Optical computing using optical flip-flops in Fourier processors: use
in matrix multiplication and discrete linear transforms

Shigeru Ando; Seishi Sekine; Motohito Mita; Satoshi Katsuo

doi:10.1364/AO.28.005363

Applied Optics
Vol. 28,
Issue 24,
pp. 5363-5373
(1989)
•https://doi.org/10.1364/AO.28.005363

Optical computing using optical flip-flops in Fourier processors: use in matrix multiplication and discrete linear transforms

Shigeru Ando, Seishi Sekine, Motohito Mita, and Satoshi Katsuo

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Shigeru Ando, Seishi Sekine, Motohito Mita, and Satoshi Katsuo, "Optical computing using optical flip-flops in Fourier processors: use in matrix multiplication and discrete linear transforms," Appl. Opt. 28, 5363-5373 (1989)

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Abstract

An architecture and the algorithms for matrix multiplication using optical flip-flops (OFFs) in optical processors are proposed based on residue arithmetic. The proposed system is capable of processing all elements of matrices in parallel utilizing the information retrieving ability of optical Fourier processors. The employment of OFFs enables bidirectional data flow leading to a simpler architecture and the burden of residue-to-decimal (or residue-to-binary) conversion to operation time can be largely reduced by processing all elements in parallel. The calculated characteristics of operation time suggest a promising use of the system in a real time 2-D linear transform.

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Modulus (m_r)	Residue (R_{m_r})	Residue in binary notation	Bias	Residue in binary with bias notation
3	0	00	01	01
	1	01		10
	2	10		11

4	0	00	00	00
	1	01		01
	2	10		10
	3	11		11

5	0	000	011	011
	1	001		100
	2	010		101
	3	011		110
	4	100		111

	m_N		w_mN		$w_{m N}^{2} / w^{2}$
n	50	500	50	500	50	500
w	50	500	50	500	50	500
4	13	17	4	5	1. 000	1. 563
8	19	23	5	5	0. 391	0. 391
16	37	37	6	6	0. 141	0. 141
32	59	61	6	6	0. 035	0. 035


χ in decimal notation	χ in residue notation{R_m₁, R_m₂, R_m₃}

	m₁ = 3	m₂ = 4	m₃ = 5

59	2	3	4
58	1	2	3
:	:	:	:
:	:	:	:

35	2	3	0	$\begin{array}{l} - {1, 1, 1} & \times 2 \\ ∥ \\ a_{1} \end{array}$

34	1	2	4
33	0	1	3
32	2	0	2
:	:	:	:
:	:	:	:	$\begin{array}{l} - \underset{︸}{{0, 3, 3}} & \times 3 \\ ∥ & ∥ \\ m_{1} & a_{2} \end{array}$
25	1	1	0
24	0	0	4
23	2	3	3
22	1	2	2
21	0	1	1
20	2	0	0
19	1	3	4
18	0	2	3
17	2	1	2
16	1	0	1
15	0	3	0
14	2	2	4
13	1	1	3
12	0	0	2	$\begin{array}{l} - \underset{︸}{{0, 0, 2}} & \times 2 \\ ∥ & ∥ \\ m_{1} \cdot m_{2} & a_{3} \end{array}$
11	2	3	1
10	1	2	0
9	0	1	4
8	2	0	3	$\begin{array}{r} {1, 1, 1} & \times 2 = & 2 \\ {0, 3, 3} & \times 3 = & 9 \\ {0, 0, 2} & \times 2 = & 24 \\ 2 \\ 9 \\ \frac{+ 24}{35} \end{array}$
7	1	3	2
6	0	2	1
5	2	1	0
4	1	0	4
3	0	3	3
2	2	2	2
1	1	1	1
0	0	0	0

Scheme of operation	Operation time
(1) Matrix Multiplication in residue number using Fourier processor and OFFs	Binary-to-residue conversion ≃ w(T_SLM + T_CO)
	Multiplication in residue number $≃ w_{m_{N}^{2}} (T_{SLM} + T_{co n})$
	Residue-to-binary conversion and truncation $≃ 2 {∑_{r = 1}^{N} (m_{r} - 1)} (T_{SLM} + T_{co})$

(2) Matrix multiplication in binary number using all electronic system	≃ w²T_cen

Modulus (m_r)	Residue (R_{m_r})	Residue in binary notation	Bias	Residue in binary with bias notation
3	0	00	01	01
	1	01		10
	2	10		11

4	0	00	00	00
	1	01		01
	2	10		10
	3	11		11

5	0	000	011	011
	1	001		100
	2	010		101
	3	011		110
	4	100		111

Modulus (m_r)	Residue (R_{m_r})	Residue in binary notation	Bias	Residue in binary with bias notation
3	0	00	01	01
	1	01		10
	2	10		11

4	0	00	00	00
	1	01		01
	2	10		10
	3	11		11

5	0	000	011	011
	1	001		100
	2	010		101
	3	011		110
	4	100		111

Modulus (m_r)	Residue (R_{m_r})	Residue in binary notation	Bias	Residue in binary with bias notation
3	0	00	01	01
	1	01		10
	2	10		11

4	0	00	00	00
	1	01		01
	2	10		10
	3	11		11

5	0	000	011	011
	1	001		100
	2	010		101
	3	011		110
	4	100		111

Modulus (m_r)	Residue (R_{m_r})	Residue in binary notation	Bias	Residue in binary with bias notation
3	0	00	01	01
	1	01		10
	2	10		11

4	0	00	00	00
	1	01		01
	2	10		10
	3	11		11

5	0	000	011	011
	1	001		100
	2	010		101
	3	011		110
	4	100		111

Modulus (m_r)	Residue (R_{m_r})	Residue in binary notation	Bias	Residue in binary with bias notation
3	0	00	01	01
	1	01		10
	2	10		11

4	0	00	00	00
	1	01		01
	2	10		10
	3	11		11

5	0	000	011	011
	1	001		100
	2	010		101
	3	011		110
	4	100		111

Abstract

Cited By

Figures (10)

Tables (4)

Equations (14)

Applied Optics

Modulus (m_r)	Residue (R_{m_r})	Residue in binary notation	Bias	Residue in binary with bias notation
3	0	00	01	01
	1	01		10
	2	10		11

4	0	00	00	00
	1	01		01
	2	10		10
	3	11		11

5	0	000	011	011
	1	001		100
	2	010		101
	3	011		110
	4	100		111

Modulus (m_r)	Residue (R_{m_r})	Residue in binary notation	Bias	Residue in binary with bias notation
3	0	00	01	01
	1	01		10
	2	10		11

4	0	00	00	00
	1	01		01
	2	10		10
	3	11		11

5	0	000	011	011
	1	001		100
	2	010		101
	3	011		110
	4	100		111