
© 2011 Emona Instruments
15-15
Part 2 - IIR with feedforward: second-order filters
In this part we implement and investigate the system in Fig 9. Note that the system with
feedforward simply builds upon the previous system with feedback only. It also provides a new
output point. The system response x0 for the feedback only, all-pole system is still available as a
subset within this new arrangement and is unchanged by the additional feedforward elements. The
feedforward elements simply add numerator terms to the overall transfer function which becomes
y/u.
b0
x0
b1
x1
x2
b2
1/Z
y
1/Z
u
-a
1
-a2
Figure 9: block diagram of 2nd-order Direct-form 2 structure with feedforward.
13.
Use ADDER B in a z-TRANSFORM module to convert the SIGEx model of Part 1 to the
system in Fig 9.
Figure 10: patching diagram for block diagram above
14.
Implement case (i) in Prep (Q7).
Settings are as follows:
ADDER GAINS: b0=1; b1=2; b2=1; a0=1; a1=+1.6; a2= -0.902
PULSE GENERATOR: 20kHz, DUTY CYCLE=0.5 (50%)
FUNCTION GENERATOR: Sinewave selected, FREQUENCY=1k; Amplitude= 1V pp
15-16
© 2011 Emona Instruments
Experiment 15 – Poles and Zeros in the z plane V1.2
SCOPE: Timebase = 4ms, Trigger level = 0V, Trigger on input signal
Figure 11: Using the Function Generator to sweep a sinusoid across the spectrum of interest.
Observation: the high pass band gain due to the selection of coefficients resulting in poles very close
to the unit circle, as shown in the figure below.
Figure 12: details of the PZ PLOT TAB for the current settings
(NB: 2 zeros at (-1 +/- 0i) are difficult to see in figure.)
NOTE: The poles are very close to the unit circle. In fact, the pole radius is 0.95. Hence the gain
close to the poles is very large. You can use the PZ PLOT to visualize the poles and zeros for any
“live” coefficient settings. Zeros are also present at z = -1.
Experiment 15 – Poles and Zeros in the z plane
© 2011 Emona Instruments
15-17
Measure the magnitude response |y/u| and confirm that it is a lowpass filter.
Graph 5: plot of responses
Question 15
In the model of step 14, adjust a2 to reduce the peaking to a minimum. As well you will need to
reduce the amplitude of the input signal to 0.5Vpp to reduce saturation. Confirm this for yourself.
Plot the resulting response and measure the new value of a2. Calculate and plot the new poles. Obtain
an estimate of the theoretical magnitude response with these poles and compare this with the
measured curve. Why was a2 used for this rather than a1?
Question 16
Change the polarity of b1 in the lowpass of step 19 and show that this produces a highpass. Compare
with your findings in Question 7.
15-18
© 2011 Emona Instruments
Experiment 15 – Poles and Zeros in the z plane V1.2
NOTE: the following three questions refer to the transfer function coefficient values. Remember to
negate the a1 and a2 values when settings them up as ADDER GAINS.
Question 17
Repeat for case (iii) in Question 7, that is: b0 = 1, b1 = 0; b2 = -1 ; a0 = 1; a1 =-1.6; a2 =0.902; Confirm
this is a bandpass filter. Tune a1 and a2 to obtain a peak at 3.1 kHz and 3dB bandwidth 500Hz.
Measure the resulting a1 and a2 and plot the new poles. Compare this with your findings in Question
7.
Question 18
Implement the following case: a0 = 1, a1 = 0, a2 = 0.8, b0 = 0.8, b1 = 0, b2 = 1. Note that b0=a2 and b1=a1.
Measure the magnitude response. Confirm it is allpass. Locate the positions of the poles and zeros.
Plot them below for your records.
Question 19
Change a1 and b1 to - 1.6 and confirm the response is still allpass. Examine the behaviour of the phase
response. Look for the frequency of most rapid phase variation, and confirm this occurs near a pole.
Plot the poles and zeros below for your records.
Experiment 15 – Poles and Zeros in the z plane
© 2011 Emona Instruments
15-19
Viewing spectrum of system with broadband noise input & FFT
As well as sweeping a single frequency signal from the FUNCTION GENERATOR across the
spectrum of interest it is also convenient to input a broad range of frequencies at once and view the
overall output frequency response of the system. Creating a broadband analog noise signal was
covered in Experiment 9. That methodology is shown in the figures below. You can revisit this
experiment with this setup in place and see the relationships of poles and zeros to system response
in real time.
Figure 13: IIR filter with flat noise input
Figure 14: experiment setup with noise input signal instead of Function Generator signal. Uses
SEQUENCE GENERATOR and TUNEABLE LPF, clocked from PULSE GENERATOR.
15-20
© 2011 Emona Instruments
Experiment 15 – Poles and Zeros in the z plane V1.2
15.
Ensure that SEQUENCE GENERATOR DIP switches are set to positions DOWN:DOWN for
the long sequence. Set TLPF knobs to fully clockwise for now. Switch to TAB “ZOOM FFT” to view
time and frequency domains simultaneously. Change scope timebase to 100ms.
Set up the coefficients as per step 14.
Setting up the input noise signal:
16.
i) Reduce the TLPF GAIN by rotating counter clockwise until the output Ch1 signal (red) is no
longer saturated ie: less than 12V peak.
ii) Reduce the noise bandwidth to around 4khz by rotating the TLPF block’s “Fc” control-counter
clockwise. View the noise spectrum as the white trace on the SCOPE & FFT windows.
The SFP should be similar to Figure 15 below for a peaky LPF as shown.
Figure 15: ZOOM FFT TAB used to view Experiment 15 setup with flat noise input.
Due to high gain, input noise level is very small. Note the limited bandwidth of the input noise to
maintain a flat input response. (SEQUENCE GENERATOR must be set to long sequence.)
