EE 1105: Introduction to EE
Freshman Seminar
Lecture 10: Introduction to signals
and systems
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Signals and Systems
– Signal:
• Any time dependent physical quantity
• Constant – DC, Variable - AC
• Electrical, Optical, Mechanical
– System:
• Object in which input signals interact to
produce output signals.
• Linear System has some have fundamental
properties that make it predictable:
– Sinusoid in, sinusoid out of same frequency
(when transients settle)
– Double the amplitude in, double the amplitude
out (when initial state conditions are zero)
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
u(t)
?
x(t)
y(t)
Signal Classification
– Continuous Time vs.
Discrete Time
• Telephone line
signals, Neuron
synapse potentials
• Stock Market, GPS
signals
– Analog vs. Digital
• Radio Frequency (RF)
waves, battery power
• Computer signals,
HDTV images
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Image Sources: Internet
• Unit Step Function u(t)
• Ramp function r(t)
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
• Below signal depends on independent variable t with
ω and  .
s= A Sin(ωt+)=A Im{ej(ωt+)}
parameters A,
A=Amplitude real
ω =angular frequency
 =phase
Sin( ~ )=function
ref: http://radarproblems.com/chapters/ch05.dir/ch05pr.dir/c05p1.dir/c05p1.htm
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Signal Classification
– Deterministic vs. Random
• FM Radio Signals
• Background Noise Speech
Signals
– Periodic vs. Aperiodic
• Sine wave
• Sum of sine waves with nonrational frequency ratio
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
System Classification
– Linear vs. Nonlinear
• Linear systems have the property of
superposition
d 2
– If U →Y, U1 →Y1, U2 →Y2 then
» U1+U2 → Y1+Y2
» A*U →A*Y
dt 2
• Nonlinear systems do not have this
property, and the I/O map is represented
by a nonlinear mapping.
d 2
dt
2

