In this paper, a novel widely linear complex-valued estimated-input adaptive filter (WLC-EIAF) is first proposed for processing noisy input and output data in the complex domain. The WLC-EIAF consists of two steps: (i) estimation of noise-free input and (ii) update of the weight vector, which is realized by alternating the minimization of an instantaneous perturbation with both input and output data. Based on the WLC-EIAF method and adopting the least mean-square (LMS) scheme, a widely linear complex-valued estimated-input LMS (WLC-EILMS) algorithm is developed. It is able to achieve an unbiased parameter estimation and, thus, outperforms the widely linear complex-valued LMS (WL-CLMS) algorithm in the presence of noisy input and output. In particular, for Gaussian signals, closed-form expressions are derived for its steady-state excess mean-square error performance. Furthermore, the linear complex-valued estimated-input LMS and linear real-valued estimated-input LMS algorithms are presented, which are two simplified versions of the WLC-EILMS for circular and real-valued signals, respectively. Simulation results demonstrate that the proposed methods achieve significantly improved performance in terms of mean-square deviation and mean-square error when compared to the WL-CLMS and CLMS algorithms.

In vehicle hands-free communication systems and video teleconferencing systems, echoes are typically encountered. Adaptive filters are usually employed to remove the echoes in these applications. However, large outliers and highly correlated speech input signals are two key factors that limit the performance of the adaptive filters. In this paper, we propose an affine projection Versoria (APV) algorithm, which is obtained by maximizing the summation of Versoria-cost reusing with a constraint on the square of the L2-norm of the filter weight vector difference. In this way, the features of the Versoria-cost maximization and data-reusing are combined and, consequently, the proposed APV algorithm obtains robustness to the large outliers and accelerates the convergence rate for the correlated input signals. The complexity of the proposed APV algorithm is analyzed and then a fast recursive filtering technique is introduced to reduce its complexity. A stability analysis proves that the proposed APV algorithm is convergent. In addition, an analytical expression of the steady-state excess mean-square error for the proposed APV algorithm has been derived and simulated results are in agreement with the theoretical analysis result. Simulations on echo channel estimation and echo cancellation show that the proposed APV algorithm performs much better than the maximum Versoria criterion, affine projection sign, and affine projection algorithms.