## 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 ...