Adaptive Detection of a Subspace Signal in Structured Random Interference Plus Thermal Noise

Abstract

This paper studies adaptive radar detection of a subspace signal embedded in two disturbance sources. The former is thermal noise with known power. The latter is a Gaussian subspace interference with zero mean and unknown covariance matrix (CM). It is assumed that the signal and interference subspaces are known and partially related. As customary, several secondary data containing only interference and thermal noise are used to estimate this interference CM. This paper derives the generalized likelihood ratio test (GLRT), and theoretically deduces the probabilities of false alarm (PFA) and detection of the new detector. This PFA shows that the new detector has the constant false alarm rate (CFAR) property against the interference CM. Several numerical experiments are performed to evaluate the detection performance of the new detector. The results show that the performance of the new detector is better than the natural counterparts in some scenarios.

Appendices

Appendix A: Derivations of the GLRT

In this section, first we expand $\ln f_{1} \left( {\mathbf{X}} \right)$ and derive the MLEs of ${\mathbf{p}}$ and ${\mathbf{R}}$. Then, we derive $\mathop {\hbox{max} }\limits_{{{\mathbf{p}},{\mathbf{R}}}} \ln f_{1} \left( {\mathbf{X}} \right)$ with ${\mathbf{p}}$ and ${\mathbf{R}}$ being replaced by their MLEs. $\mathop {\hbox{max} }\limits_{{\mathbf{R}}} \ln f_{0} \left( {\mathbf{X}} \right)$ can be obtained in a similar approach.

Applying the Corollaries 18.1.2 in [22], we expand $\ln \left| {\mathbf{C}} \right|$ as

$$ \ln \left| {\mathbf{C}} \right| = \left( {M - Q} \right)\ln \sigma^{2} + \ln \left| {{\mathbf{J}}^{H} {\mathbf{J}}} \right| + \ln \left| {{\mathbf{R}} + \sigma^{2} \left( {{\mathbf{J}}^{H} {\mathbf{J}}} \right)^{ - 1} } \right|. $$

(20)

Resorting to the Corollaries 18.2.9 in [22], we rewrite ${\mathbf{C}}^{ - 1}$ as

$$ {\mathbf{C}}^{ - 1} = \frac{1}{{\sigma^{2} }}{\mathbf{P}}_{{\mathbf{J}}}^{ \bot } + {\mathbf{U}}^{H} \left[ {{\mathbf{R}} + \sigma^{2} \left( {{\mathbf{J}}^{H} {\mathbf{J}}} \right)^{ - 1} } \right]^{ - 1} {\mathbf{U}}, $$

(21)

where ${\mathbf{U}} = \left( {{\mathbf{J}}^{H} {\mathbf{J}}} \right)^{ - 1} {\mathbf{J}}^{H}$.

Substituting (20) and (21) into (4) leads to the expended $\ln f_{1} \left( {\mathbf{X}} \right)$, which is

$$ \begin{aligned} \ln f_{1} \left( {\mathbf{X}} \right) & = - M\left( {K + 1} \right)\ln \pi - \left( {K + 1} \right)\left( {M - Q} \right)\ln \sigma^{2} \\ & - \left( {K + 1} \right)\ln \left| {{\mathbf{J}}^{H} {\mathbf{J}}} \right| - \left( {K + 1} \right)\ln \left| {{\mathbf{R}} + \sigma^{2} \left( {{\mathbf{J}}^{H} {\mathbf{J}}} \right)^{ - 1} } \right| \\ & - \frac{1}{{\sigma^{2} }}tr\left( {{\mathbf{P}}_{{\mathbf{J}}}^{ \bot } {\mathbf{T}}_{1} } \right) - tr\left\{ {\left[ {{\mathbf{R}} + \sigma^{2} \left( {{\mathbf{J}}^{H} {\mathbf{J}}} \right)^{ - 1} } \right]^{ - 1} {\mathbf{UT}}_{1} {\mathbf{U}}^{H} } \right\}. \\ \end{aligned} $$

(22)

