Page 542 - Introduction to Information Optics

9.4. Parallel Signed-Digit Arithmetic

listed as follows:

if -ID < 2w (-_ j < 0, then q { = 1
if - D < 2w,-_, < D, then g,. = 0 (9.62)
if 0 < 2w, ! < 2D, then q t = I.

From Eq. (9.62), one can see that there exists one overlap region ( —D, 0) for
the ranges of 2w,-_ t for g f = 1 and q t = 0, and another overlap region (0, £>) for
q. = 0 and q t — 1. In these overlap regions, selection of either of the quotient
digits is valid due to the redundant nature of 2w i^ i. Figure 9.21 plots the
relationship of the shifted partial remainder 2w,-_ 1 versus the divisor D. It
clearly shows the change of the range that 2w,-_ 1 can represent with the
normalized D varied from 1/2 to 1. The shaded regions 1 and 2 are the overlap
regions where more than one quotient digit may be selected. If we can find a
constant value C t (C 2), for all the values of £>e[l/2,1), within the overlap
region, then 2w,-_ ] in any iteration can be compared with this constant value
to select either q t = 1(0) if 2w i _ 1 > C^(C 2), or q t = 0(1) otherwise. Such a
constant should be independent of the value of the divisor D so that the
quotient selection will only depend on 2w,-_ t in any iteration. From Fig. 9.21,
one can observe that C, is independent of D only when it lies in the range

0.5 1

Fig. 9.21. Shifted partial remainder versus divisor plot for quotient-selected MSD division [145].