Bài giảng Xử lý tín hiệu số - Lecture 4: Z - transform - Ngô Quốc Cường

Xử lý tín hiệu số Z - transform Ngô Quốc Cường Ngô Quốc Cường ngoquoccuong175@gmail.com sites.google.com/a/hcmute.edu.vn/ngoquoccuong Z - transform • Z- transform • Properties of Z-transform • Inversion of Z- transform • Analysis of LTI systems in Z domain 2 4.1. Z - transform • Given a discrete-time signal x(n), its z-transform is defined as the following series: where z is a complex variable. • Writing explicitly a few of the terms: • Z-transform is an infinite power series, it exists only for those values of z for this series converges. • The region of convergence (ROC) of X(z) is the set of all values of z for which X(z) attains a finite value. 3 4.1. Z - transform • Example: Determines the z-transform of the following finite duration signals 4 4.1. Z - transform • Solution 5 4.1. Z - transform • Example 6 4.1. Z - transform Recall that 7 4.1. Z - transform • Example • Solution 8 4.1. Z - transform • Example • We have (l = -n), • Using the formula (when A<1) 9 10 4.1. Z - transform • We have identical closed-form expressions for the z transform • A closed-form expressions for the z transform does not uniquely specify the signal in time domain. • The ambiguity can be resolved if the ROC is specified. • Z – transform = closed-form expressions + ROC 11 4.1. Z - transform • Example • Solution – The first power series converges if |z| > |a| – The second power series converges if |z| < |b| 12 • Case 1 13 • Case 2 14 • Characteristics families of signals with their corresponding ROC 15 16 4.1. Z - transform • The z-transform of the impulse response h(n) is called the transfer function of a digital filter: • Determine the transfer function H(z) of the two causal filters 17 4.2. Properties of Z-transform 18 • Example 19 20 • Example 21 22 23 Exercise 24 25 4.2. Properties of Z-transform • The ROC of z-k X(z) is the same as that of X(z) except for z=0 if k>0 and z=∞ if k<0. 26 • Solution 27 4.2. Properties of Z-transform 28 • Example 29 30 4.2. Properties of Z-transform 31 • Example 32 4.2. Properties of Z-transform 33 4.2. Properties of Z-transform • Example 34 4.2. Properties of Z-transform 35 4.2. Properties of Z-transform • Example 36 37 4.2. Properties of Z-transform • Convolution in Z domain 38 4.3. RATIONAL Z-TRANSFORM 39 4.3.1. Poles and Zeros • An important family of z-transforms are those for which X(z) is a rational function. • Poles and Zeros – Zeros: value of z for which X(z) = 0; – Poles: value of z for which X(z) = ∞ 40 4.3.1. Poles and Zeros • Example • Solution 41 4.3.2. Causality and Stability • A causal signal of the form will have z-transform • the common ROC of all the terms will be 42 4.3.2. Causality and Stability • if the signal is completely anticausal • the ROC is in this case 43 4.3.2. Causality and Stability • Causal signals are characterized by ROCs that are outside the maximum pole circle. • Anticausal signals have ROCs that are inside the minimum pole circle. • Mixed signals have ROCs that are the annular region between two circles—with the poles that lie inside the inner circle contributing causally and the poles that lie outside the outer circle contributing anticausally. 44 4.3.2. Causality and Stability • Stability can also be characterized in the z-domain in terms of the choice of the ROC. • A necessary and sufficient condition for the stability of a signal x(n) is that the ROC of the corresponding z-transform contain the unit circle. • A signal or system to be simultaneously stable and causal, it is necessary that all its poles lie strictly inside the unit circle in the z-plane. 45 4.3.2. Causality and Stability 46 4.3.3. System function of LTI • System function • From a linear constant coefficient equation • We have, 47 4.3.3. System function of LTI • Or equivalently 48 4.3.3. System function of LTI • Example 49 4.3.3. System function of LTI • Solution • The unit sample response 50 4.4. Inverse Z- transform • By contour integration. • By power series expansion. • By partial fraction expansion. 51 • The partial fraction expansion method can be applied to z- transforms that are ratios of two polynomials • The partial fraction expansion of X(z) is given by 52 • Example • The two coefficients are obtained as follows: 53 • If the degree of the numerator polynomial N(z) is exactly equal to the degree M of the denominator D(z), then the PF expansion must be modified 54 • Example • Compute all possible inverse z-transforms of 55 • Solution • Where • |z| > 0.5: • |z|<0.5: 56 • Example • Determine all inverse z-transforms of 57 • Solution 58 • there are only two ROCs I and II: 59 MORE ABOUT INVERSE Z-TRANSFORM 60 • Distinct poles 61 • Multiple order poles • Solution • In such a case, the partial fraction expansion is: 62 63 Exercise 64 • a) • b) • c) 65 66 67