According to the Lemma 3.2.2 in [1], the MLE of ${\mathbf{R}}$ under $H_{1}$, noted as ${\hat{\mathbf{R}}}_{1}$, is derived as

$$ {\hat{\mathbf{R}}}_{1} = \frac{1}{K + 1}{\mathbf{UT}}_{1} {\mathbf{U}}^{H} - \sigma^{2} \left( {{\mathbf{J}}^{H} {\mathbf{J}}} \right)^{ - 1} . $$

(23)

Let ${\mathbf{R}} = {\hat{\mathbf{R}}}_{1}$, the maximum of $\ln f_{1} \left( {\mathbf{X}} \right)$ w.r.t. ${\mathbf{R}}$ is obtained as

$$ \begin{aligned} \mathop {\hbox{max} }\limits_{{\mathbf{R}}} \ln f_{1} \left( {\mathbf{X}} \right) & = - M\left( {K + 1} \right)\ln \pi - \left( {K + 1} \right)\left( {M - Q} \right)\ln \sigma^{2} \\ & - \left( {K + 1} \right)\ln \left| {{\mathbf{J}}^{H} {\mathbf{J}}} \right| + \left( {K + 1} \right)Q\ln \left( {K + 1} \right) - \left( {K + 1} \right)Q \\ & - \left( {K + 1} \right)\ln \left| {{\mathbf{UT}}_{1} {\mathbf{U}}^{H} } \right| - \frac{1}{{\sigma^{2} }}tr\left( {{\mathbf{P}}_{{\mathbf{J}}}^{ \bot } {\mathbf{T}}_{1} } \right). \\ \end{aligned} $$

(24)

Nulling the partial derivative of (24) w.r.t. ${\mathbf{x}}$ leads to the MLE of ${\mathbf{p}}$ under $H_{1}$ (denoted by ${\hat{\mathbf{p}}}_{1}$)

$$ {{\partial \left[ {\mathop {\hbox{max} }\limits_{{\mathbf{R}}} \ln f_{1} \left( {\mathbf{X}} \right)} \right]} \mathord{\left/ {\vphantom {{\partial \left[ {\mathop {\hbox{max} }\limits_{{\mathbf{R}}} \ln f_{1} \left( {\mathbf{X}} \right)} \right]} {\partial {\mathbf{p}}}}} \right. \kern-0pt} {\partial {\mathbf{p}}}} = {\mathbf{0}}_{P} \Rightarrow {\mathbf{x}} - {\mathbf{Hp}} = {\mathbf{0}}_{M} \Rightarrow {\hat{\mathbf{p}}}_{1} = \left( {{\mathbf{H}}^{H} {\mathbf{H}}} \right)^{ - 1} {\mathbf{H}}^{H} {\mathbf{x}}. $$

(25)

Let ${\mathbf{p}} = {\hat{\mathbf{p}}}_{1}$, the maximum of (24) w.r.t. ${\mathbf{p}}$ is expressed as

$$ \begin{aligned} \mathop {\hbox{max} }\limits_{{{\mathbf{p}},{\mathbf{R}}}} \ln f_{1} \left( {\mathbf{X}} \right) = - M\left( {K + 1} \right)\ln \pi - \left( {K + 1} \right)\left( {M - Q} \right)\ln \sigma^{2} \\ - \left( {K + 1} \right)\ln \left| {{\mathbf{J}}^{H} {\mathbf{J}}} \right| + \left( {K + 1} \right)Q\ln \left( {K + 1} \right) \\ - \left( {K + 1} \right)Q - \left( {K + 1} \right)\ln \left| {{\mathbf{USU}}^{H} } \right| \\ - \left( {K + 1} \right)\varPhi \left( {\mathbf{x}} \right), \\ \end{aligned} $$

