A Quaternionic Extension Algorithm of the Nonlinear Transformation for Active Impulsive Noise Control

The classical filtered-x least-mean-square (FxLMS) algorithm, which only accepts real-valued parameter operations, is insufficient in active impulsive noise control systems. The quaternion algorithms have better geometric properties than the real-valued and complex-valued algorithms. The paper proposes a filtered-x quaternion-valued least-mean-square algorithm of the arctangent nonlinear transformation, which is defined as the FxatanQLMS algorithm. FxatanQLMS uses the quaternion-valued least mean square (QLMS) as the core filter to operate multi-dimensional signals as single quaternion entities, and it uses the arctangent function to nonlinearly transform the error signal to smoothly adjust and constrain the update of the weight coefficients. To optimize the convergence capability, a variable-step-size FxatanQLMS (SHVSS-FxatanQLMS) algorithm containing a composite relationship that combines the sigmoid (S) function and harmonic (H) mean of the error signal, is presented to compress the error signal. Simulation studies on an impulsive-noise environment with a symmetric \(\alpha \) $\alpha$ -stable (S \(\alpha \) $\alpha$ S) distribution validate the proposed algorithms compared to other methods in terms of the convergence rate and noise reduction performance.