Page 509 - Introduction to Information Optics

494 9. Computing with Optics

Table 9.4
Truth Table for Negabinary
Half-Subtraction [55]

0 0 0 0
0 1 1 1
1 0 0 1
1 1 0 0

9.3.1.2.2. Negabinary Symbolic Addition/Subtraction
Consider the symbolic substitution rules necessary for negabinary addition. In
the first iteration, the corresponding digits of a,- and b t are added in parallel. So the
truth table of half-addition is available for each bit position. The four substitution
rules can be obtained from Table 9.3 and are illustrated in Fig. 9.12(a). The two
digits are placed one over the other, and each column including a { and 6 f is
replaced by a new two-column pattern including the sum s t and the negative carry
c^+ j to the next higher bit position. The numerical values rather than their spatial
(1)
encoding are used. This operation will generate a sum string s and a negative
(1)
carry string c~ (the number in the parenthesis defines the iteration time).
In the second iteration, the two digits to be added in each column come
(1)
1}
l)
from s\ and cf . In fact, the addition between the sum s\ and the negative
(l )
carry c-' is a negabinary half-subtraction. According to Table 9.4, the four
substitution rules are listed in Fig. 9.12(b). Consequently, another sum string
(2>
s and a positive carry string c +<2) are produced.
+
C i+i

c
f+l
0
0 1- 0 0
1
1
1 1
0 1

(a) (b)

Fig. 9.12. Substitution rules for negabinary (a) half-addition, and (b) half-subtraction.