When the additive Gaussian noise in the sensors' outputs is
colored (
i.e. , the noise values are
correlated in some fashion), the linearity of beamformingalgorithms means that the array processing output
also contains colored noise.
The solution to the colored-noise, binary detection problemremains the likelihood ratio, but differs in the form of the
a priori densities. The noise will again be
assumed zero mean, but the noise vector has non-trivialcovariance matrix
:
.
In this case, the logarithm of the likelihood ratio is
which, after the usual simplifications, is written
The sufficient statistic for the colored Gaussian noise
detection problem is
The quantities computed for each signal have a similar, but
more complicated interpretation than in the white noise case.
is a dot product, but with respect to the so-called
kernel
. The effect of the kernel is to weight certain
components more heavily than others. A positive-definitesymmetric matrix (the covariance matrix is one such example) can
be expressed in terms of its eigenvectors and eigenvalues.
The sufficient statistic can thus be written as the complicated
summation
where
and
denote the
eigenvalue and eigenvector of the covariance matrix
. Each of the
constituent dot products is largest when the signal and theobservation vectors have strong components parallel to
. However, the product of these dot
products is weighted by the reciprocal of the associatedeigenvalue. Thus, components in the observation vector parallel
to the signal will tend to be accentuated; those componentsparallel to the eigenvectors having the
smaller eigenvalues will receive greater
accentuation than others. The usual notions of parallelism andorthogonality become "skewed" because of the presence of the
kernel. A covariance matrix's eigenvalue has "units" ofvariance; these accentuated directions thus correspond to small
noise variance. We can therefore view the weighted dot productas a computation that is simultaneously trying to select
components in the observations similar to the signal, butconcentrating on those where the noise variance is small.
The second term in the expressions consistuting the optimal
detector are of the form
. This quantity is a special case of the dot product
just discussed. The two vectors involved in this dot productare identical; they are parallel by definition. The weighting
of the signal components by the reciprocal eigenvalues remains.Recalling the units of the eigenvectors of
,
has the units of a signal-to-noise ratio, which is
computed in a way that enhances the contribution of those signalcomponents parallel to the "low noise" directions.
To compute the performance probabilities, we express the
detection rule in terms of the sufficient statistic.
The distribution of the sufficient statistic on the left side of
this equation is Gaussian because it consists as a lineartransformation of the Gaussian random vector
. Assuming the
model to be true,
The false-alarm probability for the optimal Gaussian colored
noise detector is given by
As in the white noise case, the important signal-related
quantity in this expression is the signal-to-noise ratio of thedifference signal. The distance interpretation of this quantity
remains, but the distance is now warped by the kernel's presencein the dot product.