At this point we can see and explore the issues relating to :
-controlling our input signal level and bandwidth
-viewing the response in both time and frequency domains
-setting up a transfer function with appropriate internal gains
We can also confirm that the peak of the response is correct according to the position of the poles.
ie: PZ PLOT tells us that poles are at 32 degrees, hence we expect a peak close to 32/360*20,000 =
1777 Hz. You can use the cursors in ZOOM FFT window to confirm this.
Experiment 15 – Poles and Zeros in the z plane
© 2011 Emona Instruments
15-21
Question 20
Show your calculation of the where you expect the peak frequency to be using the pole position and
sampling frequency.
Dynamically varying the poles and zeros to adjust response using GAIN
ADJUST manual control
17.
Use the SIGEx board’s GAIN ADJUST knob to vary one of the coefficients by hand while
viewing the frequency response. Leave the default settings. Turn the knob until it reads +1.6 (located
in the COEFFICIENT SELECTOR window), then select radio button “ a1 ”. View the frequency
response while slowly varying the value of a1. You will find that the peak frequency changes.
18.
Find a range of a1 settings that work well and then view PZ PLOT while varying across that
range. You will see the poles moving and reflecting the changing a1 coefficient. (Theory states that
a1=-2σ, which is the real part of the pole and its conjugate.)
Question 21
Confirm this relationship from values displayed on PZ PLOT and show your working here:
19.
Set a1 back to +1.6, select OFF, then set GAIN ADJUST to -0.9, and select a2 radio button.
For more resolution vary the setup parameters as required.
Question 22
Varying a2 will vary the gain or peak level of the filter. Notice what happens in the time domain when
a2 = -1.0. The filter breaks into oscillation. View the poles again using PZ PLOT while varying a2.
(Theory states that a2 = r2).
20.
Set a2 back to -0.9, then select OFF again at the COEFFICIENT SELECTOR to disable the
GAIN ADJUST control.
15-22
© 2011 Emona Instruments
Experiment 15 – Poles and Zeros in the z plane V1.2
Using Digital Filter Design toolkit to implement Highpass filters
21.
Switch to the DFD TAB. This will allow you to automatically load the computed coefficients
for the selected filter onto the SIGEx board. You can see the values on the TAB setup in the GAIN
input controls on the SIGEx SFP.
NOTE: Maintain the order of your filter structure <= 2, to match the structure you have built.
22.
Connect CH0 to “ x0 ”, and CH1 to “Y” and view the internal signal levels at “x0” by switching to
the ZOOM FFT tab. Set the TLPF GAIN higher but avoid saturating.These filters have lower
internal gains than the previous ones. Vary the filter design type (at the DFD tab by clicking on
“DESIGN METHOD” to select) and view the output responses using ZOOM FFT.
NOTE: Press the button to transfer the coefficients into the ADDER gains when you are ready to do
so.
Set timebase to 100ms.
Again you can use the cursors to compare the actual performance to theory and design.
You can expect to see a display like so:
Figure 16: Highpass filter response using FFT; x0 (grey); Y (white)
Experiment 15 – Poles and Zeros in the z plane
© 2011 Emona Instruments
15-23
Figure 17: DFD TAB used to design filters and setup coefficients to the patched SIGEx board
Default values are chosen to enable students to see textbook like responses which they can easily
measure.
Question 23
Confirm that the SIGEx hardware performs as designed by theory in terms of notch positions etc.
You will have to use the zero positions mostly in these cases. Why ?
Notches are implemented by placement of zeros on or near the unit circle.
Question 24
Try varying design values and take note of the ORDER of the filter designed. NOTE that the SIGEx
experiment we have implemented can only support a 2nd order structure. Note your observations.
15-24
© 2011 Emona Instruments
Experiment 15 – Poles and Zeros in the z plane V1.2
Tutorial Questions
Q1.
Why do the complex poles and zeros occur in conjugate pairs in the cases covered
in this lab?
Q2.
Why is polar notation for complex valued poles and zeros preferred in the
discrete-time context? Using examples from the lab, explain the importance of
the position of complex poles/zeros relative to the unit circle when estimating
frequency responses.
Q3.
Re Eqn 3, keeping a1 constant, plot the locus of the upper half plane pole with
respect to a2. Do this for several suitable values of a1. Holding a2 constant, repeat
this with respect to a1. Use the resulting contours to explain your observations in
Q11 and Q12 .
Q4.
Determine the conditions on a1 and a2 for the poles to be complex. Display this
graphically on a plane (i.e. with a2 = 0 as horizontal axis and a1=0 as vertical axis).
Q5.
Calculate and plot the poles and zeros in Q19. Satisfy yourself that they share
the same radial line. Show that z1 = 1/p1* .
Q6.
Prove that the values of the coefficients in Q18 and Q19 generate a constant
magnitude response over all frequencies. Write down the coefficient relationship
in the transfer function of a fourth-order allpass.
Q7.
Consider the unit pulse response in Step 12. What is the effect on the decay rate
as the bandwidth is decreased? Find a simple formula or rule of thumb to express
this relationship.
Q8.
Show that the magnitude responses at nodes x1 and at x2 are the same as at x0
(can be demonstrated without math).
Q9.
Consider a bandpass filter realized with a2 = 0.98. What is the maximum deviation
allowable in a2 to maintain a bandwidth tolerance of 5 percent?
Q10.
Consider the following assertion: "Continuous-time filters can be considered as a
limiting case of discrete-time filters, as the sampling frequency to bandwidth ratio
gets very large". Hint: show that the poles and zeros migrate to the area near
(1,0) as the Nyquist ratio increases and compare the shapes of the unit circle and
of the j axis in that region.
Q11.
Find out the meaning of the term "maximally flat". Is this description applicable
to the filter produced in Q15 by reducing the value of a2?
Experiment 15 – Poles and Zeros in the z plane
© 2011 Emona Instruments
15-25
15-26
© 2011 Emona Instruments
Experiment 15 – Poles and Zeros in the z plane V1.2
Experiment 16 – Discrete-time filters – practical applications
Achievements in this experiment
You will gain an understanding of the importance of sensitivity and dynamic range issues that
arise in the practical implementation of discrete-time filters. As well you will explore the
issues relating to optimum sampling rate selection.
Preliminary discussion
In the design of systems the primary focus initially is the signal at the output interface.
