Signals and Systems
Fall 2003
Lecture #23
4 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
Geometric Evaluation of a Rational z-Transform
Example #1:
Example #3:
Example #2:
All same as
in s-plane
Geometric Evaluation of DT Frequency Responses
First-Order System
— one real pole
Second-Order System
Two poles that are a complex conjugate pair (z
1
= re
jθ
=z
2
*
)
Clearly,|H| peaks near ω = ±θ
Demo,DT pole-zero diagrams,frequency response,vector
diagrams,and impulse- & step-responses
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
Feedback System
(causal systems)
System Function Algebra and Block Diagrams
Example #1:
negative feedback
configuration
z
-1
D
Delay
Example #2:
— Cascade of
two systems
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,
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 x
1
[n<0] = x
2
[n<0] = 0)
Derivation:
Initial condition
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
β = 0? System is initially at rest:
ZSR
Example (continued)
α = 0? Get response to initial conditions
ZIR