Modified Nested Barker Codes for Ultra-Wideband Signal–Code Constructions
Abstract
:1. Introduction
2. Modified Barker Codes
3. The Analysis of Side Lobe Level in Modified Nested Barker Code Constructions, Having an Asymmetrical Alphabet
3.1. Nested Code Constructions Based on the Modified Barker Codes
- −
- “BB, BM”: 4 × 3, 4 × 5, 4 × 7 and 4 × 13;
- −
- “BM”: 2 × 3, 3 × 3, 3 × 5, 3 × 13, 7 × 3, 7 × 13, 11 × 3, 11 × 5, 11 × 13 and 13 × 5;
- −
- “MB”: 3 × 4, 5 × 2, 5 × 11, 7 × 4, 13 × 2, 13 × 7 and 13 × 11;
- −
- “MB, MM”: 5 × 3, 5 × 7, 5 × 13 and 13 × 3;
- −
- “MM”: 5 × 5, 7 × 5 and 13 × 13.
3.2. The Search for Optimal Values of Nested Code Components via Their Analytical Expressions
4. The Generation of Conventional and Modified Ultra-Wideband Signal–Code Constructions and Their Nested Representations
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
References
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Code Length, N | Barker Sequence with Symmetric Alphabet | SL NACF Levels in dB | SL NACF Levels | ||
---|---|---|---|---|---|
2 | +1 −1 | or +1 +1 | −6.02 | −1/2 | 1/2 |
3 | +1 +1 −1 | −9.54 | −1/3 | ||
4 | +1 +1 −1 +1 | or +1 +1 +1 −1 | −12.04 | ±1/4 | |
5 | +1 +1 +1 −1 +1 | −13.98 | 1/5 | ||
7 | +1 +1 +1 −1 −1 +1 −1 | −16.90 | −1/7 | ||
11 | +1 +1 +1 −1 −1 −1 +1 −1−1 +1 −1 | −20.83 | −1/11 | ||
13 | +1 +1 +1 +1 +1 −1 −1 +1 +1 −1 +1 −1 +1 | −22.28 | 1/13 |
Lobe Number ACF | N = 2 | N = 3 | N = 4 | N = 5 | N = 7 | N = 11 | N = 13 |
---|---|---|---|---|---|---|---|
R1 | b2 + 1 | b2 + 2 | b2 + 3 | b2 + 4 | 3b2 + 4 | 6b2 + 5 | 4b2 + 9 |
R2 | −b | −b + 1 | −2b + 1 or −b + 2 | −2b + 2 | b2 − 3b + 2 | 3b2 − 5b + 2 | b2 − 6b + 5 |
R3 | −b | −b + 1 | −b + 2 | b2 − 3b + 1 | 3b2 − 5b + 1 | b2 − 5b + 5 | |
R4 | 1 or −b | −b + 1 | (b − 1)2 | 3b2 − 4b + 1 | b2 − 5b + 4 | ||
R5 | 1 | −2b + 1 | 2b2 − 4b + 1 | (b − 2)2 | |||
R6 | −b + 1 | 2b2 − 3b + 1 | b2 − 4b + 3 | ||||
R7 | −b | b2 − 3b + 1 | b2 − 3b + 3 | ||||
R8 | (b − 1)2 | −3b + 3 | |||||
R9 | −2b + 1 | −2b + 3 | |||||
R10 | −b + 1 | −2b + 2 | |||||
R11 | −b | −b + 2 | |||||
R12 | −b + 1 | ||||||
R13 | 1 |
Code Length, N | Estimating Code Parameters | |||
---|---|---|---|---|
The Value of −b for the Modified Barker Pair {1, −b} | SL Level of the Classical Barker Code | SL Level of Modified Barker Code | Difference between Classic and Modified | |
2 | −1 | −6.02 dB | - | - |
3 | −0.5 | −9.54 dB | −13.06 dB | 3.52 dB |
