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Digital Filters Design for Signal and Image Processing by Mohamed Najim

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2.2. The z-transform

2.2.1. Representations and summaries

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

images

Chapter written by Mohamed NAJIM and Eric GRIVEL.

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

images

with r = −∞ and R = +∞, r and R potentially characterizing the existence of limits of Xs, (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 Xz(z) of the sequence {x(k)} is represented as follows:

images

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 Xz(z) has a finite value. The set of all values of z satisfying these properties is ...

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