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A normalized robust mixed-norm (NRMN) algorithm for system identification in the presence of impulsive noise is introduced. The standard robust mixed-norm (RMN) algorithm, despite its ability to cope with impulsive noise by virtue of combining the first and second error norm in the cost function it minimizes, exhibits slow convergence, requires a stationary operating environment, and employs a constant step-size which needs to be determined a-priori. To overcome these limitations, the proposed NRMN algorithm introduces a time varying learning rate which is derived based upon the dynamics of the input signal, and thus no longer requires a stationary environment, a major drawback of the RMN algorithm. The normalized step-size is bounded from above and a parameter is introduced within its upper-bound, which provides a trade-off between the convergence rate and the steady-state coefficient error. The analysis and experimental results show that the proposed NRMN exhibits increased convergence rate and substantially reduces the steady-state coefficient error, as compared to the least absolute deviation (LAD) and RMN algorithms.