Multi-rank Range-spread Target Detection Method for Space/Time Symmetric Array Radar under Non-Gaussian Clutter Background
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摘要: 针对多通道阵列雷达从实际杂波中检测目标场景,该文提出了一种面向多通道阵列雷达非高斯杂波背景的多秩距离扩展目标检测方法。首先,利用秩大于1的子空间矩阵和相应距离单元的坐标向量,建立了多秩距离扩展目标模型;然后,利用雷达接收单元空间或时间中心对称探测场景下杂波协方差矩阵的反对称结构信息,通过酉变换,采取广义似然比、Rao、Wald检验准则,构建待解参数小样本估计策略,设计了面向非高斯杂波背景的多秩距离扩展目标检测方法。最后,通过理论推导证明了所提检测方法相对于杂波协方差矩阵具有恒虚警特性。基于仿真数据和实测数据的实验结果表明,所提检测方法能够保证对杂波协方差矩阵具有恒虚警特性,此外,相较于现有检测方法,所提检测方法提升了小训练支持的目标检测性能,并且在导向矢量失配条件下,有效地改善目标检测的稳健性。Abstract: This study proposes a multi-rank range-spread target detection method for multi-channel array radar under a non-Gaussian clutter background. The method aims to detect the target from real clutter using the multi-channel array radar. First, a multi-rank range-spread target model was formulated using a subspace matrix with a rank greater than one and the coordinate vectors of corresponding range bins. Then, by exploiting the persymmetric structure information of the clutter covariance matrix under the detection scenario, wherein the radar receiver units were central symmetric in space or time, a small sample estimation strategy for the parameters to be solved through the unitary transformation was constructed. Further, a non-Gaussian clutter background multi-rank range-spread target detection method was designed based on the generalized likelihood ratio, Rao, and Wald tests. Finally, a theoretical derivation proved that the proposed detection method has the constant false alarm rate property. The experimental results based on both the simulated and measured data showed that the proposed detection method can ensure the constant false alarm rate property of the clutter covariance matrix. Additionally, compared with the existing detection methods, the proposed detection method improves the target detection performance under small sample support. Besides, the proposed detection method effectively improves the robustness of target detection under the condition of steering vector mismatch.
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Key words:
- Adaptive detection /
- Range-spread target /
- Multi-rank subspace /
- Persymmetry /
- Non-Gaussian
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1. 引言
多通道阵列雷达具备探测范围广、机动性好和生存能力强等优点,在机载、星载等运动平台系统得到广泛应用[1-3]。对于多通道阵列雷达检测目标问题,传统的自适应检测方法是假设均匀高斯杂波背景,即训练数据与待检测单元杂波具有相同统计特性且服从高斯分布,再进行自适应检验判决[4]。然而,随着雷达分辨率的提高及电磁环境复杂性的增强,杂波表现出大拖尾的非高斯特性[5-7]。非高斯杂波的局部特征满足中心极限定理,呈现出复合高斯尖峰,可用复合高斯过程刻画,表征为一个慢变纹理分量与一个快变散斑的乘积。因此,研究阵列雷达非高斯杂波背景下目标检测极为必要。
为解决非高斯杂波背景下点目标检测问题,国内外学者提出了一系列算法[8-19],例如基于广义似然比(Generalized Likelihood Ratio Test, GLRT)的检测方法[8]、基于Rao与Wald准则的检测方法[9],以及相应的性能分析、降维、降秩等拓展方法[10-15]。雷达带宽增大得到高距离分辨率,目标占据多个距离单元,使得点目标检测方法不再适用,需要研究合适的距离扩展目标检测方法[20-28]。文献[20]在非高斯杂波的目标检测场景下,采用GLRT准则,提出了恒虚警距离扩展目标检测算法。在此基础上,文献[21]考虑了Rao与Wald准则,推导了非高斯杂波背景下的距离扩展目标检测器。文献[22]提出了基于动态阈值的距离目标检测器,该方法能根据实际观测值动态调整检测阈值,进而提高检测性能。文献[23]推导了杂波分组距离扩展目标检测器,该方法的核心是在复合高斯杂波满足局部均匀的背景下,建立了广义杂波分组模型进而实现GLRT检测。
需要指出,上述研究成果为提升多通道阵列雷达中距离扩展目标检测的性能奠定了基础,但是,这些工作都是针对目标信号的秩为一的情况,秩一目标模型虽较为简便,但对目标信号失配[29,30]缺乏稳健性。传统自适应检测假定目标的导向矢量是确定且唯一的,则将其导向矢量列数设置为1,即目标为秩一信号。但是,实际目标检测场景中常常存在阵列误差和波形变化,不可避免地产生目标信号失配的情况,此时导向矢量存在不确定性,无法将其表示为一个单列的向量。为了克服导向矢量失配带来的检测性能的急剧下降,采用一个多秩子空间对导向矢量进行约束[31]。涉及子空间类自适应检测的研究有子空间GLRT[32]、子空间自适应匹配滤波器(Adaptive Matched Filter, AMF)[33]、子空间Rao检测器[33]等。
在多通道阵列雷达自适应检测的研究中,选择充足训练数据是非常重要的步骤。根据RMB准则[34],所选取的训练样本数至少大于系统维数的两倍才能获得较好的检测性能,这对于实际非高斯杂波背景下的距离扩展目标检测要求较为严苛。这是由于实际中平台抖动、快变的地形\海面环境、高分辨目标距离扩展导致的目标污染等因素,使得可用的训练样本数据较为有限,进而难以获得较高的检测性能。多通道阵列雷达接收单元的空间或时间中心对称探测场景下,杂波协方差矩阵的反对称结构信息为改善目标检测性能提供了又一维的信息[35-38]。反对称结构信息,指协方差矩阵关于其主对角线共轭对称且关于其副对角线置换对称,该结构信息意味着仅需使用原本所需一半的参数就可表征未知的杂波协方差矩阵,大大降低了对训练数据数量的依赖。
受以上研究启发,本文提出了一种面向多通道阵列雷达非高斯杂波背景的多秩距离扩展目标检测方法。本方法建立了多秩距离扩展目标模型,采取GLRT, Rao, Wald检验准则,通过酉变换构建待解参数小样本估计策略,旨在挖掘和充分利用阵列雷达接收单元空间或时间中心对称的杂波结构信息,设计了面向非高斯杂波背景的多秩距离扩展目标检测方法,并理论证明了所提检测方法相对于杂波协方差矩阵具有恒虚警率(Constant False Alarm Rate, CFAR)特性。基于仿真数据和实测数据的实验均证明了所提方法能提升训练数据不足的检测性能、改善检测的稳健性。
2. 问题描述
考虑N个阵元的均匀接收阵列,假定目标的距离扩展范围为L个距离单元,第
l 个距离单元的接收回波信号表示为xl∈CN×1 ,则[x1,x2,⋯,xl,⋯,xL] 构成待检测数据。选取待检测数据邻近距离单元的K个杂波数据˜xk∈CN×1,k=1,2,⋯,K 作为训练样本,且K≥N 以保证杂波协方差矩阵的非奇异性。然后,xl 内的目标信号分量建模为多秩模型Hal ,其中H∈CN×q 表示距离扩展目标所属的子空间,q 表示子空间的秩,al∈Cq×1 表示第l 个距离单元的目标信号在子空间矩阵H下对应的坐标向量。多秩距离扩展目标模型包含了多个自由度,可将不精确的目标导向矢量包含在H张成的子空间内。因此,相应的目标检测问题可表述为在待测数据里判断目标存在与否的二元假设检验:{H0:xl=nl,˜xk=nkH1:xl=Hal+nl,˜xk=nkl=1,2,⋯,L,k=1,2,⋯,K (1) 其中,
nl 和nk 分别表示第l 个待测数据分量及第k 个训练数据里的杂波信号。实际中,随着雷达带宽的增加以及电磁环境复杂性的增强,非高斯杂波的局部特征满足中心极限定理,呈现出复合高斯尖峰。因此,我们将杂波建模为复合高斯过程,用球不变随机向量来刻画:nl=√τlg,l=1,2,⋯,Lnk=√τkg,k=1,2,⋯,K} (2) 其中,
g 为复合高斯杂波的散斑分量,用均值为零,协方差矩阵为M的N维随机向量刻画,记作g∼CN(0,M) ,τl 和τk 为在空间和时间上慢变的纹理分量,代表杂波的局部功率水平,服从伽马分布:f(τ)=βaΓ(a)τa−1exp(−βτ) (3) 其中,
a 为形状参数,β 为尺度参数,Γ(⋅) 为伽马函数。需要注意的是,纹理分量与散斑分量是相互独立的。3. 检测方法设计
为求解式(1)中的假设检验问题,下面分别采用GLRT, Rao和Wald准则设计空/时对称阵列雷达非高斯杂波背景下多秩距离扩展目标检测方法,所设计的对应检测器简称为PMR-GLRT (Persymmetric Multi-Rank GLRT), PMR-Rao (Persymmetric Multi-Rank Rao)和PMR-Wald (Persymmetric Multi-Rank Wald)。为提升训练数据不足情况下的检测性能,我们将联合待检测数据和训练样本实现检测统计量的推导、利用雷达接收单元空间或时间中心对称探测场景下杂波协方差矩阵的反对称结构信息、构建多个未知参数估计策略。为推导方便,我们令
A=[aT1,aT2,⋯,aTL]T∈CqL×1 表示所有距离单元的坐标构成的向量,T=[τ1,τ2,⋯,τL]T 表示所有距离单元的慢变纹理构成的向量,X=[xT1,xT2,⋯,xTL]T 表示所有距离单元的待测数据构成的向量。3.1 PMR-GLRT检测器推导
对于所构建的假设检验问题(1),GLRT准则下的目标检测表达式可描述为
max (4) 其中,
{\rm{d}}\bar \tau = {\rm{d}}{\tau _1}{\rm{d}}{\tau _2}\cdots {\rm{d}}{\tau _L} ,\lambda 表示相应的检测阈值,{f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{A}},{\boldsymbol{T}}) 和{f_0}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{T}}) 分别表示X在{{\text{H}}_1} 和{{\text{H}}_0} 假设下的概率密度函数(Probability Density Function, PDF):\begin{split} & {f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{A}},{\boldsymbol{T}}) = \prod\limits_{l = 1}^L \frac{1}{{{\pi ^N}\left| {\boldsymbol{M}} \right|\tau _l^N}}\\ & \quad \cdot\exp \left\{ { - \frac{{{\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{x}}_l} - {\boldsymbol{H}}{{\boldsymbol{a}}_l}{\text{)(}}{{\boldsymbol{x}}_l} - {\boldsymbol{H}}{{\boldsymbol{a}}_l}{{\text{)}}^{\text{H}}}\right]}}{{{\tau _l}}}} \right\} \end{split} (5) \begin{split} {f_0}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{T}}) =& \prod\limits_{l = 1}^L \frac{1}{{{\pi ^N}\left| {\boldsymbol{M}} \right|\tau _l^N}}\\ & \cdot\exp \left\{ { - \frac{{{\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{x}}_l}{\boldsymbol{x}}_l^{\text{H}})\right]}}{{{\tau _l}}}} \right\} \end{split} (6) 其中,
{\text{tr}}(\cdot ) {\text{}} 代表求矩阵的迹,{(\cdot )^{ - 1}} 代表求矩阵的逆;{( \cdot )^{\text{H}}} 代表求矩阵或矢量的共轭转置;\left| { \cdot } \right| 代表求矩阵的行列式。在雷达接收单元空间或时间中心对称探测场景下,杂波协方差矩阵具有反对称结构,即{\boldsymbol{M}} = {\boldsymbol{J}}{{\boldsymbol{M}}^*}{\boldsymbol{J}} ,其中{( \cdot )^{\text{*}}} 表示共轭操作,{\boldsymbol{J}} 表示置换矩阵,具体为一个反对角线上的元素为1,其余元素为0的N \times N 维矩阵,可以表示为{\boldsymbol{J}}{\text{ = }}{\left[ {\begin{array}{*{20}{c}} 0&{}&1 \\ {}& {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} &{} \\ 1&{}&0 \end{array}} \right]_{N \times N}} (7) 于是,我们利用置换矩阵
{\boldsymbol{J}} 对雷达回波数据进行反对称变换,其本质是协方差矩阵的酉变换,将式(5)和式(6)中求矩阵迹的部分表示:\left. \begin{split} & {\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{x}}_l} - {\boldsymbol{H}}{{\boldsymbol{a}}_l}{\text{)(}}{{\boldsymbol{x}}_l} - {\boldsymbol{H}}{{\boldsymbol{a}}_l}{{\text{)}}^{\text{H}}}\right] \\ & \quad = {\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{{\boldsymbol{a}}_{pl}}{\text{)(}}{{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{{\boldsymbol{a}}_{pl}}{{\text{)}}^{\text{H}}}\right] \\ & {\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{x}}_l}{\boldsymbol{x}}_l^{\text{H}})\right] = {\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{\text{)}}\right] \end{split} \right\} (8) 其中,
{{\boldsymbol{X}}_{pl}} = [{{\boldsymbol{x}}_{el}},\;{{\boldsymbol{x}}_{ol}}] ,{{\boldsymbol{x}}_{el}} = ({{\boldsymbol{x}}_l} + {\boldsymbol{Jx}}_l^*)/2 ,{{\boldsymbol{x}}_{ol}} = ({{\boldsymbol{x}}_l} - {\boldsymbol{Jx}}_l^*)/2 ,{{\boldsymbol{a}}_{pl}} = [{{\boldsymbol{a}}_{el}},\;{{\boldsymbol{a}}_{ol}}] ,{{\boldsymbol{a}}_{el}} = ({{\boldsymbol{a}}_l} + {\boldsymbol{a}}_l^*)/2 ,{{\boldsymbol{a}}_{ol}} = ({{\boldsymbol{a}}_l} - {\boldsymbol{a}}_l^*)/2 。基于式(8),式(5)和式(6)可分别变为\begin{split} &{f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{A}},{\boldsymbol{T}}) = \frac{1}{{{\pi ^{NL}}{{\left| {\boldsymbol{M}} \right|}^L}\displaystyle\prod\limits_{l = 1}^L {\tau _l^N} }}\\ & \quad\cdot \exp \left\{ { - \sum\limits_{l = 1}^L {\frac{{{\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{{\boldsymbol{a}}_{pl}}{\text{)(}}{{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{{\boldsymbol{a}}_{pl}}{{\text{)}}^{\text{H}}}\right]}}{{{\tau _l}}}} } \right\} \end{split} (9) {f_0}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{T}}) = \frac{1}{{{\pi ^{NL}}{{\left| {\boldsymbol{M}} \right|}^L}\displaystyle\prod\limits_{l = 1}^L {\tau _l^N} }}\ \exp \left\{ { - \sum\limits_{l = 1}^L {\frac{{{\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}\left({{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}\right)\right]}}{{{\tau _l}}}} } \right\} (10) 回到我们的待解检测问题(4),最大化式(4)等价于最大化X在
{{\text{H}}_1} 假设下的PDF。因此,通过最大化{f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{A}},{\boldsymbol{T}}) 可求得A中每个元素的最大似然估计(Maximum Likelihood Estimate, MLE):{\boldsymbol{\hat a}}_{pl}^{} = {\left({{\boldsymbol{H}}^{\text{H}}}{{\boldsymbol{M}}^{ - 1}}{\boldsymbol{H}}\right)^{ - 1}}{{\boldsymbol{H}}^{\text{H}}}{{\boldsymbol{M}}^{ - 1}}{{\boldsymbol{X}}_{pl}},\;\; l = 1,2, \cdots ,L (11) 则
{{\hat {\boldsymbol{A}}}} = {[{\boldsymbol{\hat a}}_{p1}^{\text{T}},\;\;{\boldsymbol{\hat a}}_{p2}^{\text{T}},\;\; \cdots ,\;\;{\boldsymbol{\hat a}}_{pL}^{\text{T}}]^{\text{T}}} 。然后,将估计所得的{{\hat {\boldsymbol{A}}}} 代入式(9)得到{f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{{\hat {\boldsymbol{A}}}},{\boldsymbol{T}}) 。检测规则(4)的分子部分可以具体表示为
\begin{split} \int {{f_1}\left( {{\boldsymbol{X}}|{\boldsymbol{M}},{{\hat {\boldsymbol{A}}}},{\boldsymbol{T}}} \right)f\left( {{\tau _1}} \right)\, \cdots f\left( {{\tau _L}} \right){\rm{d}}\bar \tau } =& \int \prod\limits_{l = 1}^L \frac{{{\beta ^a}\tau _l^{a - 1 - N}}}{{{\pi ^N}\left| {\boldsymbol{M}} \right|\Gamma \left( a \right)}}\\ & \cdot\exp \left\{ - \frac{{{\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}({{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{{\hat {\boldsymbol{a}}}_{pl}}){{({{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{{\hat {\boldsymbol{a}}}_{pl}})}^{\text{H}}}\right]}}{{{\tau _l}}} - \beta {\tau _l} \right\} {\rm{d}}\bar \tau \end{split} (12) 令