(26)

where

$$ \varPhi \left( {\mathbf{x}} \right) = \frac{{{\mathbf{x}}^{H} {\mathbf{P}}_{{\mathbf{H}}}^{ \bot } {\mathbf{P}}_{{\mathbf{J}}}^{ \bot } {\mathbf{P}}_{{\mathbf{H}}}^{ \bot } {\mathbf{x}}}}{{\left( {K + 1} \right)\sigma^{2} }} + \ln \left[ {1 + {\mathbf{x}}^{H} {\mathbf{P}}_{{\mathbf{H}}}^{ \bot } {\mathbf{J}}\left( {{\mathbf{J}}^{H} {\mathbf{SJ}}} \right)^{ - 1} {\mathbf{J}}^{H} {\mathbf{P}}_{{\mathbf{H}}}^{ \bot } {\mathbf{x}}} \right]. $$

(27)

As ${\mathbf{Hp}}$ does not exist under $H_{0}$, $\mathop {\hbox{max} }\limits_{{\mathbf{R}}} \ln f_{0} \left( {\mathbf{X}} \right)$ can be obtained from (26) by setting ${\mathbf{H}} = {\mathbf{0}}_{M \times P}$, which is

$$ \begin{aligned} \mathop {\hbox{max} }\limits_{{\mathbf{R}}} \ln f_{0} \left( {\mathbf{X}} \right) & = \left. {\mathop {\hbox{max} }\limits_{{{\mathbf{p}},{\mathbf{R}}}} \ln f_{1} \left( {\mathbf{X}} \right)} \right|_{{{\mathbf{H}} = {\mathbf{0}}_{M \times P} }} \\ & = - M\left( {K + 1} \right)\ln \pi - \left( {K + 1} \right)\left( {M - Q} \right)\ln \sigma^{2} \\ & - \left( {K + 1} \right)\ln \left| {{\mathbf{J}}^{H} {\mathbf{J}}} \right| + \left( {K + 1} \right)Q\ln \left( {K + 1} \right) \\ & - \left( {K + 1} \right)Q - \left( {K + 1} \right)\ln \left| {{\mathbf{USU}}^{H} } \right| \\ & - \left( {K + 1} \right)\left( {\lambda_{1} + \lambda_{2} } \right), \\ \end{aligned} $$

(28)

where

$$ \left\{ \begin{aligned} \lambda_{1} = \frac{{{\mathbf{x}}^{H} {\mathbf{P}}_{{\mathbf{J}}}^{ \bot } {\mathbf{x}}}}{{\left( {K + 1} \right)\sigma^{2} }} \\ \lambda_{2} = \ln \left[ {1 + {\mathbf{x}}^{H} {\mathbf{J}}\left( {{\mathbf{J}}^{H} {\mathbf{SJ}}} \right)^{ - 1} {\mathbf{J}}^{H} {\mathbf{x}}} \right] \\ \end{aligned} \right.. $$

(29)

Appendix B: PDFs of $\lambda_{1}$, $\lambda_{2}$ and $\lambda = \lambda_{1} + \lambda_{2}$

2.1 B.1 PDFs of $\lambda_{1}$ in (29)

According to the descriptions on page 65 of [49], the PDFs of $\lambda_{1}$ under $H_{0}$ and $H_{1}$ can be derived as

$$ \left\{ {\begin{array}{*{20}c} {2\left( {K + 1} \right)\lambda_{1} \sim\chi_{{2\left( {M - Q} \right)}}^{2} \;\;\;\;\;\;} & {{\text{under}}\;H_{0} } \\ {2\left( {K + 1} \right)\lambda_{1} \sim\chi_{{2\left( {M - Q} \right)}}^{2} \left( {\mu_{1} } \right)} & {{\text{under}}\;H_{1} } \\ \end{array} } \right., $$

(30)

where $\mu_{1}$ is a non-centrality parameter

$$ \mu_{1} = \frac{{2{\mathbf{p}}^{H} {\mathbf{H}}^{H} {\mathbf{P}}_{{\mathbf{J}}}^{ \bot } {\mathbf{Hp}}}}{{\sigma^{2} }}. $$

(31)

Therefore, the PDFs of $\lambda_{1}$ under $H_{0}$ and $H_{1}$ can be written as

$$ h_{1} \left( {\left. {\lambda_{1} } \right|H_{0} } \right) = \frac{K + 1}{{\varGamma \left( {\delta_{1} + 1} \right)}}\gamma_{1}^{{\delta_{1} }} e^{{ - \gamma_{1} }} , $$

