南京航空航天大学学报  2017, Vol. 49 Issue (5): 635-644 PDF

Research Progress on Spatial Spectrum Estimation Based on Coprime Array
ZHANG Xiaofei, LIN Xinping, ZHENG Wang, ZHAI Hui
College of Electronic and Information Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing, 210016, China
Abstract: Array signal processing technology is an important part in the field of signal processing, which is widely used in radar, sonar and hydropower astronomy. Spatial spectrum estimation is a major research hotspot in array signal processing and is confined to the arrays with inter-element spacing smaller than half wavelength of the carrier wave to avoid ambiguous angles. Generally, the inter-element spacing of coprime arrays is larger than half wavelength. As a result, with fixed number of sensors, the coprime arrays can obtain extended array aperture and hence the corresponding spectrum estimation methods can achieve better direction of arrival estimation performance and resolution, where the targets can be detected uniquely with coprime property. Therefore, the investigations of spatial spectrum estimation with coprime arrays have aroused considerable attentions in array signal processing. In this paper, we summarize the existing researches based on coprime linear array and coprime planar array and analysis the advantages of the coprime array relative to the uniform array in expanding array aperture and increasing spatial degree of freedom. Simulation results validate the effectiveness and superiority of the methods with coprime arrays.
Key words: coprime linear array     coprime planar array     spatial spectrum estimation     direction of arrival estima tion

1 互质线阵空间谱估计

1.1 互质线阵拓扑结构与数据模型

 图 1 互质线阵基础结构拓扑图 Figure 1 Topological structure of coprime linear array

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{L}}s = \left\{ {\left( {{m_1}{d_1}\left| {{m_1} = 0,1,2, \cdots ,{M_1} - 1} \right.} \right.} \right\} \cup }\\ {\left\{ {{m_2}{d_2}\left| {{m_2} = 0,1,2, \cdots ,{M_2} - 1} \right.} \right\}} \end{array}$ (1)

 $\mathit{\boldsymbol{x}}\left( t \right) = \mathit{\boldsymbol{As}}\left( t \right) + \mathit{\boldsymbol{n}}\left( t \right)$ (2)

 $\mathit{\boldsymbol{A}} = \left[ {\mathit{\boldsymbol{a}}\left( {{\theta _1}} \right),\mathit{\boldsymbol{a}}\left( {{\theta _2}} \right), \cdots ,\mathit{\boldsymbol{a}}\left( {{\theta _K}} \right)} \right]$ (3)

 $\begin{array}{l} {\mathit{\boldsymbol{a}}_1}\left( {{\theta _k}} \right) = \left[ {1,{{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{ \mathit{ π} }}}}{\lambda }{M_2} \cdot d \cdot \sin \left( {{\theta _k}} \right)}}, \cdots ,} \right.\\ \;\;\;\;\;\;\;\;\;\;\;{\left. {{{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{ \mathit{ π} }}}}{\lambda }\left( {{M_1} - 1} \right) \cdot {M_2} \cdot d \cdot \sin \left( {{\theta _k}} \right)}}} \right]^{\rm{T}}} \end{array}$ (4)
 $\begin{array}{l} {\mathit{\boldsymbol{a}}_2}\left( {{\theta _k}} \right) = \left[ {1,{{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{ \mathit{ π} }}}}{\lambda }{M_1} \cdot d \cdot \sin \left( {{\theta _k}} \right)}}, \cdots ,} \right.\\ \;\;\;\;\;\;\;\;\;\;\;{\left. {{{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{ \mathit{ π} }}}}{\lambda }\left( {{M_2} - 1} \right) \cdot {M_1} \cdot d \cdot \sin \left( {{\theta _k}} \right)}}} \right]^{\rm{T}}} \end{array}$ (5)

 $\mathit{\boldsymbol{a}}\left( {{\theta _k}} \right) = {\left[ {\mathit{\boldsymbol{a}}_1^{\rm{T}}\left( {{\theta _k}} \right),\mathit{\boldsymbol{a}}_{21}^{\rm{T}}\left( {{\theta _k}} \right)} \right]^{\rm{T}}}$ (6)

