On forward-backward mode for direction of arrival estimation

We apply the MODE (method of direction estimation) principle to the forward-backward (FB) covariance of the output vector of a sensor array to obtain what we call the FB-MODE procedure. The derivation of FB-MODE is an interesting exercise in matrix analysis, the outcome of which was somewhat unexpected: FB-MODE simply consists of applying the standard MODE approach to the eigenelements of the FB sample co-variance matrix. By using an asymptotic expansion technique we also establish the surprising result that FB-MODE is outperformed, from a statistical standpoint, by the standard MODE applied to the forward-only sample covariance (F-MODE). We believe this to be an important result that shows that the FB approach, which proved quite useful for improving the performance of many suboptimal array processing methods, should not be used with a statistically optimal method such as F-MODE.