工程科学与技术   2017, Vol. 49 Issue (5): 156-163

Underdetermined Direction of Arrival Estimation with Coprime Array Based on Matrix Completion
WU Chenxi, ZHANG Min, WANG Keren
Dept. of Network,Electronic Eng. Inst. of PLA,Hefei 230037,China
Abstract: In the existing DOA estimation methods based on coprime array,discarding the non-uniform virtual elements in the difference coarray leads to the decrease of the number of the resolvable sources.In order to solve the problem,a DOA estimation method based on matrix completion was proposed.Firstly,according to the correspondence between the difference coarray and the wave path difference,a Toeplitz array covariance matrix was constructed,of which some elements are zero.Additionally,the DOA estimation model was established based on matrix completion and it was proved that the proposed model satisfied null space property.Secondly,according to the low rank matrix completion theory,the DOA estimation was transformed into the minimization of the nuclear norm,and then the Toeplitz matrix was recovered to the full covariance matrix by the fixed-point iterative algorithm.Finally,the DOA estimation was transformed into the determination of polynomial roots by singular value decomposition on the Toeplitz matrix.Simulation results showed the effectiveness and superiority of the proposed method.The experimental results showed that it can effectively fill the holes of the difference coarray,increase the available DOFs and the number of the resolvable signals.Meanwhile,it can eliminate the influence of the basis mismatch on the estimation performance in traditional sparse reconstruction methods due to the discretization in angel domain.Compared with the existing methods,the proposed method improved the DOA estimation accuracy and resolution.
Key words: array signal processing    underdetermined DOA estimation    coprime array    matrix completion    nuclear norm

1 互质阵列模型

 图1 互质阵列结构 Fig. 1 Structure of coprime array

 $\mathit{\boldsymbol{X}}(t) = \mathit{\boldsymbol{As}}(t) + \mathit{\boldsymbol{n}}(t)$ (1)

 $A(i,j) = {{\rm e} ^{{\rm{j}}2{{\text{π} }}{d_i}\sin {\theta _j}/\lambda }}$ (2)

 ${\mathit{\boldsymbol{R}}_{X}} = E[\mathit{\boldsymbol{X}}(t){\mathit{\boldsymbol{X}}^{\rm H}}(t)] = \mathit{\boldsymbol{A}}{\mathit{\boldsymbol{R}}_{s}}{\mathit{\boldsymbol{A}}^{\rm H}} + \mathit{\boldsymbol{n}}$ (3)

 ${\overline{R}_{X}} = (1/L)\sum\limits_{t = 1}^L {\mathit{\boldsymbol{X}}(t){\mathit{\boldsymbol{X}}^{\rm H}}(t)}$ (4)
2 基于矩阵填充的DOA估计方法

 $D = \{ {d_i} - {d_j}\} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {d_i},{d_j} \in C$ (5)

 \begin{aligned}& \quad\quad S = \{ \pm (nMd - mNd)\} ,\\& 0 \le n \le N - 1,0 \le m \le 2M - 1\end{aligned} (6)

 图2 差联合阵列结构 Fig. 2 Structure of difference coarray

 $\left\{\!\!\!\! \begin{array}{l}{\rm card}(C) = N + 2M - 1,\\{\rm card}({D_u}) = 3MN + M - N,\\{\rm card}(\varPsi) = 2MN + 2M - 1,\\{\rm card}(\varGamma) = 4MN - 2N + 1\end{array} \right.$ (7)

 $F = (\left| C \right| - 1)/2 = o(MN)$ (8)

 \begin{aligned}[b]{\mathit{\boldsymbol{R}}_{\mathit{\boldsymbol{X}}}} = & \left| {\begin{array}{*{20}{c}}{\displaystyle\sum\limits_{k = 1}^K {p_k^2} }&{\displaystyle\sum\limits_{k = 1}^K {p_k^2{{\rm e}^{\mathit{\boldsymbol{ - }}{\rm j}O}}} }& \cdots &{\displaystyle\sum\limits_{k = 1}^K {p_k^2{{\rm e}^{\mathit{\boldsymbol{ - }}{\rm j}P}}} }\\{\displaystyle\sum\limits_{k = 1}^K {p_k^2{{\rm e}^{{\rm j}O}}} }&{\displaystyle\sum\limits_{k = 1}^K {p_k^2} }& \cdots &{\displaystyle\sum\limits_{k = 1}^K {p_k^2{{\rm e}^{\mathit{\boldsymbol{ - }}{\rm j}Q}}} }\\ \vdots & \vdots &{}& \vdots \\{\displaystyle\sum\limits_{k = 1}^K {p_k^2{{\rm e}^{{\rm j}P}}} }&{\displaystyle\sum\limits_{k = 1}^K {p_k^2{{\rm e}^{{\rm j}Q}}} }& \cdots &{\displaystyle\sum\limits_{k = 1}^K {p_k^2} }\end{array}} \right| + \mathit{\boldsymbol{n}}{\kern 1pt} = \\& \left| {\begin{array}{*{20}{c}}{R(0)}&{R( - M)}& \cdots &{R( - (2M - 1)N)}\\{R(M)}&{R(0)}& \cdots &{R(M - (2M - 1)N)}\\ \vdots & \vdots &{}& \vdots \\{R((2M - 1)N)}&{R((2M - 1)N - M)}& \cdots &{R(0)}\end{array}} \right| + \mathit{\boldsymbol{n}}\end{aligned} (9)

