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In this paper, we propose a robust sparsity-aware adaptive filtering algorithm under impulsive noise environment, by using the Huber loss function in the frame of adaptive proximal forward-backward splitting (APFBS). The APFBS attempts to suppress a time-varying cost function which is the sum of a smooth function and a nonsmooth function. As the smooth function, we employ the weighted sum of the Huber loss functions of the output residuals. As the nonsmooth function, we employ the weighted ℓ1 norm. The use of the Huber loss function robustifies the estimation under impulsive noise and the use of the weighted ℓ1 norm effectively exploits the sparsity of the system to be estimated. The resulting algorithm has low computational complexity with order, where is the tap length. Numerical examples in sparse system identification demonstrate that the proposed algorithm outperforms conventional algorithms by achieving robustness against impulsive noise.