The University of Southampton
Engineering and the Environment

Research project: Fractional Calculus modelling of non-Gaussian noisy signals

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Many engineering signals are contaminated with interfering noise that is generally assumed to have Gaussian distribution characteristics. However, in real physical examples, many noise distributions have very long decaying tails, indicating their deviation from Gaussianity and can be modeled more accurately as non-Gaussian. Examples besides the focus of this work (i.e., thick-film sensors) include, electronic devices, neon lights, relay switching noise in telephone channels, automatic ignition systems, etc. Non- Gaussian measurement noise is usually non-stationary, very much dependent on the physical environment, and may be of infrequent occurrence.

Project Overview

In this project, we introduce a method of point-wise noise estimation and suppression by extracting fractional orders of local singularities of a signal. By designing and employing digital fractional differentiators a signal is decomposed into components which represent variable fractional orders of regular events or singularities. The signal de-noising procedure can then be realised by tackling it with appropriate fractional order filters. The proposed fractional operation modelling here is a posteriori without requirements for any advanced information of the input signals. Local information is extracted from a signal by fractional calculus operation through the change of its fractional order (i.e., Denoising by filtering different order singularities). In other words, fractional calculus can stationarise and filter the non-Gaussian noise by designing an appropriate rational fractional order transform function. Its stable inverse system can then be used to generate the non-Gaussian noise.

In our preliminary studies, we have designed fractional order digital differentiation filters by employing an exponential basis neural networks optimisation technique. The amplitude responses of the filters were obtained in the form of a sum of the exponential based functions. Then, the exponential basis function neural network was used to approximate the filters’ amplitude response coefficients. This approach has produced a good amplitude response approximation with a relatively constant phase, see Fig. 1.


Amplitude and phase responses of our fractional digital differentiator filter of order 0.4 compared with ideal curves
Fig. 1 Amplitude adn phase response

Associated research themes

Mechatronics, signal processing and control


Key Publication


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