 ${D_u} = \{ - (2M - 1)N, \cdots ,0, \cdots ,(2M - 1)N\}$ (10)

 $\hat{R}({d_u}) = \frac{1}{{w({d_u})}}\sum\limits_{i = 1}^{w({d_u})} {{R_i}({d_u})}$ (11)

 ${R_{\rm T}}(i,j) = \left\{ \!\!\!\! \begin{array}{l}\hat R({d_u}), {\kern 1pt} i - j = {\kern 1pt} {d_u};\\[10pt]0,{\kern 1pt} {\kern 1pt} {\kern 1pt} \text{其他}\end{array} \right.$ (12)

${\mathit{\boldsymbol{R}}_{\rm T}}$ 可看作是一个阵元数为 $(2M - 1)N + 1$ 的均匀线阵输出协方差矩阵，但其某些位置元素为零。

 ${\mathit{\boldsymbol{R}}_{\mathit{\boldsymbol{X}}}} = \left[ {\begin{array}{*{20}{c}}{R(0)}&{R( - 2)}&{R( - 3)}&{R( - 4)}&{R( - 6)}&{R( - 9)}\\[3pt]{R(2)}&{R(0)}&{R( - 1)}&{R( - 2)}&{R( - 4)}&{R( - 7)}\\[3pt]{R(3)}&{R(1)}&{R(0)}&{R( - 1)}&{R( - 3)}&{R( - 6)}\\[3pt]{R(4)}&{R(2)}&{R(1)}&{R(0)}&{R( - 2)}&{R( - 5)}\\[3pt]{R(6)}&{R(4)}&{R(3)}&{R(2)}&{R(0)}&{R( - 3)}\\[3pt]{R(9)}&{R(7)}&{R(6)}&{R(5)}&{R(3)}&{R(0)}\end{array}} \right]$ (13)
 ${\mathit{\boldsymbol{R}}_{\rm T}} = \left[ {\begin{array}{*{20}{c}}{R(0)}&{R( - 1)}&{R( - 2)}&{R( - 3)}&{R( - 4)}&{R( - 5)}&{R( - 6)}&{R( - 7)}&0&{R( - 9)}\\[3pt]{R(1)}&{R(0)}&{R( - 1)}&{R( - 2)}&{R( - 3)}&{R( - 4)}&{R( - 5)}&{R( - 6)}&{R( - 7)}&0\\[3pt]{R(2)}&{R(1)}&{R(0)}&{R( - 1)}&{R( - 2)}&{R( - 3)}&{R( - 4)}&{R( - 5)}&{R( - 6)}&{R( - 7)}\\[3pt]{R(3)}&{R(2)}&{R(1)}&{R(0)}&{R( - 1)}&{R( - 2)}&{R( - 3)}&{R( - 4)}&{R( - 5)}&{R( - 6)}\\[3pt]{R(4)}&{R(3)}&{R(2)}&{R(1)}&{R(0)}&{R( - 1)}&{R( - 2)}&{R( - 3)}&{R( - 4)}&{R( - 5)}\\[3pt]{R(5)}&{R(4)}&{R(3)}&{R(2)}&{R(1)}&{R(0)}&{R( - 1)}&{R( - 2)}&{R( - 3)}&{R( - 4)}\\[3pt]{R(6)}&{R(5)}&{R(4)}&{R(3)}&{R(2)}&{R(1)}&{R(0)}&{R( - 1)}&{R( - 2)}&{R( - 3)}\\[3pt]{R(7)}&{R(6)}&{R(5)}&{R(4)}&{R(3)}&{R(2)}&{R(1)}&{R(0)}&{R( - 1)}&{R( - 2)}\\[2pt]0&{R(7)}&{R(6)}&{R(5)}&{R(4)}&{R(3)}&{R(2)}&{R(1)}&{R(0)}&{R( - 1)}\\[3pt]{R(9)}&0&{R(7)}&{R(6)}&{R(5)}&{R(4)}&{R(3)}&{R(2)}&{R(1)}&{R(0)}\end{array}} \right]$ (14)

 \begin{aligned}& \min \ {\rm rank}({\mathit{\boldsymbol{R}}_{\rm c}})\\& {\rm s.t}. \ {P_\varOmega }{\kern 1pt} ({\mathit{\boldsymbol{R}}_{\rm c}}) = {P_\varOmega }({\mathit{\boldsymbol{R}}_{\rm T}})\end{aligned} (15)

