Page 543 - Introduction to Information Optics

528 9. Computing with Optics

(0,0.5]. Similarly, the constant C 2 should be in the range [ — 0.5,0). Therefore
a number of values in the above ranges can be selected for Cj and C 2. Note
that the constants should also result in a simple quotient selection function. To
compare 2w l -_ 1 with C { or C 2, one can approximate the result according to
several of their most significant digits instead of performing a subtraction
operation. With the observation of several most significant digits, the quotient
selection is thus independent of the length of the divisor operand. These most
significant digits should not only guarantee the correct choice of the quotient
digits but also be as few as possible to simplify the physical implementation.
Here three most significant digits w,- 0 • w ; j w,- 2 of partial remainder are used for
quotient selection since they are sufficient to represent a constant value in
range (0,0.5] or [-0.5,0). For example, C v = (0.10) MSD = (UOW = (0.5) 10
is selected as the reference constant value for selection of q i = 0 and q t = I,
while constant value C 2 = (0.10) MSD = (I.10) MSD = ( — 0.5) 10 is selected for q, =
0 and q { — \. If H> ;>0 w i5l w ii2 is equal to or greater than (0.10) MSD or (1.10) MSD,
then the quotient digit is set to 1; if w,- (0w uw,- >2 is equal to or lower than
(O.IO) MSD or (T.10) MSD, then the quotient digit is set to —1; otherwise, the
quotient digit is set to 0. The selection rules are listed in Table 9.24. As
an example, we compute the MSD division with X = (0.101 ll01) MSD =
(0.6640625)! 0 and D = (0.1110lIl) MSD = (0.8359375), 0. The partial remainder
and the quotient digit in each iteration are shown in Table 9.25. After eight
iterations, the division results are generated with Q = (0.1 1001 10) MSD =
(0.796875) 10.
Using the Karnaugh map or Quine-McCluskey method, the reduced logic
mmterrns for outputs 1 and I of the quotient digits can be obtained. We used
symbol d xv to imply a partial don't-care for digits x and y and symbol d to
imply a complete don't-care for digits 1, 0, and 1. As a result, to determine the

Table 9.24
Truth Table for the Quotient Digit q t According to w^ 0w Llw L2 of the Partial Remainder [145]

1 0 1 1
1 0 0 1
t o l l
1 I 1 1
1 T 0 l
i I T o
o i I i
0 1 0 1
O l i o
o o i o
0 0 0 0