Page 479 - DSP Integrated Circuits
P. 479
464 Chapter 11 Processing Elements
Let x denote the number obtained by complementing each digit in the corre-
sponding number x. It is easy to show that
If this number is stored in a register holding k integer digits and W<j - k frac-
tional digits, the register overflows. The most significant digit is discarded and the
result is zero. This is equivalent to taking the remainder after dividing by r*.
Now, if we select R = r^ we obtain the so-called radix complement, or two's-
complement representation in the binary case
No correction of Equation (11.4) is needed since the error term R is discarded
when computing R + (x—y). Another possible choice is
This choice is called diminished-radix complement, or one's-complement rep-
resentation in the binary case. We obtain the complement
The computation of the complement is much simpler in this case. All digits can be comple-
mented in parallel. However, a correction of the obtained result is required if# >y [22].
11.2.3 One's-Complement Representation
One's-complement representation as just discussed is a diminished-radix comple-
ment representation with JR = r^-Q = l-Qfor r = 2 and k = 0. A normalized
W^-bit binary word in one's-complement representation is interpreted as
The values lie in the range -l + Q<x<l-Q where Q = 2 d . Also, in one's-
complement representation, the zero value has a redundant binary representation
which reduces the value range and complicates the checking for zero. The values
+1 and -1 can not be represented. Numbers having the same magnitude, but differ-
ent signs, are represented by different binary words. For example, we have
For x > 0, one's-complement representation has the same binary words as, for
example, signed-magnitude representation. For x < 0, the values are the bit-com-
plement of the corresponding positive values. Addition and subtraction in one's-
complement representation are more complicated to implement than in binary off-
set and two's-complement representations, which will be discussed later.