However, at implementation, whether analog or digital, it is often discovered that the quality
of the output may be adversely affected by problems at intermediate points. An important
example is dynamic range at internal nodes. Unless properly managed, this can cause
significant SNR degradation. In digital systems, the use of floating-point arithmetic is a
solution in some instances. However in many applications, fixed-point implementation makes
possible considerable cost savings.
In this lab we will examine dynamic range performance in the filter structure used in
Experiment 15, the "Direct Form 2" and compare this with an alternative, the "Transposed
Direct Form 2".
In many applications of digital signal processing, e.g., digital audio, there is a general
perception that a high sampling rate is the key to superior performance, and that the
motivation for moderation is to reduce demands on memory and processing load. This lab
reveals another side of the coin, i.e., that an excessively generous sampling rate may unleash
sensitivity challenges, and compromise performance.
16 2
© 2011 Emona Instruments
Experiment 16 – Discrete time filters V1.2
Pre-requisite work
Question 1
Using the method in Lab15 Question 5, show that the transfer function for the system in Fig.1
is
H_y (z-1) = y/u = (b0 + b1.z-1 + b2.z-2 ) /(1 + a1.z-1 + a2.z-2 )
(Eqn1).
u
x
b
b1
2
2
1
x
1
x0
y
-a
2
- a
1
Fig 1: block diagram of 2nd-order Transposed Direct-Form2 feedback structure
Question 2
Consider a filter with a1 = -1.84 , a2 = 0.90 , b0 = 1 , b2 = b0 , b1 = -1.7. Calculate and plot the
zeros of the transfer functions in (Q1).
Question 3
From the results in (Q1) and (Q2) obtain the ratios x1/y and x2/y expressed as transfer
functions in z. Use these to calculate |y/x2| and |y/x1| at the peak of the response of the
filter in (Q2).
Question 4
Consider the implementation of the filter in (Q2) using the Direct Form 2 structure in Lab 15
Fig 2. Satisfy yourself using only a quick inspection of the diagram, that with this structure
the magnitude responses at the internal nodes are identical. Repeat (Q3) for this case, and
compare the outcomes. This comparison will be applied in the Lab, hence it's important to have
the analysis ready to use.
Question 5
Consider a transfer function with the coefficients in Question 2 and sampling rate
10ksamples/sec.
(a) Sketch the gain response versus frequency and note the peak and null frequencies. Repeat
this with sampling rate 20ksamples/sec. Note that the general shape of the response is
virtually unchanged, but the frequency axis has been rescaled.
(b) The outcome in (a) is useful in some applications, however suppose we want to use the
faster sampling rate without frequency axis rescaling. This will require relocating the poles
and zeros so that their distance from the zero frequency point on the unit circle (1,0) is
suitably reduced - by a factor of about 2, in this case. The pole should slide on a line joining
(1,0) and its original position. The zero should remain on the unit circle. Use a computer to
plot and compare the new and original responses. Suggest possible adjustments to the poles
and zeros to reduce any differences.
Experiment 16 – Discrete time filters
© 2011 Emona Instruments
16-3
Question 6
Look up a suitable reference to confirm that the the bilinear transformations are as follows (T
is the sampling interval):
s = (2/T) .(z - 1)/(z + 1)
z = (1 + (T/2)s )/(1 - (T/2)s )
These formulas are used to convert continuous time (CT) transfer functions to discrete time
(DT), and vice versa. In this exercise we obtain the CT transfer function for the case in
Question 2, and reverse the process with a new value of T to produce the DT transfer function
for a higher sampling rate.
(a) Find or write a program for implementing the bilinear transformations.
(b) Use this to obtain the transfer function and the poles and zeros corresponding to the
increased sampling rate in Question 5. Confirm that the zeros have remained on the unit circle
(optional extra: prove theoretically that z plane unit circle zeros always transform to the j
axis in the s plane, and vice versa).
(c) Obtain a plot of the gain frequency response with the new sampling rate and compare this
with the original and with the approximate case in Question xxx (b).
(d) Compare the positions of the poles and zeros generated with the bilinear transformations
versus the approximate case in Question xxx (b).
Question 7
This question is about the effect of errors in coefficient values that may be encountered as a
result of limited arithmetic word length. The errors proposed here are of the order that could
occur with a 12-bit wordlength.
(a) Consider the transfer function obtained in Q.6(c). Change the value of a2 by 0.1 percent.
Plot the gain frequency response and compare with the original response.
(b) Repeat (a) for coefficient a1, and then for both coefficients together
(c) Examine the shift in the pole and zero positions for the coefficient errors in (a)
and (b). Are these consistent with the gain response errors?
(d) Plot the locus of the movement of a pole as a1 and a2 are varied, respectively. Point to
aspects of these loci in the region near the point (1,0) that exacerbate the sensitivity issues
relating to coefficient quantization.
(e) Is there any significant advantage with floating point arithmetic compared with fixed point
for the effects of coefficient quantization?
16 4
© 2011 Emona Instruments
Experiment 16 – Discrete time filters V1.2
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should
observe the USB light turn on (top right corner of ELVIS unit).The PC may make a sound to
indicate that the ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that
all three power LEDs are on. If not call the instructor for assistance.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with
the NI ELVIS that you’re using.
9.
You’re now ready to work with the NI ELVIS/SIGEx bundle.
10.
Select the Lab 16 tab on the SIGEx SFP.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the
STOP button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the
top of the window. This will allow the program to conduct an orderly shutdown and close the
various DAQmx channels it has opened.
Ask the instructor to check
your work before continuing.
Experiment 16 – Discrete time filters
© 2011 Emona Instruments
16-5
Experiment
Part 1 Dynamic range at internal nodes
In this part we examine and compare magnitude responses at internal nodes for the DF2
(Direct Form 2) and the transposed DF2 structures. First, we review the behaviour of the
DF2 structure used in Lab 15. We carry out all the measurements for this case first, before
patching up the transposed structure.
Direct Form 2 IIR
11.
Patch up the SIGEx model of Lab 15 Part 2. Set the gains and polarity settings for the
values given in Prep (Q2) above. Use an input signal amplitude of 400mv pp. This is quite small
but necessary due to the large internal gains you will encounter. Use the sample rate from the
PULSE GENERATOR set to 10kHz.
