Expectation maximization DOA estimation algorithm based on four-order cumulant
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摘要: 针对低信噪比（signal-to-noise ratio，SNR）下，经典波达方向估计性能下降的问题，提出将信号的四阶累积量与期望最大化（expectation maximization，EM）算法相结合的波达方向估计算法.该方法引入隐含变量进行更新迭代，并求隐含变量的四阶累积量，构造关于待估波达方向的极大似然函数从而求解出信号的波达方向角.仿真结果表明：本文算法能有效地抑制高斯噪声对信号参数估计的影响，同时能利用迭代来提高估计精度.在低SNR时其估计性能优良，具有很好的稳定性和分辨率，有利于高分辨地估计信号的波达方向.Abstract: Aiming at the problem of low estimation performance of classical direction of arrival (DOA) estimation under low signal-to-noise ratio (SNR), this paper proposes a DOA estimation algorithm which combines the fourth-order cumulant of signal with the expectation maximization algorithm. The method introduces an implicit variable to update and iterate, and finds the fourth-order cumulant of the implicit variable. The maximum likelihood function of the estimated direction of arrival is constructed so as to obtain the direction-of-arrival of the signal. The simulation results show that the proposed algorithm can effectively suppress the influence of Gaussian noise on the signal parameter estimation. Meanwhile, the algorithm using iterative updating can improve the estimation accuracy. It has good performance in low signal-to-noise ratio(SNR) estimation, good stability and resolution, which is beneficial to the high-resolution estimation of DOA.
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引言
信号的波达方向估计是阵列信号处理领域的一个研究热点, 其广泛运用于通信、雷达、被动声纳、生物医学以及地震勘探等军事和民用领域.传统的波达方向估计方法, 如Schmidt等人提出的多重信号分类(multiple signal classification, MUSIC)算法[1], 其利用信号子空间和噪声子空间的正交性, 在高信噪比(signal-to-noise ratio, SNR)情况下具有优良的估计性能, 但在低SNR、少快拍数时, 其估计分辨率严重下降[2]; 文献[3-4]提出的最大似然(maximum likelihood, ML)算法, 其相比于子空间类算法稳定性有所提高, 但是该方法需要进行多维非线性最大值的全局搜索, 计算复杂度高且容易产生误差导致其分辨率下降[5-6].常规的波达方向估计算法运用信号的二阶累积量(协方差矩阵), 其对噪声没有抑制作用, 估计性能受SNR的影响较大.因此, 研究在低SNR、少快拍数情况下的高分辨波达方向估计算法具有现实意义.
期望最大化(expectation maximization, EM)算法是存在数据缺失时进行极大似然估计的一种迭代算法[7], 主要用于观测数据不完整或似然函数不是显式和似然函数求解困难的情况. EM算法具有实现简单、稳定性好的特点[8], 将其运用于信号参数估计中能有效提高信号未知参数估计的性能[9-10].信号的四阶累积量包含了信号的幅度信息和相位信息, 同时其对高斯噪声具有抑制作用, 能有效提取出信号中的非高斯成分[11-13].因此, 本文考虑将信号的四阶累积量与EM算法结合, 运用于窄带信号的波达方向估计中.仿真表明, 该算法能提高MUSIC等子空间类算法的估计性能, 同时避免了最大似然函数估计对接收信号空间谱的多维搜索时产生的误差, 有效提高波达方向估计的分辨率和稳定性.
