# Introduction to DSP - IIR filters: design by impulse variance

 Direct digital IIR filter design is rarely used, for one very simple reason: nobody knows how to do it While it is easy to calculate the filter's frequency response, given the filter coefficients, the inverse problem - calculating the filter coefficients from the desired frequency response - is so far an insoluble problem. Not many text books admit this. Because we do not know how to design digital IIR filters, we have to fall back on analogue filter designs (for which the mathematics is well understood) and then transform these designs into the sampled data z plane Argand diagram. Note that the filter's impulse response defines it just as well as does its frequency response. Here is a recipe for designing an IIR digital filter: decide upon the desired frequency response design an appropriate analogue filter calculate the impulse response of this analogue filter sample the analogue filter's impulse response use the result as the filter coefficients This process is called the method of impulse invariance. The method of impulse invariance seems simple: but it is complicated by all the problems inherent in dealing with sampled data systems. In particular the method is subject to problems of aliasing and frequency resolution.