1.2 DOA估计算法 1.2.1 解模糊方法

 $\mathit{\boldsymbol{E}}_{s1,i}^ + {\mathit{\boldsymbol{E}}_{s2,i}} = {\mathit{\boldsymbol{T}}^{ - 1}}\mathit{\boldsymbol{ \boldsymbol{\varPhi} T}}$ (7)

 $\mathit{\boldsymbol{a}}\left( \theta \right) = \mathit{\boldsymbol{a}}\left( {{\theta ^a}} \right)$ (8)

 $2{\rm{ \mathit{ π} }}{d_i}\sin \theta /\lambda - 2{\rm{ \mathit{ π} }}{d_i}\sin {\theta ^a}/\lambda = 2k{\rm{ \mathit{ π} }}\;\;\;\;\;i = {\rm{1,2}}$ (9)

 $\sin \left( {{\theta _k}} \right) - \sin \left( {{{\theta '}_k}} \right) = 2{k_1}/{M_2}$ (10)

 $\sin \left( {{\theta _k}} \right) - \sin \left( {{{\theta '}_k}} \right) = 2{k_2}/{M_1}$ (11)

 $\frac{{2{k_1}}}{{{M_2}}} = \frac{{2{k_2}}}{{{M_1}}}$ (12)

1.2.2 虚拟化阵元方法

 $\mathit{\boldsymbol{x}}\left( t \right) = {\left[ {\mathit{\boldsymbol{x}}_1^{\rm{T}}\left( t \right),\mathit{\boldsymbol{x}}_2^{\rm{T}}\left( t \right)} \right]^{\rm{T}}}$ (13)

 $\mathit{\boldsymbol{z}} = {\rm{vec}}\left( {{\mathit{\boldsymbol{R}}_{xx}}} \right) = \mathit{\boldsymbol{B}}\left( {{\theta _1},{\theta _2}, \cdots ,{\theta _K}} \right)\mathit{\boldsymbol{P}} + \sigma _n^2\mathit{\boldsymbol{H}}$ (14)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{B}}\left( {{\theta _1},{\theta _2}, \cdots ,{\theta _K}} \right) = \left[ {{\mathit{\boldsymbol{a}}^ * }\left( {{\theta _1}} \right) \otimes \mathit{\boldsymbol{a}}\left( {{\theta _1}} \right),{\mathit{\boldsymbol{a}}^ * }\left( {{\theta _2}} \right) \otimes } \right.}\\ {\left. {\mathit{\boldsymbol{a}}\left( {{\theta _2}} \right), \cdots ,{\mathit{\boldsymbol{a}}^ * }\left( {{\theta _K}} \right) \otimes \mathit{\boldsymbol{a}}\left( {{\theta _K}} \right)} \right]} \end{array}$ (15)

 $\begin{array}{l} {{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{ \mathit{ π} }}}}{\lambda }d\sin \left( {{\theta _k}} \right)\left( {{M_1}{m_2} - {M_2}{m_1}} \right)}}\\ \;\;\;0 \le {m_2} \le {M_2} - 1,\;\;\;\;0 \le {m_1} \le {M_1} - 1 \end{array}$ (16)
 $\begin{array}{l} {{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{ \mathit{ π} }}}}{\lambda }d\sin \left( {{\theta _k}} \right)\left( {{M_1}{m_{21}} - {M_1}{m_{22}}} \right)}}\\ \;\;\;0 \le {m_{21}},\;\;\;\;{m_{22}} \le {M_2} - 1 \end{array}$ (17)
 $\begin{array}{l} {{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{ \mathit{ π} }}}}{\lambda }d\sin \left( {{\theta _k}} \right)\left( {{M_2}{m_{11}} - {M_2}{m_{12}}} \right)}}\\ \;\;\;0 \le {m_{11}},{m_{12}} \le {M_1} - 1 \end{array}$ (18)

 $\begin{array}{l} {S_{{M_1}{M_2}}} = \left\{ {\left( {{M_1}{m_2} - {M_2}{m_1}} \right),0 \le {m_2} \le {M_2} - 1,} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\left. {0 \le {m_1} \le {M_1} - 1} \right\} \end{array}$ (19)

 图 2 增广互质线阵拓扑结构 Figure 2 Topological structure of a ugmented coprime linear array

 ${\mathit{\boldsymbol{R}}_i} = {\mathit{\boldsymbol{z}}_{1i}}\mathit{\boldsymbol{z}}_{1i}^{\rm{H}}$ (20)

