Experimental Methods
Ahmed Rashed
Aerospace Engineering Department
Cairo University
Monday 9th March, 2015
1
1
1
Contents
Contents
1 Contents
2
2 Description
4
3 Grading
5
4 References
6
5 Assignments
7
1
8
Introduction
1 Problems
12
2
13
Static Measurements
1 Definitions
13
2 Measurement Error
16
3 Problems
22
3
24
Electronic DAQ
1 Introduction
24
2 ADC
29
2.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Sig. Conditioning
33
3.1 Special Circuits\Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Other Comp’s
40
5 Multi-channel DAQ
41
6 Further Details
44
7 Problems
46
4
48
Dynamic Signal Analysis
1 Introduction
48
2
2 The FT
49
3 FT Properties
58
4 Convolution
63
5 Problems
66
5
Appendices
68
A Basic FT Pairs
68
6
70
Problem’s Answers
1 Part 2: Static Measurements
70
3
2
Course Description
Title Experimental Methods
Course Code ME 241
Instructor
Assistant prof. Ahmed Mohamed Rashed Desoki
Hours 3 (2 Lecture + 1 Tutorial)
(1 lecture + 1 section)/week
Grade 100
2
Course Aims
• To understand modern engineering experimentation including experiment design,
system calibration, data acquisition, analysis and presentation.
• To understand how to quantify error and uncertainty in physical measurements.
• To gain hands-on experience with modern instrumentation and systems-level experimentation.
• To improve written and oral communication skills, to develop the ability to write
engineering reports of high quality, and to improve the student’s ability to function
as a member of an engineering team.
3
Course Objectives
At the end of the course, the student who has mastered the course material will be
able to:
• Draw a concept map for a generalized measurement system that identifies the most
important concepts.
• Apply basic statistical methods to design experiments, to analyze, and to present
the results of experiments.
• Identify and describe the elements making up computer-based data acquisition systems, including alternative configurations and technologies.
• Identify and describe the various types of mechanical measurements including temperature, pressure, sound, motion and position, force and torque, stress and strain,
flow visualization and measurement (e.g., volume flow rate, velocity, etc.) and explain the transduction principles that underlie them.
• Operate modern instrumentation systems that include mechanical and electro-optical
technologies and computer-based data acquisition systems.
• Work productively and effectively in an engineering team.
4
4
Course Outline(tentative)
Week 1: Introduction
Week 2: Generalized Measuring System
Week 3: Data Acquisition, Signals, Signal Conditioning, Sampling
Week 4: Lab View – Lab View Tutorial
Week 5: Back ground and Introduction to thermal experiments
Week 6: Background and Introduction Fluid mechanics experiments
Week 7: Background and introduction to Material experiments
Week 8: Background and Introduction to solid mechanics experiments
Week 9: Presentation & communication skills
Week 10: Accuracy, Precision, Error in Measurement, Calibration - Lab Work
Week 11: Uncertainty Analysis – Exercise
Week 12: Displacement and Dimensional Measurement – Lab work
Week 13: Library Exercise
Week 14: Oral Presentation for Selected Topic
Week 15: Oral Presentation for Selected Topic
Week 16: Final Exam
5
3
Grading
Assessment
Section work (quizzes, assignments, . . . )
7th week Assignment/Midterm
Lecture Assignments
12th week Assignment/Midterm
Project
Final Exam
Total
Grade
10
15
5
10
10
10
40
100
Announced as
Grade
7th week Assignment/Midterm
30
12th week Assignment/Midterm
20
Year work
Final Exam
10
40
Additionally, every absence costs you -0.5 degree
6
5
4
References
Main Textbook
1. R. S. Figliola and D. E. Beasley, Theory and Design for Mechanical Measurements, John Wiley and Sons, 5th ed., 2010.
Reference Books
1. National Instruments, LabView data acquisition basics manual, January 2000.
2. K. Shin and J. K. Hammond, Fundamentals of Signal Processing for Sound
and Vibration Engineers, John Wiley & Sons, 2008. (Very good explanation of Signal
Processing. Comprehensible for mechanical engineering students. Assisted with very interesting examples
and Matlab codes. Wider and more practical than Brigham’s)
3. A. Brandt, Noise And Vibration Analysis, Wiley, 2011.
(Seems very comprehensive
and very good for both theoretical and experimental modal analyses.)
4. National Instruments, X series user manual
5. Yaser El-Shaer, Experimental Methods, undergraduate lecture notes, Arab
Academy for Science, Technology & Maritime Transport
7
6
5
Assignments
#
1
2
3
4
5
6
7
8
9
10
Assignment Name
Problem (1.1); only one item, (1.3); only (f), (2.1),
(2.2), (2.3), (2.4), (2.5); load cell A only
Problem (2.6)
Problem (3.2)-a and (3.2)-b
Problem (3.1)-a & c
Problem (3.2)-c to m
Problems (4.1c), (4.1e) & (4.1f)
Due date
2015-Feb.-23
2015-Mar.-2
2015-Mar.-9
2015-Mar.-16
8
7
Part 1
Introduction
What Is Measurement?
Measurement is the process of estimating a physical variable by a measuring device
Measuring
Instrument
V
C
Resistance
Measurand
9
Example Measuring Systems
10
70
Detector and
feedback
electronics
60
Laser
Photodiodes
Display scale
Stem
20
30
40
50
Output stage
Cantilever and tip
Sample surface
Bulb
Sensor–transducer stage
Atomic-force microscope; a
complicated measuring system
Sensor
Bulb thermometer; a simple
measuring system
8
Sensor and Transducer
11
Sensor is a physical element that employs some natural phenomenon to sense the variable being
measured
• e.g.: the cantilever beam senses the height
of the surface
Transducer converts the sensed information into
a detectable signal that can be meaningfully
recorded (mechanically, electrically, optically,
numerically, . . . )
• e.g.: the laser and the light sensors (photodiodes)
Detector and
feedback
electronics
Laser
Photodiodes
Cantilever and tip
Sample surface
Atomic-force microscope; a
complicated measurement system
Generalized Measuring System
Calibration
Optional
Signal conditioning stage
Sensor stage
Signal
path
Transducer
stage
Amplifier
Output
stage
Filter
Process
Control signal
Control
stage
12
9
Signal Domains Examples
Mechanical
Length, area, volume, all time derivatives such as
linear/angular velocity/acceleration, mass flow, force , torque,
pressure, acoustic wavelength and intensity
Thermal
Temperature, (specific) heat, entropy, heat flow, state of
matter
Electrical
Voltage, current,charge, resistance, inductance, capacitance,
dielectric constant, polarization, electric field, frequency, dipole
moment
Field intensity, flux density, magnetic moment, permeability
Magnetic
Radiant
Intensify, phase, wavelength, polarization, reflectance,
transmittance, refractive index
Chemical
Composition, concentration, reaction rate, pH,
oxidation/reduction potential
13
Assignment
• Solve problem (1.1)
14
Noise and Interference
15
6
Noise is a random variation of the value of the measured signal
Signal: y (t ) = 2 + sin (2 π t)
5
4
Interference imposes undesirable
trends on the measured value
Signal y(t)
• increases data scatter
deterministic
Signal + interference
3
Signal + noise
2
1
• For ex.; in electrical instruments interference comes from AC power and is seen as
a sinusoidal wave superimposed onto the
measured signal
0
0 .0
0 .5
1 .0
1 .5
2 .0
Time (s)
Effects of noise and interference
superimposed on the signal
y(t) = 2 + sin (2πt)
Test Standards and Codes
Test standard refers to well defined test procedures, technical terminology, methods to
construct test specimens or test devices, and/or methods for data reduction
The goal of a test standard is to provide consistency in the conduct and reporting
of a certain type of measurement between test facilities
10
For example;
ASME Power Test Code 19.5 provides detailed designs and operation procedures for flow meters. See [1, problem 1.34] for details.
ASTM Test Standard F558-88 provides detailed procedures for evaluating vacuum cleaner cleaning effectiveness and controls the language for product performance claims.
16
Assignment
• Solve problem (1.2)
17
Static versus Dynamic Measurements
Static measurement is performed when the measured quantity is not (or very slowly)
changing with time
• An example is beam deflection under a static load
Dynamic measurements is vise versa performed
• An example is plane wing vibration due to engine vibration and flutter induced
by high speed flight
18
Assignment
• Solve problem (1.3)
19
Measuring System Characteristics
Static characteristics may include
• Accuracy
• Linearity
• Precision
• Hysteresis
• Sensitivity
• ...
