A Competitive Mean-Squared Error Approach to Beamforming

Abstract

Abstract-We treat the problem of beamforming for signal estimation where the goal is to estimate a signal amplitude from a set of array observations. Conventional beamforming methods typically aim at maximizing the signal-to-interference-plus-noise ratio (SINR). However, this does not guarantee a small mean-squared error (MSE), so that on average the resulting signal estimate can be far from the true signal. Here, we consider strategies that attempt to minimize the MSE between the estimated and unknown signal waveforms. The methods we suggest all maximize the SINR but at the same time are designed to have good MSE performance. Since the MSE depends on the signal power, which is unknown, we develop competitive beamforming approaches that minimize a robust MSE measure. Two design strategies are proposed: minimax MSE and minimax regret. We demonstrate through numerical examples that the suggested minimax beamformers can outperform several existing standard and robust methods, over a wide range of signal-to-noise ratio (SNR) values. Finally, we apply our techniques to subband beamforming and illustrate their advantage in estimating a wideband signal.Index Terms-Beamforming, minimax mean-squared error, minimax regret, robust beamforming, subband beamforming.

Rethinking Biased Estimation: Improving Maximum Likelihood and the Cramér–Rao Bound

“…In these tests, a value of b = −1 was used for the parameter set (4.35) of the EBME. Application of the minimax ideas presented here to beamforming in the context of array processing can be found in [55,57,125].…”

“…In the special case in which θ 0 = θ 0 is a scalar so that x = hθ 0 + w for some known vector h, the minimax regret estimate over the interval L ≤ |θ 0 | ≤ U is given by [55,57] …”

“…The minimax regret concept has recently been used to develop competitive beamforming approaches [55,57]. It has also been applied to linear models in which θ 0 is random with unknown covariance [42,46,53,54].…”

Asymptotically Optimal Linear Shrinkage of Sample LMMSE and MVDR Filters

“…This model reflects a huge number of statistical signal processing problems, e.g., in the context of beamforming in array signal processing can be obtained after measuring the output of an antenna array, is a signal to be estimated, is the steering vector which depends on the direction of arrival of the wavefront plane associated to and contains the noise plus interference signals [5]. Finally, it is important to mention that for the MVDR methods, could be modeled as a deterministic parameter as in [7], and the results presented herein still would be valid.…”

“…In order to circumvent this problem one can resort to the MVDR or Capon method [5], [6], which imposes a constraint that removes the dependence on the unknown second moment of the parameter of interest. This constraint is interpreted as a distortionless constraint in the direction of interest in array processing, and as an unbiasedness constraint when the parameter of interest is modeled as deterministic [7]. The price to pay, for the lack of knowledge about the prior information of the parameter to be estimated, is that the performance of the MVDR is worse than the one of the LMMSE in terms of MSE.…”