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Transcript of Signals and System 6.003Fall 2003
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1
1) Administrative details
2) Signals3) Systems
4) For examples ...“Figures and images used in these lecture notes by permission,
copyright 1997 by Alan V. Oppenheim and Alan S. Willsky”
Signals and SystemsFall 2003
Lecture #1
Prof. Alan S. Willsky
4 September 2003
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THE INDEPENDENT VARIABLES
• Can be continuous — Trajectory of a space shuttle
— Mass density in a cross-section of a brain
• Can be discrete
— DNA base sequence
— Digital image pixels
• Can be 1-D, 2-D, ••• N-D
• For this course: Focus on a single (1-D) independent variable
which we call “time”.
Continuous-Time (CT) signals: x(t ), t — continuous values
Discrete-Time (DT) signals: x[n], n — integer values only
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CT Signals
• Most of the signals in the physical world are CTsignals—E.g. voltage & current, pressure,
temperature, velocity, etc.
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Many human-made DT Signals
Ex.#1 Weekly Dow-Jones
industrial average
Why DT? — Can be processed by modern digital computers
and digital signal processors (DSPs).
Ex.#2 digital image
Courtesy of Jason Oppenheim.
Used with permission.
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7
SYSTEMS
For the most part, our view of systems will be from an
input-output perspective:
A system responds to applied input signals, and its response
is described in terms of one or more output signals
x(t ) y(t )CT System
DT System x[n] y[n]
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• An RLC circuit
• Dynamics of an aircraft or space vehicle
• An algorithm for analyzing financial and economic
factors to predict bond prices
• An algorithm for post-flight analysis of a space launch
• An edge detection algorithm for medical images
EXAMPLES OF SYSTEMS
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SYSTEM INTERCONNECTIOINS
• An important concept is that of interconnecting systems
— To build more complex systems by interconnecting
simpler subsystems
— To modify response of a system
• Signal flow (Block) diagram
Cascade
Feedback
Parallel +
+
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SYSTEM EXAMPLES
x(t ) y(t )CT System DT System x[n] y[n]
Ex. #1 RLC circuit
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Force Balance:
Observation: Very different physical systems may be modeled
mathematically in very similar ways.
Ex. #2 Mechanical system
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Ex. #3 Thermal system
Cooling Fin in Steady State
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Ex. #3 (Continued)
Observations
• Independent variable can be something other than
time, such as space.
• Such systems may, more naturally, have boundary
conditions, rather than “initial” conditions.
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Ex. #4 Financial system
Observation: Even if the independent variable is time, there
are interesting and important systems which have boundary
conditions.
Fluctuations in the price of zero-coupon bonds
t = 0 Time of purchase at price y0
t = T Time of maturity at value yT
y(t) = Values of bond at time t
x(t) = Influence of external factors on fluctuations in bond price
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• A rudimentary “edge” detector
• This system detects changes in signal slope
Ex. #5
0 1 2 3
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Observations
1) A very rich class of systems (but by no means all systems of
interest to us) are described by differential and difference
equations.2) Such an equation, by itself, does not completely describe the
input-output behavior of a system: we need auxiliary
conditions (initial conditions, boundary conditions).
3) In some cases the system of interest has time as the natural
independent variable and is causal. However, that is not
always the case.
4) Very different physical systems may have very similar
mathematical descriptions.
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SYSTEM PROPERTIES
(Causality, Linearity, Time-invariance, etc.)
• Important practical/physical implications
• They provide us with insight and structure that we
can exploit both to analyze and understand systemsmore deeply.
WHY ?
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CAUSALITY
• A system is causal if the output does not anticipate future
values of the input, i.e., if the output at any time depends
only on values of the input up to that time.
• All real-time physical systems are causal, because time
only moves forward. Effect occurs after cause. (Imagine
if you own a noncausal system whose output depends on
tomorrow’s stock price.)
• Causality does not apply to spatially varying signals. (Wecan move both left and right, up and down.)
• Causality does not apply to systems processing recordedsignals, e.g. taped sports games vs. live broadcast.
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• Mathematically (in CT): A system x(t ) → y(t ) is causal if
CAUSALITY (continued)
when x1(t ) → y1(t ) x2(t ) → y2(t )
and x1(t ) = x2(t ) for all t ≤ t o
Then y1(t ) = y2(t ) for all t ≤ t o
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CAUSAL OR NONCAUSAL
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TIME-INVARIANCE (TI)
• Mathematically (in DT): A system x[n] → y[n] is TI if for
any input x[n] and any time shift n0,
Informally, a system is time-invariant (TI) if its behavior does not
depend on what time it is.
• Similarly for a CT time-invariant system,
If x[n] → y[n]
then x[n - n0] → y[n - n0] .
If x(t ) → y(t )
then x(t - t o)→
y(t - t o) .
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TIME-INVARIANT OR TIME-VARYING ?
TI
Time-varying (NOT time-invariant)
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NOW WE CAN DEDUCE SOMETHING!
These are the
same input!
Fact: If the input to a TI System is periodic, then the output is
periodic with the same period.
“Proof”: Suppose x(t + T ) = x(t )
and x(t ) → y(t )
Then by TI
x(t + T ) → y(t + T ).
↑ ↑
So these must be
the same output,
i.e., y(t ) = y(t + T ).
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LINEAR AND NONLINEAR SYSTEMS
• Many systems are nonlinear. For example: many circuit
elements (e.g., diodes), dynamics of aircraft, econometric
models,…
• However, in 6.003 we focus exclusively on linear systems.
• Why?
• Linear models represent accurate representations of behavior of many systems (e.g., linear resistors,
capacitors, other examples given previously,…)
• Can often linearize models to examine “small signal” perturbations around “operating points”
• Linear systems are analytically tractable, providing basis
for important tools and considerable insight
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A (CT) system is linear if it has the superposition property:
If x1(t ) → y1(t ) and x2(t ) → y2(t )
then ax1(t ) + bx2(t ) → ay1(t ) + by2(t )
LINEARITY
y[n] = x2[n] Nonlinear, TI, Causal
y(t ) = x(2t ) Linear, not TI, Noncausal
Can you find systems with other combinations ?- e.g. Linear, TI, Noncausal
Linear, not TI, Causal
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PROPERTIES OF LINEAR SYSTEMS
• Superposition
If
Then
• For linear systems, zero input → zero output
"Proof" 0 = 0 ⋅ x[n]→ 0 ⋅ y[n]= 0
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LINEAR TIME-INVARIANT (LTI) SYSTEMS
• Focus of most of this course
- Practical importance (Eg. #1-3 earlier this lectureare all LTI systems.)
- The powerful analysis tools associatedwith LTI systems
• A basic fact: If we know the response of an LTIsystem to some inputs, we actually know the response
to many inputs
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Example: DT LTI System
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Signals and SystemsFall 2003
Lecture #3
11 September 2003
1) Representation of DT signals in terms of shifted unit samples
2) Convolution sum representation of DT LTI systems
3) Examples4) The unit sample response and properties
of DT LTI systems
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Exploiting Superposition and Time-Invariance
Question: Are there sets of “basic” signals so that:
a) We can represent rich classes of signals as linear combinations of
these building block signals.
b) The response of LTI Systems to these basic signals are both simple
and insightful.
Fact: For LTI Systems (CT or DT) there are two natural choices for
these building blocks
Focus for now: DT Shifted unit samples
CT Shifted unit impulses
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Representation of DT Signals Using Unit Samples
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That is ...
