O'Reilly logo

Digital Filters Design for Signal and Image Processing by Mohamed Najim

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

2.3. The inverse z-transform

2.3.1. Introduction

The purpose of this section is to present the methods that help us find the expression of a discrete-time signal from its z-transform. This often presents problems that can be difficult to resolve. Applying the residual theorem often helps to determine the sequence {x(k)}, but the application can be long and cumbersome. So in practice, we tend to use simpler methods, notably those based on development by division, according to increasing the powers in z−1, which constitutes a decomposition of the system into subsystems. Nearly all the z-transforms that we see in filtering are, in effect, rational fractions.

2.3.2. Methods of determining inverse z-transforms

2.3.2.1. Cauchy's theorem: a case of complex variables

If we acknowledge that, in the ROC, the z-transform of {x(k)}, written Xz(z), has a Laurent serial development, we have:

images

The coefficients τk and νk are the values of the discrete sequence {x(k)} that are to be determined. They can be obtained by calculating the integral images (where C is a closed contour in the interior of the ROC), by the residual method as follows:

images

where ρ belongs to the ROC.

DEMONSTRATION 2.8.– let us look at a ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required