1 信号模型
假设接收天线阵由M个阵元的均匀线阵组成, 阵元间距为d, P个来自空间远场窄带的独立目标信号源入射到该等距均匀线阵上, 信号的入射方向为θ1, θ2, …, θP, 其中, M-1≥P, d=12λ, λ为入射信号的波长.第m(m=1, 2, …, M)个阵元在t时刻接收到的信号可以表示为
xm(t)=P∑i=1si(t−τm(θi))+nm(t)=P∑i=1am(θi)si(t)+nm(t), (1) 则阵元在t时刻接收到的信号可以表示为
\begin{array}{l} \mathit{\boldsymbol{X}}\left( t \right) = {\left[ {\begin{array}{*{20}{c}} {{x_1}\left( t \right)}&{{x_2}\left( t \right)}& \cdots &{{x_M}\left( t \right)} \end{array}} \right]^{\rm{T}}}\\ \;\;\;\;\;\;\;\; = \left[ {\begin{array}{*{20}{c}} {{a_1}\left( {{\theta _1}} \right)}&{{a_1}\left( {{\theta _2}} \right)}& \cdots &{{a_1}\left( {{\theta _P}} \right)}\\ {{a_2}\left( {{\theta _1}} \right)}&{{a_2}\left( {{\theta _2}} \right)}& \cdots &{{a_2}\left( {{\theta _P}} \right)}\\ \vdots&\vdots&\cdots&\vdots \\ {{a_M}\left( {{\theta _1}} \right)}&{{a_M}\left( {{\theta _2}} \right)}& \cdots &{{a_M}\left( {{\theta _P}} \right)} \end{array}} \right] \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\left[ {\begin{array}{*{20}{c}} {{S_1}\left( t \right)}\\ {{S_2}\left( t \right)}\\ \vdots \\ {{S_M}\left( t \right)} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {{n_1}\left( t \right)}\\ {{n_2}\left( t \right)}\\ \vdots \\ {{n_M}\left( t \right)} \end{array}} \right]\\ \;\;\;\;\;\;\;\; = {\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{a}}^{\rm{T}}}\left( {{\theta _1}} \right)}\\ {{\mathit{\boldsymbol{a}}^{\rm{T}}}\left( {{\theta _2}} \right)}\\ \vdots \\ {{\mathit{\boldsymbol{a}}^{\rm{T}}}\left( {{\theta _P}} \right)} \end{array}} \right]^{\rm{T}}} \cdot \left[ {\begin{array}{*{20}{c}} {{S_1}\left( t \right)}\\ {{S_2}\left( t \right)}\\ \vdots \\ {{S_M}\left( t \right)} \end{array}} \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). \end{array} (2) 式中:τm(θi)表示第i个信源到达第m个阵元的传播时延; am(θi)=exp[-j(m-1)2πd/λsin θi]表示第i个信源在第m个阵元处相对于参考阵元的相位延迟; si(t)表示第i个信源信号; A(θ)=[a(θ1) a(θ2) … a(θP)]表示导向矢量矩阵; S(t)表示信号矢量; N(t)表示阵列的加性噪声, 假设噪声为零均值的高斯白噪声.
阵列输出的四阶累积量可表示为[11]
\begin{array}{l} {\rm{cum}}\left( {{x_{m1}},{x_{m2}},x_{m3}^ * ,x_{m4}^ * } \right)\\ = E\left\{ {{x_{m1}}{x_{m2}}x_{m3}^ * x_{m4}^ * } \right\} - \\ \;\;\;E\left\{ {{x_{m1}}{x_{m2}}} \right\}E\left\{ {x_{m3}^ * x_{m4}^ * } \right\} - \\ \;\;\;E\left\{ {{x_{m1}}x_{m3}^ * } \right\}E\left\{ {{x_{m2}}x_{m4}^ * } \right\} - \\ \;\;\;E\left\{ {{x_{m1}}x_{m4}^ * } \right\}E\left\{ {{x_{m2}}x_{m3}^ * } \right\}. \end{array} (3) 式中:m1, m2, m3, m4∈{1, 2, 3, …, M}.
信号源的四阶累积量可表示为[11]
\begin{array}{l} {\rm{cum}}\left( {{s_{k1}},{s_{k2}},s_{k3}^ * ,s_{k4}^ * } \right)\\ = E\left\{ {{s_{k1}}{s_{k2}}s_{k3}^ * s_{k4}^ * } \right\} - \\ \;\;\;E\left\{ {{s_{k1}}{s_{k2}}} \right\}E\left\{ {{s_{k3}}{s_{k4}}} \right\} - \\ \;\;\;E\left\{ {{s_{k1}}s_{k3}^ * } \right\}E\left\{ {{s_{k2}}s_{k4}^ * } \right\} - \\ \;\;\;E\left\{ {{s_{k1}}s_{k4}^ * } \right\}E\left\{ {{s_{k2}}s_{k3}^ * } \right\}. \end{array} (4) 式中:k1, k2, k3, k4∈{1, 2, 3, …, P}.