 ${\mathit{\boldsymbol{R}}_{ss}} = \frac{1}{{{M_1}{M_2} + 1}}\sum\limits_{i = 1}^{{M_1}{M_2} + 1} {{\mathit{\boldsymbol{R}}_i}}$ (21)

 $\mathit{\boldsymbol{\hat R}} = \frac{1}{{\sqrt {{M_1}{M_2} + 1} }}\left( {{\mathit{\boldsymbol{A}}_{11}}\mathit{\boldsymbol{ \boldsymbol{\varLambda} A}}_{11}^{\rm{H}} + \sigma _n^2{\mathit{\boldsymbol{I}}_{{M_1}{M_2}}}} \right)$ (22)

 图 3 互质线阵中不同方法下的DOA估计性能 Figure 3 DOA estimation perfor mance under different methods in coprime linear array

 图 4 互质线阵中不同快拍数下算法性能 Figure 4 Algorithm performance under different snapshots in coprime line ar array

2 互质面阵空间谱估计 2.1 互质面阵拓扑结构与数据模型

 图 5 互质面阵结构拓扑图 Figure 5 Topological structure of coprime planar array

 $\begin{array}{*{20}{c}} {{L_s} = \left\{ {\left( {m{d_1},n{d_1}} \right)\left| {0 \le m,n \le {M_1} - 1} \right.} \right\} \cup }\\ {\;\;\;\;\left\{ {\left( {p{d_2},q{d_2}} \right)\left| {0 \le p,q \le {M_2} - 1} \right.} \right\}} \end{array}$ (23)

 $\begin{array}{l} {\mathit{\boldsymbol{a}}_{ix}}\left( {{\theta _k},{\phi _k}} \right) = \left[ {\begin{array}{*{20}{c}} 1\\ {{{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{ \mathit{ π} }}}}{\lambda }{d_i}\sin {\theta _k}\cos {\phi _k}}}}\\ \vdots \\ {{{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{ \mathit{ π} }}}}{\lambda }\left( {{M_i} - 1} \right){d_i}\sin {\theta _k}\cos {\phi _k}}}} \end{array}} \right]\\ {\mathit{\boldsymbol{a}}_{iy}}\left( {{\theta _k},{\phi _k}} \right) = \left[ {\begin{array}{*{20}{c}} 1\\ {{{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{ \mathit{ π} }}}}{\lambda }{d_i}\sin {\theta _k}\sin {\phi _k}}}}\\ \vdots \\ {{{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{ \mathit{ π} }}}}{\lambda }\left( {{M_i} - 1} \right){d_i}\sin {\theta _k}\sin {\phi _k}}}} \end{array}} \right] \end{array}$ (24)

 ${\mathit{\boldsymbol{x}}_{i1}}\left( t \right) = {\mathit{\boldsymbol{A}}_{ix}}\mathit{\boldsymbol{s}}\left( t \right) + {\mathit{\boldsymbol{n}}_{i1}}\left( t \right)$ (25)

 ${\mathit{\boldsymbol{x}}_{im}}\left( t \right) = {\mathit{\boldsymbol{A}}_{ix}}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_{iy}^{m - 1}\mathit{\boldsymbol{s}}\left( t \right) + {\mathit{\boldsymbol{n}}_{im}}\left( t \right)$ (26)

 ${\mathit{\boldsymbol{x}}_i}\left( t \right) = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{x}}_{i1}}\left( t \right)}\\ {{\mathit{\boldsymbol{x}}_{i2}}\left( t \right)}\\ \vdots \\ {{\mathit{\boldsymbol{x}}_{i{M_i}}}\left( t \right)} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{A}}_{ix}}}\\ {{\mathit{\boldsymbol{A}}_{ix}}{\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_{iy}}}\\ \vdots \\ {{\mathit{\boldsymbol{A}}_{ix}}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_{iy}^{{M_i} - 1}} \end{array}} \right]\mathit{\boldsymbol{s}}\left( t \right) + \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{n}}_{i1}}\left( t \right)}\\ {{\mathit{\boldsymbol{n}}_{i2}}\left( t \right)}\\ \vdots \\ {{\mathit{\boldsymbol{n}}_{i{M_i}}}\left( t \right)} \end{array}} \right]$ (27)

