Instead of breaking up a filter into a series of second-order sections,
as discussed in the previous section, we can break the filter up into a
parallel sum of second-order sections. Parallel sections are based
directly on the partial fraction expansion (PFE) of the filter
transfer function discussed in §6.8. As discussed in
§6.8.3, there is additionally an FIR part when the
order of the transfer-function denominator does not exceed that of the
numerator (i.e., when the transfer function is not strictly
proper). The most general case of a PFE, valid for any finite-order
transfer function, was given by Eq. (6.19), repeated here for convenience:

where denotes the number of distinct poles, and
denotes the multiplicity of the th pole. The polynomial is
the transfer function of the FIR part, as discussed in §6.8.3.

The FIR part is typically realized as a tapped delay line, as shown in
Fig.5.5. For distinct poles, the recursive terms may be realized
as a parallel sum of complex
one-pole filter sections.