# A few things about the IIR filter

The IIR filter or Infinite Impulse Response Filter is a pure digital implementation of the classical passive filters. Commonly derived from a passive filter design the IIR has many of the characteristics of the passive design.

## IIR filter design

The stereo generator employs IIR designs in two places. The pre-emphasis filter and the audio processing band filters.

The audio processing filters split the audio band into three parts so that each audio stream can be processed in the optimum way for a given part of the audio spectrum. For example the attack-decay parameters will be much slower for frequencies below 300Hz to those above 3KHz.

## IIR filter formulas

Recursive equation characterisation of realisable IIR digital filters. Where h(k) is the theoretically infinite duration impulse response. a and b are the k filter coefficients. x(n) and y(n) are the input and output of the filter.

The filter coefficients (a and b) determine the filter characteristics and as such are the objective of the filter design. Unlike the FIR filter the IIR is not inheritantly stable. Due to the variety of feedback possible and to output being dependent upon both past and present samples. This can be likened a positive feedback loop where the feedback level can be increased, thereby narrowing the response until oscillation occurs.

All the filters used here are a realisation of the second order IIR equation. A basic IIR second order building block can be implemented as a canonic form or direct form. Both of which are shown here.

The direct form is used for the pre-emphasis filter. From the general equation: We derive the direct format second order IIR equation: Where a0, a1, a2, b0, b1, b2 are the coefficients with b0 = 1

## IIR Multi band filter

The audio processor incorporated in the stereo generator divides the audio spectrum into three bands for optimum processing of each frequency range. The bands used were <300Hz, ≥300Hz to <3000Hz and ≥3000Hz.

The filters employed were a Low Pass at 300Hz, a High Pass of 300Hz, a Low Pass of 3000Hz and a High Pass of 3000Hz. The simple direct form Bi-quad is used for each filter block.

Note: The filter prototypes are derived from Maximally Flat Butterworth synthesis so that the -6db points of each coincide. This means when the the three pos-processed audio streams are re-combined the overall response will be flat if there is no expansion or compression. Obviously if any of the audio streams are "treated" in any way the transition levels will not coincide.

The DSP calculation of each first order bi-quad is the same as for the pre-emphasis filter with the coefficients as appropiate for the band in question.