Array Observation Model

We assume the standard narrowband observation model for D far-field sources impinging on an array of M sensors. Letting $\theta$ denote the vector of source location parameters to be estimated, the array observation is

$${\bf x}_k = {\bf A}(\theta) \; {\bf s}_k + {\bf n}_k , \hbox{ }k = 1,...,N,$$ (1)

where ${\bf s}_k$ denotes the $D \times 1$ vector of signal amplitudes at time k, ${\bf A} ( \theta )$ is the $M \times D$ matrix with (assumed linearly independent) array response vector columns ${\bf a} (\theta_{i}),$ for i=1,...,D.The $M \times 1$ noise vector, ${\bf n}_k$, is modeled as a zero-mean, temporally and spatially white Gaussian process with standard deviation $\sigma .$ Let ${\bf X} = [ {\bf x}_1 , {\bf x}_2 , ... , {\bf x}_N ]$, ${\bf N} = [ {\bf n}_1 , {\bf n}_2 , ... , {\bf n}_N ]$, and ${\bf S} = [ {\bf s}_1 , {\bf s}_2 , ... , {\bf s}_N ]$. Then

$${\bf X} = {\bf A} (\theta) \; {\bf S} + {\bf N} \; \; .$$ (2)

The complex signal amplitudes in ${\bf S}$ are treated as a set of ND unknown nuisance or secondary parameters. Our primary interest is the estimation of the location parameters $\theta$. With this array observation model and under the CML model assumption, the probability density function of the received data (conditioned on ${\bf S}$ as well as $\theta$) is:

$$f({\bf X}\vert{\bf S}, \theta) = \frac {1} {(\pi \sigma^2)^{... ..._k - {\bf A} ( \theta ) {\bf s}_k\vert\vert^2} {\sigma^2} } .$$ (3)

The basic equations above describe the source localization problem under consideration, based only on the fact that the additive noise is spatially/temporally white and Gaussian. In the succeeding sections, additional known properties of the random signal amplitudes will be considered, starting from the most general joint pdf case and proceeding to specific examples of signal amplitude probability distribution functions.