of Microprocessors

502 MUSICAL ApPLICATIONS OF MICROPROCESSORS
impulse is equally simple: an infinite number of frequency components all of
equal amplitude and all with zero phase angles.
The filter's output in response to an impulse input tells everything
about its amplitude and phase response but in an easy to use form. Since the
input spectrum is flat and has zero phase, the output spectrum will directly
represent the amplitude and phase response of the filter under test! This
means that the characteristics of a digital filter can be plotted merely by
taking the Fourier transform of its impulse response. This, in fact, is how
many of the filter response curves in this book were plotted. Conversely any
amplitude and phase response may be inverse Fourier transformed to get the
corresponding impulse response. These are very important points and are
illustrated in Fig. 14-14.
If two widely separated impulses are fed to a filter, one would expect to
see two copies of the impulse response. If the first impulse is twice the height
of the second, the first response will have twice the amplitude as well but the
same basic shape. If the impulses are moved closer together, they start to
overlap. Since filters are linear, the composite output is simply the pointby-point
sum of the responses due to each impulse. This applies to any
number of impulses at any spacing.
Although it is already being taken for granted, sampled analog signals
are really strings of impulses, the height of each being proportional to the
sample values. According to the previous paragraph, the output of a filter
receiving such impulses as input is equal to the sum of the responses to each
individual sample impulse as illustrated in Fig. 14-15. This then is the crux
of direct convolution digital filtering.
I mplementatwn
The first step in implementing the algorithm is to obtain the impulse
response in usable form. For digital filtering, samples of the impulse response
are needed, The samples must be taken at the same sample rate as the signal
to be processed. If the rate is different, all filter response frequencies will be
altered in direct proportion to the ratio of the sample rates. Any stable
impulse response will be bounded on both sides by zeroes, which can then be
discarded. Unfortunately, most responses do not suddenly cut off, so a
truncation decision must be made. Since the computation time is directly
proportional to the number of response samples retained, there is an incentive
to retain as few as possible. Very broadly speaking, if all of the response
samples omitted are less than 0.01 of the largest response sample, then only
filter attenuation beyond 40 dB is likely to be disturbed to any significant
degree. Accordingly, retaining samples down to 0.001 will not disturb
attenuation curves less than 60 dB. In any case, the impulse response becomes
a string of numbers.
The calculations involved are quite simple. Assuming that some data
has already gone through the filter, the Ith output sample is equal to the Ith