### Inverting the Z Transform

The partial fraction expansion (PFE) provides a simple means for inverting the z transform of rational transfer functions. The PFE provides a sum of first-order terms of the form

It is easily verified that such a term is the z transform of

Thus, the inverse z transform of is simply

Thus, the impulse response of every strictly proper LTI filter (with distinct poles) can be interpreted as a linear combination of sampled complex exponentials. Recall that a uniformly sampled exponential is the same thing as a geometric sequence. Thus, is a linear combination of geometric sequences. The term ratio of the th geometric sequence is the th pole, , and the coefficient of the th sequence is the th residue, .

In the improper case, discussed in the next section, we additionally obtain an FIR part in the z transform to be inverted:

The FIR part (a finite-order polynomial in ) is also easily inverted by inspection.

The case of repeated poles is addressed in §6.8.5 below.

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FIR Part of a PFE
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PFE to Real, Second-Order Sections