FIR Part of a PFE
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 . 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.
The general second-order case with (the so-called biquad section) can be written when as
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 :
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.
Alternate PFE Methods
Inverting the Z Transform