Coefficients Basic Signals
The Sifting Property of the Unit Sample
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DT System x[n] y[n]
• Suppose the system is linear, and define hk [n] as the
response to δ [n - k ]:
From superposition:
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DT System x[n] y[n]
• Now suppose the system is LTI, and define the unit
sample response h[n]:
From LTI:
From TI:
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Convolution Sum Representation of
Response of LTI Systems
Interpretation
n n
n n
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Visualizing the calculation of
y[0] = ∑ prod of
overlap for
n = 0
y[1] = ∑ prod of
overlap for
n = 1
Choose value of n and consider it fixed
View as functions of k with n fixed
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Calculating Successive Values: Shift, Multiply, Sum
-11 × 1 = 1
(-1) × 2 + 0 × (-1) + 1 × (-1) = -3
(-1) × (-1) + 0 × (-1) = 1
(-1) × (-1) = 1
4
0 × 1 + 1 × 2 = 2
(-1) × 1 + 0 × 2 + 1 × (-1) = -2
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Properties of Convolution and DT LTI Systems
1) A DT LTI System is completely characterized by its unit sample
response
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Unit Sample response
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The Commutative Property
Ex: Step response s[n] of an LTI system
“input” Unit Sample response
of accumulator
step
input
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The Distributive Property
Interpretation
The Associative Property
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The Associative Property
Implication (Very special to LTI Systems)
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Properties of LTI Systems
1) Causality ⇔
2) Stability ⇔
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Signals and SystemsFall 2003
Lecture #4
16 September 2003
1. Representation of CT Signals in terms of shifted unit impulses
2. Convolution integral representation of CT LTI systems
3. Properties and Examples
4. The unit impulse as an idealized pulse that is
“short enough”: The operational definition of δ (t)
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Representation of CT Signals
• Approximate any input x(t ) as a sum of shifted, scaled
pulses
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has unit area
The Sifting Property of the Unit Impulse
Response of a CT LTI System
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Response of a CT LTI System
LTI⇒
Operation of CT Convolution
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Example: CT convolution
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-1
-1 0
0 1
1 2
2
PROPERTIES AND EXAMPLES
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PROPERTIES AND EXAMPLES
1) Commutativity:
2)
4) Step response:
3) An integrator:
S
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DISTRIBUTIVITY
ASSOCIATIVITY
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ASSOCIATIVITY
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The impulse as an idealized “short” pulse
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Consider response from initial rest to pulses of different shapes and
durations, but with unit area. As the duration decreases, the responses
become similar for different pulse shapes.
p p
The Operational Definition of the Unit Impulse (t)
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The Operational Definition of the Unit Impulse δ(t )
δ(t ) — idealization of a unit-area pulse that is so short that, for
any physical systems of interest to us, the system responds
only to the area of the pulse and is insensitive to its duration
Operationally: The unit impulse is the signal which when
applied to any LTI system results in an output equal to theimpulse response of the system. That is,
— δ(t ) is defined by what it does under convolution.
The Unit Doublet — Differentiator
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The Unit Doublet Differentiator
Impulse response = unit doublet
The operational definition of the unit doublet:
Triplets and beyond!
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Triplets and beyond!
n is number of
differentiations
Integrators
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“-1 derivatives" = integral⇒ I.R. = unit step
Integrators (continued)
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g ( )
Notation
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Define
Then
E.g.
Sometimes Useful Tricks
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Differentiate first, then convolve, then integrate
Example
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1 21 2
Example (continued)
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Signals and SystemsFall 2003
Lecture #5
18 September 2003
1. Complex Exponentials as Eigenfunctions of LTI Systems
2. Fourier Series representation of CT periodic signals
3. How do we calculate the Fourier coefficients?
4. Convergence and Gibbs’ Phenomenon
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Signals & Systems, 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 1997, p. 179.
Portrait of Jean Baptiste Joseph Fourier
Image removed due to copyright considerations.
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The eigenfunctions φk ( t ) and their properties
(Focus on CT systems now but results apply to DT systems as well )
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(Focus on CT systems now, but results apply to DT systems as well.)
eigenvalue eigenfunction
Eigenfunction in → same function out with a “gain”
From the superposition property of LTI systems:
Now the task of finding response of LTI systems is to determine λk .
Complex Exponentials as the Eigenfunctions of any LTI Systems
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eigenvalue eigenfunction
eigenvalue eigenfunction
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DT:
What kinds of signals can we represent as
“sums” of complex exponentials?
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sums of complex exponentials?
For Now: Focus on restricted sets of complex exponentials
CT & DT Fourier Series and Transforms
CT:
DT:
⇓
Magnitude 1
Periodic Signals
Fourier Series Representation of CT Periodic Signals
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ω o =2π
T
- smallest such T is the fundamental period
- is the fundamental frequency
- periodic with period T
- ak are the Fourier (series) coefficients
- k = 0 DC
- k = ±1 first harmonic
- k = ±2 second harmonic
Question #1: How do we find the Fourier coefficients?
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First, for simple periodic signals consisting of a few sinusoidal terms
0 – no dc component
0
0
Euler's relation
(memorize!)
• For real periodic signals, there are two other commonly used
forms for CT Fourier series:
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• Because of the eigenfunction property of e jω t , we will usually
use the complex exponential form in 6.003.
- A consequence of this is that we need to include terms for
both positive and negative frequencies:
Now, the complete answer to Question #1
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Convergence of CT Fourier Series
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• How can the Fourier series for the square wave possibly makesense?
• The key is: What do we mean by
• One useful notion for engineers: there is no energy in the
difference
(just need x(t ) to have finite energy per period)
Under a different, but reasonable set of conditions
(the Dirichlet conditions)
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Condition 1. x(t ) is absolutely integrable over one period, i. e.
Condition 3. In a finite time interval, x(t ) has only a finite
number of discontinuities.
Ex. An example that violates
Condition 3.
And
Condition 2. In a finite time interval, x(t ) has a finite number
of maxima and minima.
Ex. An example that violates
Condition 2.
And
• Dirichlet conditions are met for the signals we will
encounter in the real world. Then
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- The Fourier series = x(t ) at points where x(t ) is continuous
- The Fourier series = “midpoint” at points of discontinuity
- As N → ∞, x N (t ) exhibits Gibbs’ phenomenon at points of discontinuity
Demo: Fourier Series for CT square wave (Gibbs phenomenon).
• Still, convergence has some interesting characteristics:
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Signals and SystemsFall 2003
Lecture #6
23 September 2003
1. CT Fourier series reprise, properties, and examples
2. DT Fourier series
3. DT Fourier series examples and
differences with CTFS
CT Fourier Series Pairs
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Skip it in future
for shorthand
Another (important!) example: Periodic Impulse Train
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— All components have:
(1) the same amplitude,
&
(2) the same phase.
(A few of the) Properties of CT Fourier Series
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• Linearity
Introduces a linear phase shift ∝ t o
• Conjugate Symmetry
• Time shift
Example: Shift by half period
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• Parseval’s Relation
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Energy is the same whether measured in the time-domain or thefrequency-domain
• Multiplication Property
Periodic Convolution
x(t ), y(t ) periodic with period T
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Periodic Convolution (continued)
P i di l ti I t t i d ( T/2 t T/2)
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Periodic convolution: Integrate over any one period (e.g. -T /2 to T /2)
Periodic Convolution (continued) Facts
1) z (t ) is periodic with period T (why?)
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2) Doesn’t matter what period over which we choose to integrate:
3)
Periodic
convolution
in time
Multiplication
in frequency!
Fourier Series Representation of DT Periodic Signals
• x[n] - periodic with fundamental period N , fundamental frequency
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• Only e jω n which are periodic with period N will appear in the FS
• So we could just use
• However, it is often useful to allow the choice of N consecutive
values of k to be arbitrary.
⇓
• There are only N distinct signals of this form
DT Fourier Series Representation
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= Sum over any N consecutive values of k
k =< N >
∑
— This is a finite series
ak - Fourier (series) coefficients
Questions:
1) What DT periodic signals have such a representation?
2) How do we find ak ?
Answer to Question #1:
Any DT periodic signal has a Fourier series representation
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A More Direct Way to Solve for ak
Finite geometric series
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So, from
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Example #1: Sum of a pair of sinusoids
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0
1/2
1/2
e jπ /4/2
e-jπ /4/2
0
0
a-1+16= a-1 = 1/2
a2+4×16= a2 = e jπ /4/2
Example #2: DT Square Wave
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Using n = m - N 1
Example #2: DT Square wave (continued)
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Convergence Issues for DT Fourier Series:
Not an issue, since all series are finite sums.
