信号与系统课件第10章 Z变换.ppt

CHAPTER 10THE Z-TRANSFORM10.0 INTRODUCTION z-transform is the discrete-time counterpart of the Laplace transform, however, they have some important distinctions that arise from the fundamental differences between continuous-time and discrete-time signals and systems. z-transform expand the application in which Fourier analysis can be used. 10.1 THE Z-TRANSFORM Then A problem: from the ROC, what can we know is just the x[n] is right sided, but not causal, why we finish the discussion at n=0? In fact, if we examine the cases when n takes values of -1,-2,…, we will find that x[n]=0 for n=-1,-2, … Because Consequently, For , z=0 is (n+1)th-order pole; for , z=0 is not the singularity of any more, thus in this case, we have 10.4 GEOMETRIC EVALUATION OF THE FOURIER TRANSFORM FROM THE POLE-ZERO PLOT In the discrete-time case, the Fourier transform can again be evaluated geometrically by considering the pole and zero vectors in the z-plane. However, since in this case the rational function is to be evaluated on the contour |z| = 1, we consider the vectors from the poles and zeros to the unit circle rather than to the imaginary axis. Consider a first-order causal discrete-time system with a impulse response: Its z-transform is For |a| 1, the ROC includes the unit circle, and consequently, the Fourier transform of h[n] converges and is equal to H(z) for . the pole-zero plot for H(z), including the vectors from the pole (at z = a) and zero (at z = 0) to the unit circle. Magnitude of the frequency response for a = 0.95 and a = 0.5 Phase of the frequency response for a = 0.95 and a = 0.5 10.5 PROPERTIES OF THE z-TRANSFORM 10.5.1 Linearity 10.5.2 Time Shifting Note: ROC is at least the intersection of R1 and R2, which could be empty, also can be larger than the intersection. If and then If then Except for the possible addition or deletion of the origin or infinity. Example 10.12 Consider the signal