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“A Projection Approach for Adaptive Beamforming Under Imperfect Conditions”
by David Daniel Feldman
May 1993
Beamforming is used in communications and surveillance applications to process data obtained from an array of sensors. This powerful technique enables the interference and noise components of the data to be attenuated while the desired signal component is preserved. A common measure of the beamformer performance is the ratio of the signal to interference-plus-noise (SINR) at the beamformer output. When the interference and noise characteristics change with time, adaptation of the beamformer processor can result in a higher SINR than that obtained with a fixed processor. However, under imperfect conditions, this SINR improvement can be seriously degraded when the input signal-to-noise ratio or the number of sensors is large. Imperfect conditions occur when sensor miscalibration or other perturbation errors cause the actual and presumed steering vectors for the desired signal to differ. Such imperfect conditions also occur in sample matrix inversion processing when the desired signal is present in the snapshot data. A performance loss, which is due to errors in the sample covariance matrix estimate, results even when the steering vector is known exactly. This dissertation presents an analysis showing that sample covariance error can be viewed as a particular type of perturbation error. The analogy is used to derive a simple approximation for the expected performance of the adaptive beamformer as a function of the number of data snapshots.
In the dissertation, an approach is proposed for overcoming problems resulting from both perturbation and sample covariance errors. The approach involves projecting the presumed signal steering vector onto the subspace spanned by the principal eigenvectors of the sample covariance matrix. This projection method is proved to increase the SINR when perturbation errors occur, assuming that the data covariance matrix is perfectly known. A second-order Taylor series expansion is used to derive an accurate estimate of the projection method's mean SINR. In the analysis, the combined effects of randomly modeled perturbation and sample covariance errors are considered. In order to apply the projection method, the number of principal eigenvectors must be determined. To this end, a scaled-weight norm technique is developed; the technique is demonstrated with both field-recorded and simulated data and shown to be highly effective as long as mainlobe interference is absent. An extension of the technique which requires an estimate of the perturbation error variances can be applied when the presence of mainlobe interference is a possibility.