2.3. The inverse z-transform – 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 Xz(z), has a Laurent serial development, we have:

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 (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 discrete-time causal signal {x(k)} of the z-transform Xz (z). We have, by definition:

By integrating these qualities the length of a closed contour C to the interior of the region of convergence of the transform Xz(z) by turning around 0 once in the positive direction, we get:

By taking an expression of z in the form of z = ρ ej, we easily arrive at:

Now, using the residual theorem, this sum corresponds to the sum of the residuals of Xz (z)zk-l surrounded by C.

Reminders: when pn is a rth order pole of the expression Xz (z)zk−1, we can express Xz(z)zk−1 in the form of a rational fraction of the type . The residual taken in pn is then equal to:

With a pole of the order of multiplicity 1, the expression is reduced to:

EXAMPLE 2.6.– we determine that the discrete-time causal signal whose z-transform equals .

Calculating this integral involves the one pole e-2 of the order in multiplicity 1. From this we get:

2.3.2.2. Development in rational fractions

With linear systems, the expression of the z-transform is presented in the form of a rational fraction; so we will present a decomposition of X(z) into basic elements.

Let . The decomposition into basic elements helps us express Xz (z) in the following form:

where r is the number of poles of Xz(z), βi the multiplicity order of the complex pole aj We then get:

The z-transform is written as a linear combination of simple fractions of the order 1 or 2, with which we can easily determine the inverse transforms.

EXAMPLE 2.7.-

Let . We then write that:

from this, the inverse transform corresponds to:

Figure 2.5. Decomposition into subsystems of the system represented by Xz(z)

EXAMPLE 2.8.– here, our purpose is to find the inverse z-transform of Xz(z) represented by the relation for |z} > 2.

The decomposition into basic elements allows us to express Xz (z) as follows:

from which

2.3.2.3. Development by algebraic division of polynomials

When the expression of the z-transform appears in the form of rational fractions, , we can also obtain an approximate development by carrying out the polynomial division of N(z) by D(z), on condition that the ROC contains 0 or infinity.

A division will be done according to the positive powers of z if the convergence region contains 0 and according to the negative powers of z if the convergence region contains infinity.

EXAMPLE 2.9.-let which corresponds to the expression of the transfer function of a filter used for voice signal analysis. Since the ROC contains infinity, we then carry out the polynomial division according to the negative powers of z.

We obtain:

The corresponding sequence is represented by: