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近年来,高斯干扰背景下的多通道点目标自适应检测技术在机载雷达领域受到越来越多的关注。早在1973年,Brennan等人[1]就开始了多通道信号检测的研究,随后王永良等人[2]也开展了相关研究工作,并首次在机载雷达领域提出空时自适应处理理论。该技术通过在空-时域进行联合滤波,有效地补偿了平台的运动效应,具有出色的杂波抑制性能。在此基础上,以杂波抑制和目标检测为主要目的的空时自适应检测(Space-Time Adaptive Detection, STAD)技术[3]得到快速发展,许多经典的检测方法被不断提出,主要包括广义似然比检测(Generalized Likelihood Ratio Test, GLRT)[4,5]、自适应匹配滤波检测(Adaptive Matched Filter, AMF)[6]、Rao[7,8]检测和Wald[8,9]检测等。特别地,这些方法均可以自适应地调整检测阈值以维持虚警概率(Probability of false alarm,
${{{P}}_{\rm{fa}}}$ )不随外界均匀干扰环境变化,具有恒虚警(Constant False Alarm Rate, CFAR)特性[10]。然而受信号能量起伏、杂波散射等因素的影响,实际应用中均匀辅助数据数量十分有限,由此部分均匀环境(Partially Homogeneous Environment, PHE)下的CFAR检测方法提出,如自适应相干估计器(Adaptive Coherence Estimator, ACE)[11]等。这里部分均匀指待检测数据与辅助数据的干扰协方差矩阵结构相同仅差一个未知的能量比例因子。以上STAD方法需要对高维度的协方差矩阵进行估计和求逆,对辅助数据的需求量大且计算复杂度高。为解决这一问题,许多改进方法被提出。文献[12,13]指出当雷达系统使用空间对称分布的线阵时,得到的干扰协方差矩阵具有先验的斜对称结构,即矩阵的元素关于主对角线共轭对称,关于副对角线对称。该先验知识意味着只需使用原来一半的参数就可以表征未知的干扰协方差矩阵,大大降低了对辅助数据数量的依赖。基于该特性,一系列斜对称检测方法提出。例如,文献[14]基于GLRT准则设计了部分均匀环境下的斜对称GLRT(Persymmetric GLRT, P-GLRT)方法,文献[15]基于两步GLRT准则提出了斜对称ACE (Persymmetric ACE, P-ACE)方法,这些方法均提高了经典STAD在辅助数据数量受限时的检测性能。针对辅助数据数量不足时的检测问题,基于降秩技术[16]、信号子空间变换技术[17]、Krylov子空间技术[18]以及利用杂波谱的对称特性、贝叶斯模型等其他先验知识[19-26]的STAD方法也可以作为降低辅助数据数量的有效手段。
此外,现有的STAD技术大都基于理想的目标采样模型,即采样点恰好落在目标回波匹配滤波输出的峰值处,不存在任何能量泄漏。然而,当一个点目标跨坐于两个连续的距离单元时,采样时刻很难完全对准峰值位置。为弥补能量泄漏损失,中国科学院声学研究所郝程鹏课题组与意大利教授Orlando等人[27,28]基于目标能量泄漏采样模型提出了一系列修正方法,主要包括均匀环境下的修正GLRT (Modified GLRT, M-GLRT)、修正AMF (Modified AMF, M-AMF)以及部分均匀环境下的定位GLRT (GLRT with Localization Capabilities for PHE, GLRT-LC-PHE)、修正ACE (Modified ACE, M-ACE)等检测方法,这些检测方法除了提高目标检测能力外,还兼具对待检测单元内目标距离的估计能力。文献[29-31]在这些方法的基础上联合使用干扰协方差矩阵的斜对称特性,提出了部分均匀环境下的斜对称修正AMF (Persymmetric Modified AMF for PHE, PM-AMF-PHE)和改进的PM-AMF-PHE (Modified PM-AMF-PHE, MPM-AMF-PHE)等检测方法,进一步提高了小样本辅助数据下的检测性能。
本文在MPM-AMF-PHE[31]方法的工作基础上,在部分均匀高斯干扰背景下提出了一种适用于空间对称线阵的斜对称修正GLRT (PM-GLRT for PHE, PM-GLRT-PHE)检测方法。首先给出了与MPM-AMF-PHE相似的离散接收信号模型,即对目标信号采用目标能量泄漏采样模型,干扰信号采用斜对称先验结构。在检测方法的设计阶段,相比MPM-AMF-PHE方法,PM-GLRT-PHE方法在一步GLRT准则下联合使用待检测数据和辅助数据得到部分均匀环境中能量比例因子
$\gamma $ 的数值解法,代替了MPM-AMF-PHE方法中仅使用待检测数据的次优估计结果,进一步提高了参数的估计精度以及对回波数据的利用率,进而获得更优的目标检测性能。最后,将所有未知参数的最大似然估计(Maximum Likelihood Estimate, MLE)结果代替理论值代入GLRT检测统计量中,得到最终的PM-GLRT-PHE检测方法。仿真结果显示,PM-GLRT-PHE的${{{P}}_{\rm{fa}}}$ 相对于背景参数的变化并不敏感,具有CFAR特性,并且相比于其同类型的检测方法在辅助数据数量受限时有着稳健的检测性能优势。 -
本节对接收回波的离散时间信号模型进行介绍。考虑一个由
${N_a}$ 个阵元组成的均匀线列阵,发射的时域脉冲数为${N_p}$ ,每个阵元的发射信号可以简单写为$${\rm{Re}}\left\{ {A\sum\limits_{i = 1}^{{N_p}} {p(t - (i - 1)T){{\rm{e}}^{{\rm{j}}2\pi {f_c}t}}} } \right\},\,\,t \in [0,{N_p}T)$$ (1) 其中,
${\rm{Re}}()$ 表示取实部运算,$A$ 表示复振幅,$p(t)$ 表示矩形脉冲,$T$ 为脉冲重复间隔,${f_c}$ 为载波频率。接收信号经过下变频到基带、匹配滤波、采样等一系列预处理后,得到${{N}} = {N_a}{N_p}$ 维复矢量。假设${N_p} = 1$ ,则单脉冲下第$i$ 个距离单元的回波数据可描述为$$\begin{split} & {H_0}:{{{z}}_i} = {{{n}}_i} \in {C^{N \times 1}} \\ & {H_1}:{{{z}}_i} = {{{s}}_i} + {{{n}}_i} \in {C^{N \times 1}} \end{split} $$ (2) 其中,
