Page 563 - Introduction to Information Optics

548 9. Computing with Optics

c+ +1 =a, + ^ = G f , (*=0,1,2,...,N + 1) (9,93)
cf = P 0> (9.94)

c 2 - = P 1G 0 + (P 1 + G 0)P;, (9.95)
f J - P 2G! + (P 2 + GJPTGo" + (P 2 + G.XPT + Co'W (9-96)

cZ = P 3G 2 + (P 3 + G 2)P7G7 + (P 3 + G 2XP2~ + G7)P,G 0
+ (P 3 + G 2}(P~ 2 + G~)(P , + G 0)Po".

If we rewrite Q as

c
c-4 = l/o + KoPVir + (^ + ^ 'o ]. (9-98)
where

t/ 0 = P 3G 2 + (P 3 + G 2)P^G; + (P 3 + G 2XPl + GJPjGo,

It is valid to apply the aforementioned methodology to higher bit positions
yielding

C4* +4 = V, + V^^ + (J\ S + l^c^l i - 0, 1, 2. (9.99)

Therefore, with a four-bit module, it is possible to generate the carries in the
lookahead mode for any length of operands. For long operands, a multistage
carry-lookahead architecture can be used where all carries can be generated in
parallel. Finally, the negabinary equivalence of x is given by

z^a.-e^e^ecf. (9.ioo>

The above operation is performed bitwise in parallel, as shown in the following
example:

Input: III01lTlO(-190)
a,.: 111011110
b t: 1110001000
P,.: 0110101001
G,-: 0000100001
+
c; : 00001000010
c f ~: 00101010010
Output: 01 1010001 10 (-190).