Page 521 - Introduction to Information Optics

506 9. Computing with Optics

As a result, the logical operations required in MSD addition can be considered
as the signed version of that in traditional binary addition.

9.4.2.2.2. Two-Step MSD Addition/Subtraction
The previous algorithm used two steps to guarantee that the third-step
addition is carry free. The first two steps can be combined into a single step
operation. For example, Li and Eichman [74] suggested the conditional
symbolic substitution rules, where in addition to the digit pairs 0,6,- to be
substituted, a lower-level digit pair a i_ 1b i_ l is used as the reference digits. In
the second step, the transfer and weight digits are added to yield carry-free
addition. The data flow diagram for this two-step addition is shown in Fig.
9.17. To implement the two-step algorithm with a content-addressable memory
(CAM), the truth tables are first classified in terms of the nonzero outputs 1
and 1. The minterms for the nonzero outputs are then logically minimized
using a suitable minimization technique such as the Karnaugh map or the
Quine-McClusky's algorithm [6]. The minimized minterms are listed in Table
9.7 where, in each four-variable minterm, the two first- (second) column digits
denote the variables a^b^a^^b^^^}, respectively. The symbol d implies a
complete don't-care digit for 1, 0, and 1, whereas the symbol d x implies a
partial don't-care for digits x and v.

Example: 10TTl010TOlT(1433 lo) + iTTlOOTlOl 10(758, 0)
Stage 1: iTTOlOOOOllT^ (T)
^010010011101 (W)
Stage 2: lTOOlT00100Tl(2191 10),

where the symbol (/> indicates a padded zero.

a b
Input i i
Operands

Stage 1 I T.W I I T. w T.W T. w

Stage 2

Fig. 9.17. Five-digit MSD addition configuration in two steps.