Page 523 - Introduction to Information Optics

508 9. Computing with Optics

Table 9.9
Reduced Minterms for MSD Addition Rules Shown in Table 9.3 [131]

Minterms Minterms
Function Output (a,W Function Output (a&Fi)
T 1 d 01 ll, ld oj l, Ild 01 W 1 Od, T0, d ir OO
i d oTlO, ld oT0, lld ()] 1 Od lT l,dj T O I
S 1 01, 10
1 01, 10

operations with signed digits, a closer look at the truth tables reveals that it is
necessary to introduce another reference bit, h {. For addition, /? f is true if both
df and bf are negative. For subtraction, h f is true if both a t and the complement
of b i are negative. A key feature of this definition is that we only need to
consider the unsigned binary values of a i and b t, 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 binary logic equations. In the first step, the required binary logic
expressions at ith(i = 0, 1, . . . , N — 1) digit position for the two arithmetic
operations are given by:

for f,.,

output "1": aib idi + (a, 0 ^0,-- 1, (9.27)
output "T": h t + fo ® b^iHi _ , . (9.28)

output 'T'lfoefcftei-i, (9.29)
output "T": fo e&;)0i_i. (9-30)

where 0_ l — 1. In the second step, the transfer and weight digits are added at

Table 9.10
Modified Conditional Truth Table for the First Step MSD Subtraction [134]
Wi W',. t ; 4. j W; <, + 1 ^ ,
t i+l 'i+ I VV'j ',+ , f tl w/
9i- i 0 0 0 0 0 0 l T l 1 0 T o
1 0 0 0 0 1 1 0 1 1 0 1 0
11(0)/11(0) 00(1) 01(1)/10(1) J0(0)/01(0) 11(1) 11(0)