In this book, a novel algorithm, called the signed regressor least mean fourth (SRLMF) adaptive algorithm, that reduces the computational cost and complexity while maintaining good performance is presented. Expressions are derived for the steady-state excess-mean-square error (EMSE) of the SRLMF algorithm in a stationary environment. Also, expressions are obtained for the tracking EMSE of the SRLMF algorithm in a non-stationary environment. An optimum value of the step-size is also derived. Moreover, the weighted variance relation has been extended in order to derive expressions for the mean-square error (MSE) and the mean-square deviation (MSD) of the proposed algorithm during the transient phase. Computer simulations are carried out to corroborate the theoretical findings. It is shown that there is a good match between the theoretical and simulated results. It is also shown that the SRLMF algorithm has no performance degradation when compared with the least mean fourth (LMF) algorithm. The results in this study emphasize the usefulness of this algorithm in applications requiring reduced implementation costs for which the LMF algorithm is too complex.