﻿ 一种在相干信源下的快速DOA估计算法
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 哈尔滨工程大学学报  2019, Vol. 40 Issue (2): 318-322  DOI: 10.11990/jheu.201807050 0

### 引用本文

YU Zhilong, SHEN Feng. Fast direction of the arrival estimation algorithm for coherent sources[J]. Journal of Harbin Engineering University, 2019, 40(2), 318-322. DOI: 10.11990/jheu.201807050.

### 文章历史

1. 哈尔滨理工大学 自动化学院, 黑龙江 哈尔滨 150080;
2. 哈尔滨工程大学 自动化学院, 黑龙江 哈尔滨 150000;
3. 哈尔滨工业大学 自动化学院, 黑龙江 哈尔滨 150001

Fast direction of the arrival estimation algorithm for coherent sources
YU Zhilong 1,2, SHEN Feng 3
1. College of Automation, Harbin University of Science and Technology, Harbin 150080, China;
2. College of Automation, Harbin Engineering University, Harbin 150001, China;
3. College of Automation, Harbin Engineering University, Harbin 150080, China
Abstract: To solve the complex problem in calculating the DOA of a signal by decomposing the eigenvalue of signal covariance, this study presents an algorithm that estimates the arrival direction of the spatial smoothing technique based on the propagator method. Compared with the traditional subspace method, which transforms to estimate the noise subspace by simple linear segmentation, the algorithm avoids characteristic decomposition that involves a large amount of computation in traditional feature subspace algorithms, thereby reducing the computational complexity. The spatial smoothing technique for a coherent signal processing source exhibits a good performance. By making use of the difference between the bidirectional spatial smoothing matrix and its complex conjugate, a generalized covariance matrix is constructed, which can completely eliminate the spatially nonuniform noise. The simulation results show that the combined algorithm of spatial smoothing and spreading factor can effectively maintain the good performance of the smoothing technique, which further reduces the computational complexity.
Keywords: DOA    propagation factor algorithm    coherent signal source    spatial smoothing technique    array signal

1 波达方向估计模型

 $\mathit{\boldsymbol{X}}\left( t \right) = \sum\limits_{n = 1}^N {\mathit{\boldsymbol{a}}\left( {{\theta _i}} \right){\mathit{\boldsymbol{s}}_n}\left( t \right)} + \mathit{\boldsymbol{N}}\left( t \right) = \mathit{\boldsymbol{A}}\left( \theta \right)\mathit{\boldsymbol{S}}\left( t \right) + \mathit{\boldsymbol{N}}\left( t \right)$ (1)

 $\mathit{\boldsymbol{R}} = {\rm{E}}\left\{ {\mathit{\boldsymbol{X}}\left( t \right){\mathit{\boldsymbol{X}}^{\rm{H}}}\left( t \right)} \right\} = \mathit{\boldsymbol{A}}{\mathit{\boldsymbol{R}}_S}\mathit{\boldsymbol{A + }}{\mathit{\boldsymbol{R}}_N}$ (2)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{X}}\left( t \right) = \sum\limits_{n = 1}^N {\mathit{\boldsymbol{a}}\left( {{\theta _i}} \right){\mathit{\boldsymbol{s}}_n}\left( t \right)} + \mathit{\boldsymbol{N}}\left( t \right) = }\\ {\mathit{\boldsymbol{A}}\left( \theta \right)\left[ {\begin{array}{*{20}{c}} {{s_1}\left( t \right)}\\ {{s_2}\left( t \right)}\\ \vdots \\ {{s_N}\left( t \right)} \end{array}} \right] + \mathit{\boldsymbol{N}}\left( t \right) - \mathit{\boldsymbol{G}}{\mathit{\boldsymbol{s}}_0}\left( t \right) + \mathit{\boldsymbol{N}}\left( t \right)} \end{array}$ (3)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{R}} = {\rm{E}}\left\{ {\mathit{\boldsymbol{X}}\left( t \right){\mathit{\boldsymbol{X}}^{\rm{H}}}\left( t \right)} \right\} = \mathit{\boldsymbol{G}} \cdot \mathit{\boldsymbol{E}}\left\{ {{\mathit{\boldsymbol{s}}_0}\left( t \right){\mathit{\boldsymbol{s}}_0}^{\rm{H}}\left( t \right)} \right\} \cdot }\\ {{\mathit{\boldsymbol{G}}^{\rm{H}}} + {\mathit{\boldsymbol{R}}_N} = \mathit{\boldsymbol{G\rho }}{\mathit{\boldsymbol{G}}^{\rm{H}}} + {\mathit{\boldsymbol{R}}_N}} \end{array}$ (4)

