Page 546 - Introduction to Information Optics

9.4. Parallel Signed-Digit Arithmetic 53

Table 9.28
Reduced Minterms for TSD Receding [149] where the Minterms Generating a i Output are
Digit-by-Digit Complement of that Generating a 1 Output

MJnterms «,-«,•_ i
d, 2 22d)oi2. djj-Tfdfo^ 012dyoi2, 02idi 012 , di 2 2!dj 0 i 2 , d, 2 lld 10 T2'
d, 2 12d, 0 i 2 , d^ 222d 10j 2, d^Tdo^d, dnld on d, d l2 2d 0 i 2 d, 02d 012d.
d ]2 T22, d, 2 122, d 12 2T2, d T2222, d T2 TT2, d, 2 0dd. 0222, 0112

receding operation can be performed at all digit positions in parallel. To
a
generate the 1 and T outputs of c t, 42 four-variable (a ia i_ la i_ 2 i-3) minterms
are required. Notice that the minterms for generating the 1 and I outputs are
exact complements of each other. Table 9.28 lists the 21. minterms required for
generating the 1 output in the receding step. In the second step, 6 minterms
are required for generating the sum output. In the second step, only 2
minterms, 10 and 01 (K) and Ol) are required for generating the 1 (T) output,
and only 1 minterm 11 (IT) is needed for generating the 2 (2) output of s f. In
addition, the minterms for s t — 1(2) are digit-by-digit complement of the
minterms for s t = 1(2). Therefore, the algorithm requires 21 minterms for
generating the 1 output in the receding step, and 3 minterms for generating the
1 and 2 outputs of s ; in the addition step.
The two-step TSD addition [150] was further simplified by restricting the
/th intermediate sum s t and the (i + l)th intermediate carry c i+l values to the
set {T, 0, 1). Each sum of the two TSD digits from the set (2,1,0, 1,2} has at
least one representation that satisfies the requirement. The reduced minterms
for this operation are listed in Table 9.29. Also note that the minterms for
generating the 1 output of s,-(c i+1 ) are digit-by-digit complement of the
corresponding minterms for the T minterms. Thus, only three minterms are
required for generating the 1 output of s, and three minterms are required for

Table 9.29
Reduced Minterms for the First Step Operation
of the Simplified Nonrecoded TSD
Addition [150]
Output Minterms a/b,