Dynamic characteristics usually describe
• Transient response
• Steady state response
20
11
1
Problems
Important, for any research question,
• you must mention your references
• If the reference is electronic and accessible, save it and send me a copy
1.1. Solve problem 1.8 of [1]
1.2. Solve problem 1.53 of [1]
1.3. Solve problem 2.24 of [1]
21
12
Part 2
Static Measurements
1
Definitions
Range
10
Measured values
Curve fit, y =f(x)
Output value, y (units)
8
Eo
y =f(x)
6
4
K (x1) =
dy
dx
x = x1
2
Ei
0
0
1
x1
2
3
4
5
Input value, x (units)
Measurand range (or operating range or full-scale range or span)
It is the range of the input variable, x,
ri ≡ xmax − xmin
(2.1)
Full Scale Output (FSO) range
It is the range of the output variable, y,
ro ≡ ymax − ymin
(2.2)
22
Assignment
• Solve problem (2.1)
23
Calibration
10
Measured values
Curve fit, y =f(x)
Output value, y (units)
8
y =f(x)
6
4
K (x1) =
dy
dx
x = x1
2
0
0
1
2
x1
3
Input value, x (units)
13
4
5
A calibration applies a known input value to a measurement system for the purpose
of observing its output value
• It establishes the relationship between the input and output values
Static calibration implies that the values of the variables involved remain constant with
time
• In static calibrations, only the magnitudes of the known input and the measured output are important
Dynamic calibration implies that variables are time (or space) dependent
• A dynamic calibration measures the measurement system response due to an
input of known dynamic behavior
• Usually, such calibrations involve applying either a sinusoidal signal or a step
change as the known input signal
24
Static Sensitivity
• Simply can be called Sensitivity
– Also called Gain
• It is the change of an instrument/transducer output per unit change in the measured
quantity. That is
dy
(2.3)
K(x) ≡
dx
10
Measured values
Curve fit, y =f(x)
Output value, y (units)
8
y =f(x)
6
4
K (x1) =
dy
dx
x = x1
2
0
0
1
2
x1
3
4
5
Input value, x (units)
– A more sensitive instrument’s reading changes significantly in response to
smaller change in the measured quantity
– If the output is a linear function of the input,
K(x) = constant
=K
(2.4)
25
14
Resolution
• It is quantified by the smallest scale increment of the output readout indicator
26
15
Measurement Error
Actual data trend
(2.5)
Output value
Downscale
Hysteresis
Best linear curve fit
Upscale
where
y is the measured value
and
ytrue is the true value
Input value
Input value
(a) Hysteresis error
(b) Linearity error
Maximum for
typical device
Output value
Nominal curve
for typical device
K
Minimum for
typical device
Typical shift
(high)
Output value
e ≡ y − ytrue
e
e(%) ≡
× 100
ytrue
e
× 100
e(% FSO) ≡
ro
Output value
2
Nominal
Typical shift
(low)
Input value
(d) Zero shift (null) error
Output value
Input value
(c) Sensitivity error
Probable (±2s x)
data scatter band on
successive measurements
Input value
(e) Repeatability error
• A primary objective in designing and executing an experiment is to minimize the
measurement error
• Usually, the true value, ytrue , is rarely known exactly, e(%)
27
Types of Error
Errors are usually classified as
• Systematic error
– Bias/Accuracy error
• Random error
– Precision error
• Other (illegitimate) errors
28
16
Uncertainty
• Since the true value is rarely known exactly, the error is not known exactly
• Uncertainty is a numerical estimate of the possible range of error in a measurement
• Based on available information, the operator might feel confident that the error is
within certain bounds
– e.g.: a plus or minus range of the indicated reading
– This is the assigned uncertainty
– For example, from the figure below, we might assign an estimate to the random
error based on the data scatter of the measurements
Typical pressure transducer manufacturer’s
Actual data trend
specifications
Output value
Output value
Downscale
Hysteresis
Operation
Best linear curve fit
Upscale
Input value
Input value
(a) Hysteresis error
(b) Linearity error
Maximum for
typical device
K
Minimum for
typical device
Excitation
±15V DC
Output range
0−5 V
Temperature range
0 − 50◦ C
Performance Errors
Nominal
Typical shift
(low)
Input value
Input value
(c) Sensitivity error
(d) Zero shift (null) error
Output value
0 − 1000 cm H2 O
Typical shift
(high)
Output value
Output value
Nominal curve
for typical device
Input range
1
Linearity
0.5% FSO
Hysteresis
Less than ±0.15% FSO
Sensitivity
0.25% of reading
Thermal sensitivity
±0.02%/◦ C of reading
Thermal zero drift
±0.02%/◦ C FSO
Probable (±2s x)
data scatter band on
successive measurements
Input value
(e) Repeatability error
29
Overall Uncertainty
q
u = u21 + u22 + u23 + · · ·
(2.6)
• More details are in [1, ch. 5]
30
17
Assignment
• Solve problem (2.2)
• Solve problem (2.3)
31
Precision (Random Error)
Sensor reading
Acuracy
(Bias Error)
Precision
(Random error)
Frequency
of readings
Measurement trial
Precision is the difference between (or scattering of) measured values during repeated
measurements of the same quantity.
• Calculated through statistical analysis on the results of several experiments
Precision(% FSO) ≡
max (|y − ȳ|)
× 100
ro
(2.7)
• e.g.: 100kPa ± 1%.
• represents the ability to reproduce a reading (not necessarily correct)
– also called reproducibility or repeatability
32
Accuracy (Bias Error)
Sensor reading
Acuracy
(Bias Error)
Precision
(Random error)
Frequency
of readings
Measurement trial
Accuracy is the difference between the true value and the average measured value
1
In fact, this is uncertainty. It is customary to call uncertainty as error.
18
• calculated through statistical analysis on the results of several experiments
Accuracy(% FSO) ≡
|ȳ − ytrue |
× 100
ro
(2.8)
33
Accuracy vs Precision
Accurate
Precise
Not accurate
Precise
Not accurate
Not precise
34
Example 2.1.
Sensor reading
Acuracy
(Bias Error)
Precision
(Random error)
Frequency
of readings
Measurement trial
• Four readings of 100 V yields 104, 102, 105 and 105 V;
– ytrue = 100 V
– ȳ = 104 V
– Precision = max (|y − ȳ|) = max (ymax − ȳ, ȳ − ymin ) = 2 V
– Accuracy = |ȳ − ytrue | = 4 V
35
19
Assignment
• Solve problem (2.4)
• Solve problem (2.5)
36
Hysteresis
maximum
hysteresis
upscale
output (%FSO)
100
downscale
0
0
100
measurand (% range)
• It is the difference in a test output value, y, for the same input value x, when x is
sequentially increased and then decreased; or vice versa
eh (% FSO) =
yupscale − ydownscale
× 100
ro
(2.9)
• Usually the maximum hysteresis is reported in a transducer data sheet
37
Sensitivity and Zero Shift Errors
Sensitivity error
100
output (%FSO)
Nominal
response
0
0
100
measurand (% range)
Zero shfit error
20
Zero Shift error is the change in the output level of a sensor or instrument for zero
input
Sensitivity error is the change in the sensitivity of a sensor or instrument
• Possible reasons are aging, temperature variation, sensor damage, operating conditions change, . . .
38
Assignment
• Solve problem (2.6)
39
21
3
Problems
2.1. Solve problem 1.9 of [1]
2.2. Determine the overall uncertainty of the following transducer
Typical pressure transducer manufacturer’s specifications
Operation
Input range
0 − 1000 cm H2 O
Excitation
±15V DC
Output range
0−5 V
Temperature range
0 − 50◦ C
Performance Error
Linearity
0.5% FSO
Hysteresis
Less than ±0.15% FSO
Sensitivity
0.25% of reading
Thermal sensitivity ±0.02%/◦ C of reading
Thermal zero drift
±0.02%/◦ C FSO
2.3. Solve problem 1.23 of [1]
2.4. Multiple Choice Question
(a) The difference between the instrument’s reported values during repeated measurements of the same quantities is known as
i. Accuracy
ii. Precision
iii. Resolution
iv. Sensitivity
(b) The difference between the measured and the true value is known as
i. Accuracy
ii. Resolution
iii. Precision
iv. Sensitivity
(c) The smallest increment of change in the measured value that can be determined
from the instrument’s readout scale is known as
i. Accuracy
ii. Precision
iii. Resolution
iv. Sensitivity
(d) The change of an instrument or transducer’s output per unit change in the
measured quantity is known as
i. Accuracy
ii. Precision
iii. Resolution
iv. Sensitivity
(e) Errors that occur the same way each time a measurement is made known as
22
i. Bias error
ii. Computational error
iii. Precision error
iv. Random errors
(f) Error that are different for each successive measurement but have an average
value of zero are know as
i. Bias error
ii. Computational error
iii. Precision error
iv. Random errors
2.5. Three load cells are tested for repeatability. The same 50 kg weight is placed on
each load cell 10 times. The load cells’
output along with their minimum, maximum and mean are given in the following
table. The expected (true) output is 10
mV. The maximum weight the load cell
can measure is 100 kg.
