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 <inline-formula><tex-math notation="LaTeX">$2Lth$</tex-math></inline-formula> moment of the estimation error, with <inline-formula><tex-math notation="LaTeX">$L$</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">$L > 1$</tex-math> </inline-formula>. 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 tradeoff on the value of <italic>L</italic> 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 <italic>L</italic>, initial mean-square deviation and signal-to-noise ratio. Analytical results are supported by simulations.