Generalized convolution

https://doi.org/10.1016/j.amc.2014.09.082Get rights and content

Highlights

  • Application of fractional calculus.

  • Generalization of correlation operator.

  • New concept in signal processing.

Abstract

The paper revisits the convolution operator and addresses its generalization in the perspective of fractional calculus. Two examples demonstrate the feasibility of the concept using analytical expressions and the inverse Fourier transform, for real and complex orders. Two approximate calculation schemes in the time domain are also tested.

Introduction

Fractional calculus (FC) is the scientific area that generalizes the standard integral and derivative operators up to real and complex orders. The historical origin of FC goes back to 1695, when L’Hospital asked about the meaning of D12y and Leibniz replied with some comments about the apparent paradox. Many important mathematician developed the topic during three centuries [1], [2], [3], [4], [5], [6], but only in the last decades applied sciences recognized the importance of the tool to model dynamical phenomena with long range memory effects [7], [8], [9], [10], [11], [12], [13]. Significant progresses took place in numerical analysis and signal processing [14], [15], [16], [17], [18], [19], but many important aspects still remain to be explored. For example, the interpretation of the fractional derivative or integral is still the object of strong debate and several perspectives were formulated [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32].

The concepts behind the generalization of the concept of derivatives and integrals can be applied with other operators. Several studies addressed the fractional convolution in the scope of the fractional Fourier transform [33], [34], [35], [36], [37], [38], [39], [40]. However, the study of convolution in signal analysis having in mind the FC tools has not received significant attention. This paper addresses the topic of generalizing that operator and evaluating its practical implementation in signal processing.

Bearing these ideas in mind, the paper is organized as follows. Section 2 discusses the convolution in the perspective of FC. Several examples are studied and two approximate techniques for its calculation are proposed. Finally, Section 3 outlines the main conclusions and points towards future work.

Section snippets

Convolution in the perspective of fractional calculus

The convolution of two signals f(t) and g(t) is defined as:C(f,g)(t)=(f*g)(t)=-+f(τ)g(t-τ)dτ=-+f(t-τ)g(τ)dτ,where t denotes time, is the standard symbol for the operator and C(·) is the notation adopted in the sequel.

When f and g are defined for t0 expression (1) reduces to:C(f,g)(t)=(fg)(t)=0tf(τ)g(t-τ)dτ.

Convolution is usually interpreted as the overlapping area between the two functions when one of them is flipped and shifted by t.

The product of the Laplace transforms of the

Conclusions and future work

The paper investigated the generalization of the convolution in the time domain in the perspective of FC. It was verified a relationship between generalized convolution and fractional integrals. Several examples were studied, based on analytical expressions and the numerical inversion of the Fourier transform, both for real and complex orders. Moreover, two simple interpolation algorithms for the calculation in the time domain were evaluated. The results reveal the feasibility of the proposed

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