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 first and/or 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.
First-Order Complex Resonators
For distinct poles, the recursive terms in the complete partial fraction expansion of Eq.(9.2) can be realized as a parallel sum of complex one-pole filter sections, thereby producing a parallel complex resonator filter bank. Complex resonators are efficient for processing complex input signals, and they are especially easy to work with. Note that a complex resonator bank is similarly obtained by implementing a diagonalized state-space model [Eq.(G.22)].
In practice, however, signals are typically real-valued functions of
time. As a result, for real filters (§5.1),
it is typically more efficient computationally to combine
complex-conjugate one-pole sections together to form real second-order
sections (two poles and one zero each, in general). This process was
discussed in §6.8.1, and the resulting transfer function of
each second-order section becomes
where is one of the poles, and is its corresponding residue. This is a special case of the biquad section discussed in §B.1.6.
When the two poles of a real second-order section are complex, they form a complex-conjugate pair, i.e., they are located at in the plane, where is the modulus of either pole, and is the angle of either pole. In this case, the ``resonance-tuning coefficient'' in Eq.(9.3) can be expressed as
Implementation of Repeated Poles
Fig.9.5 illustrates an efficient implementation of terms due to a repeated pole with multiplicity three, contributing the additive terms
Formant Filtering Example
Series Second-Order Sections