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Nonlinear adaptive filters, such as Volterra and functional link adaptive filter (FLAF), face increased computational complexity with rising input dimensionality or expansion order. To enhance computational efficiency, this paper introduces a nonlinear adaptive algorithm based on Rectified Linear Unit (ReLU). Applying ReLU to a tonal signal generates components corresponding to the original frequency and even harmonics. Additionally, applying ReLU to the sum of two tonal signals yields a spectrum encompassing intermodulation distortion frequencies. These traits enable the algorithm to mitigate harmonic and intermodulation distortions while maintaining judicious computational complexity. The numerical simulation results for the identification of nonlinear systems validate the algorithm's ability to fit harmonic and intermodulation nonlinear distortions. In simulations of nonlinear echo cancellation with diverse test signals and various distortion models, our proposed algorithm consistently demonstrates competitive performance, notably excelling in music and speech scenarios. [ABSTRACT FROM AUTHOR]

Highlights • In this paper, we focus on the recursive least-squares algorithm for bilinear forms, i.e., RLS-BF. • The bilinear term is defined with respect to the impulse responses of a spatiotemporal model. • Low-complexity versions of the RLS-BF algorithm are proposed, based on the DCD method. • A robust variable-regularized RLS-BF (VR-RLS-BF) algorithm is developed, using an estimate of the SNR. • Low-complexity versions of the VR-RLS-BF algorithm are derived, involving the DCD method. Abstract Due to its fast convergence rate, the recursive least-squares (RLS) algorithm is very popular in many applications of adaptive filtering, including system identification scenarios. However, the computational complexity of this algorithm represents a major limitation in applications that involve long filters. Moreover, when the parameter space becomes large, the system identification problem is more challenging and the adaptive filters should be able to cope with this aspect. In this paper, we focus on the identification of bilinear forms, where the bilinear term is defined with respect to the impulse responses of a spatiotemporal model. From this perspective, the solution requires a multidimensional adaptive filtering technique. Recently, the RLS algorithm tailored for bilinear forms (namely RLS-BF) was developed for this purpose. In this framework, the contribution of this paper is mainly twofold. First, in order to reduce the computational complexity of the RLS-BF algorithm, two versions based on the dichotomous coordinate descent (DCD) method are proposed; due to its arithmetic features, the DCD algorithm represents one of the most attractive alternatives to solve the normal equations. However, in the bilinear context, we need to consider the particular structure of the input data and the additional related challenges. Second, in order to improve the robustness of the RLS-BF algorithm in noisy environments, a regularized version is developed, together with a method to find the regularization parameters, which are related to the signal-to-noise ratio (SNR). Furthermore, using a proper estimation of the SNR, a variable-regularized RLS-BF algorithm is designed and two DCD-based low-complexity versions are proposed. Due to their nature, these variable-regularized algorithms have good robustness features against additive noise, which make them behave well in different noisy condition scenarios. Simulation results indicate the good performance of the proposed low-complexity RLS-BF algorithms, with appealing features for practical implementations. [ABSTRACT FROM AUTHOR]

Abstract The affine projection (AP) algorithm is one of the most celebrated adaptive filtering algorithms, and it achieves a good tradeoff between the convergence rate and computational complexity. However, the complexity of the AP algorithm increases with the projection order. A wealth of fast AP algorithms have been proposed to reduce the complexity in the last two decades. However, those low-complexity methods have not been well analyzed and compared. To fill this gap, this paper reviews the fast AP algorithms, including both fast filtering approaches and efficient solutions of the linear system of equations. The advantages and disadvantages of each fast implementation version are clarified based on an extensive performance evaluation, which could be useful to engineers when selecting a suitable algorithm for their specific applications and could also be a starting point for experts in this field to develop better solutions. [ABSTRACT FROM AUTHOR]