2 空间平滑技术波达方向估计算法 2.1 空间平滑算法

MUSIC算法在理想状态下具有良好的性能，但在信号源相关时，算法的性能变得很坏; 在相干信号源的情况下，正确估计信号方向的核心问题是，如何通过一系列处理或变换使得信号协方差矩阵得到有效恢复，从而正确估计信号源的方向。空间平滑MUSIC[15-16]可以利用子阵平滑恢复数据协方差矩阵，再处理相干信号源时，有良好的性能。将系统模型里M个阵元的均匀线阵分成相互交错的p个子阵，每个子阵的阵元数为m, 即有M=p+m-1。前向平滑技术方法，一般取第1个子阵为参考矩阵，则对于第k个子阵数据模型满足:

 $\mathit{\boldsymbol{x}}_k^t = {\mathit{\boldsymbol{Z}}_k}\mathit{\boldsymbol{X}}\left( t \right)$ (5)

 ${\mathit{\boldsymbol{R}}_f} = \frac{1}{p}\sum\limits_{k = 1}^p {{\mathit{\boldsymbol{Z}}_k}\mathit{\boldsymbol{RZ}}_k^{\rm{H}}}$ (6)

 ${\mathit{\boldsymbol{R}}_b} = \frac{1}{p}\sum\limits_{k = 1}^p {{\mathit{\boldsymbol{Q}}_k}{\mathit{\boldsymbol{R}}^ * }\mathit{\boldsymbol{Q}}_k^H}$ (7)

 ${\mathit{\boldsymbol{R}}_b} = \mathit{\boldsymbol{JR}}_f^ * \mathit{\boldsymbol{J}}$ (8)

Jm×m是反对角线为1的置换矩阵，即：

 $\mathit{\boldsymbol{J}} = \left[ {\begin{array}{*{20}{c}} 0& \cdots &1\\ \vdots &{}& \vdots \\ 1& \cdots &0 \end{array}} \right]$ (9)

 ${\mathit{\boldsymbol{R}}_{fb}} = \frac{1}{{2p}}\sum\limits_{k = 1}^p {{\mathit{\boldsymbol{Z}}_k}\left( {\mathit{\boldsymbol{R}} + \mathit{\boldsymbol{J}}{\mathit{\boldsymbol{R}}^ * }\mathit{\boldsymbol{J}}} \right)\mathit{\boldsymbol{Z}}_k^{\rm{H}}}$ (10)

2.2 传播因子类快速算法

 $\mathit{\boldsymbol{A}}\left( \theta \right) = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{A}}_1}\left( \theta \right)}\\ {{\mathit{\boldsymbol{A}}_2}\left( \theta \right)} \end{array}} \right]$ (11)

 $\mathit{\boldsymbol{X}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{X}}_1}}\\ {{\mathit{\boldsymbol{X}}_2}} \end{array}} \right]$ (12)

 $\mathit{\boldsymbol{P}} = {\left( {{\mathit{\boldsymbol{X}}_1}\mathit{\boldsymbol{X}}_1^{\rm{H}}} \right)^{ - 1}}{\mathit{\boldsymbol{X}}_1}\mathit{\boldsymbol{X}}_2^{\rm{H}}$ (13)

 ${\mathit{\boldsymbol{F}}_{PM}}\left( \theta \right) = {\mathit{\boldsymbol{a}}^{\rm{H}}}\left( \theta \right)\mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{Q}}^{\rm{H}}}\mathit{\boldsymbol{a}}\left( \theta \right)$ (14)

 ${\mathit{\boldsymbol{F}}_{{\rm{OPM}}}}\left( \theta \right) = {\mathit{\boldsymbol{a}}^{\rm{H}}}\left( \theta \right){\mathit{\boldsymbol{Q}}_0}\mathit{\boldsymbol{Q}}_0^{\rm{H}}\mathit{\boldsymbol{a}}\left( \theta \right)$ (15)

2.3 空间平滑算法的改进

1) 利用OPM算法进行空间平滑技术分解，从而得到波达方向估计，由式(10)可知前后向平滑技术协方差矩阵为:

 ${\mathit{\boldsymbol{R}}_{fb}} = \frac{1}{{2p}}\sum\limits_{k = 1}^p {{\mathit{\boldsymbol{Z}}_k}\left( {\mathit{\boldsymbol{R}} + \mathit{\boldsymbol{J}}{\mathit{\boldsymbol{R}}^ * }\mathit{\boldsymbol{J}}} \right)\mathit{\boldsymbol{Z}}_k^{\rm{H}}}$ (16)

