﻿ 交替方向乘子法的矢量水听器DOA估计方案
 舰船科学技术  2024, Vol. 46 Issue (12): 140-143    DOI: 10.3404/j.issn.1672-7649.2024.12.024 PDF

1. 上海海事大学 信息工程学院，上海 201306;
2. 同济大学 电子与信息工程学院，上海 201804

DOA estimation scheme for vector hydrophone based on alternating direction multiplier method
LIU Yonghao1, XU Ming1,2
1. College of lnformation Engineering, Shanghai Maritime University, Shanghai 201306, China;
2. College of Electronics and Information Engineering, Tongji University, Shanghai 201804, China
Abstract: The time required by the traditional grid-less compressive sensing approach for Direction of Arrival estimation, which utilizes a convex optimization toolbox (e.g. CVX) to solve the Semi-Definite Programming problem, gradually increases with the vector hydrophone array size grows. To improve the convergence speed of the algorithm, the Alternative Direction Method of Multiplier is applied to the DOA estimation of vector hydrophone arrays. Taking ocean environmental noise into account, the Atomic Norm Soft Thresholding is used to estimate the line spectral parameters, transform the Atomic Norm Minimization problem into a SDP problem, solve the SDP problem using the ADMM algorithm, and finally estimate the angle using dyadic polynomials. In order to verify the performance of the ADMM algorithm, comparative simulation experiments are conducted with the Root-Multiple Signal Classification(ROOTMUSIC) algorithm and the CVX algorithm under different signal-to-noise ratios and number of vector array elements, and the results show that the ADMM algorithm enhances the computational efficiency of the algorithm while ensuring convergence of the DOA estimation model.
Key words: DOA     vector hydrophone     AST     ADMM
0 引　言

1 系统模型

 $\left\{ \begin{gathered} p = s ，\\ {v_x} = \cos \theta \cdot s，\\ {v_y} = \sin \theta \cdot s 。\\ \end{gathered} \right.$ (1)

 ${a_v}(\theta ) ={\left[{e^{ j2\text{π} \frac{{0d\cos \theta }}{\lambda }}},{e^{ j2\text{π} \frac{{1d\cos \theta }}{\lambda }}}, \ldots ,{e^{ j2\text{π} \frac{{\left( {M - 1} \right)d\cos \theta }}{\lambda }}}\right]^{\text T}}。$ (2)

$M$个矢量阵元的接收水声信号为：

 $Y = AS + N 。$ (3)

2 本文方案 2.1 原子范数

 $X = \left[ \begin{gathered} \sum\limits_{k = 1}^K {{s_k} \cdot {e^{ \displaystyle j2\text{π} \frac{{0d\cos {\theta _k}}}{\lambda }}} \cdot {h_k}} \\ \sum\limits_{k = 1}^K {{s_k} \cdot {e^{ \displaystyle j2\text{π} \frac{{1d\cos {\theta _k}}}{\lambda }}} \cdot {h_k}} \\ \cdots \\ \sum\limits_{k = 1}^K {{s_k} \cdot {e^{ \displaystyle j2\text{π} \frac{{\left( {M - 1} \right)d\cos {\theta _k}}}{\lambda }}} \cdot {h_k}} \\ \end{gathered} \right] 。$ (4)

$X(t)$中的第$m$个矢量水听器的单快拍接收数据模型可表示：

 $\begin{split} {X_m}=\ &\sum\limits_{k = 1}^K {\left| {{s_k}} \right| \cdot {e^{j\left[ \displaystyle {2\text{π} \frac{{(m - 1)d\cos {\theta _k}}}{\lambda } - {\varphi _k}} \right]}} \cdot {h_k}} = \\ &\sum\limits_{k = 1}^K {\left| {{s_k}} \right|\cdot a({\theta _k},{\varphi_{{k}}})} 。\end{split}$ (5)

 ${ A} = \left\{ {a(\theta ,\varphi ):\theta \in \left[ {0,2{\text π} } \right),\varphi \in \left[ {0,2{\text π} } \right)} \right\} 。$ (6)

 ${\left\| X \right\|_{ A}} = \mathop {\inf }\limits_{a({\theta _k},{\varphi _k}) \in { A}} \left\{ {\sum\limits_{k = 1}^K {\left| {{s_k}} \right|:X = \sum\limits_{k = 1}^K {\left| {{s_k}} \right|a({\theta _k},{\varphi _k})} } } \right\}。$ (7)

 $\left\| Q \right\|_A^ * = \mathop {\sup }\limits_{a(\theta ,\varphi ) \in {\rm A}} {Re} \left( {\left\langle {Q,a({\theta _k},{\varphi _k})} \right\rangle } \right) 。$ (8)

2.2 AST

 $\mathop {\min imize}\limits_X \left\| {X - Y} \right\|_2^2 + 2\tau {\left\| X \right\|_{\rm A}}。$ (9)

 $\tau = \sigma (1 + \frac{1}{{\log M}})\sqrt {M\log M + M\log (4\text{π} \log M)} 。$ (10)

 $\begin{gathered} \mathop {\max imize}\limits_X = \left\| Y \right\|_2^2 - \left\| {Y - Q} \right\|_2^2，\\ {\mathrm{subject}}\;{\mathrm{to}}\quad \left\| Q \right\|_{\rm A}^ * \leqslant 1 。\\ \end{gathered}$ (11)

 $\begin{split} & \mathop {\min imize}\limits_{X,t,u} \left\| {X - Y} \right\|_2^2 + \tau \left( {t + {\omega ^{\rm T}}u} \right) ，\\ & {\mathrm{subject}}\;{\mathrm{to}}\quad \left[ {\begin{array}{*{20}{c}} {T(u)}&X \\ {{X^H}}&t \end{array}} \right] \geqslant 0 。\end{split}$ (12)

