signal and system lecture 22

8/14/2019 signal and system Lecture 22

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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


2/20

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)


3/20

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


4/20

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.


5/20

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).


6/20

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:


7/20

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.


8/20

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.


9/20

Example #1


10/20

Example #1 continued

Clearly, ROC does notexist ifb > 1 No z-transform forb|n|.


11/20

Inverse z-Transforms

for fixed r:


12/20

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.


13/20

ROC I:

ROC III:

ROC II:


14/20

Inversion by Identifying Coefficients

in the Power Series

A finite-duration DT sequence

Example #3:

3

-12

0 for all other ns


15/20

Example #4:

(a)

(b)


16/20

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:


17/20

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


18/20

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=


19/20

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


20/20

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.

signal and system lecture 22

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