Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

Nonlinear Filtering Using Statistical Signal Models 26-11GGD family, and as such the linear and WM filters perform well in this case, with the Laplacian noiseoptimal WM returning the best performance. The a-Stable distribution, however, has significantlyheavier tails and both linear and WM filters breakdown in this environment. This indicates thatGGD-based methods are not well suited to extremely impulsive environments, and that more sophisticatedmethods for addressing samples characterized by very heavy (algebraic) tailed distributions must bedeveloped and employed.26.3 LM-EstimationMany contemporary applications contain samples with very heavy tailed statistics including the aforementionedpowerline communications, economic forecasting, network traffic processing, and biologicalsignal processing problems [15,37–44]. The GGD family of distributions, while representing a broad classof statistics with varying tail parameters, is, nevertheless, restricted to distributions with an exponentialrate of tail decay. Distributions with exponential rates of tail decay are generally considered light tailed,and do not accurately model the prevalence or magnitude of outliers in true heavy-tailed processes. Suchprocesses are often modeled utilizing the a-Stable family of distributions [16–19]. While a-Stabledistributions do possess tails with algebraic decay rates, and are thus appropriate models for impulsivesequences, the distribution lacks a full-family closed form expression and it is therefore not easily coupledwith estimation techniques such as ML.To overcome the drawbacks of GGD and a-Stable-based techniques, we derive a generalization of theM-estimation framework that exhibits a spectrum of optimality characteristics including greater robustness.This generalization is referred to as LM-estimation, the general form of which is given in thefollowing definition.Definition 26.2: Given the set of independent observations fx(i) : i ¼ 1, 2, ..., Ng formed asx(i) ¼ s(i; u) þ n(i), the LM-estimate of u is defined asX^u NLM ¼ arg minu i¼1logfd þ r(x(i) s(i; u))g, (26:15)where d > 0 and r() are the robustness parameter and cost function, respectively.In the following, we show that LM-estimation is statistically related to the GCD. The GCD familyconsists of algebraic detailed distributions with closed form expressions, and is therefore an appropriatemodel for heavy tailed sequences and a family from which estimation and filtering techniques can bederived. We consider GCD and LM-estimation based filters, focusing on the Cauchy and Meridiandistribution special cases and their resulting filter structures. Properties of the filters are detailed alongwith optimization procedures. While the GGD-based results are well established and reported innumerous works, the LM-estimation and GCD material presented represents the newest developmentsin this area, and as such proofs for many results are included.26.3.1 Generalized Cauchy Density and Maximum Likelihood EstimationAs the previous section covering GGD-based methods shows, the robustness of error norms, estimationtechniques, and filtering algorithms derived from a density is directly related to the density tail decay rate. Therobustness of LM-estimation derives from its statistical relation to the GCD. The GCD function is defined byf GCD (u) ¼lG(2=l)2[G(1=l)] 2n(n l þjuj l : (26:16)2=l)As in the GGD case, n is referred to as the scale parameter while l is called the shape parameter.