Adaptive Beamforming for Slow Fading Rayleigh Signals

R. DeLap Dept of EENG Air Force Institute of Technology Wright-Patterson AFB, OH 45433 A. Hero Dept of EECS The University of Michigan Ann Arbor, MI 48109 ABSTRACT In this paper we develop an adaptive beamsummer for direction of arrival (DOA) estimation of slow fading Raleigh signals, using a new design approach, termed \Adaptive Detection/Estimation for sPeci c Tasks" (ADEPT). For DOA estimation, the ADEPT method yields a weight adaptation criterion which is optimized for those weights that minimize the Cramer-Rao (CR) lower bound on achievable mean-square-error of any unbiased DOA estimator constructed on the beamsummer outputs. Simulation results are provided which show that the ADEPT DOA beamsummer yields DOA estimates whose meansquared error (MSE) approaches that of the more complex maximum likelihood implementation. 1. PROBLEM STATEMENT Figure 1 shows an m-element linear beamsumming array on which impinge p signals arriving from directions 1; : : : ; p. At the k-th time instant a snapshot Y k = [Y k 1 ; : : : ; Y k m]T of the m-sensors is acquired, k = 1; : : : ; n. We assume the following model for slow Rayleigh fading [1] in the received signals: for each signal the amplitudes are perfectly coherent over the array for any particular snapshot but vary randomly for di erent snapshots. In addition we assume that the ambient noise Nk = [Nk 1 ; : : : ;Nk m]T is spatially incoherent and temporally uncorrelated. Under the above assumptions we have the model for the beamsummer output xk: xk = WHV( )Sk +WHNk (1) where S = [s1; : : : ; sp]T is a vector of complex signal amplitudes, and V is an m p matrix of steering vectors. With this model the mean and covariance of the kth snapshot are: x = E[xk] = WHV S x = cov(x) = WHV SVHW+W NW (2) where N = 2 N I is the covariance of the array noise. 2. OPTIMAL BEAMSUMMING FOR DOA-ESTIMATION Here we review the ADEPT-DOA method of optimal weight design for DOA estimation using Fisher information and the Cramer-Rao (CR) bound [2]. k