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.

中文翻译：

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一种用于端射阵列的鲁棒波数域超定向波束成形

在各种阵列配置中，基于线性差分阵列的超指向波束形成备受关注。众所周知，由于有限差分逼近的限制，其指向性性能在较高频率下会下降，并且由于实际系统的失配，其鲁棒性会恶化。在本文中，我们开发了一种用于端射阵列的波数域超指向波束形成，这是一种不能直接应用于线性阵列的相位模式波束形成的替代方法。这里，“波数域”是指通过乘法运算在波数域中计算空间导数。所提出方法的一个重要特征是它提供了一种新方法，可以利用阵列的所有传感器从重建的声场中获得高阶导数，这在满足 Nyquist 准则时有效，从而产生宽带频率无关波束形成器。另一方面，由于提议$N\rm {th}$ 阶超定向波束形成器没有传感器数量上限，可以通过增加传感器数量和使用最佳截断参数来增强其在较低频率下的鲁棒性，该截断参数起到去除对阵列失配敏感的小奇异值的作用。虽然证明了所提出的方法和加权最小二乘优化方法的性能几乎相同，但所提出方法的主要优点是采用了由实验确定的最佳截断参数来增加波束形成器的鲁棒性，其中利用了较高的本征光束对随机误差的高灵敏度。