NB: The values given in Q2 are transfer function coefficient values. Remember too negate the
a1 & a2 values when setting up the ADDER gains.
b0
x0
b1
x1
x2
b2
1/Z
y
1/Z
u
-a
1
-a2
Figure 2: Direct Form 2 IIR structure (from Lab 15)
16 6
© 2011 Emona Instruments
Experiment 16 – Discrete time filters V1.2
Figure 3: patching diagram for DF2 structure (from Lab 15)
12.
Measure and plot the magnitude responses at the output y and also at the internal
nodes x1, x2. Calculate and record the ratios |y/x2| and |y/x1| at the peaks of the responses.
Ensure that you capture enough information to determine the 3dB bandwidth of the filter.
REMINDER: To measure the 3dB bandwidth of the filter, find the difference between the
upper -3dB frequency and the lower -3dB frequency.
Graph 1: Magnitude responses
Experiment 16 – Discrete time filters
© 2011 Emona Instruments
16-7
Figure 4: coefficient to pole-zero plot for the DF2 structure
13.
Enter your measurements into the following table.
Table 1: Signal magnitudes for Direct form and Transposed Direct form IIR
Direct Form 2
(for u = 400mv)
Transposed Direct
Form 2
( for u = 400mv)
Transposed Direct
Form 2
( for u = 1.6V)
Peak (Hz)
u (Vpp)
y (Vpp)
y/u gain
x1 (Vpp)
x2 (Vpp)
Upper
3dB freq.
Lower
3dB freq.
BW 3dB
16 8
© 2011 Emona Instruments
Experiment 16 – Discrete time filters V1.2
Question 8
What is the maximum level of internal gain you have measured in this filter ?
Question 9
Why is it essential to keep the input signal at a low level ie: 400 mv pp ?
Question 10
Keeping in mind that the SIGEx circuits maximum signal range is +/- 12V and the maximum gain
of ADDER gain stages is +/- 2, what is the maximum level of observable signal you must keep
within ?
Experiment 16 – Discrete time filters
© 2011 Emona Instruments
16-9
Transposed Direct Form 2 IIR
14.
Patch up the SIGEx model of the Transposed DF2 system in Fig 5. Repeat the steps
above and enter your measurements into the above table also. Plot the response on the graph
as well.
Reminder: the responses at x1 and x2 are not the same with this structure. This should be
obvious from the block diagram of the structure. Use the two input levels as listed in the table
to avoid overload.
NOTE: The transposed form uses 3 adder junctions. What is available on the SIGEx board is a
triple ADDER a, triple ADDER b, and dual ADDER f+g. These have been labeled to suit the nontransposed structure so some reorientation is needed when using these for transpose
structures. The following figures show how to wire up the modules, as well as how to enter the
coefficient values into the SFP. You are advised to closely follow the suggested wiring in order
to get around the difficulty of the modified mapping for this structure. You may wish to
confirm for yourself that the block diagram matches the patching diagram.
u
x
b
b1
2
2
1
x
1
x0
y
-a
2
- a
1
Fig 5: block diagram of 2nd-order Transposed Direct-Form2 feedback structure
Fig 6: patching diagram of 2nd-order Transposed Direct-Form2 feedback structure
In Figure 6, the coefficient labels in the diagram do not correspond to those in the block
diagram due to the varied connections needed. Follow the patching diagram and use the Table
2 below for guidance on entering the values correctly.
16
10
© 2011 Emona Instruments
Experiment 16 – Discrete time filters V1.2
Table 2: Implementation table for mapping coefficients
Theoretical value as
per block diagram
Implementation label
as per patching
diagram
Implementation value
F
1 (fixed)
b0=1
G
1 (fixed)
b1=-1.7
B2
-1.7
b2=1
A2
-1
a1=-1.84
B0
1.84
a2=0.9
A0
-0.9
A1
0
B1
1
Figure 7: Screenshot of correctly implemented coefficients for Transposed DF2 structure
shown above
15.
We now have two sets of ratios of the values of y/x2 and y/x1 at the peaks of the
responses for each of the two structures. Confirm that with the transposed structure we can
realize greater output voltages without causing saturation of the signals at nodes x2 and x1.
Express the comparisons as ratios in dB units.
Question 11
What is the difference in internal gain between the non-transposed and transposed structures
(in dB) ?
Tutorial Q 1 guides you through a theoretical interpretation of the results.
Experiment 16 – Discrete time filters
© 2011 Emona Instruments
16-11
Part 2 Implementations with high sampling ratio
The higher the sampling rate, the greater the demands on the hardware, i.e. increased memory
storage capacity and faster switching speeds. However, in some applications this may not
represent a significant additional cost. Hence, if the opportunity is there, why not take
advantage of it - after all, faster sampling will ease the demands on the complexity of
antialiasing and analogue reconstruction filters. It turns out that there are some other issues
that come into play, as the following exercise will reveal.
For simplicity, reconstruct the DF2 filter from early in Part 1, as shown in Figure 3. This allows
us to easily relate to the coefficients as entered in to the SIGEx SFP.
Use the same coefficient values as per that part. Set the sampling clock rate to 10,000 Hz as
per before. Use an input amplitude of 400 mv as before.
16.
In the earlier part 1 of this lab you measured and noted the bandwidth of this filter
structure. Quickly confirm your currently reconstructed filter matches the filter you built
previously and confirm that its bandwidth is also the same.
17.
Switch to the PZPLOT tab to view the pole and zero positions for this filter.
Question 12
Document the transfer function and the poles and zeros for this original filter.
Question 13
What do you expect will happen to the pole and zero positions for a sampling rate of 20,000
samples/sec ?
Question 14
What do you expect this filter response to be like with a sampling clock rate of 20,000
samples/sec ?
18.
Change the sampling rate at the PULSE GENERATOR to 20,000 Hz and review the
filters response characteristic.
Question 15
What are the -3dB points and bandwidth for this filter at 20,000 samples/sec ?
16
12
© 2011 Emona Instruments
Experiment 16 – Discrete time filters V1.2
19.
