TABLE OF CONTENTS The counterpart of the Laplace transform for discrete-time systems is the z-transform. The Laplace transform converts integro-differential equations into algebraic equations. In the same way, the z-transforms changes difference equations into algebraic equations, thereby simplifying the analysis of discrete-time systems. The z-transform method of analysis of discrete-time systems parallels the Laplace transform method of analysis of continuous-time systems, with some minor differences. In fact, we shall see that the z-transform is the Laplace transform in disguise.
The behavior of discrete-time systems is similar to that of continuous-time systems (with some differences). The frequency-domain analysis of discrete-time systems is based on the fact (proved in Section 3.8-3) that the response of a linear, time-invariant, discrete-time (LTID) system to an everlasting exponential z n is the same exponential (within a multiplicative constant) given by H [ z] z n. We then express an input x [ n] as a sum of (everlasting) exponentials of the form z n. The system response to x [ n] is then found as a sum of the system's responses to all these exponential components. The tool that allows us to represent an arbitrary input x [ n] as a sum of (everlasting) exponentials of the form z n is the z-transform.
We define X[ z], the direct z-transform of x[ n], as
where z is a complex variable. The...
