# PN Codes Estimation of Binary Phase Shift Keying Signal Based on Sparse Recovery for Radar Jammer

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## Abstract

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## 1. Introduction

- A novel PN codes estimation method based on NCTVR is proposed, and its corresponding optimization function is established.
- An iterative algorithm based on the forward–backward splitting algorithm is proposed to solve the NCTVR.
- The proposed method is verified by numeric simulations and semiphysical tests.

## 2. Mathematical Model

## 3. PN Code Estimation Based on NCTVR

#### 3.1. Pretreatment

#### 3.2. PN Code Estimation Based on NCTVR

**z**play a role in enhancing sparsity. The extra computational complexity of MC penalty is (18), which includes twice matrix multiplications and a soft threshold function. The flow chart of NCTVR is shown in Figure 6.

#### 3.3. The Motivation behind the NCTVR

**z**is equal to zero, (19) is a classical TVR problem. The classical TVR problem can be solved with an L1-norm-based regularization penalty term, but it has a limitation that tends to underestimate the amplitudes of signal discontinuities. Therefore, we introduce a minimax-concave penalty function to improve the TVR problem. The sequence

**z**, resulting from the minimax-concave penalty function, enhances the sparsity of the ZIF signal and makes it more robust to noise.

**z**can be considered as the response of a Laplacian operator (LAPO) on the PN codes $c$. The Laplacian operator takes the second derivative of the ZIF signal: When the PN codes do not change, the Laplacian operator outputs zero, corresponding to jumps with a minimal jump height lower than 1/a caused by noise. As illustrated in Figure 7b, if PN codes change from −1 to +1 ($\left|Dc\left(n\right)\right|>0$), z shows a positive impact first and then a negative impact.

## 4. Performance of NCTVR

#### 4.1. Variance of the Estimated Centre Frequency

#### 4.2. Accuracy of The Estimated PN Codes

## 5. Simulations and Experiments

#### 5.1. Simulations

#### 5.2. Experiments

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 8.**Results of the pretreatment: (

**a**) standard deviations of estimated frequency; (

**b**) standard deviations of estimated initial phase; (

**c**) the ZIF signal after pretreatment under SNR = −5 dB.

**Figure 9.**The estimated PN codes under different SNRs. (

**a**) Original sequences; (

**b**) SNR = 5 dB; (

**c**) SNR = 0 dB; (

**d**) SNR = −5 dB; (

**e**) SNR = −10.

**Figure 12.**Results of the semiphysical tests. (

**a**) The captured BPSK signal in the time domain and frequency domain. (

**b**) The ZIF signal obtained from pretreatment. (

**c**) The estimated PN codes.

Existing Methods | Principle | Limitations |
---|---|---|

Methods proposed in [10,11,12] and [18,19,20,21] | They adopt time frequency analysis to estimate the chip rate and carrier frequency of the BPSK signal | They are unable to estimate the PN codes. |

Two-stage method proposed in [24] | It adopts the cross correlation to estimate the PN codes of the BPSK signal in serious SNR. | It is only suitable for Barker codes 7, 11 and 13. |

Method proposed in [23] | It adopts matrix eigen decomposition to estimate the PN codes of the BPSK signal. | It needs to know the chip rate and period of the PN code as a priori |

Method proposed in [16] | It uses the state changes of the duffing oscillator to estimate the PN codes of the BPSK signal in serious SNR | It only detects the polarity changes of the PN codes and needs to know the polarity of starting symbol as a priori. |

Our two-stage method. | It uses the sparsity of the PN codes in time domain to estimate the PN codes of the BPSK signal in serious SNR | \ |

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**MDPI and ACS Style**

Peng, B.; Chen, Q.
PN Codes Estimation of Binary Phase Shift Keying Signal Based on Sparse Recovery for Radar Jammer. *Sensors* **2023**, *23*, 554.
https://doi.org/10.3390/s23010554

**AMA Style**

Peng B, Chen Q.
PN Codes Estimation of Binary Phase Shift Keying Signal Based on Sparse Recovery for Radar Jammer. *Sensors*. 2023; 23(1):554.
https://doi.org/10.3390/s23010554

**Chicago/Turabian Style**

Peng, Bo, and Qile Chen.
2023. "PN Codes Estimation of Binary Phase Shift Keying Signal Based on Sparse Recovery for Radar Jammer" *Sensors* 23, no. 1: 554.
https://doi.org/10.3390/s23010554