Page 537 - Introduction to Information Optics

9. Computing with Optics

Table 9.23
Encoding Scheme and Computation Rules to be Used for the
Digit-Set-Restricted MSD Addition by Logic Operation [144]

1 l 1 0 0 0 0 I 0
1 T 1
T 0 0 0 0 0 0 I 1
0 T 1 0 0
0 0 0 0 0 0 0 0 0
1 0 1
1 T i 0 1 i 0 T 0
T i 1 T 1
l 0 0 1 1 1 0 T 1
0 1 i 0 0
1 1 0 0 1 1 0 0 0
1 0 1

So the encoding contains the information of the input digit combinations and
the corresponding reference digits. From Table 9.23, by using a Karnaugh map
to minimize the logical minterms for the nonzero outputs, the binary logic
expressions for the intermediate carry c i+ 1 and sum s, can be obtained:
c i+ i = u i + v i7 i, (9.49)

(9.50)

It should be mentioned that for the intermediate carry digits, which are
restricted to the set (1, 0), the true logic denotes negative digit T. Inserting Eq.
(9.48) in Eqs. (9.49), and (9.50) results in

(9.51)
(9.52)

The first and the second steps are thus combined into a single step since from
the encoding of the input operand pairs, the intermediate carry and sum digits
can be obtained directly through binary logic operations on the inputs. We
+
employ variable Z to denote all of the occurrences of positive digit 1 in the
final sum string Z by a 1 symbol and Z~ to denote all of the occurrences of
negative digit T in Z by a 1 symbol. The final result Z can thus be yielded by
:
+
combination of Z and Z~. The binary logic expressions for z* and z, are