### 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

*z*transform of

*z*transform of is simply

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

**Next Section:**

FIR Part of a PFE

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PFE to Real, Second-Order Sections