View the PZPLOT TAB again for this 20kHz filter and use your understanding of this
plot to compute the frequency of the peak for this filter.
As you would expect, the same poles and zero locations for a different sampling rate result in
a filter with similar characteristics but which no longer meets the intended frequency
specifications. The next part of this experiment is to consider how the same filter, with the
same response as in Part 1, can be achieved at 20kHz.
This will require poles and zeros to be moved towards the zero frequency point on the unit
circle, i.e. the point (1,0).
Question 16
Approximately how close to the origin will the poles and zeros need to be moved to ?
20.
Apply your answer to the question above to determine the approximate position of the
new poles and zeros, as well as the new coefficients. Implement this and measure the response
to see how well it works.
21.
Now in this step, using trial and error attempt to improve upon your previous attempt if
you feel this is needed.
Question 17
What was the best result you were able to achieve in this manner ?
22.
A mathematical solution to this problem is the use of the Bilateral Transformation. The
discrete filter design is transformed back to the continuous time domain representation as a
prototype design, and then transformed back to the discrete domain at 20kHz. In this way the
poles and zeros created are specifically for the filter required at the sampling rate specified.
Question 18
What are the new poles and zeros using the bilateral transformation approach ?
What is the new transfer function for this transformed filter ?
NB: This was covered in the pre-lab preparatory questions.
23.
Implement the coefficients from the new transfer function into the hardware via the
SFP and measure the frequency response of the filter. How well does it match your design ?
Experiment 16 – Discrete time filters
© 2011 Emona Instruments
16-13
Sketch the response of the new filter below. Measure and note the 3 dB frequencies, and
bandwidth.
Graph 2: Response of new filter at 20kHz
16
14
© 2011 Emona Instruments
Experiment 16 – Discrete time filters V1.2
Screenshot of 20ks/s filter response using PZPLOT TAB and noise input
Question 19
What can you say about this new filter in terms of its sensitivity. What are positive and
negatives of running this filter design at 20ksamples/sec ?
If the poles and zeros are too close to the origin, then you encounter sensitivity issues,
whereas if they are too distant from the frequency origin (1,0), ie: around 90 degrees, then
you have a low number of samples per period of signal.
Question 20
Can you suggest a range of angles, in which the poles and zeros would be optimally placed in
order to avoid the challenges discovered above ? This may require experimentation or
further reading.
Experiment 16 – Discrete time filters
© 2011 Emona Instruments
16-15
Tutorial Questions
16
16
Q1.
Discuss possible advantages of the transposed DF2 structure in dealing with
high sampling rate sensitivity issues.
Q2.
Look up the terms "decimation" and "interpolation" in the context of digital
filters. How would these operations help to reduce the sensitivity problems
investigated in Part 2 for filtering digital signals that have high ratios of
Nyquist frequency to bandwidth?
Q3.
Consider the realization of a sixth order filter. One method is to use a
cascade of three second order stages. An alternative is the use of a single
stage DF2 or transposed DF2 realization. Discuss the factors that would
determine your choice.
Q4.
Look up the poles and zeros for a sixth order elliptic filter (discrete-time or
continuous-time).
Suppose you decide to use a three-stage cascade
implementation. There are many possibilities for the pairing of the poles and
zeros. Show that there is an optimum pairing. Use an example to explain how
it is done.
Q5.
Consider the transposed DF2 structure in the continuous-time context
(Lab 11). Repeat the analysis of the internal nodes for that case and find the
corresponding zeros. Carry out an analysis as in Q1 to see if the transposed
structure is able to provide a dynamic range advantage in that application.
© 2011 Emona Instruments
Experiment 16 – Discrete time filters V1.2
Appendix A: SIGEx Lab to Textbook chapter table
This table aims to direct users to sections of relevant texts which contain theory and exercises
related to experiments currently documented and implemented with the SIGEx/NI ELVIS bundle.
Given that SIGEx is by design an open-ended modeling system it is possible to build many more
experiments than is currently documented.
Users will find that many exercises from the texts which are currently undocumented in this Lab
Manual can also be implemented directly with minimum extra documentation.
Students can easily be directed to implement exercises from texts on the SIGEx board once they
become familiar with the block diagram modeling approach to building experiments.
The texts which are currently referred to in this Appendix are:
Lathi.B.P. , “Signal processing & Linear Systems”, Oxford University Press
Oppenheim.A.V.,Wilsky.A.S., “Signals & Systems”, Prentice Hall, 2nd edition
Haykin, Van Veen, “Signals and Systems”, Wiley, 2nd edition
Ziemer.R.E,Tranter.W.H, Fannin.D.R, “Signals & Systems: Continuous and Discrete”, Prentice
Hall, 4th edition
Boulet.B.: “Fundamentals of Signals & Systems”, Thomson/Delmar Learning
McClellan.J.H, Schafer.R.W, Yoder.M.A, “DSP First”, Prentice Hall
Lathi.B.P. , “Signal processing & Linear Systems”, Oxford University Press
SIGEx Lab Manual
Lathi: text book correlation
S1-03: Special signals – characteristics and
applications
1 Introduction to Signals and Systems
B.2 Sinusoids
2.4 System response to external input: zero-state response
S1-04: Systems: Linear and non-linear
1 Introduction to Signals and Systems
S1-05: Unraveling convolution
9.4-1 Graphical procedure for the convolution sum
S1-06: Integration, convolution, correlation &
matched filters
2.4-1 The convolution integral
3.2 Signal comparison: Correlation
S1-07: Exploring complex numbers and
exponentials
