Two solutions are presented to the problem of efficiently approximating a family of noises parameterized by a scalar\Upsilon, 0 \leq \Upsilon \leq \infty. The noises are represented in the form of vectors withmrandom components, and their covariance matrices are such that the number of significant eigenvalues increases with\Upsilon. The noise sample vector is to be approximated, within a specified error\epsilon, by a linear combination of vectors taken from a fixed set ofmvectors that are independent of\Upsilon. Furthermore, for each\Upsilonthe number of approximating vectors is to be mlnlmlzed while keeping the error below\epsilon. This number increases with\Upsilonas does the number of significant eigenvalues. The problem is to find a sequence of parameter values\Upsilon{1} < \cdots < \Upsilon_{m},andasetofvectorsu_{1}, \cdots ,u_{m}such that, for eachj, \Upsilon_{j}is the maximum value of\Upsilonfor which the noise can be approximated within the error of\epsilonby using onlyjvectors, andu_{1}, \cdots , u_{j}are the approximatingjvectors corresponding to\Upsilon_{j}The critical constraint is that the set ofmapproximating vectors be independent of\Upsilon. In the first solution, the root-mean-square error is used for the error that is to remain below\epsilon. In the second, the sample error is used but the\epsilon-approximation is limited to only those noise samples which have nonnegligible average power. In both solutions a recursive scheme is given for obtaining\Upsilon_{1}, \cdots , \Upsilon_{m}andu_{1} , \cdots , u_{m}, the resultant\Upsilon-sequence andu-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.

The optimum detector of Gaussian signal fields in the multipath-anisotropic noise environment is derived. The optimum detector consists of a series of matched quadratic filters and a decisionmaker. The filters are spatially and temporally matched to the eigenwaves of the signal-to-noise covariance, and the decisionmaker compares a quadratic form with a preassigned threshold. The optimum detector for single-path signals in isotropic white noise is the well-known delay-and-sum beamformer, which is widely used because of its simple structure. In the case of multipath signals in anisotropic noise, a substantial improvement in detection performance is possible over the beamformer. This is shown by numerically comparing the detection probability of the beamformer with that of the optimum detector for this case. In addition, we numerically evaluate the sensitivity of such an improvement to accurate knowledge of the signal and noise parameters that are required by the optimum detector. We also show the possibility of further improvement by the addition of a vertical segment to the commonly used horizontal linear array to provide sensitivity to the elevation angles of multipath signals.