Fourier models for non-linear signal processing

Abstract

This paper proposes a trigonometric functional extension, hereafter named the Fourier model, as an alternative framework to the Volterra approach for non-linear systems modelling. This work is focused on the general advantages that trigonometric functionals show in adaptive implementations and also on the possibility they provide to reuse well-known linear processing tools in a non-linear context. The performance of the Fourier model is compared in a set of simulations that cover companders for audio and radio frequency amplifiers, probability density function (PDF) whitening and PDF estimation.

Zusammenfassung

In diesem Artikel wird eine trigonometrische funktionelle Erweiterung – im Folgenden Fouriermodell genannt – als alternatives Grundgerüst zum Volterra-Ansatz für eine nichtlineare Systemmodellierung vorgeschlagen. Diese Arbeit Konzentriert sich sowohl auf allgemeine Vorteile, die trigonometrische Funktionen in adaptiven Realisierungen zeigen, als auch auf die Möglichkeit, bekannte lineare Verarbeitungswerkzeuge in einem nichtlinearen Zusammenhang wiederzuverwenden. Die Leistungsfähigkeit des Fouriermodells wird in Simulation verglichen, wobei verschiedene Anwendungen wie Kompander für Audio- und Radiofrequenzverstärker, formgefilterte Wahrscheinlichkeitsdichtefunktion und Schätzungen von Wahrscheinlichkeitsdichtefunktionen abgedeckt werden.

Résumé

Cet article propose une extension fonctionnelle trigonométrique, que nous appellerons ci-après Modèle de Fourier, comme étant un cadre de travail alternatif à l'approche de Volterra pour la modélisation de systèmes non-linéaires. Ce travail est centré sur les avantages généraux que les fonctionnelles trigonométriques présentent dans des implémentations adaptatives et également sur la possibilité qu'elles offrent de réutiliser des outils bien connus de traitement linéaire dans un contexte non-linéaire. Les performances du modèle de Fourier sont comparées dans un ensemble de simulations qui couvent des companders pour amplificateurs audio et radio, le blanchiment de fonctions de densité de probabilité (FDP) et pestimation de FDP.

Introduction

Non-linear signal processing includes a wide range of applications where existing linear processing tools fail to provide appropriate results. The search of a `suitable’ functional extension in terms of the input signal is of paramount importance in the design of a non-linear system (NLS). The non-linear models more widely reported are usually based on polynomial functional extensions, such as the Volterra model or the G-functional Wiener model [11], [23]. More recent functional extensions are those approached in a neural network framework, particularly the so-called radial basis functions successfully applied in modern communication receivers [17], [26]. In addition, the contributions of fuzzy logic to the non-linear field are worth mentioning [5], [6], [12], [27].

As a contribution to all these NLS models based on the mapping of the input data, this work reports the advantages of using a trigonometric functional extension of the NLS input/output relationship [7], [9]. Previous miscellaneous applications of trigonometric functionals can be found for instance in [28] where they are useful for synchronous communications, in [16] to approximate the angular distribution in a nuclear reaction, and in [20] to develop an architecture that follows, as much as possible, the guidelines stated in Kolmogorov's theorem. A recent work on NLS identification and prediction is found in [24] where the trigonometric functionals are normalised by the input PDF in order to become orthogonal.

The major contributions of this work, derived from the use of the Fourier model over other existing alternatives in the fields of non-linear processing, are three:

• First, it is proved that, in a minimum mean square error (MMSE) design of the weights of the Fourier model, the uniform PDF plays the same role as the white spectral density in linear processing. Furthermore, it is shown that the data covariance matrix, involved in the MMSE design, is formed by samples of the characteristic function (CF), whence a parallel is drawn between non-linear processing based on the CF and linear processing based on the autocorrelation function.

• Second, the constant power of trigonometric kernels together with their high bandwidth offers, in adaptive algorithms, absolute control of misadjustment, fast trace convergence, and direct matrix inversion (DMI) performance from instantaneous gradient (LMS) updates.

• Third, the mentioned parallelism is further exploited introducing new PDF estimation procedures, which, acting over the CF, are formulated in the same manner as classical spectral density estimators.

These properties are described hereafter, ending with a set of simulations supporting the claimed superiority of the Fourier model over other models using different functional extensions. Since polynomial functionals are the most widely used, they will be taken as a baseline along the presentation, henceforth referred to as Volterra models.