Stabilization of High-Order Stochastic Gradient Adaptive Filtering Algorithms

The paper is concerned with stabilizing the family of adaptive filtering algorithms based on minimizing the 2Lth moment of the estimation error, with L being an integer greater than 1. Stabilization is attained via a proposed normalization of the algorithm. Mean square stability of the normalized algorithm is proved for a Markov plant for all L > 1. Transient and steady-state performances of the algorithm are analyzed for a time-invariant plant. Expressions are derived for the steady-state misadjustment and convergence time. Tradeoff between the transient and steady-state performances is evaluated. Dependence of this tradeoffon the value of L is studied. This tradeoff is compared with the tradeoff of the NLMS algorithm. The proposed algorithm is dramatically superior to the NLMS algorithm for sub-Gaussian noise. The superiority increases with L, initial mean-square deviation and signal-to-noise ratio. Analytical results are supported by simulations.