A Novel Fractional Order Normalized LMS Algorithm with Direction Optimization

A generalized fractional order least mean squares (LMS) algorithm with direction optimization is proposed in this paper. Firstly, the LMS algorithm with 0 < α < 2 is investigated, and it is shown that the larger iteration order, the faster convergence speed can be achieved. Besides, to improve the convergence speed further, the direction optimization method is introduced for the fractional order case, which extracts an optimal direction in a given data block as the iteration direction. In addition, for decreasing the weight noise in steady state, a more general variable step method with more design freedom is developed. Finally, the effectiveness and applicability of the proposed method are demonstrated in three numerical examples.