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

26-2 Passive, Active, and Digital Filtersdevelopment of hundreds of nonlinear signal processing algorithms. These algorithms range fromtheoretically derived broad filter classes, such as polynomial and rank–order based methods [1–9], toboutique methods tailored to specific applications. Thus the dynamic growth of nonlinear methods andlack of unifying theory makes covering the entirety of such operators in a single chapter impossible. Still,large classes of nonlinear filtering algorithms can be derived and studied through fundamentals that arewell founded.The fundamental approach adopted in this chapter is that realized through the coupling of statisticalsignal modeling with optimal estimation-based filter development. This general approach leads to anumber of well-established filtering families, with the specific filtering scheme realized depending on theestimation methodology adopted and the particular signal model deployed. Particularly amenable to filterdevelopment is the maximum likelihood estimation (M-estimation) approach. Originally developed inthe theory of robust statistics [10], M-estimation provides a framework for the development of statisticalprocess location estimators, which, when employed with sliding observation windows, naturally extend tostatistical filtering algorithms.The characteristics of a derived family of filtering operators depend not only on the estimationmethodology upon which the family is founded, but also on the statistical model employed to characterizea sequence of observations. The most commonly employed statistical models are those based on theGaussian distribution. Utilization of the Gaussian distribution is well founded in many cases, for instancedue to the central limit theorem, and leads to computationally simple linear operations that are optimalfor the assumed environment. There are many applications, however, in which the underlying processesare decidedly non-Gaussian. Included in this broad array of applications are important problems inwireless communications, teletrafic, networking, hydrology, geology, economics, and imaging [11–15].The element common to these applications, and numerous others, is that the underlying processestend to produce more large magnitude observations, often referred to as outliers or impulses, than ispredicted by Gaussian models. The outlier magnitude and frequency of occurrence predicted by a modelis governed by the decay rate of the distribution tail. Thus, many natural sequences of interest aregoverned by distributions that have heavier tails (e.g., lower tail decay rates) than that exhibited bythe Gaussian distribution. Modeling such sequences as Gaussian processes leads not only to a poorstatistical fit, but also to the utilization of linear operators that suffer serious degradation in the presenceof outliers.Couplings, an estimation (filtering) methodology with a statistical model appropriate for the observedsequence, significantly improves performance. This is particularly true in heavy tailed environments. Asan illustrative example, consider the restoration of an image corrupted by (heavy tailed) salt and peppernoise. Typical sources of salt and pepper include flecks of dust on the lens or inside the camera, or, indigital cameras, faulty CCD elements. Figure 26.1 shows a sample corrupted image, the results of twodimensionallinear and nonlinear filtering, and the true underlying (desired) image. It is clear that thelinear filter, unable to exploit the characteristics of the corrupting noise, provides an unacceptable result.On the other hand, the nonlinear filter, utilizing the statistics of the image, provides a very good result.The nonlinear filtering utilized in this example is the median, which is derived to be optimal for certainheavy tailed processes. This appropriate statistical modeling results in performance that is far superior tolinear processing, which is inherently based on the processing of light tailed samples.To formally address the processing of heavy tailed sequences, this chapter first considers sequences ofsamples drawn from the generalizes Gaussian distribution (GGD). This family generalizes the Gaussiandistribution by incorporating a parameter that controls the rate of exponential tail decay. Setting thisparameter to 2 yields the standard Gaussian distribution, while for values less than two the GGD tailsdecay slower than in the standard Gaussian case, resulting in heavier tailed distributions. Of particularinterest is the first order exponential decay case, which yields the double exponential, or Laplacian,distribution. The Gaussian and Laplacian GGD special cases warrant particular attention due to theirtheoretical underpinnings, widespread use, and resulting classes of operators when deployed in anM-estimation framework. Specifically, it is shown here that M-estimation of Gaussian distributed