Residue number system holographic truth-table look-up processing: detector threshold setting and probability of error due to amplitude and phase variations
M. M. Mirsalehi, C. C. Guest, and T. K. Gaylord, "Residue number system holographic truth-table look-up processing: detector threshold setting and probability of error due to amplitude and phase variations," Appl. Opt. 22, 3583-3592 (1983)
The use of a holographic content-addressable memory system for parallel truth-table look-up digital data processing is analyzed. For binary-coded residue numbers, the operations of 4-, 8-, 12-, and 16-bit addition and multiplication are treated. The minimum probability of error that can be achieved and the corresponding detector threshold settings are determined in each case allowing for the effects of Gaussian distributions in the amplitude and the phase in the recording beams. Resultant probabilities of error for practical conditions are found to be very competitive with those from state-of-the-art nonparallel technologies.
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Operational Characteristics for Addition with Residue Number System Holographic Truth-Table Look-Up Processora
Modulus
N
σa = σθ = 0.01
σa = σθ = 0.02
ao,c
Pe,c
Pe,s
ao,c
Pe,c
Pe,s
2
2
0.451
3.0 × 10−221
3.0 × 10−221
0.454
1.5 × 10−56
1.5 × 10−56
3
6
0.586
2.0 × 10−126
1.1 × 10−126
0.586
5.7 × 10−33
3.0 × 10−33
4
16
0.628
1.4 × 10−45
1.2 × 10−45
0.626
4.0 × 10−13
3.3 × 10−13
5
25
0.628
9.0 × 10−46
7.6 × 10−46
0.626
2.5 × 10−13
2.1 × 10−13
6
42
0.628
2.4 × 10−45
1.6 × 10−45
0.625
6.7 × 10−13
4.4 × 10−13
7
63
0.628
6.0 × 10−45
4.6 × 10−45
0.626
1.7 × 10−12
1.3 × 10−12
8
96
0.659
1.8 × 10−25
1.5 × 10−25
0.650
3.6 × 10−8
2.9 × 10−8
9
117
0.659
1.4 × 10−25
1.2 × 10−25
0.650
2.8 × 10−8
2.3 × 10−8
10
150
0.659
6.4 × 10−26
4.7 × 10−26
0.647
1.3 × 10−8
1.0 x 10−8
11
187
0.659
9.6 × 10−26
7.8 × 10−26
0.647
2.0 × 10−8
1.6 × 10−8
12
240
0.659
2.9 × 10−25
2.3 × 10−25
0.649
5.7 × 10−8
4.5 × 10−8
13
286
0.659
2.5 × 10−25
2.0 × 10−25
0.647
5.0 × 10−8
3.9 × 10−8
14
350
0.659
4.4 × 10−25
3.1 × 10−25
0.647
8.3 × 10−8
6.3 × 10−8
15
420
0.659
9.5 × 10−25
7.5 × 10−25
0.650
1.8 × 10−7
1.4 × 10−7
16
512
0.682
2.3 × 10−17
1.8 × 10−17
0.657
3.7 × 10−6
3.2 × 10−6
17
561
0.682
2.0 × 10−17
1.6 × 10−17
0.657
3.2 × 10−6
2.9 × 10−6
18
630
0.682
1.2 × 10−17
9.5 × 10−18
0.657
2.2 × 10−6
1.9 × 10−6
19
703
0.682
1.6 × 10−17
1.3 × 10−17
0.657
2.6 × 10−6
2.3 × 10−6
20
800
0.682
9.7 × 10−18
7.7 × 10−18
0.657
2.1 × 10−6
1.8 × 10−6
21
882
0.682
9.1 × 10−18
7.0 × 10−18
0.657
2.0 × 10−6
1.7 × 10−6
22
990
0.681
4.8 × 10−18
3.2 × 10−18
0.653
1.7 × 10−6
1.4 × 10−6
23
1104
0.682
1.1 × 10−17
8.7 × 10−18
0.657
2.7 × 10−6
2.3 × 10−6
Number of reference patterns (N), standard deviation in aaplitude (σa), standard deviation in phase (σθ), optimum amplitude threshold detector setting when all detectors for a given modulus have a common setting (ao,c), probability of error when all detectors have ao,c threshold detector setting (Pe,c), and probability of error when each detector has its own separate optimum threshold setting (Pe,s) are given.
