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Z-Transform
Book chapter

Z-Transform

Fernand Cohen
Signals, Systems, and Transforms
2025
DOI: https://doi.org/10.1002/9781394300877.ch6

Abstract

Laplace Domain Complex Frequency Continuous System Communication systems Digital systems Image processing Transfer Function Sum Of Terms Differential Equations Laplace Transform Table lookup MATLAB Rational Map Time-reversal Directional antennas Rational Function Mathematics computing Rational System Causal System Contour Integration Unit Circle Time-reversal Symmetry Complex Integration Continuous Signal Continuous-time Transfer Function Of System Inverse System Complex Zeros Data structures Roots Of Polynomial Poles And Zeros Communication system signaling Discrete System Impulse Response Discrete-time Causal Sequence Publishing
Summary The Z‐transform (ZT) is a generalized discrete‐time Fourier transform (DTFT), where the signal in the time domain is modulated by a decaying exponent that forces the function to have the DTFT for values of the coefficients in the exponent function. This chapter examines the region of convergence for rational signals that are causal, anticausal, and noncausal, both of infinite as well as of finite duration. It describes the major properties of the ZT, and shows how these could be used and combined in obtaining ZT and inverse ZT pairs. The chapter explains rational systems, their transfer function, which is the ZT of the impulse response, the poles and zeros associated with the system. It derives a relationship between the Laplace transform of a continuous signal and the ZT of the sampled discrete, and explores that the ZT of the signal obtained by sampling a rational continuous signal is no longer rational.

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