• Calculate the precision and accuracy of each of the three load cells.
Trail
no.
1
2
3
4
5
6
7
8
9
10
Maximum
Average
Minimum
A
10.02
10.96
11.20
9.39
10.50
10.94
9.02
9.47
10.08
9.32
11.20
10.09
9.02
Load cell output
(mV)
B
11.50
11.53
11.52
11.47
11.42
11.51
11.58
11.50
11.43
11.48
11.58
11.49
11.42
C
10.00
10.03
10.02
9.93
9.92
10.01
10.08
10.00
9.97
9.98
10.08
9.99
9.92
2.6. Solve problem 1.12 of [1]
40
23
Part 3
Electronic Data Acquisition (DAQ)
1
Introduction
A data acquisition (DAQ) system is the portion of a measurement system that quantifies
and stores data
Examples
• Experimenter
– reads a transducer dial
– records the measurement
• Electronic DAQ system
– automates data quantification and storage
We focus herein on electronic DAQ systems
41
Electronic DAQ system
Physical
variable
Voltage
Current
Power
Force
etc
Proportional
Voltage
ADC
Transducer Signal conditioning
DAQ system
24
Digital
form
Storage
Sample DAQ’s
• Sound card of a PC
• National Instruments products
– e.g., USB line, PXI line, compact DAQ line, compact Rio line, PCI line, PCI
express line, . . .
• Brüel & Kjær products
– e.g., Brüel & Kjær Multichannel Analysis System Type 3550
Multichannel
Data Acquisition
Interface 7520
25 kHz Zoom Processor
3156
100 kHz/Multichannel
Zoom Processor 3157
Dual Channel Signal Analyzer
Type 2032
Signal Analyzer
Unit 2035
Dual-channel
Analysis Software 7649
B
^
¨
ESC
CTRL
!
1
¤
2
TAB
Q
ALPHA
LOCK
A
>
<
↑
#
3
W
$
4
E
S
Z
%
5
R
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X
^
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4/8-channel
Analysis
Software 7674
CLR
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FEED
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SHIFTS
←
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RETURN
REP
→
↓
Keyboard
NP 0028
Preamp. Input
Acc. Input
Charge
Pol. Voltage
Direct Input
1MΩ / / 100pF
200V
Signal Analysis
Software 7671
Multichannel
Analysis Software 7640
Preamp. Input
Acc. Input
Charge
Pol. Voltage
Direct Input
1MΩ / / 100pF
4–channel Input Module Type 3023
Input 1
Sampling
Input/Output
Input 4
Input 3
Input 2
Vibration Diagnostic
Software 7672
200V
28V
Tracking Analysis
Software 7670
K
Bruël & Kjær
SHIFTS
Extended
Analysis
Software 7639
Trigger
Input
Intensity Probe/
Remote Control
Sampling
Input/Output
Trigger
Input
Signal Generator
Output
28V
0V
0V
Channel A
!
Floating / Signal Ground
25 kHz
Input Module
3019
Channel B
!
!
B
K
7/6-'89
Floating / Signal Ground
!
Brüel & Kjær
!
4-channel
Input Module
3023
100 kHz
Input Module
3020
Multichannel
Data Acquisition
Unit 2816
Unit 1
B
Unit 2
Direct Input
1MΩ / / 100pF Active
Direct Input
1MΩ / / 100pF Active
Direct Input
1MΩ / / 100pF Active
Active
Active
Sampling
Module
3018
Blank Panel
FA 1009
Active
K
7/6-'89
Direct Input
1MΩ Ground
/ / 100pF Active
Signal
Brüel & Kjær
Signal Ground
Sampling Input
Delay Adj.
!
Signal Ground
!
Direct Input
1MΩ / / 100pF Active
Floating
Trigger Input
On
B
Off
Input Module
3015
Acc. Input
200V 28V 0VCharge
Pol. Voltage
Input Module
Preamp. Input
B
Preamp. Input
B
K
7/6-'89
7/6-'89
Brüel & Kjær
Signal Ground
Signal Ground
!
B
7/6-'89
Intensity Probe/
Remote Control
1
Intensity Probe/
Remote Control
!
Floating
Floating
Generator Module Generator Module
3107
3107
Acc. Input
Charge
Acc. Input
Charge
1
3015
Input Module
3015
Input Module
3016
Sound Intensity
Module
3017
200V 28V 0V
Pol. Voltage
200V 28V 0V
Pol. Voltage
200V 28V 0V
Pol. Voltage
200V 28V 0V
Pol. Voltage
Preamp. Input
Preamp. Input
Signal Generator
Output
Preamp. Input
K
Brüel & Kjær
Active
Floating
Signal Ground
!
Floating
Acc. Input
Charge
Input Module
3015
200V 28V 0V
Pol. Voltage
K
7/6-'89
Acc. Input
Charge
Input Module
3015
200V 28V 0V
Pol. Voltage
Brüel & Kjær
Signal Ground
Floating
Signal Ground
Power
!
Floating
!
Acc. Input
Charge
Sampling Output
Signal Analyzer
Interface Module
Type 7521
Direct Input
1MΩ / / 100pF Active
!
Floating
Multichannel
Data Acquisition
Unit
Type 2816
B
K
7/6-'89
Brüel & Kjær
B
B
K
7/6-'89
& Kjær
K
Brüel & Kjær
Preamp. Input
7/6-'89
Brüel & Kjær
B
K
7/6-'89
Brüel & Kjær
Signal Generator
Output
B
Sound Intensity
Module
3017
200V 28V 0V
Pol. Voltage
2
2
K
7/6-'89
Intensity Probe/
Remote Control
B
7/6-'89
Brüel & Kjær
K
Brüel & Kjær
B
7/6-'89
K
Brüel & Kjær
Intensity Probe/
Remote Control
B
K
K
7/6-'89
Brüel & Kjær
Brüel & Kjær
Signal Analyzer
Interface Module
7521
Sound Intensity
Module 3017
25 kHz or 100 kHz
Input Modules
3015 or 3016
Generator Module
3107
Unit 1
B
Unit 2
Direct Input
1MΩ / / 100pF Active
Direct Input
1MΩ / / 100pF Active
Direct Input
1MΩ / / 100pF Active
Active
Signal Ground
Sampling Input
Delay Adj.
!
Direct Input
1MΩ Ground
/ / 100pF Active
Signal
Signal Ground
!
Direct Input
1MΩ / / 100pF Active
Floating
Floating
Acc. Input
Charge
Signal Ground
!
Input Module
3015
Input Module
3015
Input Module
3015
200V 28V 0V
Pol. Voltage
200V 28V 0V
Pol. Voltage
Acc. Input
200V 28V 0VCharge
Pol. Voltage
Input Module
Signal Ground
Signal Ground
!
Signal Ground
Active
Active
Floating
Noise
Floating
Floating
500 Hz
Floating
Active
Input Module
3016
200V 28V 0V
Pol. Voltage
Active
Floating
Acc. Input
Charge
Input Module
3015
200V 28V 0V
Pol. Voltage
Σ
Σ
Floating
Generator Module Generator Module
3107
3107
Acc. Input
Charge
3015
200V 28V 0V
Pol. Voltage
Noise
Active
Floating
!
Floating
Acc. Input
Charge
!
Floating
Signal Ground
!
Acc. Input
Charge
Direct Input
1MΩ / / 100pF Active
!
Floating
Sampling Output
Signal Analyzer
Interface Module
Type 7521
Signal Output
Module
Type 5955
Active
Signal Output
Module
Type 5955
K
7/6-'89
Brüel & Kjær
Multichannel
Data Acquisition
Unit
Type 2816
Floating / Signal Ground
Generator &
Sampling Module
3106
3 ms
Delay
x0,5
x 0.5
Σ
Ch.B
Gen.
Active
Σ
Floating
Noise
Active
Power
On
Off
Trigger Input
B
7/6-'89
K
Brüel & Kjær
Preamp. Input
B
7/6-'89
K
Brüel & Kjær
Preamp. Input
B
7/6-'89
K
Brüel & Kjær
B
7/6-'89
B
K
7/6-'89
Brüel & Kjær
K
Brüel & Kjær
Preamp. Input
B
7/6-'89
& Kjær
Floating
Signal Generator
Output
Preamp. Input
K
Brüel & Kjær
Preamp. Input
B
7/6-'89
Active
Signal Generator
Output
B
7/6-'89
Ch. A
Floating
B
K
7/6-'89
Brüel & Kjær
Preamp. Input
K
Brüel & Kjær
B
7/6-'89
K
Brüel & Kjær
B
7/6-'89
ZZ 0220
Simulator Unit
Simulator Unit
ZZ 0220
K
Brüel & Kjær
Floating
K
Brüel &
& Kjær
Signal Output
Module 5955
910704/3e
Hardware and software components of M ultichannel Analysis System Type 3550
• ...