4 | −1 | −12.04 dB | - | - |
5 | −1.5 | −13.98 dB | −15.92 dB | 1.94 dB |
7 | −16.90 dB | −18.68 dB | 1.78 dB | |
11 | −1 | −20.83 dB | - | - |
13 | −22.28 dB | −23.77 dB | 1.49 dB |
Barker Code View | Barker Code |
---|---|
BB | [{1, 1, −1}, {1, 1, −1}, {1, 1, −1}, {1, 1, −1}, {1, 1, −1}, {−1, −1, 1}, {−1, −1, 1}, {1, 1, −1}, {1, 1, −1}, {−1, −1, 1}, {1, 1, −1}, {−1, −1, 1}, {1, 1, −1}] |
BM | [{1, 1, −b}, {1, 1, −b}, {1, 1, −b}, {1, 1, −b}, {1, 1, −b}, {−1, −1, b}, {−1, −1, b}, {1, 1, −b}, {1, 1, −b}, {−1, −1, b}, {1, 1, −b}, {−1, −1, b}, {1, 1, −b}] |
MB | [{1, 1, −1}, {1, 1, −1}, {1, 1, −1}, {1, 1, −1}, {1, 1, −1}, {−b, −b, b}, {−b, −b, b}, {1, 1, −1}, {1, 1, −1}, {−b, −b, b}, {1, 1, −1}, {−b, −b, b}, {1, 1, −1}] |
MM | [{1, 1, −b}, {1, 1, −b}, {1, 1, −b}, {1, 1, −b}, {1, 1, −b}, {−b, −b, b2}, {b, −b, b2}, {1, 1, −b}, {1, 1, −b}, {−b, −b, b2}, {1, 1, −b}, {−b, −b, b2}, {1, 1, −b}] |
Code Length, N × M | BB | Best Type | Code Length, N × M | BB | Best Type | Code Length, N × M | BB | Best Type |
---|---|---|---|---|---|---|---|---|
2 × 2 | 0.25 | BB | 4 × 2 | 0.2500 | BB | 11 × 2 | 0.0455 | BB |
−12.0412 | −12.0412 | −26.8485 | ||||||
2 × 4 | 0.3750 | BB | 4 × 4 | 0.3125 | BB | 11 × 4 | 0.2500 | BB |
−8.5194 | −10.1030 | −12.0412 | ||||||
2 × 11 | 0.0455 | BB | 4 × 11 | 0.25 | BB | 11 × 11 | 0.0083 | BB |
−26.8485 | −12.0412 | −41.6557 |
Code Length, N × M | BB | BM | Differences between BB-BM | Best Type | Code Length, N × M | BB | BM | Differences between BB-BM | Best Type |
---|---|---|---|---|---|---|---|---|---|
2 × 3 | 0.1667 | 0.1111 | 0.0556 | BM | 4 × 7 | 0.25 | 0.25 | 0 | BB, BM |
−15.5630 | −19.0849 | −3.5219 | −12.0412 | −12.0412 | 0 | ||||
2 × 5 | 0.2000 | 0.2400 | −0.04 | BB | 4 × 13 | 0.25 | 0.25 | 0 | BB, BM |
−13.9794 | −12.3958 | 1.5836 | −12.0412 | −12.0412 | 0 | ||||
2 × 7 | 0.0714 | 0.1235 | −0.0521 | BB | 11 × 3 | 0.0303 | 0.0202 | 0.0101 | BM |
−22.9226 | −18.1667 | 4.7559 | −30.3703 | −33.8930 | −3.5227 | ||||
2 × 13 | 0.0769 | 0.0972 | −0.0203 | BB | 11 × 5 | 0.2000 | 0.1600 | 0.04 | BM |
−22.2789 | −20.2459 | 2.033 | −13.9794 | −15.9176 | −1.9382 | ||||
4 × 3 | 0.25 | 0.25 | 0 | BB, BM | 11 × 7 | 0.0130 | 0.0824 | −0.0694 | BB |
−12.0412 | −12.0412 | 0 | −37.7298 | −21.6815 | 16.0483 | ||||
4 × 5 | 0.25 | 0.25 | 0 | BB, BM | 11 × 13 | 0.0769 | 0.0648 | 0.0121 | BM |
−12.0412 | −12.0412 | 0 | −22.2789 | −23.7678 | −1.4889 |