t = \beta {\tau _l} ,z = 2\sqrt {\beta {\text{tr}}[{{\boldsymbol{M}}^{ - 1}} ({{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{{\hat {\boldsymbol{a}}}_{pl}}){{({{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{{\hat {\boldsymbol{a}}}_{pl}})}^{\text{H}}}]} ,v = N - a ,根据第2类修正贝塞尔函数{K_v}\left( z \right) = 1/2 \cdot {\left( {z/2} \right)^v}\displaystyle\int_0^\infty {{t^{ - \left( {v + 1} \right)}}\exp ( - t - {z^2}/4t)} {\rm{d}}t ,对式(12)的第l 项进行化简,可得\begin{split} & \displaystyle \int \frac{{\beta }^{a}{\tau }_{l}^{a-1-N}}{{\pi }^{N}\left|{\boldsymbol{M}}\right|\Gamma \left(a\right)} \mathrm{exp}\left\{-\frac{\text{tr}\left[{{\boldsymbol{M}}}^{-1}({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{\boldsymbol{a}}}_{pl}){({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{\boldsymbol{a}}}_{pl})}^{\text{H}}\right]}{{\tau }_{l}}-\beta {\tau }_{l}\right\}{\rm{d}}{\tau }_{l}\\ & =\frac{2{\beta }^{\textstyle\frac{{{a}}+N}{2}}{\left\{\text{tr}\left[{{\boldsymbol{M}}}^{-1}({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{{\boldsymbol{a}}}}_{pl}){({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{\boldsymbol{a}}}_{pl})}^{\text{H}}\right]\right\}}^{\textstyle\frac{{{a}}-{{N}}}{2}}}{{\pi }^{N}\left|{\boldsymbol{M}}\right|\Gamma \left({{a}}\right)} {K}_{N-{{a}}}\left(2\sqrt{\beta \text{tr}\left[{{\boldsymbol{M}}}^{-1}({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{{\boldsymbol{a}}}}_{pl}){({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{\boldsymbol{a}}}_{pl})}^{\text{H}}\right]}\right)\\ & \propto \frac{{\left\{\text{tr}\left[{{\boldsymbol{M}}}^{-1}({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{{\boldsymbol{a}}}}_{pl}){({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{{\boldsymbol{a}}}}_{pl})}^{\text{H}}\right]\right\}}^{\textstyle\frac{{{a}}-N}{2}}}{\left|{\boldsymbol{M}}\right|} {K}_{{{N}}-{{a}}}\left(2\sqrt{\beta \text{tr}\left[{{\boldsymbol{M}}}^{-1}({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{{\boldsymbol{a}}}}_{pl}){({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{{\boldsymbol{a}}}}_{pl})}^{\text{H}}\right]}\right)\\[-21pt] \end{split} (13) 则检测规则(4)的分子部分与下式呈正比:
\begin{split} {\displaystyle \int {f}_{1}\left({\boldsymbol{X}}|{\boldsymbol{M}},{{\hat {\boldsymbol{A}}}},{\boldsymbol{T}}\right)f\left({\tau }_{1}\right)\cdots f\left({\tau }_{L}\right){\rm{d}}\overline{\tau }} \propto & {\displaystyle \prod _{l=1}^{L}\frac{{\left\{\text{tr}\left[{{\boldsymbol{M}}}^{-1}({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{{\boldsymbol{a}}}}_{pl}){({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{{\boldsymbol{a}}}}_{pl})}^{\text{H}}\right]\right\}}^{\textstyle\frac{{{a}}-N}{2}}}{\left|{\boldsymbol{M}}\right|}}\\ & \cdot {K}_{N-{{a}}}\left(2\sqrt{\beta \text{tr}\left[{{\boldsymbol{M}}}^{-1}({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{{\boldsymbol{a}}}}_{pl}){({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{{\boldsymbol{a}}}}_{pl})}^{\text{H}}\right]}\right) \end{split} (14) 同理,令
z = 2\sqrt {\beta {\text{tr[}}{{\boldsymbol{M}}^{ - 1}}({{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}){\text{]}}} ,我们可以推导出:{\displaystyle \int {f}_{0}\left({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{T}}\right)f\left({\tau }_{1}\right)\cdots f\left({\tau }_{L}\right){\rm{d}}\overline{\tau }} \propto {\displaystyle \prod _{l=1}^{L}\frac{{\left\{\text{tr}\left[{{\boldsymbol{M}}}^{-1}({{\boldsymbol{X}}}_{pl}{{\boldsymbol{X}}}_{pl}^{\text{H}})\right]\right\}}^{\textstyle\frac{{{a}}-N}{2}}}{\left|{\boldsymbol{M}}\right|}} {K}_{N-{{a}}}\left(2\sqrt{\beta \text{tr}\left[{{\boldsymbol{M}}}^{-1}({{\boldsymbol{X}}}_{pl}{{\boldsymbol{X}}}_{pl}^{\text{H}})\right]}\right) (15) 将式(14)和式(15)代入检测规则(4),可得
{\displaystyle \prod _{l=1}^{L}{\left\{\frac{\text{tr}[{{\boldsymbol{M}}}^{-1}({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{{\boldsymbol{a}}}}_{pl}){({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{{\boldsymbol{a}}}}_{pl})}^{\text{H}}]}{\text{tr[}{{\boldsymbol{M}}}^{-1}({{\boldsymbol{X}}}_{pl}{{\boldsymbol{X}}}_{pl}^{\text{H}})\text{]}}\right\}}^{\textstyle\frac{{{a}}-N}{2}}} \frac{{K}_{N-{{a}}}\left(2\sqrt{\beta \text{tr}[{{\boldsymbol{M}}}^{-1}({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{{\boldsymbol{a}}}}_{pl}){({{\boldsymbol{X}}}_{pl}-{\boldsymbol{H}}{\hat{{\boldsymbol{a}}}}_{pl})}^{\text{H}}]}\right)}{{K}_{N-{{a}}}\left(2\sqrt{\beta \text{tr[}{{\boldsymbol{M}}}^{-1}({{\boldsymbol{X}}}_{pl}{{\boldsymbol{X}}}_{pl}^{\text{H}})\text{]}}\right)}\\ \begin{array}{c}{\text{H}}_{1}\\ > \\ < \\ {\text{H}}_{0}\end{array}\lambda (16) 将式(11)所示的
