Modified signed-digit trinary arithmetic by using optical symbolic substitution

A. A. S. Awwal; M. N. Islam; M. A. Karim

doi:10.1364/AO.31.001687

Applied Optics
Vol. 31,
Issue 11,
pp. 1687-1694
(1992)
•https://doi.org/10.1364/AO.31.001687

Modified signed-digit trinary arithmetic by using optical symbolic substitution

A. A. S. Awwal, M. N. Islam, and M. A. Karim

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A. A. S. Awwal, M. N. Islam, and M. A. Karim, "Modified signed-digit trinary arithmetic by using optical symbolic substitution," Appl. Opt. 31, 1687-1694 (1992)

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Abstract

Carry-free addition and borrow-free subtraction of modified signed-digit trinary numbers with optical symbolic substitution are presented. The proposed two-step and three-step algorithms can be easily implemented by using phase-only holograms, optical content-addressable memories, a multichannel correlator, or a polarization-encoded optical shadow-casting system.

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	Addend	Augend	Digits at Next Lower-Order Position		Intermediate

Type	A_i	B_i	A_i₋₁	B_i₋₁	Carry	Sum
1	0	0	Irrelevant		0	0
	1	$\bar{1}$
	$\bar{1}$	1
	2	$\bar{2}$
	$\bar{2}$	2
2	0	1	Both positive		1	$\bar{2}$
	1	0
	2	$\bar{1}$			0	1
	$\bar{1}$	2	Otherwise
3	0	$\bar{1}$	Both negative		$\bar{1}$	2
	$\bar{1}$	0
	1	$\bar{2}$			0	$\bar{1}$
	$\bar{2}$	1	Otherwise
4	0	2	Both negative		0	2
	2	0
	1	1	Otherwise		1	$\bar{1}$
5	0	$\bar{2}$	Both positive		0	$\bar{2}$
	$\bar{2}$	0
	$\bar{1}$	$\bar{1}$	Otherwise		$\bar{1}$	1
6	1	2	Irrelevant		1	0
	2	1
7	$\bar{1}$	$\bar{2}$	Irrelevant		$\bar{1}$	0
	$\bar{2}$	$\bar{1}$
8	2	2	Both positive		2	$\bar{2}$
			Otherwise		1	1
9	$\bar{2}$	$\bar{2}$	Both negative		$\bar{2}$	2
			Otherwise		$\bar{1}$	$\bar{1}$

	Addend	Augend	Digits at Next Lower-Order Position		Intermediate

Type	A_i	B_i	A_i₋₁	B_i₋₁	Carry	Sum
1	0	0	Irrelevant		0	0
	1	$\bar{1}$
	$\bar{1}$	1
	2	$\bar{2}$
	$\bar{2}$	2
2	0	1	2,2		1	$\bar{2}$
	1	0
	2	$\bar{1}$	Otherwise		0	1
	$\bar{1}$	2
3	0	$\bar{1}$	$\bar{2}$ , $\bar{2}$		$\bar{1}$	2
	$\bar{1}$	0
	1	$\bar{2}$	Otherwise		0	$\bar{1}$
	$\bar{2}$	1
4	0	2	$\bar{2}$ , $\bar{2}$		0	2
	2	0
	1	1	Otherwise		1	$\bar{1}$
5	0	$\bar{2}$	2,2		0	$\bar{2}$
	$\bar{2}$	0
	$\bar{1}$	$\bar{1}$	Otherwise		$\bar{1}$	1
6	1	2	Irrelevant		1	0
	2	1
7	$\bar{1}$	$\bar{2}$	Irrelevant		$\bar{1}$	0
	$\bar{2}$	$\bar{1}$
8	2	2	2,2		2	$\bar{2}$
			Otherwise		1	1
9	$\bar{2}$	$\bar{2}$	$\bar{2}$ , $\bar{2}$		$\bar{2}$	2
			Otherwise		$\bar{1}$	$\bar{1}$

