logo

SCIENTIA SINICA Informationis, Volume 49, Issue 1: 87-103(2019) https://doi.org/10.1360/N112017-00164

Three-dimensional target localization method based on the tensor subspace FDS-MIMO radar with a co-prime frequency offset

More info
  • ReceivedNov 21, 2017
  • AcceptedFeb 13, 2018
  • PublishedJan 8, 2019

Abstract

Target localization utilizing frequency diversity array (FDA) is one of the most popular research directions in the study of radars. Considering the problems of range ambiguity and localization accuracy, this paper investigates the issue of three-dimensional (3D) target localization based on the proposed frequency diverse-subaperturing multiple-input multiple-output (FDS-MIMO) radar. First, a fifth-order tensor signal model is established by exploiting the inherent multidimensional structure of the matched filter output. To alleviate the difficulty of range ambiguity, the idea of employing co-prime frequency offsets along the two directions in the planer array is provided. This strategy can resolve the contradiction between the range resolution and maximum unambiguous range in beam domain. The tensor-based complex- and real-valued rotational invariance technique (tensor- and unitary tensor-ESPRIT) algorithms are developed based on frequency offset structure. Given that the inherent multidimensional structure is utilized, the proposed methods can resolve range ambiguity with improved localization performance as compared with the existing ESPRIT algorithm and frequency offset manner. Theoretical analysis and numerical results demonstrate the effectiveness of the proposed approaches.


Funded by

国家自然科学基金(61179015)

国家自然科学基金(61401503)


Supplement

Appendix A

本附录讨论4.2小节中基于张量的最小二乘解, 利用${\left\|~{{\cal~A}}~\right\|_{\rm{H}}}~=~{\left\|~{{{{\cal~A}}_{(n)}}}~\right\|_{\rm{F}}}$, 式(44)可以重写为如下的最小二乘问题 \begin{equation}\hat \Psi _{\rm TE}^{\left( r \right)} = \arg \mathop {\min }\limits_{{\Psi ^{\left( r \right)}}} {\left\| {{{\left( {{{{\cal U}}^{\left[ s \right]}}{ \times _r}{{\boldsymbol J}}_1^{\left( r \right)}{ \times _{R + 1}}{\Psi ^{\left( r \right)}} - {{{\cal U}}^{\left[ s \right]}}{ \times _{R + 1}}{{\boldsymbol J}}_2^{\left( r \right)}} \right)}_{R + 1}}} \right\|_{\rm{F}}}, \tag{A1}\end{equation} 从而有 \begin{equation}\hat \Psi _{\rm TE}^{\left( r \right)} = \arg \mathop {\min }\limits_{{\Psi ^{\left( r \right)}}} {\left\| {{{\left[ {{{{\cal U}}^{\left[ s \right]}}{ \times _r}{{\boldsymbol J}}_1^{\left( r \right)}{ \times _{R + 1}}{\Psi ^{\left( r \right)}}} \right]}_{(R + 1)}} - {{\left[ {{{{\cal U}}^{\left[ s \right]}}{ \times _r}{{\boldsymbol J}}_2^{\left( r \right)}} \right]}_{(R + 1)}}} \right\|_{\rm{F}}}. \tag{A2}\end{equation} 利用张量的$n$-模式积性质${\left[~{{{\cal~A}}{~\times~_n}{{\boldsymbol~B}}}~\right]_{(n)}}~=~{{\boldsymbol~B}}~\cdot~{{{\cal~A}}_{(n)}}$可得 \begin{equation}{\left[ {{{{\cal U}}^{\left[ s \right]}}{ \times _r}{{\boldsymbol J}}_1^{\left( r \right)}{ \times _{R + 1}}{\Psi ^{\left( r \right)}}} \right]_{(R + 1)}} = {\Psi ^{\left( r \right)}} \cdot {\left[ {{{{\cal U}}^{\left[ s \right]}}{ \times _r}{{\boldsymbol J}}_1^{\left( r \right)}} \right]_{(R + 1)}}. \tag{A3}\end{equation} 利用${{\cal~A}}{~\times~_n}{{{\boldsymbol~I}}_n}~=~{{\cal~A}}$可得 \begin{equation}{\left[ {{{{\cal U}}^{\left[ s \right]}}{ \times _r}{{\boldsymbol J}}_1^{\left( r \right)}} \right]_{(R + 1)}} = {\left[ {{{{\cal U}}^{\left[ s \right]}}{ \times _1}{{{\boldsymbol I}}_{{M_1}}}{ \times _2} \ldots { \times _r}{{\boldsymbol J}}_1^{\left( r \right)}{ \times _{r + 1}}{{{\boldsymbol I}}_{{M_{r + 1}}}} \ldots { \times _{R + 1}}{{{\boldsymbol I}}_{{M_{R + 1}}}}} \right]_{(R + 1)}}. \tag{A4}\end{equation} 运用张量性质 \begin{equation}{\left[ {{{\cal A}}{ \times _1}{{{\boldsymbol X}}_1}{ \times _2}{{{\boldsymbol X}}_2} \times \ldots { \times _R}{{{\boldsymbol X}}_R}} \right]_{(n)}} = {{{\boldsymbol X}}_n} \cdot {\left[ {{\cal A}} \right]_{(n)}} \cdot \left( {{{{\boldsymbol X}}_{n + 1}} \otimes {{{\boldsymbol X}}_{n + 2}} \cdots \otimes {{{\boldsymbol X}}_R} \otimes {{{\boldsymbol X}}_1} \cdots \otimes {{{\boldsymbol X}}_{n - 1}}} \right). \tag{A5}\end{equation} 式(A4)可以简化为 \begin{equation} {\left[ {{{{\cal U}}^{\left[ s \right]}}{ \times _r}{{J}}_1^{\left( r \right)}} \right]_{(R + 1)}} = {{{I}}_{{M_{R + 1}}}} \cdot {\left[ {{{{\cal U}}^{\left[ s \right]}}} \right]_{(R + 1)}} \cdot {\left( {{{\tilde J}}_1^{\left( r \right)}} \right)^{\rm T}}, \tag{A6} \end{equation} 同理有 \begin{equation} {\left[ {{{{\cal U}}^{\left[ s \right]}}{ \times _r}{{J}}_2^{\left( r \right)}} \right]_{(R + 1)}} = {{{I}}_{{M_{R + 1}}}} \cdot {\left[ {{{{\cal U}}^{\left[ s \right]}}} \right]_{(R + 1)}} \cdot {\left( {{{\tilde J}}_2^{\left( r \right)}} \right)^{\rm T}}. \tag{A7}\end{equation} 将式(A6)和(A7)带入式(A2)中可得 \begin{equation} \hat \Psi _{\rm TE}^{\left( r \right)} = \arg \mathop {\min }\limits_{{\Psi ^{\left( r \right)}}} {\left\| {{\Psi ^{\left( r \right)}} \cdot {{\left[ {{{{\cal U}}^{\left[ s \right]}}} \right]}_{(R + 1)}} \cdot {{\left( {{{\tilde J}}_1^{\left( r \right)}} \right)}^{\rm T}} - {{\left[ {{{{\cal U}}^{\left[ s \right]}}} \right]}_{(R + 1)}} \cdot {{\left( {{{\tilde J}}_1^{\left( r \right)}} \right)}^{\rm T}}} \right\|_{\rm{F}}}, \tag{A8}\end{equation} 此时上式已经转化为矩阵形式, 因此可以直接得到 \begin{equation} \hat \Psi _{\rm TE}^{{{\left( r \right)}^{\rm T}}} = {\left( {{{\tilde J}}_1^{\left( r \right)} \cdot \left[ {{{{\cal U}}^{\left[ s \right]}}} \right]_{(R + 1)}^{\rm T}} \right)^ + } \cdot {{\tilde J}}_2^{\left( r \right)} \cdot \left[ {{{{\cal U}}^{\left[ s \right]}}} \right]_{(R + 1)}^{\rm T}, \tag {A9}\end{equation} 令$R~=~4$, 可得式(45).