42
25
DAQ System Components
Digital Inputs
Process
feedback
Analog
Transducers
Signal
Conditioning
Display
Multiplexer
Buffer
Storage
ADC
Clock
External
Computer
Controller
Trigger
Multiplexed DAQ
The main DAQ components are:
1. Analog to Digital Converter (ADC)
2. Signal Conditioning (optional)
• Amplifier, Filters, . . .
3. Other Components
• Clock, Trigger, . . .
43
Single-Ended/Floating Signals
+
_
+
_
Vs
Ground
Vs
Ground
Single ended (grounded) signal
Floating signal
44
Floating (Differential) signal
Single-ended (Grounded) signal
• referenced to a system ground
• Examples are devices that plug into
a building ground through wall outlets, such as:
• not connected to an absolute reference
• Examples include:
– batteries
– signal generators and
– battery-powered sources
– power supplies
– thermocouples
– transformers
– isolation amplifiers
26
Single-ended/Differential Measurement Systems
Multiplexer
Channel 1
+
Transducer 1
+ CH1 Hi
–
ADC
Channel N
+
Transducer N
Vm
+ CH N Hi
–
GND
GND
Single-ended DAQ system
Multiplexer 1
Channel 1
+
Transducer 1
+ CH1 Hi
–
+ CH N/2 Hi
ADC
Channel N/2
Transducer N/2
+
- CH1 Lo
Vm
–
- CH N/2 Lo
Multiplexer 2
GND
Differential DAQ system
Single-ended measurement system
Differential measurement system
can be used when all input signals meet
• Preferred way to measure low-level
the following criteria
signals, because it rejects:
• Input signals are high level (greater
than 1 V as a rule of thumb)
– ground loop-induced errors
• Signal cabling is short and travels
through a noise-free environment or
is properly shielded
• All input signals can share a common reference signal at the source
27
– the noise picked up in the environment (to a certain degree)
• By using twisted pairs, noise effects
are greatly reduced
• One differential channel consumes
two single ended channels
• More details are in
– “Ground Loops and Returns” (http://www.ni.com/white-paper/3394/en/)
∗ Chapter 4 of [4]
45
The Oscilloscope
It is a very useful diagnostic tool
• It is used to measure and display voltage magnitude versus time for dynamic
signals over a wide range of frequencies extending commonly to megahertz (or
even gigahertz)
• It can display the signal magnitude, frequency, distortion, and DC and AC components
– Typical oscilloscopes can also display several more signals (e.g. x(t) and y(t)),
perform addition and subtraction (x(t) ± y(t))and other features such as fast
Fourier transform (FFT)
• This visual display can help the user to detect noise/interference superimposed on
a signal, something nonvisual metering devices cannot do
• The LabView interactive program “Oscilloscope.vi” with the software accompanying
reference [1]
– It features a basic two-channel oscilloscope. The user can vary channel displayed and the time sweep and gain using two built-in signals (sine wave and
square wave) and an active signal trigger
46
28
Signal
Signal
The Analog to Digital Converter (ADC)
Signal
2
Time
Quantization levels
Time
Analog signal
Time
Discrete signal
Digital signal
• It converts analog voltage into a binary number through a process called quantization
• The ADC has both an analog side and a digital side.
47
The Analog side
ADC
• It is specified in terms of the full-scale voltage range, EFSR
ADC
defines the voltage range over which the ADC operates
– The EFSR
ADC
Example 3.1. An ADC with EFSR
= 10 V can accept voltages in the range 0 ∼ 10 V
(unipolar) or −5 ∼ 5 V (bipolar), depending on the ADC setting
48
The Digital side
• It is specified in terms of the number of bits of its register
• An M -bit ADC outputs an M -bit binary number
– It can represent 2M different numbers
Example 3.2.
• An 3-bit ADC represents analog voltages with 23 = 8 different values
Amplitude (volts)
• A 16-bit ADC represents analog voltages with 216 = 65, 536 different values
10.00
8.75
7.50
6.25
5.00
3.75
2.50
1.25
0
0
16-Bit Versus 3-Bit Resolution
(5 kHz Sine Wave)
111
16-bit
110
101
100
011
010
3-bit
001
000
50
100
Time (µs)
150
200
49
29
2.1
ADC Characteristics
Amplitude (volts)
Resolution
16-Bit Versus 3-Bit Resolution
(5 kHz Sine Wave)
10.00
8.75
7.50
6.25
5.00
3.75
2.50
1.25
0
0
111
16-bit
110
101
100
011
010
3-bit
001
000
50
100
Time (µs)
150
200
• It is defined in terms of the smallest voltage change that causes a change of the least
significant bit
– Same as was defined earlier in section 1
Q
ADC
ADC
EFSR
= M
2
(3.1)
Example 3.3. For a 3-bit ADC
• setting the range to 0 ∼ 10 V yields
QADC =
ADC
10
EFSR
= 3 = 1.25 V
M
2
2
(3.2)
• setting the range to –10 ∼ 10 V yields
QADC =
ADC
EFSR
20
= 3 = 2.5 V
M
2
2
(3.3)
The figure below shows a 0 ∼ 10 V sine signal digitized by both settings
10.00
8.75
7.50
6.25
5.00
3.75
2.50
1.25
0
0
Range = 20 V [ -10 to 10 V ]
111
Amplitude (vo lts )
Amplitude (vo lts )
Range = 10 V [ 0 to 10 V ]
110
101
100
011
010
001
000
50
100
150
Time (µs )
200
10.00
7.50
5.00
2.50
0
-2.50
-5.00
-7.50
-10.00
0
111
110
101
100
011
010
001
000
50
100
150
Time (µs )
200
50
30
Clipping
ADC
• Signals should use as much as possible of the EFSR
range, but without overloading
– Otherwise, clipping will occur
51
Amplitude (volts)
Dynamic Range
10.00
8.75
7.50
6.25
5.00
3.75
2.50
1.25
0
0
16-Bit Versus 3-Bit Resolution
(5 kHz Sine Wave)
111
16-bit
110
101
100
011
010
3-bit
001
000
50
100
Time (µs)
150
200
• It is the ratio between the largest and smallest number that can be represented by
a measuring system
– calculated in dB
• For an M -bit ADC, the dynamic range is calculated as
ADC EFSR
ADC
D
(dB) ≡ 20 log10
QADC
≡ 20 log10 2M
≈ 6M
(3.4)
• More details are in [2; section 5.3]
52
Practical DAQ System Dynamic Range
• A DAQ system includes several components other than the ADC
– Hence, the useful dynamic range of the whole system is usually less than that
of the ADC
31
Example 3.4. For a typical 16-bit B&K DAQ system,
• DADC = 96 ≈ 100 dB
• but DSYS ≈ 80 dB for the whole system
Example 3.5. For a modern 24-bit B&K DAQ systems,
• DADC = 6 × 24 = 144 dB
• but DSYS ≈ 100 ∼ 110 dB for the whole system
53
Quantization Error
• If the input voltage falls between two adjacent output codes, it is erroneously
digitized
• This error is referred to as the quantization error eQ
Amplitude (volts)
• It behaves as noise imposed on the digital signal
10.00
8.75
7.50
6.25
5.00
3.75
2.50
1.25
0
0
16-Bit Versus 3-Bit Resolution
(5 kHz Sine Wave)
111
16-bit
110
101
100
011
010
3-bit
001
000
50
100
Time (µs)
150
200
54
32
Demonstration
• “Chapter 7 > Bits of Resolution” in the Mechanical Measurements LabView demo
companion of [1]
• “Chapter 7 > Digitizing for Spectral Analysis” in the Mechanical Measurements
LabView demo companion of [1]
55
Assignment
• Solve problem (3.2)-a and b
56
3
Signal Conditioning
Signal Conditioning
AC
Analog
transducer
DC
Special circuit
Transducer specific
signal conditioning
AC/DC- Amplification/
Coupling Attenuation
ADC
Analog
filters
Sampling
• Transducer outputs must be ‘conditioned ’ to accommodate cabling, environmental
considerations, features of the recording instrumentation, . . .
Conditioning usually includes:
• amplification,
• filtering,
• enhance power supplies and cabling
– For example, some transducers, such as strain gauges, require power supply
∗ For this case, the signal conditioner must provide stabilized the power
supply with little ripple, low noise, . . .
• ...