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{R}}_{fb}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{A}}_1}}\\ {{\mathit{\boldsymbol{A}}_2}} \end{array}} \right]{\mathit{\boldsymbol{R}}_S}\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{A}}_1}}&{{\mathit{\boldsymbol{A}}_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{A}}_1}{\mathit{\boldsymbol{R}}_S}\mathit{\boldsymbol{A}}_1^{\rm{H}}}&{{\mathit{\boldsymbol{A}}_1}{\mathit{\boldsymbol{R}}_S}\mathit{\boldsymbol{A}}_2^{\rm{H}}}\\ {{\mathit{\boldsymbol{A}}_2}{\mathit{\boldsymbol{R}}_S}\mathit{\boldsymbol{A}}_1^{\rm{H}}}&{{\mathit{\boldsymbol{A}}_2}{\mathit{\boldsymbol{R}}_S}\mathit{\boldsymbol{A}}_2^{\rm{H}}} \end{array}} \right] = }\\ {\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{A}}_1}{\mathit{\boldsymbol{R}}_S}\mathit{\boldsymbol{A}}_1^{\rm{H}}}&{{\mathit{\boldsymbol{A}}_1}{\mathit{\boldsymbol{R}}_S}\mathit{\boldsymbol{A}}_1^{\rm{H}}\mathit{\boldsymbol{P}}}\\ {{\mathit{\boldsymbol{A}}_2}{\mathit{\boldsymbol{R}}_S}\mathit{\boldsymbol{A}}_1^{\rm{H}}}&{{\mathit{\boldsymbol{A}}_\mathit{\boldsymbol{R}}}_S\mathit{\boldsymbol{A}}_1^{\rm{H}}\mathit{\boldsymbol{P}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{G}}&\mathit{\boldsymbol{H}} \end{array}} \right]} \end{array}$ (17)

 $\mathit{\boldsymbol{P}} = {\left( {{\mathit{\boldsymbol{G}}^{\rm{H}}}\mathit{\boldsymbol{G}}} \right)^{ - 1}}{\mathit{\boldsymbol{G}}^{\rm{H}}}\mathit{\boldsymbol{H}}$ (18)
 $\mathit{\boldsymbol{Q}} = {\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{P}}^{\rm{T}}}}&{ - {\mathit{\boldsymbol{I}}_{M - N}}} \end{array}} \right]^{\rm{T}}}$ (19)

Q进行正交变换Q0= Q(Q H Q)-1/2可以得到前后向平滑技术OPM空间谱：

 ${F_{{\rm{fb - OPM}}}}\left( \theta \right) = \frac{1}{{{{\left\| {{\mathit{\boldsymbol{Q}}_0}\mathit{\boldsymbol{a}}\left( \theta \right)} \right\|}^2}}}$ (20)

2) 对前后向平滑协方差Rfb取共轭矩阵:

 $\mathit{\boldsymbol{R}}_{fb}^ * = \frac{1}{{2p}}\sum\limits_{k = 1}^p {{\mathit{\boldsymbol{Z}}_k}\left( {{\mathit{\boldsymbol{R}}^ * } + \mathit{\boldsymbol{JRJ}}} \right)\mathit{\boldsymbol{Z}}_k^{\rm{H}}}$ (21)

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{R}}_{{\rm{GCD}}}} = {\mathit{\boldsymbol{R}}_{fb}} - \mathit{\boldsymbol{R}}_{fb}^ * = \frac{1}{{2p}}\sum\limits_{k = 1}^p {{\mathit{\boldsymbol{Z}}_k}\left( {\mathit{\boldsymbol{R}} + \mathit{\boldsymbol{J}}{\mathit{\boldsymbol{R}}^ * }\mathit{\boldsymbol{J}} - } \right.} }\\ {\left. {{\mathit{\boldsymbol{R}}^ * } - \mathit{\boldsymbol{JRJ}}} \right)\mathit{\boldsymbol{Z}}_k^{\rm{T}}} \end{array}$ (22)

 ${F_{{\rm{GCD - OPM}}}}\left( \theta \right) = \frac{1}{{{{\left\| {{\mathit{\boldsymbol{Q}}_0}\mathit{\boldsymbol{a}}\left( \theta \right)} \right\|}^2}}}$ (23)
3 仿真分析

 Download: 图 1 3种MUSIC算法对比曲线 Fig. 1 Comparison curves of three MUSIC algorithms

 Download: 图 2 2种双向平滑算法对比曲线 Fig. 2 Comparison curves of two kinds of spatial smoothing techniques
 Download: 图 3 2种GCD平滑算法对比曲线 Fig. 3 Comparison curves of two kinds of GCD-spatial smoothing techniques

 Download: 图 4 4种MUSIC算法的RMSE随SNR变化曲线 Fig. 4 RMSE curves of four kinds of MUSIC algorithms with SNR
 Download: 图 5 3种平滑算法的RMSE随SNR变化曲线 Fig. 5 RMSE curves of three kinds of smoothing algorithms with SNR
4 结论

1) 结合传播因子类方法的双向平滑技术、GCD平滑技术，在不影响良好的精确度的情况下，进一步降低了运算复杂度。

2) 算法存在不足之处，如其他环境因素的限制，精度方面没有太大提高，因此还需要在这些方面加以改进。

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