Appendix B

根据式(47)和(49)可得 \begin{equation}{{\cal T}{\cal Y}} = {{\cal Y}}{ \times _1}{{\boldsymbol Q}}_{{J_1}}^{\rm H}{ \times _2}{{\boldsymbol Q}}_{{J_2}}^{\rm H}{ \times _3}{{\boldsymbol Q}}_{{J_3}}^{\rm H}{ \times _4}{{\boldsymbol Q}}_{{J_4}}^{\rm H}{ \times _5}{{\boldsymbol Q}}_{2L}^{\rm H} = {{{\cal E}}^{\left[ s \right]}}{ \times _5}{{\boldsymbol E}}_5^{\left[ s \right]}, \tag{B1}\end{equation} 复值张量${{\cal~Y}}$的HOSVD可以表示为 \begin{equation}{{\cal Y}} = {{{\cal U}}^{\left[ s \right]}}{ \times _5}{{\boldsymbol U}}_5^{\left[ s \right]}, \tag{B2}\end{equation} 这里张量${{\cal~Y}}$的子空间${{{\cal~U}}^{\left[~s~\right]}}$的获得与张量${{\cal~X}}$的类似. 将式(B2)代入式(B1), ${{{\cal~E}}^{\left[~s~\right]}}$与${{{\cal~U}}^{\left[~s~\right]}}$的关系可以表示为 \begin{equation}\left( {{{{\cal U}}^{\left[ s \right]}}{ \times _5}{{\boldsymbol U}}_5^{\left[ s \right]}} \right){ \times _1}{{\boldsymbol Q}}_{{J_1}}^{\rm H}{ \times _2}{{\boldsymbol Q}}_{{J_2}}^{\rm H}{ \times _3}{{\boldsymbol Q}}_{{J_3}}^{\rm H}{ \times _4}{{\boldsymbol Q}}_{{J_4}}^{\rm H}{ \times _5}{{\boldsymbol Q}}_{2L}^{\rm H} = {{{\cal E}}^{\left[ s \right]}}{ \times _5}{{\boldsymbol E}}_5^{\left[ s \right]}. \tag{B3}\end{equation} 利用${{\boldsymbol~E}}_5^{{{\left[~s~\right]}^{\rm~H}}}{{\boldsymbol~E}}_5^{\left[~s~\right]}~=~{{{\boldsymbol~I}}_D}$和${{\cal~A}}{~\times~_1}{{{\boldsymbol~X}}_1}{~\times~_1}{{{\boldsymbol~Y}}_1}~=~{{\cal~A}}{~\times~_1}\left(~{{{{\boldsymbol~Y}}_1}~\cdot~{{{\boldsymbol~X}}_1}}~\right)$可得 \begin{equation}{{{\cal U}}^{\left[ s \right]}}{ \times _1}{{\boldsymbol Q}}_{{J_1}}^{\rm H}{ \times _2}{{\boldsymbol Q}}_{{J_2}}^{\rm H}{ \times _3}{{\boldsymbol Q}}_{{J_3}}^{\rm H}{ \times _4}{{\boldsymbol Q}}_{{J_4}}^{\rm H}{ \times _5}\left( {{{\boldsymbol E}}_5^{{{\left[ s \right]}^{\rm H}}} \cdot {{\boldsymbol Q}}_{2L}^{\rm H} \cdot {{\boldsymbol U}}_5^{\left[ s \right]}} \right) = {{{\cal E}}^{\left[ s \right]}}. \tag{B4}\end{equation} 根据${{{\cal~U}}^{\left[~s~\right]}}~=~{{\cal~A}}{~\times~_5}{{{\boldsymbol~T}}_{\rm~TE}}$有 \begin{equation}{{\cal A}}{ \times _1}{{\boldsymbol Q}}_{{J_1}}^{\rm H}{ \times _2}{{\boldsymbol Q}}_{{J_2}}^{\rm H}{ \times _3}{{\boldsymbol Q}}_{{J_3}}^{\rm H}{ \times _4}{{\boldsymbol Q}}_{{J_4}}^{\rm H}{ \times _5}\left( {{{\boldsymbol E}}_5^{{{\left[ s \right]}^{\rm H}}} \cdot {{\boldsymbol Q}}_{2L}^{\rm H} \cdot {{\boldsymbol U}}_5^{\left[ s \right]} \cdot {{{\boldsymbol T}}_{\rm TE}}} \right) = {{{\cal E}}^{\left[ s \right]}}. \tag{B5}\end{equation} 进一步可知 \begin{equation}{{\cal B}}{ \times _5}\left( {{{\boldsymbol E}}_5^{{{\left[ s \right]}^{\rm H}}} \cdot {{\boldsymbol Q}}_{2L}^{\rm H} \cdot {{\boldsymbol U}}_5^{\left[ s \right]} \cdot {{{\boldsymbol T}}_{\rm TE}}} \right) = {{{\cal E}}^{\left[ s \right]}}, \tag{B6}\end{equation} 记${{{\boldsymbol~T}}_{\rm~UTE}}~=~(~{{{\boldsymbol~E}}_5^{{{[~s~]}^{\rm~H}}}~\cdot~{{\boldsymbol~Q}}_{2L}^{\rm~H}~\cdot~{{\boldsymbol~U}}_5^{\left[~s~\right]}~\cdot~{{{\boldsymbol~T}}_{\rm~TE}}}~)$, 很明显由于矩阵${{\boldsymbol~E}}_5^{\left[~s~\right]}$, ${{\boldsymbol~Q}}_{2L}^{\rm~H}$, ${{\boldsymbol~U}}_5^{\left[~s~\right]}$都是列满秩矩阵, 因此${{{\boldsymbol~T}}_{\rm~UTE}}$是非奇异矩阵, 定理1证明完毕.

