Signal and Systems Prof. H. Sameti

1. Signal and SystemsProf. H. Sameti Chapter 10: Introduction to the z-Transform Properties of the ROC of the z-Transform Inverse z-Transform Examples Properties of the z-Transform System Functions of DT LTI Systems a. Causality b. Stability Geometric Evaluation of z-Transforms and DT Frequency Responses First- and Second-Order Systems System Function Algebra and Block Diagrams Unilateral z-Transforms

2. Book Chapter10: Section 1 The z-Transform Motivation: Analogous to Laplace Transform in CT We now do not restrict ourselves just to z = ejω The (Bilateral) z-Transform

3. Book Chapter10: Section 1 The ROC and the Relation Between ZT and DTFT • Unit circle (r = 1) in the ROC ⇒DTFT X(ejω) exists —depends only on r = |z|, just like the ROC in s-plane only depends on Re(s)

4. Book Chapter10: Section 1 Example #1 • This form for PFE and inverse z- transform This form to find pole and zero locations That is, ROC |z| > |a|, outside a circle

5. Book Chapter10: Section 1 Example #2: Same X(z) as in Ex #1, but different ROC.

6. Book Chapter10: Section 1 Rational z-Transforms x[n] = linear combination of exponentials for n > 0 and for n < 0 Polynomials in z — characterized (except for a gain) by its poles and zeros

7. Book Chapter10: Section 1 The z-Transform • Last time: • Unit circle (r = 1) in the ROC ⇒DTFT 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)

8. Book Chapter10: Section 1 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

9. Book Chapter10: Section 1 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.

10. Book Chapter10: Section 1 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).

11. Book Chapter10: Section 1 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

12. Book Chapter10: Section 1 ROC Properties Continued • (4) If x[n] is a right-sided sequence, and if |z| = rois in the ROC, then all finite values of z for which |z| > roare also in the ROC.

13. Book Chapter10: Section 1 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

14. Book Chapter10: Section 1 Example #1

15. Book Chapter10: Section 1 Example #1 continued Clearly, ROC does not exist if b > 1 ⇒No z-transform for b|n|.

16. Book Chapter10: Section 1 Inverse z-Transforms for fixed r:

17. Book Chapter10: Section 1 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.

18. Book Chapter10: Section 1 ROC III: ROC II: ROC I:

19. Book Chapter10: Section 1 • Inversion by Identifying Coefficients in the Power Series Example #3: 3 -1 2 0 for all other n’s — A finite-duration DT sequence

20. Book Chapter10: Section 1 Example #4: (a) (b)

21. Book Chapter10: Section 1 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:

22. Book Chapter10: Section 1 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 the system

23. Book Chapter10: Section 1 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 = ∞

24. Book Chapter10: Section 1 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

25. Book Chapter10: Section 1 Stability • LTI System Stable⇔ ROC of H(z) includes the unit circle |z| = 1 • A causal LTI system with rational system function is stable ⇔ all poles are inside the unit circle, i.e. have magnitudes < 1 ⇒ Frequency Response (DTFT of h[n]) exists.

26. Book Chapter10: Geometric Evaluation of a Rational z-Transform • Example 1 • Example 2 • Example 3

27. Book Chapter10: Geometric Evaluation of DT Frequency Responses First-Order System one real pole

28. Book Chapter10: Second-Order System • Two poles that are a complex conjugate pair (z1= rejθ=z2*) Clearly, |H| peaks near ω = ±θ

29. Book Chapter10: Demo: DT pole-zero diagrams, frequency response, vector diagrams, and impulse- & step-responses

30. Book Chapter10: DT LTI Systems Described by LCCDEs Use the time-shift property —Rational ROC: Depends on Boundary Conditions, left-, right-, or two-sided. For Causal Systems ⇒ ROC is outside the outermost pole

31. Book Chapter10: System Function Algebra and Block Diagrams Feedback System (causal systems) • negative feedback configuration Example #1: D Delay

32. Book Chapter10: Example 2 • — Cascade of two systems

33. Book Chapter10: Unilateral z-Transform Note: (1) If x[n] = 0 for n < 0, then • UZT of x[n] = BZT of x[n]u[n]⇒ROC always outside a circle • and includes z = ∞ (3) For causal LTI systems,

34. Book Chapter10: Properties of Unilateral z-Transform • Many properties are analogous to properties of the BZT e.g. • Convolution property (for x1[n<0] = x2[n<0] = 0) • But there are important differences. For example, time-shift Derivation: Initial condition

35. Book Chapter10: Use of UZTs in Solving Difference Equations with Initial Conditions UZT of Difference Equation ZIR — Output purely due to the initial conditions, ZSR — Output purely due to the input.

36. Book Chapter10: Example (continued) β = 0 ⇒ System is initially at rest: ZSR α = 0 ⇒ Get response to initial conditions ZIR