	Minuend	Subtrahend	Digits at Next Lower-Order Position		Intermediate

Type	A_i	B_i	A_i₋₁	B_i₋₁	Borrow	Difference
1	0	0	Irrelevant		0	0
	1	1
	$\bar{1}$	$\bar{1}$
	2	2
	$\bar{2}$	$\bar{2}$
2	0	$\bar{1}$	Both positive		1	$\bar{2}$
	1	0
	2	1	Otherwise		0	1
	$\bar{1}$	$\bar{2}$
3	0	1	Both negative		$\bar{1}$	2
	$\bar{1}$	0
	1	2	Otherwise		0	$\bar{1}$
	$\bar{2}$	$\bar{1}$
4	0	$\bar{2}$	Both negative		0	2
	2	0
	1	$\bar{1}$	Otherwise		1	$\bar{1}$
5	0	2	Both positive		0	$\bar{2}$
	$\bar{2}$	0
	$\bar{1}$	1	Otherwise		$\bar{1}$	1
6	1	$\bar{2}$	Irrelevant		1	0
	2	$\bar{1}$
7	$\bar{1}$	2	Irrelevant		$\bar{1}$	0
	$\bar{2}$	$\bar{1}$
8	2	$\bar{2}$	Both positive		2	$\bar{2}$
			Otherwise		1	1
9	$\bar{2}$	2	Both negative		$\bar{2}$	2
			Otherwise		$\bar{1}$	$\bar{1}$

	Addend	Augend	Digits at Next Lower-Order Position		Intermediate

Type	A_i	B_i	A_i₋₁	B_i₋₁	Borrow	Difference
1	0	0	Irrelevant		0	0
	1	1
	$\bar{1}$	$\bar{1}$
	2	2
	$\bar{2}$	$\bar{2}$
2	0	$\bar{1}$	2,2		1	$\bar{2}$
	1	0
	2	1	Otherwise		0	1
	$\bar{1}$	$\bar{2}$
3	0	1	$\bar{2}$ , $\bar{2}$		$\bar{1}$	2
	$\bar{1}$	0
	1	2	Otherwise		0	$\bar{1}$
	$\bar{2}$	$\bar{1}$
4	0	$\bar{2}$	$\bar{2} \bar{2}$		0	2
	2	0
	1	$\bar{1}$	Otherwise		1	$\bar{1}$
5	0	2	2 2		0	$\bar{2}$
	$\bar{2}$	0
	$\bar{1}$	1	Otherwise		$\bar{1}$	1
6	1	$\bar{2}$	Irrelevant		1	0
	2	$\bar{1}$
7	$\bar{1}$	2	Irrelevant		$\bar{1}$	0
	$\bar{2}$	$\bar{1}$
8	2	$\bar{2}$	2 2		2	$\bar{2}$
			Otherwise		1	1
9	$\bar{2}$	2	$\bar{2} \bar{2}$		$\bar{2}$	2
			Otherwise		$\bar{1}$	$\bar{1}$

