Converting Infinite Impulse Response Filters to Parallel Form [Tips & Tricks]

Discrete-time rational transfer functions are often converted to parallel second-order sections due to better numerical performance compared to direct form infinite impulse response (IIR) implementations. This is usually done by performing partial fraction expansion over the original transfer function. When the order of the numerator polynomial is greater or equal to that of the denominator, polynomial long division is applied before partial fraction expansion resulting in a parallel finite impulse response (FIR) path. This article shows that applying this common procedure can cause a severe dynamic range limitation in the filter because the individual responses can be much larger than the net transfer function. This can be avoided by applying a delayed parallel form where the response of the second-order sections is delayed in such a way that there is no overlap between the IIR and FIR parts. In addition, a simple least-squares procedure is presented to perform the conversion that is numerically more robust than the usual Heaviside partial fraction expansion. Finally, the possibilities of converting series second-order sections to the delayed parallel form are discussed.