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

26-32 Passive, Active, and Digital FiltersGGD are not effective in the processing of very impulsive sequences. The GCD family, in contrast,consists of distributions with algebraic tail decay rates that do accurately modeled these most demandingimpulsive environments. Within this family we again focus on two special cases, the Cauchy andMeridian distributions and their resulting myriad and meridian filtering operations. As these filtersare derived from heavy tailed distributions, they are well suited for applications dominated by veryimpulsive statistics.Properties and optimization procedures are presented for each of the filters covered. In particular, weshow that the operators can be ordered in terms of their robustness, from least to most robust, as linear,median, myriad, and meridian. Moreover, myriad filters contain linear filters as special cases, whilemeridian filters contain median operators as special cases. Thus the myriad and meridian operatorsare inherently more efficient than their traditional (subset) counterparts. Simulations presented incommunications and frequency selective filtering applications show and contrast the performances ofthe filters in applications with varying levels of heavy tailed statistics. As expected, linear operatorsbreakdown in these environments while the robust, nonlinear operators yield desirable results.Although the presentation in this chapter ranges from theoretical development through properties,optimization, and applications, the coverage is, in fact, simply an overview of one segment within thebroad array of nonlinear filtering algorithms. To probe further, the interested reader is referred to thecited articles, as well as numerous other works in this area. Additionally, there are many other areas ofresearch in nonlinear methods that are under active investigation. Research areas of importance include(1) order-statistic based signal processing, (2) mathematical morphology, (3) higher order statistics andpolynomial methods, (4) radial basis function and kernel methods, and (5) emerging nonlinear methods.Researchers and practitioners interested in the broader field of nonlinear signal processing are encouragedto see the many good books, monographs, and research papers covering these, and other, areas innonlinear signal processing.References1. B. I. Justusson, Median filtering: statistical properties, Two Dimensional Digital Signal Processing II.New York: Springer Verlag, 1981.2. I. Pitas and A. Venetsanopoulos, Nonlinear Digital Filters: Principles and Application. New York:Kluwer Academic, 1990.3. K. E. Barner and G. R. Arce, Eds., Nonlinear Signal and Image Processing: Theory, Methods, andApplications. Boca Raton, FL: CRC Press, 2004.4. M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems. New York: Wiley, 1980.5. C. L. Nikias and A. P. Petropulu, Higher Order Spectra Analysis: A Nonlinear Signal ProcessingFramework. Englewood Cliffs, NJ: Prentice-Hall, 1993.6. V. J. Mathews and G. L. Sicuranza, Polynomial Signal Processing. New York: John Wiley & Sons, Inc.,2000.7. M. B. Priestrley, Non-linear and Non-stationary Time Series Analysis. New York: Academic, 1988.8. G. R. Arce, Nonlinear Signal Processing: A Statistical Approach. New York: John Wiley & Sons, Inc.,2005.9. H. A. David and H. N. Nagaraja, Order Statistics. New York: John Wiley & Sons, Inc., 2003.10. P. Huber, Robust Statistics. New York: John Wiley & Sons, Inc., 1981.11. H. Hall, A new model for impulsive phenomena: application to atmospheric noise communicationschannels, Stanford University Electronics Laboratories Technical Report, no. 3412–8 and 7050–7,1966, sU-SEL-66-052.12. J. Ilow, Signal proceesing in alpha–stable noise environments: Noise modeling, detection andestimation, PhD dissertation, University of Toronto, Toronto, ON, 1995.13. B. Mandelbrot, Long-run linearity, locally Gaussian processes, H-spectra, and infinite variances,International Economic Review, 10, 82–111, 1969.