(32)

$$ h_{1} \left( {\left. {\lambda_{1} } \right|H_{1} } \right) = \frac{K + 1}{{e^{{{{\left( {2\gamma_{1} + \mu_{1} } \right)} \mathord{\left/ {\vphantom {{\left( {2\gamma_{1} + \mu_{1} } \right)} 2}} \right. \kern-0pt} 2}}} }}\left( {{{2\gamma_{1} } \mathord{\left/ {\vphantom {{2\gamma_{1} } {\mu_{1} }}} \right. \kern-0pt} {\mu_{1} }}} \right)^{{{{\delta_{1} } \mathord{\left/ {\vphantom {{\delta_{1} } 2}} \right. \kern-0pt} 2}}} {\mathcal{I}}_{{\delta_{1} }} \left( {\sqrt {2\gamma_{1} \mu_{1} } } \right), $$

(33)

respectively, where $\delta_{1} = M - Q - 1$ and $\gamma_{1} = \left( {K + 1} \right)\lambda_{1}$.

2.2 B.2 PDFs of $\lambda_{2}$ in (29)

According to the Theorem 5.2.2 in [1], we have

$$ \left\{ {\begin{array}{*{20}c} {e^{{\lambda_{2} }} - 1\sim\frac{{\chi_{2Q}^{2} }}{{\chi_{{2\left( {K - Q + 1} \right)}}^{2} }}} & {{\text{under}}\;H_{0} } \\ {e^{{\lambda_{2} }} - 1\sim\frac{{\chi_{2Q}^{2} \left( {\mu_{2} } \right)}}{{\chi_{{2\left( {K - Q + 1} \right)}}^{2} }}} & {{\text{under}}\;H_{1} } \\ \end{array} ,} \right. $$

(34)

where $\mu_{2}$ is a non-centrality parameter

$$ \mu_{2} = 2{\mathbf{p}}^{H} {\mathbf{H}}^{H} {\mathbf{J}}\left( {{\mathbf{J}}^{H} {\mathbf{CJ}}} \right)^{ - 1} {\mathbf{J}}^{H} {\mathbf{Hp}}, $$

(35)

where ${\mathbf{C}}$ is defined in (5).

According to (34), we have

$$ \left\{ {\begin{array}{*{20}c} {e^{{ - \lambda_{2} }} \sim\frac{{\chi_{{2\left( {K - Q + 1} \right)}}^{2} }}{{\chi_{{2\left( {K - Q + 1} \right)}}^{2} + \chi_{2Q}^{2} }}} & {{\text{under}}\;H_{0} } \\ {1 - e^{{ - \lambda_{2} }} \sim\frac{{\chi_{2Q}^{2} \left( {\mu_{2} } \right)}}{{\chi_{2Q}^{2} \left( {\mu_{2} } \right) + \chi_{{2\left( {K - Q + 1} \right)}}^{2} }}} & {{\text{under}}\;H_{1} } \\ \end{array} } \right.. $$

(36)

Referring to [48], we know that $e^{{ - \lambda_{2} }}$ follows a Beta distribution with two shape parameters $\left\{ {K - Q + 1,Q} \right\}$ under $H_{0}$. $1 - e^{{ - \lambda_{2} }}$ follows a Beta distribution with two shape parameters $\left\{ {Q,K - Q + 1} \right\}$ and a non-centrality parameter $\mu_{2}$ under $H_{1}$. Namely,

$$ \left\{ {\begin{array}{*{20}l} {e^{{ - \lambda_{2} }} \sim{\mathcal{B}\mathcal{E}\mathcal{T}\mathcal{A}}\left( {K - Q + 1,Q} \right)} \hfill & {{\text{under}}\;H_{0} } \hfill \\ {1 - e^{{ - \lambda_{2} }} \sim{\mathcal{B}\mathcal{E}\mathcal{T}\mathcal{A}}\left( {Q,K - Q + 1,\mu_{2} } \right)} \hfill & {{\text{under}}\;H_{1} .} \hfill \\ \end{array} } \right. $$

