Download presentation

Published byCole Schwartz Modified over 4 years ago

1
**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 System Functions of DT LTI Systems a. Causality b. Stability

2
**Unit circle (r = 1) in the ROC ⇒DTFT X(ejω) exists **

The z-Transform -depends only on r = |z|, just like the ROC in s-plane only depends on Re(s) Last time: Unit circle (r = 1) in the ROC ⇒DTFT X(ejω) exists Rational transforms correspond to signals that are linear combinations of DT exponentials

3
**Some Intuition on the Relation between zT and LT**

Let t=nT The (Bilateral) z-Transform Can think of z-transform as DT version of Laplace transform with

4
**More intuition on zT-LT, s-plane - z-plane relationship**

LHP in s-plane, Re(s) < 0⇒|z| = | esT| < 1, inside the |z| = 1 circle. Special case, Re(s) = -∞ ⇔|z| = 0. RHP in s-plane, Re(s) > 0⇒|z| = | esT| > 1, outside the |z| = 1 circle. Special case, Re(s) = +∞ ⇔|z| = ∞. A vertical line in s-plane, Re(s) = constant⇔| esT| = constant, a circle in z-plane.

5
**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).

6
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? Examples: CT counterpart

7
**ROC Properties Continued**

(4) If x[n] is a right-sided sequence, and if |z| = ro is in the ROC, then all finite values of z for which |z| > ro are also in the ROC.

8
Side by Side If x[n] is a left-sided sequence, and if |z| = ro is in the ROC, then all finite values of z for which 0 < |z| < ro are also in the ROC. (6) If x[n] is two-sided, and if |z| = ro is in the ROC, then the ROC consists of a ring in the z-plane including the circle |z| = ro. What types of signals do the following ROC correspond to? right-sided left-sided two-sided

9
Example #1

10
Example #1 continued Clearly, ROC does not exist if b > 1 ⇒ No z-transform for b|n|.

11
Inverse z-Transforms for fixed r:

12
Example #2 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. 2) When doing inverse z-transform using PFE, express X(z) as a function of z-1.

13
ROC III: ROC II: ROC I:

14
**Inversion by Identifying Coefficients in the Power Series**

Example #3: 3 -1 2 0 for all other n’s — A finite-duration DT sequence

15
Example #4: (a) (b)

16
**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 no> 0 ⇒ ROC z ≠ 0 (maybe) no< 0 ⇒ ROC z ≠ ∞ (maybe) (2) z-Domain Differentiation same ROC Derivation:

17
**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

18
CAUSALITY (1) h[n] right-sided⇒ROC is the exterior of a circle possibly including z = ∞: No zm terms with m>0 =>z=∞ ROC A DT LTI system with system function H(z) is causal ⇔ the ROC of H(z) is the exterior of a circle including z = ∞

19
**then 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

20
**Stability LTI System Stable⇔ ROC of H(z) includes**

the unit circle |z| = 1 ⇒ Frequency Response H(ejω) (DTFT of h[n]) exists. A causal LTI system with rational system function is stable ⇔ all poles are inside the unit circle, i.e. have magnitudes < 1

Similar presentations

Presentation is loading. Please wait....

OK

CHAPTER 4 Laplace Transform.

CHAPTER 4 Laplace Transform.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on circular linked list Ppt on various categories of software Ppt on amplitude shift keying circuit Ppt on market friendly state for retirement Ppt on indian politics quoted Ppt on paintings and photographs related to colonial period dates Ppt on australian continental drift Ppt on solid waste management in india Ppt on diversity in living organisms class Ppt on mind reading computer download