Processing for compression - The MPEG - page 58

Infinite-impulse response (IIR) filters respond to an impulse indefinitely and are not necessarily stable, as they have
a return path from the output to the input. For this reason they are also called recursive filters. As the impulse
response in not symmetrical, IIR filters are not phase linear. Audio equalizers often employ recursive filters.
3.5 FIR filters
A FIR filter performs convolution of the input waveform with its own impulse response. It does this by graphically
constructing the impulse response for every input sample and superimposing all these responses. It is first
necessary to establish the correct impulse response. Figure 3.9 (a) shows an example of a low-pass filter which
cuts off at 14 of the sampling rate. The impulse response of an ideal low-pass filter is a sin x / x curve, where the time
between the two central zero crossings is the reciprocal of the cut-off frequency. According to the mathematics, the
waveform has always existed, and carries on for ever.
Figure 3.9: (a) The impulse response of an LPF is a sin x / x curve which stretches from - to + in time. The ends of
the response must be neglected, and a delay introduced to make the filter causal. (b) The structure of an FIR LPF.
Input samples shift across the register and at each point are multiplied by different coefficients. (c) When a single
unit sample shifts across the circuit of Figure 3.9(b), the impulse response is created at the output as the impulse is
multiplied by each coefficient in turn.
The peak value of the output coincides with the input impulse. This means that the filter cannot be causal, because
the output has changed before the input is known. Thus in all practical applications it is necessary to truncate the
extreme ends of the impulse response, which causes an aperture effect, and to introduce a time delay in the filter
equal to half the duration of the truncated impulse in order to make the filter causal. As an input impulse is shifted
through the series of registers in Figure 3.9 (b), the impulse response is created, because at each point it is
multiplied by a coefficient as in Figure 3.9 (c).
These coefficients are simply the result of sampling and quantizing the desired impulse response. Clearly the
sampling rate used to sample the impulse must be the same as the sampling rate for which the filter is being
designed. In practice the coefficients are calculated, rather than attempting to sample an actual impulse response.
The coefficient wordlength will be a compromise between cost and performance. Because the input sample shifts
across the system registers to create the shape of the impulse response, the configuration is also known as a
transversal filter. In operation with real sample streams, there will be several consecutive sample values in the filter
registers at any time in order to convolve the input with the impulse response.