{\boldsymbol{\hat a}}_{pl}^{} 代入式(16)的{\text{tr}}[{{\boldsymbol{M}}^{ - 1}}({{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{\hat {\boldsymbol{a}}_{pl}}){({{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{\hat {\boldsymbol{a}}_{pl}})^{\text{H}}}] 中,可以将其改写为\begin{split} {\text{tr}}[{{\boldsymbol{M}}^{ - 1}}({{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{\hat {\boldsymbol{a}}_{pl}}){({{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{\hat {\boldsymbol{a}}_{pl}})^{\text{H}}}] &= {\text{ tr}}[{{\boldsymbol{M}}^{ - 1}}({{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{\hat {\boldsymbol{a}}_{pl}})({\boldsymbol{X}}_{pl}^{\text{H}} - \hat {\boldsymbol{a}}_{pl}^{\text{H}}{{\boldsymbol{H}}^{\text{H}}})] \\ &= {\text{tr}}[{{\boldsymbol{M}}^{ - 1}}({{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}} - {{\boldsymbol{X}}_{pl}}\hat {\boldsymbol{a}}_{pl}^{\text{H}}{{\boldsymbol{H}}^{\text{H}}} - {\boldsymbol{H}}{\hat {\boldsymbol{a}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}} + {\boldsymbol{H}}{\hat {\boldsymbol{a}}_{pl}}\hat {\boldsymbol{a}}_{pl}^{\text{H}}{{\boldsymbol{H}}^{\text{H}}})] \\ & = {\text{tr}}[{{\boldsymbol{M}}^{ - 1}}({{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}} - {{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{{\boldsymbol{M}}^{ - 1}}{\boldsymbol{H}} {\left( {{{\boldsymbol{H}}^{\text{H}}}{{\boldsymbol{M}}^{ - 1}}{\boldsymbol{H}}} \right)^{ - 1}}{{\boldsymbol{H}}^{\text{H}}})] \end{split} (17) 令
{{\boldsymbol{P}}_{\boldsymbol{H}}} = {{\boldsymbol{M}}^{ - 1}}{\boldsymbol{H}}{({{\boldsymbol{H}}^{\text{H}}}{{\boldsymbol{M}}^{ - 1}}{\boldsymbol{H}})^{ - 1}}{{\boldsymbol{H}}^{\text{H}}} 指代式(17)中H的投影矩阵,可以将式(16)进一步化简,得\displaystyle \prod _{l=1}^{L}\left\{\frac{\text{tr}\left[{{\boldsymbol{M}}}^{-1}\text{(}{{\boldsymbol{X}}}_{pl}{{\boldsymbol{X}}}_{pl}^{\text{H}}-{{\boldsymbol{X}}}_{pl}{{\boldsymbol{X}}}_{pl}^{\text{H}}{{\boldsymbol{P}}}_{{\boldsymbol{H}}})\right]}{\text{tr}\left[{{\boldsymbol{M}}}^{-1}\text{(}{{\boldsymbol{X}}}_{pl}{{\boldsymbol{X}}}_{pl}^{\text{H}})\right]}\right\}^{\textstyle\frac{{{a}}-N}{2}} \frac{{K}_{N-a}\left(2\sqrt{\beta \text{tr}\left[{{\boldsymbol{M}}}^{-1}\text{(}{{\boldsymbol{X}}}_{pl}{{\boldsymbol{X}}}_{pl}^{\text{H}}-{{\boldsymbol{X}}}_{pl}{{\boldsymbol{X}}}_{pl}^{\text{H}}{{\boldsymbol{P}}}_{{\boldsymbol{H}}}\text{)}\right]}\right)}{{K}_{N-a}\left(2\sqrt{\beta \text{tr}\left[{{\boldsymbol{M}}}^{-1}\text{(}{{\boldsymbol{X}}}_{pl}{{\boldsymbol{X}}}_{pl}^{\text{H}}\text{)}\right]}\right)} \begin{array}{c}{\text{H}}_{1}\\ > \\ < \\ {\text{H}}_{0}\end{array}\lambda (18) 由于在式(18)两边同时取单调递增操作不会影响检测性能,我们将不等式(18)两边同时取对数得到:
\sum\limits_{l = 1}^L \left[ \frac{{a - N}}{2}\ln \left\{ {\frac{{{\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}} - {{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{{\boldsymbol{P}}_{\boldsymbol{H}}}{\text{)}}\right]}}{{{\text{tr[}}{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{\text{)]}}}}} \right\} + \ln \left\{ {\frac{{{K_{N - a}}\left( {2\sqrt {\beta {\text{tr}} \left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}} - {{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{{\boldsymbol{P}}_{\boldsymbol{H}}}{\text{)}}\right]} } \right)}}{{{K_{N - a}}\left( {2\sqrt {\beta {\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{\text{)}}\right]} } \right)}}} \right\} \right] \begin{array}{*{20}{c}} {{{\text{H}}_1}} \\ > \\ < \\ {{{\text{H}}_0}} \end{array} \lambda\\[-21pt] (19) 注意到,检测统计量(19)中的杂波协方差M实际上是未知的,因此需要对其进行可靠的估计以实现检测器的自适应检测。本文采用定点(FP)迭代法估计杂波协方差矩阵M的最大似然估计
{{\boldsymbol{\hat M}}_p} :{{\boldsymbol{\hat M}}_{p,i + 1}} = \frac{N}{K}\sum\limits_{k = 1}^K {\frac{{{{{\boldsymbol{\tilde x}}}_k}{\boldsymbol{\tilde x}}_k^{\text{H}}}}{{{\boldsymbol{\tilde x}}_k^{\text{H}}{\boldsymbol{\hat M}}_{p,i}^{ - 1}{{{\boldsymbol{\tilde x}}}_k}}}} (20) 其中,
{{\boldsymbol{\hat M}}_p} 的迭代计算初始值为{{\boldsymbol{\hat M}}_{p,{\text{ini}}}} = {N/ K}\displaystyle\sum\nolimits_{k = 1}^K ({{{\boldsymbol{\tilde x}}}_k} {\boldsymbol{\tilde x}}_k^{\text{H}})/ {({\boldsymbol{\tilde x}}_k^{\text{H}}{{{\boldsymbol{\tilde x}}}_k})} 。基于定点迭代估计法,我们对{{\boldsymbol{\hat M}}_p} 进行变换,利用上空/时对称阵列协方差的反对称结构,通过以下计算得到杂波协方差矩阵的估计值:{\hat {\boldsymbol{S}}_p} = \frac{{{{\hat {\boldsymbol{M}}}_p} + {\boldsymbol{J}}\hat {\boldsymbol{M}}_p^*{\boldsymbol{J}}}}{2} (21) 最终,将
{{\boldsymbol{\hat S}}_p} 代入式(19)可得PMR-GLRT检测器:\sum\limits_{l = 1}^L \left[ \frac{{a - N}}{2}\ln \left\{ {\frac{{{\text{tr}}\left[{\boldsymbol{\hat S}}_p^{ - 1}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}} - {{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{{\boldsymbol{P}}_{{\boldsymbol{\hat H}}}}{\text{)}}\right]}}{{{\text{tr}}\left[{\boldsymbol{\hat S}}_p^{ - 1}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{\text{)}}\right]}}} \right\} + \ln \left\{ {\frac{{{K_{N - a}} \left( {2\sqrt {\beta {\text{tr}}\left[{\boldsymbol{\hat S}}_p^{ - 1}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}} - {{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{{\boldsymbol{P}}_{{\boldsymbol{\hat H}}}}{\text{)}}\right]} } \right)}}{{{K_{N - a}}\left( {2\sqrt {\beta {\text{tr}}\left[{\boldsymbol{\hat S}}_p^{ - 1}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{\text{)}}\right]} } \right)}}} \right\} \right]\begin{array}{*{20}{c}} {{{\text{H}}_1}} \\ > \\ < \\ {{{\text{H}}_0}} \end{array} \lambda\\[-21pt] (22) 其中,
