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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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,

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

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.

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