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Next: The Exact Analytical Expression Up: Approximation of distribution function Previous: Introduction

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:

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

    (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

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

    (2)

where

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

    (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


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

    (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


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

    (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:


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

    (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 $P_{FA} = \alpha$.


next up previous
Next: The Exact Analytical Expression Up: Approximation of distribution function Previous: Introduction