AST是通过在式（12）中选择$w = 2{e_0}$来获得的，其中${e_0}$为第一个元素为1，其他都为0的向量。另外，${\boldsymbol u} \in {{C}^{3M \times 1}} $${\boldsymbol {T(u)}} \in {{C}^{3M \times 3M}} 为Toe-plitz矩阵。  {\boldsymbol{T(u)}} = \left[ {\begin{array}{*{20}{c}} {{u_1}}&{{u_2}}& \ldots &{{u_{3M}}} \\ {u_2^{\mathrm{H}}}&{{u_1}}& \ldots &{{u_{3M - 1}}} \\ \vdots & \vdots & \ddots & \vdots \\ {u_{3M}^{\mathrm{H}}}&{u_{3M - 1}^{\mathrm{H}}}& \ldots &{{u_1}} \end{array}} \right] 。 (13) 式中： {u_j}$$ u$的第$j$个值；${\left| \cdot \right|^{\mathrm{H}}}$为共轭转置。

 $\begin{split} &{L_\rho }(t,u,X,Z,\Lambda ) = \left\| {X - Y} \right\|_2^2 + \tau \left( {t + {\omega ^{\rm T}}u} \right)+\\ & 2\left\langle {\Lambda ,Z - \underbrace {\left[ {\begin{array}{*{20}{c}} {T(u)}&X \\ {{X^H}}&t \end{array}} \right]}_T} \right\rangle + \rho \left\| {Z - \left[ {\begin{array}{*{20}{c}} {T(u)}&X \\ {{X^H}}&t \end{array}} \right]} \right\|_F^2 。\end{split}$ (14)

$l$次迭代，对各个变量的更新如下：

 $({t}^{l+1},{u}^{l+1},{X}^{l+1})\leftarrow \mathrm{arg}\underset{t,u,x}{\mathrm{min}}{L}_{\rho }（t,u,X,{Z}^{l},{\Lambda }^{l}），$ (15)
 ${Z}^{l+1}\leftarrow \mathrm{arg}\underset{Z\geqslant 0}{\mathrm{min}}{L}_{\rho }（{t}^{l+1},{u}^{l+1},{X}^{l+1},Z,{\Lambda }^{l}），$ (16)
 ${\Lambda ^{l + 1}} \leftarrow {\Lambda ^l} + \rho \left( {{Z^{l + 1}} - {T^{l + 1}}} \right) 。$ (17)

2.4 对偶多项式恢复角度

 $Y = \widehat X + \tau \widehat Q ，$ (18)

 $\widehat q(\theta ,\varphi ) = \left\langle {\widehat Q,a(\theta ,\varphi )} \right\rangle ，$ (19)

 $\left| {\widehat q(\theta ,\varphi )} \right| = 1 。$ (20)

 图 2 网格上对偶多项式的值 Fig. 2 Values of dual polynomials on the grid
3 仿真实验

RMSE的计算方式为：

 ${RMSE}=\sqrt{\frac{1}{m}{\displaystyle \sum _{i=1}^{m}({\widehat{\theta }}_{i}-{\theta }_{i}{)}^{2}}} 。$ (21)

 图 3 不同信噪比下均方根误差，M=32 Fig. 3 RMSE at different SNR, M=32

 图 4 不同矢量阵元个数下运行时间，SNR=10 Fig. 4 Running time with different number of v-ector array elements, SNR=10

4 结　语

 [1] 燕慧超. MEMS矢量水听器信号的去噪与定向技术研究[D]. 太原: 中北大学, 2021. [2] ZISHU H, LI Y, XIANG J C. A Modified root−MUSIC algorithm for signal DOA estimation[J]. Journal of Systems Engineering and Electr−onics, 1999, 10(4): 42-47. [3] 井岩. 基于压缩感知的水声信号波达方向估计方法研究[D]. 哈尔滨: 哈尔滨工业大学, 2016. [4] TANG G, BHASKAR B N, SHAH P, et al. Compressed sensing off the grid[J]. IEEE Transactions on Information Theory, 2013, 59(11): 7465−7490. [5] BOYD S, PARIKH N, CHU E, et al. Distributed opti−mization and statistical learning via the alternating direction method of multipliers[J]. Foundations and Trends® in Machine learning, 2011, 3(1): 1-122. [6] BHASKAR B N , TANG G , RECHT B . Atomic norm denoising with applications to line spectral estim−ateion[J]. IEEE Transactions on Signal Processing, 2013, 61(23): 5987−5999. [7] 陈涛, 李敏行, 郭立民, 等. 基于原子范数最小化的极化敏感阵列DOA估计[J]. 电子学报, 2023, 51(04): 835-842. CHEN Tao, LI Minxing, GUO Limin, et al. DOA estimation of polarization sensitive array based on atomic norm minimization[J]. Journal of Electronics, 2023, 51(04): 835-842. [8] ZHU Z, TANG G, SETLUR P, et al. Super−resolution in SAR imaging: analysis with the atomic norm[C]//2016 IEEE Sensor Array and Multichannel S−ignal Processing Workshop (SAM). IEEE, 2016: 1−5. [9] HANSEN T L, JENSEN T L. A fast interiorpoint method for atomic norm soft thresholding[J]. Signal Processing, 2019, 165: 7-19. DOI:10.1016/j.sigpro.2019.06.023 [10] 朱昭华 . 阵列误差条件下基于无网格压缩感知的DOA估计方法[D]. 长春: 吉林大学, 2022.