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Sparse system identification (SSI) is one of the most important problems in broadband wireless communication systems. At present, the sparse aware adaptive filtering algorithms have been developed for SSI. Considering the noisy input and output noise with non-Gausssian feature case, we propose a new sparse normalized least mean fourth (NLMF) algorithm to address this problem in SSI. The bias-compensated NLMF algorithm (BCNLMF) using the unbiasedness criterion can reduce the interference of the input noise. We introduce the correntropy induced metric (CIM) as an approximation of the $l_{0}$ -norm into the BCNLMF to develop a novel sparse BCNLMF (BC-CIM-NLMF) algorithm which can efficiently solve the SSI under the noisy input and output noise environments. Simulation results show that the proposed algorithm outperforms other existing sparse aware algorithms for SSI.