of Microprocessors

MUSIC SYNTHESIS SOFTWARE 651
operands. This means that the number of fraction bits in the product is
the sum of the number of fraction bits in the operands.
From the foregoing we can conclude that if the operands are of equal
length, the result length is twice as great. Actually, it is one bit shorter when
both operands are signed (provided that largest negative numbers are
avoided). It is also easy to see that when both operands are fractions that the
range of the result is unchanged, although the resolution is increased. Thus,
fractional arithmetic is attractive because a chain of multiplications will not
increase the range of numbers to be handled. Note that, if both operands are
signed binary fractions, the binary point in the result will be two positions to
the right of the sign bit of the result, that is, the sign bit will be duplicated.
Thus, one would normally shift the double-length fractional product left one
position before using it.
One may be tempted to conclude that the excess resolution that may
result can always be discarded without any loss of significant data. This is not
generally true if one of the factors is small. SiN ratio will be lost if the excess
resolution is discarded in such a situation, but the absolute noise level
relative to full scale will remain constant. Ideally, multiplication and other
arithmetic operations are done with word lengths longer than the initial
input samples or final output samples in order to eliminate "computational
noise" from the results. Of course, this must be tempered by a possible
increase in computation time, particularly when using machines like the
6502.
Division
Division is by far the most difficult fixed-point arithmetic operation to
control. Fortunately, it is not needed very often in signal-processing work
and in many of the remaining cases it can still be eliminated. Division
operands must be carefully distinguished. The numerator is called the dil!idend
and the denominator is called the divisor. Like multiplication, there is
no restriction on the position of the operands' binary points, although there
is definitely a practical restriction. Unlike multiplication, nothing concrete
can be said in general about the range and resolution of a division result. This
should be obvious, since division by very small numbers gives rise to a very
large quotient, while for most operands there is no limit to how far the
fractional part of the quotient can be carried.
Fixed-point binary division is customarily done with a dividend word
length precisely double the divisor word length. The quotient is restricted to
a word length equal to the divisor. With these restrictions, the rules of
binary division are as follows:
1. The upper half of the dividend must be numerically smaller than the
numerical value of the divisor to avoid overflow.