Appendix C

在这一附录中, 我们讨论各子空间之间的关系. 限于篇幅, 我们仅推导式(54), 同理可得式(53).根据式(48), 截尾核心矩阵${{\cal~S}}_Y^{\left[~s~\right]}$可以计算为 \begin{equation}{{\cal S}}_Y^{\left[ s \right]} \approx {{\cal T}{\cal Y}}{ \times _1}{{\boldsymbol E}}_1^{{{\left[ s \right]}^{\rm H}}}{ \times _2}{{\boldsymbol E}}_2^{{{\left[ s \right]}^{\rm H}}}{ \times _3}{{\boldsymbol E}}_3^{{{\left[ s \right]}^{\rm H}}}{ \times _4}{{\boldsymbol E}}_4^{{{\left[ s \right]}^{\rm H}}}{ \times _5}{{\boldsymbol E}}_5^{{{\left[ s \right]}^{\rm H}}}, \tag{C1}\end{equation} 将式(C1)代入式(49)可得 \begin{equation}{{{\cal E}}^{\left[ s \right]}}={{\cal T}{\cal Y}}{ \times _1}\left( {{{\boldsymbol E}}_1^{\left[ s \right]} \cdot {{\boldsymbol E}}_1^{{{\left[ s \right]}^{\rm H}}}} \right) \ldots { \times _4}\left( {{{\boldsymbol E}}_4^{\left[ s \right]} \cdot {{\boldsymbol E}}_4^{{{\left[ s \right]}^{\rm H}}}} \right){ \times _5}\left( {\sum _5^{{{\left[ s \right]}^{ - 1}}} \cdot {{\boldsymbol E}}_5^{{{\left[ s \right]}^{\rm H}}}} \right), \tag{C2}\end{equation} 这里增加了$\sum~_5^{\left[~s~\right]}$的逆矩阵这一项, 其中$\sum~_5^{\left[~s~\right]}$包含了张量${{\cal~Y}}$的5-模式展开矩阵的$D$个主特征值, 但这并不影响实值子空间${{{\cal~E}}^{\left[~s~\right]}}$. 对上式进行5-模式展开并转, 利用式(A5)可得 \begin{equation}\left[ {{{{\cal E}}^{\left[ s \right]}}} \right]_5^{\rm T}{\rm{ = }}\left( {\left( {{{\boldsymbol E}}_1^{\left[ s \right]} \cdot {{\boldsymbol E}}_1^{{{\left[ s \right]}^{\rm H}}}} \right) \otimes \ldots \otimes \left( {{{\boldsymbol E}}_4^{\left[ s \right]} \cdot {{\boldsymbol E}}_4^{{{\left[ s \right]}^{\rm H}}}} \right)} \right) \cdot \left[ {{{\cal T}{\cal Y}}} \right]_{\left( 5 \right)}^{\rm T} \cdot {{\boldsymbol E}}_5^{{{\left[ s \right]}^ * }} \cdot \sum _5^{{{\left[ s \right]}^{ - 1}}}, \tag{C3}\end{equation} 根据式(33)有 \begin{equation}{{\boldsymbol Y}}{\rm{ = }}\left[ {{{\cal T}{\cal Y}}} \right]_{(5)}^{\rm T} = {{{\boldsymbol Q}}^{\rm H}}\left[ {{\cal Y}} \right]_{(5)}^{\rm T}{{\boldsymbol Q}}_{2L}^ *, \tag{C4}\end{equation} 其中 ${{\boldsymbol~Q}}~=~\left(~{{{{\boldsymbol~Q}}_{{J_1}}}~\otimes~{{{\boldsymbol~Q}}_{{J_2}}}~\otimes~{{{\boldsymbol~Q}}_{{J_3}}}~\otimes~{{{\boldsymbol~Q}}_{{J_4}}}}~\right)$, 又有 \begin{equation}\left[ {{\cal Y}} \right]_{(5)}^{\rm T}{\rm{ = }}\left[ {{{\cal X}}_{(5)}^{\rm T}{ \sqcup _2}\left[ {{{{\cal X}}^ * }{ \times _1}{{{\boldsymbol \Pi }}_{{J_1}}}{ \times _2}{{{\boldsymbol \Pi }}_{{J_2}}}{ \times _3}{{{\boldsymbol \Pi }}_{{J_3}}}{ \times _4}{{{\boldsymbol \Pi }}_{{J_4}}}{ \times _5}{{{\boldsymbol \Pi }}_L}} \right]_{(5)}^{\rm T}} \right]{\rm{ = }}\left[ {{{\cal X}}_{(5)}^{\rm T}{ \sqcup _2}{{\boldsymbol \Pi }}{{\cal X}}_{(5)}^{\rm H}{{{\boldsymbol \Pi }}_L}} \right]{\rm{ = }}\left[ {\begin{array}{*{20}{c}} {{\boldsymbol X}}&{{{\boldsymbol \Pi }}{{{\boldsymbol X}}^ * }{{{\boldsymbol \Pi }}_L}} \end{array}} \right], \tag{C5}\end{equation} 从而可得 \begin{equation}{{\boldsymbol Y}}{\rm{ = }}\left[ {{{\cal T}{\cal Y}}} \right]_{(5)}^{\rm