Bias‐compensated FX‐LMS algorithm

Abstract

Active noise control is an expanding field that requires a suitable synthesis of secondary perturbations. Unfortunately, most schemes for noise cancelling do not take into account that the input signal that drives the adaptive filter can be noisy. In this Letter, it is theoretically shown that noise perturbations in the excitation data degrade the performance of the standard filtered‐x least mean squares (FX‐LMS) algorithm. Furthermore, a method that compensates such an issue is devised, and a first‐order stochastic analysis of the resulting algorithm is performed. The results reveal that the proposed scheme outperforms the standard FX‐LMS algorithm, even when the variance of the additive noise in the input is not accurately estimated.

Conjugate gradient‐based active noise cancellation

“…where the step size β ∈ R + is a user-defined parameter whose choice should account for convergence rate, asymptotic performance and tracking capabilities [6]. Assuming a constant vector w ∈ R N , the optimal solution in the mean square sense (i.e.…”

“…The minimization of Equation ( 9) can be obtained by enforcing ∇ w [ξ (w)] = 0, which implies that the optimal solution w opt ∈ R N also solves the following system of linear equations [6]:…”

“…The FX‐LMS algorithm iteratively updates the adaptive weight vector by an adaptation of the least mean squares (LMS) rule: $$$$\begin{equation} \bm{w}(k+1) = \bm{w}(k) + \beta e(k)\hat{\bm{x}}_{\text{f}}(k), \end{equation}$$$$where the step size $$\beta \in \mathbb {R}_+$$ is a user‐defined parameter whose choice should account for convergence rate, asymptotic performance and tracking capabilities [6].…”

“…The minimization of Equation () can be obtained by enforcing $$\nabla _{\bm{w}}\left[\xi (\bm{w})\right]=\bm{0}$$, which implies that the optimal solution $$\bm{w}_{\text{opt}} \in \mathbb {R}^N$$ also solves the following system of linear equations [6]: $$$$\begin{equation} \tilde{\bm{R}}\bm{w}_{\text{opt}}=\tilde{\bm{p}}, \end{equation}$$$$where $$\tilde{\bm{p}} \triangleq \sum _{m=0}^{M-1}s_m\bm{p}_m$$ and $$\tilde{\bm{R}} \triangleq \sum _{m=0}^{M-1}\sum _{n=0}^{M-1}s_ms_n\bm{R}_{n-m}$$. The results of Equation () imply that the FX‐LMS algorithm in a stationary setting solves a linear system of equations with low computational burden.…”

“…In such a context, CG schemes inherit the properties of both least mean squares (LMS) and recursive least squares (RLS) algorithms, in the sense that they present a faster convergence rate than the former and lower computational complexity than the latter (which requires the inversion of the input auto-correlation matrix) [3,5]. Adaptive filtering techniques are also state-of-the-art solutions for online active noise cancellation (ANC) tasks, which have been a research topic of great interest due to the inherent disadvantages of passive noise and vibration cancellation methods [6].…”