Digital Filters Design for Signal and Image Processing

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 ...