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Properties of DT Fourier Series: Lots, just as with CT Fourier Series
Example:
Si l d S t
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Signals and SystemsFall 2003
Lecture #7
25 September 2003
1. Fourier Series and LTI Systems
2. Frequency Response and Filtering
3. Examples and Demos
The Eigenfunction Property of Complex Exponentials
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DT:
CT:
CT"System Function"
DT"System Function"
Fourier Series: Periodic Signals and LTI Systems
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The Frequency Response of an LTI System
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CT notation
Frequency Shaping and Filtering
• By choice of H(jω ) (or H(e jω
)) as a function of ω , we can shapethe frequency composition of the output
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the frequency composition of the output
- Preferential amplification
- Selective filtering of some frequencies
Example #1: Audio System
Adjustable
FilterEqualizer Speaker
Bass, Mid-range, Treble controls
For audio signals, the amplitude is much more important than the phase.
Example #2: Frequency Selective Filters
L Fil
— Filter out signals outside of the frequency range of interest
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Lowpass Filters:
Only show
amplitude here.
lowfrequency
lowfrequency
Highpass Filters
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Remember:
high
frequency
high
frequency
Bandpass Filters
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Demo: Filtering effects on audio signals
Idealized Filters
CT
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ωc — cutoff
frequency
DT
Note: | H | = 1 and ∠ H = 0 for the ideal filters in the passbands,
no need for the phase plot.
Highpass
CT
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DT
Bandpass
CT
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DT
lower cut-off upper cut-off
Example #3: DT Averager/Smoother
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LPF
FIR (Finite Impulse
Response) filters
Example #4: Nonrecursive DT ( FIR) filters
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Rolls off at lower
ω as M+N+1
increases
Example #5: Simple DT “Edge” Detector
— DT 2-point “differentiator”
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Passes high-frequency components
Demo: DT filters, LP, HP, and BP applied to DJ Industrial average
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Example #6: Edge enhancement using DT differentiator
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Courtesy of Jason Oppenheim.
Used with permission.
Courtesy of Jason Oppenheim.
Used with permission.
Example #7: A Filter Bank
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Demo: Apply different filters to two-dimensional image signals.
HPFace of a monkey.
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Note: To really understand these examples, we need to understand
frequency contents of aperiodic signals⇒ the Fourier Transform
LP
BP
BP
LP
HP
Image removed do to
copyright considerations
Signals and Systems
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g yFall 2003
Lecture #8
30 September 2003
1. Derivation of the CT Fourier Transform pair
2. Examples of Fourier Transforms
3. Fourier Transforms of Periodic Signals4. Properties of the CT Fourier Transform
Fourier’s Derivation of the CT Fourier Transform
• x(t ) - an aperiodic signal
view it as the limit of a periodic signal as T → ∞
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- view it as the limit of a periodic signal as T → ∞
• For a periodic signal, the harmonic components arespaced ω0 = 2π/T apart ...
• As T → ∞, ω 0 → 0, and harmonic components are spaced
closer and closer in frequency
⇓
Fourier series Fourier integral ⎯ → ⎯
Motivating Example: Square wave
increases
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Discrete
frequency
points become
denser in
ω as T
increases
kept fixed
So, on with the derivation ...
For simplicity, assume
(t) h fi it d ti
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x(t ) has a finite duration.
Derivation (continued)
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Derivation (continued)
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a) Finite energy
For what kinds of signals can we do this?
(1) It works also even if x(t ) is infinite duration, but satisfies:
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In this case, there is zero energy in the error
E.g. It allows us to consider FT for periodic signals
c) By allowing impulses in x(t ) or in X(jω ), we can represent
even more signals
b) Dirichlet conditions
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Example #2: Exponential function
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Even symmetry Odd symmetry
Example #3: A square pulse in the time-domain
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Useful facts about CTFT’s
Note the inverse relation between the two widths ⇒ Uncertainty principle
Example #4: x(t ) = e−at 2 — A Gaussian, important in
probability, optics, etc.
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Also a Gaussian! Uncertainty Principle! Cannot make
both ∆t and ∆ω arbitrarily small.
(Pulse width in t )•(Pulse width in ω)
⇒ ∆t• ∆ω ~ (1/a1/2)•(a1/2) = 1
CT Fourier Transforms of Periodic Signals
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— periodic in t withfrequency ωo
— All the energy is
concentrated in one
frequency — ωo
Example #4:
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“Line spectrum”
— Sampling functionExample #5:
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Same function in
the frequency-domain!
Note: (period int ) T
⇔ (period in ω) 2π/T
Inverse relationship again!
Properties of the CT Fourier Transform
1) Linearity
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2) Time Shifting
FT magnitude unchanged
Linear change in FT phase
Properties (continued)
3) Conjugate Symmetry
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Even
Odd
Even
Odd
The Properties Keep on Coming ...
4) Time-Scaling
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a) x(t ) real and even
b) x(t ) real and odd
c)
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The CT Fourier Transform Pair
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Last lecture: some properties
Today: further exploration
(Synthesis Equation)
(Analysis Equation)
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The Frequency Response Revisited
impulse response
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The frequency response of a CT LTI system is simply the Fourier
transform of its impulse response
Example #1:
frequency response
Example #2: A differentiator
Differentiation property:
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1) Amplifies high frequencies (enhances sharp edges)
Larger at high ωo phase shift
Example #3: Impulse Response of an Ideal Lowpass Filter
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2) What is h(0)?
No.
Questions:
1) Is this a causal system?
3) What is the steady-state value of
the step response, i.e. s(∞)?
Example #4: Cascading filtering operations
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H(jω)
Example #5:
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Gaussian × Gaussian = Gaussian⇒ Gaussian ∗ Gaussian = Gaussian
Example #6:
Example #2 from last lecture
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Example #7:
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Example #8: LTI Systems Described by LCCDE’s
(Linear-constant-coefficient differential equations)
Using the Differentiation Property
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Using the Differentiation Property
1) Rational, can use
PFE to get h(t )
2) If X ( jω ) is rationale.g.
then Y ( jω ) is also rational
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Examples of the Multiplication Property
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⇓For any s(t) ...
Example (continued)
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The Discrete-Time Fourier Transform
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DTFT Derivation (Continued)
DTFS synthesis eq.
DTFS analysis eq.
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Define
DTFT Derivation (Home Stretch)
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Signals and SystemsFall 2003
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Lecture #10
7 October 2003
1. Examples of the DT Fourier Transform
2. Properties of the DT Fourier Transform
3. The Convolution Property and its
Implications and Uses
DT Fourier Transform Pair
Analysis Equation
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– Analysis Equation
– FT
– Synthesis Equation – Inverse FT
Convergence Issues
Synthesis Equation: None, since integrating over a finite interval
Analysis Equation: Need conditions analogous to CTFT, e.g.
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— Absolutely summable
— Finite energy
ExamplesParallel with the CT examples in Lecture #8
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More Examples
Infinite sum formula
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Still More
4) DT Rectangular pulse (Drawn for N 1 = 2)
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5)
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DTFTs of Sums of Complex Exponentials
Recall CT result:
What about DT:
a) We expect an impulse (of area 2π) at ω = ω
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Note: The integration in the synthesis equation is over 2π period,
only need X (e jω ) in one 2π period. Thus,
a) We expect an impulse (of area 2π) at ω = ω
o b) But X (e jω ) must be periodic with period 2π
In fact
DTFT of Periodic Signals
DTFSsynthesis eq.
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Linearity
of DTFT
Example #1: DT sine function
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Example #2: DT periodic impulse train
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— Also periodic impulse train – in the frequency domain!
Properties of the DT Fourier Transform
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— Different from CTFT
More Properties
— Important implications in DT because of periodicity
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Example
Still More Properties
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Yet Still More Properties
7) Time Expansion
Recall CT property:
Time scale in CT is
infinitely fine
But in DT: x[n/2] makes no sense
x[2n] misses odd values of x[n]
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Insert two zeros
in this example
(k=3)
But we can “slow” a DT signal down by inserting zeros:k — an integer ≥ 1
x(k )[n] — insert (k - 1) zeros between successive values
Time Expansion (continued)
— Stretched by a factor
of k in time domain
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-compressed by a factor
of k in frequency domain
Is There No End to These Properties?
