A Simple Example on the Application of the Bilinear Transform

Let's consider the following polynomial of the complex variable z

It could be the characteristic polynomial of a discrete-time LTI system.

We aim to infer the location of the polynomial

's roots, depending on the parameters a and b. Does any set of values for the parameters exist that guarantees obtaining polynomial

roots with modulus (the magnitude) less than one? Or with the modulus equal to one? Or with the magnitude higher than unity?

A suitable strategy to obtain the answers to those questions relies on the application of the Bilinear Transform and then on the application of the Routh-Hurwitz theorem on the transformed new polynomial.

The Bilinear Transformation: a Recap

Substitute

into

, thus obtaining

and hence one gets

with suitable coefficients

Given the expression of

, how to compute the transformed polynomial

using MATLAB?

% Initialisation
clear
clc

How to define the polynomial

in MATLAB?

syms z a b w

assume(a, 'real');

assumeAlso(b, 'real');

assumptions

ans =

Now, let's define the polynomial

as a symbolic expression depending on the (symbolic) variable z

pz = z^2+a*z+b

pz =

whos pz z w a b

  Name      Size            Bytes  Class    Attributes

  a         1x1                 8  sym                
  b         1x1                 8  sym                
  pz        1x1                 8  sym                
  w         1x1                 8  sym                
  z         1x1                 8  sym                

How to perform the Bilinear Transform?

According to the general formulation above, the polynomial

can be determined as follows

Using MATLAB, the polynomial is

q(w) = collect(simplify(expand(((w-1)^2) *subs(pz, z, ((w+1)/(w-1))))),w)

q(w) =