Array processing with known waveform and steering vector but unknown diagonal noise covariance matrix

The problem of estimating the complex amplitude of a signal which is known only to an unknown scaling factor with noise present is a well studied problem. Maximum likelihood (ML) and Capon estimates of the complex amplitude in the case where the noise vectors are circularly symmetric complex Gaussian with an unknown arbitrary covariance matrix have been proposed in previous literatures in closed form. In this paper, we consider the special case of the covariance matrix being diagonal (whose entries are still unknown). We reduce the ML estimation problem in this case to a non-linear optimization problem, and the optimal solution to the amplitude which maximizes the likelihood function can be obtained. We compare the performance of this method against the previously proposed method, which does not assume any structure of the covariance matrix. We show that the performance in the case wherein we use the fact that the covariance matrix is diagonal is only better in situations where the number of data samples is small. Both of the methods have the same Cramer-Rao bound (CRB), and both of them are asymptotically optimal. For large number of measurements, the mean squared errors (MSE) of both methods approach the CRB. Lastly, we provide an approximate ML solution to the problem, which has performance almost the same as the optimal solution, but computationally much more efficient.