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 [85]. 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.

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Subsections

• Example: The General Biquad PFE

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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.