of Microprocessors

554 MUSICAL ApPLICATIONS OF MICROPROCESSORS
tion is straightforward as well. Even when implemented in software, it is
reasonably efficient if the number of frequency bands is small.
A general block diagram of the structure to be implemented is shown
in Fig. 16-6. This is essentially an extension of the analog channel diagram
into digital form. The input signal is a string of samples at the normal audio
sample rate, which will be assumed to be 20 ksls in this discussion. The
sampled signal passes through a bandpass filter, rectifier, and low-pass filter
in succession, all of which operate at the audio sample rate. The output of the
low-pass filter, however, changes very slowly and therefore can be resampled
at a much lower frequency to provide spectral frames at a reasonable rate such
as 100/sec. As a computer program, the analyzer would essentially accept
samples continuously and return a spectrum frame for every 200 input
samples.
The first element to consider is the bandpass filter. Since only a
bandpass response is needed and the center frequencies are fixed, the cannonical
form will be used because it requires only two multiplications per
sample.
The next task is to select the center frequencies of the filters. The
spectrum sample period of 10 msec suggests that 100-Hz bands are optimum,
although wider or narrower bands can be used as well. A bandwidth
narrower than 100 Hz simply means that the bandpass filter will respond
slowly to signal changes and, as a result, the channel output will be oversampled.
A bandwidth greater than 100 Hz means that the low-pass filter and
sampler following the rectifier will limit the channel response speed rather
than the bandpass filter.
Let's assume, then, that we wish to minimize the number of frequency
bands yet retain enough data to provide a good representation of significant
audible features. For illustration there will be 30 frequency bands scaled on a
quasiexponential scale with bandwidths ranging from 50 Hz at the low end
to 500 Hz at 7.5 kHz. Frequencies above 7.5 kHz are not considered, since
the low-pass filter used in A-to-D conversion has probably attenuated them
anyway.
It is convenient to specify lower and upper cutoff frequencies for the
filters, especially when using a mixrure of bandwidths. The center frequency,
however, is what is needed to design the filter and it is not exactly midway
between the upper and lower cutoff points. It is, in fact, the geometric mean
of Ph and PI and is given by Pc = VPIFh. where P, is the center frequency.
This formula holds for any definition of cutoff frequency as long as the same
definition applies to both PI and Ph.
Given the center frequencies and bandwidths, the last task is to compute
the filter Q factors. Since the percentage bandwidths vary from band to
band as well as the bandwidths themselves, each filter will have a different Q.
But before the Qs can be determined, the inevitable overlap between bands
must be considered. Overlap is bound to occur because the cutoff slope at the
band edges is finite. If the Qs are made high in order to minimize overlap as