Performance analysis of MVDR beamformer in finite data: with and without spacial smoothing.

Abstract

In an MVDR beamformer, the array weights are chosen to pass the desired directional (look direction) signal without distortion while maximally rejecting interfering signals. The only assumption made is that the desired signal direction is known a priori.

The performance of the MVDR beamformer has been extensively studied for the case when the true or asymptotic array covariance matrix is available. In practical situations, however, only a finite number of snapshots are available from which the weight vector must be determined. In such cases, the array covariance matrix is estimated using the maximum likelihood estimate. The estimated matrix is perturbed from the asymptotic matrix, and consequently, the beamformer performance degrades.

In this thesis, the finite-data performance of the MVDR beamformer is analyzed using first-order perturbation analysis and the Wishart distribution property of the estimated covariance matrix. In particular, expressions for the mean power gain in any direction of interest, the mean output power, and the mean weight-error vector norm, as a function of the number of snapshots, are derived. These results are accurate to first order in the data size. No assumptions on signal correlations or array characteristics are made in the analysis. The results are simplified for a single interference case to show explicitly the effect of signal-to-noise ratio, spacing of the interference from the desired signal, and the correlation between them on finite-data performance.

How close the mean power gain expression predicts the result in any given experiment depends on its variance. This motivated the development of an expression for the variance. Since the result obtained using first-order approximations was not accurate enough, the variance expression was developed using second-order approximations.

It is well known that correlation between the look-direction signal and the interferences severely degrades the asymptotic performance of the beamformer. For the case of a uniform linear array, spatial smoothing has recently been proposed to alleviate problems due to correlation. To study how finite-data performance is affected by spatial smoothing, the above analysis has been extended to this case. The results show that spatial smoothing, in addition to decorrelating the sources, also reduces the effects due to finite-data perturbations. Thus, we can compensate for a low number of snapshots by using more smoothing steps (more subarrays).

Simulations are used to verify the usefulness of the analytical expressions, and the results show excellent agreement with the theoretically predicted values.