The problem sought to be solved by the present invention is one which arises out of a plane wavefront being intercepted by an array of sensors producing sensor outputs which are time related as a function of the angle at which the wavefront arrives at each sensor and as a function of frequency. The resultant difficulty is essentially that of detecting the direction of the origin of the wavefront in the presence of noise or other unwanted interference. The sensors may be acoustic such as hydrophones employed in a sonar system for the detection of underwater targets, radar antennae, or any comparable sensor array which produces time related outputs containing directional information in terms of phase and frequency. The outputs of a set of N sensors may be recorded as time functions X.sub.i(t), where i=1, 2, 3, . . . , N. It is desired to draw inferences about the environment of the array from the N time sequences. It is presumed that the X.sub.i(t) may be coordinated in some manner to produce environmental information.
This problem has usually been approached by applying linear transformations to the X.sub.i 's, summing the results, and then squaring and time averaging the result. The linear transformations are chosen to correspond to a far field source of signal in a particular direction. As an example, the linear transformations may be assumed to be represented as convolutions of the X.sub.i 's with a set of H.sub.i 's, the summed output is then ##EQU1##
When this output is squared and integrated the result becomes ##EQU2##
This expression can be simplified by letting x(f) and v(f) denote the Fourier transforms of x(t) and v(T), respectively, the output is therefore by parseval's theorem, ##EQU3##
If V.sup.T (f)=[v.sub.1 (f) - - - v.sub.N (f)] and X.sup.T (f)=[x.sub.1 (f) - - - x.sub.N (f)] then this expression may be written as ##EQU4##
If a large number of these time samples of output power are averaged: ##EQU5##
Where C(f)=&lt;-X*(f)X.sup.T (f)&gt; it should be noted that C is an N.times.N positive definite matrix.
This integrand may be considered one frequency at a time. The function of interest then is a function of .phi. which is representative of the angles describing the direction of interest in terms of azimuth and elevation, and the noise covariance, C. The V contains the dependence on .phi.. Specifically, the linear operators used are usually time delays chosen so that noise from a far field signal source will add coherently at the summation point. The v's are v.sub.i(t) =.delta.(T-T.sub.i) and v.sub.i (f)=exp (.sqroot.-1 2.pi.fT.sub.i), where T.sub.i is the travel time to the i.sup.th sensor from some surface of equal phase of the wavefront of interest. Therefore, at a frequency of interest, the function may be expressed as, p(.phi.,C)=V.sup.H (.phi.)CV(.phi.).
This function will generally be large if there is considerable noise coming from the (.phi.) direction and will generally be significantly smaller if there is considerably lesser noise coming from that direction. From it it is possible to derive a fairly accurate indication of the acoustic environment involved, however, in principle it cannot be claimed to be optimal.
Most optimum processing for sensor arrays is formulated for the detection and/or recovery of significant signal information from one specific direction (.phi.) in the presence of ambient noise. The solutions generally require noise covariance matrix, C.sub.o, to be measured at a time when the one specific signal of interest is known either to be present or to be absent. Any fault or error in identification of the circumstances under which C.sub.o is determined, may lead to subtle and not readily identifiable problems which are nonetheless very serious. Typically, a prior knowledge of C.sub.o is not possible. In the face of this state of facts, the effort must still be made to derive significant and reliable information from the observed C, because usually odd order moments are either absent or have no physical significance, and moments of higher order than two are prohibitively difficult to obtain because of the limitation in instrumentation to reliably develop information relative to the second order moments of interest. The problem is aggrevated considerably by the absence of a good and reliable criterion for determining if any other function q(.phi.,C) is better or worse than p(.phi.,C).
Accordingly, it is highly desirable that the criteria, methods, and equipment for data analysis currently in use be improved upon to provide more reliable and significant information as to the direction, i.e., elevation and azimuth origin of a wavefront which impinges upon a multi-sensor array so as to produce time related signals in terms of phase and frequency.