Code Length, N × M | BB | MB | Differences between BB-MB | Best Type | Code Length, N × M | BB | MB | Differences between BB-MB | Best Type |
---|---|---|---|---|---|---|---|---|---|
3 × 2 | 0.1667 | 0.2222 | −0.0555 | BB | 7 × 2 | 0.0714 | 0.1165 | −0.0451 | BB |
−15.5630 | −13.0643 | 2.4987 | −22.9226 | −18.6756 | 4.247 | ||||
3 × 4 | 0.2500 | 0.2222 | 0.0278 | MB | 7 × 4 | 0.25 | 0.2209 | 0.0291 | MB |
−12.0412 | −13.0643 | −1.0231 | −12.0412 | −13.1168 | −1.0756 | ||||
3 × 11 | 0.0303 | 0.2222 | −0.1919 | BB | 7 × 11 | 0.0130 | 0.1165 | −0.1035 | BB |
−30.3703 | −13.0643 | 17.306 | −37.7298 | −18.6756 | 19.0542 | ||||
5 × 2 | 0.2000 | 0.1600 | 0.04 | MB | 13 × 2 | 0.0769 | 0.0648 | 0.0121 | MB |
−13.9794 | −15.9176 | −1.9382 | −22.2789 | −23.7678 | −1.4889 | ||||
5 × 4 | 0.25 | 0.2900 | −0.04 | BB | 13 × 4 | 0.25 | 0.2662 | −0.0162 | BB |
−12.0412 | −10.7520 | 1.2892 | −12.0412 | −11.4958 | 0.5454 | ||||
5 × 11 | 0.2000 | 0.1600 | 0.04 | MB | 13 × 11 | 0.0769 | 0.0648 | 0.0121 | MB |
−13.9794 | −15.9176 | −1.9382 | −22.2789 | −23.7678 | −1.4889 |
Code Length, N × M | BB | BM | MB | MM | Differences between BB-MB | Differences between BB-MB | Differences between BB-MM | Best Type |
---|---|---|---|---|---|---|---|---|
3 × 3 | 0.1111 | 0.0741 | 0.2222 | 0.2222 | 0.037 | −0.1111 | −0.1111 | BM |
−19.0849 | −22.6036 | −13.0643 | −13.0643 | −3.5187 | 6.0206 | 6.0206 | ||
3 × 5 | 0.2000 | 0.1600 | 0.2222 | 0.2222 | 0.04 | −0.0222 | −0.0222 | BM |
−13.9794 | −15.9176 | −13.0643 | −13.0643 | −1.9382 | 0.9151 | 0.9151 | ||
3 × 7 | 0.0476 | 0.0824 | 0.2222 | 0.5742 | −0.0348 | −0.1746 | −0.5266 | BB |
−26.4444 | −21.6815 | −13.0643 | −4.8188 | 4.7629 | 13.3801 | 21.6256 | ||
3 × 13 | 0.0769 | 0.0648 | 0.2222 | 0.2222 | 0.0121 | −0.1453 | −0.1453 | BM |
−22.2789 | −23.7678 | −13.0643 | −13.0643 | −1.4889 | 9.2146 | 9.2146 | ||
5 × 3 | 0.2000 | 0.2000 | 0.1600 | 0.1600 | 0 | 0.04 | 0.04 | MB, MM |
−13.9794 | −13.9794 | −15.9176 | −15.9176 | 0 | −1.9382 | −1.9382 | ||
5 × 5 | 0.2000 | 0.2000 | 0.2000 | 0.1856 | 0 | 0 | 0.0144 | MM |
−13.9794 | −13.9794 | −13.9794 | −14.6284 | 0 | 0 | −0.649 | ||
5 × 7 | 0.2000 | 0.2000 | 0.1600 | 0.1600 | 0 | 0.04 | 0.04 | MB, MM |
−13.9794 | −13.9794 | −15.9176 | −15.9176 | 0 | −1.9382 | −1.9382 | ||
5 × 13 | 0.2000 | 0.2000 | 0.1600 | 0.1600 | 0 | 0.04 | 0.04 | MB, MM |
−13.9794 | −13.9794 | −15.9176 | −15.9176 | 0 | −1.9382 | −1.9382 | ||
