# Search and Study of Marked Code Structures for a Spatially Distributed System of Small-Sized Airborne Radars

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Analysis of the Level of Side Lobes of Modified Barker Codes with Asymmetric Alphabet

_{2}and R

_{3}, which is shown in Figure 1. In this case, the optimal value of the maximum SL is R

_{2}= R

_{3}= −13.06 dB.

_{2}= R

_{5}= −15.92 and b = 1.5a for the pair {a, −b}.

_{2}= R

_{7}= −18.68 dB and, accordingly, for the pair {a, −b} associated with the expression $b=\left(2-\sqrt{2}\right)a$. The expressions for searching for the modified Barker code structures of length N = 11 are given in Table 4.

_{2}= R

_{13}= −23.77 dB at $b=3-\sqrt{3}$ for the pair {1, −b} and, accordingly, the pair {a, −b} is related through the expression $b=\left(3-\sqrt{3}\right)a$.

_{2}and R

_{N}.

_{2}and R

_{N}, after normalizing the ACF to unity; that is, after dividing them by R

_{1}.

^{2}, and the number of positive elements, as it can be replaced, corresponds to the square of a number one greater than the previous one (q + 1)

^{2}. These code structures exist on lengths N(q) = q

^{2}+(q + 1)

^{2}, N = 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, etc. These code lengths are of less interest as the value of the code length grows faster; hence, there are large gaps among the possible lengths N, in contrast to the codes formed on the calculation of quadratic residues. The possibility of a smaller choice of code lengths during its generation is less attractive in the implementation of the SD AR as it may impose additional requirements when generating a probing signal.

^{L}− 1, L is an integer, N = 3, 7, 15, 31, 63, 127, 255, 511, 1023, etc. M-sequences are also proposed to be investigated for the expediency of their applicability as they have several implementations within the same length (several generating polynomials), which is required for the effective functioning of the SD; thus, they should be considered in more detail. In this case, the selected code structures should provide high compression characteristics [2,4,7] of broadband code structure systems, as well as a low degree of mutual influence during joint reception in the common channel of the SD of the small-sized AR.

## 3. Generation of M-Sequences

_{1}(x) = x

^{5}+ x

^{4}+ x

^{2}+ x + 1 and g

_{2}(x) = x

^{5}+ x

^{4}+ x

^{3}+ x + 1—each of which generates different M-sequences of length N = 31.

## 4. Method for Searching for Modified M-Sequences

- (1)
- From Table 6, you need to choose several different polynomials, g
_{1}(x), g_{2}(x), …, g_{n}(x), of one order. The number of polynomials for the generation depends on the number of positions in the SD of the small-sized AR. - (2)
- Generate the required number of M-sequences based on the diagram (see Figure 7), M
_{1}, M_{2}, …, M_{n}sequences over polynomials g_{1}(x), g_{2}(x), …, g_{n}(x), and bring their values to the symmetrical pair {1, −1}.

- (3)
- Construct a normalized ACF for each of the M-sequences obtained at step 2.
- (4)
- Determine the maximum modulo value of the SL of two ACFs obtained at step 3.
- (5)
- In the generated code sequences at step 2, it is required to replace the code element from the value −1 to the value −b.
- (6)
- Obtain the expressions for each lobe (main and side) of ACF depending on b.
- (7)
- Obtain the expressions for each BL for the normalized ACF, thereby forming a system of SL expressions.
- (8)
- Find the parameter b for which the SL value of the normalized ACF will be the smallest possible by solving the system of expressions obtained in step 7.
- (9)
- Verify that the ACF SL values found in step 8 are lower than the maximum SL level determined in step 4.
- (10)
- Make sure that the values of the CCF of the modified M-sequences M
_{1}, M_{2}, ..., M_{n}are uniformly distributed.

## 5. Computer Experiments on the Search for Labeled Code Structures based on Modifications of M-Sequences

- (1)
- Polynomial: g
_{1}(x) = x^{5}+ x^{2}+ 1 initial conditions for generation: [0 0 0 1 0]. - (2)
- Polynomial: g
_{2}(x) = x^{5}+ x^{3}+ 1 initial conditions for generation: [0 0 0 0 1].

