Page 531 - Introduction to Information Optics

516 9. Computing with Optics

Table 9.16
Truth Table for the Three-Step Digit-Set-Restricted
MSD Addition Based on the Nonrestricted
Reference Digits: the Second Step Rules [144]

x y, r ; c l+t s
i i
T T I T 1
0 0 0
I 0 1
T 0 T T 0
or 0 T 1
0 T 1 0 0
0 0 T T I
0 0 0
1 0 1
1 I T 1 i
or 0 0 0
T i 1 0 1
1 0 T T 0
or 0 I 1
0 1 i 0 0
1 1 I T 1
0 0 0
1 0 1

l
1
as £f=o 2 c : since c 0 = 0. The second term XfLo 2's,- can be expressed as
£ 2's,= X 2's,-2%. (9.39)
( = 0

The third term of Eq. (9.38) is equivalent to the word r N . . . r 2r 10 and the fourth
term is equivalent to the word r N_ i...r 2r lr 0, where r 0 = 0. Therefore, the
difference of the third and fourth terms is

2l+ 2 2
Vi- ^ = V (9-40)
i = 0 / = 0
As a result, Eq. (9.38) becomes

2i c
2N
X + Y = X ( < + s t) + (r N ~ S K). (9.41)
i = 0
Note that X N = 0 and y N — 0. The reference digit r N has three possible states;