Z Transform (1)

1. Z Transform (1) Hany Ferdinando Dept. of Electrical Eng. Petra Christian University

2. Overview • Introduction • Basic calculation • RoC • Inverse Z Transform • Properties of Z transform • Exercise Z Transform (1) - Hany Ferdinando

3. Introduction • For discrete-time, we have not only Fourier analysis, but also Z transform • This is special for discrete-time only • The main idea is to transform signal/system from time-domain to z-domain  it means there is no time variable in the z-domain Z Transform (1) - Hany Ferdinando

4. Introduction • One important consequence of transform-domain description of LTI system is that the convolution operation in the time domain is converted to a multiplication operation in the transform-domain Z Transform (1) - Hany Ferdinando

5. Introduction • It simplifies the study of LTI system by: • Providing intuition that is not evident in the time-domain solution • Including initial conditions in the solution process automatically • Reducing the solution process of many problems to a simple table look up, much as one did for logarithm before the advent of hand calculators Z Transform (1) - Hany Ferdinando

6. Basic Calculation • They are general formula: • Index ‘k’ or ‘n’ refer to time variable • If k > 0 then k is from 1 to infinity • Solve those equation with the geometrics series or Z Transform (1) - Hany Ferdinando

7. Basic Calculation Calculate: Z Transform (1) - Hany Ferdinando

8. Basic Calculation • Different signals can have the same transform in the z-domain  strange • The problem is when we got the representation in z-domain, how we can know the original signal in the time domain… Z Transform (1) - Hany Ferdinando

9. Region of Convergence (RoC) • Geometrics series for infinite sum has special rule in order to solve it • This is the ratio between adjacent values • For those who forget this rule, please refer to geometrics series Z Transform (1) - Hany Ferdinando

10. Region of Convergence (RoC) Z Transform (1) - Hany Ferdinando

11. Region of Convergence (RoC) Z Transform (1) - Hany Ferdinando

12. Region of Convergence (RoC) Z Transform (1) - Hany Ferdinando

13. RoC Properties • RoC of X(z) consists of a ring in the z-plane centered about the origin • RoC does not contain any poles • If x(n) is of finite duration then the RoC is the entire z-plane except possibly z = 0 and/or z = ∞ Z Transform (1) - Hany Ferdinando

14. RoC Properties • If x(n) is right-sided sequence and if |z| = ro is in the RoC, then all finite values of z for which |z| > ro will also be in the RoC • If x(n) is left-sided sequence and if |z| = ro is in the RoC, then all values for which 0 < |z| < ro will also be in the RoC Z Transform (1) - Hany Ferdinando

15. RoC Properties • If x(n) is two-sided and if |z| = ro is in the RoC, then the RoC will consists of a ring in the z-plane which includes the |z| = ro Z Transform (1) - Hany Ferdinando

16. Inverse Z Transform • Direct division • Partial expansion • Alternative partial expansion Use RoC information Z Transform (1) - Hany Ferdinando

17. Direct Division • If the RoC is less than ‘a’, then expand it to positive power of z • a is divided by (–a+z) • If the RoC is greater than ‘a’, then expand it to negative power of z • a is divided by (z-a) Z Transform (1) - Hany Ferdinando

18. Partial Expansion • If the z is in the power of two or more, then use partial expansion to reduce its order • Then solve them with direct division Z Transform (1) - Hany Ferdinando

19. Properties of Z Transform General term and condition: • For every x(n) in time domain, there is X(z) in z domain with R as RoC • n is always from –∞ to ∞ Z Transform (1) - Hany Ferdinando

20. Linearity • a x1(n) + b x2(n) ↔ a X1(z) + b X2(z) • RoC is R1∩R2 • If a X1(z) + b X2(z) consist of all poles of X1(z) and X2(z) (there is no pole-zero cancellation), the RoC is exactly equal to the overlap of the individual RoC. Otherwise, it will be larger • anu(n) and anu(n-1) has the same RoC, i.e. |z|>|a|, but the RoC of [anu(n) – anu(n-1)] or d(n) is the entire z-plane Z Transform (1) - Hany Ferdinando

21. Time Shifting • x(n-m) ↔ z-mX(z) • RoC of z-mX(z) is R, except for the possible addition or deletion of the origin of infinity • For m>0, it introduces pole at z = 0 and the RoC may not include the origin • For m<0, it introduces zero at z = 0 and the RoC may include the origin Z Transform (1) - Hany Ferdinando

22. Frequency Shifting • ej(Wo)nx(n) ↔ X(ej(Wo)z) • RoC is R • The poles and zeros is rotated by the angle of Wo, therefore if X(z) has complex conjugate poles/zeros, they will have no symmetry at all Z Transform (1) - Hany Ferdinando

23. Time Reversal • x(-n) ↔ X(1/z) • RoC is 1/R Z Transform (1) - Hany Ferdinando

24. Convolution Property • x1(n)*x2(n) ↔ X1(z)X2(z) • RoC is R1∩R2 • The behavior of RoC is similar to the linearity property • It says that when two polynomial or power series of X1(z) and X2(z) are multiplied, the coefficient of representing the product are convolution of the coefficient of X1(z) and X2(z) Z Transform (1) - Hany Ferdinando

25. Differentiation • RoC is R • One can use this property as a tool to simplify the problem, but the whole concept of z transform must be understood first… Z Transform (1) - Hany Ferdinando

26. Next… For the next class, students have to read Z transform: • Signals and Systems by A. V. Oppeneim ch 10, or • Signals and Linear Systems by Robert A. Gabel ch 4, or • Sinyal & Sistem (terj) ch 10 Z Transform (1) - Hany Ferdinando