### FIR Part of a PFE

When in Eq.(6.7), we may perform a step of *long division*
of to produce an *FIR part* in parallel with a
strictly proper IIR part:

where

When , we define . This type of decomposition is
computed by the `residuez` function (a matlab function for
computing a complete partial fraction expansion, as illustrated in
§6.8.8 below).

An alternate FIR part is obtained by performing long division on the
*reversed* polynomial coefficients to obtain

where is again the order of the FIR part. This type of decomposition is computed (as part of the PFE) by

`residued`, described in §J.6 and illustrated numerically in §6.8.8 below.

We may compare these two PFE alternatives as follows:
Let denote ,
, and
.
(*I.e.*, we use a subscript to indicate polynomial order, and `' is
omitted for notational simplicity.) Then for
we have two cases:

In the first form, the
coefficients are ``left
justified'' in the reconstructed numerator, while in the second form
they are ``right justified''. The second form is generally more
efficient for *modeling* purposes, since the numerator of the IIR
part (
) can be used to match additional
terms in the impulse response after the FIR part has
``died out''.

In summary, an arbitrary digital filter transfer function with
distinct poles can always be expressed as a parallel combination
of *complex one-pole filters*, together with a parallel FIR part
when . When there is an FIR part, the strictly proper IIR
part may be delayed such that its impulse response begins where that
of the FIR part leaves off.

In artificial reverberation applications, the FIR part may correspond
to the *early reflections*, while the IIR part provides the
*late reverb*, which is typically dense, smooth, and
exponentially decaying [86]. The
*predelay* (``pre-delay'') control in some commercial reverberators
is the amount of pure delay at the beginning of the reverberator's
impulse response. Thus, neglecting the early reflections, the order of
the FIR part can be viewed as the amount of predelay for the IIR part.

#### Example: The General Biquad PFE

The general second-order case with (the so-called
*biquad* section) can be written when as

yielding

The delayed form of the partial fraction expansion is obtained by leaving the coefficients in their original order. This corresponds to writing as a ratio of polynomials in :

giving

Numerical examples of partial fraction expansions are given in §6.8.8 below. Another worked example, in which the filter is converted to a set of parallel, second-order sections is given in §3.12. See also §9.2 regarding conversion to second-order sections in general, and §G.9.1 (especially Eq.(G.22)) regarding a state-space approach to partial fraction expansion.

**Next Section:**

Alternate PFE Methods

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Inverting the Z Transform