#
DOA Estimation Based on Weighted l_{1}-norm Sparse Representation for Low SNR Scenarios

^{*}

## Abstract

**:**

_{1}-norm is proposed in a l

_{1}-norm-based singular value decomposition (L1-SVD) algorithm, which can suppress spurious peaks and improve accuracy of direction of arrival (DOA) estimation for the low signal-to-noise (SNR) scenarios. The weighted matrix is determined by optimizing the orthogonality of subspace, and the weighted l

_{1}-norm is used as the minimum objective function to increase the signal sparsity. Thereby, the weighted matrix makes the l

_{1}-norm approximate the original l

_{0}-norm. Simulated results of orthogonal frequency division multiplexing (OFDM) signal demonstrate that the proposed algorithm has s narrower main lobe and lower side lobe with the characteristics of fewer snapshots and low sensitivity of misestimated signals, which can improve the resolution and accuracy of DOA estimation. Specifically, the proposed method exhibits a better performance than other works for the low SNR scenarios. Outdoor experimental results of OFDM signals show that the proposed algorithm is superior to other methods with a narrower main lobe and lower side lobe, which can be used for DOA estimation of UAV and pseudo base station.

## 1. Introduction

_{0}-norm as the minimized objective function, which is a problem of non-deterministic polynomial hard (NP-hard). Therefore, researchers use the l

_{p}(0 < p ≤ 1) norm to approximate [12]. Many algorithms such as FOCUSS use the l

_{p}(0 < p < 1) norm as the objective function, which is optimized by an iterative approximation method. The computation would become complicated with the increase in snapshots [9,10,13,14] by using the iterative method, which would be troubled by local extrema during optimization.

_{1}-norm is able to satisfy the sparsity constraint and eliminate the possibility of local convergence of the objective function, which is beneficial for the solution. Therefore, the l

_{1}-norm-based singular value decomposition (L1-SVD) algorithm [12] proposed by Malioutov et al. is a classical DOA estimation method. The authors of [15] present a covariance matrix sparse representation method for DOA estimation. These methods are effective in high signal-to-noise (SNR) scenarios. However, most of these algorithms use l

_{1}-norm instead of l

_{0}-norm to obtain an approximate result. When the value of the SNR becomes low, the sparsity of the solution would become worse, and more spurious peaks will appear in the spatial spectrum.

_{1}-norm constraint minimization to increase the recoverable sparsity threshold and improve the recovery accuracy in the noise case. The orthogonality weighting of the noise subspace and the signal subspace is proposed, which enhances the robustness of the L1-SVD algorithm and improves the resolution of DOA estimation [17,18]. The literature [19] proposes a weighted norm penalty function from the capon spectrum. These methods can improve accuracy of DOA estimation at low SNR to some extent. However, when SNR becomes lower, especially when the SNR is lower than −12 dB, these methods will become worse.

_{1}-norm is used as the objective function to minimize the signal sparsity, thereby improving the accuracy of DOA estimation and suppressing spurious peaks at low SNR.

## 2. System Model and the Proposed Method

#### 2.1. Sparse Representation of Narrowband Array Signal

_{n}

^{2}variance. For the convenience of description, Formula (2) can be simplified as follows:

**X**can recover

**S**, then the DOA estimation of the source can be determined according to the position of the non-zero line in

**S**. Formula (3) can be considered as a l

_{0}-norm problem, but the optimization of the l

_{0}-norm is a NP-hard problem. We use l

_{1}-norm instead of l

_{0}-norm to attain the approximate, and l

_{1}-norm is usually approximated [12] as follows:

_{2}-norm of each row in

**S**, namely ${S}^{{l}_{2}}=\left[\begin{array}{cccc}{s}_{1}^{{l}_{2}},& {s}_{2}^{{l}_{2}},& \dots ,& {s}_{P}^{{l}_{2}}\end{array}\right]$. h is the regularization parameter affected by noise, which is usually a small constant. ${\Vert X-AS\Vert}_{F}$ is the result of straightening matrix $X-AS$ by column and calculating l

_{2}-norm, that is:

**X**

_{sv}. The SVD of

**X**is:

**R**

_{x}is the covariance matrix of the original signal vector

**X**. At this time, Formula (7) can be transformed into

#### 2.2. Weighted l_{1}-norm Method

_{0}-norm is replaced by l

_{1}-norm in the L1-SVD algorithm, it would be difficult to guarantee the sparsity, especially when the value of the SNR is low. The constraint under l

_{1}-norm is the solution with the smallest modulus value. Here, the sparse signal

**S**has a large modulus corresponding to a large coefficient, and a small coefficient corresponds to a small modulus. Therefore, the sparse signal

**S**can be weighted to improve the sparsity of the solution.

_{s}of the noise array steering vector is relatively small in the signal subspace E

_{s}, while the projection N

_{n}is relatively large in the noise subspace E

_{n}. On the contrary, the projection S

_{s}of the signal array steering vector is large in the signal subspace E

_{s}, while the projection S

_{n}is small in the noise subspace E

_{n}. When the algorithm is weighted by the literature [17], its peak ratio is N

_{n}/S

_{n}, while the proposed algorithm’s peak ratio is N

_{s}N

_{n}/S

_{s}S

_{n}(N

_{s}/S

_{s}< 1). Therefore, the weighted value decreases as the peak ratio decreases, and a small coefficient corresponds to a small modulus, which improves the sparsity of the solution. According to Formulas (8) and (12), we can obtain:

- (1)
- Decompose the data matrix to reduce dimension by the singular value, and preprocess
**X**_{sv}and**A**to obtain**X**_{svw}and**A**_{w}; - (2)
- Calculate the weight
**W**according to Formula (12); - (3)
- Estimate the spectrum by using Formula (14).

