Direction-of-arrival estimation of near-field sources based on compressed symmetric nested array and sparse signal reconstruction

LI Shuang, ZHENG Da-qing, LIU Wei, HU Shun-ren, HE Wei

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Direction-of-arrival estimation of near-field sources based on compressed symmetric nested array and sparse signal reconstruction

LI Shuang1, ZHENG Da-qing1, LIU Wei1, HU Shun-ren1, HE Wei2

(1. School of Electrical and Electronic Engineering, Chongqing University of Technology, Chongqing 400054, China;2. Key Labarotory of Wireless Sensor Networks and Communication, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences,Shanghai 200180, China)

In this paper, a novel underdetermined direction-of-arrival (DOA) estimation method based on compressed symmetric nested array and sparse signal representation is proposed in the near-field. Firstly, the forth order cumulants are employed to transform the original two-dimensional parameter estimation problem into a one-dimensional one and to obtain the difference co-array of the physical array. Then, in order to further increase the angular resolution and reduce the estimate error, the received signals of the virtual array are sparsely represented in spatial domain. Finally, the DOAs of the sources are founded through the use of the L1-regularized least square method. Compared to the existing methods, the proposed approach can process more sources and have lower variance and higher resolution. Simulation results are given to demonstrate the effectiveness and efficiency of the proposed method.

array signal processing; underdetermined direction-of-arrival estimation; near-field; sparse signal recovery; fourth order cumulant

0 Introduction

Source localization using sensor arrays plays an important role in many applications such as radar, sonar, wireless sensor networks, wireless communication and so on. According to the distance between the sources and the reference sensor of an array, source localization can be categorized into two types: far-field and near-field. In the far-field, the signal at the array can be approximated as plane wave and only the direction-of-arrival (DOA) estimation is required. However, the wavefront of the signal must be characterized by both azimuth and range in the near-field, where a joint azimuth and range estimation should be conducted. Therefore, the performances of those source localization methods making a far-field assumption will severely degrade in the near-field.

In this paper, based on compressed symmetric nested array and sparse signal reconstruction, a new source localization method is proposed in the near-field. The degrees of freedom are further increased by exploiting sparse signal reconstruction so that higher resolution can be achieved. Compared to the existing methods, the proposed method has a superior performance including higher resolution and lower variance.

The rest of the paper is organized as follows. Section 2 describes the DOA estimation model in the near-field. How the proposed method works is given in Section 3, which is followed by simulation results in section 4. Section 5 concludes the paper.

1 Data model

where denotes the source, represents the noise received by thesensor, refers to the wavelength of the sources, denotes the number of snapshots, stands for the distance between the source and the reference sensor of the array and is the distance from the sensor to the source.

According to Fig.1, we have

Through the use of second order Taylor expansion of (2), we can obtain the so-called Fresnel approximation

Substituting (3) into (1), we obtain

where

and

To simplify the derivation, we make some assumptions as follows.

(A1) - the sources are non-Gaussian, and independent with each other.

(A2) - the noises are Gaussian, either white or colored.

(A3) - the noises are independent of the sources; and

2 The proposed method

2.1 Nested array and compressed symmetric nested array(CSNA)

Fig.2 Nested array and compressed symmetric nested array

2.2 The proposed method

To acquire the difference co-array of a physical array, we exploit the fourth order cumulants of the received signals, which can be defined as

Under the assumptions (A2) and (A3), by substituting (4) into (8) and employing the properties of cumulants, we have.

1: regarding the number of sources that the proposed method can process, we give a theorem to explain it.

3:As to array geometry, since the proposed method only requires the array is symmetric, it will work well when the number of the array is odd as we mentioned before.

3 Simulation results

3.1 Spatial spectra

Hence, according to the two figures, we can draw a conclusion that our method achieves higher resolution compared to the other two methods, which will be verified in the subsequent section. Besides, both the proposed method and the method in[13] can identify more sources than sensors.

3.2 RMSE

In this subsection, the RMSE of the above three methods are evaluated. The RMSE is defined as

wheredenotes the number of Monte Carlo trials, and represent theestimate and real DOA in the run, respectively.

Fig.4 Angular spatial spectrum for underdetermined case

Then we keep all the parameters the same as the last experiment except that the SNR is fixed 10dB and make a comparison to the three methods when the number of snapshots ranges from 200 to 4 000. Fig.6 illustrates RMSE of the three methods with respect to the number of snapshots. It can be noted that the proposed method has a higher RMSE when the number of snapshot is relatively small while lower variance can be obtained by the proposed method when a great number of snapshots are available compared to the other methods.

Fig.5 RMSE versus SNR

Fig.6 RMSE as a function of the number of snapshot

3.3 Resolution

Then, the resolution probabilities of the three methods are investigated for different numbers of snapshots, which varies from 200 to 2 000 with a step of 300. How the resolution ability changing with the number of snapshots is exhibited in Fig.8. It can be obviously noted that the proposed method outperforms the other two methods.

Fig.7 Resolution ability with respect to SNR

Fig.8 Resolution ability with respect to the number of snapshots

4 Conclusions

In this paper, a novel DOA estimation method based on compressed symmetric nested array and sparse signal recovery is proposed in the near-field. The proposed method has several advantages. First, it can process more sources than sensors. Furthermore, compared to the existing methods, the proposed method can obtain higher resolution and lower variance. Future research will involve the design of more efficient array for near-field source localization.

