The general second-order case with (the so-called biquad section) can be written when as
To perform a partial fraction expansion, we need to extract an order 0 (length 1) FIR part via long division. Let and rewrite as a ratio of polynomials in :
Then long division gives
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 :
Long division now looks like
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.