Active noise control algorithms aimed at reducing Gaussian noise have been studied. However, the performance of these algorithms can deteriorate if impulsive noise or other non-Gaussian noise such as piling noise exists. To solve this problem, an algorithm named the filtered-x affine projection sign algorithm with a post-adaptive filter is proposed to reduce impulsive noise. The proposed algorithm is based on the filtered-x affine projection sign algorithm and the post-adaptive filter structure. In addition, an algorithm with a variable step size is proposed to improve the performance of the filtered-x affine projection sign algorithm with a post-adaptive filter by using a convex combination strategy. The fixed step size in the convex combination is replaced with a linear function of the ratio between the estimated error signal and the estimated desired signal. The detailed derivation processes of the proposed algorithms are given. The computational complexities are analyzed. Numerical simulations are carried out to demonstrate the performance of the proposed algorithms. The results illustrate that the proposed algorithms can effectively reduce impulsive noise. At the same time, the algorithm with a variable step size has better convergence performance than the filtered-x affine projection sign algorithm with a post-adaptive filter.