Z Transform, Inverse by partial fraction expansion.

Cuthbert Nyack

In many cases the Z Transform can be written as a ratio of polynomials as shown below.

If the denominator B(z) can be factorised then X(z)/z can be expanded into partial fractions as shown below.

For simple poles, the coefficients are given by:-

and for multiple poles the coefficients can be determined from the following expression as for the Laplace Transform.

Consider the following Z Transform and its denominator factorisation

Expanding as indicated above gives:-

Which simplifies to:-

The Inverse Z transform of each term gives x(k).

Consider the following Z Transform X(z):-

Factorising the denominator

and Expanding in partial fractions gives:-

The inverse of each term can be taken separately

and rearranged to:-

Substituting the numerical values for a and b gives the following sampled damped sinusoid.

The definition of the Z transform meant that for relatively simple signals, the Z transform can be written as a polynomial thereby facilitating the above process. Provided the signal is not too complicated, then this method of finding the inverse Z transform is often the easiest and most convenient to apply.
Compared with the inverse Laplace transform we see that the exponent terms in the inverse Laplace Transform is replaced by power terms in the inverse Z Transform. The reason is indicated in the following equation.

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