Uncertainty, fuzzy logic, and signal processing

Abstract

In this paper we focus on model-based statistical signal processing and how some problems that are associated with it can be solved using fuzzy logic. We explain how uncertainty (which is prevalent in statistical signal processing applications) can be handled within the framework of fuzzy logic. Type-1 singleton and non-singleton fuzzy logic systems (FLSs) are reviewed. Type-2 FLSs, which are relatively new, and are very appropriate for signal processing problems, because they can handle linguistic and numerical uncertainties, are overviewed in some detail. The output of a type-2 FLS is a type-2 fuzzy set. Using a new operation called type-reduction, the type-2 set can be reduced to a type-1 set } the type-reduced set } which plays the role of a con"dence interval for linguistic uncertainties. No such result can be obtained for a type-1 FLS. We demonstrate, by means of examples, that a type-2 FLS can outperform a type-1 FLS for one-step prediction of a Mackey}Glass chaotic time series whose measurements are corrupted by additive noise, and equalization of a nonlinear time-varying channel.2000 Elsevier Science B.V. All rights reserved. ZusammenfassungIn diesem Artikel setzen wir den Schwerpunkt auf modellbasierte statistische Signalverarbeitung und zeigen MoK glichkeiten zur LoK sung von Problemen dieses Umfeldes mit Hilfe der Fuzzy Logik auf. Wir erlaK utern, wie Entscheidungsunsicherheit (die in saK mtlichen Anwendunger der statistischen Signalverarbeitung vorherrscht) im Rahmen der Fuzzy Logik behandelt werden kann. ZunaK chst werden Typ-1 Singleton und Nicht-Singleton Fuzzy Logik Systeme (FLS) besprochen. Typ-2 FLS, die relativ neu sind, werden etwas detaillierter behandelt. Sie eignen sich sehr zur LoK sung von Signalverarbeitungsproblemen, da sie linguistische und numerische Unsicherheit handhaben koK nnen. Die Ausgabe eines Typ-2 FLS stellt die Typ-2 Fuzzy-Menge dar. Mit Hilfe einer neuen, als Typ-Reduktion bezeichneten Operation, kann die Typ-2 Menge auf eine Typ-1 Menge } die Typ-reduzierte Menge } uK berfuK hrt werden. Sie entspricht einem Kon"denzintervall fuK r linguistische Unsicherheit. Kein derartiges Resultat kann fuK r eine Typ-1 FLS abgeleitet werden. Anhand von Beispielen zeigen wir, da{ eine Typ-2 FLS eiher Typ-1 FLS zum einen als Einschritt-PraK diktion uK berlegen sein kann, wenn, sie auf eine chaotische Mackey}Glass Zeitreihe angwandt wird, deren Me{werte durch additives Rauschen gestoK rt sind oder wenn sie zum anderen bei der Entzerrung eines nichtlinearen zeitvarianten Kanals angewandt wird.2000 Elsevier Science B.V. All rights reserved. Re2 sume2Dans cet article, nous nous concentrons sur le traitement statisque de signaux a`base de mode`les et sur la fac7 on dont certains proble`mes qui lui sont associeH s peuvent e( tre reH solus en utilisant de la logique #oue. l'incertitude (qui preH vaut en traitement statistique des signaux) peut e( tre manipuleH oe dans le cadre de la logique #oue. Nous passons en revue des systeH mes a`logique #oue (SLF) a`singletones de type 1 et sans s...

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