Page 532 - Introduction to Information Optics

9.4. Parallel Signed-Digit Arithmetic 5 1 7
f, 0, and 1. In the following, we discuss these three cases based on the
computation rule for the input digit pair (0,0) in Table 9.16.
1, % = 1. The second term of Eq. (9.41)
Case I —r N T: there exist c N+l
is written as

(9.42)

Case II — r N = 0: there exist C N+ , =0, S N = 0. The second term of Eq. (9.41)
is written as
(9.43)

Case III — r N = 1: there exist c N + 1 = 0, S N = 1. The second term of Eq.
(9.41) is written as

(9.44)

Consequently, Eq. (9.14) can be simplified as

(9.45)
N+l N+I
I 2'Ci + £
i = 0 i = 0
where s N+i = 0. The first and second terms represent the (N + 2) -bit inter-
mediate carry word C and sum word S, respectively. That is, Eq. (9.45) can be
rewritten as

X + Y = C + S. (9.46)

From Eq. (9.46), we conclude that the sum of two N-digit MSD words is equal
to the sum of the (N + 2) -bit intermediate carry and sum words, regardless of
the reference word.
Figure 9.19 is the diagram of the three-step MSD adder based on this
scheme. Using the Karnaugh map or the Quine-McCluskey method, the
corresponding reduced-logic minterms in each step are summarized in Table
9.17. The first two steps can be combined into a single step. An example is
shown below for addition of two 8-digit MSD numbers. It is necessary to pad
one zero preceding the most significant digit of the operand digits, as shown