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Representation of Numbers 259
negative integers between −2 B−1 and 2 B−1 − 1 are given by
x (e)
∆
=2 B−1 + x (6.35)
For example, using 3 bits, we can represent the numbers from −4 to
3as
-4 -3 -2 -1 0 1 2 3
-+----+----+----+----+----+----+----+-
000 001 010 011 100 101 110 111
Notice that this format is very similar to the two’s-complement format,
but the sign bit is complemented. The arithmetic for this format is similar
to that of the two’s-complement format. It is used in the exponent of
floating-point number representation.
6.6.2 GENERAL FIXED-POINT ARITHMETIC
Using the discussion of integer arithmetic from the last section as a guide,
we can extend the fixed-point representation to arbitrary real (integer
and fractional) numbers. We assume that a given infinite-precision real
number, x, isapproximated by a binary number, ˆx, with the following bit
arrangement:
ˆx = ±
↑
xx ···x
} {{ }
“L”
Sign bit
Integer bits
xx ···x
} {{ }
“B”
Fraction bits
(6.36)
where the sign bit ± is 0 for positive numbers and 1 for negative numbers,
x represents either a 0 or a 1, and represents the binary point. This
representation is in fact the sign-magnitude format for real numbers, as
we will see. The total word length of the number ˆx is then equal to L+B+1
bits.
□ EXAMPLE 6.14 Let L =4and B =5,which means ˆx is a 10-bit number. Represent 1101001110
in decimal.
Solution
ˆx = −(1 × 2 3 +0× 2 2 +1× 2 1 +0× 2 0 +0× 2 −1 +1× 2 −2 +1× 2 −3 +1× 2 −4 +0× 2 −5 )
= −10.4375
in decimal.
□
In many A/D converters and processors, the real numbers are scaled
so that the fixed-point representation is in the (−1, 1) range. This has
the advantage that the multiplication of two fractions is always a fraction
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