Digital optical processing based on higher-order modified signed-digit symbolic substitution

Mohammad S. Alam; Abdul Ahad S. Awwal; Mohammad A. Karim

doi:10.1364/AO.31.002419

Applied Optics
Vol. 31,
Issue 14,
pp. 2419-2425
(1992)
•https://doi.org/10.1364/AO.31.002419

Digital optical processing based on higher-order modified signed-digit symbolic substitution

Mohammad S. Alam, Abdul Ahad S. Awwal, and Mohammad A. Karim

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Mohammad S. Alam, Abdul Ahad S. Awwal, and Mohammad A. Karim, "Digital optical processing based on higher-order modified signed-digit symbolic substitution," Appl. Opt. 31, 2419-2425 (1992)

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Abstract

An efficient higher-order modified signed-digit optical symbolic substitution technique for addition and subtraction has been derived. This two-step symbolic substitution technique, which is independent of the number of bits to be processed, saves a tremendous amount of computation time. Two separate approaches for optical implementation that use content-addressable memory and the joint transform correlator are also suggested.

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Decimal Number	Modified Signed-Digit Representation
−6	$\bar{1} \bar{1} 0$
−5	$\bar{1} 0 \bar{1}, \bar{1} \bar{1} 1$
−4	$\bar{1}$ 00
−3	$0 \bar{1} \bar{1}, \bar{1} 01, \bar{1} 1 \bar{1}$
−2	0 $\bar{1}$ 0, $\bar{1}$ 10
−1	00 $\bar{1}$ , 0 $\bar{1}$ 1, $\bar{1}$ 11
0	000
1	001, $01 \bar{1}, 1 \bar{1} \bar{1}$
2	010, 1 $\bar{1}$ 0
3	011, 10 $\bar{1}$ , 1 $\bar{1}$ 1
4	100
5	101, 11 $\bar{1}$
6	110

Group	Addend A_iA_i₋₁	Augend B_iB_i₋₁	A_i₋₂B_i₋₂	Carry C_i	Sum S_iS_i₋₁
1	$\bar{1} \bar{1}$	$\bar{1} \bar{1}$	Don’t care	$\bar{1}$	$\bar{1}$ 0
2	$\bar{1} \bar{1}$	$\bar{1}$ 0	All negative	$\bar{1}$	$\bar{1}$ 1
	$\bar{1}$ 0	$\bar{1} \bar{1}$	Otherwise	$\bar{1}$	0 $\bar{1}$
3	$\bar{1} \bar{1}$	0 $\bar{1}$	Don’t care	$\bar{1}$	0 0
	$\bar{1}$ 0	$\bar{1}$ 0
	0 $\bar{1}$	$\bar{1} \bar{1}$
	$\bar{1} \bar{1}$	$\bar{1}$ 1
	$\bar{1}$ 1	$\bar{1} \bar{1}$
4	$\bar{1} \bar{1}$	0 0	All negative	$\bar{1}$	0 1
	0 0	$\bar{1} \bar{1}$
	$\bar{1}$ 0	0 $\bar{1}$
	0 $\bar{1}$	$\bar{1}$ 0	Otherwise	0	$\bar{1} \bar{1}$
	$\bar{1}$ 0	$\bar{1}$ 1
	$\bar{1}$ 1	$\bar{1}$ 0
5	$\bar{1}$ 0	0 0	Don’t care	0	$\bar{1}$ 0
	0 0	$\bar{1}$ 0
	0 $\bar{1}$	0 $\bar{1}$
	0 $\bar{1}$	$\bar{1}$ 1
	$\bar{1}$ 1	0 $\bar{1}$
	$\bar{1}$ 1	$\bar{1}$ 1
6	0 $\bar{1}$	0 0	All negative	0	$\bar{1}$ 1
	0 0	0 $\bar{1}$
	$\bar{1}$ 1	0 0	Otherwise	0	0 $\bar{1}$
	0 0	$\bar{1}$ 1
7	0 0	0 0	Don’t care	0	0 0
8	0 1	0 0	All negative	0	0 1
	0 0	0 1
	1 $\bar{1}$	0 0	Otherwise	0	1 1
	0 0	1 $\bar{1}$
9	1 0	0 0	Don’t care	0	1 0
	0 0	1 0
	0 1	0 1
	0 1	1 $\bar{1}$
	1 $\bar{1}$	0 1
	1 $\bar{1}$	1 $\bar{1}$
10	1 1	0 0	All negative	0	1 1
	0 0	1 1
	1 0	0 1
	0 1	1 0	Otherwise	1	0 $\bar{1}$
	1 0	1 $\bar{1}$
	1 $\bar{1}$	1 0
11	1 1	0 1	Don’t care	1	0 0
	1 0	1 0
	0 1	1 1
	1 1	1 $\bar{1}$
	1 $\bar{1}$	1 1
12	1 1	1 0	All negative	1	0 1
	1 0	1 1	Otherwise	1	1 $\bar{1}$
13	1 1	1 1	Don’t care	1	1 0