57
33
Amplifier/Attenuator
• Usually sensor output is a very small signal
– For best digitization, the sensor output has to be amplified/attenuated to fill
ADC
the EFSR
∗ This is accomplished using both amplifier and offset circuit
Range = 20 V [ -10 to 10 V ]
111
Amplitude (vo lts )
Amplitude (vo lts )
Range = 10 V [ 0 to 10 V ]
10.00
8.75
7.50
6.25
5.00
3.75
2.50
1.25
0
0
110
101
100
011
010
001
000
50
100
150
Time (µs )
200
10.00
7.50
5.00
2.50
0
-2.50
-5.00
-7.50
-10.00
0
111
110
101
100
011
010
001
000
50
100
150
Time (µs )
200
• Take care; if you amplified the signal too much, clipping will occur
DAQ system
Analog
transducer
G
ADC
Amplification/
Attenuation
• Using low noise amplifier/attenuator with gain G, the usable range of the DAQ
system is expanded/contracted as
sys
EFSR
=
ADC
EFSR
G
(3.5)
– By using selectable variable amplifier gains, the ADC can be adjusted to
wider/smaller ranges for optimum signal acquisition
58
34
ADC
Example 3.6.
• A typical 12 bit DAQ system has EFSR
= 10 V. The ADC resolution
is thus calculated as
ADC
EFSR
ADC
Q
= M = 2.44 mV
2
• Using low noise amplifier with adjustable gains from G = 0.5 ∼ 1000;
– when set at maximum gain G = 1000, the input voltage range contracts to
sys
EFSR
ADC
EFSR
=
= 0.01 V
1000
and the resolution becomes
Qsys =
sys
ADC
EFSR
QADC
EFSR
= 2.44 µV
=
=
2M
G 2M
G
– when set at minimum gain (maximum attenuation) G = 0.5, the input voltage
range expands to
E ADC
sys
EFSR
= FSR = 20 V
0.5
and the resolution becomes
Qsys =
QADC
= 4.88 mV
G
59
More Information
• Analog amplifiers; Typical op amp circuits are in [1; sec. 6.6]
• Basic amplifier for digital DAQ is in [1; sec. 7.14]
60
Offset Circuit
10.00
8.75
7.50
6.25
5.00
3.75
2.50
1.25
0
0
Range = 20 V [ -10 to 10 V ]
111
Amplitude (vo lts )
Amplitude (vo lts )
Range = 10 V [ 0 to 10 V ]
110
101
100
011
010
001
000
50
100
150
Time (µs )
200
10.00
7.50
5.00
2.50
0
-2.50
-5.00
-7.50
-10.00
0
111
110
101
100
011
010
001
000
50
100
150
Time (µs )
200
• For best digitization, the sensor output has to be expanded/contracted to fill the
ADC
EFSR
range
– This is accomplished using both amplifier and offset circuit
• Usually the offset circuit is included in the ADC amplifier
• Typical offset circuit is in [1; Fig. 7.16]
61
35
More Information
• Check the “Calibration Circuitry” section in NI user manual of X series or M series
and learn about
– EEPROM installed in Integrated Electronics Piezo Electric (IEPE) sensors and
– device self calibration
• More information about IEPE are in [3, sec. 7.3]
62
Filters
• Analog filters are used to control the frequency content of the signal being sampled.
Examples include:
– low-pass filter to remove high frequency noise
– anti-alias filter which is a low pass filter that removes frequencies higher than
fs /2
More details are in [1; sec. 6.8]
63
Anti-Aliasing Filter
folding
Frequency spectrum of anti-aliasing filter
Usable range
(Passband)
Aliasing
0 dB
-40 dB
-80 dB
• This must be an analog filter, since it appears before the ADC
• Not all DAQ boards contain analog anti-aliasing filter, so be careful when you buy
a DAQ board
• In early equipment for noise and vibration analysis, filters with oversampling factor
β ≈ 1.28 (fc = fsβ/2 ≈ 0.39 fs ) was established
• In more recent systems utilizing sigma–delta ADCs, this factor is somewhat smaller,
say down to β ≈ 1.1 (fc ≈ 0.45 fs )
• More information are in [1; section 6.8]
64
36
AC Coupling Circuit
DC coupling
AC coupling
35
2 .5
Fluctuating voltage
30
20
15
y(t) – y
y(t) Voltage
25
10
5
0
0 .5
1
1 .5
2
Time (s)
2 .5
3
3 .5
2
1 .5
1
0 .5
0
–0 .5
–1
–1 .5
–2
–2 .5
0
0 .5
Signal prior to subtracting DC offset
1
1 .5
2
Time (s)
2 .5
3
3 .5
Fluctuating component of signal
AC coupling removes the DC component from the signal
• desired when the AC component we want to analyze is relatively small in
comparison to a superimposed DC component
• Most dynamic signals should preferably be acquired with AC coupling, since
we are in most cases only interested in the dynamic part of the signal
• AC coupling can be implemented by passing the signal through a very low frequency
high-pass filter
65
More Information
• http://www.ni.com/white-paper/8734/en/
66
3.1
Special Circuits\Modules
Signal Conditioning
AC
Analog
transducer
Special circuit
Transducer specific
signal conditioning
DC
AC/DC- Amplification/
Coupling Attenuation
ADC
Analog
filters
Sampling
• Several modules exist for different types of transducers
• They convert the transducer output to a form suitable for ADC (usually a volt)
67
37
Resistance bridge (Wheatstone bridge)
Gauge 3
S3
R
Amplifier
+
R
E supply
–
R
Ei
Gauge 2
Gauge 4
S2
S4
G
Channel 1
M
u
l
t
i
p
l
e
Channel 2 x
e
r
A/D
P
C
b
u
s
I
n
t
e
r
f
a
c
e
Gauge 1
• Used to measure small resistance changes as in strain gauges
• This module allows direct interface of strain gauges or other resistance-based sensors
68
Charge Amplifier
• Piezoelectric transducers produce charge2 rather than voltage
Coulomb / sec
– Also charge signals are sensitive to noise/disturbances
– Hence, the measured signal must be converted to a proportional voltage suitable for a normal DAQ system
• The conversion is accomplished using a charge amplifier , see [3; Figure 7.2]
– Charge amplifiers are relatively expensive
• More information are in
– Triboelectric (static electricity) effect; [3; sec. 7.2] and [2; sec. 5.4]
– Mark Serridge and Torben R. Licht, Piezoelectric Accelerometers and Vibration
Preamplifiers, Theory and Application Handbook, Brüel & Kjær documentation
69
2
1 A = 1 C/1 s
38
Thermocouple module
Thermocouple
wires
Copper connecting
wires
Cu
A
1
Cu
B
A
2
1′
Ice bath
(reference junction)
B
A
N
2′
Measuring
junctions
A
Potentiometer
Measuring
junctions
B
N'
Ice bath
(reference junction)
Thermocouples arranged to sense temperature differences. (From Benedict, R. P. Fundamentals of
Temperature, Pressure and Flow Measurements, 3rd ed. Copyright 1984 by John Wiley and Sons, New
York.)
• It allows for cold junction compensation for signal linearization for reasonably accurate (down to 0.5 C) temperature measurements
• More details are in [1; p. 347]
70
Shunt Resistor
• Used to measure current
• It converts current signals into proportional voltage signals
• Typical shunt resistor circuit is in [1; Fig. 7.15]
71
Assignment
• Solve problem (3.1)
72
Demonstration
• “Chapter 2 > Sound in from Microphone” in the Mechanical Measurements LabView
demo companion of [1]
73
More Information
• Field Wiring and Noise Considerations for Analog Signals (go to ni.com/info and
enter the Info Code “analogwiring”)
74
39
4
Other Components
Multiplexer
Digital Inputs
Process
feedback
Analog
Transducers
Signal
Conditioning
Display
Multiplexer
Buffer
Storage
ADC
Clock
External
Computer
Controller
Trigger
Multiplexed DAQ
• It is used to switch between connections when multiple input signal lines are connected by a common throughput line to a single ADC
75
Clock
Single channel sampling
Channel 1
Single channel
sample clock
• More information are in http://zone.ni.com/reference/en-XX/help/370466AA-01/
mxcncpts/clocks/#GUID-8A10ED1D-A7A9-42EA-8987-2F8225D7500D
76
Trigger
77
40
Digital to Analog Converter (DAC)
• It converts digital numbers into analog voltages, which might be used
– for process control
– to activate a device
– to drive a sensor positioning motor
– ...
