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.
Dealing with Repeated Poles Analytically
Series Combination is Commutative