Exact Equation,
g
sin( )  0 nonlinear
L

g
 0
L
Approximation
around vertical
equilibrium, linear
– Examples: Diode, Dry Friction, Robot Arm
at High Speeds.
– Memoryless vs. Dynamical
• A memoryless system is represented
by a static (non-time dependent) I/O
map: Y=f(U).
– Example: Amplifier – Y=A*U, A- amplification
factor.
• A dynamical system is represented by
a time-dependent I/O map, usually a
differential equation:
– Example: dY/dt=A*u, Integrator with Gain A.
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Mandelbrot set, a fractal image,
result of a Nonlinear Discrete
System Zn+1=Zn²+C
System Classification
– Time-Invariant vs. Time Varying
• Time-invariant system parameters do not change over time. Example: pendulum, low
power circuit
• Time-varying systems perform differently over time. Example: human body during
exercise.
– Causal vs. Non-Causal
• For a causal system, outputs depend on past inputs but not future inputs.
Examples: most engineered and natural systems
• A non-causal system, outputs depend on future inputs. Example: computer
simulation where we know the inputs a-priori, digital filter with known images or
signals.
– Stable vs. Unstable
• For a stable system the output to bounded inputs is also bounded. Example:
pendulum at bottom equilibrium
• For an unstable system the ouput diverges to infinity or to values causing
permanent damage. Example: short circuit on AC line.
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
An electronic amplifier is a device for increasing the
power of a signal.
Amplifier
It does this by taking energy from a power supply and
controlling the output to match the input signal shape but
with a larger amplitude.
There are various types of amplifier.
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
A time shifter system shifts the function f(t) forward
or backward by a specific time.
Time shifter
t
t
f(t)
f(t – t0)
The above system is a forward time shifter. It adds
a delay (t0) to the signal.
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
sampling is the reduction of a continuous-time signal to a
discrete-time signal
Sampler
The sampling frequency must be higher than the frequency
of the signal to be sampled. (minimum twice as high)
ref: http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
An analog-to-digital converter (ADC, A/D) is a device that converts a
continuous quantity to a discrete time digital representation.
Analog to
Digital
\
The system that does the opposite is called DAC
DAC
ref: http://pictureofgoodelectroniccircuit.blogspot.com/2010/04/phase-and-function-of-analog-signal-or.html
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Conversion of Analog to digital is done in two step.
Sampler
A/D
Continuous analog Sampled signal
Sampled signal  Quantized digital signal
ref: http://en.wikipedia.org/wiki/Quantization_%28signal_processing%29
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
A low-pass filter is a filter that passes low-frequency
signals but attenuates (reduces the amplitude of)
signals with frequencies higher than the cutoff
frequency.
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3.5
3.5
3
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2.5
2.5
2
Low Pass
Filter
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1
0.5
2
1.5
1
0.5
0
0
-0.5
-1
-0.5
0
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ref: http://en.wikipedia.org/wiki/Low-pass_filter
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
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A high-pass filter (HPF) is a device that passes high
frequencies and attenuates (i.e., reduces the amplitude
of) frequencies lower than its cutoff frequency.
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3.5
3.5
3
3
2.5
2.5
2
Low Pass
Filter
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0.5
1.5
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-1
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ref: http://en.wikipedia.org/wiki/High-pass_filter
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
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A band-pass filter is a device that passes frequencies
within a certain range and rejects (attenuates)
frequencies outside that range.
3.5
7
3
6
2.5
5
2
4
Band Pass
Filter
3
2
1.5
1
1
0.5
0
0
-1
-0.5
0
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ref: http://en.wikipedia.org/wiki/Band-pass_filter
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
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band-stop filter or band-rejection filter is a filter that
passes most frequencies unaltered, but attenuates
those in a specific range to very low levels
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4
Band –stop
Filter
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2
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1
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ref: http://en.wikipedia.org/wiki/Band-stop_filter
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
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System Modeling
• Building mathematical models based on
observed data, or other insight for the system.
– Parametric models (analytical): ODE, PDE
– Non-parametric models: graphical models plots, look-up cause-effect tables
– Mental models – Driving a car and using the
cause-effect knowledge
– Simulation models – Many interconnect
subroutines, objects in video game
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Types of Models
• White Box
– derived from first principles laws: physical,
chemical, biological, economical, etc.
– Examples: RLC circuits, MSD mechanical
models (electromechanical system models).
• Black Box
– model is entirely derived from measured data
– Example: regression (data fit)
• Gray Box – combination of the two
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
White Box Systems: Electrical
• Defined by Electro-Magnetic Laws of
Physics: Ohm’s Law, Kirchoff’s Laws,
Maxwell’s Equations
• Example: Resistor, Capacitor, Inductor
i
i
i
u
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
L
C
R
u
u
Physics of an Inductor
Core flux, f
Coil current, i
i
+
l
Flux
linkage, l
Coil of
N turns
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
f l
m
N
A
Truncated hollow
cylinder of
permeability, m,
area, A, and
length lm.
Voltage Drop Across Inductor
i
L
+ v + l -
Note Passive Sign
Convention
d l d  m N 2 Ai  m N 2 A di
v
 

dt dt  lm 
lm dt
m N2A
di
L
vL
lm
dt
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Physics of a Capacitor
Plate area, A
Plate separation
distance, g
g
i, q
A
e
Current, i, and
Charge, q.
+
v
-
Dielectric material of
permittivity, e.
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Voltage
across
plates
Physics of a Capacitor
dq d  e Av  e A dv
 

