Optical implementation of an efficient modified signed-digit trinary ...

Optics & Laser Technology 30 (1998) 49±55

Optical implementation of an ecient modi®ed signed-digit trinary addition M.M. Hossain a, J.U. Ahmed b, A.A.S. Awwal c, *, H.E. Michel c a

Spectra Precision Software, Atlanta, GA, USA b Hewlett Packard, Fort Collins, CO, USA c Wright State University, Computer Science and Engineering Department, Dayton, OH 45435, USA Received 20 November 1997; accepted 17 February 1998

Abstract Ecient parallel schemes for carry-propagation-free addition of modi®ed signed-digit trinary numbers are presented. The necessary minterms for implementation using an optical programmable logic array area are derived. The proposed schemes require only a truth table of 25 entries compared with an earlier scheme of 625 entries. The proposed schemes are amenable to optical implementation. Experimental results using an optical programmable logic array are demonstrated. The experiments show the problems of noise and crosstalk. This suggests some dc bias is necessary to increase the signal-to-noise ratio of the optical circuit. # 1998 Published by Elsevier Science Ltd. All rights reserved. Keywords: Optical computing; Modi®ed signed digit; Multivalued logic

1. Introduction Harnessing the full advantage of optics for parallel computation requires one to exploit algorithms which are inherently parallel in nature. While ripple-carry addition leads to simple circuitry suitable for electronic implementation, the requirement of delaying the carry creates additional diculty in optics. Carry-propagation-free addition using modi®ed signed-digit (MSD) numbers alleviates [1] such a problem and is therefore well suited for optics [2, 3]. Higher radix number representations, such as trinary based arithmetic, is even more attractive because of its higher storage density and reduced logic requirement [4, 5]. As the number of to-be-processed bits is increased, carry-look-ahead or ripple-carry addition [6], in addition to the architectural limitations, leads to increased fan-in and the resultant higher level of gate complexity. Computing systems built using MSD logic, on the other hand, are scalable to higher sizes, by simple replication of the hardware. * Corresponding author. Tel.: +937-775-5111; fax: +937-7755133.

Binary MSD based schemes have already been proposed for optical symbolic substitution [7±18]. Recently, these schemes were extended to trinary logic based optical symbolic substitution [19]. However, the huge size of the trinary truth table with 625 entries makes the scheme impractical for optical realization. In this paper, we propose an extremely ecient trinary carry-free addition scheme, which signi®cantly reduces the size of the truth table by 96% of that in Ref. [19].

2. Modi®ed signed-digit (MSD) trinary number Any signed-digit trinary number, X, may be represented by Xˆ

n X x i Ri

…1†

iˆ0

where the radix, R = 3, and the numeral xi can take any value from the set {2, 1, 0, 1, 2}. Multiple representation of the same number leads to redundancy in the MSD system. The inherent redundancy in MSD number representation is exploited to derive rules for carry-propagation-free addition.

0030-3992/98/$19.00 # 1998 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 9 8 ) 0 0 0 1 5 - 2

50

M.M. Hossain et al. / Optics & Laser Technology 30 (1998) 49±55

3. Rules for carry-free addition When two single-digit trinary MSD numbers are added, a carry is generated whenever the sum exceeds the radix. Thus the combinations (1, 2), or (2, 2), or (2, 1) or (2, 2) results in a carry. The generation of carry may be avoided by mapping the two digits in question into an intermediate sum and an intermediate carry (also known as weight and transfer bits [1, 11]) such that the nth intermediate sum and the (n ÿ 1)th intermediate carry never form any one of the aforementioned four combinations. The following steps are systematically utilized in deriving the required truth table for realizing carry-free addition. 1. The pairs of trinary addend and augend digits are classi®ed according to their sum into multiple classes and form the input columns of the truth table. 2. All possible two-digit MSD trinary representations for each of the classes are identi®ed. In these MSD trinary representations, the lesser signi®cant digit is termed the intermediate sum while the other digit is called the intermediate carry. 3. The substitution rule for a group of paired numbers is determined by identifying the (n ÿ 1)th pair of digits; so that the nth intermediate sum when added to the (n ÿ 1)th intermediate carry does not generate a

Table 1 Truth table for trinary MSD addition Type

1

2

3

4 5 6 7 8 9

Addend

Augend

Intermediate

Ai

Bi

carry

sum

0 1 1 2 2 0 1 2 1 0 1 1 2 0 2 1 0 2 1 1 2 1 2 2 2

0 1 1 2 2 1 0 1 2 1 0 2 1 2 0 1 2 0 1 2 1 2 1 2 2

0

0

0

1

0

1

1

1

1

1

1

0

1

0

1 1

1 1

carry, i.e. the intermediate sum Sn and intermediate carry Cn ÿ 1 generated satisfy the condition: jSn ‡ Cnÿ1 j