根据文献[11], 可构成阵列输出和信号源的四阶累积矩阵如下:
\begin{array}{l} {\mathit{\boldsymbol{C}}_s} = E\left\{ {\left( {\mathit{\boldsymbol{S}} \otimes {\mathit{\boldsymbol{S}}^ * }} \right){{\left( {\mathit{\boldsymbol{S}} \otimes {\mathit{\boldsymbol{S}}^ * }} \right)}^{\rm{H}}}} \right\} - \\ \;\;\;\;\;\;\;E\left\{ {\left( {\mathit{\boldsymbol{S}} \otimes {\mathit{\boldsymbol{S}}^ * }} \right)} \right\} \cdot E\left\{ {{{\left( {\mathit{\boldsymbol{S}} \otimes {\mathit{\boldsymbol{S}}^ * }} \right)}^{\rm{H}}}} \right\} - \\ \;\;\;\;\;\;\;E\left\{ {\left( {\mathit{\boldsymbol{S}} \cdot {\mathit{\boldsymbol{S}}^ * }} \right)} \right\} \otimes E\left\{ {{{\left( {\mathit{\boldsymbol{S}} \cdot {\mathit{\boldsymbol{S}}^{\rm{H}}}} \right)}^ * }} \right\}, \end{array} (5) \begin{array}{l} {\mathit{\boldsymbol{C}}_X} = E\left\{ {\left( {\mathit{\boldsymbol{X}} \otimes {\mathit{\boldsymbol{X}}^ * }} \right){{\left( {\mathit{\boldsymbol{X}} \otimes {\mathit{\boldsymbol{X}}^ * }} \right)}^{\rm{H}}}} \right\} - \\ \;\;\;\;\;\;\;\;E\left\{ {\left( {\mathit{\boldsymbol{X}} \otimes {\mathit{\boldsymbol{X}}^ * }} \right)} \right\} \cdot E\left\{ {{{\left( {\mathit{\boldsymbol{X}} \otimes {\mathit{\boldsymbol{X}}^ * }} \right)}^{\rm{H}}}} \right\} - \\ \;\;\;\;\;\;\;\;E\left\{ {\left( {\mathit{\boldsymbol{X}} \cdot {\mathit{\boldsymbol{X}}^ * }} \right)} \right\} \otimes E\left\{ {{{\left( {\mathit{\boldsymbol{X}} \cdot {\mathit{\boldsymbol{X}}^{\rm{H}}}} \right)}^ * }} \right\}\\ \;\;\;\;\; = \mathit{\boldsymbol{B}}\left( \theta \right){\mathit{\boldsymbol{C}}_s}{\mathit{\boldsymbol{B}}^{\rm{H}}}\left( \theta \right). \end{array} (6) 式中:B(θ)=A(θ)⊗AH(θ); ⊗表示Kronecker乘积; *表示共轭; H表示矩阵的共轭转置.
信号源与阵元之间是相互独立的, 我们将接收信号按照信源划分, 定义第i个信源t时刻在阵列上产生的输出为
{\mathit{\boldsymbol{y}}_i}\left( t \right) = \mathit{\boldsymbol{a}}\left( {{\theta _i}} \right){\mathit{\boldsymbol{s}}_i}\left( t \right) + {\mathit{\boldsymbol{N}}_i}\left( t \right). (7) 式中:a(θi)=[a1(θi) a2(θi)… aM(θi)]; Ni(t)是从N(t)中抽取的部分噪声, 其表示第i个信源在阵列上产生的噪声, 假设为零均值的高斯白噪声.
因此可将式(2)变形为
\mathit{\boldsymbol{X}}\left( t \right) = \sum\limits_{i = 1}^P {{y_i}\left( t \right)} . (8) 2 算法原理
我们知道EM算法是一种数据添加算法, 其重点在于构造一个与观测数据有关的隐含变量, 利用这个隐含变量迭代求解待估参数[8].同时研究表明, 将信号的四阶累积量用于阵列信号处理中可以不损失信号的方向信息[14].
