Designs of optoelectronic trinary signed-digit multiplication by use of
joint spatial encodings and optical correlation

Abdallah K. Cherri

doi:10.1364/AO.38.000828

Applied Optics
Vol. 38,
Issue 5,
pp. 828-837
(1999)
•https://doi.org/10.1364/AO.38.000828

Designs of optoelectronic trinary signed-digit multiplication by use of joint spatial encodings and optical correlation

Abdallah K. Cherri

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Abdallah K. Cherri, "Designs of optoelectronic trinary signed-digit multiplication by use of joint spatial encodings and optical correlation," Appl. Opt. 38, 828-837 (1999)

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Abstract

Trinary signed-digit (TSD) symbolic-substitution-based (SS-based) optical adders, which were recently proposed, are used as the basic modules for designing highly parallel optical multiplications by use of cascaded optical correlators. The proposed multiplications perform carry-free generation of the multiplication partial products of two words in constant time. Also, three different multiplication designs are presented, and new joint spatial encodings for the TSD numbers are introduced. The proposed joint spatial encodings allow one to reduce the SS computation rules involved in optical multiplication. In addition, the proposed joint spatial encodings increase the space–bandwidth product of the spatial light modulators of the optical system. This increase is achieved by reduction of the numbers of pixels in the joint spatial encodings for the input TSD operands as well as reduction of the number of pixels used in the proposed matched spatial filters for the optical multipliers.

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Equations (5)

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Digit Group	D _i D _i-1	D _i-2	D _i-3	C _i
1	2̅2̅, 01, 12̅	d _01̅2̅	d	0
		1	d _101̅2̅	0
		2	2̅	0
		1	2	1
		2	d _1012̅	1
2	11̅, 02, 2̅1̅	2̅	d _101̅2̅	0
		1̅	2̅	0
		2̅	2	1
		1̅	d _1012̅	1
		d ₀₁₂	d	1
3	21̅, 1̅2̅, 11, 22̅	d _01̅2̅	d	1
		1	d _101̅2̅	1
		2	2̅	1
		1	2	1̅
		2	d _1012̅	1̅
4	20̅, 10	d	d	1
5	00	d	d	0
6	20, 10̅	d	d	1̅
7	21̅, 12, 1̅1̅, 22̅	d ₀₁₂	d	1̅
		1̅	d _1012̅	1̅
		2̅	2	1̅
		1̅	2̅	1
		2̅	d _101̅2̅	1
8	11̅, 02̅, 21	d _01̅2̅	d	1̅
		1	d _101̅2̅	1̅
		2	2̅	1̅
		1	2	0
		2	d _1012̅	0
9	22, 01̅, 12̅	d ₀₁₂	d	0
		1̅	d _1012̅	0
		2̅	2	0
		1̅	2̅	1̅
		2̅	d _101̅2̅	1̅

Digit Group	Addend A _i	Augend B _i	Final Sum S _i
1	1	1	2
2	0	1	1
	1	0
3	0	0
4	1	1̅	0
	1̅	1
5	0	1̅	1̅
	1̅	0
6	1̅	1̅	2̅

Digit Group	Multiplicand a _i	Multiplier b _i	Product P _i
1	1	1	1
2	1̅	1̅	1
3	0	1	0
	1	0
4	0	0	0
5	0	1̅	0
	1̅	0
6	1	1̅	1̅
	1̅	1

Digit Group	Multiplicand a _i	Multiplier b _i	Product P _i
1	2	1	2
2	2̅	1̅	2
3	1	1	1
4	1̅	1̅	1
5	2	0	0
6	2̅	0	0
7	0	1	0
	1	0
8	0	0	0
9	0	1̅	0
	1̅	0
10	1	1̅	1̅
	1̅	1
11	2	1̅	2̅
12	2̅	1	2̅

Digit Group	Multiplicand a _i	Multiplier b _i	Product c _i+1	P _i
1	2	2	1	1
2	2̅	2̅	1	1
3	1	1	0	1
4	1̅	1̅	0	1
5	2	1̅	1̅	1
	1̅	2
6	2̅	1	1̅	1
	1	2̅
7	0	2	0	0
	2	0
8	0	1	0	0
	1	0
9	0	0	0	0
10	0	1̅	0	0
	1̅	0
11	0	2̅	0	0
	2̅	0
12	2	1	1	1̅
	1	2
13	1̅	2̅	1	1̅
	2̅	1̅
14	1	1̅	0	1̅
	1̅	1
15	2	2̅	1̅	1̅
	2̅	2

Digit Group	Multiplicand a _i	Multiplier b _i	Product P _i
1	2	1	2
2	2̅	1̅	2
3	1	1	1
4	1̅	1̅	1
5	2	0	0
6	2̅	0	0
7	0	1	0
	1	0
8	0	0	0
9	0	1̅	0
	1̅	0
10	1	1̅	1̅
	1̅	1
11	2	1̅	2̅
12	2̅	1	2̅

Digit Group	Multiplicand a _i	Multiplier b _i	Product c _i+1	P _i
1	2	2	1	1
2	2̅	2̅	1	1
3	1	1	0	1
4	1̅	1̅	0	1
5	2	1̅	1̅	1
	1̅	2
6	2̅	1	1̅	1
	1	2̅
7	0	2	0	0
	2	0
8	0	1	0	0
	1	0
9	0	0	0	0
10	0	1̅	0	0
	1̅	0
11	0	2̅	0	0
	2̅	0
12	2	1	1	1̅
	1	2
13	1̅	2̅	1	1̅
	2̅	1̅
14	1	1̅	0	1̅
	1̅	1
15	2	2̅	1̅	1̅
	2̅	2

