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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

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

Abstract


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 ${{{\boldsymbol~I}}_{\Gamma~_1^{(r)}}}~\otimes~{\mathop{\rm~Re}\nolimits}~\{~{2Q_{{m_r}}^{\rm~H}{{\boldsymbol~J}}_2^{(~r~)}Q_M^{\rm~H}}~\}~\otimes~{{{\boldsymbol~I}}_{\Gamma~_2^{(r)}}}$ $(~{{{\boldsymbol~E}}_1^{[~s~]}~\cdot~{{\boldsymbol~E}}_1^{{{[~s~]}^{\rm~H}}}}~)~\otimes~~\ldots~~\otimes~(~{{{\boldsymbol~E}}_4^{[~s~]}~\cdot~{{\boldsymbol~E}}_4^{{{[~s~]}^{\rm~H}}}}~)$
    $\cdot~{\cal~P}\{~{{\mathop{\rm~Re}\nolimits}~\{~{{{{\boldsymbol~Q}}^{\rm~H}}{{{\boldsymbol~U}}_s}~\cdot~{{{\boldsymbol~\Lambda~}}_s}~\cdot~{{(~{{{{\boldsymbol~Q}}^{\rm~H}}{{{\boldsymbol~U}}_s}}~)}^{\rm~H}}}~\}}~\}$
qqqq

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