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

Thus, the inverse

Thus, the impulse response of every strictly proper LTI filter (with distinct poles) can be interpreted as a

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University