In this work, the ε-normalized sign regressor least mean mixed-norm (NSRLMMN) adaptive algorithm is proposed. The proposed algorithm exhibits increased convergence rate as compared to the least mean mixed-norm (LMMN) and the sign regressor least mean mixed-norm (SRLMMN) algorithms. Also, the steady-state analysis and convergence analysis are presented. Moreover, the proposed ε-NSRLMMN algorithm substantially reduces the computational load, a major drawback of the ε-normalized least mean mixed-norm (NLMMN) algorithm. Finally, simulation results are presented to support the theoretical findings.