{{\boldsymbol{P}}_{\hat {\boldsymbol{H}}}} = \hat {\boldsymbol{S}}_p^{ - 1}{\boldsymbol{H}}{({{\boldsymbol{H}}^{\text{H}}}\hat {\boldsymbol{S}}_p^{ - 1}{\boldsymbol{H}})^{ - 1}}{{\boldsymbol{H}}^{\text{H}}} 。3.2 PMR-Rao检测器推导
令
{{\boldsymbol{\theta}} _r} = {[{\boldsymbol{a}}_1^{\text{T}},\;{\boldsymbol{a}}_2^{\text{T}},\; \cdots ,\;{\boldsymbol{a}}_L^{\text{T}}]^{\text{T}}} ,{{\boldsymbol{\theta}} _s} = {[{\tau _1},\;{\tau _2},\; \cdots ,\;{\tau _L}]^{\text{T}}} ,{\boldsymbol{\theta}} = {[{\boldsymbol{\theta}} _r^{\text{T}},\;{\boldsymbol{\theta}} _s^{\text{T}}]^{\text{T}}} ,对于所构建的假设检验问题(1),我们将Rao准则下的目标检测描述如下\frac{{\partial \ln {f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{\theta}} )}}{{\partial {{\boldsymbol{\theta}} _r}}}|_{{\boldsymbol{\theta}} = {{\hat {\boldsymbol{\theta}} }_0}}^{\text{T}} {\left[ {{{\boldsymbol{I}}^{ - 1}}({{\hat {\boldsymbol{\theta }}}_0})} \right]_{{{\boldsymbol{\theta}} _r},{{\boldsymbol{\theta}} _r}}} \frac{{\partial \ln {f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{\theta }})}}{{\partial {\boldsymbol{\theta}} _r^*}}|_ {{\boldsymbol{\theta}} = {{\hat {\boldsymbol{\theta}} }_0}}\;\begin{array}{*{20}{c}} {{{\text{H}}_1}} \\ > \\ < \\ {{{\text{H}}_0}} \end{array}\;\eta (23) 其中,
\eta 表示相应的检测阈值;{\boldsymbol{I}}({\boldsymbol{\theta}} ) 为Fisher信息矩阵;{\hat {\boldsymbol{\theta}} _0} 为{\boldsymbol{\theta }} 在{{\text{H}}_0} 假设下的最大似然估计;{f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{\theta}} ) 表示{{\text{H}}_1} 下X的PDF,{f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{\theta }}) = \prod\limits_{l = 1}^L {\frac{1}{{{\pi ^N}\left| {\boldsymbol{M}} \right|\tau _l^N}}} \exp \left\{ { - \frac{{{\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{{\boldsymbol{a}}_{pl}}{\text{)(}}{{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{{\boldsymbol{a}}_{pl}}{{\text{)}}^{\text{H}}} \right]}}{{{\tau _l}}}} \right\} (24) 为简洁起见,式(23)的详细求解和推导过程详见附录(1)。从而,我们得到空/时对称阵列雷达非高斯杂波背景下多秩距离扩展目标PMR-Rao检测器:
\sum\limits_{l = 1}^L \frac{{{\text{tr}}\left[{\boldsymbol{\hat S}}_p^{ - 1}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{{\boldsymbol{P}}_{{\boldsymbol{\hat H}}}}{\text{)}}\right]}}{b + \sqrt {{b^2} + 4\beta {\text{tr}}\left[{\boldsymbol{\hat S}}_p^{ - 1}\left({{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}\right)\right] }} \;\begin{array}{*{20}{c}} {{{\text{H}}_1}} \\ > \\ < \\ {{{\text{H}}_0}} \end{array}\;\eta (25) 3.3 PMR-Wald检测器推导
对于所构建的假设检验问题(1),Wald准则下的目标检测问题可表示为
{({\hat {\boldsymbol{\theta}} _{r,1}} - {\hat {\boldsymbol{\theta}} _{r,0}})^{\text{H}}}{\left\{ {{{[{{\boldsymbol{I}}^{ - 1}}({{\hat {\boldsymbol{\theta}} }_1})]}_{{{\boldsymbol{\theta}} _r},{{\boldsymbol{\theta}} _r}}}} \right\}^{ - 1}}({\hat {\boldsymbol{\theta}} _{r,1}} - {{\boldsymbol{\theta}} _{r,0}})\;\begin{array}{*{20}{c}} {{{\text{H}}_1}} \\ > \\ < \\ {{{\text{H}}_0}} \end{array}\;\gamma (26) 其中,
\gamma 表示相应的检测阈值;{\hat {\boldsymbol{\theta }}_{r,1}} = {[\hat {\boldsymbol{a}}_{p1}^{\text{T}},\; \hat {\boldsymbol{a}}_{p2}^{\text{T}},\; \cdots ,\; \hat {\boldsymbol{a}}_{pL}^{\text{T}}]^{\text{T}}} 为{{\text{H}}_1} 下{{\boldsymbol{\theta }}_r} 的最大似然估计,{\hat {\boldsymbol{a}}_{pl}} 的表达式为式(11)。同样地,为简洁起见,式(26)的详细求解和推导过程详见附录(2)。从而,我们得到空/时对称阵列雷达非高斯杂波背景下多秩距离扩展目标PMR-Wald检测器:
\sum\limits_{l = 1}^L {\frac{{{\text{tr}}\left[{\boldsymbol{\hat S}}_p^{ - 1}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{{\boldsymbol{P}}_{{\boldsymbol{\hat H}}}}{\text{)}}\right]}}{{b + \sqrt {{b^2} + 4\beta {\text{tr}}\left[{\boldsymbol{\hat S}}_p^{ - 1}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}} - {{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{{\boldsymbol{P}}_{{\boldsymbol{\hat H}}}}{\text{)}}\right]} }}} \begin{array}{*{20}{c}} {{{\text{H}}_1}} \\ > \\ < \\ {{{\text{H}}_0}} \end{array}\;\gamma (27) 4. CFAR特性分析
CFAR特性是自适应目标检测的一个重要性质。因此,在验证本文所提的空/时对称阵列雷达非高斯杂波背景下多秩距离扩展目标检测方法的性能效果之前,首先对所提检测器的CFAR特性进行分析,即证明检测器所对应的检测统计量是否对杂波协方差矩阵结构具有CFAR特性。已知PMR-GLRT, PMR-Rao, PMR-Wald检测器如式(22)、式(25)和式(27)所示,通过分析这3个检测统计量的表达式,我们要证明检测器的CFAR特性即要验证
{\boldsymbol{\hat S}}_p^{ - 1}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}} 和{\boldsymbol{\hat S}}_p^{ - 1}({{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{{\boldsymbol{P}}_{{\boldsymbol{\hat H}}}}) 与杂波协方差矩阵M无关。令
{{\boldsymbol{\bar X}}_{pl}} = {{\boldsymbol{M}}^{ - 1/2}}{{\boldsymbol{X}}_{pl}} ,{\boldsymbol{\bar H}}{\text{ = }}{{\boldsymbol{M}}^{ - 1/2}}{\boldsymbol{H}} ,{\boldsymbol{\bar S}}_p^{ - 1} = {{\boldsymbol{M}}^{1/2}}{\boldsymbol{\hat S}}_p^{ - 1}{{\boldsymbol{M}}^{1/2}} ,则有{\boldsymbol{\hat S}}_p^{ - 1}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}} = {\boldsymbol{\bar S}}_p^{ - 1}{{\boldsymbol{\bar X}}_{pl}}{\boldsymbol{\bar X}}_{pl}^{\text{H}} (28) {\boldsymbol{\hat S}}_p^{ - 1}({{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{{\boldsymbol{P}}_{{\boldsymbol{\hat H}}}}) = {\boldsymbol{\bar S}}_p^{ - 1}{{\boldsymbol{\bar X}}_{pl}}{\boldsymbol{\bar X}}_{pl}^{\text{H}}{\boldsymbol{\bar H}}{({{\boldsymbol{\bar H}}^{\text{H}}}{\boldsymbol{\bar S}}_p^{ - 1}{\boldsymbol{\bar H}})^{ - 1}}{{\boldsymbol{\bar H}}^{\text{H}}} (29) 经过上述白化处理后,易看出在
