Block-diagonal normalization for robust multichannel underwater narrowband active noise control using parametric-array secondary sources

Localized underwater narrowband active noise control (ANC) requires secondary sources that balance compact size, directivity, and multichannel control stability. Underwater parametric arrays, which generate highly directional difference-frequency sound, provide a promising secondary-source solution for underwater noise control. However, in such multichannel control systems, secondary-path coupling and filtered-reference energy imbalance may degrade the multichannel normalized filtered-x least-mean-square (MNFxLMS) algorithm. Its global normalization factor can be dominated by high-energy filtered-reference components, reducing the effective step size of weak-energy branches and causing slow convergence or insufficient attenuation at certain target tones. Motivated by the near-block-diagonal secondary-path structure enabled by directional parametric-array secondary sources, this paper develops a branch-wise block-diagonal normalized MNFxLMS formulation. The proposed method replaces global normalization with branch-wise normalization for individual subfilters, thereby reducing the suppression of weak-energy updates. Compared with multichannel filtered-x affine projection algorithms (MFxAPA), it avoids projection matrix inversion while achieving comparable performance at lower computational complexity. A mean-square performance model is further developed to predict the transient behavior of the proposed and reference algorithms. Simulations show good agreement between theoretical predictions and Monte Carlo learning curves, and BD-MNFxLMS is less sensitive to filtered-reference imbalance. Experiments show that, under imbalanced conditions, BD-MNFxLMS outperforms MNFxLMS in 11 out of 12 electrical-domain target-tone comparisons, with particularly pronounced advantages for the weak-energy target branch. For BD-MNFxLMS, acoustic-domain measurements yield positive attenuation at all tested target spectral lines, ranging from approximately 8.16 to 20.63 dB, confirming the practical effectiveness of the proposed system. • A branch-wise normalized multichannel FxLMS formulation is developed. • Parametric-array secondary sources can produce approximately block-diagonal paths. • Global normalization is shown to suppress weak-energy branch updates. • Mean-square analysis accurately predicts Monte Carlo learning curves. • Water-tank tests confirm effective controller-side and acoustic-domain attenuation.