 $\begin{array}{l} {\mathit{\boldsymbol{x}}_i}\left( t \right) = \left[ {{\mathit{\boldsymbol{A}}_{iy}} \odot {\mathit{\boldsymbol{A}}_{ix}}} \right]\mathit{\boldsymbol{s}}\left( t \right) + {\mathit{\boldsymbol{n}}_i}\left( t \right) = \left[ {{\mathit{\boldsymbol{a}}_{iy}}\left( {{\theta _1},{\phi _1}} \right) \otimes } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left. {{\mathit{\boldsymbol{a}}_{ix}}\left( {{\theta _1},{\phi _1}} \right), \cdots ,{\mathit{\boldsymbol{a}}_{iy}}\left( {{\theta _K},{\phi _K}} \right) \otimes {\mathit{\boldsymbol{a}}_{ix}}\left( {{\theta _K},{\phi _K}} \right)} \right] \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\mathit{\boldsymbol{s}}\left( t \right) + {\mathit{\boldsymbol{n}}_i}\left( t \right) = {\mathit{\boldsymbol{A}}_i}\mathit{\boldsymbol{s}}\left( t \right) + {\mathit{\boldsymbol{n}}_i}\left( t \right) \end{array}$ (28)

2.2 DOA估计方法

2.2.1 解模糊方法

 $\Delta {\varphi _x} + 2{k_x}{\rm{ \mathit{ π} }} = 2{\rm{ \mathit{ π} }}/\lambda {d_i}\sin {\phi _k}\cos {\theta _k}$ (29)
 $\Delta {\varphi _y} + 2{k_y}{\rm{ \mathit{ π} }} = 2{\rm{ \mathit{ π} }}/\lambda {d_i}\sin {\phi _k}\sin {\theta _k}$ (30)

 ${k_x} \in \left[ {\frac{{ - {d_i}}}{\lambda } - \frac{{\Delta {\varphi _x}}}{{2{\rm{ \mathit{ π} }}}},\frac{{{d_i}}}{\lambda } - \frac{{\Delta {\varphi _x}}}{{2{\rm{ \mathit{ π} }}}}} \right]$ (31)
 ${k_y} \in \left[ {\frac{{ - d}}{\lambda } - \frac{{\Delta {\varphi _y}}}{{2{\rm{ \mathit{ π} }}}},\frac{d}{\lambda } - \frac{{\Delta {\varphi _y}}}{{2{\rm{ \mathit{ π} }}}}} \right]$ (32)

 $\sin {\phi _p}\cos {\theta _p} - \sin {\phi _{i,a}}\cos {\theta _{i,a}} = \frac{{2{k_{i,x}}}}{{{M_j}}}$ (33)
 $\sin {\phi _p}\cos {\theta _p} - \sin {\phi _{i,a}}\sin {\theta _{i,a}} = \frac{{2{k_{i,y}}}}{{{M_j}}}$ (34)

 $\sin {{\hat \phi }_1}\cos {{\hat \theta }_1} - \sin {{\hat \phi }_2}\cos {{\hat \theta }_2} = \frac{{2{k_{1,x}}}}{{{M_2}}}$ (35)
 $\sin {{\hat \phi }_1}\sin {{\hat \theta }_1} - \sin {{\hat \phi }_2}\sin {{\hat \theta }_2} = \frac{{2{k_{1,y}}}}{{{M_2}}}$ (36)

 $\sin {{\hat \phi }_1}\cos {{\hat \theta }_1} - \sin {{\hat \phi }_2}\cos {{\hat \theta }_2} = \frac{{2{k_{2,x}}}}{{{M_1}}}$ (37)
 $\sin {{\hat \phi }_1}\sin {{\hat \theta }_1} - \sin {{\hat \phi }_2}\sin {{\hat \theta }_2} = \frac{{2{k_{2,y}}}}{{{M_1}}}$ (38)

 $\frac{{{k_{1,x}}}}{{{M_2}}} = \frac{{{k_{2,x}}}}{{{M_1}}}\;\;\;\;\;\frac{{{k_{1,y}}}}{{{M_2}}} = \frac{{{k_{2,y}}}}{{{M_1}}}$ (39)

