of Microprocessors

DIGITAL TONE GENERATION TECHNIQUES 421
full scale is sent out. Rectangular waves with the width specified by a
parameter are ne-d.rly as easy; simply compare the sawtooth samples with the
parameter rather than with zero.
Conversion to a triangular waveform requires a little more manipulation,
although it still parallels the analog operation. Full-wave rectification is
equivalent to an absolute value function, thus the first step is to test the
sawtooth sample and negate it if it is negative. A potential problem exists,
however, because there is no positive equivalent of negative full scale in
twos-complement arithmetic. If this value is seen, simply convert to the
largest positive numbet which results in an ever so slightly clipped triangle
wave. The next step is to center the triangle, which is accomplished by
subtracting one-half of full scale from the absolute value. The result now is a
triangle wave but with one-half of the normal amplitude. A final shift left by
1 bit doubles the amplitude to full scale.
Conversion to a sine wave is most difficult. The analog-rounding circuits
used to do the job have as their digital equivalent either the evaluation
of an equation representing the rounding curve or a table lookup. Of these
two, table lookup is far faster but does require some memory for the table.
The brute-force way to handle table lookup is to simply take a sawtooth
sample and treat it as all integer index into a table of sines. The table entry
would be fetched and returned as the sine wave satnple. Memory usage can be
cut by a factor of four by realizing that the sine function is redundant. Thus,
if the sawtooth sample, S, is between zero and one-halffull scale use the table
entry directly. If it is between one-half and full scale, look into the table at
1.°- S. If S is negative, negate the table entry before using it.
Still, if the sawtooth samples have very many significant bits, the table
size can become quite large. One could truncate S by just ignoring the less
significant bits and looking up in a smaller table. Rounding is another
possibility that is implemented simply by adding the value of the most
significant bit ignored to the sample before truncation. As we shall see later,
rounding has no effect on the audible portion of the error. Generally, truncation
is accurate enough if a reasonable size table is used. For example, a
256-entry table, which would require 512 bytes, using symmetry would be
the equivalent of 1,024 entries. The distortion incurred by using this table
would be approximately 54 dB below th~ signal level, or the equivalent of
0.2% distortion. Actually, the kind of distortion produced will sound more
like noise so a SiN ratio is the more appropriate measure. At very low
frequencies, the noise amplitude will appear to be modulated by the signal.
Linear Interpolation
The most accurate approach, however, is interpolation between the sine
table entries. Linear interpolation gives good results for very gentle curves
such as sine waves and, if done perfectly, could be expected to reduce the
noise level to -103 dB based on a 256-entry table with symmetry, which is
the limit of 16-bit samples anyway. Figure 13-1 shows generalized linear