B.1 Complex numbers
B.3-1 Monotonic exponentials
B.3-2 The exponentially varying sinusoid
S1-08: Build a Fourier series analyzer
3.4 Trigonometric fourier series
S1-09: Spectrum analysis of various signal types
4 Continuous-time signal analysis: The fourier transform
S1-10: Time domain analysis of an RC circuit
1.8 System model: Input-output description
S1-11: Poles and zeros in the Laplace domain
6 Continuous-time system analysis using the Laplace
transform
S1-12: Sampling and Aliasing
5 Sampling
8.3 Sampling continuous-time sinusoid and aliasing
S1-13: Getting started with analog-digital
conversion
5.1-3 Applications of the sampling theorem (Pulse code
modulation PCM)
S1-14: Discrete-time filters with FIR systems
11 Discrete-time system analysis using the z-transform
12.1 Frequency response of discrete-time systems
12.2 Frequency response from pole-zero location
12 Frequency response and digital filters
S1-15: Poles and zeros in the z plane with IIR
systems
S1-16: Discrete-time filters – issues in practical
applications
Not covered
Oppenheim.A.V.,Wilsky.A.S., “Signals & Systems”, Prentice Hall, 2nd edition
SIGEx Lab Manual
Oppenheim, text book correlation
S1-03: Special signals – characteristics and
applications
1 Signals and Systems
S1-04: Systems: Linear and non-linear
1 Signals and Systems
2 Linear time-invariant systems
S1-05: Unraveling convolution
2.1 Discrete-time LTI systems: The convolution sum
S1-06: Integration, convolution, correlation &
matched filters
2.2 Continuous-time LTI systems: The convolution integral
2 Linear time-invariant systems; Problem 2.67
S1-07: Exploring complex numbers and
exponentials
1 Signal and systems: Mathematical review
1.3 Exponentials and sinusoidal signals
S1-08: Build a Fourier series analyzer
3.3 Fourier series representation of continuous-time
periodic signals
S1-09: Spectrum analysis of various signal types
4.1.3 Examples of Continuous-Time Fourier transforms
S1-10: Time domain analysis of an RC circuit
3.10.1 A simple RC lowpass filter
3.10.2 A simple RC highpass filter
S1-11: Poles and zeros in the Laplace domain
9 The Laplace transform
9.4 Geometric evaluation of the Fourier transform from the
pole-zero plot
S1-12: Sampling and Aliasing
7 Sampling
S1-13: Getting started with analog-digital
conversion
8.6.3 Digital Pulse-Amplitude (PAM) and Pulse-Code
modulation (PCM)
S1-14: Discrete-time filters with FIR systems
6.6 First-order and second-order discrete time systems
6.7.2 Examples of discrete-time nonrecursive filters
S1-15: Poles and zeros in the z plane with IIR
systems
10.4 Geometric evaluation of the Fourier transform from
the pole-zero plot
S1-16: Discrete-time filters – issues in practical
applications
Not covered
Haykin, Van Veen, “Signals and Systems”, Wiley, 2nd edition
SIGEx Lab Manual
Haykin, Van Veen, text book correlation
S1-03: Special signals – characteristics and
applications
1.6 Elementary signals
S1-04: Systems: Linear and non-linear
1.8 Properties of systems
S1-05: Unraveling convolution
2.2 The convolution sum
S1-06: Integration, convolution, correlation &
matched filters
2.5 Convolution integral evaluation procedure
S1-07: Exploring complex numbers and
exponentials
1.6.3 Relation between sinusoidal and complex exponential
signals
A.2 Complex numbers
S1-08: Build a Fourier series analyzer
3.5 Continuous-time periodic signals: The Fourier series
S1-09: Spectrum analysis of various signal types
4.2 Fourier Transform representations of Periodic signals
S1-10: Time domain analysis of an RC circuit
6.7 Laplace transform methods in circuit analysis
S1-11: Poles and zeros in the Laplace domain
6 Representing signals by using continuous-time complex
exponentials: the Laplace transform
6.13 Determining the Frequency response from poles & zeros
S1-12: Sampling and Aliasing
4.5 Sampling
4.6 Reconstruction of continuous-time signals from samples
S1-13: Getting started with analog-digital
conversion
4.6.3 A practical reconstruction: the zero order hold
5.2 Types of modulation (PCM)
S1-14: Discrete-time filters with FIR systems
7 Representing signals by using continuous-time complex
exponentials: the z- transform
8.9 Digital FIR filters
7.8 Determining the Frequency response from poles & zeros
8.10 IIR Digital filters
S1-15: Poles and zeros in the z plane with IIR
systems
S1-16: Discrete-time filters – issues in practical
applications
7.9 Computational structures for implementing discretetime LTI systems
Ziemer.R.E,Tranter.W.H, Fannin.D.R, “Signals & Systems : Continuous and Discrete”, Prentice Hall,
4th edition
SIGEx Lab Manual
Ziemer, Tranter, Fannin, text book
correlation
S1-03: Special signals – characteristics and
applications
1-3 Signal models
S1-04: Systems: Linear and non-linear
2-2 Properties of systems
S1-05: Unraveling convolution
8-4 Difference equations and discrete-time systems;
Example 8-12 Discrete convolution
10-6 Convolution
10-6 Energy spectral density and autocorrelation function
S1-06: Integration, convolution, correlation &
matched filters
S1-07: Exploring complex numbers and
exponentials
1-3 Phasor signals and spectra
S1-08: Build a Fourier series analyzer
3-3 Obtaining trigonometric Fourier series representations
for periodic signals
3-4 The complex exponential Fourier series
4.5 Fourier transform theorems
S1-09: Spectrum analysis of various signal types
S1-10: Time domain analysis of an RC circuit
2-2:2-7 System modeling concepts
6-2 Network analysis using the Laplace transform
S1-11: Poles and zeros in the Laplace domain
6-4 Transfer functions
S1-12: Sampling and Aliasing
8-2 Sampling
8-2 Impulse-train sampling model
S1-13: Getting started with analog-digital
conversion
8-2 Quantizing and encoding
S1-14: Discrete-time filters with FIR systems
9-5 Design of finite-duration impulse response (FIR) digital
filters
S1-15: Poles and zeros in the z plane with IIR
systems
9-4 Infinite Impulse Response (IIR) filter design
S1-16: Discrete-time filters – issues in practical
applications
9-2 Structures of digital processors
Boulet.B.: “Fundamentals of Signals & Systems”, Thomson/Delmar Learning
SIGEx Lab Manual
Boulet, text book correlation
