Digital Filters Design for Signal and Image Processing

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 X_z(z), has a Laurent serial development, we have:

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

where ρ belongs to the ROC.

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