Slepian Beamforming: Broadband Beamforming using Streaming Least Squares

In this paper we revisit the classical problem of estimating a signal as it impinges on a multi-sensor array. We focus on the case where the impinging signal's bandwidth is appreciable and is operating in a broadband regime. Estimating broadband signals, often termed broadband (or wideband) beamforming, is traditionally done through filter and summation, true time delay, or a coupling of the two. Our proposed method deviates substantially from these paradigms in that it requires no notion of filtering or true time delay. We use blocks of samples taken directly from the sensor outputs to fit a robust Slepian subspace model using a least squares approach. We then leverage this model to estimate uniformly spaced samples of the impinging signal. Alongside a careful discussion of this model and how to choose its parameters we show how to fit the model to new blocks of samples as they are received, producing a streaming output. We then go on to show how this method naturally extends to adaptive beamforming scenarios, where we leverage signal statistics to attenuate interfering sources. Finally, we discuss how to use our model to estimate from dimensionality reducing measurements. Accompanying these discussions are extensive numerical experiments establishing that our method outperforms existing filter based approaches while being comparable in terms of computational complexity.

Broadband Beamforming via Linear Embedding

In modern applications multi-sensor arrays are subject to an ever-present demand to accommodate signals with higher bandwidths. Standard methods for broadband beamforming, namely digital beamforming and true-time delay, are difficult and expensive to implement at scale. In this work, we explore an alternative method of broadband beamforming that uses a set of linear measurements and a robust low-dimensional signal subspace model. The linear measurements, taken directly from the sensors, serve as a method for dimensionality reduction and serve to limit the array readout. From these embedded samples, we show how the original samples can be recovered to within a provably small residual error using a Slepian subspace model. Previous work in multi-sensor array subspace models have largely analyzed performance from a qualitative or asymptotic perspective. In contrast, we give quantitative estimates of how well different dimensionality reduction strategies preserve the array gain. We also show how spatial and temporal correlations can be used to relax the standard Nyquist sampling criterion, how recovery can be achieved through fast algorithms, and how "hardware friendly" linear measurements can be designed.