Since the existing block-sparse system identification algorithm based on Mean Square Error (MSE) shows poor performance under impulsive interference, an Improved Block Sparse-Normalization Least Mean Square (IBS-NLMS) algorithm was proposed by introducing the inverse hyperbolic sine cost function instead of MSE. A new cost function was constructed and the additive value was obtained by steepest-descent method. Furthermore, a new vector updating equation for filter coefficients was deduced. The adaptive update of the weight vector was close to zero in the presence of impulsive interference, which eliminated the estimation error of adaptive updating based on the wrong information. Meanwhile, mean convergence behavior was analyzed theoretically and then the simulation results demonstrate that in comparison with the Block Sparse-Normalization Least Mean Square (BS-NLMS) algorithm, the proposed algorithm has higher convergence rate and less steady-state error under non-Gaussion noise impulsive interference and abrupt change.