_{1}(x) and g

_{2}(x) with respect to b (see Table 7).

_{1}(x) and g

_{2}(x).

^{5}+ x

^{4}+ x

^{3}+ x

^{2}+ 1, generating an M-sequence of length N = 31 with the value b = −0.6835, and the smallest SL decrease was 0.1959 dB for the 9th degree polynomial x

^{9}+ x

^{8}+ x

^{6}+ x

^{5}+ x

^{4}+ x

^{3}+ x

^{2}+ x + 1, generating an M-sequence of length N = 511 with value b = −0.9186. The remaining similar numerical results are between these two indicated values of the SL levels.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Graphs of the expressions SL 2 and 3 with normalized ACF for searching for b in the decibel measurement scale.

**Figure 2.**Graphs of expressions SL 2–5 with a normalized ACF for finding b in the decibel measurement scale.

**Figure 3.**Graphs of expressions SL 2–7 with a normalized ACF for finding b in the decibel measurement scale.

**Figure 4.**Graphs of the expressions SL 2–11 with a normalized ACF for finding b in the decibel measurement scale.

**Figure 5.**Graphs of expressions SL 2–13 with a normalized ACF for finding b in the decibel measurement scale.

**Figure 8.**SL plots of the modified M-sequence: (

**a**) SL expressions corresponding to g

_{1}(x); (

**b**) SL expressions corresponding to g

_{2}(x).

Lobe Number ACF | Search Expressions b | Expressions to Search for a and b |
---|---|---|

R_{1} | b^{2} + 2 | 2a^{2} + b^{2} |

R_{2} | −b + 1 | a^{2} − ab |

R_{3} | −b | −ab |

Lobe Number ACF | Search Expressions b | Expressions to Search for a and b |
---|---|---|

R_{1} | b^{2} + 4 | 4a^{2} + b^{2} |

R_{2} | −2b + 2 | 2a^{2} − 2ab |

R_{3} | −b + 2 | 2a^{2} − ab |

R_{4} | −b + 1 | a^{2} − ab |

R_{5} | 1 | a^{2} |

Lobe Number ACF | Search Expressions b | Expressions to Search for a and b |
---|---|---|

R_{1} | 3b^{2} + 4 | 4a^{2} + 3b^{2} |

R_{2} | b^{2} − 3b + 2 | 2a^{2} − 3ab + b^{2} |

R_{3} | b^{2} − 3b + 1 | a^{2} − 3ab + b^{2} |

R_{4} | (b − 1)^{2} | (a − b)^{2} |

R_{5} | −2b + 1 | a^{2} − 2ab |

R_{6} | −b + 1 | a^{2} − ab |

R_{7} | −b | −ab |

Lobe Number ACF | Search Expressions b | Expressions to Search for a and b |
---|---|---|

R_{1} | 6b^{2} + 5 | 5a^{2} + 6b^{2} |

R_{2} | 3b^{2} − 5b + 2 | 2a^{2} − 5ab + 3b^{2} |

R_{3} | 3b^{2} − 5b + 1 | a^{2} − 5ab + 3b^{2} |

R_{4} | 3b^{2} − 4b + 1 | a^{2} − 4ab + 3b^{2} |

R_{5} | 2b^{2} − 4b + 1 | a^{2} − 4ab + 2b^{2} |

R_{6} | 2b^{2} − 3b + 1 | a^{2} − 3ab + 2b^{2} |

R_{7} | b^{2} − 3b + 1 | a^{2} − 3ab + b^{2} |

R_{8} | (b − 1)^{2} | (a − b)^{2} |

R_{9} | −2b + 1 | a^{2} − 2ab |

R_{10} | −b + 1 | a^{2} − ab |

R_{11} | −b | −ab |

Lobe Number ACF | Search Expressions b | Expressions to Search for a and b |
---|---|---|

