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

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

Z Transform (1) - Hany Ferdinando 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

Z Transform (1) - Hany Ferdinando 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

Z Transform (1) - Hany Ferdinando 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

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

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

Z Transform (1) - Hany Ferdinando 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

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

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

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

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

Z Transform (1) - Hany Ferdinando 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

Z Transform (1) - Hany Ferdinando 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

Z Transform (1) - Hany Ferdinando 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

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

Z Transform (1) - Hany Ferdinando 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

Z Transform (1) - Hany Ferdinando 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

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

Z Transform (1) - Hany Ferdinando 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

Z Transform (1) - Hany Ferdinando 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

Z Transform (1) - Hany Ferdinando 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

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

Z Transform (1) - Hany Ferdinando 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

Z Transform (1) - Hany Ferdinando 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

Z Transform (1) - Hany Ferdinando 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