$C$ 表示复数域,${H_0}$ 和${H_1}$ 分别表示无目标和有目标的假设,${{{n}}_i}$ 表示仅包含白噪声和杂波的复高斯干扰矢量,${{{s}}_i}$ 表示目标信号矢量。由文献[31]可知,当存在能量泄漏时,目标能量会泄漏到相邻距离单元中,由此得到由邻近距离单元组成的目标能量泄漏采样模型$$ {{{s}}_i} = \left\{ {\begin{aligned} & {\alpha {\chi _p}( - \varepsilon ,f){{v}}, \,\;\;\quad i = l} \\ & {\alpha {\chi _p}({T_p} - \varepsilon ,f){{v}}, \; i = l + 1} \\ & {0,\,\qquad\qquad\qquad\;\; i \ne l,l + 1} \end{aligned}} \right. $$ (3) 其中,
$\alpha $ 表示目标信号的复幅值因子,${\chi _p}()$ 表示复模糊度函数,${T_p}$ 为信号脉宽,$\varepsilon \in [ - \,{T_p}/2,\,{T_p}/2]$ 为导致目标能量泄漏的剩余时间延迟,$f$ 为多普勒频移,${{v}} = \dfrac{1}{{\sqrt {{N_a}} }} \times {[1\, {{\rm{e}}^{{\rm{j}}2\pi {\nu _s}}}\, ···\, {{\rm{e}}^{{\rm{j}}2\pi ({N_a} - 1){\nu _s}}}]^{\rm{T}}}$ 表示归一化空域导向矢量,${\nu _s}$ 表示归一化空间频率,${()^{\rm{T}}}$ 表示转置运算,$l$ 为待检测单元序号。 -
基于离散时间模型,目标能量泄漏采样模型下的目标检测问题可表述为以下2元假设检验
$$\left. \begin{aligned} &{{H_0}:\left\{ {\begin{aligned} &{{{{z}}_i} = {{{n}}_i},\,\,i = l - 1,l,l + 1} \\ & {{{{z}}_k} = {{{n}}_k},k = l + 2,···,l + K + 1} \end{aligned}} \right.} \\ & {{H_1}:\left\{ \begin{aligned} & - {T_p}/2 \le \varepsilon < 0, \\ &\quad{{{z}}_i} =\left\{ {\begin{aligned} &{\alpha {\chi _p}( - {T_p} - \varepsilon ,f){{v}} + {{{n}}_i},i = l - 1} \\ & {\alpha {\chi _p}( - \varepsilon ,f){{v}} + {{{n}}_i},\qquad i = l} \\ &{{{{n}}_i},\qquad\qquad\qquad\qquad\;\; i = l + 1} \end{aligned}} \right. \\ & 0 \le \varepsilon \le {T_p}/2, \\ &\quad{{{z}}_i} =\left\{ {\begin{aligned} &{{{{n}}_i},\qquad\qquad\qquad\qquad i = l - 1} \\ & {\alpha {\chi _p}( - \varepsilon ,f){{v}} + {{{n}}_i},\quad\; i = l} \\ &{\alpha {\chi _p}({T_p} - \varepsilon ,f){{v}} + {{{n}}_i},i = l + 1} \end{aligned}} \right. \\ & {{{{z}}_k} = {{{n}}_k},k = l + 2,···,l + K + 1} \end{aligned} \right.} \end{aligned}\right\}\!\!\! $$ (4) 其中,
${{{z}}_i} \in {C^{N \times 1}}$ 表示待检测数据矢量。${{{n}}_k}$ ,${{{z}}_k} \in {C^{N \times 1}}$ 表示从待检测数据相邻的距离单元收集到的$K$ 个独立同分布的辅助数据,为保证干扰协方差矩阵的非奇异性,需满足$K \ge N$ 。${{{n}}_i}$ 为待检测数据中的干扰成分,它与${{{z}}_k}$ 之间统计独立,且均服从零均值的多元复高斯分布,即${{{n}}_i} \sim {\rm{C}}{{\rm{N}}_N}({\bf{0}},\gamma {{M}})$ 和${{{z}}_k} \sim {\rm{C}}{{\rm{N}}_N} ({\bf{0}},{{M}})$ ,唯一不同的是两者协方差矩阵相差一个未知的能量比例因子$\gamma $ ($\gamma > 0$ )。此外在使用空间对称线阵的主动雷达检测系统中,其干扰协方差矩阵
${{M}}$ 和导向矢量${{v}}$ 均具有斜对称特性。即满足${{M}} = {{{J}}_N}{{{M}}^ * }{{{J}}_N},\,\,{{v}} = {{{J}}_N}{{{v}}^ * }$ ,其中${()^ * }$ 表示共轭运算,${{{J}}_N}$ 为N维置换矩阵,有$$ {{{J}}_N}(i,j) = \left\{ {\begin{aligned} &{1,\,\;\;\;\;i + j = N + 1}\\ &{0,\,\;\;\;\;{\text{其他}}} \end{aligned}} \right. $$ (5) -
为求解式(6)中的假设检验问题,下面采用GLRT准则设计斜对称泄漏模型①下的自适应检测方法PM-GLRT-PHE。MPM-AMF-PHE方法单独使用待检测数据估计未知参数