Group	Minuend P_iP_i _{− 1}	Subtrahend P_iP_i _{− 1}	P_i _{− 2}Q_i _{− 2}	Borrow B_i	Difference D_iD_i _{− 1}
1	$\bar{1} \bar{1}$	1 1	Don’t care	$\bar{1}$	$\bar{1}$ 0
2	$\bar{1} \bar{1}$	1 0	All negative	$\bar{1}$	$\bar{1}$ 1
	$\bar{1}$ 0	1 1	Otherwise	$\bar{1}$	0 $\bar{1}$
3	$\bar{1} \bar{1}$	0 1	Don’t care	$\bar{1}$	0 0
	$\bar{1}$ 0	1 0
	0 $\bar{1}$	1 1
	$\bar{1} \bar{1}$	1 $\bar{1}$
	$\bar{1}$ 1	1 1
4	$\bar{1} \bar{1}$	0 0	All negative	$\bar{1}$	0 1
	0 0	1 1
	$\bar{1}$ 0	0 1
	0 $\bar{1}$	1 0	Otherwise	0	$\bar{1} \bar{1}$
	$\bar{1}$ 0	1 $\bar{1}$
	$\bar{1}$ 1	1 0
5	$\bar{1}$ 0	0 0	Don’t care	0	$\bar{1}$ 0
	0 0	1 0
	0 $\bar{1}$	0 1
	0 $\bar{1}$	$\bar{1}$ 1
	$\bar{1}$ 1	0 $\bar{1}$
	$\bar{1}$ 1	1 $\bar{1}$
6	0 $\bar{1}$	0 0	All negative	0	$\bar{1}$ 1
	0 0	0 1
	$\bar{1}$ 1	0 0	Otherwise	0	0 $\bar{1}$
	0 0	1 $\bar{1}$
7	0 0	0 0	Don’t care	0	0 0
8	0 1	0 0	All negative	0	0 1
	0 0	$\bar{1} \bar{1}$
	1 $\bar{1}$	0 0	Otherwise	0	1 1
	0 0	$\bar{1}$ 1
9	1 0	0 0	Don’t care	0	1 0
	0 0	$\bar{1}$ 0
	0 1	0 $\bar{1}$
	0 1	$\bar{1}$ 1
	1 $\bar{1}$	0 $\bar{1}$
	1 $\bar{1}$	$\bar{1}$ 1
10	1 1	0 0	All negative	0	1 1
	0 0	$\bar{1} \bar{1}$
	1 0	0 $\bar{1}$
	0 1	$\bar{1}$ 0	Otherwise	1	0 $\bar{1}$
	1 0	$\bar{1}$ 1
	1 $\bar{1}$	$\bar{1}$ 0
11	1 1	0 $\bar{1}$	Don’t care	1	0 0
	1 0	$\bar{1}$ 0
	0 1	$\bar{1} \bar{1}$
	1 1	$\bar{1}$ 1
	1 $\bar{1}$	$\bar{1} \bar{1}$
12	1 1	$\bar{1}$ 0	All negative	1	0 1
	1 0	$\bar{1} \bar{1}$	Otherwise	1	1 $\bar{1}$
13	1 1	$\bar{1} \bar{1}$	Don’t care	1	1 0

