Number representation effects in truth-table look-up processing: 8-bit addition example

Mir M. Mirsalehi; Thomas K. Gaylord; Daniel C. Fielder; Clark C. Guest

doi:10.1364/AO.28.001931

Applied Optics
Vol. 28,
Issue 10,
pp. 1931-1939
(1989)
•https://doi.org/10.1364/AO.28.001931

Number representation effects in truth-table look-up processing: 8-bit addition example

Mir M. Mirsalehi, Thomas K. Gaylord, Daniel C. Fielder, and Clark C. Guest

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Mir M. Mirsalehi, Thomas K. Gaylord, Daniel C. Fielder, and Clark C. Guest, "Number representation effects in truth-table look-up processing: 8-bit addition example," Appl. Opt. 28, 1931-1939 (1989)

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Abstract

The parallelism and interconnectivity of optical systems may provide important advantages for these systems in massively parallel processing applications. Electronic systems, however, retain all the advantages of a highly developed technology that has been widely applied with excellent success. In both of these technologies, the methods of direct truth-table look-up processing are becoming increasingly important as the need grows for increased speed and throughput. A major issue in truth-table look-up processing is the number representation used for data. In this paper, the effects of number representation are investigated for the important case of 8-bit addition as a specific example. The inputs are two 8-bit binary numbers together with an input carry. The output is a full precision 9-bit binary sum. For the intermediate processing three number representations are treated: binary, residue, and modified signed-digit. The numbers in all three representations are in binary-coded form throughout the processing. The critically important steps of encoding the numbers into the residue and modified signed-digit systems and then decoding the results back into direct binary are also performed using truth-table look-up methods. For the direct binary representation, a total of 2545 gates (2519 holograms) are required. For the residue representation, a total of 1764 gates (1686 holograms) are required. For the modified signed-digit representation, a total of 4142 gates (4052 holograms) are required.

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INPUT:	Two 8-bit numbers and an input carry bit.
	Total number of input bits − (2 × 8) + 1 − 17.
REFERENCE PATTERNS:	Number of reference patterns is equal to the number of input combinations.
	Number of input combinations − 2¹⁷ − 131,072.
CONSTRUCTION OF TRUTH	Express each bit of answer as sum-of-products logical expression.
TABLE:	Number of input combinations to produce a “one” for each output bit − 2¹⁶ − 65,536 (number of terms in each sum-of-products expression).
	Total number of input combinations for all output bits − 9 × 2¹⁶ − 589,824 (total number of terms in all sum-of-products expressions).
LOGICAL REDUCTION:	For each output bit:
	Determine prime implicants.
	Construct table-of-choice.
	Obtain minimal sum.
	Determine essential prime implicants.
	Eliminate dominated rows.
	Eliminate dominating columns.
	Construct new table of choice.
	Repeat above steps.
REDUCED REFERENCE	Number of reference patterns (number of and gates in PAL or number of holograms in optical CAM) after logical minimization.
PATTERNS:	Output	Reference
	Bit	Patterns
	0 (LSB)	4
	1	12
	2	28
	3	60
	4	124
	5	252
	6	508
	7 (MSB)	1020
	8 (carry)	511
	Total No. Reference Patterns − 2,519
OUTPUT:	One 8-bit number and an output carry bit.
	Number of output bits − 8 + 1 − 9.
	Number of possible outputs − 2⁹ − 512.

b₂ = 1		b₁ = 1		b₀ = 1
00000100	10010101	000X0010	10010011	00X00001	10011001
01000000	10011010	0X001000	10110001	X0001000
	10101001	001X0000	11000110	00X10000	0001X111
00001001	10111000	1X000000	11100100	1000X000	X0010111
00011000	11001100				X0101011
00100010	11010001	00010001	0X010111	0000X011	X0111010
10000001		01000100	00101X11	X0000011	01X00111
10010000	00111011		0X101011	00X00110	0101X011
	01001111	00000X11	0X110101	0001X010	X1010011
00001110	01011110	0X000011	010X1101	X0010010	01X10110
00010011	01101101	0000110X	0110011X	001001X0	011001X1
00101100	01111100	000X1100	011X0110	01X00010	011101X0
00110001	10101110	00100X01	1000111X	X1000100	100011X1
01000101	10110011	0X100001	100X1110	011000X0	10X01101
01001010	11000111	1000010X	10011X01	100010X0	100111X0
01010100	11010110	100X0100	1X011001	10X01000	10X11100
01101000	11100101		10100X11		1100X110
10000110	11101010	0001011X	1X100011	X0001101	110010X1
10100100	11110100	0X100110	1010110X	00X10101	11X01001
11000010		001X1010	101X1100	X0011100	110110X0
11100000	01110111	0011010X	110X0101	0010X110	11X11000
	10011111	01001X01		X0100110	1110X100
00011101	10111101	0101001X	0011111X	001010X1
00100111	11011011	01011X00	0111101X	001110X0	001111X1
00110110	11111001	01100X10	1011011X	X1001001	X1011101
01011001		0111000X	11001X11	010011X0	0110X111
01100011	11101111	100X1001	11011X10	01X01100	X1100111
01110010	11111110	1X001010	11101X01	01X10001	0111X110
10001011		1001100X	1111001X	X1011000	X1110110
		101X0010	11111X00	0110X010	011110X1
		1X101000		X1100010	101101X1
		11000X01	10111011	1000X101	11X01110
		11010X00	11101110	1001X100	1101X101
				101000X1	111100X1
		00011011	0X111111	101100X0
		00111001	1X110111	1100X001	X0111111
		01001110	110X1111	1101X000	X1111011
		01101100	111X1101		110111X1
				00110011	11X11101

b₂ = 1		b₁ = 1		b₀ = 1
0001000	0010110	00X0100	00100X1	0000X01	1001000
1000000	0100011	01000X0	001001X	0X00001	0010011
0100100	0110001	1001000	010x001	0000X10	0010011
0100100	0110010	1001000	011X000	0X00010	0X11001
0000111	1001001	00X0011	011100X	0010X00	1000X11
0010101		X000101	100011X	0X10000
		X000110	0110111		0100111
				0101000	0110101
				1000100	0110110

c₁	x₀	y₀	W²
0	0	0	0
0	0	1	$\bar{1}$
0	1	0	$\bar{1}$
0	1	1	0
1	0	0	1
1	0	1	0
1	1	0	0
1	1	1	1

	Number System
	Direct Binary	Residue	Modified Signed-Digit
Encoder
NOT	0	16	0
AND	0	722	0
OR	0	12	0
Adder
NOT	17	21	17
AND	2519	215	374
OR	9	10	18
NOR	0	0	9
Decoder
NOT	0	10	0
AND	0	749	3678
OR	0	9	45
NOR	0	0	1
Total Gates	2545	1764	4142
Number of Holograms (AND gates)	2519	1686	4052

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Applied Optics