$\gamma $ ,得到的参数估计精度有限,限制了该方法的检测性能。为进一步提升辅助数据数量受限情况下的目标检测性能,PM-GLRT-PHE将联合待检测数据和辅助数据实现检测统计量的推导和未知参数${{M}}$ ,${\alpha}$ 和$\gamma $ 的MLE,以获得更高的检测性能。为便于推导,令${{{Z}}_K} = \left[ {{{{z}}_1}\,{{{z}}_2}\cdots\,{{{z}}_K}} \right] \in {C^{N \times K}}$ 表示辅助数据矩阵,${{Z}} = [ {{{z}}_{l - 1}}\, {{{z}}_l}\,{{{z}}_{l + 1}}\,{{{Z}}_K} ] \in {C^{N \times (3 + K)}}$ 表示联合数据矩阵。GLRT准则下的检测表达式为$$\frac{{\mathop {\max}\limits_{\varepsilon ,{{\alpha}} ,\gamma ,{{M}}} {f_1}({{Z}};\varepsilon ,{{\alpha}} ,\gamma ,{{M}})}}{{\mathop {\max }\limits_{\gamma ,{{M}}} {f_0}({{Z}};\gamma ,{{M}})}}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{H_1}} \\ > \end{array}} \\ {\begin{array}{*{20}{c}} < \\ {{H_0}} \end{array}} \end{array}\eta $$ (6) 其中,
$\eta $ 表示一定${{{P}}_{\rm{fa}}}$ 下的检测阈值,根据文献[31]可知,${f_j}({{Z}};j\varepsilon ,j{{\alpha}} ,\gamma ,{{M}})$ 表示待检测数据和辅助数据在$H{}_j,j = 0,1$ 假设下的概率密度函数$$ \begin{split} {f_j}({{Z}};j\varepsilon ,j{{\alpha}} ,\gamma ,{{M}}) =\,& {\left( {\frac{{{\gamma ^{ - 3N/(K + 3)}}}}{{{\pi ^N}{\rm{det}}({{M}})}}} \right)^{K + 3}} \exp\{ - {\rm{tr}}\\ & \left[ {{{{M}}^{ - 1}}\!\left(\! {\frac{1}{\gamma }{{F}}(j{{\alpha}} ){{{F}}^{\rm{H}}}(j{{\alpha}} ) \!+\! {{S}}} \!\right)} \right] \}\\ \end{split} $$ (7) 其中,
${\rm{det}}()$ ,${\rm{tr}}()$ 和${()^{\rm{H}}}$ 分别表示矩阵的行列式、求迹和共轭转置运算,${{S}} = \left( {{{{Z}}_K}{{Z}}_K^{\rm{H}} + {{{J}}_N}{{({{{Z}}_K}{{Z}}_K^{\rm{H}})}^ * }{{{J}}_N}} \right)\!\Big/2$ ,${{F}}(j{{\alpha}} ) = {{X}} - j{{v\alpha}} {{{D}}^{\rm{T}}}$ ,${{X}} = [ {{{z}}_{{e_{_{l - 1}}}}}\,{{{z}}_{{o_{_{l - 1}}}}}\,{{{z}}_{{e_{_l}}}}\,{{{z}}_{{o_{_l}}}}\,{{{z}}_{{e_{_{l + 1}}}}} \,{{{z}}_{{o_{_{l + 1}}}}} ]$ ,${{{z}}_{{e_i}}} = ({{{z}}_i} + {{{J}}_N}{{z}}_i^ * )/2$ ,$\,{{{z}}_{{o_i}}} = ({{{z}}_i} - {{{J}}_N}{{z}}_i^ * )/2$ ,$i = l - 1,\,l,\,l + 1$ ,${{\alpha}} = \left[ {\alpha \,\,{\alpha ^ * }} \right]$ ,且有$$\begin{split} & {{D}} = \\ & \left\{ {\begin{aligned} & {\left[\!\! {\begin{array}{*{20}{l}} {{\chi _p}({t_1},f)\,\,\,\,{\chi _p}({t_1},f)\;\,\,\,\,\,{\chi _p}({t_2},f)\,\,\,\,{\chi _p}({t_2},f)\;\,\,\,\,\,\,0\;\,\,0} \\ {\chi _p^ * ({t_1},f)\,\, - \chi _p^ * ({t_1},f)\,\,\chi _p^ * ({t_2},f)\,\, - \chi _p^ * ({t_2},f)\,\,0\,\,\,0} \end{array}}\!\! \right]^{\rm{T}}}, \\ \\ & \quad\;\, - {T_p}/2\,\, \le \,\,\varepsilon \,\, < \,0 \\ & {\left[\!\! {\begin{array}{*{20}{l}} {0\,\,\,0\,\,\,{\chi _p}({t_2},f)\,\,\,\;\,{\chi _p}({t_2},f)\;\,\,\,{\chi _p}({t_3},f)\;\,\,\,\,\;{\chi _p}({t_3},f)} \\ {0\,\,\,0\,\,\,\chi _p^ * ({t_2},f)\,\, - \chi _p^ * ({t_2},f)\,\,\chi _p^ * ({t_3},f)\,\, - \chi _p^ * ({t_3},f)} \end{array}} \!\!\!\right]^{\rm{T}}}, \\ & \quad\; 0\,\, \le \,\,\varepsilon \,\, \le \,{T_p}/2 \end{aligned}} \right. \end{split}$$ (8) 其中,