T} = {{{\boldsymbol Q}}^{\rm H}}\left[ {\begin{array}{*{20}{c}} {{\boldsymbol X}}&{{{\boldsymbol \Pi }}{{{\boldsymbol X}}^ * }{{{\boldsymbol \Pi }}_L}} \end{array}} \right]{{\boldsymbol Q}}_{2L}^ *. \tag{C6}\end{equation} 矩阵${\boldsymbol~Y}$的SVD可以表示为 \begin{equation}{{\boldsymbol Y}}{\rm{ = }}\left[ {{{\cal T}{\cal Y}}} \right]_{(5)}^{\rm T} \approx {{{\boldsymbol E}}_s} \cdot {\sum _s} \cdot {{\boldsymbol F}}_s^{\rm H}, \tag{C7}\end{equation} 其中, ${{{\boldsymbol~U}}_s}~\in~{{\mathbb~C}^{J~\times~D}}$, ${\sum~_s}~\in~{{\mathbb~C}^{D~\times~D}}$和${{{\boldsymbol~V}}_s}~\in~{{\mathbb~C}^{L~\times~D}}$. 注意到${\left[~{{{\cal~T}{\cal~Y}}}~\right]_{(5)}}{\rm{~=~}}{{\boldsymbol~E}}_5^{\left[~s~\right]}\sum~_5^{\left[~s~\right]}{{\boldsymbol~F}}_5^{{{\left[~s~\right]}^{\rm~H}}}$, 我们可得 \begin{equation}{{\boldsymbol E}}_5^{{{\left[ s \right]}^ * }} = {{{\boldsymbol F}}_s}, \tag{C8}\end{equation} 因此实值张量子空间与对应的矩阵子空间有如下的关系: \begin{equation}\left[ {{{{\cal E}}^{\left[ s \right]}}} \right]_5^{\rm T}{\rm{ = }}\left( {\left( {{{\boldsymbol E}}_1^{\left[ s \right]} \cdot {{\boldsymbol E}}_1^{{{\left[ s \right]}^{\rm H}}}} \right) \otimes \cdots \otimes \left( {{{\boldsymbol E}}_4^{\left[ s \right]} \cdot {{\boldsymbol E}}_4^{{{\left[ s \right]}^{\rm H}}}} \right)} \right) \cdot {{{\boldsymbol E}}_s}, \tag{C9}\end{equation} 下面讨论实值子空间${{{\boldsymbol~E}}_s}$和复值子空间${{{\boldsymbol~U}}_s}$的关系. 除了式(C7), 实值子空间${{{\boldsymbol~E}}_s}$也能从回波${\boldsymbol~Y}$的协方差矩阵${{{\boldsymbol~R}}_{{\boldsymbol~Y}}}$的特征值分解获得, 其中${{{\boldsymbol~E}}_s}$可以表示为 \begin{equation}{{{\boldsymbol R}}_{{\boldsymbol Y}}} = {{\boldsymbol Y}}{{{\boldsymbol Y}}^{\rm H}}/2L{\rm{ = }}{{{\boldsymbol Q}}^{\rm H}}\left[ {{\cal Y}} \right]_{(5)}^{\rm T}{{\boldsymbol Q}}_{2L}^ * {{\boldsymbol Q}}_{2L}^{\rm T}\left[ {{\cal Y}} \right]_{(5)}^ * {{\boldsymbol Q}} = {{{\boldsymbol Q}}^{\rm H}}\left[ {{\cal Y}} \right]_{(5)}^{\rm T}\left[ {{\cal Y}} \right]_{(5)}^ * {{\boldsymbol Q}}, \tag{C10}\end{equation} 其中 ${{\boldsymbol~\Pi~}}{\rm{~=~}}{{{\boldsymbol~\Pi~}}_{{J_1}}}~\otimes~{{{\boldsymbol~\Pi~}}_{{J_2}}}~\otimes~{{{\boldsymbol~\Pi~}}_{{J_3}}}~\otimes~{{{\boldsymbol~\Pi~}}_{{J_4}}}$. 将式(C5)代入式(C10)可得 \begin{equation}\begin{array}{l} {{{\boldsymbol R}}_{{\boldsymbol Y}}} = {{{\boldsymbol Q}}^{\rm H}}\left[ {{\cal Y}} \right]_{(5)}^{\rm T}\left[ {{\cal Y}} \right]_{(5)}^ * {{\boldsymbol Q}} = {{{\boldsymbol Q}}^{\rm H}}\left[ {\begin{array}{*{20}{c}} {{\boldsymbol X}}&{{{\boldsymbol \Pi }}{{{\boldsymbol X}}^ * }{{{\boldsymbol \Pi }}_L}} \end{array}} \right]{\left[ {\begin{array}{*{20}{c}} {{\boldsymbol X}}&{{{\boldsymbol \Pi }}{{{\boldsymbol X}}^ * }{{{\boldsymbol \Pi }}_L}} \end{array}} \right]^{\rm H}}{{\boldsymbol Q}} \\ = {{\boldsymbol Q}}_J^{\rm H}\left( {{{\boldsymbol X}}{{{\boldsymbol X}}^{\rm H}} + {{\boldsymbol \Pi }}{{{\boldsymbol X}}^ * }{{{\boldsymbol X}}^{\rm T}}{{\boldsymbol \Pi }}} \right){{{\boldsymbol