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.
['Jose Francisco de Andrade', 'Marcello L. R. de Campos', 'José Antonio Apolinário']
${{\bf L}_1}$ -Constrained Normalized LMS Algorithms for Adaptive Beamforming