# 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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© 2000 Cuthbert A. Nyack.