Function	Output	Minterms A_iB_iF_i	Function	Output Bit	Minterms A_iB_iF_i
Carry	1	$\begin{array}{l} 01 (10) 1, 2 \bar{1} (\bar{1} 2) 1, 02 (20) d_{10}, \\ 11 d_{10}, 12 (21) d, 22 d_{\bar{1} 0} \end{array}$	Sum	1	$\begin{array}{l} 01 (10) d_{\bar{1} 0}, 2 \bar{1} (\bar{1} 2) d_{\bar{1} 0} \\ 0 \bar{2} (\bar{2} 0) d_{\bar{1} 0}, \bar{1} \bar{1} d_{10}, 22 d_{\bar{1} 0} \end{array}$
	$\bar{1}$	$\begin{array}{l} 0 \bar{1} (\bar{1} 0) \bar{1}, 1 \bar{2} (\bar{2} 1) \bar{1}, \bar{2} \bar{2} d_{10} \\ 0 \bar{2} (\bar{2} 0) d_{\bar{1} 0}, \bar{1} \bar{1} d_{\bar{1} 0}, \bar{1} \bar{2} (\bar{2} \bar{1}) d \end{array}$		$\bar{1}$	$\begin{array}{l} 0 \bar{1} (\bar{1} 0) d_{10}, 1 \bar{2} (\bar{2} 1) d_{10}, 02 (20) d_{10} \\ 11 d_{10}, \bar{2} \bar{2} d_{10} \end{array}$
	2	221		2	$0 d_{\bar{1} 2} (d_{\bar{1} 2} 0) \bar{1}, d_{1 \bar{2}} d_{1 \bar{2}} \bar{1}$
	$\bar{2}$	$\bar{2} \bar{2} 1$		$\bar{2}$	$0 d_{1 \bar{2}} (d_{1 \bar{2}} 0) 1, d_{\bar{1} 2} d_{\bar{1} 2} 1$
Borrow	1	$\begin{array}{l} d_{\bar{1} 01} \bar{2} 1, d_{12} \bar{2} d_{\bar{1} 0}, 2 \bar{1} d \\ d_{12} \bar{1} d_{10}, 2 d_{0 \bar{1}} d_{10}, 2 d_{01 \bar{1}} 1 \\ 0 d_{\bar{1} \bar{2}} 1, 1 d_{0 \bar{1} \bar{2}} 1, 0 \bar{2} d_{10} \end{array}$	Difference	1	$0 d_{\bar{1} 2} d_{\bar{1} 0}, d_{1 \bar{2}} 0 d_{\bar{1} 0}, d_{\bar{1} 2} d_{1 \bar{2}} d_{\bar{1} 0}$
	$\bar{1}$	$\begin{array}{l} \bar{2} d_{01 \bar{1}} 1, \bar{2} d_{12} d_{10}, d_{01 \bar{1}} 2 \bar{1} \\ d_{\bar{1} \bar{2}} 2 d_{10}, \bar{1} d_{012} \bar{1}, 0 d_{12} \bar{1} \\ \bar{2} d_{01} d_{\bar{1} 0}, d_{0 \bar{1}} 2 d_{\bar{1} 0}, \bar{1} d_{12} d_{\bar{1} 0} \end{array}$		$\bar{1}$	$0 d_{1 \bar{2}} d_{10}, d_{\bar{1} 2} 0 d_{10}, d_{1 \bar{2}} d_{1 \bar{2}} d_{10}$
	2	$22 \bar{1}$		2	$0 d_{1 \bar{2}} \bar{1}, d_{\bar{1} 2} 0 \bar{1}, d_{1 \bar{2}} d_{\bar{1} 2} \bar{1}$
	$\bar{2}$	$\bar{2} 2 \bar{1}$		$\bar{2}$	$0 d_{\bar{1} 2} 1, d_{1 \bar{2}} 01, d_{\bar{1} 2} d_{\bar{2} 1} 1$

