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In this paper, we show that the sparse channel estimation problem can be formulated as a regularization problem between mean squared error (MSE) and the L1-norm constraint of the channel impulse response. A simple adaptive method to solve regularization problem using the convolution inequality for entropy is proposed. Performance of this proposed regularization method is compared to the Wiener filter, the matching pursuit (IMP) algorithm and the information criterion based method. The results show that the estimate of the sparse channel using the MSE criterion with the L1-norm constraint outperforms the Wiener filter and the conventional sparse solution methods in terms of MSE of the estimates and the generalization performance.