本文将阵列接收数据按信号源的不同进行一个划分, 将信号的四阶累积量与EM算法结合, 迭代分别估计每个信号源的来波方向.
在本文信号模型的基础上, 我们将第i个信源t时刻在阵列上产生的输出yi(t)看做隐含变量.
构造隐含变量yi(t)关于待估参数的对数似然函数如下[10]:
\begin{array}{l} \ln L\left( {\mathit{\Theta }\left| {{y_i}} \right.} \right) = \ln p\left( {{y_i}\left| \mathit{\Theta } \right.} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = - \frac{1}{2}\ln \left( {2{\rm{ \mathsf{ π} }}\sigma _i^2} \right) - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{{2\sigma _i^2}}{\left| {{y_i}\left( t \right) - \sum\limits_{m = 1}^M {{a_m}\left( {{\theta _i}} \right){s_i}\left( t \right)} } \right|^2} \propto - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{\sigma }{\left\| {{y_i}\left( t \right) - \sum\limits_{m = 1}^M {\left[ {{a_m}\left( {{\theta _i}} \right){s_i}\left( t \right)} \right]} } \right\|^2}. \end{array} (9) 式中:θ为信号参数, ∝表示成正比例; ‖‖表示矩阵的Frobenius范数; σ表示噪声标准差; p(yi|Θ)表示在信号参数的条件下阵列输出的概率密度函数.
联合式(8)可以得到阵列输出关于信号参数Θ的对数似然函数:
\begin{array}{l} \ln L\left( {\mathit{\Theta }\left| \mathit{\boldsymbol{X}} \right.} \right) = \ln \prod\limits_{i = 1}^P {p\left( {{y_i}\left| \mathit{\Theta } \right.} \right)} \\ \;\;\;\;\;\;\;\;\; = \sum\limits_{i = 1}^P {\ln L\left( {\mathit{\Theta }\left| {{y_i}} \right.} \right)} . \end{array} (10) 利用四阶累积量的EM算法迭代计算每个信源的波达方向:
1) E-step
当接收信号X和信号参数Θk已知时, 信源在阵列上的输出可以估计为[10]
\begin{array}{l} {{\hat y}_i}\left( {t,{\mathit{\Theta }^k}} \right) = E\left[ {{y_i}\left| {\mathit{\boldsymbol{X}},{\mathit{\Theta }^k}} \right.} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; = a\left( {\theta _i^k} \right)s_i^k\left( t \right) + \left[ {X\left( t \right) - \sum\limits_{i - 1}^P {a\left( {\theta _i^k} \right)s_i^k\left( t \right)} } \right], \end{array} (11) \begin{array}{l} Q\left( {{\mathit{\Theta }^{k + 1}}\left| {{\mathit{\Theta }^k}} \right.} \right) = E\left[ {\ln L\left( {{\mathit{\Theta }^{k + 1}}\left| {{y_i}} \right.} \right)\left| {\mathit{\boldsymbol{X}},{\mathit{\Theta }^k}} \right.} \right] \propto - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{\sigma }E\left\{ {{{\left\| {{y_i}\left( {t,{\mathit{\Theta }^k}} \right) - a\left( {\theta _i^k} \right)s_i^k\left( t \right)} \right\|}^2}} \right\}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = - \frac{1}{\sigma }{\left\| {{{\hat y}_i}\left( {t,{\mathit{\Theta }^k}} \right) - a\left( {\theta _i^k} \right)s_i^k\left( t \right)} \right\|^2}. \end{array} (12) 式中:E[·]表示求数学期望; 上标k表示第k步迭代过程中所得到的各变量的估计值.
