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THIRD ANNUAL
ASAP '95 WORKSHOP

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Beamforming in Slow
Rayleigh Fading
Environments: Narrowband and Wideband Results

Alfred Hero
Department of Electrical Engineering and Computer Science
University of Michigan
Ann Arbor, MI 48109-2122
email: hero@eecs.umich.edu

Ron Delap
Air Force Institute of Technology
Wright-Patterson AFB, OH 45433
email: rdelap@afit.af.mil

Abstract
A transmission medium with Rayleigh fading produces severe variations in received signal amplitudes at the receiving array. Rayleigh fading is characteristic of optical and radio frequency tropospheric and ionospheric propagation, land, sea, and free-space radar backscatter, multipath propagation in mobile radio channels, and volume and surface reverberation in underwater acoustic channels. While statistical models for fading signals have been available for some time, including, for example, the Rayleigh, Rician, and Nakagami models, they are generally intractable in terms of developing optimal beamformer structures for DOA estimation or signal detection. In this presentation we derive optimal beamformers under a simplified model which is based on a slow fading assumption: amplitudes are coherent over space but incoherent over time with unknown mean and variance. We derive optimal beamformers under two different adaptation criteria: 1) maximization of a detectability index for detection; and 2) minimization of a monotone function of the CR lower bound for DOA estimation. These lead to optimal beamformers that can be implemented as a split beam adaptive algorithm. The algorithm controls and combines a signal nulling beam, which has intrinsically high angular resolution, and a signal enhancement beam, which has intrinsically high gain robustness. We show experimental results which indicate that our split beam adaptive array compares very favorably to other low complexity algorithms, e.g., Frost and Applebaum algorithms, while coming close to the performance of a clairvoyant maximum likelihood algorithm implemented with an Akaike signal selection criterion. Simulations of both narrowband and wideband arrays for Rayleigh fading signals are presented.

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