## 3. Results

#### 3.1. Simulation Results and Analysis

#### 3.1.1. Simulation 1

#### 3.1.2. Simulation 2

_{c}is the number of Monte Carlo simulations, ${\theta}_{p}$ is the real angle of signal, and $\stackrel{\wedge}{{\theta}_{p}}\left({n}_{c}\right)$ is the DOA estimation of the n

_{c}times Monte Carlo of the signal source.

#### 3.1.3. Simulation 3

#### 3.1.4. Simulation 4

_{1}-norm is closer to the l

_{0}-norm, and the main lobe of the spectrum is sharper and spurious peaks are suppressed. Therefore, the proposed algorithm can achieve a higher accuracy with fewer snapshots, which is conducive to saving calculation time.

#### 3.1.5. Simulation 5

#### 3.2. Experiment Results and Analysis

## 4. Discussion

_{1}-norm sparse representation. Compared with the traditional array signal processing method, the sparse representation DOA estimation algorithm requires fewer snapshots and fewer array elements. OFDM is often used in communication systems, such as WiFi, long term evolution (LTE), 5th generation mobile communication technology (5G) and unmanned aerial vehicle (UAV) video signals [20,21,22,23,24,25]; therefore, the DOA estimation of OFDM signal is discussed in this paper.

## 5. Conclusions

_{1}-norm. The weighted l

_{1}-norm is used as the minimum objective function to increase the signal sparsity, which is able to improve the accuracy of DOA estimation and suppress spurious peaks for the low SNR scenarios. Additionally, the OFDM signal of communication is taken as the simulated object. The simulated and experimental results show that the proposed algorithm has a sharper main lobe and lower side lobe, which can improve the resolution and estimate DOA accurately. Due to the above characteristics, the proposed algorithm also has an important guiding role in engineering, such as anti-UAV technology and pseudo base station positioning.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**Sensitivity of the 4 algorithms to the assumed number of signals: (

**a**) L1-SVD; (

**b**)W-L1-SVD; (

**c**) C-L1-SVD; (

**d**) the proposed algorithm.

**Figure 7.**Photo taken during the measurement campaigns of the outdoor scenario pointing out the positions of the transmitter and the antenna array-based receiver.

**Figure 9.**Normalized spectra during the measurement of outdoor scenario when the incident angle is 117.2°: (

**a**) the SNR of emission signal is 0 dB; (

**b**) the SNR of emission signal is 15 dB.

**Figure 10.**Normalized spectra during the measurement of outdoor scenario when the incident angle is 90°: (

**a**) the SNR of emission signal is 0 dB; (

**b**) the SNR of emission signal is 15 dB.

Carrier Frequency | Bandwidth | Number of the Subcarriers | Number of Effective Subcarriers | Subcarrier Spacing | Time of Useful Symbol Length | Number of Symbols |
---|---|---|---|---|---|---|

2038 MHz | 20 MHz | 256 | 192 | 78.125 KHz | 12.8 us | 10 |

**Table 2.**DOA estimates in Figure 4.

SNR (dB) | L1-SVD | W-L1-SVD | C-L1-SVD | The Proposed |
---|---|---|---|---|

−12 | (79°, 100°) | (78°, 100°) | (78°, 100°) | (79°, 100°) |

0 | (80°, 100°) | (79°, 101°) | (79°, 101°) | (80°, 101°) |

**Table 3.**The accuracy and average calculation time of the four algorithms over 100 times Monte Carlo.

Algorithm | RMSE (deg) | Average Calculation Time (s) |
---|---|---|

W-L1-SVD | 3.57 | 2.993 |

L1-SVD | 22.9 | 2.992 |

The proposed | 0.98 | 2.997 |

C-L1-SVD | 26.03 | 2.972 |

**Table 4.**DOA estimates in Figure 6.

Algorithm | p = 1 | p = 2 | p = 4 |
---|---|---|---|

L1-SVD | (81°, 101°) | (80°, 101°) | (80°, 100°) |

W-L1-SVD | (81°, 100°) | (79°, 101°) | (79°, 101°) |

C-L1-SVD | (81°, 103°) | (79°, 101°) | (80°, 100°) |

The proposed | (81°, 101°) | (80°, 101°) | (79°, 100°) |

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**MDPI and ACS Style**

Zuo, M.; Xie, S.; Zhang, X.; Yang, M.
DOA Estimation Based on Weighted l_{1}-norm Sparse Representation for Low SNR Scenarios. *Sensors* **2021**, *21*, 4614.
https://doi.org/10.3390/s21134614

**AMA Style**

Zuo M, Xie S, Zhang X, Yang M.
DOA Estimation Based on Weighted l_{1}-norm Sparse Representation for Low SNR Scenarios. *Sensors*. 2021; 21(13):4614.
https://doi.org/10.3390/s21134614

**Chicago/Turabian Style**

Zuo, Ming, Shuguo Xie, Xian Zhang, and Meiling Yang.
2021. "DOA Estimation Based on Weighted l_{1}-norm Sparse Representation for Low SNR Scenarios" *Sensors* 21, no. 13: 4614.
https://doi.org/10.3390/s21134614