A Robust Wavenumber-Domain Superdirective Beamforming for Endfire Arrays

Abstract

Among various array configurations, superdirective beamforming based on linear differential arrays has attracted much attention. It is well known that its directivity performance degrades at higher frequencies due to the limitation of finite difference approximation, and its robustness deteriorates due to the mismatch of practical systems. In this paper, we develop a wavenumber-domain superdirective beamforming for endfire arrays, an alternative way of phase mode beamforming that cannot be applied directly to linear arrays. Here, “wavenumber-domain” means the spatial derivatives are calculated in the wavenumber domain through multiplication operations. An important feature of proposed method is that it provides a novel way to obtain the higher order derivatives from the reconstructed sound field utilizing all sensors of array, which is valid when the Nyquist criterion is satisfied, resulting in a broadband frequency-independent beamformer. On the other hand, since the proposed $N\rm {th}$ -order superdirective beamformer has no upper limit of number of sensors, its robustness at lower frequencies can be enhanced through increasing the number of sensors and using an optimum truncation parameter which plays a role to remove small singular values sensitive to the mismatch of array. Although it is demonstrated that the proposed method and the weighted least-squares optimization method lead to almost identical performances, the main advantage of the proposed method is that an optimum truncation parameter determined by the experiment is adopted to increase the robustness of beamformers, where the high sensitivity of the higher eigenbeams to random errors is exploited.