78
Digital Input/Output
Digital I/O signals may be used as
single state (HIGH or LOW; 5 V or 0 V)
• might be used to operate a switch or relay, light a led, signal an alarm, . . .
series of pulses of HIGH/LOW states
• Pulse stepping
– used to drive stepper motors and servos
– sends a predetermined number of pulses in a series
– counting pulses that occur over a specified period of time enables frequency
determination (number of pulses/unit time) and counting/timing applications
79
5
Multi-channel DAQ
Single, Aggregate and Multiplexed Sample Rates
Digital Inputs
Process
feedback
Analog
Transducers
Signal
Conditioning
Display
Multiplexer
Buffer
Storage
ADC
Clock
Controller
Trigger
Multiplexed DAQ
41
External
Computer
• A multiplexed DAQ device may have several sampling rates
– Single-channel sampling rate, fssingle
∗ It is the fastest sampling rate attainable from a single channel
Single channel sampling
Channel 1
Single channel
sample clock
– Aggregate sampling rate, fsA
∗ It is the fastest sampling rate attainable from all the multiplexed channels
∗ Due to the multiplexer hardware limitations
fsA < fssingle
Multiplexed sampling of 3 channels on a 4 channel DAQ device
Channel 1
Channel 2
Channel 3
Sample clock
Convert clock
– Multiplexed sampling rate, fsMux
∗ It is the fastest sampling rate attainable from each multiplexed channel
∗ For a K channel DAQ device,
1
fsMux
≥K·
1
fsA
⇐⇒
fsMux ≤
fsA
K
Example 3.7. The NI 6351 has 16 analog inputs, fssingle = 1.25 MS/s and fsA = 1 MS/s.
• Thus it can sample
– one channel at fssingle = 1.25 MS/s
42
– 8 channels at fsMux =
fsA
8
– 16 channels at fsMux =
= 0.125 MS/s each
fsA
16
= 0.0625 MS/s each
• Check http://www.ni.com/white-paper/9376/en/ for typical examples
80
Multi-channel Sampling
Multi-channel DAQ devices use either multiplexed or simultaneous sampling;
• Simultaneous sampling devices have an ADC for each analog input channel
Digital Inputs
Process
feedback
Analog
Transducers
Signal
Conditioning
ADC
Buffer
ADC
Buffer
Buffer
ADC
Display
Storage
Buffer
Clock
ADC
External
Computer
Controller
Trigger
Simultaneous DAQ
Simultaneous Sampling
Channel 1
Channel 2
Channel 3
Single channel
sample clock
– It can sample all channels at the same time (simultaneous or synchronous
sampling)
43
• Multiplexed sampling devices have a single ADC for all analog input channels
Digital Inputs
Process
feedback
Analog
Transducers
Signal
Conditioning
Display
Multiplexer
Buffer
Storage
ADC
Clock
External
Computer
Controller
Trigger
Multiplexed DAQ
Multiplexed sampling of 3 channels on a 4 channel DAQ device
Channel 1
Channel 2
Channel 3
Sample clock
Convert clock
81
6
Further Details
Sample DAQ Systems
Sampling clock
ADC
Signal conditioning
Multichannel sampling
Basic
none
single
none
multiplexed
44
Dedicated
exist
single
none
multiplexed
Highly Dedicated
exist
multiple
exist
simultaneous
Basic
• PIC
Dedicated
• Sound card
• Arduino
• MBed
– DC
???
coupling
• LabJack
• ...
– No anti aliasing
filter
Highly Dedicated
• Brüel & Kjær Multichannel Analysis
System Type 3550
• Many National Instruments products
as well
• ...
– No AC coupling
• NI USB-6251
– No anti aliasing
filter
– No AC coupling
• ...
82
Hardware Specifications
Check [3; sec. 11.2.4] for further topics as:
• Cross channel match
• Cross-channel talk
83
Build or buy DAQ System
• Of course it is possible to build a system based on general ADC components
– but once all necessary parts are put together, the price will likely be higher
than using dedicated hardware
– For example, a good system for noise and vibration applications, requires very
expensive components such as:
∗ highly accurate anti-aliasing filters
∗ matched between the channels, and
∗ electronics with relatively low noise floor
84
45
7
Problems
3.1. Study the specifications of the following multi-channel DAQ devices and describe
sys
(i) channels input range (EFSR
)
(ii) maximum number of analog input channels
(iii) single-ended or floating input channels
(iv) AC or DC coupling
(v) sampling rate/channel
(vi) aggregate (all channels) analog input throughput
(vii) whether there is anti-aliasing filter, or not
• In NI documents, the cutoff frequency of the low-pass filter is also called
the “small signal bandwidth”
(viii) available other signal conditioning modules; if any
(ix) ADC buffer size
(x) and whether they have simultaneous channel sampling, or not
(a)
(b)
(c)
(d)
B&K input module type 3015
NI-USB-6251-BNC
NI USB 6009
NI 9237 Strain gauge module
(e) NI 9234 dynamic signal acquisition
module
(f) NI myRIO-1900
3.2. A typical DAQ system has a 16-bit ADC and a low noise amplifier. The ADC range
is −1 ∼ 1 V. The amplifier has adjustable gains G = 0.1, 0.2, 0.5, 1, 10, 100 & 1000.
Calculate:
(a) Range of the ADC
(b) Resolution of the ADC
(c) Maximum range of the DAQ system
(d) Minimum rang of the DAQ system
(e) Maximum resolution of the DAQ system
(f) Minimum resolution of the DAQ system
(g) Determine the best amplifier gain to use for measuring a bipolar −10 ∼ 10 V
signal
(h) Do you need any additional circuit to improve the measurement of this signal?
(i) If yes, what is this circuit? Explain how/why it improves the measurement.
(j) Determine the best amplifier gain to use for measuring a unipolar 0 ∼ 10 V
signal
(k) Do you need any additional circuit to improve the measurement of this signal?
(l) If yes, what is this circuit? Explain how/why it improves the measurement.
46
(m) Determine the best amplifier gain to use for measuring a bipolar −10 ∼ 10 mV
signal
3.3. Create 1 s wav file with the sound of the telephone tone number 7
(a) Choose a suitable sampling frequency
(b) Calculate the bit rate of your file
(c) Plot one period of both the continuous and discretized sound signal
(d) Calculate the size of your file
(e) How can you reduce the file size?
i. Explain the advantages/drawbacks of your answer
(f) If you read your wav file, will there be any difference as compared to the original
data?
(g) Plot one period of both the original continuous and read sound signal
(h) Explain the difference if any
Hints
• Tone information are in http://en.wikipedia.org/wiki/Telephone_keypad
• “wavwrite” Matlab function can create wav files
• “wavread” Matlab function can read wav files
3.4. Plot a schematic chart showing the internal modules inside of sound card of your
PC.
85
47
Part 4
Dynamic Signal Analysis
1
Introduction
Signal Properties
Amplitude
PeakPeak
Peak
RMS
Average
Time
86
Signal Types
Stationary
Properties do not vary with time
Non-stationary
Properties vary with time
Deterministic
Random
Transient
Predictable
Described using
its statistical
properties
Starts and stops
within the
analysis time
Continuous
• Example real signals are in [2; sec. 1.2]
87
48
2
The Fourier Transform
The Fourier Transform of a Signal
The Fourier Integral
Z
∞
x(t) e−jωt dt ∀ω
,: ω ∈ R
(4.1)
x(t) e−j2πf t dt ∀f
,: f ∈ R
(4.2)
X(ω) =
−∞
or
Z
∞
X(f ) =
−∞
where ω = 2πf
• Signals, x(t), are a function of time (usually real data)
• Frequency (Fourier) spectrum, X(f ), is an equivalent representation of the signal
– Complex domain (Real, imaginary) or (Magnitude, phase)
The Inverse Fourier Transform
Z
∞
x(t) =
X(f )ej2πf t df
∀t
,: t ∈ R
(4.3)
−∞
Fourier Transform Pair
F
x(t) ⇐⇒ X(f )
88
49
Vi
ol
et
Physical Meaning of The Fourier Transform
ue
Bl
White light
n
ee
Gr
Yellow
Spectrum
ge
Oran
Red
Prism
Separation of white light into its color spectrum. Color corresponds to a particular
frequency or wavelength; light intensity corresponds to varying amplitudes
• Any signal can be represented as a summation of harmonic (sinusoidal) signals
89
50
Demonstrations
51
Gibb’s phenomenon
• Square wave
52
• Sawtooth wave
53
Matlab Illustration
• Study example 3.1 of [2]
Wolframe demonstrations (http://demonstrations.wolfram.com)
• Approximation of Discontinuous Functions by Fourier Series
90
Frequency Spectrum of a Signal
Input
Output
x(t)
y(t)
FFT
IFT
INPUT SPECTRUM
OUTPUT SPECTRUM
• Frequency spectrum of a signal can highlight characteristics of the signal
– numerous applications
• Transformation is done using Fourier Transform
– Frequency Spectrum ≡ Fourier Spectrum≡ Spectrum
91
54
Typical Applications of Fourier Analysis[??]