7 × 3 | 0.0476 | 0.0318 | 0.5387 | 0.1165 | 0.0158 | −0.4911 | −0.0689 | BM |
−26.4444 | −29.9515 | −5.3726 | −18.6735 | −3.5071 | 21.0718 | 7.7709 | ||
7 × 5 | 0.2000 | −0.1600 | 0.2000 | 0.1414 | 0.04 | 0 | 0.0586 | MM |
−13.9794 | −15.9176 | −13.9794 | −16.9932 | −1.9382 | 0 | −3.0138 | ||
7 × 7 | 0.0204 | 0.0824 | 0.1165 | 0.1165 | −0.062 | −0.0961 | −0.0961 | BB |
−33.8039 | −21.6815 | −18.6756 | −18.6756 | 12.1224 | 15.1283 | 15.1283 | ||
7 × 13 | 0.0769 | 0.0648 | 0.1165 | 0.1165 | 0.0121 | −0.0396 | −0.0396 | BM |
−22.2789 | −23.7678 | −18.6756 | −18.6756 | −1.4889 | 3.6033 | 3.6033 | ||
13 × 3 | 0.0769 | 0.0769 | 0.0648 | 0.0648 | 0 | 0.0121 | 0.0121 | MB, MM |
−22.2789 | −22.2815 | −23.7678 | −23.7678 | −0.0026 | −1.4889 | −1.4889 | ||
13 × 5 | 0.2000 | 0.1600 | 0.2000 | 0.1704 | 0.04 | 0 | 0.0296 | BM |
−13.9794 | −15.9176 | −13.9794 | −15.3722 | −1.9382 | 0 | −1.3928 | ||
13 × 7 | 0.0769 | 0.0824 | 0.0648 | 0.0877 | −0.0055 | 0.0121 | −0.0108 | MB |
−22.2789 | −21.6815 | −23.7678 | −21.1400 | 0.5974 | −1.4889 | 1.1389 | ||
13 × 13 | 0.0769 | 0.0769 | 0.0769 | 0.0690 | 0 | 0 | 0.0079 | MM |
−22.2789 | −22.2789 | −22.2789 | −23.2224 | 0 | 0 | −0.9435 |
ACF Petal Number | Expressions for Searching b in the Case of the MM Option | Expressions for Searching b1 and b2 in the Case of the MM Option |
---|---|---|
R1 | b4 + 4b2 + 4 | b12b22 + 2b12 + 2b22 + 4 |
R2 | b3 + 2b2 + 3b + 2 | 3b2 + b1b2 + b12b2 + b12 + 2 |
R3 | b3 + b2 + 4b + 1 | b1 + 3b2 + b1b2 + b12b2 + 1 |
R4 | b3 + b2 + 2b + 2 | 2b1 + b1b22 + b22 + 2 |
R5 | 2b2 + 2b + 1 | b1 + b2 + 2b1b2 + 1 |
R6 | 2b2 + 2b | b1 + b2 + 2b1b2 |
R7 | b3 + 2b | b1b22 + 2b1 |
R8 | b2 + b | b1 + b1b2 |
R9 | b2 | b1b2 |
Code Length, N × M | Meaning [b1, b2] in Nested Type Code MM2: {1, b1} × {1, b2} | Maximum SL Level of NACF for Types BB/BM/MB/MM and MM2 | Difference “Best Type” of Table 5, Table 6, Table 7 and Table 8 | Best Type | Code Length, N × M | Meaning [b1, b2] in Nested Type Code MM2: {1, b1} × {1, b2} | Maximum SL Level of NACF for Types BB/BM/MB/MM and MM2 | Difference “Best Type” of Table 5, Table 6, Table 7 and Table 8 | Best Type |
---|---|---|---|---|---|---|---|---|---|
2 × 2 | BB {1, −1} × {1, 1} MM2 {1, −1} × {1, 1} | −0.25 −0.25 | 0 | BB | 5 × 7 | MM {1, −1.5} × {1, −0.5858} MM2 {1, −10} × {1, −1} | 0.1600 0.0247 | 0.1353 | MM2 |
