Simulation of a Four-Element “Optimum” Sonar Array
Charles E. Schmid · The Journal of the Acoustical Society of America · 1968
Various optimization criteria for multichannel estimation and detection have led to solutions dependent upon measuring and then inverting the noise cross-spectral density matrix [Q(ω)]. This digital-computer simulation involves generating random-noise samples with known stationary statistics for four hydrophones, and thereafter, estimating the cross-spectral density matrices and the inverses at sample frequencies. The matrices are estimated by successively averaging periodograms [S(ωn)]*[S(ωn)]T, where [S(ωn)]4×1 is obtained with a fast Fourier transform of the four generated hydrophone time series. The inverse is initially calculated and subsequently updated via a standard matrix equation: [Q+S*ST]4×4−1 = Q−1−(1+STQ−1S*)−1Q−1S*STQ−1. The array gain, which is proportional to the sum of the elements of the inverse matrix, is calculated for the simulation and compared to theoretical gains using Gaussian samples and combinations of plane-wave coherent, incoherent, and isotropic noise fields. The difference between the theoretical and simulation is determined as a function of number of periodograms averaged.