Page 548 - Introduction to Information Optics

9.4. Parallel Signed-Digit Arithmetic

a 0 = A = Multiplicand
b Q = fl = Multiplier

0 0 oo
0 0
0 Partial Products
0 I2 10 0
0 II Pro • ( and
0 C 0 0 0 Partial Carries
0 ^ P23 P22 P2I P20 20 0 0
0 0 0 0
32 30
0 0 0 0 >
P33 P32 P»
Fig. 9.22. Four-digit nonrecoded TSD multiplication [153]. p and c are for the partial product and
the carry, respectively.

can be limited to the set (1, 0, 1}. When A is multiplied by b h both the partial
product and partial carry can be arranged as separate numbers, as shown in
Fig. 9.22. As a result, we have N partial-carry words, in addition to N
partial-product words to be added.

(iv) Nonrecoded Multiplication without Carries As described above, when
two TSD digits are multiplied, the carry c t + 1 and the partial product p f can be
limited to the set {1,0,1). Therefore, partial-carry generation can be avoided
by considering the product of two consecutive digits of the multiplicand
A(a ia i_ {) and one digit of the multiplier B(bj). The truth table can be obtained
directly from the multiplication of two digits.

For each of the above-mentioned cases, when all partial products are
generated in parallel, they can be added in a tree structure using the nonrecod-
ing addition technique. But for the third case, an extra addition step is
required.

9.4.3.3. Division
Based on the TSD addition and multiplication algorithms, recently TSD
division through the convergence approach [154] has been studied. Table 9.30
shows an illustration of the TSD division algorithm using X = (O.l2) 3 =
(0.5555) 10 and Y - (0.20) 3 = (0.6666), 0.