The z-transform is
a useful tool in linear systems analysis. However, just as important as techniques
for finding the z-transform of a sequence are methods that may be used to
invert the z-transform and recover the sequence x(n) fromX(z). Three possible approaches are described below.
a simple and
straightforward approach to find the inverse z-transform is to perform a
partial fraction expansion of X(z). Assuming that p > q, and
that all of the roots in the denominator are simple, a i ¹ ak for i ¹ k, X(z) may be expanded as
follows:
If p £ q,
the partial fraction expansion must include a polynomial in z-1of
order (p-q). The coefficients of this polynomial may be found by long
division (i.e., by dividing the numerator polynomial by the denominator). For multiple-order
poles, the expansion must be modified. For example, if X(z) has a
second-order pole at z = a k, the expansion will include two terms,
Thus, we may find x(n)
using a partial fraction expansion of X(z) and then evaluate the
sequence at n = 4. With this approach, however, we are finding the
values of x(n) for all n. Alternatively, we could perform long
division and divide the numerator of X(z) by the denominator. The
coefficient multiplying z -4 would then be the value of x(n)
at n = 4, and the value of the integral. However, because we are only
interested in the value of the sequence at n = 4, the easiest approach
is to evaluate the integral directly using the Cauchy integral theorem. The
value of the integral is equal to the sum of the residues of the poles of X(z)z3inside the unit circle. Because