用克雷洛夫子空间加速波束成形中的对角加载，提升大麦克风阵列稳定性。
Adaptive Diagonal Loading using Krylov Subspaces for Robust Beamforming
- 通过兰彻斯迭代构建小规模克雷洛夫子空间，快速估算相关矩阵极值特征值。
- 计算复杂度从O(M³)降至O(kM²)，性能与精确特征分解一致。
- 适合高动态声学环境下需要实时鲁棒波束成形的系统应用。
在高度动态的声学环境中，大规模麦克风阵列的自适应波束成形面临挑战。当说话者和干扰源快速移动时，用于估计空间相关矩阵的样本数量往往严重不足，导致白噪声增益（WNG）下降，引发目标信号严重衰减。为确保波束成形的稳定性和鲁棒性，我们此前提出一种基于Kantorovich不等式的自适应对角加载方法，可保证WNG严格落在指定范围内。然而，准确确定最小必要加载量需计算相关矩阵的极值特征值，这对大阵列而言是复杂度高达O(M³)的计算任务。本文提出一种高效的O(kM²)估计技术，利用兰彻斯迭代构建维度k ≪ M的克雷洛夫子空间。将相关矩阵投影至三对角矩阵后，通过瑞兹值快速收敛逼近真实极值特征值。实验表明，该兰彻斯加速方法性能与精确特征分解（EVD）完全一致，实现最优干扰抑制并严格满足WNG约束，同时计算开销大幅降低。
原文摘要 · Abstract (English)
Reliable adaptive beamforming is critical for large microphone arrays operating in highly dynamic acoustic environments. In scenarios characterized by fast-moving talkers and interferers, the available sample support for estimating the spatial correlation matrix is often snapshot-deficient. This deficiency degrades the White Noise Gain (WNG), leading to severe target signal cancellation. To ensure stable and robust beamforming, we previously proposed an adaptive diagonal loading method that leverages the Kantorovich inequality to guarantee the WNG remains strictly within specified bounds. However, accurately determining the smallest necessary loading level requires calculating the extreme eigenvalues of the spatial correlation matrix, a computationally expensive $\mathcal{O}(M^3)$ operation for large arrays. In this paper, we introduce a highly efficient $\mathcal{O}(kM^2)$ estimation technique using Lanczos iterations to build a small Krylov subspace. By projecting the correlation matrix onto a tridiagonal matrix of dimension $k \ll M$, we extract Ritz values that rapidly converge to the exact extreme eigenvalues. Our evaluations demonstrate that this Lanczos-accelerated approach achieves performance identical to exact Eigenvalue Decomposition (EVD), ensuring optimal interference suppression and strict WNG adherence at a fraction of the computational cost.
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