S1-03: Special signals – characteristics and
applications
1 Elementary continuous-time and discrete-time signals and
systems
S1-04: Systems: Linear and non-linear
2 Linear Time-invariant systems
S1-05: Unraveling convolution
2 Discrete-time systems: The convolution sum
S1-06: Integration, convolution, correlation &
matched filters
2 The convolution integral
S1-07: Exploring complex numbers and
exponentials
1 Complex exponential signals
S1-08: Build a Fourier series analyzer
4 Determination of the Fourier series representation of a
continuous-time periodic signal
S1-09: Spectrum analysis of various signal types
4 Fourier series representation of periodic continuous-time
signals
S1-10: Time domain analysis of an RC circuit
9 Application of Laplace transform techniques to electric
circuit analysis
S1-11: Poles and zeros in the Laplace domain
6 Poles and zeros of rational Laplace transforms
S1-12: Sampling and Aliasing
15 Sampling systems
S1-13: Getting started with analog-digital
conversion
16 Modulation of a pulse-train carrier
15 Signal reconstruction
S1-14: Discrete-time filters with FIR systems
14 Geometric evaluation of the DTFT from the pole-zero
plot
S1-15: Poles and zeros in the z plane with IIR
systems
14 Infinite Impulse Response and Finite Impulse Response
filters
S1-16: Discrete-time filters – issues in practical
applications
Not covered
McClellan.J.H, Schafer.R.W, Yoder.M.A, “DSP First”, Prentice Hall
SIGEx Lab Manual
“DSP First”, text book correlation
S1-03: Special signals – characteristics and
applications
1 Mathematical representation of signals
S1-04: Systems: Linear and non-linear
2 Thinking about systems
S1-05: Unraveling convolution
5.3.3 Convolution and FIR filters
S1-06: Integration, convolution, correlation &
matched filters
Not covered
S1-07: Exploring complex numbers and
exponentials
2.5 Complex exponentials and phasors
S1-08: Build a Fourier series analyzer
3.4.1 Fourier series analysis
S1-09: Spectrum analysis of various signal types
3 Spectrum representation
S1-10: Time domain analysis of an RC circuit
Not covered
S1-11: Poles and zeros in the Laplace domain
Not covered
S1-12: Sampling and Aliasing
4 Sampling and aliasing
S1-13: Getting started with analog-digital
conversion
4.4 Discrete to continuous conversion
S1-14: Discrete-time filters with FIR systems
5 FIR filters
S1-15: Poles and zeros in the z plane with IIR
systems
8 IIR filters
S1-16: Discrete-time filters – issues in practical
applications
8 IIR filters
Appendix B: Quickstart guide to using SIGEx
Using “Experiment 15–Poles and zeros in the z-plane” as a basis for
a demonstration of SIGEx
Achievements in this experiment
Students learn better when they can see the theory in action with real systems. You will be able to
visualize the effect of pole zero placement on the system response for the time and frequency
domain. You can confirm the theory and compare with real system performance. You can vary
parameters in real time and confirm your understanding. You will be able to develop knowledge to
intuitively design transfer function responses.
Firstly browse through the sample manual to see the breadth and depth of possible hands-on
laboratory work possible with the NI ELVIS/SIGEx bundle.
Equipment
PC with LabVIEW Runtime Engine software appropriate for the version being used.
NI ELVIS 2 or 2+ and USB cable to suit
EMONA SIGEx Signal & Systems add-on board
Assorted patch leads
Two BNC – 2mm leads
Follow setup Procedure as listed on page 11 of Experiment 15 to run the SIGEx board. Reproduced
here:
Procedure
Part A – Setting up the NI ELVIS/SIGEx bundle
1.
Turn off the NI ELVIS unit and its Prototyping Board switch.
2.
Plug the SIGEx board into the NI ELVIS unit.
Note: This may already have been done for you.
3.
Connect the NI ELVIS to the PC using the USB cable.
4.
Turn on the PC (if not on already) and wait for it to fully boot up (so that it’s ready to
connect to external USB devices).
5.
Turn on the NI ELVIS unit but not the Prototyping Board switch yet. You should observe
the USB light turn on (top right corner of ELVIS unit).The PC may make a sound to indicate that
the ELVIS unit has been detected if the speakers are activated.
6.
Turn on the NI ELVIS Prototyping Board switch to power the SIGEx board. Check that all
three power LEDs are on.
7.
Launch the SIGEx Main VI.
8.
When you’re asked to select a device number, enter the number that corresponds with the
NI ELVIS that you’re using. You’re now ready to work with the NI ELVIS/SIGEx bundle.
9.
Select the EXPT 15 tab on the SIGEx SFP.
Note: To stop the SIGEx VI when you’ve finished the experiment, it’s preferable to use the STOP
button on the SIGEx SFP itself rather than the LabVIEW window STOP button at the top of the
window. This will allow the program to conduct an orderly shutdown and close the various DAQmx
channels it has opened.
Part B – Setting up the experiment
Run “NI ELVISmx Instrument Launcher” and select the FGEN “ Function Generator”. Run it and
minimise the FGEN panel for now.
Viewing SIGEx SFP, select Lab 15, which relates to laboratory Experiment 15.
Patch together the experiment from “Part 2-IIR with feedforward : second order filters” on page
16 using Figure 1 . You will be building a 2nd order IIR filter and reviewing its response by sweeping
it with a single frequency sinusoid from the FGEN across its range.
Figure 1: IIR filter with sinusoidal input
Settings are as listed in step 19. Reproduced here:
ADDER GAINS: b0=1; b1=2; b2=1; a0=1; a1=+1.6; a2= -0.902
PULSE GENERATOR: 20kHz, DUTY CYCLE=0.5 (50%)
FUNCTION GENERATOR: Sinewave selected, FREQUENCY=1k; Amplitude= 1V pp
SCOPE: Timebase = 4ms, Trigger level = 0V, Trigger on input signal
Note that you are entering into the SIGEx SFP the coefficient values for the transfer function
which are immediately setup in the hardware experiment structure you have patched up.
Connect CH0 scope lead to the S/H analog input terminal, and CH1 scope lead to the output of the B
TRIPLE ADDER, which is the output of the IIR filter. (Remember to also connect the scope’s black
leads to GND terminals.)