对ŷi(t, Θk)进行四阶累积扩展, 可得到扩展后第i个信源在第m个阵元处相对于参考阵元的相位延迟
{b_m}\left( {{\theta _i}} \right) = {a_m}\left( {{\theta _i}} \right) \otimes {a_m}\left( {{\theta _i}} \right), (13) 根据高阶累积量的性质可得
\begin{array}{l} {\mathit{\boldsymbol{C}}_X} = \left[ {{\mathit{\boldsymbol{C}}_{{y_1}}}\;{\mathit{\boldsymbol{C}}_{{y_2}}} \cdots {\mathit{\boldsymbol{C}}_{{y_P}}}} \right]\\ \;\;\;\;\; = \mathit{\boldsymbol{B}}\left( \theta \right){\mathit{\boldsymbol{C}}_s}{\mathit{\boldsymbol{B}}^{\rm{H}}}\left( \theta \right)\\ \;\;\;\;\; = \left[ {\mathit{\boldsymbol{b}}\left( {{\theta _1}} \right){\mathit{\boldsymbol{C}}_{{s_1}}}{\mathit{\boldsymbol{b}}^{\rm{H}}}\left( {{\theta _1}} \right)\mathit{\boldsymbol{b}}\left( {{\theta _2}} \right){\mathit{\boldsymbol{C}}_{{s_2}}}{\mathit{\boldsymbol{b}}^{\rm{H}}}\left( {{\theta _2}} \right)} \right.\\ \;\;\;\;\;\;\;\;\left. { \cdots \mathit{\boldsymbol{b}}\left( {{\theta _P}} \right){\mathit{\boldsymbol{C}}_{{s_P}}}{\mathit{\boldsymbol{b}}^{\rm{H}}}\left( {{\theta _P}} \right)} \right]. \end{array} (14) 式中:Csi表示第i个信源的四阶累积矩阵; Cyi表示隐含变量的四阶累积矩阵.
2) M-step
\begin{array}{l} \theta _i^{k + 1} = \arg \mathop {\max }\limits_\theta \left\{ {{b_m}\left( {\theta _i^{k + 1}} \right) \cdot } \right.\\ \;\;\;\;\;\;\;\;\;{\left[ {b_m^ * \left( {\theta _i^{k + 1}} \right){b_m}\left( {\theta _i^{k + 1}} \right)} \right]^1} \cdot \\ \;\;\;\;\;\;\;\;\;\left. {b_m^ * \left( {\theta _i^{k + 1}} \right){C_{{{\hat y}_i}}}\left( {{\mathit{\Theta }^k}} \right)} \right\}. \end{array} (15) EM是一种迭代算法, 我们假设信号参数Θk已知, 阵列输出X已知, 利用式(12)求出在该条件下信源i在阵列上产生的输出ŷi(t, Θk); 然后利用四阶累积量将ŷi(t, Θk)扩展为Cŷi, am(θi)扩展为bm(θi); 将其代入式(16)中可求出该信源波达方向; 再结合式(8)、式(9)可对参数ŷi(t, Θk)进行更新.
本文利用上述基于四阶累积量的EM算法迭代分别计算每个信源的波达方向, 能有效避免误差, 提高估计的分辨率和精度.
3 仿真实验
为了验证本文提出方法的估计性能、收敛效果及分辨率, 我们进行以下仿真实验, 且仿真实验的窄带信号源均采用如下形式[16]:
S\left( t \right) = I\left( t \right){{\rm{e}}^{{\rm{j}}\left[ {{w_0}t + \theta \left( t \right)} \right]}}. (16) 式中:I(t)表示慢变幅度调制信号; θ(t)表示慢变相位调制信号; ω0为信号载频.
实验1 考虑8个全向阵元组成的均匀线阵, 阵元间距为0.5, 空域角度搜索范围为[-90° 90°]. 3个等功率的信号源分别以角度-30°、30°、60°入射到等距均匀线阵上.当SNR为-8 dB, 快拍数为50时, 对传统算法、改进算法及本文算法进行仿真.
图 1、图 2、图 3、图 4、图 5分别为ML方法、MUSIC方法、基于四阶累积量ML方法、二阶EM算法和本文算法的波达方向估计图.从图中可以看出:在低SNR、少快拍数的条件下, 传统的ML方法和MUSIC方法均具有较大的估计误差; 基于四阶累积量的ML方法和二阶EM算法相对于传统算法估计性能有所改善, 但其依旧不能实现准确估计; 而本文算法能准确估计出信号源的来波方向.
