文档介绍：Chapter 4 The Z Transform
§4-1 z-Transform of A Discrete-Time Signal
§4-2 Properties of the z-Transform
§4-putation of the Inverse z-Transform
Problems
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Chapter 4 The Z Transform
The chapter deals with the z-transform, which is the discrete-time counterpart of the Laplace transform. The z-transform operates on discrete-time signal x[n] in contrast to Laplace transform, which operates on a continuous-time or analog signal x[t].
The theory of the z-transform and its application to causal linear time-invariant discrete-time systems closely resembles the theory of the Laplace transform and its application to causal linear time-invariant continuous-time systems. In particular, results and techniques in this chapter closely parallel the results and techniques on Laplace transform. However, there are some differences between the transform theory in the continuous-time case and the transform theory in the discrete-time case, although for the most part, these differences are minor.
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§4-1 z-Transform of A Discrete-Time Signal
Given the discrete-time signal x[n], the discrete-time Fourier transform (DTFT) was defined by
Recall that X(Ω) is in general plex-valued function of the frequency variable Ω.
The z-transform of the signal x[n] is generated by adding the factor ρ–n to the summation in (), where ρ is a real number. The factor ρ–n plays the same role as the exponential factor e–σt that was added to the Fourier transform to generate the Laplace transform in the continuous-time case. Inserting ρ–n in () gives
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which can be rewritten as
()
The function X(Ω) given by () is now a function of plex number
()
()
so X should be rewritten as a function of z, which gives
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the function X(z) given by () is the two-sided z-transform of the discrete-time signal x[n]. The one-sided z-transform of x[n], also denoted by X(z), is defined by
As seen from () the one-sided z-transform is a power series in z–n whose coefficients are the val