This work presents a Kronecker product based methodology of frequency-domain beamforming of large sensor arrays for far-field broadband speech signals. The principal idea involves splitting up a given uniform linear array (ULA) into two smaller virtual ULAs (VULAs), using the Kronecker product. The linear system of the original ULA is bifurcated into two smaller linear systems of the VULAs. Henceforth, traditional adaptive beamformers such as the minimum-variance-distortionless-response (MVDR) beamformer may be obtained for each of the VULAs, using lesser data to estimate the statistics. The short-length beamformers, obtained from the VULAs, are finally combined by the Kronecker product to derive the full-length Kronecker product beamformer. Additionally, the VULAs allow fixed and adaptive beamforming to be implemented separately on each of them. As fixed beamformers do not employ statistical information, the Kronecker product hybrid beamformers reduce the original linear system to just a small linear system involving one VULA. Accordingly, hybrid beamformers may be implemented using traditional fixed beamformers, such as the delay-and-sum (DS) beamformer, on one VULA, and traditional adaptive beamformers, such as the MVDR, on the other. The proposed Kronecker product beamformers are observed to provide faster convergence and superior robustness with respect to the traditional beamformers.

这项工作提出了一种用于远场宽带语音信号的大型传感器阵列的频域波束形成基于Kronecker产品的方法。主要思想涉及使用Kronecker产品将给定的均匀线性阵列（ULA）分成两个较小的虚拟ULA（VULA）。原始ULA的线性系统分为两个较小的VULA线性系统。今后，可以使用较少的数据来估计统计量，从而为每个VULA获得传统的自适应波束形成器，例如最小方差无失真响应（MVDR）波束形成器。从VULA获得的短长度波束形成器最终由Kronecker产品合并，以得出全长Kronecker产品波束形成器。另外，VULA允许分别在它们上分别实现固定和自适应波束成形。由于固定波束形成器不使用统计信息，因此Kronecker产品混合波束形成器将原始线性系统缩减为仅包含一个VULA的小型线性系统。因此，可以在一个VULA上使用传统的固定波束形成器（例如，延迟与和（DS）波束形成器），在另一个上使用传统的自适应波束形成器（例如MVDR）来实现混合波束形成器。与传统的波束形成器相比，观察到建议的Kronecker产品波束形成器可提供更快的收敛性和出色的鲁棒性。例如，一个VULA上的延迟与和（DS）波束形成器，另一个上的传统自适应波束形成器，例如MVDR。与传统的波束形成器相比，观察到建议的Kronecker产品波束形成器可提供更快的收敛性和出色的鲁棒性。例如，一个VULA上的延迟与和（DS）波束形成器，另一个上的传统自适应波束形成器，例如MVDR。与传统的波束形成器相比，观察到建议的Kronecker产品波束形成器可提供更快的收敛性和出色的鲁棒性。