Table II
Operational Characteristics for Multiplication with Residue Number System Holographic Truth-Table Look-Up Processora
Operational Characteristics for 4-, 8-, 12-, and 16-Bit Addition and Multiplication with Residue Number System Holographic Truth-Table Look-Up Processora
Operation
Nb
Moduli Set
N
σa = σθ = 0.01
σa = σθ = 0.02
ao,c
Pte,c
Pte,c
ao,c
Pte,c
Pte,s
Addition
4
3.4.5
47
0.628
2.3 × 10−45
1.9 × 10−45
0.626
6.5 × 10−13
5.4 × 10−13
8
3,5,7,8
190
0.659
1.8 × 10−25
1.5 × 10−25
0.650
3.6 × 10−8
2.9 × 10−8
12
3,5,7,8,11
377
0.659
2.8 × 10−25
2.3 × 10−25
0.649
5.6 × 10−8
4.5 × 10−8
16
4,5,7,9,11,13
694
0.659
4.9 × 10−25
3.9 × 10−25
0.648
9.8 × 10−8
7.8 × 10−8
Multiplication
4
3,4,5,7
88
0.627
9.2 × 10−45
5.4 × 10−45
0.623
2.5 × 10−12
1.5 × 10−12
8
3,5,7,8,11,13
580
0.659
4.9 × 10−25
3.5 × 10−25
0.647
9.6 × 10−8
7.2 × 10−9
12
3,4,5,7,11,13,17,19
1716
0.680
1.5 × 10−17
9.7 × 10−18
0.653
3.9 × 10−6
3.3 × 10−6
16
5,7,9,11,13,16,17,19,23
3252
0.681
3.3 × 10−17
2.2 × 10−17
0.653
8.3 × 10−6
7.0 × 10−6
Number of bits in each word (Nb), total number of reference patterns (N), standard deviation in amplitude (σa), standard deviation in phase (σθ), optlmua amplitude threshold detector setting when all detectors for all moduli have a common setting (ao,c), total probability of error when all detectors have ao,c threshold detector setting (Pte,c), and total probability of error when each detector has its own separate optimum threshold setting (Pte,c) are given.
Table VI
Operational Characteristics for Logically Reduced 4-, 8-, 12-, and 16-Bit Addition and Multiplication with Residue Number System Holographic Truth-Table Look-Up Processora
Operational Characteristics for Addition with Residue Number System Holographic Truth-Table Look-Up Processora
Modulus
N
σa = σθ = 0.01
σa = σθ = 0.02
ao,c
Pe,c
Pe,s
ao,c
Pe,c
Pe,s
2
2
0.451
3.0 × 10−221
3.0 × 10−221
0.454
1.5 × 10−56
1.5 × 10−56
3
6
0.586
2.0 × 10−126
1.1 × 10−126
0.586
5.7 × 10−33
3.0 × 10−33
4
16
0.628
1.4 × 10−45
1.2 × 10−45
0.626
4.0 × 10−13
3.3 × 10−13
5
25
0.628
9.0 × 10−46
7.6 × 10−46
0.626
2.5 × 10−13
2.1 × 10−13
6
42
0.628
2.4 × 10−45
1.6 × 10−45
0.625
6.7 × 10−13
4.4 × 10−13
7
63
0.628
6.0 × 10−45
4.6 × 10−45
0.626
1.7 × 10−12
1.3 × 10−12
8
96
0.659
1.8 × 10−25
1.5 × 10−25
0.650
3.6 × 10−8
2.9 × 10−8
9
117
0.659
1.4 × 10−25
1.2 × 10−25
0.650
2.8 × 10−8
2.3 × 10−8
10
150
0.659
6.4 × 10−26
4.7 × 10−26
0.647
1.3 × 10−8
1.0 x 10−8
11
187
0.659
9.6 × 10−26
7.8 × 10−26
0.647
2.0 × 10−8
1.6 × 10−8
12
240
0.659
2.9 × 10−25
2.3 × 10−25
0.649
5.7 × 10−8
4.5 × 10−8
13
286
0.659
2.5 × 10−25
2.0 × 10−25
0.647
5.0 × 10−8
3.9 × 10−8
14
350
0.659
4.4 × 10−25
3.1 × 10−25
0.647
8.3 × 10−8
6.3 × 10−8
15
420
0.659
