### 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

`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:

*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

giving

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Alternate PFE Methods

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Inverting the Z Transform