Q}}_J}/2L = {{{\boldsymbol Q}}^{\rm H}}{{{\boldsymbol R}}_{{\boldsymbol X}}}{{\boldsymbol Q}}/2 + {{{\boldsymbol Q}}^{\rm H}}{{\boldsymbol \Pi R}}_{{\boldsymbol X}}^ * {{\boldsymbol \Pi Q}}/2. \end{array} \tag{C11}\end{equation} 利用${{{\boldsymbol~\Pi~}}_p}{{\boldsymbol~Q}}_p^~*~=~{{{\boldsymbol~Q}}_p}$可得 \begin{equation}\begin{array}{l} {{{\boldsymbol Q}}^{\rm H}}{{\boldsymbol \Pi }} = \left( {{{\boldsymbol Q}}_{{J_1}}^{\rm H} \otimes {{\boldsymbol Q}}_{{J_2}}^{\rm H} \otimes {{\boldsymbol Q}}_{{J_3}}^{\rm H} \otimes {{\boldsymbol Q}}_{{J_4}}^{\rm H}} \right)\left( {{{{\boldsymbol \Pi }}_{{J_1}}} \otimes {{{\boldsymbol \Pi }}_{{J_2}}} \otimes {{{\boldsymbol \Pi }}_{{J_3}}} \otimes {{{\boldsymbol \Pi }}_{{J_4}}}} \right) \\ = \left( {{{\boldsymbol Q}}_{{J_1}}^{\rm H}{{{\boldsymbol \Pi }}_{{J_1}}}} \right) \otimes \left( {{{\boldsymbol Q}}_{{J_2}}^{\rm H}{{{\boldsymbol \Pi }}_{{J_2}}}} \right) \otimes \left( {{{\boldsymbol Q}}_{{J_3}}^{\rm H}{{{\boldsymbol \Pi }}_{{J_3}}}} \right) \otimes \left( {{{\boldsymbol Q}}_{{J_4}}^{\rm H}{{{\boldsymbol \Pi }}_{{J_4}}}} \right) \\ = {{\boldsymbol Q}}_{{J_1}}^{\rm T} \otimes {{\boldsymbol Q}}_{{J_1}}^{\rm T} \otimes {{\boldsymbol Q}}_{{J_1}}^{\rm T} \otimes {{\boldsymbol Q}}_{{J_1}}^{\rm T}{\rm{ = }}{{{\boldsymbol Q}}^{\rm T}}. \end{array} \tag{C12}\end{equation} 将式(C12)代入式(C11)可得 \begin{equation}\begin{array}{l} {{{\boldsymbol R}}_{{\boldsymbol Y}}} = {{{\boldsymbol Q}}^{\rm H}}{{{\boldsymbol R}}_{{\boldsymbol X}}}{{{\boldsymbol Q}}_J}/2 + {{{\boldsymbol Q}}^{\rm T}}{{\boldsymbol R}}_{{\boldsymbol X}}^ * {{{\boldsymbol Q}}^ * }/2{{\boldsymbol X}} = {{{\boldsymbol Q}}^{\rm H}}{{{\boldsymbol R}}_{{\boldsymbol X}}}{{\boldsymbol Q}}/2 + {\left( {{{{\boldsymbol Q}}^{\rm H}}{{{\boldsymbol R}}_{{\boldsymbol X}}}{{\boldsymbol Q}}} \right)^ * }/2 \\ = {\mathop{\rm Re}\nolimits} \left\{ {{{{\boldsymbol Q}}^{\rm H}}{{{\boldsymbol R}}_{{\boldsymbol X}}}{{\boldsymbol Q}}} \right\} = {\mathop{\rm Re}\nolimits} \left\{ {{{{\boldsymbol Q}}^{\rm H}}{{{\boldsymbol U}}_s} \cdot {{{\boldsymbol \Lambda }}_s} \cdot {{\left( {{{{\boldsymbol Q}}^{\rm H}}{{{\boldsymbol U}}_s}} \right)}^{\rm H}}} \right\}. \end{array} \tag{C13}\end{equation} 因此式(C3)能够进一步写成 \begin{equation}\left[ {{{{\cal E}}^{\left[ s \right]}}} \right]_5^{\rm T}{\rm{ = }}\left( {{{\boldsymbol E}}_1^{\left[ s \right]} \cdot {{\boldsymbol E}}_1^{{{\left[ s \right]}^{\rm H}}}} \right) \otimes \ldots \otimes \left( {{{\boldsymbol E}}_4^{\left[ s \right]} \cdot {{\boldsymbol E}}_4^{{{\left[ s \right]}^{\rm H}}}} \right) \cdot {{\cal P}}\left\{ {{\mathop{\rm Re}\nolimits} \left\{ {{{{\boldsymbol Q}}^{\rm H}}{{{\boldsymbol U}}_s} \cdot {{{\boldsymbol \Lambda }}_s} \cdot {{\left( {{{{\boldsymbol Q}}^{\rm H}}{{{\boldsymbol U}}_s}} \right)}^{\rm H}}} \right\}} \right\}, \tag{C14}\end{equation} 其中${{\cal~P}}\left\{~{{\boldsymbol~A}}~\right\}$表示选择矩阵$A$的$D$个主特征向量. 从而讨论2证明完毕.


