Techniques based on conventional higher-order statistics fail when the underlying processes become impulsive. Although methods based on fractional lower-order statistics (FLOS) have proven successful in dealing with heavy-tailed processes, they fail in general when the noise distribution has very heavy algebraic tails, i.e., when the algebraic tail constant is close to zero. In this paper we introduce a signal processing framework that we call Zero-Order Statistics (ZOS). ZOS are well dejined for any process with algebraic or lighter tails, including the full class of a-stable distnbutions. We introduce zero-order scale and location statistics and study several of their properties. The intimate link between ZOS and FLOS is presented. We also show that ZOS are the optimal framework when the underlying processes are very impulsive.All jigures, simulations and source code utilized in this paper are reproducible and freely accessible in the Intemet at http:Nwww.ee.udel.ed~gonzalez/PUBS/HOS97a
The adaptation of Volterra lters by one particular method, the method of least mean squares (LMS), while easily implemented, is complicated by the fact that upper bounds for the values of step sizes employed by a parallel update LMS scheme are di cult to obtain. In this paper, we propose a modi cation of the Volterra lter in which the lter weights of a given order are optimized independently of those weights of higher order. Using this approach, we then solve the MMSE ltering problem as a series of constrained optimization problems, which produce a partially decoupled normal equation for the Volterra lter. From this normal equation, we are able to develop an adaptation routine which uses the principles of partial decoupling which is similar in form to the Volterra LMS algorithm, but with important structural di erences that allow a straightforward derivation of bounds on the algorithm's step sizes; these bounds can be shown to depend on the respective diagonal blocks of the Volterra autocorrelation matrix. This produces a reliable set of design guidelines which allow more rapid convergence of the lower-order weight sets.
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