${t_1} = - {T_p} - \varepsilon ,\,\,{t_2} = - \varepsilon ,\,\,{t_3} = {T_p} - \varepsilon $ 为剩余时间延迟。根据文献[31],参数
${{{M}}_j}$ 在$H{}_j,j = 0,1$ 假设下的MLE结果为$${\tilde {{M}}_j} = \frac{1}{{K + 3}}\left( {\frac{1}{{{\gamma _j}}}{{F}}(j{{\alpha}} ){{{F}}^{\rm{H}}}(j{{\alpha}} ) + {{S}}} \right)$$ (9) 其中,
${\gamma _j},j = 0,1$ 表示$H{}_j$ 假设下的能量比例因子。将式(9)代入式(6),此时的GLRT检测等价为
$$ \mathop {\max}\limits_{\varepsilon ,\gamma ,{{\alpha}} } \frac{{\gamma _0^{\tfrac{{3N}}{{3 + K}}}\det \left( {\dfrac{1}{{{\gamma _0}}}{{X}}{{{X}}^{\rm{H}}} + {{S}}} \right)}}{{\gamma _1^{\tfrac{{3N}}{{3 + K}}}\det\left( {\dfrac{1}{{{\gamma _1}}}{{F}}({{\alpha}} ){{{F}}^{\rm{H}}}({{\alpha}} ) + {{S}}} \right)}}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{H_1}} \\ > \end{array}} \\ {\begin{array}{*{20}{c}} < \\ {{H_0}} \end{array}} \end{array}\eta $$ (10) 由文献[29]可知,对
${{\alpha}} $ 的MLE为$$ \begin{split} \tilde {{\alpha}} =& \mathop {\arg \min }\limits_{{\alpha}} \gamma _1^{\tfrac{{3N}}{{3 + K}}}\det\left( {\frac{1}{{{\gamma _1}}}{{F}}({{\alpha}} ){{{F}}^{\rm{H}}}({{\alpha}} ) + {{S}}} \right) \\ =& \mathop {\arg \min }\limits_{{\alpha}} \gamma _1^{\tfrac{{3N}}{{3 + K}} - 6}\det\left( {{S}} \right) \\ & \cdot\det\left(\! {{\gamma _1}{{{I}}_6} + {{{F}}^{\rm{H}}}({{\alpha}} ){{{S}}^{ - 1/2}}\left( {{{P}}_s^ \bot \!+\! {{{P}}_s}} \right){{{S}}^{ - 1/2}}{{F}}\!({{\alpha}} )} \!\right) \end{split} $$ (11) 其中,
${{{P}}_s} = {{{S}}^{ - 1/2}}{{v}}{{{v}}^{\rm{H}}}{{{S}}^{ - 1/2}}/({{{v}}^{\rm{H}}}{{{S}}^{ - 1}}{{v}})$ 表示由${{{S}}^{ - 1/2}}{{v}}$ 张成的子空间的投影矩阵,${{P}}_s^ \bot = {{{I}}_N} - {{{P}}_s}$ 表示矩阵${{{P}}_s}$ 的正交补,${{{I}}_6}$ 和${{{I}}_N}$ 分别表示6维和$N$ 维的单位矩阵。为了化简式(11),将第2个行列式展开后的前两项写作$$ \begin{split} & {\gamma _1}{{{I}}_6} + {{{F}}^{\rm{H}}}({{\alpha}} ){{{S}}^{ - 1/2}}{{P}}_s^ \bot {{{S}}^{ - 1/2}}{{F}}({{\alpha}} ) \\ & \quad = {\gamma _1}{{{I}}_6} + {\left( {{{X}} - {{v}}{{\alpha}} {{{D}}^{\rm{T}}}} \right)^{\rm{H}}}{{{S}}^{ - 1/2}}{{P}}_s^ \bot {{{S}}^{ - 1/2}}\\ & \qquad \cdot \left( {{{X}} - {{v\alpha}} {{{D}}^{\rm{T}}}} \right) \\ & \quad = {\gamma _1}{{{I}}_6} + {{{X}}^{\rm{H}}}{{{S}}^{ - 1/2}}{{P}}_s^ \bot {{{S}}^{ - 1/2}}{{X}} - {{{X}}^{\rm{H}}}{{{S}}^{ - 1/2}}{{P}}_s^ \bot \\ & \quad\quad \cdot{{{S}}^{ - 1/2}}{{v\alpha}} {{{D}}^{\rm{T}}} - {{{D}}^*}{{{\alpha}} ^{\rm{H}}}{\left( {{{{S}}^{ - 1/2}}{{v}}} \right)^{\rm{H}}}{{P}}_s^ \bot {{{S}}^{ - 1/2}}{{X}}\\ & \quad\quad + {{{D}}^*}{{{\alpha}} ^{\rm{H}}}{\left( {{{{S}}^{ - 1/2}}{{v}}} \right)^{\rm{H}}}{{P}}_s^ \bot {{{S}}^{ - 1/2}}{{v\alpha}} {{{D}}^{\rm{T}}}\\[-17pt] \end{split} $$ (12) 在式(11)中已经提到,矩阵
${{P}}_s^ \bot $ 为由${{{S}}^{ - 1/2}}{{v}}$ 张成子空间的正交投影矩阵,因此${{P}}_s^ \bot {{{S}}^{ - 1/2}}{{v}} = {\bf{0}}$ 。则式(12)可进一步化简得$$ \begin{split} & {\gamma _1}{{{I}}_6} + {{{F}}^{\rm{H}}}({{\alpha}} ){{{S}}^{ - 1/2}}{{P}}_s^ \bot {{{S}}^{ - 1/2}}{{F}}({{\alpha}} ) \\ & \quad = {\gamma _1}{{{I}}_6} + {{{X}}^{\rm{H}}}{{{S}}^{ - 1/2}}{{P}}_s^ \bot {{{S}}^{ - 1/2}}{{X}} \end{split} $$ (13) 其中,令