R_{1} | 4b^{2} + 9 | 9a^{2} + 4b^{2} |

R_{2} | b^{2} − 6b + 5 | 5a^{2} − 6ab + b^{2} |

R_{3} | b^{2} − 5b + 5 | 5a^{2} − 5ab + b^{2} |

R_{4} | b^{2} − 5b + 4 | 4a^{2} − 5ab + b^{2} |

R_{5} | (b − 2)^{2} | (2a − b)^{2} |

R_{6} | b^{2} − 4b + 3 | 3a^{2} − 4ab + b^{2} |

R_{7} | b^{2} − 3b + 3 | 3a^{2} − 3ab + b^{2} |

R_{8} | −3b + 3 | 3a^{2} − 3ab |

R_{9} | −2b + 3 | 3a^{2} − 2ab |

R_{10} | −2b + 2 | 2a^{2} − 2ab |

R_{11} | −b + 2 | 2a^{2} − ab |

R_{12} | −b + 1 | a^{2} − ab |

R_{13} | 1 | a^{2} |

Degree of Polynomial L | The Length of the Generated Sequence N = 2^{L} − 1 | Type of Polynomials g(x) for Generating the M-Sequence |
---|---|---|

3 | 7 | x^{3} + x + 1x ^{3} + x^{2} + 1 |

4 | 15 | x^{4} + x + 1x ^{4} + x^{3} + 1 |

5 | 31 | x^{5} + x^{2} + 1x ^{5} + x^{3} + 1x ^{5} + x^{3} + x^{2} + x + 1x ^{5} + x^{4}+ x^{3} + x + 1x ^{5} + x^{4} + x^{3} + x^{2} + 1x ^{5} + x^{4} + x^{2} + x + 1 |

6 | 63 | x^{6} + x + 1x ^{6} + x^{4} + x^{3} + x + 1x ^{6} + x^{5} + 1x ^{6} + x^{5} + x^{2} + x+ 1x ^{6} + x^{5} + x^{3} + x^{2}+ 1x ^{6} + x^{5} + x^{4} + x + 1 |

7 | 127 | x^{7} + x + 1x ^{7} + x^{3} + 1x ^{7} + x^{3} + x^{2} + x + 1x ^{7} + x^{4} + 1x ^{7} + x^{4} +x^{3} + x^{2} + 1x ^{7} + x^{5} + x^{2} + x + 1x ^{7} + x^{5} + x^{3} + x + 1x ^{7} + x^{5} + x^{4} +x^{3} + 1x ^{7} + x^{5} + x^{4} +x^{3}+ x^{2} + x + 1x ^{7} + x^{6} + 1x ^{7} + x^{6} + x^{3} + x + 1x ^{7} + x^{6} + x^{4} + x^{2} + 1x ^{7} + x^{6} + x^{4} + x + 1x ^{7} + x^{6} + x^{5} + x^{2} + 1x ^{7} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1x ^{7} + x^{6} + x^{5} + x^{4} + 1x ^{7} + x^{6} + x^{5} + x^{4} + x^{2} + x + 1x ^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + 1 |

8 | 255 | x^{8} + x^{4} + x^{3} + x^{2} + 1x ^{8} + x^{5} + x^{3} + x + 1x ^{8} + x^{5} + x^{3} + x^{2} + 1x ^{8} + x^{6} + x^{3} + x^{2} + 1x ^{8} + x^{6} + x^{5} + x + 1x ^{8} + x^{6} + x^{5} + x^{2} + 1x ^{8} + x^{6} + x^{5} + x^{3} + 1x ^{8} + x^{6} + x^{5} + x^{4} + 1x ^{8} + x^{6} + x^{4} + x^{3} + x^{2} + x + 1x ^{8} + x^{6} + x^{5} + x + 1x ^{8} + x^{7} + x^{2} + x + 1x ^{8} + x^{7} + x^{3} + x^{2} +1x ^{8} + x^{7} + x^{5} + x^{3} +1x ^{8} + x^{7} + x^{6} + x^{5} + x^{2} + x +1x ^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{2} +1x ^{8} + x^{7} + x^{6} + x +1 |

9 | 511 | x^{9} + x^{4} + 1x ^{9} + x^{4} + x^{3} + x +1x ^{9} + x^{5} + 1x ^{9} + x^{5} + x^{3} + x^{2} + 1x ^{9} + x^{5} + x^{4} + x + 1x ^{9} + x^{6} + x^{4} + x^{3} + 1x ^{9} + x^{7} + x^{2} + x +1x ^{9} + x^{7} + x^{5} + x +1x ^{9} + x^{7} + x^{5} + x^{2} +1x ^{9} + x^{7} + x^{6} + x^{4} + 1x ^{9} + x^{8} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} +x +1x ^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x +1 |