8) Differentiation in Frequency
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Total energy in
time domain
Total energy in
frequency domain
9) Parseval’s Relation
Differentiation
in frequency
Multiplication
by n
The Convolution Property
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Example #1:
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Example #3:
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Signals and SystemsFall 2003
L #11
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Lecture #11
9 October 2003
1. DTFT Properties and Examples
2. Duality in FS & FT
3. Magnitude/Phase of Transforms
and Frequency Responses
Convolution Property Example
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DT LTI System Described by LCCDE’s
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— Rational function of e-jω,
use PFE to get h[n]
Example: First-order recursive system
with the condition of initial rest⇔ causal
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DTFT Multiplication Property
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Calculating Periodic Convolutions
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Example:
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Duality in Fourier AnalysisFourier Transform is highly symmetric
CTFT: Both time and frequency are continuous and in general aperiodic
Same except for
these differences
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Suppose f (•) and g (•) are two functions related by
Then
Example of CTFT dualitySquare pulse in either time or frequency domain
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DTFS
Duality in DTFS
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Duality in DTFS
Then
Duality between CTFS and DTFT
CTFS
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DTFT
CTFS-DTFT Duality
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Magnitude and Phase of FT, and Parseval Relation
CT:
Parseval Relation:
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Energy density in ω
DT:
Parseval Relation:
Effects of Phase
• Not on signal energy distribution as a function of frequency
• Can have dramatic effect on signal shape/character
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g p
— Constructive/Destructive interference
• Is that important?
— Depends on the signal and the context
Demo: 1) Effect of phase on Fourier Series
2) Effect of phase on image processing
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Log-Magnitude and Phase
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Easy to add
Plotting Log-Magnitude and Phase
Plot for ω ≥ 0, often with alogarithmic scale for
frequency in CT
b) In DT, need only plot for 0≤ ω ≤ π (with linear scale)
a) For real-valued signals and systems
c) For historical reasons log-magnitude is usually plotted in units
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So… 20 dB or 2 bels:
= 10 amplitude gain
= 100 power gain
c) For historical reasons, log magnitude is usually plotted in units
of decibels (dB):
power magnitude
A Typical Bode plot for a second-order CT system20 log| H ( jω )| and ∠ H ( jω ) vs. log ω
40 dB/decade
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Changes by -π
A typical plot of the magnitude and phase of a second-
order DT frequency response
20log| H (e jω )| and ∠ H (e jω ) vs. ω
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For real signals,
0 to π is enough
Signals and SystemsFall 2003
Lecture #12
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1. Linear and Nonlinear Phase
2. Ideal and Nonideal Frequency-Selective
Filters
3. CT & DT Rational Frequency Responses
4. DT First- and Second-Order Systems
16 October 2003
Linear Phase
Result: Linear phase ⇔ simply a rigid shift in time, no distortionNonlinear phase ⇔ distortion as well as shift
CT
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Nonlinear phase ⇔ distortion as well as shift
Question:
DT
All-Pass Systems
CT
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DT
Demo: Impulse response and output of an all-pass
system with nonlinear phase
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How do we think about signal delay when the phase is nonlinear?
Group Delay
φ
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Ideal Lowpass Filter
CT
← → ⎯
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• Noncausal h(t <0) ≠ 0
• Oscillatory Response — e.g. step response Overshoot by 9%,
Gibbs phenomenon
Nonideal Lowpass Filter
• Sometimes we don’t want a sharp cutoff, e.g.
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• Often have specifications in time and frequency domain ⇒ Trade-offs
Step responseFreq. Response
CT Rational Frequency Responses
CT: If the system is described by LCCDEs, then
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Prototypical
Systems — First-order system, has only oneenergy storing element, e.g. L or C
— Second-order system, has two
energy storing elements, e.g. L and C
DT Rational Frequency Responses
If the system is described by LCCDE’s (Linear-Constant-Coefficient
Difference Equations), then
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DT First-Order Systems
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Demo: Unit-sample, unit-step, and frequency response
of DT first-order systems
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DT Second-Order System
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oscillations
decaying
Demo: Unit-sample, unit-step, and frequency response of
DT second-order systems
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Signals and SystemsFall 2003
Lecture #13
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1. The Concept and Representation of PeriodicSampling of a CT Signal
2. Analysis of Sampling in the Frequency Domain
3. The Sampling Theorem — the Nyquist Rate
4. In the Time Domain: Interpolation
5. Undersampling and Aliasing
21 October 2003
We live in a continuous-time world: most of the signals we
encounter are CT signals, e.g. x(t ). How do we convert them into DTsignals x[n]?
SAMPLING
— Sampling, taking snap shots of x(t ) every T seconds.
T – sampling period x[n] ≡ x(nT ), n = ..., -1, 0, 1, 2, ... — regularly spaced samples
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How do we perform sampling?
Applications and Examples
— Digital Processing of Signals
— Strobe
— Images in Newspapers
— Sampling Oscilloscope
— …
Why/When Would a Set of Samples Be Adequate?
• Observation: Lots of signals have the same samples
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• By sampling we throw out lots of information – all values of x(t ) between sampling points are lost.
• Key Question for Sampling:
Under what conditions can we reconstruct the original CT signal x(t ) from its samples?
Impulse Sampling — Multiplying x(t ) by the sampling function
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Analysis of Sampling in the Frequency Domain
I t t t
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Important to
note: ωs∝1/T
Illustration of sampling in the frequency-domain for a
band-limited ( X ( j ω)=0 for |ω |> ωM) signal
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No overlap between shifted spectra
Reconstruction of x (t ) from sampled signals
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If there is no overlap between
shifted spectra, a LPF can
reproduce x(t ) from x p(t )
The Sampling Theorem
Suppose x(t ) is bandlimited, so that
Then x(t ) is uniquely determined by its samples x(nT ) if
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Observations on Sampling
(1) In practice, we obviously
don’t sample with impulsesor implement ideal lowpass
filters.
— One practical example:
The Zero-Order Hold
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Time-Domain Interpretation of Reconstruction of Sampled Signals — Band-Limited Interpolation
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The lowpass filter interpolates the samples assuming x(t ) contains
no energy at frequencies ≥ ωc
T
h(t)
Graphic Illustration of Time-Domain Interpolation
Original
CT signal
After sampling
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After passing the LPF
Interpolation Methods
• Bandlimited Interpolation
• Zero-Order Hold
• First-Order Hold — Linear interpolation
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Undersampling and Aliasing
When ωs ≤ 2 ωM⇒ Undersampling
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Undersampling and Aliasing (continued)
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— Higher frequencies of x(t ) are “folded back” and take on the“aliases” of lower frequencies
— Note that at the sample times, xr (nT ) = x(nT )
X r ( jω
)≠
X ( jω
)Distortion because
of aliasing
A Simple Example
Picture would be
Modified…
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Demo: Sampling and reconstruction of cosω ot
Modified…
Signals and SystemsFall 2003
Lecture #1423 October 2003
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1. Review/Examples of Sampling/Aliasing
2. DT Processing of CT Signals
Sampling Review
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Demo: Effect of aliasing on music.
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DT Processing of Band-Limited CT Signals
Why do this? — Inexpensive, versatile, and higher noise margin.
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How do we analyze this system?
— We will need to do it in the frequency domain in both CT and DT — In order to avoid confusion about notations, specify
ω — CT frequency variable
Ω — DT frequency variable (Ω = ωΤ)
Step 1: Find the relation between xc(t ) and xd[n], or X c( jω ) and X d(e jΩ)
Time-Domain Interpretation of C/D Conversion
Note: Not full
analog/digital
(A/D) conversion
– not quantizingthe x[n] values
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Frequency-Domain Interpretation of C/D Conversion
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Note: ωs ⇔ 2π
CT DT
Illustration of C/D Conversion in the Frequency-Domain
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)(eX jΩ
d)(eX jΩ
d
1ωTΩ = 2ωTΩ =
D/C Conversion yd[n] → yc(t )Reverse of the process of C/D conversion
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Now the whole picture
• Overall system is time varying if sampling theorem is not satisfied
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• Overall system is time-varying if sampling theorem is not satisfied
• It is LTI if the sampling theorem is satisfied , i.e. for bandlimitedinputs xc(t ), with
• When the input xc(t ) is band-limited ( X ( jω ) = 0 at |ω | > ω Μ) and the
sampling theorem is satisfied (ω s > 2ω M), then
ω M <ω s
2
DT omege needs to changed
Frequency-Domain Illustration of DT Processing of CT Signals
Sampling
DT filter
DT freq→ CT freq
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Interpolate
(LPF)
⇓
equivalent
CT filter
CT freq→ DT freq
Assuming No Aliasing
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In practice, first specify the desired H c( jω ), then design H d(e jΩ).
