Page 513 - Introduction to Information Optics

498 9. Computing with Optics

For addition:

(9.15)

For subtraction:

a, ® bf © c © cr
fl,-fr,-e7 + (a-b, + afi^cf cf , (9.16)

93.1.2.4. Higher-Order Negabinary Addition
In the above-mentioned 1-bit negabinary addition, the addition of two Is
generates a negative carry to the next higher bit position. If the carry
generation is limited to be positive, twin carries will be generated to the next
two higher bit positions according to the following identity [42,95]:

+ 2
I +1
f
I -(-2)'+ l-(-2)'' = 0-(-2) + 1 •(-2) ' + l-(-2)' . (9.17)
The logical function is given by

(9.18)

The nature of the twin-carry generation in negabinary is different from other
number system algorithms and matches well the representation of higher-order
symbolic addition.
Consider a 2-bit-wide module with the inputs a i+^a { and h i+lb t (Fig.
9.15 [a]). The outputs contain a 2-bit sum s i+ js,- and a 2-bit carry c, + 3c ! + 2 . The
output-input relation can be represented by a higher-order substitution rule,
as shown in the right side of the figure. The operands «,•+[«,• are placed over
the other operands b i+1b t, and in the output pattern, the lower row corre-
sponds to the sum s i+ is,- and the upper row corresponds to the carry c ! + 3c,- +2.
There are 16 total combinations for the four input bits; the 16 substitution rules
are listed in Fig. 9.15(b). With dual-rail spatial encoding, the number of
reference patterns can be reduced [95]. An optical implementation based on