Z Transform 1 Hany Ferdinando Dept of Electrical

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

Overview n n n Introduction Basic calculation Ro. C Inverse Z Transform Properties of Z transform Exercise 2 Z Transform (1) - Hany Ferdinando

Introduction n 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 zdomain it means there is no time variable in the z-domain 3 Z Transform (1) - Hany Ferdinando

Introduction n 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 4 Z Transform (1) - Hany Ferdinando

Introduction n 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 5 Z Transform (1) - Hany Ferdinando

Basic Calculation or n 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 6 Z Transform (1) - Hany Ferdinando

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

Basic Calculation n n 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… 8 Z Transform (1) - Hany Ferdinando

Region of Convergence (Ro. C) n n n 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 9 Z Transform (1) - Hany Ferdinando

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

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

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

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

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

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

Inverse Z Transform Use Ro. C information n Direct division Partial expansion Alternative partial expansion 16 Z Transform (1) - Hany Ferdinando

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

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

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

Linearity n n a x 1(n) + b x 2(n) ↔ a X 1(z) + b X 2(z) Ro. C is R 1∩R 2 If a X 1(z) + b X 2(z) consist of all poles of X 1(z) and X 2(z) (there is no pole-zero cancellation), the Ro. C is exactly equal to the overlap of the individual Ro. C. Otherwise, it will be larger anu(n) and anu(n-1) has the same Ro. C, i. e. |z|>|a|, but the Ro. C of [anu(n) – anu(n-1)] or d(n) is the entire z-plane 20 Z Transform (1) - Hany Ferdinando

Time Shifting n n x(n-m) ↔ z-m. X(z) Ro. C of z-m. X(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 Ro. C may not include the origin For m<0, it introduces zero at z = 0 and the Ro. C may include the origin 21 Z Transform (1) - Hany Ferdinando

Frequency Shifting n n n ej(Wo)nx(n) ↔ X(ej(Wo)z) Ro. C 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 22 Z Transform (1) - Hany Ferdinando

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

Convolution Property n n x 1(n)*x 2(n) ↔ X 1(z)X 2(z) Ro. C is R 1∩R 2 The behavior of Ro. C is similar to the linearity property It says that when two polynomial or power series of X 1(z) and X 2(z) are multiplied, the coefficient of representing the product are convolution of the coefficient of X 1(z) and X 2(z) 24 Z Transform (1) - Hany Ferdinando

Differentiation n Ro. C 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… 25 Z Transform (1) - Hany Ferdinando

Next… For the next class, students have to read Z transform: n n n 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 26 Z Transform (1) - Hany Ferdinando