${{Q}} = {\gamma _1}{{{I}}_6} + {{{X}}^{\rm{H}}}{{{S}}^{ - 1/2}}{{P}}_s^ \bot {{{S}}^{ - 1/2}}{{X}}$ ,由此,式(11)等号右边可进一步表示为$$ \begin{split} & \gamma _1^{\tfrac{{3N}}{{3 + K}}}\det\left( {\frac{1}{{{\gamma _1}}}{{F}}({{\alpha}} ){{{F}}^{\rm{H}}}({{\alpha}} ) + {{S}}} \right) \\ &\quad = \gamma _1^{\tfrac{{3N}}{{3 + K}} - 6}\det\left( {{S}} \right)\\ & \qquad \cdot\det \left( {{{Q}} + {{{F}}^{\rm{H}}}({{\alpha}} ){{{S}}^{ - 1/2}}{{{P}}_s}{{{S}}^{ - 1/2}}{{F}}({{\alpha}} )} \right) \\ &\quad = \gamma _1^{\tfrac{{3N}}{{3 + K}} - 6}\det\left( {{S}} \right)\det\left( {{Q}} \right) \\ & \qquad \cdot\left( {1 + \frac{{{{{v}}^{\rm{H}}}{{{S}}^{ - 1}}{{F}}({{\alpha}} ) {{{Q}}^{ - 1}}{{{F}}^{\rm{H}}}({{\alpha}} ){{{S}}^{ - 1}}{{v}}}}{{{{{v}}^{\rm{H}}}{{{S}}^{ - 1}}{{v}}}}} \right) \!\!\!\!\! \end{split} $$ (14) 参考文献[29,31]可得
$ {{\alpha}} $ 的估计结果为$$ {{\tilde{{\alpha}}}} = \frac{{{{{v}}^{\rm{H}}}{{{S}}^{ - 1}}{{X}}{{{Q}}^{ - 1}}{{{D}}^ * }{{\left[ {{{{D}}^{\rm{T}}}{{{Q}}^{ - 1}}{{{D}}^ * }} \right]}^{ - 1}}}}{{{{{v}}^{\rm{H}}}{{{S}}^{ - 1}}{{v}}}} $$ (15) 将式(15)代入式(14),式(14)等价为
$$ \begin{split} & \gamma _1^{\tfrac{{3N}}{{3 + K}}}\det\left( {\frac{1}{{{\gamma _1}}}{{F}}(\tilde {{\alpha}} ){{{F}}^{\rm{H}}}(\tilde {{\alpha}} ) + {{S}}} \right) \\ & \quad = \gamma _1^{\tfrac{{3N}}{{3 + K}} - 6}\det\left( {{S}} \right)\det \left( {{Q}} \right)\left( {1 + {{B}} - {{C}}} \right) \\ & \qquad \cdot\propto \gamma _1^{\tfrac{{3N}}{{3 + K}} - 6}\det\left( {{Q}} \right)\left( {1 + {{B}} - {{C}}} \right) \end{split} $$ (16) 其中,
$$ {{B}} = \frac{{{{{v}}^{\rm{H}}}{{{S}}^{ - 1}}{{X}}{{{Q}}^{ - 1}}{{{X}}^{\rm{H}}}{{{S}}^{ - 1}}{{v}}}}{{{{{v}}^{\rm{H}}}{{{S}}^{ - 1}}{{v}}}} $$ (17) $$ \begin{split} & {{C}} \\ & = \frac{{{{{v}}^{\rm{H}}}{{{S}}^{ - 1}}{{X}}{{{Q}}^{ - 1}}{{{D}}^ * }{{\left(\! {{{{D}}^{\rm{T}}}{{{Q}}^{ - 1}}{{{D}}^ * }} \!\right)}^{ - 1}}\!{{{D}}^{\rm{T}}}\!{{{Q}}^{ - 1}}{{{X}}^{\rm{H}}}{{{S}}^{ - 1}}{{v}}}}{{{{{v}}^{\rm{H}}}{{{S}}^{ - 1}}{{v}}}} \end{split} $$ (18) 接下来求
$H{}_1$ 假设下$\gamma $ 的MLE,得到$${\tilde \gamma _1} = \mathop {\arg \min }\limits_{{\gamma _1}} \gamma _1^{\tfrac{{3N}}{{3 + K}} - 6}\det\left( {{Q}} \right)\left( {1 + {{B}} - {{C}}} \right)$$ (19) 对
${{Q}}$ 特征分解得${{Q}} = {{U}}\left( {{\gamma _1}{{{I}}_6} + {{\varLambda}} } \right){{{U}}^{\rm{H}}}$ ,其中${{U}} \in {C^{6 \times 6}}$ 为酉矩阵,${{\varLambda}} $ 为特征值为${\lambda _1},{\lambda _2},···,{\lambda _6}$ 的对角阵。将特征分解代入式(19)中$$ \begin{split} & \gamma _1^{\tfrac{{3N}}{{3 + K}} - 6}\det\left( {{Q}} \right)\left( {1 + {{B}} - {{C}}} \right) \\ &\quad = \gamma _1^{\tfrac{{3N}}{{3 + K}} - 6}\det\left( {{{U}}\left( {{\gamma _1}{{{I}}_6} + {{\varLambda}} } \right){{{U}}^{\rm{H}}}} \right)\\ & \qquad \cdot \left( {1 + {{E}} - {{G}}} \right) \end{split} $$ (20) 其中,