Example: Digital DifferentiatorApplications: Edge Enhancement
Courtesy of Jason Oppenheim.
Used with permission.
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Courtesy of Jason Oppenheim.
Used with permission.
Construction of Digital Differentiator
Bandlimited Differentiator
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Band-Limited Digital Differentiator (continued)
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CT DT
Signals and SystemsFall 2003
Lecture #15
28 October 2003
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1. Complex Exponential Amplitude Modulation2. Sinusoidal AM
3. Demodulation of Sinusoidal AM
4. Single-Sideband (SSB) AM
5. Frequency-Division Multiplexing
6. Superheterodyne Receivers
The Concept of Modulation
Why?
• More efficient to transmit E&M signals at higher frequencies
• Transmitting multiple signals through the same medium usingdifferent carriers
• Transmitting through “channels” with limited passbands
Others
Transmitted Signalx(t)
Carrier Signal
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• Others...
• Many methods
• Focus here for the most part on Amplitude M odulation (AM)
How?
Amplitude M odulation (AM) of a
Complex Exponential Carrier
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Demodulation of Complex Exponential AM
Corresponds to two separate modulation channels (quadratures)
with carriers 90o out of phase
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Sinusoidal AM
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Drawn assumingωc > ωM
Synchronous Demodulation of Sinusoidal AM
Supposeθ
= 0 for now,⇒ Local oscillator is in
phase with the carrier.
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Synchronous Demodulation in the Time Domain
Now suppose there is a phase difference, i.e. θ ≠ 0, then
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Two special cases:
1) θ = π/2, the local oscillator is 90o out of phase with the carrier,
⇒ r (t ) = 0, signal unrecoverable.2) θ = θ(t ) — slowly varying with time, ⇒ r (t ) ≅ cos[θ(t )] • x(t ),
⇒ time-varying “gain”.
Synchronous Demodulation (with phase error) in theFrequency Domain
Demodulating signal –
has phase difference θ w.r.t.
the modulating signal
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Again, the low-frequency signal (ω < ωM) = 0 when θ = π/2.
Alternative: Asynchronous Demodulation
• Assume ωc >> ωM, so signal envelope looks like x(t )
• Add same carrier with amplitude A to signal
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A = 0 ⇒ DSB/SC (Double Side Band, Suppressed Carrier)
A > 0 ⇒ DSB/WC (Double Side Band, With Carrier)
Time Domain
Frequency Domain
Asynchronous Demodulation (continued)Envelope Detector
In order for it to function properly, the envelope function must be positivefor all time, i.e. A + x(t ) > 0 for all t.
Demo: Envelope detection for asynchronous demodulation.
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Disadvantages of asynchronous demodulation: — Requires extra transmitting power [ Acosωct ]
2 to make sure
A + x(t ) > 0 ⇒ Maximum power efficiency = 1/3 (P8.27)
Advantages of asynchronous demodulation:
— Simpler in design and implementation.
Double-Sideband (DSB) and Single-Sideband (SSB) AM
Since x(t ) and y(t ) are
real , from conjugatesymmetry both LSB
and USB signals carry
exactly the same
information.
DSB, occupies
2ωM bandwidth
inω
> 0.
Each sidebandUSB
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Each sideband
approach only
occupies ωM
bandwidth in
ω > 0.LSB
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Frequency-Division Multiplexing (FDM)(Examples: Radio-station signals and analog cell phones)
All the channels
can share the same
medium.
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air
FDM in the Frequency-Domain
“Baseband”
signals
Channel a
Channel b
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Channel c
Multiplexed
signals
Demultiplexing and Demodulation
ωa needs to be tunable
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• Channels must not overlap ⇒ Bandwidth Allocation
• It is difficult (and expensive) to design a highly selective
bandpass filter with a tunable center frequency
• Solution – Superheterodyne Receivers
The Superheterodyne Receiver
AM,ω c
2π
= 535 −1605 kHz — RF
FCC:ω IF
2π= 455 kHz — IF
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Operation principle: — Down convert from ωc to ωIF, and use a coarse tunable BPF for the front end.
— Use a sharp-cutoff fixed BPF at ωIF to get rid of other signals.
Signals and SystemsFall 2003
Lecture #16
30 October 2003
1. AM with an Arbitrary Periodic Carrier
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2. Pulse Train Carrier and Time-Division Multiplexing
3. Sinusoidal Frequency Modulation
4. DT Sinusoidal AM
5. DT Sampling, Decimation,and Interpolation
AM with an Arbitrary Periodic Carrier
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Modulating a Rectangular Pulse Train Carrier, cont’d
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for rectangular pulse
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Sinusoidal F requency M odulation (FM)
FM
x(t) is signal
to be
transmitted
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FM
Sinusoidal FM (continued)
• Transmitted power does not depend on x(t ): average power = A2/2
• Bandwidth of y(t ) can depend on amplitude of x(t )
• Demodulationa) Direct tracking of the phase θ (t ) (by using phase-locked loop)
b) Use of an LTI system that acts like a differentiator
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H ( jω ) — Tunable band-limited differentiator, over the bandwidth of y(t )
… looks like AM
envelope detection
DT Sinusoidal AM
Multiplication ↔ Periodic convolution
Example #1:
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Example #2: Sinusoidal AM
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No overlap of
shifted spectra
Example #2 (continued): Demodulation
Possible as long as there is
no overlap of shifted replicas
of X (e jω ):
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Misleading drawing – shown for a
very special case of ωc = π/2
Example #3: An arbitrary periodic DT carrier
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Example #3 (continued):
2πa3 = 2πa0
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No overlap when: ωc > 2ωM (Nyquist rate) ⇒ ωM < π/N
DT Sampling
Motivation: Reducing the number of data points to be stored or
transmitted, e.g. in CD music recording.
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DT Sampling Theorem
We can reconstruct x[n]
if ωs = 2π/ N > 2ωM
Drawn assuming
ωs > 2ωM
Nyquist rate is met
⇒ ωM < π/N
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Drawn assuming
ωs < 2ωM
Aliasing!
/ N
Decimation — Downsampling
x p[n] has (n - 1) zero values between nonzero values:
Why keep them around?
Useful to think of this as sampling followed by discarding the zero values
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compressed in
time by N
Illustration of Decimation in the Time-Domain (for N = 3)
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Decimation in the Frequency Domain
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Squeeze in time
Expand in frequency
Illustration of Decimation in the Frequency Domain
After sampling
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After discarding zeros
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Signals and SystemsFall 2003
Lecture #17
4 November 2003
1. Motivation and Definition of the (Bilateral)
Laplace Transform
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p
2. Examples of Laplace Transforms and Their
Regions of Convergence (ROCs)
3. Properties of ROCs
• CT Fourier transform enables us to do a lot of things, e.g.
— Analyze frequency response of LTI systems
— Sampling — Modulation
Motivation for the Laplace Transform
• In particular, Fourier transform cannot handle large (and important)classes of signals and unstable systems i e when
• Why do we need yet another transform?
• One view of Laplace Transform is as an extension of the Fourier
transform to allow analysis of broader class of signals and systems
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classes of signals and unstable systems, i.e. when
Motivation for the Laplace Transform (continued)
• How do we analyze such signals/systems?
Recall from Lecture #5, eigenfunction property of LTI systems:
• In many applications, we do need to deal with unstable systems, e.g.