References

[1] Antonik P, Wicks M, Griffiths H, et al. Frequency diverse array radars. In: Proceedings of IEEE Radar Conference, Verona, 2006. 215--217. Google Scholar

[2] Antonik P, Wicks M, Griffiths H, et al. Multi-mission multi-mode waveform diversity. In: Proceedings of IEEE Radar Conference, Verona, 2006. 580--582. Google Scholar

[3] Wang W Q, Shao H Z, Chen H. Frequency diverse array radar: concept, principle and application. J Eletron Inf Tech, 2016, 38: 1000--1011. Google Scholar

[4] Eker T, Demir S, Hizal A. Exploitation of Linear Frequency Modulated Continuous Waveform (LFMCW) for Frequency Diverse Arrays. IEEE Trans Antenn Propag, 2013, 61: 3546-3553 CrossRef ADS Google Scholar

[5] Wang W Q. Frequency Diverse Array Antenna: New Opportunities. IEEE Antenn Propag Mag, 2015, 57: 145-152 CrossRef ADS Google Scholar

[6] Antonik P. An investigation of a frequency diverse array. Dissertation for Ph.D. Degree. London: University College London, 2009. Google Scholar

[7] Secmen M, Demir S, Hizal A, et al. Frequency diverse array antenna with periodic time modulated pattern in range and angle. In: Proceedings of IEEE Radar Conference, Boston, 2007. 427--430. Google Scholar

[8] Wang W Q. Range-Angle Dependent Transmit Beampattern Synthesis for Linear Frequency Diverse Arrays. IEEE Trans Antenn Propag, 2013, 61: 4073-4081 CrossRef ADS Google Scholar

[9] Xu Y, Shi X, Xu J. Range-Angle-Dependent Beamforming of Pulsed Frequency Diverse Array. IEEE Trans Antenn Propag, 2015, 63: 3262-3267 CrossRef ADS Google Scholar