S_i

1	$\bar{1}$	1	$\bar{1}$	1	$\bar{1}$
001X₀₁ $X_{0 \bar{1}} X_{0 \bar{1}}$	$00 \bar{1} X_{0 \bar{1}} X_{0 \bar{1}} X_{0 \bar{1}}$	X₀₁1X₀₁ $X_{0 \bar{1}} X_{0 \bar{1}}$	$X_{01} 1 X_{01} 0 \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$11 X_{01} X_{01} 0 \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$\bar{1} \bar{1} X_{0 \bar{1}} X_{0 \bar{1}} X_{0 \bar{1}} X_{0 \bar{1}}$
0X₀₁10 $X_{0 \bar{1}} X_{0 \bar{1}}$	$0 X_{0 \bar{1}} 0 \bar{1} X_{0 \bar{1}} X_{0 \bar{1}}$	X₀₁0X₀₁1 $X_{0 \bar{1}} X_{0 \bar{1}}$	$X_{01} 0 X_{01} 1 \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$X_{01} X_{01} 11 \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$X_{0 \bar{1}} X_{0 \bar{1}} \bar{1} \bar{1} X_{0 \bar{1}} X_{0 \bar{1}}$
1X₀₁00 $X_{0 \bar{1}} X_{0 \bar{1}}$	$\bar{1} X_{0 \bar{1}} \bar{1} \bar{1} X_{0 \bar{1}} X_{0 \bar{1}}$	$X_{0 \bar{1}} \bar{1} X_{0 \bar{1}} 0 X_{0 \bar{1}} X_{0 \bar{1}}$	$X_{0 \bar{1}} \bar{1} X_{0 \bar{1}} 0 \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$X_{01} 11 X_{01} \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$X_{0 \bar{1}} \bar{1} \bar{1} X_{0 \bar{1}} X_{0 \bar{1}} X_{0 \bar{1}}$
100X₀₁ $X_{0 \bar{1}} X_{0 \bar{1}}$	$\bar{1} \bar{1} \bar{1} X_{0 \bar{1}} X_{0 \bar{1}} X_{0 \bar{1}}$	$X_{0 \bar{1}} 0 X_{0 \bar{1}} \bar{1} X_{0 \bar{1}} X_{0 \bar{1}}$	$X_{0 \bar{1}} 0 X_{0 \bar{1}} \bar{1} \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$101 X_{0 \bar{1}} 1 \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$\bar{1} 0 \bar{1} X_{01} X_{0 \bar{1}} X_{0 \bar{1}}$
$1 X_{0 \bar{1}} 1 \bar{1} X_{0 \bar{1}} X_{0 \bar{1}}$	$0 X_{0 \bar{1}} \bar{1} 1 X_{0 \bar{1}} X_{0 \bar{1}}$	$X_{0 \bar{1}} 0 \bar{1} 1 X_{0 \bar{1}} X_{0 \bar{1}}$	$X_{0 \bar{1}} 0 \bar{1} 1 \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$1 X_{0 \bar{1}} 10 \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$\bar{1} X_{01} \bar{1} 0 X_{0 \bar{1}} X_{0 \bar{1}}$
$1 \bar{1} 1 X_{0 \bar{1}} X_{0 \bar{1}} X_{0 \bar{1}}$	$\bar{1} 10 X X_{0 \bar{1}} X_{0 \bar{1}}$	$\bar{1} 10 X_{0 \bar{1}} X_{0 \bar{1}} X_{0 \bar{1}}$	$\bar{1} 10 X_{0 \bar{1}} \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	11X₀₁1 $X_{0 \bar{1}} X_{0 \bar{1}}$	$\bar{1} \bar{1} X_{0 \bar{1}} \bar{1 X_{0 \bar{1}} X_{0 \bar{1}}}$
$010 X_{01} \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$00 \bar{1} X_{0 \bar{1}} \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$X_{01} 01 \bar{1} X_{0 \bar{1}} X_{0 \bar{1}}$	$X_{01} 01 \bar{1} \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	X111 $X_{0 \bar{1}} X_{0 \bar{1}}$	$X \bar{1} \bar{1} \bar{1} \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$
$0 X_{01} 01 \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$0 X_{0 \bar{1}} \bar{1} 0 \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$1 \bar{1} X_{01} 0 X_{0 \bar{1}} X_{0 \bar{1}}$	$1 \bar{1} X_{01} 0 \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	1X₀₁1X₀₁XX	$\bar{1} X_{0 \bar{1}} \bar{1} X_{0 \bar{1}} X X$
$1 X_{01} 11 \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$\bar{1} X_{0 \bar{1}} 00 \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$			1X₀₁X₀₁1XX	$\bar{1} X_{0 \bar{1}} X_{0 \bar{1}} \bar{1} X X$
$111 X_{01} \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$\bar{1} 00 X_{0 \bar{1}} \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$			$X_{0 \bar{1}}$ 111XX	X₀₁ $\bar{1} \bar{1} \bar{1}$ XX
$0 X_{01} 1 \bar{1} \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$\bar{1} 11 X_{01} \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$			11 $X_{0 \bar{1}}$ 1XX	$\bar{1} \bar{1}$ X₀₁ $\bar{1}$ XX
$1 \bar{1} 0 X \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$	$\bar{1} X_{01} \bar{1} 1 \bar{X_{0 \bar{1}} X_{0 \bar{1}}}$
0101XX	0 $\bar{1}$ 0 $\bar{1}$ XX
1111XX	$\bar{1} \bar{1} \bar{1} \bar{1} X X$
011 $\bar{1}$ XX	$0 \bar{1} \bar{1} \bar{1} X X$
1 $\bar{1}$ 1 $\bar{1}$ XX	$\bar{1}$ 1 $\bar{1}$ 1XX
1 $\bar{1}$ 01XX	$\bar{1}$ 10 $\bar{1}$ XX
1000XX	$\bar{1}$ 000XX
0010XX	00 $\bar{1}$ 0XX

Decimal Number	Modified Signed-Digit Representation
−6	$\bar{1} \bar{1} 0$
−5	$\bar{1} 0 \bar{1}, \bar{1} \bar{1} 1$
−4	$\bar{1}$ 00
−3	$0 \bar{1} \bar{1}, \bar{1} 01, \bar{1} 1 \bar{1}$
−2	0 $\bar{1}$ 0, $\bar{1}$ 10
−1	00 $\bar{1}$ , 0 $\bar{1}$ 1, $\bar{1}$ 11
0	000
1	001, $01 \bar{1}, 1 \bar{1} \bar{1}$
2	010, 1 $\bar{1}$ 0
3	011, 10 $\bar{1}$ , 1 $\bar{1}$ 1
4	100
5	101, 11 $\bar{1}$
6	110

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Applied Optics