Cramer-Rao bound and approximate maximum likelihood estimation for non-coherent direction of arrival problem

In previous work we proposed a direction of arrival (DOA) estimation method from non-coherent measurements taken by an array of sensors. Here, it is shown that the non-coherent measurements in the form of magnitude squared of array observations measured in the presence of additive white Gaussian noise are distributed according to a non-central chisquare distribution. It is further shown that, under certain conditions, the non-coherent measurements may be approximated by a Gaussian distribution. With this approximation, we develop the Cramer-Rao bound (CRB) on the non-coherent DOA estimation of a single source as well as an analytical expression of the maximum likelihood estimation (MLE) of the DOA. Numerical examples are presented to illustrate the performance of the non-coherent DOA estimator. For example, non-coherent DOA estimation outperforms coherent DOA when the standard deviation of the phase errors exceeds 15 degrees and the signal to noise ratio (SNR) exceeds 5 dB.