[10] Yao A M, Wu W, Fang D G. Frequency Diverse Array Antenna Using Time-Modulated Optimized Frequency Offset to Obtain Time-Invariant Spatial Fine Focusing Beampattern. IEEE Trans Antenn Propag, 2016, 64: 4434-4446 CrossRef ADS Google Scholar

[11] Khan W, Qureshi I M, Saeed S. Frequency Diverse Array Radar With Logarithmically Increasing Frequency Offset. Antenn Wirel Propag Lett, 2015, 14: 499-502 CrossRef ADS Google Scholar

[12] Shao H, Dai J, Xiong J. Dot-Shaped Range-Angle Beampattern Synthesis for Frequency Diverse Array. Antenn Wirel Propag Lett, 2016, 15: 1703-1706 CrossRef ADS Google Scholar

[13] Gao K, Wang W Q, Chen H. Transmit Beamspace Design for Multi-Carrier Frequency Diverse Array Sensor. IEEE Senss J, 2016, 16: 5709-5714 CrossRef ADS Google Scholar

[14] Wang T Y, Lu X F, Deng L, et al. Bayesian compressive sensing-based sparse imaging for Off-Grid target in frequency diverse MIMO radar. Acta Electron Sin, 2016, 44: 1314--1321. Google Scholar

[15] Farooq J, Temple M, Saville M. Exploiting frequency diverse array processing to improve SAR image resolution. In: Proceedings of IEEE Radar Conference, Rome, 2008. 1--5. Google Scholar

[16] Xu J, Liao G, Zhu S. Deceptive jamming suppression with frequency diverse MIMO radar. Signal Processing, 2015, 113: 9-17 CrossRef Google Scholar

[17] Sammartino P F, Baker C J, Griffiths H D. Frequency Diverse MIMO Techniques for Radar. IEEE Trans Aerosp Electron Syst, 2013, 49: 201-222 CrossRef ADS Google Scholar

[18] Xu J, Liao G, Zhu S. Joint Range and Angle Estimation Using MIMO Radar With Frequency Diverse Array. IEEE Trans Signal Process, 2015, 63: 3396-3410 CrossRef ADS Google Scholar

[19] Wang W Q, So H C. Transmit Subaperturing for Range and Angle Estimation in Frequency Diverse Array Radar. IEEE Trans Signal Process, 2014, 62: 2000-2011 CrossRef ADS Google Scholar

[20] Wang W Q, Shao H C. Range-angle localization of targets by a double-pulse frequency diverse array radar. IEEE J Sel Top Signal Process, 2014, 8: 106-114 CrossRef Google Scholar

[21] Qin S, Zhang Y D, Amin M G. Multi-target localization using frequency diverse coprime arrays with coprime frequency offsets. In: Proceedings of IEEE Radar Conference, Philadelphia, 2016. 1--5. Google Scholar

[22] Wen-Qin Wang , So H C, Huaizong Shao H C. Nonuniform frequency diverse array for range-angle imaging of targets. IEEE Senss J, 2014, 14: 2469-2476 CrossRef ADS Google Scholar

[23] Khan W, Qureshi I M, Basit A, et al. A double pulse MIMO frequency diverse array radar for improved range-angle localization of target. Wirel Pers Commun, 2013, 82: 4073--4081. Google Scholar

[24] Wang W Q. Subarray-based frequency diverse array radar for target range-angle estimation. IEEE Trans Aerosp Electron Syst, 2014, 50: 3057-3067 CrossRef ADS Google Scholar

[25] Xiong J, Cai J, Wang W Q. Decoupled frequency diverse array range-angle-dependent beampattern synthesis using non-linearly increasing frequency offsets. IET Microw Antenn Propag, 2016, 10: 880-884 CrossRef Google Scholar

[26] Wang Y X, Huang G C, Li W. Transmit beampattern design in range and angle domains for MIMO frequency diverse array radar. IEEE Antenn Wirel Propag Lett, 2016, 16: 1003--1006. Google Scholar

[27] Qin S, Zhang Y M D, Amin M G, et al. Frequency diverse coprime arrays with coprime frequency offsets for multi-target localization. IEEE J Sel Topics Signal Process, 2016, 11: 321--335. Google Scholar

[28] Wang W Q, Zhu C. Nested array receiver with time delayers for joint target range and angle estimation. IET Radar Sonar Navig, 2017, 10: 1384--1393. Google Scholar

[29] Huang L, Li X, Gong P C. Frequency diverse array radar for target range-angle estimation. COMPEL, 2016, 35: 1257-1270 CrossRef Google Scholar

[30] Jones A M, Rigling B D. Planar frequency diverse array receiver architecture. In: Proceedings of IEEE Radar Conference, Atlanta, 2012. 145--150. Google Scholar

[31] Li X X, Wang D W, Ma X Y. Three-dimensional target localization and Cramér-Rao bound for two-dimensional OFDM-MIMO radar. Int J Antenn Propag, 2017, 2017: 1--14. Google Scholar

[32] De Lathauwer L, De Moor B, Vandewalle J. A Multilinear Singular Value Decomposition. SIAM J Matrix Anal Appl, 2000, 21: 1253-1278 CrossRef Google Scholar

[33] Sidiropoulos N D, De Lathauwer L, Fu X. Tensor Decomposition for Signal Processing and Machine Learning. IEEE Trans Signal Process, 2017, 65: 3551-3582 CrossRef ADS arXiv Google Scholar