Number Lobe ACF | Expressions for Finding b by g_{1}(x) L = 5 | Expressions for Finding b by g_{2}(x) L = 5 | Number Lobe ACF | Expressions for Finding b by g_{1}(x) L = 5 | Expressions for Finding b by g_{2}(x) L = 5 |
---|---|---|---|---|---|

1 | 15b^{2} + 16 | 15b^{2} + 16 | 17 | 3b^{2} − 9b + 3 | 2b^{2} − 10b + 3 |

2 | 6b^{2} − 16b + 8 | 7b^{2} − 15b + 8 | 18 | 5b^{2} − 5b + 4 | 2b^{2} − 8b + 4 |

3 | 6b^{2} − 15b + 8 | 6b^{2} − 15b + 8 | 19 | 4b^{2} − 6b + 3 | 2b^{2} − 8b + 3 |

4 | 5b^{2} − 15b + 8 | 5b^{2} − 15b + 8 | 20 | 3b^{2} − 7b + 2 | 2b^{2} − 8b + 2 |

5 | 6b^{2} − 13b + 8 | 5b^{2} − 14b + 8 | 21 | 2b^{2} − 7b + 2 | 2b^{2} − 7b + 2 |

6 | 4b^{2} − 15b + 7 | 4b^{2} − 15b + 7 | 22 | 3b^{2} − 4b + 3 | 4b^{2} − 3b + 3 |

7 | 5b^{2} − 12b + 8 | 5b^{2} − 12b + 8 | 23 | 2b^{2} − 6b + 1 | 3b^{2} − 5b + 1 |

8 | 5b^{2} − 12b + 7 | 5b^{2} − 12b + 7 | 24 | 2b^{2} − 5b + 1 | 3b^{2} − 4b + 1 |

9 | 5b^{2} − 11b + 7 | 4b^{2} − 12b + 7 | 25 | 2b^{2} − 4b + 1 | 2b^{2} − 4b + 1 |

10 | 5b^{2} − 10b + 7 | 4b^{2} − 11b + 7 | 26 | 2b^{2} − 4b | 2b^{2} − 4b |

11 | 4b^{2} − 12b + 5 | 3b^{2} − 13b + 5 | 27 | 3b^{2} − b + 1 | 3b^{2} − b + 1 |

12 | 5b^{2} − 9b + 6 | 5b^{2} − 9b + 6 | 28 | b^{2} − 3b | 2b^{2} − 2b |

13 | 4b^{2} − 9b + 6 | 5b^{2} − 8b + 6 | 29 | 2b^{2} − b | 2b^{2} − b |

14 | 3b^{2} − 10b + 5 | 5b^{2} − 8b + 5 | 30 | b^{2} − b | b^{2} − b |

15 | 2b^{2} − 11b + 4 | 5b^{2} − 8b + 4 | 31 | b^{2} | −b |

16 | 4b^{2} − 7b + 5 | 5b^{2} − 6b + 5 |

**Table 8.**The results of the selection of polynomials for the generation of modified M-sequences for the SD AR.

N | System Polynomials | The Value of b without Modification and with it | Level of SL in dB | The Difference from the SL Level at {1, −1} | The Nature of the CCF Values |
---|---|---|---|---|---|

15 | x^{4} + x + 1 | −1 −2 | −11.4806 −13.0641 | 1.5835 | UC |

15 | x^{4} + x^{3} + 1 | −1 −0.6667 | −11.4806 −13.9793 | 2.4987 | UC |

31 | x^{5} + x^{2} + 1 | −1 −0.7387 | −14.2642 −16.0330 | 1.7688 | UC |

31 | x^{5} + x^{3}+ 1 | −1 −0.7387 | −15.8478 −17.3107 | 1.4629 | UC |

31 | x^{5} + x^{3}+ x^{2}+ x + 1 | −1 −1.5469 | −14.2642 −15.2345 | 0.9703 | UC |

31 | x^{5} + x^{4} + x^{2}+ x + 1 | −1 −0.7387 | −15.8478 −17.6782 | 1.8304 | UC |

