Optimal delay estimation in a multiple sensor array having spatially correlated noise

The maximal likelihood (ML) estimation of time-of-arrival differences for signals from a single source or target arriving at M \geq 2 sensors has been the subject of a large number of papers in recent years. These time differences or delays enable target location. Nearly all previous work has assumed noises which are independent among all sensors. Herein, noises are taken to have complex correlation between sensors. A set of nonlinear equations in the unknown delays is derived and the Fisher information matrix (FIM) for the estimates is also derived. The Cramer-Rao matrix bound (CRMB), which is the inverse of FIM, shows optimal estimator covariances. Computer evaluations are plotted for CRMB elements with varied SNR and noise covariance values typical of turbulent boundary layer noise in towed arrays and signal sources at infinite range (plane-wave fronts). Maximum changes in the bound are within ±3 dB for complex noise correlations with magnitudes up to 0.4, which we tested.