of Microprocessors

650 MUSICAL ApPLICATIONS OF MICROPROCESSORS
from the right. When shifting right, bits equal to the sign bit must be
brought in. Adding entire bytes to the left of a short number so that it can be
added to a long number is essentially equivalent to shifting right so these
bytes must be filled with sign bits. If an unsigned number is to be added to a
signed number, it is usually necessary to provide a high order extension of
the signed number because of the greater range of the unsigned number.
Rule 2 simply says that to avoid overflow when unknown numbers are
added up, it is necessary to allow for sums larger than the addends. When a
string of equal-length numbers is being added, the word size of the final sum
will not exceed 10gzN more bits than the word sizes of the addends, where N
is the number of numbers added. Ifit is known that the addends do not cover
the full range allowed by their word size, then the word size allowed for the
sum may be less than that given by the rule.
The usable resolution of the sum may be less than the resolution given
by Rule 2. If the addend with less resolution is an audio signal or other
"approximate" value, the usable resolution of the result cannot be any greater
because of quantization noise.
In subtraction, there is a distinction between the operands. When done
on paper, the top number is called the minuend and the number being
subtracted is called the subtrahend. Twos-complement subtraction is generally
accomplished by negating the subtrahend and adding the result to the
minuend. This may be done either by a hardware subtract instruction or by
actually complementing the subtrahend and adding. Since signed operands
were assumed for addition anyway, the properties and rules for subtraction
are the same as for addition.
Often, such as in digital filters, a quantity of numbers that partially
cancel each other may be added up. A property of twos-complement arithmetic
is that the order of addition/subtraction is immaterial. This is true even if
intermediate sums overflow! This is a handy property to keep in mind
because it may allow the word size of intermediate results to be the same as
the addends.
Multiplication
The rules of binary multiplication are quite different from addition and
subtraction. In particular, there is none governing location of the binary
points of the operands. Those governing the range and resolution of the
result are as follows:
1. The range of the result is the product of the ranges of the operands.
This means that the number of magnitude bits in the integer part of
the product is the sum of the number of magnitude bits in the integer
parts of the operands.
2. The resolution of the result is the product of the resolutions of the