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260 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
and, as such, there is no overflow. Hence we will consider the following
representation:
ˆx = A(± xxxxxx
} {{
···x
}
) (6.37)
B fraction bits
where A is a positive scaling factor.
□ EXAMPLE 6.15 Represent the number ˆx = −10.4375 in Example 6.14 using a fraction-only
arrangement.
Solution
Choose A =2 4 =16and B =9.Then
ˆx = −10.4375 = 16 (1101001110)
Hence by properly choosing A and B, one can obtain any fraction-only representation.
Note: The scalar A need not be a power of 2. In fact, by choosing any real number
A we can obtain an arbitrary range. The power-of-2 choice for A,however, makes
hardware implementation a little easier.
□
As discussed in the previous section, there are three main formats for
fixed-point arithmetic, depending on how negative numbers are obtained.
For all these formats, positive numbers have exactly the same representation.
In the following we assume the fraction-only arrangement.
Sign-magnitude format As the name suggests, the magnitude is
given by the B-bit fraction, and the sign is given by the MSB. Thus,
ˆx =
{ 0x 1 x 2 ···x B if x ≥ 0
1x 1 x 2 ···x B if x