 $\mathit{\boldsymbol{Null}}({P_\varOmega }) = \{ \varPhi \in {\mathit{\boldsymbol{R}}^{|\varGamma | \times |\varGamma |}}:{P_\varOmega }(\varPhi ) = \mathit{\boldsymbol{O}}\}$ (16)

 $\min\left(\mu {\left\| {{\mathit{\boldsymbol{R}}_{\rm c}}} \right\|_*} + \frac{1}{2}\left\| {{P_\varOmega }({\mathit{\boldsymbol{R}}_{\rm c}}) - {\mathit{\boldsymbol{R}}_{\rm T}}} \right\|_2^2\right)$ (17)

 \begin{aligned}\\[-10pt] {P_\varOmega }({R_{\rm c}}(i,j)) = \left\{ \begin{array}{l}\!\!\!\!\!{R_{\rm T}}(i,j), i - j = {d_u};\\\!\!\!\!\! 0, \text{其他}\end{array} \right.\end{aligned} (18)

 $\left\{ \begin{array}{l}\!\!\!\!\! {\mathit{\boldsymbol{Y}}^k} = \mathit{\boldsymbol{R}}_{\rm c}^k - \tau g(\mathit{\boldsymbol{R}}_{\rm c}^k),\\[8pt]\!\!\!\!\! \mathit{\boldsymbol{R}}_{\rm c}^{k + 1} = {\varGamma _{\tau \mu }}({\mathit{\boldsymbol{Y}}^k})\end{array} \right.$ (19)

 ${\varGamma _{\tau \mu }}(\mathit{\boldsymbol{Y}}) = {\mathit{\boldsymbol{U}}_{Y}}{\rm{diag(}}{s_{\tau \mu }}(\sigma ))\mathit{\boldsymbol{V}}_{Y}^*$ (20)

 ${s_{\tau \mu }}(\sigma ) = \tilde \sigma ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\tilde \sigma _i} = \left\{ \begin{array}{l}\!\!\!\!\! {\sigma _i} - \tau \mu ,{\rm{ }}{\sigma _i} - \tau \mu > 0;\\[5pt]\!\!\!\!\! 0,{\sigma _i} - \tau \mu \le 0\end{array} \right.$ (21)

 ${\mathit{\boldsymbol{R}}_{\rm c}} = {\mathit{\boldsymbol{U}}_{s}}{{\varSigma} _{s}}\mathit{\boldsymbol{U}}_{s}^{\rm H} + {\mathit{\boldsymbol{U}}_n}{{\varSigma} _n}\mathit{\boldsymbol{U}}_n^{\rm H}$ (22)

 $f({\textit z}) = {{\textit z}^{2MN - N}}{\mathit{\boldsymbol{P}}^{\rm T}}({{\textit z}^{ - 1}}){\mathit{\boldsymbol{U}}_n}\mathit{\boldsymbol{U}}_n^{\rm H}\mathit{\boldsymbol{P}}({\textit z})$ (23)

 ${\hat \theta _i} = {\rm arcsin}({\rm arg}({\hat {\textit{z}}_i})/{{\text{π} }})$ (24)

3 仿真实验

 $RMS\!\!E(\theta ) = \frac{1}{K}\sum\limits_{j = 1}^K {\sqrt {\frac{1}{J}\sum\limits_{l = 1}^J {{{({{\hat \theta }_i} - {\theta _i})}^2}} } }$ (25)

1）可行性分析

 图3 4种方法的可估计信号数比较 Fig. 3 Comparison of resolvable signals number of the four methods

2）分辨率比较

 图4 4种方法的分辨率比较 Fig. 4 Comparison of the resolution of the four methods

3）估计精度比较

 图5 角度均方根误差随信噪比变化 Fig. 5 RMSE with different SNR

 图6 角度均方根误差随快拍数变化 Fig. 6 RMSE with different snapshot number

4）分辨成功率比较

 图7 4种方法的分辨成功率比较 Fig. 7 Success rates of resolution of four methods

5）时效性比较

 图8 运算时间随信号数变化 Fig. 8 Operation time variable with numbers of signal

4 结　论

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