of Microprocessors

524 MUSICAL ApPLICATIONS OF MICROPROCESSORS
Using the table for interpolation is really quite simple. First we will
restrict the unknown point to be within one-half of a sample period, either
side, of the middle tabulated point as shown in Fig. 14-32. This middle
point will be designated So while those on the left are numbered S- 1, S- 2,
etc., and conversely for those on the right. Next, the position of the unknown
point with respect to the central point will be expressed to the nearest
1/32 of a sample interval and will be called A. Thus, in the example, A will
always be between -16/32 and + 15/32 inclusive. To compute the unknown
sample value, one simply iterates i through the 13 tabulated points from -6
through +6 multiplying each point value by the contents of the table at A +i
and adding up the products. The sum is the interpolated value! When
looking into the table the sign of the argument is ignored, which implements
the symmetry. If the argument is greater than 6 or less than -6, zero
should be returned.
Note that the location of the unknown point is quantized to 1/32 of a
sampling interval. Abiding by this restriction limits the variety of samplerate
ratios. One could reduce this restriction by increasing the size of the
table or by linear interpolation in the table, which is now allowable because
the tabulated function is densely sampled. One could also apply the table to
compute two interpolated samples, one on either side of the arbitrary unknown
sample and linearly interpolate between them, thus reducing interpolation
effort. This is permissible because now the waveform is grossly oversampled
(by a factor of 32 in the example) and linear interpolation will not
add nearly as much noise as before. The two methods are in fact exactly
equivalent but the second only requires one linear interpolation rather than
13.
Actually writing a sample-rate-conversion program is somewhat tricky
because both "future" and "past" input samples are needed and because of the
different input and output sample rates. The problem is generally solved by
using an input buffer, an output buffer, and a simulated shift register for the
samples used in the interpolation. The routine would also accept a sample
rate ratio, R, which would be a mixed number either greater than or less than
unity.
In practice, the routine would initially be given a full input buffer, an
empty output buffer, and a shift register full of zeroes. It would also keep
track ofa time variable, A , which corresponds to the A used in the interpolation
example and which is constrained to the range of - o. 5 to +o. 5. To
generate the next output sample, R is added to A. If the sum is between
-0.5 and +0.5, an interpolation is performed and the computed sample is
put into the output buffer. If the sum is greater than +0.5, the shift
register is shifted one position, the next input sample is taken from the input
buffer and put into the vacant shift register slot, and 1.0 is subtracted ftom
A. This is repeated if necessary until A is less than +0.5, at which point an
interpolation is performed generating another output sample and so forth.