华科奥本海姆讲义7.ppt

CHAPTER 7THE Z-TRANSFORM7.0 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. 7.1 THE Z-TRANSFORM 7.5.8 Differentiation in the z-Domain 7.5.9 The Initial- and Final-Value Theorems Initial-value theorem : Final -value theorem : If x[n] is a causal sequence, i.e., x[n] = 0, for n 0, then Example 7.14 Determine the inverse z-transform for From Example 7.1, and hence, 7.6 ANALYSIS AND CHARACTERIZATION OF LTI SYSTEMS USING z-TRANSFORM The z-transforms of the input and the output of an LTI system are related through multiplication by the z-transform of the impulse response of the system: Y(z) = X(z) H(z) The ROC associated with the system function for a causal system is the exterior of a circle in the z-plane. A discrete-time LTI system is causal if and only if the ROC of its system function is the exterior of a circle, including infinity. A discrete-time LTI system with rational system function H(z) is causal if and only if: (a) the ROC is the exterior of a circle outside the outermost pole; and (b) with H(z) expressed as a ratio of polynomials in z, the order of the numerator cannot be greater than the order of the denominator. system function A causal LTI system with rational system function H(z) is stable if and only if all of the poles of H(z) lie inside the unit circle ― i.e., they must all have magnitude smaller than 1. An LTI system is stable if and only if the ROC of its system function H(z) includes the unit circle, |z| = 1. For an LTI system which is described by a linear constant-coefficient difference equation of the form Its system function (transfer function) is: Example 7.15 Consider a system with system function whose algebraic expression is We can conclude that the system is not