Current: i 

dt dt  g  g dt
eA
C
Capacitance:
g
i, q
Note Passive
Sign Convention:
C
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
+
v
-
dv
iC
dt
Inductors in Series
i
L1
L2
+ v1 - + v2 +
v
-
di
di
v  v1  v2  L1  L2
dt
dt
di
di
v   L1  L2   Leq
dt
dt
Leq  L1  L2
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Capacitors in Parallel
i
C1
i1
C2
i2 +
v
-
dv
dv
i  i1  i2  C1  C2
dt
dt
dv
dv
i   C1  C2   Ceq
dt
dt
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Ceq  C1  C2
Inductors in Parallel
i
i1
+
l
-
i  i1  i2 
l
L1

i2
L1
l
L2
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015

l
Leq
L2
1
1 1
 
Leq L1 L2
Capacitors in Series
q C1
q
C2 q
+ v2 + v1 +
v
-
q
q
q
v  v1  v2  

C1 C2 Ceq
1
1
1
 
Ceq C1 C2
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
RLC Circuit as a System
R
u
u
1
u
u
2
C
u(t)
RLC
q(t)
3
L
Kirchoff’s Voltage Law (KVL):
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
i(t)
White Box Systems: Mechanical
K
B
M
F
Newton’s Law:
Mechanical-Electrical Equivalance:
F (force) ~V (voltage)
x (displacement) ~ q (charge)
M (mass) ~ L (inductance)
B (damping) ~ R (resistance)
1/K (compliance) ~ C (capacitance)
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
F(t)
MSD
x(t)
x(t)
White-Box vs. Black-Box Models
ω_r(t), ω_l(t)
Lawn
Mower
x,y,θ
X(t), Y(t) Θ(t)
Newton-Euler Law:
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Image Sources: Internet
Grey-Box Models
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Image Sources: Internet
White Box vs Black Box Models
White Box Models
Black-Box Models
Information Source
First Principle
Experimentation
Advantages
Good Extrapolation
Short time to develop
Good understanding
Little domain expertise
High reliability, scalability required
Works for not well
understood systems
Disadvantages
Time consuming and
detailed domain
expertise required
Not scalable, data
restricts accuracy, no
system understanding
Application Areas
Planning, Construction,
Design, Analysis, Simple
Systems
Complex processes
Existing systems
Start to understand simple white continuous time models which are linear
Eventually deal with grey-box or black-box models in real-life
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Application Areas for Systems Thinking
• Classical circuits & systems (1920s – 1960s) (transfer
functions, state-space description of systems).
• First engineering applications: military - aerospace
1940’s-1960s
• Transitioned from specialized topic to ubiquitous in
1980s with EE applications to:
– Electronic circuit design
– Signal and image processing
• Networks (wired, wireless), imaging, radar, optics.
– Control of dynamical systems
• Feedback control, prediction/estimation/identification of systems, robotics,
micro and nano systems
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Diagram Representation of Systems
Top
Middle
Bottom 1
Bottom 2
Bottom 3
Hierarchical Diagram: Organizations
Graph
Node 1
Graph
Node 2
Graph
Node 4
Graph
Node 3
Graph
Node 5
Undirected Graph: Networks
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Flowchart: Procedures, Software
System Simulation Software
• Matlab Simulink
– http://www.mathworks.com/support/2010b/sim
ulink/7.6/demos/sl_env_intro_web.html
• National Instruments Labview
– http://www.ni.com/gettingstarted/labviewbasic
s/environment.htm
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
EE-Specific Diagrams
•
Block Diagram Model:
–
–
–
–
Helps understand flow of information (signals) through a complex system
Helps visualize I/O dependencies
Equivalent to a set of linear algebraic equations.
U
Based on a set of primitives:
U2
U(s)
Y(s)
H(s)
Transfer Function
•
U1
+
U1+U2
+
Summer/Difference
Signal Flow Graph (SFG):
– Directed Graph alternative
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
U
Pick-off point
U
EE-Specific Diagrams: Signal Flow
Graph (SFG – Directed Graph)
2-port circuit SFG
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Multi-loop Control SFG
Image Sources: Internet
Integrator and Low Pass Filter
from http://www.electronics-tutorials.ws/rc/rc_3.html
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Differentiator and High Pass Filter
from http://www.electronics-tutorials.ws/rc/rc_3.html
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
EE 1106 Lab 9 Circuits
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
Acknowledgemengts:
Dr. Bill Dillon, Dr. Kambiz Alavi, UTA
Next Time:
Homework 8 due
Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015
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