Efficient approximation of a family of noises for application in adaptive spatial processing for signal detection

Two solutions are presented to the problem of efficiently approximating a family of noises parameterized by a scalar . The noises are represented in the form of vectors with random components, and their covariance matrices are such that the number of significant eigenvalues increases with . The noise sample vector is to be approximated, within a specified error , by a linear combination of vectors taken from a fixed set of vectors that are independent of . Furthermore, for each the number of approximating vectors is to be mlnlmlzed while keeping the error below . This number increases with as does the number of significant eigenvalues. The problem is to find a sequence of parameter values ,andasetofvectors such that, for each is the maximum value of for which the noise can be approximated within the error of by using only vectors, and are the approximating vectors corresponding to The critical constraint is that the set of approximating vectors be independent of . In the first solution, the root-mean-square error is used for the error that is to remain below . In the second, the sample error is used but the -approximation is limited to only those noise samples which have nonnegligible average power. In both solutions a recursive scheme is given for obtaining and , the resultant -sequence and -set (orthonormal) are unique. The result is applied to adaptive spatial processing for signal detection in the case where the signal wave, though temporally incoherent, has a known wavefront, the dominant noise ls spatlally localized, and the processor must be nearly opthnum for a wide range of frequencies.