of Microprocessors

MUSIC SYNTHESIS SOFTWARE 647
signed twos-complement numbers and unsigned numbers. In an unsigned
number, the weight of each bit is two raised to the bit number power. Thus,
a single byte in the 6502 can represent any number from 0 through +25 5.
Unsigned numbers are therefore always positive. In programming, they are
normally used to represent addresses but in signal processing they are most
useful as multiplying coefficients in digital filters, etc. Besides doubled
dynamic range compared with signed numbers, the use of unsigned numbers
simplifies the associated multiplications.
Signed twos-complement numbers are probably more familiar to most
readers. The bit weights are the same as in unsigned numbers except that bit
7 (the leftmost) has a weight of -128. The number is negative only if this
bit is a 1; thus, it is called the sign bit. The remaining bits are called
magnitude bits, although it must be remembered that they have been complemented
in the case of a negative number. A signed single-byte number
can therefore represent quantities between -128 and + 127. In signal processing,
signed numbers are used to represent audio signal samples that
invariably swing positive and negative.
A signed twos-complement number can be negated, that is, N converted
to -N or -N converted to N', by complementing every bit in the number
and then incrementing the result by one. This will not work for the number
-128, however, because the increment operation overflows and the result
becomes -128 again. Since negation will be required often in signal processing,
mere existence of the largest negative number in a sample stream may
cause an overflow that gives rise to a full-amplitude noise spike. Avoidance of
such overflows usually requires an extra bit of precision in the numbers that
may propagate when calculations are chained together. Thus, it is wise to
prevent the occurrence of the largest negative number in synthesized samples
and to search and correct ADC data by converting any - 128s to - 127
(which amounts to slight clipping rather than a full-scale noise spike) or the
equivalent when other word lengths are used.
Fixed-point arithmetic is often equated with integer arithmetic because
memory addresses are usually involved. In signal processing, fractions and
mixed numbers are more common. There are at least two ways to think about
such numbers. One involves the concept of a scale factor. The numbers
being manipulated are considered to be the product of the actual quantity
and a scale factor that the programmer keeps track of. The scale factor is
chosen so that when it multiplies the numbers being handled, the results are
pure integers, which are "compatible" with integer arithmetic. In the course
of a chained calculation, the scale factors change so as to maintain the largest
range of integers possible without overflowing the chosen word lengths.
The other method, which will be used here, involves the concept of a
binary point. The function, meaning, and consequences of binary-point position
are the same as rhey are with decimal points and decimal arithmetic.
The microcomputer can be likened to an old mechanical calculator or a slide