[34] Balda E R, Cheema S A, Steinwandt J, et al. First-order pertubation analysis of low-rank tensor approximations based on the truncated HOSVD. In: Proceedings of the 50th Asilomar conference on signals, systems and computers, Pacific Grove, 2016. 1723--1727. Google Scholar

[35] Haardt M, Roemer F, Del Galdo G. Higher-Order SVD-Based Subspace Estimation to Improve the Parameter Estimation Accuracy in Multidimensional Harmonic Retrieval Problems. IEEE Trans Signal Process, 2008, 56: 3198-3213 CrossRef ADS Google Scholar

[36] Domanov I, De Lathauwer L. Canonical polyadic decomposition of third-order tensors: Relaxed uniqueness conditions and algebraic algorithm. Linear Algebra its Appl, 2017, 513: 342-375 CrossRef Google Scholar

[37] Miron S, Song Y, Brie D. Multilinear direction finding for sensor-array with multiple scales of invariance. IEEE Trans Aerosp Electron Syst, 2015, 51: 2057-2070 CrossRef ADS Google Scholar

[38] Sahnoun S, Comon P. Joint Source Estimation and Localization. IEEE Trans Signal Process, 2015, 63: 2485-2495 CrossRef ADS Google Scholar

[39] Abed-Meraim K, Hua Y. A least-squares approach to joint Schur decomposition. In: Proceedings of IEEE Conference on Acoustics, Speech and Signal Processing, Seattle, 1998. 2541--2544. Google Scholar

[40] Haardt M, Nossek J A. Simultaneous Schur decomposition of several nonsymmetric matrices to achieve automatic pairing in multidimensional harmonic retrieval problems. IEEE Trans Signal Process, 1998, 46: 161-169 CrossRef ADS Google Scholar

  • Figure 1

    (Color online) Basic geometry of the FDS-MIMO radar

  • Figure 2

    (Color online) Beampattern comparison in 4D view. (a) FDS-MIMO radar; (b) traditional MIMO radar

  • Figure 3

    (Color online) Beampattern of FDS-MIMO radar in 4D view. (a) $\left(~{{N_x},{N_z}}\right)~=~\left(~{4,4}\right)$; (b) $\left(~{{N_x},{N_z}}\right)~=~\left(~{4,3}\right)$

  • Figure 4

    (Color online) RMSE vs. SNR under independent signal source. (a) Angle dimension; (b) range dimension

  • Figure 5

    (Color online) RMSE vs. SNR under related signal source. (a) Angle dimension; (b) range dimension

  • Table 1   Unified formulation of ESPRIT-type algorithms
    ESPRIT-type algorithms ${{\boldsymbol~\tilde~H}}_2^{(~r~)}$ ${{{\boldsymbol~F}}_s}$
    Classic ESPRIT ${{{\boldsymbol~I}}_{\Gamma~_1^{(r)}}}~\otimes~{{\boldsymbol~J}}_2^{(~r~)}~\otimes~{{{\boldsymbol~I}}_{\Gamma~_2^{(r)}}}$ ${{{\boldsymbol~U}}_s}$
    Tensor ESPTIT ${{{\boldsymbol~I}}_{\Gamma~_1^{(r)}}}~\otimes~{{\boldsymbol~J}}_2^{(~r~)}~\otimes~{{{\boldsymbol~I}}_{\Gamma~_2^{(r)}}}$ $(~{{{\boldsymbol~U}}_1^{[~s~]}~\cdot~{{\boldsymbol~U}}_1^{{{[~s~]}^{\rm~H}}}}~)~\otimes~~\ldots~~\otimes~(~{{{\boldsymbol~U}}_4^{[~s~]}~\cdot~{{\boldsymbol~U}}_4^{{{[~s~]}^{\rm~H}}}}~)~\cdot~{{{\boldsymbol~U}}_s}$
    Tensor-unitary ESPIRT ${{{\boldsymbol~I}}_{\Gamma~_1^{(r)}}}~\otimes~{\mathop{\rm~Re}\nolimits}~\{~{2Q_{{m_r}}^{\rm~H}{{\boldsymbol~J}}_2^{(~r~)}Q_M^{\rm~H}}~\}~\otimes~{{{\boldsymbol~I}}_{\Gamma~_2^{(r)}}}$ $(~{{{\boldsymbol~E}}_1^{[~s~]}~\cdot~{{\boldsymbol~E}}_1^{{{[~s~]}^{\rm~H}}}}~)~\otimes~~\ldots~~\otimes~(~{{{\boldsymbol~E}}_4^{[~s~]}~\cdot~{{\boldsymbol~E}}_4^{{{[~s~]}^{\rm~H}}}}~)$
    $\cdot~{\cal~P}\{~{{\mathop{\rm~Re}\nolimits}~\{~{{{{\boldsymbol~Q}}^{\rm~H}}{{{\boldsymbol~U}}_s}~\cdot~{{{\boldsymbol~\Lambda~}}_s}~\cdot~{{(~{{{{\boldsymbol~Q}}^{\rm~H}}{{{\boldsymbol~U}}_s}}~)}^{\rm~H}}}~\}}~\}$

Copyright 2019 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有

京ICP备18024590号-1