9.5 × 10−25
7.5 × 10−25
0.650
1.8 × 10−7
1.4 × 10−7
16
512
0.682
2.3 × 10−17
1.8 × 10−17
0.657
3.7 × 10−6
3.2 × 10−6
17
561
0.682
2.0 × 10−17
1.6 × 10−17
0.657
3.2 × 10−6
2.9 × 10−6
18
630
0.682
1.2 × 10−17
9.5 × 10−18
0.657
2.2 × 10−6
1.9 × 10−6
19
703
0.682
1.6 × 10−17
1.3 × 10−17
0.657
2.6 × 10−6
2.3 × 10−6
20
800
0.682
9.7 × 10−18
7.7 × 10−18
0.657
2.1 × 10−6
1.8 × 10−6
21
882
0.682
9.1 × 10−18
7.0 × 10−18
0.657
2.0 × 10−6
1.7 × 10−6
22
990
0.681
4.8 × 10−18
3.2 × 10−18
0.653
1.7 × 10−6
1.4 × 10−6
23
1104
0.682
1.1 × 10−17
8.7 × 10−18
0.657
2.7 × 10−6
2.3 × 10−6
Number of reference patterns (N), standard deviation in aaplitude (σa), standard deviation in phase (σθ), optimum amplitude threshold detector setting when all detectors for a given modulus have a common setting (ao,c), probability of error when all detectors have ao,c threshold detector setting (Pe,c), and probability of error when each detector has its own separate optimum threshold setting (Pe,s) are given.
Table II
Operational Characteristics for Multiplication with Residue Number System Holographic Truth-Table Look-Up Processora
Operational Characteristics for 4-, 8-, 12-, and 16-Bit Addition and Multiplication with Residue Number System Holographic Truth-Table Look-Up Processora
Operation
Nb
Moduli Set
N
σa = σθ = 0.01
σa = σθ = 0.02
ao,c
Pte,c
Pte,c
ao,c
Pte,c
Pte,s
Addition
4
3.4.5
47
0.628
2.3 × 10−45
1.9 × 10−45
0.626
6.5 × 10−13
5.4 × 10−13
8
3,5,7,8
190
0.659
1.8 × 10−25
1.5 × 10−25
0.650
3.6 × 10−8
2.9 × 10−8
12
3,5,7,8,11
377
0.659
2.8 × 10−25
2.3 × 10−25
0.649
5.6 × 10−8
4.5 × 10−8
16
4,5,7,9,11,13
694
0.659
4.9 × 10−25
3.9 × 10−25
0.648
9.8 × 10−8
7.8 × 10−8
Multiplication
4
3,4,5,7
88
0.627
9.2 × 10−45
5.4 × 10−45
0.623
2.5 × 10−12
1.5 × 10−12
8
3,5,7,8,11,13
580
0.659
4.9 × 10−25
3.5 × 10−25
0.647
9.6 × 10−8
7.2 × 10−9
12
3,4,5,7,11,13,17,19
1716
0.680
1.5 × 10−17
9.7 × 10−18
0.653
3.9 × 10−6
3.3 × 10−6
16
5,7,9,11,13,16,17,19,23
3252
0.681
3.3 × 10−17
2.2 × 10−17
0.653
8.3 × 10−6
7.0 × 10−6
Number of bits in each word (Nb), total number of reference patterns (N), standard deviation in amplitude (σa), standard deviation in phase (σθ), optlmua amplitude threshold detector setting when all detectors for all moduli have a common setting (ao,c), total probability of error when all detectors have ao,c threshold detector setting (Pte,c), and total probability of error when each detector has its own separate optimum threshold setting (Pte,c) are given.
Table VI
Operational Characteristics for Logically Reduced 4-, 8-, 12-, and 16-Bit Addition and Multiplication with Residue Number System Holographic Truth-Table Look-Up Processora
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