— Stabilizing an inverted pendulum
— Stabilizing an airplane or space shuttle
— Instability is desired in some applications, e.g. oscillators and
lasers
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— e st is an eigenfunction of any LTI system
— s = σ + jω can be complex in general
(2) A critical issue in dealing with Laplace transform is convergence:
— X ( s) generally exists only for some values of s,
located in what is called the region of convergence (ROC)
The (Bilateral) Laplace Transform
Basic ideas:
(1)
absolute integrability needed
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(3) If s = jω is in the ROC (i.e. σ = 0), then absoluteintegrability
condition
Example #1:
Unstable:
• no Fourier Transform• but Laplace Transform
exists
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This converges only if Re( s+a) > 0, i.e. Re( s) > - Re(a)
Example #2:
This converges only if Re( s+a) < 0, i.e. Re( s) < - Re(a)
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Key Point (and key difference from FT ): Need both X ( s) and ROC touniquely determine x(t ). No such an issue for FT .
Graphical Visualization of the ROC
Example #2Example #1
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Rational Transforms
• Many (but by no means all) Laplace transforms of interest to us
are rational functions of s (e.g., Examples #1 and #2; in general,
impulse responses of LTI systems described by LCCDEs), where
• Roots of N(s) = zeros of X ( s)
• Roots of D(s) = poles of X ( s)
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• Any x(t ) consisting of a linear combination of complex
exponentials for t > 0 and for t < 0 (e.g., as in Example #1 and #2)
has a rational Laplace transform.
Example #3
Notation:
× — pole
° — zero
BOTH required→
ROC intersection
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Q: Does x(t ) have FT ?
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Properties of the ROC
1) The ROC consists of a collection of lines parallel to the jω -axis in
the s-plane (i.e. the ROC only depends on σ).
Why?
• The ROC can take on only a small number of different forms
2) If X ( s) is rational, then the ROC does not contain any poles.Why?
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y
Poles are places where D( s) = 0
⇒ X ( s) =
N ( s)
D( s) = ∞ Not convergent.
More Properties
3) If x(t ) is of finite duration and is absolutely integrable, then the ROC
is the entire s-plane.
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ROC Properties that Depend on Which Side You Are On - I
4) If x(t ) is right-sided (i.e. if it is zero before some time), and if
Re( s) = σo is in the ROC, then all values of s for which
Re( s) > σo are also in the ROC.
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ROC is a right half plane (RHP)
ROC Properties that Depend on Which Side You Are On - II
5) If x(t ) is left-sided (i.e. if it is zero after some time), and if
Re( s) = σo is in the ROC, then all values of s for which
Re( s) < σo are also in the ROC.
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ROC is a left half plane (LHP)
Still More ROC Properties6) If x(t ) is two-sided and if the line Re( s) = σo is in the ROC,
then the ROC consists of a strip in the s-plane that includes
the line Re( s) = σo
.
ROC is
RHP
Strip =
RHP ∩ LHP
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ROC is
LHP
Example:
Intuition?
• Okay: multiply by
constant (e0t) and
will be integrable
• Looks bad: no eσ t
will dampen both
id
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sides
Example (continued):
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What if b < 0? ⇒ No overlap ⇒ No Laplace Transform
Properties, Properties
7) If X ( s) is rational, then its ROC is bounded by poles or extends to
infinity. In addition, no poles of X ( s) are contained in the ROC.
8) Suppose X ( s) is rational, then
(a) If x(t ) is right-sided, the ROC is to the right of the rightmost pole.
(b) If x(t ) is left-sided, the ROC is to the left of the leftmost pole.
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(a) (b) (c)
9) If ROC of X ( s) includes the jω -axis, then FT of x(t ) exists.
Example:
9) If ROC of X ( s) includes the jω -axis, then FT of x(t ) exists.
Three possible ROCs
Fourier
Transform
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x(t ) is right-sided ROC:
x(t ) is left-sided ROC: x(t ) extends for all time
exists?
No
NoROC: Yes
III
III
Signals and SystemsFall 2003
Lecture #18
6 November 2003
• Inverse Laplace Transforms
• Laplace Transform Properties
Th S t F ti f LTI S t
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• The System Function of an LTI System
• Geometric Evaluation of Laplace Transforms
and Frequency Responses
Inverse Laplace Transform
But s = σ + jω (σ fixed) ⇒ ds = jd ω
Fix σ ∈ ROC and apply the inverse Fourier transform
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Inverse Laplace Transforms Via Partial Fraction
Expansion and Properties
Example:
Three possible ROC’s — corresponding to three different signals
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Recall
ROC I: — Left-sided signal.
ROC III:— Right-sided signal.
ROC II: — Two-sided signal, has Fourier Transform.
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Properties of Laplace Transforms
• For example:
Linearity
ROC at least the intersection of ROCs of X 1( s) and X 2( s)
ROC can be bigger (due to pole-zero cancellation)
• Many parallel properties of the CTFT, but for Laplace transforms
we need to determine implications for the ROC
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⇒ ROC entire s-plane
Time Shift
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Time-Domain Differentiation
ROC could be bigger than the ROC of X ( s), if there is pole-zero
cancellation. E.g.,
s-Domain Differentiation
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Convolution Property
For Then
• ROC of Y ( s) = H ( s) X ( s): at least the overlap of the ROCs of H ( s) & X ( s)
• ROC could be empty if there is no overlap between the two ROCs
E.g.
• ROC could be larger than the overlap of the two. E.g.
)t ( ue )t ( h ),t ( ue )t ( x t t −−== − and
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The System Function of an LTI System
The system function characterizes the system
⇓System properties correspond to properties of H ( s) and its ROC
A first example:
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Geometric Evaluation of Rational Laplace Transforms
Example #1: A first-order zero
Graphic evaluation
of
Can reason about- vector length
- angle w/ real axis
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Example #2: A first-order pole
Example #3: A higher-order rational Laplace transform
Still reason with vector, but
remember to "invert" for poles
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First-Order System
Graphical evaluation of H ( jω ):
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Bode Plot of the First-Order System
-20 dB/decade
changes by -π/2
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Second-Order System
0 < ζ <1 ⇒ complex poles
ζ =1 ⇒
— Under damped
double pole at s = −ω n — Critically damped
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ζ >1 ⇒ 2 poles on negative real axis
— Over damped
Demo Pole-zero diagrams, frequency response, and step
response of first-order and second-order CT causal systems
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Bode Plot of a Second-Order System
-40 dB/decade
Top is flat when
ζ= 1/√2 = 0.707⇒ a LPF for
ω < ωn
changes by -π
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Unit-Impulse and Unit-Step Response of a Second-
Order System
No oscillations when
ζ ≥ 1⇒ Critically (=) and
over (>) damped.
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First-Order All-Pass System
1. Two vectors havethe same lengths
2.
a
a
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Signals and SystemsFall 2003
Lecture #19
18 November 2003
1. CT System Function Properties2. System Function Algebra and
Block Diagrams
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3. Unilateral Laplace Transform and
Applications
CT System Function Properties
2) Causality ⇒ h(t ) right-sided signal ⇒ ROC of H ( s) is a right-half plane
Question:
If the ROC of H ( s) is a right-half plane, is the system causal?
|h(t ) | dt < ∞−∞
∞
∫ 1) System is stable ⇔ ⇔ ROC of H ( s) includes jω axis
Ex.
H(s) = “system function”
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Non-causal
Properties of CT Rational System Functions
a) However, if H ( s) is rational, then
The system is causal ⇔ The ROC of H ( s) is to the
right of the rightmost pole
jω-axis is in ROC
⇔ all poles are in LHP
b) If H ( s) is rational and is the system function of a causal
system, then
The system is stable ⇔
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⇔ all poles are in LHP
Checking if All Poles Are In the Left-Half Plane
Method #1: Calculate all the roots and see!
Method #2: Routh-Hurwitz – Without having to solve for roots.
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Initial- and Final-Value Theorems
If x(t ) = 0 for t < 0 and there are no impulses or higher order
discontinuities at the origin, then
Initial value
If x(t ) = 0 for t < 0 and x(t ) has a finite limit as t → ∞, then
Final value
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Applications of the Initial- and Final-Value Theorem
• Initial value:
• Final value
For
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LTI Systems Described by LCCDEs
roots of numerator ⇒ zerosroots of denominator ⇒ poles
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ROC =? Depends on: 1) Locations of all poles.2) Boundary conditions, i.e.
right-, left-, two-sided signals.