2 × 3 | BM {1, −1} × {1, −0.5} MM2 {1, −1} × {1, 0.5} | 0.1111 −0.1111 | 0.2222 | MM2 | 5 × 11 | MB {1, −1.5} × {1, −1} MM2 {1, −10} × {1, −1} | 0.1600 0.0157 | 0.1443 | MM2 |
2 × 4 | BB {1, −1} × {1, −1} MM2 {1, 0} × {1, −2} | 0.3750 0.1429 | 0.2321 | MM2 | 5 × 13 | MM {1, −1.5} × {1, −1.2679} MM2 {1, −10} × {1, −2} | 0.1600 0.0608 | 0.0992 | MM2 |
2 × 5 | BB {1, −1} × {1, −1} MM2 {1, 0} × {1, −2} | 0.2000 0.1250 | 0.0750 | MM2 | 7 × 2 | BB {1, −1} × {1, −1} MM2 {1, −1} × {1, −10} | 0.0714 0.0141 | 0.0573 | MM2 |
2 × 7 | BB {1, −1} × {1, −1} MM2 {1, 0} × {1, −1} | 0.0714 0 | 0.0714 | MM2 | 7 × 3 | BM {1, −1} × {1, −0.5} MM2 {1, −1} × {1, −10} | 0.0318 0.0140 | 0.0178 | MM2 |
2 × 11 | BB {1, −1} × {1, −1} MM2 {1, 0} × {1, −1} | 0.0455 0 | 0.0455 | MM2 | 7 × 4 | MB {1, −0.5858} × {1, −1} MM2 {1, −1} × {1, −10} | 0.2209 0.0264 | 0.1945 | MM2 |
2 × 13 | BB {1, −1} × {1, −1} MM2 {1, 0} × {1, −2} | 0.0769 0.0376 | 0.0393 | MM2 | 7 × 5 | MM {1, −0.5858} × {1, −1.5} MM2 {1, −1} × {1, −10} | 0.1414 0.0247 | 0.1167 | MM2 |
3 × 2 | BB {1, −1} × {1, −1} MM2 {1, −2} × {1, 1} | 0.1667 −0.1667 | 0.3334 | MM2 | 7 × 7 | BB {1, −1} × {1, −1} MM2 {1, −0.89} × {1, −0.89} | 0.0204 0.0198 | 0.0006 | MM2 |
3 × 3 | BM {1, −1} × {1, −0.5} MM2 {1, −1.721} × {1, 0.1211} | 0.0741 −0.0209 | 0.0950 | MM2 | 7 × 11 | BB {1, −1} × {1, −1} MM2 {1, −0.92} × {1, −1} | 0.0130 0.0128 | 0.0002 | MM2 |
3 × 4 | MB {1, −0.5} × {1, −1} MM2 {1, −10} × {1, −10} | 0.2222 0.0260 | 0.1962 | MM2 | 7 × 13 | BM {1, −1} × {1, −1.2679} MM2 {1, −1 } × {1, −2} | 0.0648 0.0400 | 0.0248 | MM2 |
3 × 5 | BM {1, −1} × {1, −1.5} BM {1, −0.77} × {1, −2.18} | 0.1600 0.0903 | 0.0697 | MM2 | 11 × 2 | BB {1, −1} × {1, −1} MM2 {1, −1} × {1, −10} | 0.0455 0.0090 | 0.0365 | MM2 |
3 × 7 | BB {1, −1} × {1, −1} MM2 {1, −10} × {1, −1} | 0.0476 0.0140 | 0.0336 | MM2 | 11 × 3 | BM {1, −1} × {1, −0.5} MM2 [−1, −10] | 0.0202 0.0089 | 0.0113 | MM2 |
3 × 11 | BB {1, −1} × {1, −1} MM2 {1, −10} × {1, −1} | 0.0303 0.0089 | 0.0214 | MM2 | 11 × 4 | BB {1, −1} × {1, −1} MM2 {1, −1} × {1, −10} | 0.2500 0.0168 | 0.2332 | MM2 |
3 × 13 | BM {1, −1} × {1, −1.2679} MM2 {1, −1} × {1, −2} | 0.0648 0.0400 | 0.0248 | MM2 | 11 × 5 | BM {1, −1} × {1, −1.5} MM2 {1, −1} × {1, −10} | 0.1600 0.0157 | 0.1443 | MM2 |
4 × 2 | BB {1, −1} × {1, −1} MM2 {1, −10} × {1, 0} | 0.2500 0.0097 | 0.2403 | MM2 | 11 × 7 | BB {1, −1} × {1, −1} MM2 {1, −1} × {1, −0.92} | 0.0130 0.0128 | 0.0002 | MM2 |