[1] ZHI W, CHIA M W. Near-field source localization via symmetric subarrays[C]//2007. IEEE International Conference on Acoustics, Speech and Signal Processing(ICASSP 2007) IEEE, 2007: 1121-1124.

[2] XIE J, TAO H, RAO X, et al. Comments on “near-field source localization via symmetric subarrays”[J]. IEEE Signal Processing Letters, 20155(22): 643-644.

[3] LIANG J, LIU D. Passive localization of mixed near-field and far-field sources using two-stage MUSIC algorithm[J]. IEEE Transactions on Signal Processing, 201058(1): 108-120.

[4] MALIOUTOV D, ?ETIN M, WILLSKY A S. A sparse signal reconstruction perspective for source localization with sensor arrays[J]. IEEE Transactions on Signal Processing, 200553(8): 3010-3022.

[5] TIAN Y, SUN X. Passive localization of mixed sources jointly using MUSIC and sparse signal reconstruction[J]. AEU- International Journal of Electronics and Communications, 201468(6): 534-539.

[6] HU K, CHEPURI S P, LEUS G. Near-field source localization using sparse recovery techniques[C]//Signal Processing and Communications (SPCOM), 2014 International Conference on IEEE, 2014: 1-5.

[7] WANG B, LIU J, SUN X. Mixed sources localization based on sparse signal reconstruction[J]. IEEE Signal Processing Letters, 201219(8): 487-490.

[8] 李雙, 劉驍, 胡順仁, 等. 對稱子陣列的近場信號稀疏表示定位方法[J]. 信號處理, 2017, 33(1): 78-86.LI Shuang, LIU xiao, HU Shunren. et al. Localization of near- field sources using the sparse signal representation with symmetric subarrays[J]. Journal of Signal Processing, 201733(1): 78-86.

[9] 李雙, 劉驍, 胡順仁, 等. 加權(quán)稀疏信號重構(gòu)的近場源定位方法[J]. 聲學(xué)技術(shù), 2017, 36(1): 75-80. LI Shuang, LIU Xiao, HU Shunren, et al. Source localization based on sparse signal recovery using a weighted penalty in the near-field[J]. Technical Acoustics, 2017, 36 (1): 75-80.

[10] 劉亮, 陶建武, 黃家才. 基于稀疏對稱陣列的近場源定位[J]. 電子學(xué)報, 2009, 37(6): 1307-1312. LIU Liang, TAO Jianwu, HUANG Jiacai. Near-field source localization based on sparse symmetric array[J]. Acta Elctrnica Sinica, 200937(6): 1307-1312.

[11] 梁國龍, 韓博. 基于互素對稱陣的近場源定位[J]. 電子與信息學(xué)報, 2014, 1(1): 135-139. LIANG Guolong, HAN Bo. Near-field sources localization based on co-prime symmetric array[J]. Journal of Electronics & Information Technology, 20141(1): 135-139.

[12] PAL P, VAIDYANATHAN P. Nested arrays: a novel approach to array processing with enhanced degrees of freedom[J]. IEEE Transactions on Signal Processing, 201058(8): 4167-4181.

[13] LI S, XIE D. Compressed symmetric nested arrays and their application for direction-of-arrival estimation of near-field sources[J]. Sensors, 201616(11): 1939.

[14] GRANT M, BOYD S, YE Y. CVX: Matlab software for disciplined convex programming. Available: http://cvxr. com/ cvx/. 2008.

[15] LI S, JIANG X, MA S, et al., Sparse cumulants fitting for direction-of-arrival estimation without redundancy[J]. International Journal of Antennas and Propagation, 2013(3): 1-7.

壓縮對稱嵌套陣列和稀疏信號重構(gòu)的近場目標(biāo)方位估計

李雙1，鄭大青1，劉偉1，胡順仁1，何為2

(1. 重慶理工大學(xué)電氣與電子工程學(xué)院，重慶 400054； 2.中國科學(xué)院上海微系統(tǒng)與信息技術(shù)研究所無線傳感網(wǎng)與通信重點實驗室，上海 200180）

針對現(xiàn)有近場源估計算法中近場源數(shù)量受限于陣元數(shù)的問題，提出了一種基于稀疏對稱嵌套陣列和稀疏信號重構(gòu)的近場欠定波達(dá)方向估計方法。首先利用四階累積量，將二維空間參數(shù)估計問題轉(zhuǎn)化為一維參數(shù)估計問題，同時得到差分陣列；為了進(jìn)一步提高估計分辨率與減少估計誤差，對虛擬陣列的接收信號在空間域進(jìn)行稀疏表示；最后通過L1范數(shù)最小二乘法得到目標(biāo)源的波達(dá)方向。相較于現(xiàn)有算法，該方法可以估計更多的目標(biāo)源，并且有更低的均方誤差與更高的分辨率。實驗仿真驗證了算法的有效性與優(yōu)越性。

陣列信號處理；欠定波達(dá)方向估計；近場；稀疏信號重構(gòu)；四階累積量

TN911.72

A

1000-3630(2018)-01-0082-07

10.16300/j.cnki.1000-3630.2018.01.015

2017-03-17;

2017-05-03

重慶市基礎(chǔ)與前沿研究計劃項目(cstc2015jcyjA040055); 重慶市教委科學(xué)技術(shù)研究項目(KJ1500917, KJ1600936)

李雙(1986－), 男, 四川南充人, 博士, 講師, 研究方向為陣列信號處理及其應(yīng)用。

李雙, E-mail: lis@cqut.edu.cn