${{E}}{\rm{ = }}\dfrac{{{{{V}}^{\rm{H}}}{{V}}}}{{{\gamma _1}{{{I}}_6} + {{\varLambda}} }}$ ,${{G}}{\rm{ = }}\dfrac{{{{{V}}^{\rm{H}}}{{\left( {{\gamma _1}{{{I}}_6} + {{\varLambda}} } \right)}^{ - 1}}{{{WW}}^{\rm{H}}}{{\left( {{\gamma _1}{{{I}}_6} + {{\varLambda}} } \right)}^{ - 1}}{{V}}}}{{{{{W}}^{\rm{H}}}{{\left( {{\gamma _1}{{{I}}_6} + {{\varLambda}} } \right)}^{ - 1}}{{W}}}}$ ,${{W}}{\rm{ = }} {{{U}}^{\rm{H}}}{{{D}}^ * } = {\left[ {\begin{array}{*{20}{c}} {{x_1}\,\cdots\,{x_6}} \\ {{y_1}\,\cdots\,{y_6}} \end{array}} \right]^{\rm{T}}} \in {C^{6 \times 2}}$ ,${{V}}{\rm{ = }}\dfrac{{{{{U}}^{\rm{H}}}{{{X}}^{\rm{H}}}{{{S}}^{ - 1}}{{v}}}}{{\sqrt {{{{v}}^{\rm{H}}}{{{S}}^{ - 1}}{{v}}} }} = {\left[ {{v_1}\,{v_2}\cdots\,{v_6}} \right]^{\rm{T}}} \in {C^{6 \times 1}}$ 。将式(20)进一步化简得
$$ \begin{split} \!\! {h_1}({\gamma _1}) =\,& \gamma _1^{\tfrac{{3N}}{{3 + K}} - 6}\prod\limits_{i = 1}^6 ({\gamma _1} + {\lambda _i})\\ & \cdot \left[ {1 + e - \frac{{(qb - gd)m + ( - qc + ga)n}}{{ab - cd}}} \right]\!\!\!\!\! \end{split} $$ (21) 其中,
$a = \displaystyle\sum\limits_{i = 1}^6 {\frac{{{{\left| {{x_i}} \right|}^2}}}{{{\gamma _1} + {\lambda _i}}}} $ ,$b = \displaystyle\sum\limits_{i = 1}^6 {\frac{{{{\left| {{y_i}} \right|}^2}}}{{{\gamma _1} + {\lambda _i}}}} $ ,$c = \displaystyle\sum\limits_{i = 1}^6 {\frac{{x_i^ * {y_i}}}{{{\gamma _1} + {\lambda _i}}}} $ ,$d = {c^ * }$ ,$e = \displaystyle\sum\limits_{i = 1}^6 {\frac{{{{\left| {{v_i}} \right|}^2}}}{{{\gamma _1} + {\lambda _i}}}} $ ,$q = \displaystyle\sum\limits_{i = 1}^6 {\frac{{v_i^ * {x_i}}}{{{\gamma _1} + {\lambda _i}}}} $ ,$g = \displaystyle\sum\limits_{i = 1}^6 {\frac{{v_i^ * {y_i}}}{{{\gamma _1} + {\lambda _i}}}} $ ,$m = {q^ * }$ ,$n = {g^ * }$ ,$\left| {\;} \right|$ 表示求绝对值。通过对式(21)求关于
${\gamma _1}$ 的1阶导数并置0,可得到估计值${\tilde \gamma _1}$ 。需要说明的是,${\tilde \gamma _1}$ 未给出解析解的形式,因此需要采用数值方法求解,例如可以采用fsolve函数求解此非线性方程。基于式(10),在
$H{}_0$ 假设下对${\gamma _0}$ 的MLE为$$ {\tilde \gamma _0} = \mathop {\arg \min }\limits_{{\gamma _0}} \gamma _0^{\tfrac{{3N}}{{3 + K}}}\det \left( {\frac{1}{{{\gamma _0}}}{{X}}{{{X}}^{\rm{H}}} + {{S}}} \right) $$ (22) 化简得到
$$ \begin{split} \quad & \gamma _0^{\tfrac{{3N}}{{3 + K}}}\det \left( {\frac{1}{{{\gamma _0}}}{{{XX}}^{\rm{H}}} + {{S}}} \right) \\ & \quad = \gamma _0^{\tfrac{{3N}}{{3 + K}}}\det\left( {{S}} \right)\det \left( {\frac{1}{{{\gamma _0}}}{{{X}}^{\rm{H}}}{{{S}}^{ - 1}}{{X}} + {{{I}}_6}} \right) \\ & \quad = \gamma _0^{\tfrac{{3N}}{{3 + K}} - 6}\det \left( {{S}} \right)\det \left( {{\gamma _0}{{{I}}_6} + {{{X}}^{\rm{H}}}{{{S}}^{ - 1}}{{X}}} \right) \\ & \quad \propto \gamma _0^{\tfrac{{3N}}{{3 + K}} - 6}\det \left( {{\gamma _0}{{{I}}_6} + {{{X}}^{\rm{H}}}{{{S}}^{ - 1}}{{X}}} \right) \end{split} $$ (23) 对
