Competitive Mean-Squared Error Beamforming

Beamforming methods are used extensively in a variety of different areas, where one of their main goals is to estimate the source signal amplitude s(t) from the array observations y(t) = s(t)a + i(t) + e(t), t = 1,2,..., where a is the steering vector, i(t) is the interference, and e(t) is a Gaussian noise vector [1, 2]. To estimate s(t), we may use a beamformer with weights w so that s(t) = w*y(t), where s(t) is an estimate of s(t). To ensure that s(t) is close to s(t) in some sense, we may design the beamformer weights to minimize the MSE. However, since the MSE of a linear beamformer depends in general on the unknown signal power σ2, we cannot directly minimize the MSE. A possible approach is to minimize the MSE subject to a constraint on the beamformer that ensures that the MSE does not depend on σ, which, as we show below, results in the conventional Capon beamformer. However, this approach does not guarantee a small MSE, so that on average, the resulting estimate of s(t) may be far from s(t). Instead, it would be desirable to design a robust beamformer whose performance is reasonably good access all possible signal powers. In our work, we propose a minimax regret beamformer whose MSE is uniformly as close as possible to that of the optimal beamformer that knows σ2, for all possible values of σ2. Thus, we ensure that over a wide range power signal of values, our beamformer will result in a relatively low MSE. Specifically, the MSE corresponding to a beamformer w is (1) where R is the interference+noise covariance matrix, which is typically replaced by the sample covariance. Since σ2 is not known, we cannot choose an estimator to minimize the MSE directly. One approach is to force the term depending on σ2 to 0, and then minimize the MSE, i.e.,