Signal Processing
Numerical Methods
Applied Mechanics
• matched filters
• high-speed interpolation
• structural dynamics
• deconvolution
• conjugate gradient method
• real-time spectral analysis
• aircraft wing-flutter suppression
• boundary value problems
• cepstrum analysis
• Riccati and Dirichlet equations
• machinery dynamics diagnostics
• coherence function estimation
• Rayleigh’s integral
• nuclear power plant modeling
• speech synthesis and recognition
• Wiener-Hopf integral equation
• vibration analysis
• random process generation
• diffusion equation
• transfer function estimation
• numerical integration
• echo/reverbation removal
• Karhunen-Loeve transform
• elliptic differential equations
Electromagnetics
• chromatography
• microscopy
• spectroscopy
• x-ray diffraction
• micros trip line propagation
Sonics And Acoustics
• conducting bodies scattering
• acoustic imaging
• antenna radiation patterns
• passive sonar
• dielectric substrate capacitance
• ultrasonic transducers
• phased-array antenna analysis
Instrumentation
• electrochronography
Radar
• array processing
• time-domain reflectometry
• architecture acoustic measurement
• waveguide analysis
• music synthesis
• network analysis
• cross-section measurement
• moving target indicator
• synthetic aperture
• doppler processor
• pulse compression
• clutter rejection
Communications
Biomedical Engineering
• diagnosis of airways obstruction
• muscle fatigue monitoring
• assessing heart valve damage
• tissue structure characterization
Miscellaneous
• systems analysis
• transmultiplexers
• demodulators
• speech scrambler system
• multichannel filtering
• M -ary signaling
• gastric disturbances investigation
• signal detection
• cardiac patients diagnosis
• high-speed digital filters
• ECG data compression
• voice coding systems
• artery dynamics investigation
• video bandwidth compression
• magnetotellurics
• metallurgy
• electrical power systems
• image restoration
• nonlinear system analysis
• geophysics
• GaAs FET transient response
• integrated circuit modeling
• quality control
92
Signal Processing is a Tool
• In this course, Signal Processing is no more than a tool used to extract the model
properties
55
– Understanding the physical meaning of the FT and its properties is the
goal
– Theoretical derivation of the FT and its properties is assumed trivial
∗ All details can be found for example in [2; sec. 4.3] or any other signal
processing textbook
– Check sample FT pairs in appendix A
93
56
Plotting FT pairs
94
57
3
Fourier Transform Properties
F
x(t) ⇐⇒ X(f )
F
a x(t) ⇐⇒ a X(f )
F
x(t) + y(t) ⇐⇒ X(f ) + Y (f )
F
X(t) ⇐⇒ x(−f )
f
1
F
X
x(kt) ⇐⇒
|k|
k
1
t
F
x
⇐⇒ X(k f )
|k|
k
F
x(t − t0 ) ⇐⇒ X(f ) e−j2πf t0
F
x(t) ej2πf0 t ⇐⇒ X(f − f0 )
1
F
[X(f + f0 ) + X(f − f0 )]
x(t) cos (2πf0 t) ⇐⇒
2
F
x(t) = xEven (t) ⇐⇒ X(f ) = XReal (f )
F
x(t) = xOdd (t) ⇐⇒ X(f ) = j Ximaginary (f )
F
x(t) = xReal (t) ⇐⇒ X(f ) = XEven (f ) + j XOdd (f )
58
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
Linearity; equation (4.6)
Symmetry; equation (4.7)
Scale
F
x(t) + y(t) ⇐⇒ X(f ) + Y (f )
Scale
Dt & Df = 2.5 mm
T & fs = 20mm
F Dt & Df = 2.5 mm
X(t) ⇐⇒T x(−f
)
& fs = 20mm
59
FT of even function; equation (4.13)
FT of odd function; equation (4.14)
F
F
x(t) = xEven (t) ⇐⇒ X(f ) = XReal (f )
x(t) = xOdd (t) ⇐⇒ X(f ) = j Ximaginary (f )
60
FT of real function; equation (4.15)
F
x(t) = xReal (t) ⇐⇒ X(f ) = XEven (f ) + j XOdd (f )
Time shift; equation (4.10)
F
x(t − t0 ) ⇐⇒ X(f ) e−j2πf t0
61
95
Demonstrations
Wolframe demonstrations (http://demonstrations.wolfram.com)
• Rectangular Pulse and Its Fourier Transform
96
Logarithmic Scale
5
x(t) (volts)
3
1
–1
–3
–5
0
0.005
0.01
0.015
0.02
t (seconds)
0.025
0.03
0.035
0.04
(a) Time history
|X( f )| (linear scale, volts / Hz)
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0
0.2 0.3
0.4 0.5
0.6 0.7
0.8 0.9
1
1.1 1.2
1.3 1.4
1.5 1.6 1.7 1.8
1.9
2
1.5 1.6 1.7 1.8
1.9
2
Frequency (kHz)
(b) Frequency components (linear scale)
|X( f )| (log scale, volts /Hz)
100
10–2
10–4
10–6
10–8
10–10
0
0.1
0.2 0.3
0.4 0.5
0.6 0.7
0.8 0.9
1
1.1 1.2
1.3 1.4
Frequency (kHz)
(c) Frequency components (log scale)
Measured telephone tone (No. 8) signal considered as periodic
62
• More information are in [3; Appendix B]
97
Assignment
• Solve problems (4.1); 4.1a to 4.1e
98
4
Convolution
Z
∞
x(τ ) y(t − τ ) dτ
x(t) ∗ y(t) ≡
(4.16)
−∞
Corollary 4.1.
x(t) ∗ y(t) = y(t) ∗ x(t)
(4.17)
Important for
t − tstart
t
• understanding the Discrete Fourier Transform
1
0
T
x(τ )
y(τ )
y(t − τ )
(x ∗ y)(t)
• dealing with Linear Time-Invariant (LTI) systems [2; sec. 4.7]
0.5
99
0
−8
−6
−4
−2
Graphical evaluation
t − tstart
t
1
0.5
0
2
4
6
8
τ
t
0
T
1
x(τ )
y(τ )
y(t − τ )
(x ∗ y)(t)
1
0.5
0.5
x(τ ) t
y(τ
y(τ ))
y(t −
x(τ
) τ)
(x ∗−y)(t)
x(t
τ)
t − tstart
t − t0start
T
0
T
(y ∗ x)(t)
0
0
−8
−6
−4
−2
0
2
4
6
8
τ
t
1
1
0.5
0.5
x(τ ) t
y(τ
y(τ ))
y(t
x(τ −
) τ)
(x
x(t∗−y)(t)
τ)
t − tstart
t − t0start
T
0
T
0
−8
−6
−4
−2
0
2
4
6
8
−8
−6
−4
−2
τ0
τ
2
4
6
8
t
1
1
0.5
0.5
(y ∗ x)(t)
x(τ )
y(τ
y(τ ))
y(t −
x(τ
) τ)
(x ∗−y)(t)
x(t
τ)
t
t − tstart
t − t0start
T
0
T
(y ∗ x)(t)
0
0
0
−8
−6
−4
−2
0
2
4
6
8
−8
−6
−4
−2
τ0
τ
2
4
6
8
t
1
1
0.5
0.5
x(τ )
y(τ
y(τ ))
y(t −
x(τ
) τ)
(x
x(t∗−y)(t)
τ)
t
t − tstart
t − t0start
T
0
T
0
−8
−6
−4
−2
0
2
4
6
8
−8
−6
−4
−2
τ0
τ
2
4
6
8
t
1
1
0.5
0.5
(y ∗ x)(t)
t
x(τ )
y(τ
y(τ ))
y(t −
x(τ
) τ)
(x
x(t∗−y)(t)
