Improved frequency selective filters

This paper is concerned with the design and implementation of frequencyselective recursive filters in which the filtered output is incorporated on the input side, at various lags, via a series of feedback terms. Such devices are commonly described as infinite impulse response (IIR) filters. Filters that are devoid of feedback are apt to be described as finite impulse response (FIR) filters. Relatively little attention has been paid to the design of such IIR digital filters, with the result that there are few clear design principles that can be followed. Part of the reason for this deficiency lies in the success of the FIR filters which are usually preferred when one wishes to avoid inducing non linear phase effects in the output. Also, some of the digital IIR filters that have been derived by translating classical analogue designs have proved eminently successful; and this has reduced the incentive for innovation. In this paper, our purpose is to design filters with very few coefficients, which show a rapid transition from the pass band to the stop band. Improvements in the rate of transition are usually secured by increasing the number of coefficients within a given family of filters; but we wish to avoid this recourse which is liable to exacerbate the start-up problems that affect the processing of short nonstationary sequences In our case, the rapid transition is achieved via a filter that operates close to the borders of instability. This can lead to the rapid propagation of numerical rounding error when the filter is realised in low-precision arithmetic—i.e. 16 bits or less. The filters become increasingly unstable as their poles approach the