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Variable step size norm-constrained adaptive filtering algorithms are proposed in the paper. A variable step size is derived by minimizing the variance of the noise-free a posterior error. Thus, the update equation can obtain a reasonable step size at each iteration. Due to the introduction of variable step size, the proposed algorithms based on the constrained conditions of $$L_{1}$$ and $$L_{0}$$ norm have a significant advantage that the convergence rate is faster than some well-known algorithms in the sparse system. The simulation results illustrate the good performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]