Normalized least mean squares algorithms for FIR adaptive filtering with or without the reuse of past information are known to converge often faster than the conventional least mean squares (LMS) algorithm. This correspondence analyzes an LMS-like algorithm: the binormalized data-reusing least mean squares (BNDR-LMS) algorithm. This algorithm, which corresponds to the affine projection algorithm for the case of two projections, compares favorably with other normalized LMS-like algorithms when the input signal is correlated. Convergence analyses in the mean and in the mean-squared are presented, and a closed-form formula for the mean squared error is provided for white input signals as well as its extension to the case of a colored input signal. A simple model for the input-signal vector that imparts simplicity and tractability to the analysis of second-order statistics is fully described. The methodology is readily applicable to other adaptation algorithms of difficult analysis. Simulation results validate the analysis and ensuing assumptions

This paper proposes low-complexity constrained affine-projection (CAP) algorithms. The algorithms are suitable for linearly constrained filtering problems often encountered in communications systems. The CAP algorithms derived in this paper trade convergence speed and computational complexity in the same way as the conventional affine-projection (AP) algorithm. In addition, data-selective versions of the CAP algorithm are derived based on the concept of set-membership filtering. The set-membership constrained affine-projection (SM-CAP) algorithms include several constraint sets in order to construct a space of feasible solutions for the coefficient updates. The SM-CAP algorithms include a data-dependent step size that provides fast convergence and low mean-squared error. The paper also discusses important aspects of convergence and stability of constrained normalized adaptation algorithms and shows that normalization may introduce bias in the final solution.

We detail in this paper an L ₁ -norm Linearly constrained normalized least-mean-square (L ₁ -CNLMS) algorithm and its weighted version (L ₁ -WCNLMS) applied to solve problems whose solutions have some degree of sparsity, such as the beamforming problem in uniform linear arrays, standard hexagonal arrays, and (non-standard) hexagonal antenna arrays. In addition to the linear constraints present in the CNLMS algorithm, the L ₁ -WCNLMS and the L ₁ -CNLMS algorithms take into account an L ₁ -norm penalty on the filter coefficients, which results in sparse solutions producing thinned arrays. The effectiveness of both algorithms is demonstrated via computer simulations. When employing these algorithms to antenna array problems, the resulting effect due to the L ₁ -norm constraint is perceived as a large aperture array with few active elements. Although this work focuses the algorithm on antenna array synthesis, its application is not limited to them, i.e., the L ₁ -CNLMS is suitable to solve other problems like sparse system identification and signal reconstruction, where the weighted version, the L ₁ -WCNLMS algorithm, presents superior performance compared to the L ₁ -CNLMS algorithm.