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
![$ H(z)$](http://www.dsprelated.com/josimages_new/filters/img308.png)
is simply
Thus, the
impulse response of every strictly proper
LTI filter (with distinct
poles) can be interpreted as a
linear combination of sampled
complex exponentials.
Recall that a
uniformly sampled
exponential is the same thing as a
geometric
sequence. Thus,
![$ h$](http://www.dsprelated.com/josimages_new/filters/img580.png)
is a linear combination of
![$ N$](http://www.dsprelated.com/josimages_new/filters/img278.png)
geometric
sequences. The
term ratio of the
![$ i$](http://www.dsprelated.com/josimages_new/filters/img571.png)
th geometric sequence is
the
![$ i$](http://www.dsprelated.com/josimages_new/filters/img571.png)
th pole,
![$ p_i$](http://www.dsprelated.com/josimages_new/filters/img691.png)
, and the
coefficient of the
![$ i$](http://www.dsprelated.com/josimages_new/filters/img571.png)
th
sequence is the
![$ i$](http://www.dsprelated.com/josimages_new/filters/img571.png)
th residue,
![$ r_i$](http://www.dsprelated.com/josimages_new/filters/img692.png)
.
In the improper case, discussed in the next section, we
additionally obtain an FIR part in the z transform to be inverted:
The FIR part (a
finite-order polynomial in
![$ z^{-1}$](http://www.dsprelated.com/josimages_new/filters/img91.png)
) is also easily
inverted by inspection.
The case of repeated poles is addressed in §6.8.5 below.
Next Section: FIR Part of a PFEPrevious Section: PFE to Real, Second-Order Sections