${{{X}}^{\rm{H}}}{{{S}}^{ - 1}}{{X}}$ 进行特征分解,得${{{X}}^{\rm{H}}}{{{S}}^{ - 1}}{{X}} = {{{U}}_0}{{{\varLambda}} _0}{{U}}_0^{\rm{H}}$ 。其中${{{U}}_0} \in {C^{6 \times 6}}$ 为酉矩阵,${{{\varLambda}} _0}$ 为特征值为${\lambda _{0,1}},{\lambda _{0,2}},···,{\lambda _{0,6}}$ 的对角阵。将其代入式(23),化简得到$${h_0}({\gamma _0}) = \gamma _0^{\tfrac{{3N}}{{3 + K}} - 6}\prod\limits_{i = 1}^6 {({\gamma _0} + {\lambda _{0,i}})} $$ (24) 同样对式(24)求关于
${\gamma _0}$ 的1阶导数并置0,通过采用数值方法可以获得估计值${\tilde \gamma _0}$ 。将式(15)、式(21)和式(24)得到的估计结果代替式(10)中的真实值,最终得部分均匀环境下的PM-GLRT-PHE检测方法
$$ \mathop {\max }\limits_\varepsilon \frac{{\tilde \gamma _0^{\tfrac{{3N}}{{3 + K}}}\det \left( {\dfrac{1}{{{{\tilde \gamma }_0}}}{{{XX}}^{\rm{H}}} + {{S}}} \right)}}{{\tilde \gamma _1^{\tfrac{{3N}}{{3 + K}}}\det \left( {\dfrac{1}{{{{\tilde \gamma }_1}}}{{F}}(\tilde {{\alpha}} ){{{F}}^{\rm{H}}}(\tilde {{\alpha}} ) + {{S}}} \right)}}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{H_1}} \\ > \end{array}} \\ {\begin{array}{*{20}{c}} < \\ {{H_0}} \end{array}} \end{array}\eta $$ (25) 需要说明的是,由于文中未能给出对剩余时延
$\varepsilon $ 估计的解析解,进而采用格搜索方法进行估计。格搜索精度定义为$\varDelta = {{{T}}_p}/(2{N_\varepsilon })$ ,其中$2{N_\varepsilon }$ 为均匀分布在$[ - \,{T_p}/2,\,{T_p}/2]$ 上的取值。由估计结果$\tilde \varepsilon $ 可以得到目标距离估计为$\left( {{t_{\min }} + l{T_p} + \tilde \varepsilon } \right){{{c_0}}}/{2}$ ,其中${t_{\min }}$ 为采样初始时刻,${c_0} = 3 \times {10^8}$ m/s为电磁波传播速度,且估计结果$\tilde \varepsilon $ 的精确度最终体现为待检测距离单元内的目标距离估计均方根误差上,即${\delta _{\rm{rms}}} = \sqrt {\displaystyle\sum\nolimits_{i = 1}^{{n_t}} {{{\left( {(\tilde \varepsilon - \varepsilon ) \times {{10}^{ - 6}} \times {c_0}/2} \right)}^2}} /{n_t}} $ ,${n_t}$ 为仿真次数。 -
下面采用蒙特卡罗方法分析PM-GLRT-PHE在部分均匀环境下的CFAR特性以及目标检测和距离估计性能。假设系统采用等间距的均匀线列阵,阵元数
${N_a} = N = 12$ ,信号脉宽${T_p}$ =$0.2\,\,{\rm{μs}}$ ,目标的波达角度为0°,多普勒频移$f$ =0,${N_\varepsilon } = 5$ ,白噪声能量${\sigma ^2}$ =1,杂波噪声比(Clutter-to-Noise Ratio, CNR)为30 dB。最后,干扰协方差矩阵建模为${{M}} = {{{I}}_N} + {\rm{CNR}}{{{M}}_c}$ ,其中${{{M}}_c}(i,j) = {\rho ^{\left| {i - j} \right|}}$ 为基于指数相关复高斯模型的杂波协方差矩阵,$\rho = 0.9$ 为一步滞后系数,信号杂波噪声比(Signal-to-Clutter-plus-Noise Ratio, SCNR)${\rm{SCNR}} = {\left| {{\alpha}} \right|^2}{{{v}}^{\rm{H}}}{{{M}}^{ - 1}}{{v}}/\gamma$ 。 -
由于以上推导中未能给出检测方法关于
${P_{\rm{fa}}}$ 的解析表达式,为验证PM-GLRT-PHE的CFAR特性,本节给出了${P_{\rm{fa}}}$ 随着背景参数,即能量比例因子$\gamma $ 和干扰协方差矩阵M的变化趋势,通过数值变化的平稳程度来体现${P_{\rm{fa}}}$ 关于这两个参数的鲁棒性。图1给出了PM-GLRT-PHE的${P_{\rm{fa}}}$ 随$\gamma $ 的变化曲线,假设辅助数据数量受限,即$K = N + 1$ ,预设${P_{\rm{fa}}} = {10^{ - 3}}$ 。由图中所示,当$\gamma $ 由2增大至16时,${P_{\rm{fa}}}$ 始终稳定在${10^{ - 3}}$ 左右,这说明${P_{\rm{fa}}}$ 独立于$\gamma $ ,即检测方法关于$\gamma $ 是CFAR的。图 1 PM-GLRT-PHE的