2.2.2 虚拟化阵元方法

 ${\mathit{\boldsymbol{X}}_1}\left( t \right) = \sum\limits_{k = 1}^K {{\mathit{\boldsymbol{a}}_x}\left( {{\theta _k},{\phi _k}} \right)\mathit{\boldsymbol{a}}_y^{\rm{T}}\left( {{\theta _k},{\phi _k}} \right){\mathit{\boldsymbol{s}}_k}\left( t \right) + {\mathit{\boldsymbol{Z}}_1}\left( t \right)}$ (40)

 ${\mathit{\boldsymbol{x}}_1}\left( t \right) = {\rm{vec}}\left( {{\mathit{\boldsymbol{X}}_1}\left( t \right)} \right) = \left( {{\mathit{\boldsymbol{A}}_x} \odot {\mathit{\boldsymbol{A}}_y}} \right)\mathit{\boldsymbol{s}}\left( t \right) + {\mathit{\boldsymbol{z}}_1}\left( t \right)$ (41)

 ${\mathit{\boldsymbol{x}}_2}\left( t \right) = {\rm{vec}}\left( {{\mathit{\boldsymbol{X}}_2}\left( t \right)} \right) = \left( {{\mathit{\boldsymbol{B}}_x} \odot {\mathit{\boldsymbol{B}}_y}} \right)\mathit{\boldsymbol{s}}\left( t \right) + {\mathit{\boldsymbol{z}}_2}\left( t \right)$ (42)

 $\begin{array}{l} {\mathit{\boldsymbol{R}}_{12}} = E\left[ {{\mathit{\boldsymbol{x}}_1}\left( t \right)\mathit{\boldsymbol{x}}_2^{\rm{H}}\left( t \right)} \right] = \\ \;\;\;\;\;\;\;\;\left( {{\mathit{\boldsymbol{A}}_x} \odot {\mathit{\boldsymbol{A}}_y}} \right){\mathit{\boldsymbol{R}}_s}{\left( {{\mathit{\boldsymbol{B}}_x} \odot {\mathit{\boldsymbol{B}}_y}} \right)^{\rm{H}}} + {\mathit{\boldsymbol{z}}_{12}} \end{array}$ (43)
 $\begin{array}{l} {\mathit{\boldsymbol{R}}_{21}} = E\left[ {{\mathit{\boldsymbol{x}}_2}\left( t \right)\mathit{\boldsymbol{x}}_1^{\rm{H}}\left( t \right)} \right] = \\ \;\;\;\;\;\;\;\;\left( {{\mathit{\boldsymbol{B}}_x} \odot {\mathit{\boldsymbol{B}}_y}} \right){\mathit{\boldsymbol{R}}_s}{\left( {{\mathit{\boldsymbol{A}}_x} \odot {\mathit{\boldsymbol{A}}_y}} \right)^{\rm{H}}} + {\mathit{\boldsymbol{z}}_{21}} \end{array}$ (44)

 ${\mathit{\boldsymbol{r}}_{12}} = {\rm{vec}}\left( {{\mathit{\boldsymbol{R}}_{12}}} \right) = {\mathit{\boldsymbol{C}}_{12}}\left( {u,v} \right)\mathit{\boldsymbol{p}}$ (45)
 ${\mathit{\boldsymbol{r}}_{21}} = {\rm{vec}}\left( {{\mathit{\boldsymbol{R}}_{21}}} \right) = {\mathit{\boldsymbol{C}}_{21}}\left( {u,v} \right)\mathit{\boldsymbol{p}}$ (46)

