Inverting the Z Transform

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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

$\displaystyle H_i(z) \eqsp \frac{r_i}{1-p_iz^{-1}}.$

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

$\displaystyle h_i(n) \eqsp r_i p_i^n, \quad n=0,1,2,\ldots\,.$

Thus, the inverse z transform of

is simply

$\displaystyle h(n) \eqsp \sum_{i=1}^N h_i(n) \eqsp \sum_{i=1}^N r_i p_i^n, \quad n=0,1,2,\ldots\,.$

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:

$\displaystyle F(z) \eqsp f_0 + f_1z^{-1}+ f_2z^{-2}+ \cdots + f_K z^{-K} \;\longleftrightarrow\; [f_0,f_1,\ldots,f_K,0,0,\ldots].$

The FIR part (a finite-order polynomial in $z^{-1}$ ) is also easily inverted by inspection.

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

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Previous: PFE to Real, Second-Order Sections
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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Introduction to Digital Filters
    Transfer Function Analysis
       Partial Fraction Expansion
          Inverting the Z Transform