Function	Output Bit	Minterms A_iB_iA_i₋₁B_i₋₁	Function	Output Bit	Minterms A_iB_iA_i₋₁B_i₋₁
Carry	1	$\begin{array}{l} 01 (10) d_{12} d_{12}, 22 \bar{d_{12} d_{12}} \\ 02 (20) \bar{d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}}, 12 (21) d d \\ 2 \bar{1} (\bar{1} 2) d_{12} d_{12}, 11 \bar{d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}} \end{array}$	Sum	1	$\begin{array}{l} 01 (10) \bar{d_{12} d_{12}}, 22 \bar{d_{12} d_{12}} \\ 0 \bar{2} (\bar{2} 0) \bar{d_{12} d_{12}}, \bar{11 d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}} \\ 2 \bar{1} (\bar{1} 2) \bar{d_{12} d_{12}} \end{array}$
	$\bar{1}$	$\begin{array}{l} 0 \bar{1} (\bar{1} 0) d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}, \bar{22 d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}} \\ 0 \bar{2} (\bar{2} 0) \bar{d_{12} d_{12}}, \bar{11 d_{12} d_{12}}, \\ 1 \bar{2} (\bar{2} 1) d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}, \bar{1} \bar{2} (\bar{2} \bar{1}) d d \end{array}$		$\bar{1}$	$\begin{array}{l} 1 \bar{2} (\bar{2} 1) \bar{d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}}, 02 (20) \bar{d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}} \\ 11 \bar{d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}}, \bar{22 d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}} \\ 0 \bar{1} (\bar{1} 0) \bar{d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}} \end{array}$
	2	22d₁₂d₁₂		2	$0 d_{\bar{1} 2} (d_{\bar{1} 2} 0) d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}, d_{1 \bar{2}} d_{1 \bar{2}} d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}$
	$\bar{2}$	$\bar{22} d_{12}$		$\bar{2}$	$0 d_{1 \bar{2}} (d_{1 \bar{2}} 0) d_{12}, d_{\bar{1} 2} d d_{\bar{1} 2} d_{12}$
Borrow	1	$\begin{array}{l} d_{\bar{1} 01} \bar{2} d_{12} d_{12}, d_{12} \bar{2 d_{12} d_{12}}, \\ 2 d_{0 \bar{1}} d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}, 0 \bar{2 d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}} \\ 0 d_{\bar{1} \bar{2}} d_{12} d_{12}, 1 d_{0 \bar{1} \bar{2}} d_{12} d_{12}, \\ 2 d_{01 \bar{1}} d_{12} d_{12}, 2 \bar{1} d d \\ d_{12} \bar{1 d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}} \end{array}$	Difference	1	$\begin{array}{l} 0 d_{\bar{1} 2} \bar{d_{12} d_{12}}, d_{1 \bar{2}} 0 \bar{d_{12} d_{12}} \\ d_{\bar{1} 2} d_{1 \bar{2}} \bar{d_{12} d_{12}} \end{array}$
	$\bar{1}$	$\begin{array}{l} \bar{2} d_{01 \bar{1}} d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}, \bar{2} d_{12} \bar{d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}}, \\ d_{\bar{1} \bar{2}} 2 d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}, \bar{1} d_{012} d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}, \\ \bar{2} d_{01} d_{0 \bar{1}}, d_{0 \bar{1}} 2 d_{0 \bar{1}}, \bar{1} d_{12} \bar{d_{12} d_{12}} \\ d_{01 \bar{1}} 2 d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}, 0 d_{12} d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}} \end{array}$		$\bar{1}$	$\begin{array}{l} 0 d_{1 \bar{2}} \bar{d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}}, d_{\bar{1} 2} 0 \bar{d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}} \\ d_{1 \bar{2}} d_{1 \bar{2}} d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}} \end{array}$
	2	$22 d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}$		2	$\begin{array}{l} 0 d_{1 \bar{2}} d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}, d_{1 \bar{2}} d_{\bar{1} 2} d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}} \\ d_{\bar{1} 2} 0 d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}} \end{array}$
	$\bar{2}$	$\bar{2} 2 d_{\bar{1} \bar{2}} d_{\bar{1} \bar{2}}$		$\bar{2}$	$\begin{array}{l} 0 d_{\bar{1} 2} d_{12} d_{12}, d_{1 \bar{2}} 0 d_{12} d_{12}, \\ d_{\bar{1} 2} d_{\bar{2} 1} d_{12} d_{12} \end{array}$

