Page 545 - Introduction to Information Optics

530 9. Computing with Optics

Table 9.27
Encodings of w ti0w,- Avv ; 2 Generating a Quotient Digit 1

T 0 i 1 0 0 0 1 0
I 0 0 1 0 0 0 0 0
I 0 1 1 0 0 0 0 i
T 1 I 1 0 0 1 ] 0
1 1 0 1 0 0 1 0 0
0 1 I 0 0 1 0 I 0
0 T 0 0 0 1 0 0 0

9.4.3. TSD ARITHMETIC

9.4.3.1. Addition

A two-step TSD algorithm [146] was designed by mapping the two digits
a, and b ( into an intermediate sum and an intermediate carry so that the ith
intermediate sum and the (i — l)th intermediate carry do not generate a carry.
The digits a i^ lb i^. l from the next lower order position are used as the
reference. Both the intermediate sum and the intermediate carry belong to the
set (2,1,0,1,2}, and a total of 28 minterms are required. Later on, the
higher-order TSD operation [147] was developed, which involves the operand
digits a ia i._ 1 and b tb j^_ l and the reference digits a,-_ 2 fe / _ 2 to yield a two-digit
intermediate sum s ;s ;_ l and an intermediate carry. In this technique, the truth
table becomes relatively large and more minterms are needed for implementa-
tion. Recently, a two-digit trinary full-adder [148] was designed with
a,-«i- \b ib l,_! as the inputs. In this technique, the intermediate carry and one of
the intermediate sum digits s i^ 1 are restricted to {1,0, 1}, thus ensuring that
the second-step addition is carry free.
In a TSD number system, a carry will be generated for the addition of the
digit combinations 22, 2l, 1.2, 12, 21, and 22. For carry-free addition, the
aforementioned digit combinations must be eliminated from the augend and
the addend. By exploiting the redundancy of the TSD numbers, a recoding
truth table [149] has been developed to eliminate the occurrence of these digit
combinations. An JV-digit TSD number A = %_ 1 a A ,_ 2 ...a 1 a 0 is receded into
c c sucn tnat an
an (N + 1)-digit TSD number C = tVjv-i • -- \ o C d ^ are
numerically equal and the recoded TSD output c, does not include the 2 and
2 literals. Thus, this recoding operation maps the TSD set {2,1,0,1,2} to a
smaller set {T, 0,1}. This recoding naturally guarantees that the second-step
addition of the recoded numbers is carry free. The recoded digit c, depends on
the current TSD value a t and three lower-order digits a i._. la i^ 2a i^ 3 and the