19 How to inverse the Z transform?

These examples show to inverse a Z transform.

19.0.1 example 1

Problem: Given

$$F\left( z\right) =\frac{z}{z-1}%$$

find $x[n]=F^{-1}\left( z\right)$ i.e. the inverse Ztransform.

Mathematica solution. Example 1

Note, the MATLAB solution below uses the following function found in the toolbox that comes with the book Digital Signal Processing by Proakis and Ingle.

function [x,n] = impseq(n0,n1,n2)
% Generates x(n) = delta(n-n0); n1 <= n,n0 <= n2
% ----------------------------------------------
% [x,n] = impseq(n0,n1,n2)
%
if ((n0 < n1) | (n0 > n2) | (n1 > n2))
 error('arguments must satisfy n1 <= n0 <= n2')
end
n = [n1:n2];
x = [(n-n0) == 0];

Matlab solution. Example 1

19.0.2 example 2

Problem: Given

$$F\left( z\right) =\frac{5z}{\left( z-1\right) ^{2}}%$$

find $x[n]=F^{-1}\left( z\right)$ i.e. the Inverse Ztransform. We know $x[n]$ is the ramp function in this case with a weight of $k.~$i.e. $x[n]=5n$

Mathematica solution. Example 2

Matlab solution. Example 2