{{\text{H}}_0} 假设下{{\boldsymbol{\bar X}}_{pl}} ,{\boldsymbol{\bar H}} 和{\boldsymbol{\bar S}}_p^{} 均与杂波协方差矩阵M无关。令酉矩阵
{\boldsymbol{U}} = [{\boldsymbol{\bar H}}{({{\boldsymbol{\bar H}}^{\text{H}}}{\boldsymbol{\bar H}})^{ - 1/2}},{{\boldsymbol{\bar H}}_ \bot }] ,{\boldsymbol{U}}{{\boldsymbol{U}}^{\text{H}}} = {{\boldsymbol{I}}_N} ,{\boldsymbol{\bar H}}_ \bot ^{\text{H}}{\boldsymbol{\bar H}}{({{\boldsymbol{\bar H}}^{\text{H}}}{\boldsymbol{\bar H}})^{ - 1/2}}{\text{ = }}{{\boldsymbol{0}}_{(N - q) \times q}} ,则式(28)和式(29)可化简为\quad {\boldsymbol{\hat S}}_p^{ - 1}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}} = {\boldsymbol{\bar S}}_{p,{\boldsymbol{U}}}^{ - 1}{{\boldsymbol{\bar X}}_{pl,{\boldsymbol{U}}}}{\boldsymbol{\bar X}}_{pl,{\boldsymbol{U}}}^{\text{H}} (30) \begin{split} \quad & {\boldsymbol{\hat S}}_p^{ - 1}({{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{{\boldsymbol{P}}_{{\boldsymbol{\hat H}}}}) \\ & = {\boldsymbol{\bar S}}_{p,{\boldsymbol{U}}}^{ - 1}{{\boldsymbol{\bar X}}_{pl,{\boldsymbol{U}}}}{\boldsymbol{\bar X}}_{pl,{\boldsymbol{U}}}^{\text{H}}{{\boldsymbol{\bar H}}_{\boldsymbol{U}}}{({\boldsymbol{\bar H}}_{\boldsymbol{U}}^{\text{H}}{\boldsymbol{\bar S}}_{p,{\boldsymbol{U}}}^{ - 1}{{\boldsymbol{\bar H}}_{\boldsymbol{U}}})^{ - 1}}{\boldsymbol{\bar H}}_{\boldsymbol{U}}^{\text{H}} \end{split} (31) 其中,
{{\boldsymbol{\bar X}}_{pl,{\boldsymbol{U}}}} = {{\boldsymbol{U}}^{\text{H}}}{{\boldsymbol{\bar X}}_{pl}} ,{{\boldsymbol{\bar H}}_{\boldsymbol{U}}} = {{\boldsymbol{U}}^{\text{H}}}{\boldsymbol{\bar H}} ,{\boldsymbol{\bar S}}_{p,{\boldsymbol{U}}}^{ - 1} = {{\boldsymbol{U}}^{\text{H}}}{\boldsymbol{\bar S}}_p^{ - 1}{\boldsymbol{U}} 。由于在{{\text{H}}_0} 假设下{{\boldsymbol{\bar X}}_{pl,{\boldsymbol{U}}}} ,{{\boldsymbol{\bar H}}_{\boldsymbol{U}}} 和{\boldsymbol{\bar S}}_{p,{\boldsymbol{U}}}^{} 均与杂波协方差矩阵M无关,则表达式{\boldsymbol{\hat S}}_p^{ - 1}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}} 和{\boldsymbol{\hat S}}_p^{ - 1}({{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{{\boldsymbol{P}}_{{\boldsymbol{\hat H}}}}) 与杂波协方差矩阵M无关,从而证明了所提的PMR-GLRT, PMR-Rao, PMR-Wald检测器对杂波协方差矩阵具有CFAR特性。此外,所提出方法仅对于协方差结构保持恒虚警性能,并未对于杂波纹理分量保证恒虚警性能。5. 实验结果
为了验证本文所提的空/时对称阵列雷达非高斯杂波背景下多秩距离扩展目标检测方法的有效性,本节分别基于仿真数据与实测数据对所提检测器在不同训练样本支持条件下的目标检测性能及CFAR特性进行分析。
为了验证本文所提PMR-GLRT, PMR-Rao, PMR-Wald检测器的性能,我们在实验中将文献[39]提出的S-GLRT, S-AMF, S-Rao和文献[11]提出的Rao, Wald检测器、文献[40]提出的NSDD-GLRT检测器与本文所提检测器的检测性能进行对比:
{t_{{\rm{S}} {\text{-}} {\rm{GLRT}}}} = \sum\limits_{l = 1}^L {\frac{{{\boldsymbol{x}}_l^{\rm{H}}\hat {\boldsymbol{R}}_b^{ - 1}{\boldsymbol{H}}{{({{\boldsymbol{H}}^{\rm{H}}}\hat {\boldsymbol{R}}_b^{ - 1}{\boldsymbol{H}})}^{ - 1}} {{\boldsymbol{H}}^{\rm{H}}}\hat{\boldsymbol{R}}_b^{ - 1}{\boldsymbol{x}}_l^{}}}{{1 + {\boldsymbol{x}}_l^{\rm{H}}\hat {\boldsymbol{R}}_b^{ - 1}{\boldsymbol{x}}_l^{}}}} (32) {t_{{\rm{S}} {\text{-}} {\rm{AMF}}}} = \sum\limits_{l = 1}^L {{\boldsymbol{x}}_l^{\rm{H}}\hat {\boldsymbol{R}}_b^{ - 1}{\boldsymbol{H}}{{({{\boldsymbol{H}}^{\rm{H}}}\hat {\boldsymbol{R}}_b^{ - 1}{\boldsymbol{H}})}^{ - 1}}{{\boldsymbol{H}}^{\rm{H}}}\hat {\boldsymbol{R}}_b^{ - 1}{\boldsymbol{x}}_l^{}} (33) {t_{{\rm{S}} {\text{-}} {\rm{Rao}}}} = \sum\limits_{l = 1}^L {{\boldsymbol{x}}_l^{\rm{H}}\bar {\boldsymbol{R}}_b^{ - 1}{\boldsymbol{H}}{{({{\boldsymbol{H}}^{\rm{H}}}\bar {\boldsymbol{R}}_b^{ - 1}{\boldsymbol{H}})}^{ - 1}}{{\boldsymbol{H}}^{\rm{H}}}\bar {\boldsymbol{R}}_b^{ - 1}{\boldsymbol{x}}_l^{}} (34) {t_{{\rm{Rao}}}} = \sum\limits_{l = 1}^L {\frac{{{\boldsymbol{x}}_l^{\rm{H}}\hat {\boldsymbol{M}}_p^{ - 1}{\boldsymbol{H}}{{({{\boldsymbol{H}}^{\rm{H}}}\hat {\boldsymbol{M}}_p^{ - 1}{\boldsymbol{H}})}^{ - 1}}{{\boldsymbol{H}}^{\rm{H}}}\hat {\boldsymbol{M}}_p^{ - 1}{\boldsymbol{x}}_l^{}}}{{{\boldsymbol{x}}_l^{\rm{H}}\hat {\boldsymbol{M}}_p^{ - 1}{\boldsymbol{x}}_l^{}}}} (35) \begin{split} & {t_{{\rm{Wald}}}}\\ & = \sum\limits_{l = 1}^L {\frac{{{\boldsymbol{x}}_l^{\rm{H}}\hat {\boldsymbol{M}}_p^{ - 1}{\boldsymbol{H}}{{({{\boldsymbol{H}}^{\rm{H}}}\hat {\boldsymbol{M}}_p^{ - 1}{\boldsymbol{H}})}^{ - 1}}{{\boldsymbol{H}}^{\rm{H}}}\hat {\boldsymbol{M}}_p^{ - 1}{\boldsymbol{x}}_l^{}}}{{{\boldsymbol{x}}_l^{\rm{H}} \left[ {\hat {\boldsymbol{M}}_p^{ - 1} - \hat {\boldsymbol{M}}_p^{ - 1} {\boldsymbol{H}}{{({{\boldsymbol{H}}^{\rm{H}}}\hat {\boldsymbol{M}}_p^{ - 1} {\boldsymbol{H}})}^{ - 1}} {{\boldsymbol{H}}^{\rm{H}}}\hat {\boldsymbol{M}}_p^{ - 1}} \right]{\boldsymbol{x}}_l^{}}}} \end{split} (36) \begin{split} &{t_{{\rm{NSDD}} {\text{-}} {\rm{GLRT}}}} = - 2\left( {N - 1} \right)\\ &\qquad \cdot \sum\limits_{l = 1}^L {\ln \left[ {1 - \frac{{{\boldsymbol{x}}_l^{\rm{H}}{\boldsymbol{\hat R}}_b^{ - 1}{\boldsymbol{H}}{{({{\boldsymbol{H}}^{\rm{H}}}{\boldsymbol{\hat R}}_b^{ - 1}{\boldsymbol{H}})}^{ - 1}}{{\boldsymbol{H}}^{\rm{H}}}{\boldsymbol{\hat R}}_b^{ - 1}{\boldsymbol{x}}_l^{}}}{{{\boldsymbol{x}}_l^{\rm{H}}{\boldsymbol{\hat R}}_b^{ - 1}{\boldsymbol{x}}_l^{}}}} \right]} \end{split} (37) 其中,