System Function Algebra
Example: A basic feedback system consisting of causal blocks
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ROC: Determined by the roots of 1+ H 1( s) H 2( s), instead of H 1( s)
in feedback
Block Diagram for Causal LTI Systems
with Rational System Functions
— Can be viewed
as cascade of
two systems.
Example:
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Example (continued)
Instead of 1
s2 + 3 s + 2
2 s2 + 4 s − 6
H ( s)
We can construct H ( s) using:
x(t)
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Notation: 1/ s — an integrator
Note also that
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Lesson to be learned: There are many different ways to construct asystem that performs a certain function.
The Unilateral Laplace Transform
(The preferred tool to analyze causal CT systemsdescribed by LCCDEs with initial conditions)
Note:1) If x(t ) = 0 for t < 0, then
2) Unilateral LT of x(t ) = Bilateral LT of x(t )u(t-)
3) For example, if h(t ) is the impulse response of a causal LTI
system, then
4) Convolution property:If x1(t ) = x2(t ) = 0 for t < 0, then
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Same as Bilateral Laplace transform
Differentiation Property for Unilateral Laplace Transform
Note:
Derivation:
Initial condition!
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Use of ULTs to Solve Differentiation Equations
with Initial Conditions
Example:
Take ULT:
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ZIR — Response for zero input x(t )=0
ZSR — Response for zero state, β=γ= 0, initially at rest
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Signals and Systems
Fall 2003
Lecture #20
20 November 2003
1. Feedback Systems
2. Applications of Feedback Systems
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2. Applications of Feedback Systems
Why use Feedback?
• Reducing Effects of Nonidealities
• Reducing Sensitivity to Uncertainties and Variability
• Stabilizing Unstable Systems
• Reducing Effects of Disturbances
• Tracking
Sh i S t R Ch t i ti (b d idth/ d)
A Typical Feedback System
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• Shaping System Response Characteristics (bandwidth/speed)
One Motivating Example
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Open-Loop System Closed-Loop Feedback System
Analysis of (Causal!) LTI Feedback Systems: Black’s Formula
CT System
Black’s formula (1920’s)
Closed - loop system function =
forward gain
1 - loop gain
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Forward gain — total gain along the forward path from the input to the outputLoop gain — total gain around the closed loop
Applications of Black’s Formula
Example:
1)
2)
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)
The Use of Feedback to Compensate for Nonidealities
Assume KP ( jω ) is very large over the frequency range of interest.
In fact, assume
— Independent of P(s)!!
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Example of Reduced Sensitivity
10)0990)(1000(1
1000)(
0990)(1000)(
1
11
=
+
=
==
. jQ
. jG , j KP
ω
ω ω
1)The use of operational amplifiers
2)Decreasing amplifier gain sensitivity
Example:
(a) Suppose
(b) Suppose
(50% gain change)
0990)(500)( 22 . jG , j KP == ω ω
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99)0990)(500(1
500)( 2 ..
jQ ≅
+
=ω (1% gain change)
Fine, but why doesn’t G ( j ω) fluctuate ?
Note:
Needs a large loop gain to produce a steady (and linear ) gain for the
whole system.
⇒
For amplification, G( jω ) must attenuate, and it is much easier to
build attenuators (e.g. resistors) with desired characteristics
There is a price:
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⇒
Consequence of the negative (degenerative) feedback.
Example: Operational Amplifiers
If the amplitude of the loop gain
| KG( s)| >> 1 — usually the case, unless the battery is totally dead.
The closed-loop gain only depends on the passive components
Then Steady State
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( R1 & R2), independent of the gain of the open-loop amplifier K .
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Improving the Dynamics of Systems
Example: Operational Amplifier 741
The open-loop gain has a very large value at dc but very limited bandwidth
Not very useful on its own
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Stabilization of Unstable Systems
• P ( s) — unstable
• Design C ( s), G( s) so that the closed-loop system
is stable
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⇒ poles of Q( s) = roots of 1+C ( s) P ( s)G( s) in LHP
Example #1: First-order unstable systems
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Example #2: Second-order unstable systems
— Unstable for all values of K
— Physically, need damping — a term proportional to s⇔ d /dt
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Example #2 (continued):
Attempt #2: Try Proportional-Plus-Derivative (PD) Feedback
— Stable as long as K 2 > 0 (sufficient damping)
and K 1 > 4 (sufficient gain).
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Example #2 (one more time):
Why didn’t we stabilize by canceling the unstable poles?
There are at least two reasons why this is a really bad idea:
a) In real physical systems, we can never know the precisevalues of the poles, it could be 2±∆.
b) Disturbance between the two systems will cause instability.
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Demo: Magnetic Levitation
io
= current needed to balance the weight W at the rest height yoForce balance
Linearize about equilibrium with specific values for parameters
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— Second-order unstable system
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Signals and Systems
Fall 2003Lecture #21
25 November 2003
1. Feedback
a) Root Locus
b) Tracking
c) Disturbance Rejection
d) The Inverted Pendulum
2 I t d ti t th Z T f
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2. Introduction to the Z-Transform
The Concept of a Root Locus
• C (s) , G( s) — Designed with one or more free parameters
• Question: How do the closed-loop poles move as we vary
these parameters? — Root locus of 1+ C (s)G( s) H ( s)
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The “Classical” Root Locus Problem
C ( s) = K — a simple linear amplifier
Closed-loop
poles are
the same.
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A Simple Example
Becomes more stable Becomes less stable
Sketch where
pole moves
as |K| increases...
In either case, pole is at so = -2 - K
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What Happens More Generally ?
• For simplicity, suppose there is no pole-zero cancellation in G( s) H ( s)
— Difficult to solve explicitly for solutions given any specific
value of K , unless G( s) H ( s) is second-order or lower.
That is
Closed-loop poles are the solutions of
— Much easier to plot the root locus, the values of s that aresolutions for some value of K , because:
1) It is easier to find the roots in the limiting cases for
K = 0, ±∞.
2) There are rules on how to connect between these
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2) There are rules on how to connect between theselimiting points.
Rules for Plotting Root Locus
• End points
— At K = 0, G( so) H ( so) = ∞
⇒ so are poles of the open-loop system function G( s) H ( s).
— At | K| = ∞, G( so) H ( so) = 0
⇒ so are zeros of the open-loop system function G( s) H ( s). Thus:
Rule #1:A root locus starts (at K = 0) from a pole of G( s) H ( s) and ends (at
| K| = ∞) at a zero of G( s) H ( s).
Question: What if the number of poles ≠ the number of zeros?
Answer: Start or end at ±∞.
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Rule #2: Angle criterion of the root locus
• Thus, s0 is a pole for some positive value of K if:
In this case, s0 is a pole if K = 1/|G( s0) H ( s0)|.
• Similarly s0 is a pole for some negative value of K if:
In this case s0 is a pole if K = -1/|G(s0) H(s0)|
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In this case, s is a pole if K -1/|G( s ) H ( s )|.
Example of Root Locus.
One zero at -2,
two poles at 0, -1.
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In addition to stability, we may want good tracking behavior, i.e.
for at least some set of input signals.
Tracking
⇓
+= )(
)()(1
1)( s X
s H sC s E
)()()(1
1)( ω
ω ω
ω j X j H jC
j E +
=
We want to be large in frequency bands in which we)()( ω ω jPjC
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We want to be large in frequency bands in which wewant good tracking
)()( j P jC
Tracking (continued)
Using the final-value theorem
Basic example: Tracking error for a step input
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Disturbance Rejection
There may be other objectives in feedback controls due to unavoidabledisturbances.
Clearly, sensitivities to the disturbances D1( s) and D2( s) are much
reduced when the amplitude of the loop gain
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Internal Instabilities Due to Pole-Zero Cancellation
Hw(t)
)(33
1
)()()(1
)()(
)(
2)()1(
1)(
Stable
2 s X s s s X s H sC
s H sC
sY
s
s s H , s s sC
++=+=
+=+=
However
)()33(
2)(
)()(1
)()(
2 s X
s s s
s s X
s H sC
sC sW
++
+=
+=
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)()()(Unstable
+++
Inverted Pendulum
— Unstable!