Digit Group	Multiplicand a _i	Multiplier b _i	Product P _i
1	2	1	2
2	2̅	1̅	2
3	1	1	1
4	1̅	1̅	1
5	2	0	0
6	2̅	0	0
7	0	1	0
	1	0
8	0	0	0
9	0	1̅	0
	1̅	0
10	1	1̅	1̅
	1̅	1
11	2	1̅	2̅
12	2̅	1	2̅

Digit Group	Multiplicand a _i	Multiplier b _i	Product c _i+1	P _i
1	2	2	1	1
2	2̅	2̅	1	1
3	1	1	0	1
4	1̅	1̅	0	1
5	2	1̅	1̅	1
	1̅	2
6	2̅	1	1̅	1
	1	2̅
7	0	2	0	0
	2	0
8	0	1	0	0
	1	0
9	0	0	0	0
10	0	1̅	0	0
	1̅	0
11	0	2̅	0	0
	2̅	0
12	2	1	1	1̅
	1	2
13	1̅	2̅	1	1̅
	2̅	1̅
14	1	1̅	0	1̅
	1̅	1
15	2	2̅	1̅	1̅
	2̅	2

Digit Group	Multiplicand a _i	Multiplier b _i	Product P _i
1	2	1	2
2	2̅	1̅	2
3	1	1	1
4	1̅	1̅	1
5	2	0	0
6	2̅	0	0
7	0	1	0
	1	0
8	0	0	0
9	0	1̅	0
	1̅	0
10	1	1̅	1̅
	1̅	1
11	2	1̅	2̅
12	2̅	1	2̅

Digit Group	Multiplicand a _i	Multiplier b _i	Product c _i+1	P _i
1	2	2	1	1
2	2̅	2̅	1	1
3	1	1	0	1
4	1̅	1̅	0	1
5	2	1̅	1̅	1
	1̅	2
6	2̅	1	1̅	1
	1	2̅
7	0	2	0	0
	2	0
8	0	1	0	0
	1	0
9	0	0	0	0
10	0	1̅	0	0
	1̅	0
11	0	2̅	0	0
	2̅	0
12	2	1	1	1̅
	1	2
13	1̅	2̅	1	1̅
	2̅	1̅
14	1	1̅	0	1̅
	1̅	1
15	2	2̅	1̅	1̅
	2̅	2

Digit Group	Multiplicand a _i	Multiplier b _i	Product P _i
1	2	1	2
2	2̅	1̅	2
3	1	1	1
4	1̅	1̅	1
5	2	0	0
6	2̅	0	0
7	0	1	0
	1	0
8	0	0	0
9	0	1̅	0
	1̅	0
10	1	1̅	1̅
	1̅	1
11	2	1̅	2̅
12	2̅	1	2̅

Digit Group	Multiplicand a _i	Multiplier b _i	Product c _i+1	P _i
1	2	2	1	1
2	2̅	2̅	1	1
3	1	1	0	1
4	1̅	1̅	0	1
5	2	1̅	1̅	1
	1̅	2
6	2̅	1	1̅	1
	1	2̅
7	0	2	0	0
	2	0
8	0	1	0	0
	1	0
9	0	0	0	0
10	0	1̅	0	0
	1̅	0
11	0	2̅	0	0
	2̅	0
12	2	1	1	1̅
	1	2
13	1̅	2̅	1	1̅
	2̅	1̅
14	1	1̅	0	1̅
	1̅	1
15	2	2̅	1̅	1̅
	2̅	2

Abstract

Cited By

Figures (14)

Tables (5)

Equations (5)

Applied Optics

Digit Group	Multiplicand a _i	Multiplier b _i	Product P _i
1	2	1	2
2	2̅	1̅	2
3	1	1	1
4	1̅	1̅	1
5	2	0	0
6	2̅	0	0
7	0	1	0
	1	0
8	0	0	0
9	0	1̅	0
	1̅	0
10	1	1̅	1̅
	1̅	1
11	2	1̅	2̅
12	2̅	1	2̅

Digit Group	Multiplicand a _i	Multiplier b _i	Product c _i+1	P _i
1	2	2	1	1
2	2̅	2̅	1	1
3	1	1	0	1
4	1̅	1̅	0	1
5	2	1̅	1̅	1
	1̅	2
6	2̅	1	1̅	1
	1	2̅
7	0	2	0	0
	2	0
8	0	1	0	0
	1	0
9	0	0	0	0
10	0	1̅	0	0
	1̅	0
11	0	2̅	0	0
	2̅	0
12	2	1	1	1̅
	1	2
13	1̅	2̅	1	1̅
	2̅	1̅
14	1	1̅	0	1̅
	1̅	1
15	2	2̅	1̅	1̅
	2̅	2

Digit Group	Multiplicand a _i	Multiplier b _i	Product P _i
1	2	1	2
2	2̅	1̅	2
3	1	1	1
4	1̅	1̅	1
5	2	0	0
6	2̅	0	0
7	0	1	0
	1	0
8	0	0	0
9	0	1̅	0
	1̅	0
10	1	1̅	1̅
	1̅	1
11	2	1̅	2̅
12	2̅	1	2̅

Digit Group	Multiplicand a _i	Multiplier b _i	Product c _i+1	P _i
1	2	2	1	1
2	2̅	2̅	1	1
3	1	1	0	1
4	1̅	1̅	0	1
5	2	1̅	1̅	1
	1̅	2
6	2̅	1	1̅	1
	1	2̅
7	0	2	0	0
	2	0
8	0	1	0	0
	1	0
9	0	0	0	0
10	0	1̅	0	0
	1̅	0
11	0	2̅	0	0
	2̅	0
12	2	1	1	1̅
	1	2
13	1̅	2̅	1	1̅
	2̅	1̅
14	1	1̅	0	1̅
	1̅	1
15	2	2̅	1̅	1̅
	2̅	2