Editing Array processing (section)

== General model and problem formulation==
Consider a system that consists of array of '''r''' arbitrary sensors that have arbitrary locations and arbitrary directions (directional characteristics) which receive signals that generated by '''q''' narrow band sources of known center frequency ω and locations θ1, θ2, θ3, θ4 ... θq. since the signals are narrow band the propagation delay across the array is much smaller than the reciprocal of the signal bandwidth and it follows that by using a complex envelop representation the array output can be expressed  (by the sense of superposition) as :<ref name="ref2"/><ref name="ref6"/><ref name="ref5"/><br>
<math>\textstyle x(t)=\sum_{K=1}^q a(\theta_k)s_k(t)+n(t)</math>

Where:
* <math>x(t)</math> is the vector of the signals received by the array sensors,
* <math>s_k(t)</math> is the signal emitted by the kth source as received at the frequency sensor 1 of the array,
* <math>a(\theta_k)</math> is the steering vector of the array toward direction (<math>\theta_k</math>),
* τi(θk): is the propagation delay between the first and the ith sensor for a waveform coming from direction (θk), 
* <math>n(t)</math> is the noise vector.

The same equation can be also expressed in the form of vectors:<br>
<math>\textstyle \mathbf x(t) = A(\theta)s(t) + n(t)</math>

If we assume now that M snapshots are taken at time instants t1, t2 ... tM, the data can be expressed as:<br>
<math>\mathbf X = \mathbf A(\theta)\mathbf S + \mathbf N</math>

Where X and N are the r × M matrices and S is q × M:<br>
<math>\mathbf X = [x(t_{1}), ......, x(t_{M})]</math><br>
<math>\mathbf N = [n(t_{1}), ......, n(t_{M})]</math><br>
<math>\mathbf S = [s(t_{1}), ......, s(t_{M})]</math>

'''Problem definition'''<br>
'''“The target is to estimate the DOA’s θ1, θ2, θ3, θ4 …θq of the sources from the M snapshot of the array x(t1)… x(tM). In other words what we are interested in is estimating the DOA’s of emitter signals impinging on receiving array,  when given a finite data set {x(t)} observed over t=1, 2 … M. This will be done basically by using the second-order statistics of data”'''<ref name="ref6"/><ref name="ref5"/>

In order to solve this problem (to guarantee that there is a valid solution) do we have to add conditions or assumptions on the operational environment and\or the used model? Since there are many parameters used to specify the system like the number of sources, the number of array elements ...etc. are there conditions that should be met first? Toward this goal we want to make the following assumptions:<ref name="utexas1"/><ref name="ref2"/><ref name="ref6"/>
# The number of signals is known and is smaller than the number of sensors, {{math|q < r}}.
# The set of any q steering vectors is linearly independent.
# Isotropic and non-dispersive medium – Uniform propagation in all directions.
# Zero mean white noise and signal, uncorrelated.
# Far-Field.
::a. Radius of propagation >> size of array.
::b. Plane wave propagation.

Throughout this survey, it will be assumed that the number of underlying signals, q, in the observed process is considered known. There are, however, good and consistent techniques for estimating this value even if it is not known.