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Feedback System to Stabilize the Pendulum
• PI feedback stabilizes θ
– Additional PD feedback around motor / amplifier
centers the pendulum
• Subtle problem: internal instability in x(t)!
a
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p
Root Locus & the Inverted Pendulum
• Attempt #1: Negative feedback driving the motor
– Remains unstable!
• Root locus of M(s)G(s)
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after K. Lundberg
Root Locus & the Inverted Pendulum
• Attempt #2: Proportional/Integral Compensator
– Stable for large enough K
• Root locus of K(s)M(s)G(s)
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after K. Lundberg
Root Locus & the Inverted Pendulum
• BUT – x(t) unstable:
System subject to drift...
• Solution: add PD feedback
around motor and
compensator:
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after K. Lundberg
The z-Transform
The (Bilateral) z-Transform
Motivation: Analogous to Laplace Transform in CT
We now do not
restrict ourselves
just to z = e jω
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The ROC and the Relation Between z T and DTFT
• Unit circle (r = 1) in the ROC ⇒ DTFT X (e jω ) exists
•
— depends only on r = | z |, just like the ROC in s-plane
only depends on Re( s)
, r = |z|
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Example #1
That is, ROC | z | > |a|,
outside a circle
This form to find
pole and zero locations
This formfor PFE
and
inverse z-
transform
11
1−−
=az a z
z
−=
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Example #2:
Same X(z) as in Ex #1 but different ROC
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Same X ( z ) as in Ex #1, but different ROC.
Rational z-Transforms
x[n] = linear combination of exponentials for n > 0 and for n < 0
— characterized (except for a gain) by its poles and zeros
Polynomials in z
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Signals and Systems
Fall 2003Lecture #22
2 December 2003
1. Properties of the ROC of the z-Transform
2. Inverse z-Transform
3. Examples4. Properties of the z-Transform
5. System Functions of DT LTI Systems
a. Causality
b. Stability
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y
The z-Transform
• Last time:
•Unit circle (r = 1) in the ROC⇒DTFT X (e jω ) exists
•Rational transforms correspond to signals that are linear
combinations of DT exponentials
-depends only on r = | z |, just like the ROC in s-plane
only depends on Re( s)
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Some Intuition on the Relation between z T and LT
Can think of z-transform as DT
version of Laplace transform with
The (Bilateral) z-Transform
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More intuition on z T- LT, s-plane - z -plane relationship
• LHP in s-plane, Re( s) < 0 ⇒ | z | = | e sT | < 1, inside the | z | = 1 circle.
Special case, Re( s) = -∞ ⇔ | z | = 0.
• RHP in s-plane, Re( s) > 0 ⇒ | z | = | e sT | > 1, outside the | z | = 1 circle.
Special case, Re( s) = +∞ ⇔ | z | = ∞.
• A vertical line in s-plane, Re( s) = constant ⇔ | e sT
| = constant, acircle in z -plane.
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Properties of the ROCs of z -Transforms
(1) The ROC of X ( z ) consists of a ring in the z -plane centered about
the origin (equivalent to a vertical strip in the s-plane)
(2) The ROC does not contain any poles (same as in LT ).
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More ROC Properties
(3) If x[n] is of finite duration, then the ROC is the entire z -plane,
except possibly at z = 0 and/or z = ∞.
Why?
CT counterpartExamples:
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ROC Properties Continued
(4) If x[n] is a right-sided sequence, and if | z | = r o is in the ROC, then
all finite values of z for which | z | > r o are also in the ROC.
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Side by Side
(6) If x[n] is two-sided, and if | z | = r o is in the ROC, then the ROC
consists of a ring in the z -plane including the circle | z | = r o.
What types of signals do the following ROC correspond to?
right-sided left-sided two-sided
(5) If x[n] is a left-sided sequence, and if | z | = r o is in the ROC,
then all finite values of z for which 0 < | z | < r o are also in the ROC.
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Example #1
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Example #1 continued
Clearly, ROC does not exist if b > 1⇒ No z -transform for b|n|
.
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Inverse z-Transforms
for fixed r:
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Example #2
2) When doing inverse z -transform
using PFE, express X ( z ) as afunction of z -1.
Partial Fraction Expansion Algebra: A = 1, B = 2
Note, particular to z -transforms:
1) When finding poles and zeros,
express X ( z ) as a function of z .
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ROC I:
ROC III:
ROC II:
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Inversion by Identifying Coefficients
in the Power Series
— A finite-duration DT sequence
Example #3:
3
-1
2
0 for all other n’s
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Example #4:
(a)
(b)
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Properties of z-Transforms
(1) Time Shifting
The rationality of X ( z ) unchanged, different from LT. ROC unchanged
except for the possible addition or deletion of the origin or infinityno> 0⇒ ROC z ≠ 0 (maybe)
no< 0⇒ ROC z ≠ ∞ (maybe)
(2) z -Domain Differentiation same ROC
Derivation:
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Convolution Property and System Functions
Y ( z ) = H ( z ) X ( z ) , ROC at least the intersection of
the ROCs of H ( z ) and X ( z ),
can be bigger if there is pole/zero
cancellation. e.g.
H(z) + ROC tells us everything about system
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CAUSALITY
(1) h[n] right-sided ⇒ ROC is the exterior of a circle possibly
including z = ∞:
A DT LTI system with system function H ( z ) is causal⇔ the ROC of
H ( z ) is the exterior of a circle including z = ∞
⇓
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Causality for Systems with Rational System Functions
A DT LTI system with rational system function H ( z ) is causal
⇔ (a) the ROC is the exterior of a circle outside the outermost pole;
and (b) if we write H(z) as a ratio of polynomials
then
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Stability
• A causal LTI system with rational system function is stable⇔ all
poles are inside the unit circle, i.e. have magnitudes < 1
• LTI System Stable ⇔ ROC of H ( z ) includes
the unit circle | z | = 1
⇒ Frequency Response H (e jω
) (DTFT of h[n]) exists.
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Signals and Systems
Fall 2003Lecture #234 December 2003
1. Geometric Evaluation of z-Transforms and DT Frequency
Responses
2. First- and Second-Order Systems
3. System Function Algebra and Block Diagrams
4. Unilateral z-Transforms
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Geometric Evaluation of a Rational z-Transform
Example #1:
Example #3:
Example #2:
All same as
in s-plane
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Geometric Evaluation of DT Frequency Responses
First-Order System— one real pole
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Second-Order System
Two poles that are a complex conjugate pair ( z1
= re jθ = z2
*)
Clearly, | H | peaks near ω = ±θ
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Demo: DT pole-zero diagrams, frequency response, vector
diagrams, and impulse- & step-responses
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DT LTI Systems Described by LCCDEs
ROC: Depends on Boundary Conditions, left-, right-, or two-sided.
— Rational
Use the time-shift property
For Causal Systems ⇒ ROC is outside the outermost pole
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Feedback System
(causal systems)
System Function Algebra and Block Diagrams
Example #1:
negative feedback
configuration
z-1 D⇔
Delay
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Example #2:
— Cascade of
two systems
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Unilateral z-Transform
Note:
(1) If x[n] = 0 for n < 0, then
(2) UZT of x[n] = BZT of x[n]u[n] ⇒ ROC always outside a circle
and includes z = ∞
(3) For causal LTI systems,
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• But there are important differences. For example, time-shift
Properties of Unilateral z-Transform
Many properties are analogous to properties of the BZT e.g.
• Convolution property (for x1[n<0] = x2[n<0] = 0)
Derivation:Initial condition
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Use of UZTs in Solving Difference Equations
with Initial Conditions
ZIR — Output purely due to the initial conditions,
ZSR — Output purely due to the input.
UZT of Difference Equation
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β = 0 ⇒ System is initially at rest:
ZSR
Example (continued)
α = 0 ⇒ Get response to initial conditions
ZIR
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