 $\begin{array}{l} {\mathit{\boldsymbol{C}}_{12}}\left( {u,v} \right) = {\left( {{\mathit{\boldsymbol{B}}_x} \odot {\mathit{\boldsymbol{B}}_y}} \right)^ * } \odot \left( {{\mathit{\boldsymbol{A}}_x} \odot {\mathit{\boldsymbol{A}}_y}} \right) = \left[ {\mathit{\boldsymbol{b}}_x^ * \left( {{u_1}} \right) \otimes } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathit{\boldsymbol{b}}_y^ * \left( {{v_1}} \right) \otimes {\mathit{\boldsymbol{a}}_x}\left( {{u_1}} \right) \otimes {\mathit{\boldsymbol{a}}_y}\left( {{v_1}} \right), \cdots ,\mathit{\boldsymbol{b}}_x^ * \left( {{u_k}} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { \otimes \mathit{\boldsymbol{b}}_y^ * \left( {{v_k}} \right) \otimes {\mathit{\boldsymbol{a}}_x}\left( {{u_k}} \right) \otimes {\mathit{\boldsymbol{a}}_y}\left( {{v_k}} \right)} \right] \end{array}$ (47)
 $\begin{array}{l} {\mathit{\boldsymbol{C}}_{21}}\left( {u,v} \right) = {\left( {{\mathit{\boldsymbol{A}}_x} \odot {\mathit{\boldsymbol{A}}_y}} \right)^ * } \odot \left( {{\mathit{\boldsymbol{B}}_x} \odot {\mathit{\boldsymbol{B}}_y}} \right) = \left[ {\mathit{\boldsymbol{a}}_x^ * \left( {{u_1}} \right) \otimes } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathit{\boldsymbol{a}}_y^ * \left( {{v_1}} \right) \otimes {\mathit{\boldsymbol{b}}_x}\left( {{u_1}} \right) \otimes {\mathit{\boldsymbol{b}}_y}\left( {{v_1}} \right), \cdots ,\mathit{\boldsymbol{a}}_x^ * \left( {{u_k}} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { \otimes \mathit{\boldsymbol{a}}_y^ * \left( {{v_k}} \right) \otimes {\mathit{\boldsymbol{b}}_x}\left( {{u_k}} \right) \otimes {\mathit{\boldsymbol{b}}_y}\left( {{v_k}} \right)} \right] \end{array}$ (48)

 $\mathit{\boldsymbol{r}} = {\left[ {\mathit{\boldsymbol{r}}_{12}^{\rm{T}},\mathit{\boldsymbol{r}}_{21}^{\rm{T}}} \right]^{\rm{T}}} = \mathit{\boldsymbol{C}}\left( {u,v} \right)\mathit{\boldsymbol{p}}$ (49)

 $\begin{array}{l} {{S'}_{{M_1}{M_2}}} = \left\{ {\left( {{M_1}{m_2} + {M_2}{m_1}} \right),0 \le {m_2} \le {M_2} - 1,} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\left. {0 \le {m_1} \le 2{M_1} - 1} \right\} \end{array}$ (50)

 $\begin{array}{l} S = \left\{ {\left( {{M_{1x}}{m_{2x}} + {M_{1y}}{m_{2y}}} \right) - } \right.\\ \;\;\;\;\;\;\left. {\left( {{M_{2x}}{m_{1x}} + {M_{2y}}{m_{1y}}} \right)} \right\} \end{array}$ (51)

 ${\rm{DO}}{{\rm{F}}_M} = \left[ {\frac{{{{\left( {{S_{UM}} + 5} \right)}^2}}}{8} - 1} \right]$ (52)

 $\begin{array}{l} {{R'}_{{m_1}{m_2}}} = \mathit{\boldsymbol{AD}}_x^{\left( {{m_1} - 1} \right)}\mathit{\boldsymbol{D}}_y^{\left( {{m_2} - 1} \right)}{\mathit{\boldsymbol{R}}_{ss}}{\left( {\mathit{\boldsymbol{D}}_y^{\left( {{m_2} - 1} \right)}} \right)^{\rm{H}}}\\ \;\;\;\;\;\;\;\;\;\;\;\;{\left( {\mathit{\boldsymbol{D}}_x^{\left( {{m_1} - 1} \right)}} \right)^{\rm{H}}}{\mathit{\boldsymbol{A}}^{\rm{H}}} + \sigma _n^2\mathit{\boldsymbol{I}} \end{array}$ (53)

 $\mathit{\boldsymbol{\dddot R}} = \frac{1}{{M_1^2M_2^2}}\sum\limits_{{m_1} = 1}^{{M_1}{M_2}} {\sum\limits_{{m_2} = 1}^{{M_1}{M_2}} {{{\mathit{\boldsymbol{R'}}}_{{m_1}{m_2}}}} }$ (54)

 图 6 互质面阵中不同方法下的DOA估计性能 Figure 6 DOA estimation performance under different methods in coprime plan ar array

 图 7 互质面阵中不同快拍数下算法性能 Figure 7 Algorithm performance under different snapshots in coprime plan ar array

3 结束语

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