## 2.2. The z-transform

### 2.2.1. *Representations and summaries*

With analog systems, the Laplace transform *X*_{s}(*s*) related to a continuous function *x*(*t*), is a function of a complex variable *s* and is represented by:

Chapter written by Mohamed NAJIM and Eric GRIVEL.

This variable exists when the real part of the complex variable *s* satisfies the relation:

with *r* = −∞ and *R* = +∞, *r* and *R* potentially characterizing the existence of limits of *X*_{s}, (*s*).

The Laplace transform helps resolve the linear differential equations to constant coefficients by transforming them into algebraic products.

Similarly, we introduce the z-transform when studying discrete-time signals.

Let {*x*(*k*)} be a real sequence. The bilaterial or two-sided z-transform *X*_{z}(*z*) of the sequence {*x*(*k*)} is represented as follows:

where *z* is a complex variable. The relation (2.3) is sometimes called the direct z-transform since this makes it possible to transform the time-domain signal {*x*(*k*)} into a representation in the complex-plane.

The z-transform only exists for the values of z that enable the series to converge; that is, for the value of z so that *X*_{z}(*z*) has a finite value. The set of all values of z satisfying these properties is ...