Page 557 - Introduction to Information Optics

542 9. Computing with Optics

Table 9.39

Reduced Minterms for the Two-Step NSD Addition and Subtraction [162] where d. Kr
Indicates a Partial Don't-Care Digit
Step Output
number Operation Function digit Logical terms («,-£>,-/;)

Addition f, f , 1 TTd 10, oil, loi
_
Ild 10 , 100, 010
First: step H 1 d lT 01, Od, T l
'i
I d lT00, OduO
1 Tld 01 ,0!l, 101
frf i
Subtraction T Ild 01, 100, 010
1 d, T 01,Od, T l
w f
T d, T00, Od, T 0
Second step Addition and .s, l Oi, 10
subtraction T ol, lo

a t and b t are positive while h { is true if both a i and b { are negative. For
subtraction, g t is true if both a,- and the complement of b { are positive while h i
is true if both a t and the complement of b { are negative. For the reference bits,
we only need to consider the unsigned binary values of a t and b^ independent
of their signs. This greatly simplifies the logical operations. Another advantage
of this definition is that both addition and subtraction operations can be
expressed by the same logic equations. In the first step, the required binary
logic expressions for the two arithmetic operations are given by:
for
output "1": hf + (a t( (9.66)

(9.67)
output "T": (a,.fr, + (a t
for
output "1": (a,-©^)//, (9.68)
output "T": (a ;- © ^,-),/-, (9.69)

In the second step, the logic operations for the final sum S and D are the same
as W in the first step by defining a binary reference bit g t which is true for
positive t { and w,-:

for Sj and d h
output "1": (tj © (9.70)
output "T": (t t © (9.71)