\hat {\boldsymbol{R}}_b^{} = \displaystyle\sum\nolimits_{k = 1}^K {{{\boldsymbol{x}}_k}{\boldsymbol{x}}_k^{\rm{H}}} 表示采样协方差矩阵,\bar {\boldsymbol{R}}_b^{}{\text{ = }} \hat {\boldsymbol{R}}_b^{}{\text{ + }}\displaystyle\sum\nolimits_{l = 1}^L {{{{{\boldsymbol{x}}_l}{\boldsymbol{x}}_l^{\rm{H}}}/ L}} 。5.1 仿真数据
在仿真数据实验中,设置阵元数
N = 8 ,虚警概率{P_{{\rm{FA}}}} \,=\, {10^{ - 3}} ,蒙特卡罗试验次数为{\rm{Num}} \,={\rm{}} 100/{P_{{\rm{FA}}}} = 1{0^5} 次。在生成杂波数据时,杂波协方差矩阵{\boldsymbol{M}}{\text{ = }}{\rho ^{|i - j|}} ,其中\rho 为1阶迟滞系数,设置为0.9;杂波纹理分量的形状参数a = 4 ,尺度参数\beta = 3 ;FP估计中迭代次数{\rm{iter}} = 3 。在生成目标信号数据时,假定其多普勒频率{f_d} = [0.09,\;0.10,\;0.11] ,则目标信号子空间的秩q = 3 ,并假定目标信号占据L = 3 个径向距离单元,利用式(38)所示的输入信杂比生成目标信号为{\text{SCR}} = \sum\limits_{l = 1}^L {{\text{tr(}}{\boldsymbol{a}}_l^{\text{H}}{{\boldsymbol{H}}^{\text{H}}}{{\boldsymbol{M}}^{ - 1}}{\boldsymbol{Ha}}_l^{}{\text{)}}} (38) 第4节通过理论分析验证了所提的PMR-GLRT, PMR-Rao和PMR-Wald检测器关于杂波协方差矩阵的CFAR特性,本节通过蒙特卡罗实验对所提检测器的CFAR特性进行仿真验证。图1(a)—图1(c)分别显示了本文提出的检测器在不同杂波相关性下检测阈值与虚警概率的关系,其中杂波相关性通过在0至0.9之间取值的1阶迟滞系数来表述,训练样本数为16。图1表明,本文所提检测器在不同的1阶迟滞系数下的虚警概率非常接近,即所提检测器的检测性能基本不受杂波相关性的影响,验证了本文所提检测器关于杂波协方差矩阵具有CFAR特性,从仿真实验方面验证了第4节对于所提检测器CFAR特性进行的理论分析。
图2(a)—图2(d)分别是训练样本数为9, 10, 12和16时基于仿真数据的检测概率曲线,给出了不同训练支持度下各检测器的检测性能,图2(a)表示小训练支持下各检测器的检测概率曲线,可以看出,本文所提的PMR-GLRT, PMR-Rao和PMR-Wald检测器的检测概率相近,并且均高于现有检测器;其次是S-GLRT,高于NSDD-GLRT, Rao和Wald检测器;S-AMF和S-Rao的检测概率最低,并且在信杂比高于一定值后,S-Rao的检测概率曲线呈平顶状,与文献[39]所述的检测性能相符。从图2(a)到图2(d),训练支持度逐渐提高,可以看出,仿真实验中各检测器的检测概率均得到了提高,这是由于训练样本数的增加会使杂波协方差矩阵的估计值更加接近真实值,从而有效地提高检测性能。图2(d)所示的情况为高训练支持度下各检测器检测性能的对比,显示了在训练样本充足的条件下,所有检测器均表现出较好的检测性能,但现有检测器的检测性能仍低于所提检测器。
因此,综合图2可以看出,基于仿真数据,本文所提PMR-GLRT, PMR-Rao和PMR-Wald检测器与现有检测器在训练样本充足的条件下均表现出优良的检测性能,其中所提检测器的检测概率更高;而在训练样本数不足,即小样本支持的条件下,现有检测器的检测概率严重降低,检测性能较差,而所提检测器仍保持较高的检测概率。
由于在目标检测过程中会出现信号失配的情况,即子空间实际导向矢量与名义导向矢量不匹配,因此需要验证所提检测器在失配条件下的检测性能。利用
{\cos ^2}\theta 来表示实际接收回波的子空间导向矢量与检测器导向矢量的失配程度,其表达式为{\cos ^2}\theta = \frac{{{{\left| {{\text{tr(}}{{\boldsymbol{H}}^{\text{H}}}{{\boldsymbol{M}}^{ - 1}}{{\boldsymbol{H}}_m}{\text{)}}} \right|}^2}}}{{{\text{tr(}}{{\boldsymbol{H}}^{\text{H}}}{{\boldsymbol{M}}^{ - 1}}{\boldsymbol{H}}{\text{)tr(}}{\boldsymbol{H}}_m^{\text{H}}{{\boldsymbol{M}}^{ - 1}}{{\boldsymbol{H}}_m}{\text{)}}}} (39) 其中,
{{\boldsymbol{H}}_m} 代表子空间实际导向矢量矩阵,H代表检测器子空间导向矢量矩阵,当{\cos ^2}\theta = 1 即{\boldsymbol{H}} = {{\boldsymbol{H}}_m} 时代表完全适配,当{\cos ^2}\theta = 0 即H与{{\boldsymbol{H}}_m} 正交时代表完全失配。图3显示了本文所提检测器与现有检测器在不同失配程度下的检测概率曲线,其仿真的信杂比设置为{\text{SCR}} = 22\;{\text{dB}} ,训练样本数分别为12和16,其余仿真条件与图2相同。图3(a)显示了在小训练支持下,本文所提的PMR-GLRT和PMR-Rao检测器的检测性能基本一致,均保持了较高的检测概率与较好的稳健性,所提PMR-Wald检测器的检测性能次于PMR-GLRT和PMR-Rao,但仍优于现有检测器。图3(b)表明当训练样本充足时,所有检测器的检测概率均得到了提高,其中本文所提出的PMR-GLRT和PMR-Rao检测器仍具有最优的检测性能,PMR-Wald检测器与现有的S-GLRT的检测性能比较接近,并且优于其他的现有检测器。同时,可以看出本文所提的检测器对失配条件表现出更加稳健的性能,而现有的检测器的检测性能受信号失配影响较大,表现出非常敏感的特性。因此,综合图3可看出,本文所提检测器与现有检测器的检测概率均随失配程度的降低而提高,并且在小训练支持下,本文所提检测器的检测概率高于现有检测器,其中PMR-GLRT和PMR-Rao的检测性能非常接近,并且优于其他的检测器,此外,本文所提的检测器对信号失配条件具有更强的稳健性。
5.2 实测数据
本文的实测仿真实验基于加拿大McMaster大学利用IPIX雷达(Ice Multi-parameter Imaging X-Band Radar)在不同海情下采集到的实测海杂波数据,其采用的IPIX雷达是一种完全相干的X波段雷达,具有双发射/接收极化、频率捷变和凝视/监视模式等高级功能。所采集的实测海杂波数据包含两个数据集:Dartmouth 1993和Grimsby 1998,均为将天线架设在海平面上方以俯瞰角度采集海平面杂波数据。本文采用的实测海杂波数据为Dartmouth 1993数据集的第283组数据,其幅度二维图见图4,数据矩阵的尺寸为131072×14,其中131072对应脉冲维,14对应距离维。
通过蒙特卡罗实验验证所提检测器在复合高斯杂波中的小训练支持下的子空间距离扩展目标检测中具有有效性,设置阵元数
N = 5 ,虚警概率{P_{{\rm{FA}}}} = {10^{ - 2}} ,分别统计不同训练样本数(K = 6,\;8,\;10 )时检测器的检测概率。需要指出的是,此时形状参数与尺度参数是通过对每列数据进行Gamma函数拟合估计得到的,具体为利用MATLAB软件中的gamfit函数求得。实验中选取的实测数据为:设置目标占据的距离单元数L = 3 ,选取实测杂波数据中的第1, 2, 3列为待测数据。当训练样本数K = 6 时,选取第4~9列数据作为训练数据集;当K = 8 时,选取第4~11列数据作为训练数据集;当K = 10 时,选取第4~13列数据作为训练数据集。图5(a)—图5(c)分别是训练样本数为6, 8和10时本文所提检测器与现有检测器的检测概率曲线, 该图有效支持了图2所得出的结论,即随着训练样本数的增大,各检测器的检测性能均有提升,其中所提检测器PMR-GLRT, PMR-Rao和PMR-Wald表现出最优检测性能。且在小训练样本支持(
K = 6,8 )下,传统检测器的检测概率受到训练样本的严重影响,而本文所提检测器的检测概率无明显下降,仍可保持高检测性能。图6为不同失配程度下基于实测数据的检测概率曲线,其中图6(a)为基于小训练样本支持,图6(b)为基于充足的训练样本支持,可以看出,随着导向矢量失配程度的变化,无论训练样本数充足与否,本文所提的PMR-GLRT, PMR-Rao和PMR-Wald检测器的检测概率稳定在0.8~1.0,表现出对于导向矢量失配的稳健性,而传统检测器的检测性能对于导向矢量失配相对敏感,有效地支撑了图3的基于仿真数据的结论。
6. 结语
针对多通道阵列雷达从实际杂波中检测目标场景,该文提出了一种面向多通道阵列雷达非高斯杂波背景的多秩距离扩展目标检测方法。该方法通过构建多秩距离扩展目标模型,考虑了雷达接收单元空间或时间中心对称探测场景下杂波协方差矩阵的反对称结构信息,基于广义似然比、Rao、Wald检验准则,推导了多个未知参数的估计,设计了完整的目标检测方法;此外,所提检测方法的恒虚警特性也通过理论分析得到了证明。基于仿真数据和实测数据的实验结果表明,该方法相较于传统检测方法在训练样本不足时具有更好的检测性能。而且本方法性能对目标导向矢量失配不敏感。下一步的研究工作将探讨导向矢量完全未知情况下的多参数联合优化设计,以及考虑利用其他先验知识辅助检测器设计。
附录
(1) PMR-Rao检测器的相关推导
Rao准则下,检测统计量只需求
{{\text{H}}_0} 下的未知参数估计方法。已知
