Post on 30-May-2018
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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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The z-Transform
Last time:
Unit circle (r= 1) in the ROCDTFTX(ej
) existsRational transforms correspond to signals that are linear
combinations of DT exponentials
-depends only on r= |z|, just like the ROC ins-plane
only depends onRe(s)
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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
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More intuition on zT-LT, s-plane - z-plane relationship
LHP ins-plane,Re(s) < 0 |z| = | esT| < 1, inside the |z| = 1 circle.
Special case,Re(s) = -
|z| = 0. RHP ins-plane,Re(s) > 0 |z| = | esT| > 1, outside the |z| = 1 circle.
Special case,Re(s) = + |z| = .
A vertical line ins-plane,Re(s) = constant | esT| = constant, a
circle inz-plane.
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Properties of the ROCs ofz-Transforms
(1) The ROC ofX(z) consists of a ring in thez-plane centered about
the origin (equivalent to a vertical strip in thes-plane)
(2) The ROC does not contain any poles (same as inLT).
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More ROC Properties
(3) If x[n] is of finite duration, then the ROC is the entirez-plane,
except possibly atz= 0 and/orz= .
Why?
CT counterpartExamples:
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ROC Properties Continued
(4) If x[n] is a right-sided sequence, and if |z| = ro is in the ROC, then
all finite values ofzfor which |z| > ro are also in the ROC.
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Side by Side
(6) Ifx[n] is two-sided, and if |z| = ro is in the ROC, then the ROC
consists of a ring in thez-plane including the circle |z| = ro.
What types of signals do the following ROC correspond to?
right-sided left-sided two-sided
(5) Ifx[n] is a left-sided sequence, and if |z| = ro is in the ROC,then all finite values ofzfor which 0 < |z| < ro are also in the ROC.
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Example #1
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Example #1 continued
Clearly, ROC does notexist ifb > 1 No z-transform forb|n|.
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Inverse z-Transforms
for fixed r:
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Example #2
2) When doing inversez-transform
using PFE, expressX(z) as afunction ofz-1.
Partial Fraction Expansion Algebra: A = 1,B = 2
Note, particular toz-transforms:
1) When finding poles and zeros,expressX(z) as a function ofz.
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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
-12
0 for all other ns
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Example #4:
(a)
(b)
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Properties of z-Transforms
(1) Time Shifting
The rationality ofX(z) unchanged, differentfrom LT. ROC unchanged
except for the possible addition or deletion of the origin or infinity
no> 0 ROCz 0 (maybe)
no< 0 ROCz (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 ofH(z) andX(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 circlepossiblyincludingz= :
A DT LTI system with system functionH(z) is causal the ROC of
H(z) is the exterior of a circle includingz=
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Causality for Systems with Rational System Functions
A DT LTI system with rational system functionH(z) is causal
(a) the ROC is the exterior of a circle outside the outermost pole;
and (b) if we writeH(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 ofH(z) includes
the unit circle |z| = 1
Frequency ResponseH(e
j
) (DTFT ofh[n]) exists.