1. Signal and Systems Prof. 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. The z-Transform Book Chapter10: Section 1 Computer Engineering Department, Signal and Systems 2 Motivation:Analogous to Laplace Transform in CT We now do not restrict ourselves just to z = e j ω The (Bilateral) z-Transform

3. The ROC and the Relation Between ZT and DTFT  Unit circle (r = 1) in the ROC ⇒ DTFT X(e j ω ) exists Computer Engineering Department, Signal and Systems 3 —depends only on r = |z|, just like the ROC in s-plane only depends on Re(s) Book Chapter10: Section 1

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

5. Example #2: Computer Engineering Department, Signal and Systems 5 Same X(z) as in Ex #1, but different ROC. Book Chapter10: Section 1

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

7. 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 Computer Engineering Department, Signal and Systems 7 -depends only on r = |z|, just like the ROC in s-plane only depends on Re(s) Book Chapter10: Section 1

8. Some Intuition on the Relation between ZT and LT Computer Engineering Department, Signal and Systems 8 The (Bilateral) z-Transform Can think of z-transform as DT version of Laplace transform with Let t=nT Book Chapter10: Section 1

9. 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. Computer Engineering Department, Signal and Systems 9 Book Chapter10: Section 1

10. Properties of the ROCs of Z-Transforms Computer Engineering Department, Signal and Systems 10 (2) The ROC does notcontain any poles (same as in LT). (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) Book Chapter10: Section 1

11. More ROC Properties Computer Engineering Department, Signal and Systems 11 (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 Book Chapter10: Section 1

12. ROC Properties Continued Computer Engineering Department, Signal and Systems 12 (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. Book Chapter10: Section 1

13. Side by Side Computer Engineering Department, Signal and Systems 13 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. (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. Book Chapter10: Section 1

14. Computer Engineering Department, Signal and Systems 14 Example #1 Book Chapter10: Section 1

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

16. Computer Engineering Department, Signal and Systems 16 Inverse z-Transforms for fixed r: Book Chapter10: Section 1

17. Computer Engineering Department, Signal and Systems 17 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. Book Chapter10: Section 1

18. Computer Engineering Department, Signal and Systems 18 ROC III: ROC II: ROC I: Book Chapter10: Section 1

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

20. Computer Engineering Department, Signal and Systems 20 Example #4: (a) (b) Book Chapter10: Section 1

21. Computer Engineering Department, Signal and Systems 21 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: Book Chapter10: Section 1

22. Computer Engineering Department, Signal and Systems 22 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 Book Chapter10: Section 1

23. Computer Engineering Department, Signal and Systems 23 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 = ∞ No z m terms with m>0 =>z=∞ ROC Book Chapter10: Section 1

24. Computer Engineering Department, Signal and Systems 24 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 Book Chapter10: Section 1

25. 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 Computer Engineering Department, Signal and Systems 25 Book Chapter10: Section 1

26. Geometric Evaluation of a Rational z-Transform  Example 1  Example 2  Example 3 Book Chapter10: Computer Engineering Department, Signal and Systems 26

27. Geometric Evaluation of DT Frequency Responses Book Chapter10: Computer Engineering Department, Signal and Systems 27 First-Order System one real pole

28. Second-Order System  Two poles that are a complex conjugate pair (z 1 = re j θ =z 2 * ) Book Chapter10: Computer Engineering Department, Signal and Systems 28 Clearly, |H| peaks near ω = ± θ

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

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

31. System Function Algebra and Block Diagrams Book Chapter10: Computer Engineering Department, Signal and Systems 31 Feedback System (causal systems) negative feedback configuration Example #1: Delay

32. Example 2 Book Chapter10: Computer Engineering Department, Signal and Systems 32 —Cascade of two systems

33. Unilateral z-Transform Book Chapter10: Computer Engineering Department, Signal and Systems 33 Note: (1) If x[n] = 0 for n < 0, then (2)UZT of x[n] = BZT of x[n]u[n] ⇒ ROC always outside a circle and includes z = ∞ (3) For causal LTI systems,

34. Properties of Unilateral z-Transform  Many properties are analogous to properties of the BZT e.g. Book Chapter10: Computer Engineering Department, Signal and Systems 34 Convolution property (for x 1 [n<0] = x 2 [n<0] = 0) But there are important differences. For example, time-shift Derivation: Initial condition

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

36. Example (continued) Book Chapter10: Computer Engineering Department, Signal and Systems 36 β = 0 ⇒ System is initially at rest: ZSR α = 0 ⇒ Get response to initial conditions ZIR