Affine projection sign algorithm with ℓ1 minimization-based variable step-size

• A robust affine projection sign algorithm with ℓ1-norm minimization-based variable step-size is proposed.
• To solve the ℓ1-norm based minimization problem, an efficient numerical procedure is also presented.
• The proposed algorithm, not requiring any a priori knowledge, yields more robust convergence in impulsive noise environments.
• The proposed efficient numerical procedure can be further utilized to solve optimization problems of the same form.

In this paper, a new affine projection sign algorithm with a variable step-size, robust in impulsive noise environments, is proposed. For that purpose, ℓ1-norm of the a posteriori error vector is minimized under a box constraint on the step-size. Since the proposed ℓ1-norm minimization problem is non-differentiable convex one, an efficient numerical procedure is developed for the affine projection structure algorithm. Finally, it is demonstrated that the proposed algorithm, not requiring any a priori knowledge, yields more robust convergence in various impulsive noise environments than conventional algorithms with robustness to outliers and recent variable step-size algorithms with a priori information.