4 × 3 | BM {1, −1} × {1, −0.5} MM2 {1, −10} × {1, 0} | 0.2500 0.0097 | 0.2403 | MM2 | 11 × 11 | BB {1, −1} × {1, −1} MM2 {1, −1} × {1, −1} | 0.0083 0.0083 | 0 | BB |
4 × 4 | BB {1, −1} × {1, −1} MM2 {1, −10} × {1, −10} | 0.3125 0.0437 | 0.2688 | MM2 | 11 × 13 | BM {1, −1} × {1, −1.2679} MM2 {1, −1} × {1, −2} | 0.0648 0.0400 | 0.0248 | MM2 |
4 × 5 | BM {1, −1} × {1, −1.5} MM2 {1, −10} × {1, −10} | 0.2500 0.0415 | 0.2085 | MM2 | 13 × 2 | MB {1, −1.2679} × {1, −1} MM2 {1, −2} × {1, −9} | 0.0648 0.0400 | 0.0248 | MM2 |
4 × 7 | BM {1, −1} × {1, −0.5858} MM2 {1, −10} × {1, −1} | 0.2500 0.0264 | 0.2236 | MM2 | 13 × 3 | MM {1, −1.2679} × {1, −0.5} MM2 {1, −2} × {1, −1} | 0.0648 0.0400 | 0.0248 | MM2 |
4 × 11 | BB {1, −1} × {1, −1} MM2 {1, −10} × {1, −1} | 0.2500 0.0168 | 0.2332 | MM2 | 13 × 4 | BB {1, −1} × {1, −1} MM2 {1, −10} × {1, −2} | 0.2500 0.0400 | 0.2100 | MM2 |
4 × 13 | BM {1, −1} × {1, −1.2679} MM2 {1, −10} × {1, −2} | 0.2500 0.0621 | 0.1879 | MM2 | 13 × 5 | BM {1, −1} × {1, −1.5} MM2 {1, −2} × {1, −8} | 0.1600 0.0400 | 0.1200 | MM2 |
5 × 2 | MB {1, −1.5} × {1, −1} MM2 {1, −10} × {1, −1} | 0.1600 0.0096 | 0.1504 | MM2 | 13 × 7 | MB {1, −1.2679} × {1, −1} MM2 {1, −2} × {1, −1.4} | 0.0648 0.0400 | 0.0248 | MM2 |
5 × 3 | MM {1, −1.5} × {1, −0.5} MM2 {1, −10} × {1, 0} | 0.1600 0.0096 | 0.1504 | MM2 | 13 × 11 | MB {1, −1.2679} × {1, −1} MM2 {1, −2} × {1, −1} | 0.0648 0.0400 | 0.0248 | MM2 |
5 × 4 | BB {1, −1} × {1, −1} MM2 {1, −10} × {1, −10} | 0.2500 0.0416 | 0.2084 | MM2 | 13 × 13 | MM {1, −1.2679} × × {1, −1.2679} MM2 {1, −1.6} × {1, −2} | 0.0690 0.0527 | 0.0163 | MM2 |
5 × 5 | MM {1, −1.5} × {1, −1.5} MM2 {1, −10} × {1, −10} | 0.1856 0.0396 | 0.1460 | MM2 |
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Nenashev, V.A.; Bestugin, A.R.; Rabin, A.V.; Solenyi, S.V.; Nenashev, S.A. Modified Nested Barker Codes for Ultra-Wideband Signal–Code Constructions. Sensors 2023, 23, 9528. https://doi.org/10.3390/s23239528
Nenashev VA, Bestugin AR, Rabin AV, Solenyi SV, Nenashev SA. Modified Nested Barker Codes for Ultra-Wideband Signal–Code Constructions. Sensors. 2023; 23(23):9528. https://doi.org/10.3390/s23239528
Chicago/Turabian StyleNenashev, Vadim A., Aleksandr R. Bestugin, Alexey V. Rabin, Sergei V. Solenyi, and Sergey A. Nenashev. 2023. "Modified Nested Barker Codes for Ultra-Wideband Signal–Code Constructions" Sensors 23, no. 23: 9528. https://doi.org/10.3390/s23239528