Function	Output Bit	Minterms A_iB_iA_i₋₁B_i₋₁B_i₋₁	Function	Output Bit	Minterms A_iB_iA_i₋₁B_i₋₁B_i₋₁
Carry	1	$\begin{array}{l} 01 (10) 22, 2 \bar{1} (\bar{1} 2) 22, \\ 12 (21) d d, 22 \bar{22}, 02 (20) \bar{\bar{2} \bar{2}} \end{array}$	Sum	1	$\begin{array}{l} 01 (10) \bar{22}, 2 \bar{1} (\bar{1} 2) \bar{22}, 11 \bar{\bar{2} \bar{2}} \\ 0 \bar{2} (\bar{2} 0) \bar{22}, \bar{1} \bar{1} \bar{\bar{2} \bar{2}}, 22 \bar{22} \end{array}$
	$\bar{1}$	$\begin{array}{l} 0 \bar{1} (\bar{1} 0) \bar{2} \bar{2}, 1 \bar{2} (\bar{2} 1) \bar{2} \bar{2}, \bar{2} \bar{2} \bar{\bar{2} \bar{2}} \\ 0 \bar{2} (\bar{2} 0) \bar{22}, \bar{1} \bar{1} \bar{22}, \bar{1} \bar{2} (\bar{2} \bar{1}) d d \end{array}$		$\bar{1}$	$\begin{array}{l} 0 \bar{1} (\bar{1} 0) \bar{\bar{2} \bar{2}}, 1 \bar{2}, (\bar{2} 1) \bar{\bar{2} \bar{2}}, 02 (20) \bar{\bar{2} \bar{2}} \\ 11 \bar{\bar{2} \bar{2}}, \bar{2} \bar{2}, \bar{\bar{2} \bar{2}} \end{array}$
	2	2222		2	$0 d_{\bar{1} 2} (d_{\bar{1} 2} 0) \bar{2} \bar{2}, d_{1 \bar{2}} d_{1 \bar{2}} \bar{2} \bar{2}$
	$\bar{2}$	$\bar{22} 22$		$\bar{2}$	$0 d_{1 \bar{2}} (d_{1 \bar{2}} 0) 22, d_{\bar{1} 2} d_{\bar{1} 2} 22$
Borrow	1	$\begin{array}{l} d_{\bar{1} 01} \bar{2} 22, d_{12} \bar{2} \bar{22}, 2 \bar{1} d d \\ d_{12} \bar{1} \bar{\bar{2} \bar{2}}, 2 d_{0 \bar{1}} \bar{\bar{2} \bar{2}}, 2 d_{01 \bar{1}} 22 \\ 0 d_{\bar{1} \bar{2}} 1, d_{12} 022 \end{array}$	Difference	1	$0 d_{\bar{1} 2} \bar{22}, d_{1 \bar{2}} 0 \bar{22}, d_{\bar{1} 2} d_{1 \bar{2}} \bar{22}$
	$\bar{1}$	$\begin{array}{l} \bar{2} d_{01 \bar{1}} \bar{2} \bar{2}, \bar{2} d_{12} \bar{\bar{2} \bar{2}}, d_{01 \bar{1}} 2 \bar{2} \bar{2} \\ d_{\bar{1} \bar{2}} 2 \bar{\bar{2} \bar{2}}, \bar{1} d_{012} \bar{2} \bar{2}, 0 d_{12} \bar{2} \bar{2} \\ \bar{2} d_{01} \bar{22}, d_{0 \bar{1}} 2 \bar{22}, \bar{1} d_{12} \bar{22} \end{array}$		$\bar{1}$	$0 d_{1 \bar{2}} \bar{\bar{2} \bar{2}}, d_{\bar{1} 2} 0 \bar{\bar{2} \bar{2}}, d_{1 \bar{2}} d_{1 \bar{2}} \bar{\bar{2} \bar{2}}$
	2	$22 \bar{2} \bar{2}$		2	$0 d_{1 \bar{2}} \bar{2} \bar{2}, d_{\bar{1} 2} 0 \bar{2} \bar{2}, d_{1 \bar{2}} d_{\bar{1} 2} \bar{2} \bar{2}$
	$\bar{2}$	$\bar{2} 2 \bar{2} \bar{2}$		$\bar{2}$	$0 d_{\bar{1} 2} 22, d_{1 \bar{2}} 022, d_{\bar{1} 2} d_{\bar{2} 1} 22$

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