Signals and Systems
Fall 2003
Lecture #22
2 December 2003
1,Properties of the ROC of the z-Transform
2,Inverse z-Transform
3,Examples
4,Properties of the z-Transform
5,System Functions of DT LTI Systems
a,Causality
b,Stability
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)
Some Intuition on the Relation between zT and LT
Can think of z-transform as DT
version of Laplace transform with
The (Bilateral) z-Transform
More intuition on zT-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,a
circle in z-plane.
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).
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 counterpart
Examples:
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.
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.
Example #1
Example #1 continued
Clearly,ROC does not exist if b > 1? No z-transform for b
|n|
.
Inverse z-Transforms
for fixed r:
Example #2
2) When doing inverse z-transform
using PFE,express X(z) as a
function 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.
ROC I:
ROC III:
ROC II:
Inversion by Identifying Coefficients
in the Power Series
— A finite-duration DT sequence
Example #3:
3
-1
2
0 for all other n’s
Example #4:
(a)
(b)
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 infinity
n
o
> 0? ROC z ≠ 0 (maybe)
n
o
< 0? ROC z ≠∞(maybe)
(2) z-Domain Differentiation
same ROC
Derivation:
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
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 = ∞
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
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.