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图 2 MUSIC算法的波达方向估计Fig. 2 The estimation of the direction of arrival of MUSIC algorithm![]()
图 3 四阶累积量ML算法波达方向估计Fig. 3 The estimation of the direction of arrival of the four order cumulant ML algorithm![]()
图 4 二阶EM算法的波达方向估计Fig. 4 The direction of arrival estimation of the two order EM algorithm信号的四阶累积量需要在较多数据量的情况下才能体现其不损失方向信息且能滤除高斯噪声的优势[14], 因此本文利用EM算法迭代增加了数据量, 使得本算法在小快拍数下也能体现四阶累积量的优势.
为了更进一步地说明本文提出的基于四阶累积量的EM算法的估计优势, 我们分别以上面几种方法进行了50次蒙特卡洛仿真实验, 得到不同SNR下波达方向估计的均方误差, 计算公式为
{E_{{\rm{RMS}}}} = \sqrt {\sum\limits_{i = 1}^I {\sum\limits_{p = 1}^P {\frac{{{{\left( {{{\hat \theta }_{pj}} - {\theta _p}} \right)}^2}}}{{IP}}} } } . (17) 式中:I表示蒙特卡洛实验次数; P表示信源个数; {{\hat \theta }_{pi}}表示第i次试验的波达方向角; θp表示信号真实的波达方向角.
图 6为几种方法的波达方向估计均方误差对比.从图中可以看出, 当SNR较低时ML方法和MUSIC方法都有较大的估计误差, 而本文所述的方法依然能准确估计出信号源的来波方向.
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图 6 波达方向估计均方误差对比Fig. 6 Comparison of root mean square error in the direction of arrival为分析本文算法的收敛性, 改变算法EM的迭代次数并进行波达方向估计, 图 7为不同迭代次数下本文算法的波达方向估计值.从图中可以看出本文算法在第2次迭代就能收敛到来波方向附近, 具有收敛速度快的特点.
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图 7 本文算法在不同迭代次数的DOA估计值Fig. 7 The DOA estimation of this algorithm at different iterations实验2 为分析本文方法的分辨率, 参照文献[15]的方法, 当SNR为4 dB时, 考虑2个等功率的独立信号源分别以角度30°、31°入射到等距均匀线阵上, 其他条件同实验1.改变两信号源的角度间隔, 使其变化范围为[0° 20°], 且在每个角度间隔下进行50次独立的实验.
我们利用绝对误差来衡量估计的分辨率:
\Delta \theta = \left| {{{\hat \theta }_p} - {\theta _p}} \right|. (18) 式中:{{\hat \theta }_p}表示第p个信号源的波达方向角的估计值; θp表示第p个信号真实的波达方向角; Δθ表示绝对误差.
当式(19)满足\Delta \theta = \left| {{{\hat \theta }_p} - {\theta _p}} \right|≤1°, 即信源的角度估计值与真实值的误差在[-1° 1°]时, 我们认为可以准确分辨信源的波达方向角.实验统计成功分辨的次数与实验次数的比值即估计正确分辨概率.
图 8为不同角度间隔下的波达方向估计的正确分辨概率.从图中可以看出, 随着角度间隔的增大, 所有方法的正确分辨概率都在提高, 但在角度间隔仅为1°的情况下本文方法就能实现高分辨的估计, 性能优于其他对比方法.
4 结论
本文提出一种基于四阶累积量的EM波达方向估计算法, 通过仿真实验与传统方法(ML方法和MUSIC方法)及改进方法(四阶累积量ML算法和二阶EM算法)进行对比, 研究表明:本文算法在窄带信号波达方向估计中具有明显优势, 尤其是在低SNR、少快拍数情况下, 其估计性能明显优于传统方法和改进方法, 具有较高的空间分辨率和稳定性.
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图 2 MUSIC算法的波达方向估计
Fig. 2 The estimation of the direction of arrival of MUSIC algorithm
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图 3 四阶累积量ML算法波达方向估计
Fig. 3 The estimation of the direction of arrival of the four order cumulant ML algorithm
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图 4 二阶EM算法的波达方向估计
Fig. 4 The direction of arrival estimation of the two order EM algorithm
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图 6 波达方向估计均方误差对比
Fig. 6 Comparison of root mean square error in the direction of arrival
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图 7 本文算法在不同迭代次数的DOA估计值
Fig. 7 The DOA estimation of this algorithm at different iterations
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