Direct digital IIR filter design is rarely used, for one very simple reason:
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:
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.
