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Slide 1 Click to edit Master subtitle style Nguyen Thanh Tuan, M Eng Department of Telecommunications (113B3) Ho Chi Minh City University of Technology Email nttbk97@yahoo com z Transform Chapter 5 Di[.]
Chapter z-Transform Nguyen Thanh Tuan, Click M.Eng to edit Master subtitle style Department of Telecommunications (113B3) Ho Chi Minh City University of Technology Email: nttbk97@yahoo.com The z-transform is a tool for analysis, design and implementation of discrete-time signals and LTI systems Convolution in time-domain multiplication in the z-domain Digital Signal Processing z-Transform Content z-transform Properties of the z-transform Causality and Stability Inverse z-transform Digital Signal Processing z-Transform The z-transform The z-transform of a discrete-time signal x(n) is defined as the power series: X ( z) n 2 1 2 x ( n ) z x ( ) z x ( ) z x ( ) x ( ) z x ( ) z n The region of convergence (ROC) of X(z) is the set of all values of z for which X(z) attains a finite value ROC {z C | X ( z ) x ( n) z n } n The z-transform of impulse response h(n) is called the transform function of the filter: H ( z) n h ( n ) z n Digital Signal Processing z-Transform Example Determine the z-transform of the following finite-duration signals a) x1(n)=[1, 2, 5, 7, 0, 1] b) x2(n)=x1(n-2) c) x3(n)=x1(n+2) d) x4(n)=(n) e) x5(n)=(n-k), k>0 f) x6(n)=(n+k), k>0 Digital Signal Processing z-Transform Example Determine the z-transform of the signal a) x(n)=(0.5)nu(n) b) x(n)=-(0.5)nu(-n-1) Digital Signal Processing z-Transform z-transform and ROC It is possible for two different signal x(n) to have the same ztransform Such signals can be distinguished in the z-domain by their region of convergence z-transforms: and their ROCs: ROC of a causal signal is the exterior of a circle Digital Signal Processing ROC of an anticausal signal is the interior of a circle z-Transform Example Determine the z-transform of the signal x(n) a nu(n) b nu(n 1) The ROC of two-sided signal is a ring (annular region) Digital Signal Processing z-Transform Properties of the z-transform Linearity: if and z x1 (n) X ( z ) with ROC1 z x2 (n) X ( z ) with ROC2 then z x(n) x1 (n) x2 (n) X ( z) X1 ( z) X ( z) with ROC ROC1 ROC2 Example: Determine the z-transform and ROC of the signals a) x(n)=[3(2)n-4(3)n]u(n) b) x(n)=cos(w0 n)u(n) c) x(n)=sin(w0 n)u(n) Digital Signal Processing z-Transform Properties of the z-transform Time shifting: if then z x(n) X ( z) z x(n D) z D X ( z) The ROC of z D X (z ) is the same as that of X(z) except for z=0 if D>0 and z= if D