31 | x^{5} + x^{4} + x^{3} + x^{2} + 1 | −1 −0.6835 | −14.2642 −16.9058 | 2.6416 | UC |

63 | x^{6} + x + 1 | −1 −1.3334 | −20.4238 −20.7396 | 0.3158 | UC |

63 | x^{6} + x^{5} + 1 | −1 −0.7998 | −17.9250 −19.1452 | 1.2202 | UC |

63 | x^{6} + x^{5} + x^{3} + x^{2} + 1 | −1 −1.3334 | −19.0849 −19.2976 | 0.2127 | UC |

63 | x^{6} + x^{5} + x^{4} + x + 1 | −1 −1.3334 | −17.9250 −18.7051 | 0.7801 | UC |

127 | x^{7} + x + 1 | −1 −1.2146 | −21.2482 −21.7965 | 0.5483 | UC |

127 | x^{7} + x^{3} + 1 | −1 −0.8498 | −21.2482 −22.1399 | 0.8917 | UC |

127 | x^{7} + x^{3} + x^{2}+ x + 1 | −1 −1.2146 | −21.2482 −21.4885 | 0.2403 | UC |

127 | x^{7} + x^{4} + 1 | −1 −0.8498 | −20.4924 −21.0648 | 0.5724 | UC |

127 | x^{7} + x^{4} + x^{3} + x^{2} + 1 | −1 −1.2146 | −20.4924 −21.1896 | 0.6972 | UC |

127 | x^{7} + x^{5} + x^{4} + x^{3} + 1 | −1 −0.8498 | −21.2482 −22.0346 | 0.7864 | UC |

255 | x^{8} + x^{4}+ x^{3}+ x^{2} + 1 | −1 −0.8814 | −23.5218 −24.2805 | 0.7587 | UC |

255 | x^{8} + x^{6} + x^{3} + x^{2} + 1 | −1 −0.8886 | −24.0484 −24.4268 | 0.3784 | UC |

255 | x^{8} + x^{6} + x^{4} + x^{3} + x^{2} + 1 | −1 −0.8886 | −23.5218 −24.1275 | 0.6057 | UC |

255 | x^{8} + x^{6} + x^{5} + x^{4} + 1 | −1 −0.8886 | −24.0484 −24.6809 | 0.6325 | UC |

255 | x^{8} + x^{6} + x^{4} + x^{3} + x^{2} + x + 1 | −1 −0.8886 | −23.5218 −24.1275 | 0.6057 | UC |

511 | x^{9} + x^{4} + 1 | −1 −0.9186 | −27.7240 −28.0037 | 0.2797 | UC |

511 | x^{9} + x^{4} + x^{3} + x + 1 | −1 −1.0970 | −26.9339 −27.2522 | 0.3184 | UC |

511 | x^{9} + x^{5} + x^{3} + x^{2} + 1 | −1 −0.8714 | −26.9339 −27.5418 | 0.6080 | UC |

511 | x^{9} + x^{7} + x^{6} + x^{4} + 1 | −1 −0.8338 | −26.9339 −27.6126 | 0.6787 | UC |

511 | x^{9} + x^{8} + x^{6} + x^{5} + x^{4}+ x^{3} + x^{2} + x +1 | −1 −0.9186 | −26.9339 −27.1297 | 0.1959 | UC |

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## Share and Cite

**MDPI and ACS Style**

Nenashev, V.A.; Nenashev, S.A.
Search and Study of Marked Code Structures for a Spatially Distributed System of Small-Sized Airborne Radars. *Sensors* **2023**, *23*, 6835.
https://doi.org/10.3390/s23156835

**AMA Style**

Nenashev VA, Nenashev SA.
Search and Study of Marked Code Structures for a Spatially Distributed System of Small-Sized Airborne Radars. *Sensors*. 2023; 23(15):6835.
https://doi.org/10.3390/s23156835

**Chicago/Turabian Style**

Nenashev, Vadim A., and Sergey A. Nenashev.
2023. "Search and Study of Marked Code Structures for a Spatially Distributed System of Small-Sized Airborne Radars" *Sensors* 23, no. 15: 6835.
https://doi.org/10.3390/s23156835