(37)

Based on (37), we derive the PDFs of $\lambda_{2}$ under $H_{0}$ and $H_{1}$ as

$$ h_{2} \left( {\left. {\lambda_{2} } \right|H_{0} } \right) = \frac{{\left( {1 - e^{{ - \lambda_{2} }} } \right)^{Q - 1} }}{{{\mathcal{B}}\left( {\delta_{2} ,Q} \right)e^{{\gamma_{2} }} }}, $$

(38)

$$ h_{2} \left( {\left. {\lambda_{2} } \right|H_{1} } \right) = \sum\limits_{j = 0}^{ + \infty } {\frac{{\left( {{{\mu_{2} } \mathord{\left/ {\vphantom {{\mu_{2} } 2}} \right. \kern-0pt} 2}} \right)^{j} }}{{e^{{{{\left( {2\gamma_{2} + \mu_{2} } \right)} \mathord{\left/ {\vphantom {{\left( {2\gamma_{2} + \mu_{2} } \right)} 2}} \right. \kern-0pt} 2}}} \cdot j!}} \cdot \frac{{\left( {1 - e^{{ - \lambda_{2} }} } \right)^{Q + j - 1} }}{{{\mathcal{B}}\left( {Q + j,\delta_{2} } \right)}}} , $$

(39)

respectively, where $\delta_{2} = K - Q + 1$ and $\gamma_{2} = \delta_{2} \lambda_{2}$.

2.3 B.3 PDFs of $\lambda = \lambda_{1} + \lambda_{2}$ in (13)

As given in (29), $\lambda_{1} \propto {\mathbf{x}}^{H} {\mathbf{P}}_{{\mathbf{J}}}^{ \bot } {\mathbf{x}}$ and $\lambda_{2} \propto {\mathbf{x}}^{H} {\mathbf{J}}\left( {{\mathbf{J}}^{H} {\mathbf{SJ}}} \right)^{ - 1} {\mathbf{J}}^{H} {\mathbf{x}}$. Observe that ${\mathbf{P}}_{{\mathbf{J}}}^{ \bot } \times {\mathbf{J}}\left( {{\mathbf{J}}^{H} {\mathbf{SJ}}} \right)^{ - 1} {\mathbf{J}}^{H} = {\mathbf{0}}_{M \times Q}$, by applying the Theorem 2.4.2 (2) in [20], we know that $\lambda_{1}$ and $\lambda_{2}$ are independent. Thus, the PDF of $\lambda = \lambda_{1} + \lambda_{2}$ under $H_{i}$, denoted by $g\left( {\left. \lambda \right|H_{i} } \right)$, is the convolution between $h_{1} \left( {\left. {\lambda_{1} } \right|H_{i} } \right)$ and $h_{2} \left( {\left. {\lambda_{2} } \right|H_{i} } \right)$, $i = 0,1$. Namely,

$$ \begin{aligned} g\left( {\left. \lambda \right|H_{i} } \right) & = h_{1} \left( {\left. {\lambda_{1} } \right|H_{i} } \right) \otimes h_{2} \left( {\left. {\lambda_{2} } \right|H_{i} } \right) \\ & = \int_{0}^{\lambda } {h_{1} \left( {\left. t \right|H_{i} } \right)h_{2} \left( {\left. {\lambda - t} \right|H_{i} } \right){\text{d}}t} ,\quad i = 0,1. \\ & = \int_{0}^{\lambda } {h_{1} \left( {\left. {\lambda - t} \right|H_{i} } \right)h_{2} \left( {\left. t \right|H_{i} } \right){\text{d}}t} \\ \end{aligned} $$

(40)

Substituting (32) and (38) into (40) leads to the PDF of $\lambda$ under $H_{0}$, as shown in (14). Substituting (33) and (39) into (40) results in the PDF of $\lambda$ under $H_{1}$, as given in (15).