A type of norm-adaption penalized least mean square/fourth (NA-LMS/F) algorithm is proposed for sparse channel estimation applications. The proposed NA-LMS/F algorithm is realized by incorporating a p-norm-like into the cost function of the conventional least mean square/fourth (LMS/F) which acts as a combination of the l(0)- and l(1)-norm constraints. A reweighted NA-LMS/F (RNA-LMS/F) algorithm is also developed by adding a reweighted factor in the NA-LMS/F algorithm. The proposed RNA-LMS/F algorithm exhibits an improved performance in terms of the convergence speed and the steady-state error, which can provide a zero attractor to further exploit the sparsity of the channel by the use of the-norm adaption penalty and the reweighting factor. The simulation results obtained from the sparse channel estimations are given to verify that our proposed RNA-LMS/F algorithm is superior to the previously reported sparse aware LMS/F and the conventional LMS/F algorithms in terms of both the convergence speed and the steady-state behavior. (C) 2016 Elsevier B.V. All rights reserved.