{f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{\theta}} ) 如式(24)所示,令\ln {f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{\theta}} ) 分别对{\tau _l} 求导得\frac{{\partial \ln {f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{\theta}} )}}{{\partial {{\boldsymbol{\theta}} _r}}} = \left[ \frac{2}{{{\tau _1}}}{{\left\{ {{({{\boldsymbol{X}}_{p1}} - {\boldsymbol{H}}{{\boldsymbol{a}}_{p1}})}^{\text{H}}}{{\boldsymbol{M}}^{ - 1}}{\boldsymbol{H}}\right\} }^{\text{T}}},\cdots , \frac{2}{{{\tau _L}}}{{\left\{ {{({{\boldsymbol{X}}_{pL}} - {\boldsymbol{H}}{{\boldsymbol{a}}_{pL}})}^{\text{H}}}{{\boldsymbol{M}}^{ - 1}}{\boldsymbol{H}}\right\} }^{\text{T}}} \right]^{\text{T}} \tag{A-1} \frac{{\partial \ln {f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{\theta}} )}}{{\partial {\boldsymbol{\theta}} _r^*}} =\left[ \frac{2}{{{\tau _1}}}{{\left\{ {\boldsymbol{H}}^{\rm{H}}{{\boldsymbol{M}}^{ - 1}}({{\boldsymbol{X}}_{p1}} - {\boldsymbol{H}}{{\boldsymbol{a}}_{p1}})\right\} }^{\text{T}}}, \cdots ,\frac{2}{{{\tau _L}}}{{\left\{ {\boldsymbol{H}}^{\rm{H}}{{\boldsymbol{M}}^{ - 1}}({{\boldsymbol{X}}_{pL}} - {\boldsymbol{H}}{{\boldsymbol{a}}_{pL}})\right\} }^{\text{T}}} \right]^{\text{T}} \tag{A-2} 由式(A-1),式(A-2)可得
{{\boldsymbol{I}}_{{{\boldsymbol{\theta}} _r},{{\boldsymbol{\theta}} _r}}}({\hat {\boldsymbol{\theta}} _0}) = {\text{E}}\left\{ \left[ {\frac{{\partial \ln {f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{\theta }})}}{{\partial {\boldsymbol{\theta}} _r^*}}} \right] \left[ {\frac{{\partial \ln {f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{\theta}} )}}{{\partial {\boldsymbol{\theta}} _r^{\text{T}}}}} \right] \right\} = {{\boldsymbol{H}}^{\text{H}}}{{\boldsymbol{M}}^{ - 1}}{\boldsymbol{H}} \cdot {\rm{diag}}\left( {\frac{4}{{{\tau _1}}},\frac{4}{{{\tau _2}}}, \cdots ,\frac{4}{{{\tau _L}}}} \right) \tag{A-3} 由于
{{\boldsymbol{I}}_{{{\boldsymbol{\theta}} _r},{{\boldsymbol{\theta}} _s}}}({\hat {\boldsymbol{\theta}} _0}) 是零矩阵,可以得到{[{{\boldsymbol{I}}^{ - 1}}({\hat {\boldsymbol{\theta}} _0})]_{{{\boldsymbol{\theta}} _r},{{\boldsymbol{\theta}} _r}}} = {[{{\boldsymbol{I}}_{{{\boldsymbol{\theta}} _r},{{\boldsymbol{\theta}} _r}}}({\hat {\boldsymbol{\theta}} _0})]^{ - 1}} = {({{\boldsymbol{H}}^{\text{H}}}{{\boldsymbol{M}}^{ - 1}}{\boldsymbol{H}})^{ - 1}} {\rm{diag}}\left( {\frac{{{\tau _1}}}{4}, \frac{{{\tau _2}}}{4},\cdots ,\frac{{{\tau _L}}}{4}} \right) \tag{A-4} 已知
{f_0}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{\theta}} ) = \dfrac{1}{{{\pi ^{NL}}{{\left| {\boldsymbol{M}} \right|}^L}\displaystyle\prod\limits_{l = 1}^L {\tau _l^N} }} \exp \left\{ - \displaystyle\sum\limits_{l = 1}^L {\dfrac{{{\text{tr[}}{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{\text{)]}}}}{{{\tau _l}}}} \right\} ,令{f_0}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{\theta}} )f({\tau _l}) 分别对{\tau _l} 求导得{\hat{{\boldsymbol{\theta}} }}_{s,0}={[{\hat{\tau }}_{1,0},{\hat{\tau }}_{2,0},\cdots ,{\hat{\tau }}_{L,0}]}^{\text{T}} ,其中{\hat \tau _{l,0}} = \mathop {\arg \max }\limits_{{\tau _l}} {f_0}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{\theta}} )f({\tau _l}) = \frac{{b + \sqrt {{b^2} + 4\beta {\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{\text{)}}\right]} }}{{2\beta }},\quad l = 1,2, \cdots ,L \tag{A-5} 其中,
b = a - N - 1 。将
{\hat {\boldsymbol{\theta}} _{s,0}} 和式(A-4)代入Rao检测器中,并将其中的杂波协方差矩阵替换为其最大似然估计值{{\boldsymbol{\hat S}}_p} ,得\sum\limits_{l = 1}^L {\frac{{{\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{{\boldsymbol{P}}_{{\boldsymbol{\hat H}}}}{\text{)}}\right]}}{{b + \sqrt {{b^2} + 4\beta {\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{\text{)}}\right]} }}} \;\begin{array}{*{20}{c}} {{{\text{H}}_1}} \\ > \\ < \\ {{{\text{H}}_0}} \end{array}\;\eta \tag{A-6} (2) PMR-Wald检测器的相关推导
Wald准则下,检测统计量只需求
{{\text{H}}_1} 下的未知参数估计方法。为求解式(26),我们先由式(A-2)计算出{\left\{ {{{[{{\boldsymbol{I}}^{ - 1}}({{\hat {\boldsymbol{\theta}} }_1})]}_{{{\boldsymbol{\theta}} _r},{{\boldsymbol{\theta}} _r}}}} \right\}^{ - 1}} = {{\boldsymbol{H}}^{\text{H}}}{{\boldsymbol{M}}^{ - 1}}{\boldsymbol{H}} \cdot {\rm{diag}}\left( {\frac{4}{{{\tau _1}}}, \frac{4}{{{\tau _2}}},\cdots ,\frac{4}{{{\tau _L}}}} \right)\tag{A-7} {{\boldsymbol{\theta}} _s} 在{{\text{H}}_1} 下的最大似然估计为{\hat {\boldsymbol{\theta}} _{s,1}} = [{\hat \tau _{1,1}},\;\;{\hat \tau _{2,1}},\;\; \cdots , \;\;{\hat \tau _{L,1}}]^{\text{T}} ,其中{\hat \tau _{l,1}} = \mathop {\arg \max }\limits_{{\tau _l}} {f_1}({\boldsymbol{X}}|{\boldsymbol{M}},{\boldsymbol{\theta}} )f({\tau _l}) = \frac{{b + \sqrt {{b^2} + 4\beta {\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{{\hat {\boldsymbol{a}}}_{pl}}{\text{)(}}{{\boldsymbol{X}}_{pl}} - {\boldsymbol{H}}{{\hat {\boldsymbol{a}}}_{pl}}{{\text{)}}^{\text{H}}}\right]} }}{{2\beta }},\; l = 1,2, \cdots ,L \tag{A-8} 将
{\hat {\boldsymbol{\theta}} _{s,1}} ,{\hat {\boldsymbol{\theta}} _{r,1}} 和式(A-7)代入Wald检测器中,并将其中的杂波协方差矩阵替换为其最大似然估计值{{\boldsymbol{\hat S}}_p} ,得\sum\limits_{l = 1}^L {\frac{{{\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{{\boldsymbol{P}}_{{\boldsymbol{\hat H}}}}{\text{)}}\right]}}{{b + \sqrt {{b^2} + 4\beta {\text{tr}}\left[{{\boldsymbol{M}}^{ - 1}}{\text{(}}{{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}} - {{\boldsymbol{X}}_{pl}}{\boldsymbol{X}}_{pl}^{\text{H}}{{\boldsymbol{P}}_{{\boldsymbol{\hat H}}}}{\text{)}}\right]} }}} \begin{array}{*{20}{c}} {{{\text{H}}_1}} \\ > \\ < \\ {{{\text{H}}_0}} \end{array}\;\gamma \tag{A-9} -
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