signal and system lecture 23
TRANSCRIPT
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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= rej=z2
*)
Clearly, |H| peaks near =
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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) Ifx[n] = 0 for n < 0, then
(2) UZT ofx[n] = BZT ofx[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-TransformMany properties are analogous to properties of the BZT e.g.
Convolution property (for x1[n
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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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= 0 System is initially at rest:
ZSR
Example (continued)
= 0 Get response to initial conditions
ZIR