τ)
t − tstart
t − t0start
T
0
T
(y ∗ x)(t)
0
0
0
−8
−6
−4
−2
0
2
4
6
8
−8
−6
−4
−2
τ0
τ
2
4
6
8
t
1
1
0.5
0.5
t
x(τ )
y(τ
y(τ ))
y(t −
x(τ
) τ)
(x ∗−y)(t)
x(t
τ)
t − tstart
t − t0start
T
0
T
−4
−2
0
2
4
6
8
−6
−4
−2
τ0
τ
2
4
6
8
t − tstart
t
0.5
(y ∗ x)(t)
0
0
T
y(τ )
x(τ )
x(t − τ )
(y ∗ x)(t)
0
−6
−4
−2
0
2
4
6
8
−8
−6
−4
−2
τ0
τ
2
4
6
8
t − tstart
t
0.5
−6
−8
1
0
−8
1
0
−8
y(τ )
x(τ )
x(t − τ )
(y ∗ x)(t)
0
63
T
−8
−6
−4
−2
0
τ
2
4
6
8
0.5
y(τ )
y(t − τ )
(x ∗ y)(t)
0
−8
−6
−4
−2
0
2
4
6
8
τ
t − tstart
t
1
0.5
0
t
T
x(τ )
y(τ )
y(t − τ )
(x ∗ y)(t)
1
1
0.5
0.5
0
0
−8
−8
0
−8
−6
−4
−2
0
2
4
6
8
τ
t
1
1
0.5
0.5
0
0
−8
−8
t
x(τ )
y(τ ))
y(τ
x(τ −
) τ)
y(t
x(t∗−y)(t)
τ)
(x
(y ∗ x)(t)
−6
−6
t − tstart
t − tstart
0
0
−2
−2
0
τ0
2
2
4
4
6
6
x(τ )
y(τ ))
y(τ
x(τ
) τ)
y(t −
x(t∗−y)(t)
τ)
(x
(y ∗ x)(t)
−6
−6
−4
−4
−2
−2
1
1
0.5
0.5
0
0
−8
−8
t
x(τ )
y(τ ))
y(τ
x(τ
) τ)
y(t −
x(t∗−y)(t)
τ)
(x
(y ∗ x)(t)
−6
−6
t 0
0
1
1
0
0
−8
−8
8
8
−2
−2
0
τ0
τ
2
2
4
4
6
6
t − tstart
t − tstart
1
1
0.5
0.5
0
0
−8
−8
0 t
t 0
x(τ )
y(τ ))
y(τ
x(τ −
) τ)
y(t
x(t∗−y)(t)
τ)
(x
(y ∗ x)(t)
−6
−6
−2
−2
0
τ0
τ
t − tstart
t − tstart
1
1
0.5
0.5
0
0
−8
−8
0
0 t
x(τ )
y(τ ))
y(τ
x(τ −
) τ)
y(t
x(t
τ)
(x ∗−y)(t)
(y ∗ x)(t)
−6
−6
1
1
0
0
−8
−8
8
8
x(τ )
y(τ ))
y(τ
x(τ −
) τ)
y(t
x(t∗−y)(t)
τ)
(x
(y ∗ x)(t)
2
2
4
4
6
6
t
1
1
0
0
−8
−8
8
8
−2
−2
0
τ0
2
2
4
4
6
6
−6
−6
−4
−4
−2
−2
1
1
0.5
0.5
0
0
−8
−8
0
0
x(τ )
y(τ ))
y(τ
x(τ −
) τ)
y(t
x(t∗−y)(t)
τ)
(x
(y ∗ x)(t)
t
−6
−6
−4
−4
−2
−2
0
τ0
τ
0
τ0
τ
0
0 t
x(τ )
y(τ ))
y(τ
x(τ −
) τ)
y(t
x(t∗−y)(t)
τ)
(x
(y ∗ x)(t)
−6
−6
−4
−4
−2
−2
1
1
0
0
−8
−8
8
8
x(τ )
y(τ ))
y(τ
x(τ
) τ)
y(t −
x(t∗−y)(t)
τ)
(x
(y ∗ x)(t)
−6
−6
−4
−4
−2
−2
0
τ0
τ
2
2
4
4
6
6
8
8
T
T
2
2
4
4
6
6
t
t
0
2
2
8
8
T
T
2
2
0
0
τ0
τ
8
8
4
4
6
6
t
8
8
T
T
4
4
6
6
8
8
t − tstart
T
T
t
6
6
0 t
t 0
x(τ )
y(τ ))
y(τ
x(τ −
) τ)
y(t
x(t
τ)
(x ∗−y)(t)
(y ∗ x)(t)
τ
t − tstart
t − tstart
4
4
T
T
t − tstart
t − tstart
T
T
0.5
0.5
−4
−4
2
2
t 0
0
t − tstart
t − tstart
T
T
0.5
0.5
−4
−4
0
T
T
t − tstart
t − tstart
T
T
0.5
0.5
−4
−4
0
0
τ0
τ
t
τ
t − tstart
t − tstart
t − tstart
t − tstart
t − tstart
t − tstart
T
T
0.5
0.5
−4
−4
t
0
1
0.5
t
T
y(τ )
x(τ )
x(t − τ )
(y ∗ x)(t)
0
−6
−6
−4
−4
−2
−2
0
τ0
2
2
4
4
6
6
8
8
τ
−8
−6
−4
−2
0
2
4
6
8
τ
100
t − tstart
0
1
0.5
t
T
y(τ )
x(τ )
x(t − τ )
(y ∗ x)(t)
0
−8
−6
−4
−2
0
2
4
6
8
τ
64
Convolving with impulse function
101
Convolution Theorems
Theorems
Time convolution
F
x(t) ∗ y(t) ⇐⇒ X(f ) Y (f )
(4.18)
Frequency convolution
F
x(t) y(t) ⇐⇒ X(f ) ∗ Y (f )
(4.19)
Continuous LTI system
102
Assignment
• Solve problems (4.1); 4.1f to ??
103
65
5
Problems
Important: For any FT pair you sketch in the
following questions, make sure to:
• clearly write all the horizontal and vertical axes tick values on both time-domain
and frequency-domain curves; whenever possible.
• use the
and
symbols whenever applicable.
• If a frequency domain curve is real, write the vertical axis label similar to
“X(f ); X(f ) is real”. Otherwise, write the vertical axis label similar to “|X(f )|”.
4.1. Using the FT pairs in appendix A and the FT properties of section (3),
derive a mathematical expression and
sketch (magnitude of the FT is enough) the
(a) FT of
y(t) = A sin (2πf0 t)
(b) FT of
x(t) = A cos (2πf0 t + φ) + B
(c) FT of
x(t) = A sin (2πf0 t + φ) + B
(d) the IFT of the low pass filter
???
(e) IFT of
???
66
(f) IFT of band pass filter, where fc = 4fmax
???
(g) FT of
F
⇐⇒
???
104
67
Part 5
Appendices
A
Basic Fourier
Scale
Scale
Scale
Dt
& Df = 2.5 mm
Scale
Dt
=
Dt
&fsDf
Df
= 2.5
2.5 mm
mm
= 20mm
T
&&
Dt
&f Df
= 2.5 mm
=
20mm
T
&
Transform TTPairs
& fss = 20mm
& fs = 20mm
68
105
69
Part 6
Problem’s Answers
1
Part 2: Static Measurements
Problem (2.5)
• Load cells’ output is plotted in the following charts.
Load cell A
Load cell B
Load cell C
•Accurate and repeatable
• Not accurate but repeatable
• Seem to accurate but not usable
Max.
11.6
11.6
x
11.0
x
10.6
Output (mV)
Output (mV)
10.8
x
10.4
Ave.
x
x
10.0
x
x
11.2
x
10.2
x
x
x
x
x
x
x
11.4
Min.
11.2
11.0
11.0
10.8
10.8
10.6
Output (mV)
11.2
x
11.4
Max.
11.4
11.6
10.4
10.2
10.0
10.6
10.4
9.8
9.6
9.6
9.6
9.4
9.4
9.2
x
x
x
9.2
Min
9.2
9.0
x
9.0
0
1
2
3
4
5
6
7
8
9
10
x
10.0
9.8
9.4
Max.
10.2
x
x
x
x
x
x
x
x
x
8
9
10
9.8
0
1
2
3
4
5
6
7
8
9
10
Min.
9.0
0
1
Trial no.
Trial no.
2
3
4
5
6
7
Trial no.
• Inputs of the problem are summarized as
– x = 50 kg & ytrue = 10 mV
– xmax = 100 kg
• The load cell nominal sensitivity is calculated as
S=
10
= 0.2 mV/kg
50
and the FSO is calculated as
ro = S xmax = 20 mV
(Of course it is understood that xmin = 0 kg and ymin = 0 mV. Hence ro = ymax −
ymin = ymax = S xmax )
• Thus the accuracy and precision are calculated as
|ȳ − ytrue |
max (|y − ȳ|) = max (ymax − ȳ, ȳ − ymin )
70
Load cell
ymin
ȳ
ymax
Accuracy = |ȳ − ytrue |
Accuracy(%FSO)
ymax − ȳ
ȳ − ymin
Precision= max (|y − ȳ|)
Precision (%FSO)
A
9.02
10.09
11.2
0.09 mV
0.45%
1.11
1.07
1.11 mV
5.55%
B
11.42
11.49
11.58
1.49 mV
7.45%
0.09
0.07
0.09 mV
0.45%
C
9.92
9.99
10.08
0.01 mV
0.05%
0.09
0.07
0.09 mV
0.45%
Conclusion
A transducer or sensor that is repeatable but not overly accurate may still be quite
usable in a measurement or control application. As long as the transducer or sensor
is repeatable, you will get consistent results. We may correct this inaccuracy by the
recalibration this transducer or sensor
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