Problem Formulation

Consider a $p$-element antenna array with arbitrary locations of sensors. The $p$-dimensional input signal ${\vec{\bf z}}$ is assumed to be complex random Gaussian vector. We suppose that $N$ samples of the signal $z_1, z_2, ..., z_N$ are statistically independent and identically distributed (i.i.d.) zero-mean random vectors with spatial covariance matrix $\S$. By reviewing of non-singular case, let us suppose that the sample size $N$ is greater than the number of the antenna elements $(N > p)$. The problem of detection of some spatially correlated useful signal by the antenna array is formulated as a classical two-hypothesis alternative:

$$\begin{array}{l} \mbox{Null hypothesis (noise only),} \\ \medski... ...ignal plus noise),} \\ \medskip H_1 : \S\ne {\bf\sigma}^2 {\bf I}, \end{array}$$ (1)

where ${\bf I}$ is the identity matrix, and ${\bf\sigma}^2$ is the a-priori unknown noise power.

For the problem considered herein, the GLR test-statistic is given by

$$\L = \frac {\max\limits_{\S\in \omega} L(\vec{\bf0},\S )} {\max\limits_{\S\in\Omega}L(\vec{\bf0},\S )},$$ (2)

where

$$L(\vec{\bf0},\S ) = \frac{1}{\vert\S \vert^N\pi^{pN}} e^{-\sum\limits_{\alpha=1}^{N}{\vec{\bf z}}^{(\alpha)\dag }\S ^{-1}{\vec{\bf z}}^{(\alpha)}}$$ (3)

is the likelihood function for complex variables, $\omega$ is the parameter sub-region corresponding to the null hypothesis $H_0$ in the parameter space $\Omega$, $\vert.\vert$ is the determinant of a matrix and the superscript $\dag$ represents the transpose conjugate or Hermitian operator. Note that the test-statistic obtained accordingly with (2) can accept values only in interval [0,1].

It can be shown [1] that maximum value of the GLR (2) denominator is achieved by using maximum likelihood estimation $\hat\S _\Omega$ of the covariance matrix $\S$:

$\hat\S _\Omega=\frac{1}{N}{\bf A}= \frac{1}{N}\sum\limits_{\alpha=1}^{N}{\vec{\bf z}}^{(\alpha)}{\vec{\bf z}}^{(\alpha)^\dagger}$

and this value is equal to

$$\max\limits_{\S\in \Omega}L(\vec{\bf0},\S ) = \frac{1}{\vert\hat\S _\Omega\vert^N\pi^{pN}} e^{-pN}.$$ (4)

The maximum of the GLR (2) numerator should be obtained in parameter sub-region $\omega$ ($\S\in \omega$) corresponding to the null hypothesis (1). If hypothesis $H_0$ is true, the likelihood function is given as:

$L(\vec{\bf0},\S _0)=\prod\limits_{i=1}^pL_i(0,{\bf\sigma}^2)=\\ =\frac{1}{(\p... ...a}^2}\sum\limits_{\alpha=1}^N\sum\limits_{i=1}^p \vert z_i^{(\alpha)}\vert^2},$

where $L_i(0,{\bf\sigma}^2)=\prod\limits_{\alpha=1}^N\frac{1}{\pi{\bf\sigma}^2} e^{-z_i^{(\alpha)\star }{\bf\sigma}^{-2}z_i^{(\alpha)}}$ is the likelihood function of the signal from the $i$-th sensor.

From the last expression for the likelihood function the maximum value of the GLR (2) numerator is easily obtained

$$\max\limits_{\S\in \omega}L(\vec{\bf0},\S )=\max\limits_{{\bf\sig... ...{i=1}^p (L_i(0,{\bf\sigma}^2))=\frac{e^{-pN}} {\pi^{pN}(\hat{\bf\sigma})^{pN}},$$ (5)

and this maximum is achieved by using maximum likelihood estimation $\hat{\bf\sigma}^2$ of the variance ${\bf\sigma}^2$

$\hat{\bf\sigma}^2=\frac{1}{pN}\sum\limits_{\alpha=1}^N\sum\limits_{i=1}^p \vert z_i^{(\alpha)}\vert^2=\frac{Sp {\bf A}}{pN}$

where ${\bf A}=\sum\limits_{\alpha=1}^N{\vec{\bf z}}^{(\alpha)}{\vec{\bf z}}^{(\alpha)\dag }$

Taking into account the expressions (4), (5) the GLR test-statistic  (2) for the problem  (1) can be represented as:

$$\L = \frac{\vert\hat\S _{\Omega}\vert^N}{(\frac{Sp {\bf A}}{p})^{pN}(\frac{1}{N})^{pN}}= \frac{\vert{\bf A}\vert^N}{(\frac{Sp {\bf A}}{p})^{pN}}$$ (6)

The rejection region of the hypothesis $H_0$ is determined by the inequality $\L\le\L _{th}$, where $\L _{th}$ is the test-statistic threshold only dependent upon the given false alarm probability level $P_{FA} = \alpha$.