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112 Chapter 4 THE z-TRANSFORM
4.3 INVERSION OF THE z-TRANSFORM
From equation (4.3), the inverse z-transform computation requires an
evaluation of a complex contour integral that, in general, is a complicated
procedure. The most practical approach is to use the partial fraction expansion
method. It makes use of the z-transform Table 4.1 (or similar
tables available in many textbooks). The z-transform, however, must be
a rational function. This requirement is generally satisfied in digital signal
processing.
Central Idea
• When X(z) isarational function of z −1 ,itcan be expressed as a sum
of simple factors using the partial fraction expansion. The individual
sequences corresponding to these factors can then be written down
using the z-transform table.
The inverse z-transform procedure can be summarized as follows:
Method
• Given
• express it as
X(z) = b 0 + b 1 z −1 + ···+ b M z −M
1+a 1 z −1 + ···+ a N z −N ,R x− < |z|