Select “Y AUTOSCALE” OFF using the convenient toggle switch above the tabs. At times you may
need to turn the Y autoscale ON again to adjust to current signals levels.
Sweep the FGEN frequency from about 500Hz to 7kHz and observe the LPF effect you have
implemented. Note that just before the rolloff that the high q of the filter causes the output to
saturate. This is entirely due to the coefficient values selected and is discussed in the experiment
preparation.
To see the poles and zeros of the transfer function you have implemented, switch to TAB “PZ
PLOT”. This TAB will display the poles and zeros for the coefficients currently setup on the SIGEx
SFP. You can see that the poles are very close to the unit circle (r=0.95), hence the high gain at the
peak before rolloff.
Figure 1: IIR filter with broad spectrum noise input
Viewing the system response using FFT
To view the response more dynamically, you will now change your input signal. Change the patching of
the FUNCTION GENERATOR to match Figure 2. You will now be inputting a flat noise spectrum and
viewing the FFT of the filtered output of the system.
Figure 2: IIR filter with flat broadband noise input
Ensure that SEQUENCE GENERATOR DIP switches are set to positions DOWN:DOWN. Set TLPF
knobs to fully clockwise for now. Switch to TAB “ZOOM FFT” to view time and frequency domains
simultaneously.
Change scope timebase to 100ms.
Setting up the input noise signal:
i) Reduce the TLPF GAIN by rotating counter clockwise until the output Ch1 signal (red) is no longer
saturated ie: less than 12V peak. A maximum level of 6 V is optimum to avoid saturation of internal
ADDER nodes.
ii) Reduce the noise bandwidth to around 4khz by rotating the TLPF block’s “Fc” control-counter
clockwise. View the noise spectrum as the white trace on the SCOPE & FFT windows.
The SFP should be similar to Figure 3 below:
Figure 3: IIR filter response using FFT; note the very small input signal (white)
At this point we can see and explore the issues relating to :
-controlling our input signal level and bandwidth
-viewing the response in both time and frequency domains
-setting up a transfer function with appropriate internal gains
We can also confirm that the peak of the response is correct according to the position of the poles.
ie: PZ PLOT tells us that poles are at 32 degrees, hence we expect a peak close to 32/360*20,000
= 1777 Hz. You can use the cursors in ZOOM FFT window to confirm this.
Dynamically varying the poles and zeros to adjust response using GAIN ADJUST manual control
Use the SIGEx board’s GAIN ADJUST knob to vary one of the coefficients by hand while viewing
the frequency response. Leave the default settings. Turn the knob until it reads +1.6 (located in the
COEFFICIENT SELECTOR window), then select radio button “ a 1 ”. View the frequency response
while slowly varying the value of a1. You will find that the peak frequency changes.
Find a range of a1 settings that work well and then view PZ PLOT while varying across that range.
You will see the poles moving and reflecting the changing a 1 coefficient. (Theory states that a1=-2σ,
which is the real part of the pole and its conjugate.) You can confirm this relationship from values
displayed on PZ PLOT.
Set a1 back to +1.6, select OFF, then set GAIN ADJUST to -0.9, and select a 2 radio button.
Varying a2 will vary the gain or peak level of the filter. Notice what happens in the time domain
when a2 = -1.0. The filter breaks into oscillation. View the poles again using PZ PLOT while varying a2.
(Theory states that a2 = r 2).
Set a2 back to -0.9, then select OFF again at the COEFFICIENT SELECTOR to disable the GAIN
ADJUST control.
Using Digital Filter Design toolkit to implement Highpass filters
Switch to the DFD TAB. This will automatically load the computed coefficients for the selected
filter onto the SIGEx board. You can see the values on the TAB setup in the GAIN input controls on
the SIGEx SFP.
Connect CH0 to “ x0 ”, and CH1 to “Y” and view the internal signal levels at “x0” by switching to the
ZOOM FFT tab. Set the TLPF GAIN higher but avoid saturating.These filters have lower internal
gains than the previous ones. Vary the filter design type (at the DFD tab by clicking on “DESIGN
METHOD” to select) and view the output responses using ZOOM FFT.
Set timebase to 100ms.
Again you can use the cursors to compare the actual performance to theory and design.
You can expect to see a display like so:
Figure 4: Highpass filter response using FFT; x0 (white); Y (red)
Figure 5: DFD tab used to design filters and setup coefficients to the patched SIGEx board
Default values are chosen to enable students to see textbook like responses which they can easily
measure.
Conclusion
This quickstart guide is designed to demonstrate quickly, and without supporting preparation, some
of the powerful interactivity of the SIGEx board. Students can expect to gradually build up to this
level of complexity with the many documented earlier experiments from the SIGEx Lab Manual.
By building and controlling experiments using real circuits, students will develop confidence and
intuitive familiarity with the theory they have covered in class.
References
Lathi.B.P. , “Signal processing & Linear Systems”, Oxford University Press
Oppenheim.A.V.,Wilsky.A.S., “Signals & Systems”, Prentice Hall, 2nd edition
Haykin, Van Veen, “Signals and Systems”, Wiley, 2nd edition
Ziemer.R.E,Tranter.W.H, Fannin.D.R, “Signals & Systems: Continuous and
Discrete”, Prentice Hall, 4th edition
Boulet.B.: “Fundamentals of Signals & Systems”, Thomson/Delmar Learning
Frederick.D.K.,Carlson.A.B.: “Linear systems in communication and control”,
Wiley
Carlson.A.B.: “Communication systems”, McGraw Hill, 2nd edition
Smith.C.E.: “Applied mathematics for radio and communication engineers”,
Dover
Lynn.P.A., “An introduction to the analysis and processing of signals”, Macmillan
Maor.E.,”e: The story of a number”, Princeton University Press
Radzyner.R / Rakus.M: “Signals and systems experiments with Emona TIMS”,
Emona Instruments, Issue 3.0
Langton, C.: “Fourier analysis made easy”, www.complextoreal.com
McClellan.J.H, Schafer.R.W, Yoder.M.A, “DSP First”, Prentice Hall
Emona SIGEx™ Lab Manual Signals & Systems Experiments with the Emona SIGEx
Volume 1
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