# Converting Infinite Impulse Response Filters to Parallel Form

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By:
Balazs Bank

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.

Introduction

IIR digital filters are part of most signal processing algorithms. They are used not only in classic filtering applications (low-pass, high-pass, and so on) but also as tools for approximating any given transfer function, e.g., a measured frequency response that we wish to model in discrete time. Compared to FIR filters, IIR filters typically require lower computational resources for the same modeling accuracy. However, care has to be taken to assure their stability: a theoretically stable IIR filter might become unstable when implemented with finite coefficient precision. The problem becomes pronounced when the filter has high order and/or has poles near the unit circle. As a remedy, IIR filters are often implemented as a series or parallel combination of (typically, second-order) subfilters.

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