${P_{\rm{fa}}}$ 随$\gamma $ 的变化曲线Figure 1.
${P_{\rm{fa}}}$ against$\gamma $ for the PM-GLRT-PHE为验证M与
${P_{\rm{fa}}}$ 之间的CFAR关系,表1通过固定$\gamma $ 不变,加入不同数量和波达角度的类噪声干扰产生5种不同的仿真情景,以计算不同M值下检测方法PM-GLRT-PHE的阈值$\eta $ 。其中$\eta $ 的仿真次数为$100/{P_{\rm{fa}}}$ ,$K = N + 1$ ,$\gamma = 2$ ,${P_{\rm{fa}}} = {10^{ - 3}}$ ,干扰噪声比为30 dB。从表1可见,在不同仿真场景下PM-GLRT-PHE的$\eta $ 值均稳定在2附近。由于$\eta $ 与${P_{\rm{fa}}}$ 呈一一对应关系,因而可以说明检测方法的${P_{\rm{fa}}}$ 不会随着${{M}}$ 而改变,即该方法关于${{M}}$ 是CFAR的。此外,为进一步证明PM-GLRT-PHE相对于${{M}}$ 的CFAR特性,图2给出了${P_{\rm{fa}}}$ 随杂波协方差矩阵${{{M}}_c}$ 中参数$\rho $ 的变化曲线,假设$K = N + 1$ ,$\gamma = 2$ ,预设${P_{\rm{fa}}} = {10^{ - 3}}$ 。结果表明,当$\rho $ 由0.1增大至0.9时,${P_{\rm{fa}}}$
Modified Generalized Likelihood Ratio Test Detection Based on a Symmetrically Spaced Linear Array in Partially Homogeneous Environments
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摘要: 针对部分均匀高斯干扰环境下的点目标检测问题,该文基于广义似然比准则(GLRT)提出一种适用于空间对称线阵的修正GLRT检测方法。考虑到采样时存在的目标能量泄漏,在接收信号建模时采用目标能量泄漏采样模型弥补泄漏损失,并基于干扰协方差矩阵的斜对称结构降低对辅助数据的需求,最终联合待检测数据和辅助数据实现未知参数的估计,得到兼具有良好目标检测和距离估计性能的斜对称修正GLRT检测方法。仿真结果表明,该方法不仅在部分均匀环境下具有恒虚警特性,而且在辅助数据数量受限时,相比其同类型的检测方法具有1 dB以上的检测性能优势。Abstract: To address the problem of detecting point-like targets in a partially homogeneous Gaussian cluttered environment, we developed a modified Generalized Likelihood Ratio Test (GLRT) detection method based on a symmetrically spaced linear array that relies on a GLRT design criterion. Considering the target energy spillover during sampling, we use a spillover model of the target energy to decrease spillover loss. To establish the discrete-time signal mode, we use a persymmetric structure of the disturbance covariance matrix to reduce the requirement for auxiliary signals. Lastly, we estimate all of the unknown parameters based on a consideration of both primary and secondary data to derive the persymmetric modified GLRT detector, which has good target detection and range estimation performance. The performance assessment shows that the proposed method not only performs as a constant false-alarm-rate receiver in partially homogeneous environments but also guarantees superior detection performance relative to that of its competitors. In sample-starved environments, compared with other detection methods of the same type, it realizes a detection performance advantage greater than 1 dB.
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表 1 不同场景下PM-GLRT-PHE的阈值
Table 1. Thresholds of PM-GLRT-PHE in different cases
仿真场景 干扰数量 干扰波达角度 阈值$\eta $ 场景1 0 0 2.0098 场景2 1 5° 2.0484 场景3 2 5°, 10° 2.0188 场景4 3 –15°, 5°, 15° 1.9873 场景5 4 –20°, –10°, 5°, 20° 2.0308 -
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