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This paper presents a brief overview of some existing fractional order signal processing (FOSP) techniques where the developments in the mathematical communities are introduced; relationship between the fractional operator and long-range dependence is demonstrated, and fundamental properties of each technique and some of its applications are summarized. Specifically, we presented a tutorial on 1) fractional order linear systems; 2) autoregressive fractional integrated moving average (ARFIMA); 3) 1/ f α noise; 4) Hurst parameter estimation; 5) fractional order Fourier transformation (FrFT); 6) fractional order linear transforms (Hartley, Sine, Cosine); 7) fractal and 8) fractional order splines. Whenever possible, we indicate the connections between these FOSP techniques.

A fractional Fourier transform (FrFT) based estimation method is introduced in this paper to analyze the long range dependence (LRD) in time series. The degree of LRD can be characterized by the Hurst parameter. The FrFTbased estimation of Hurst parameter proposed in this paper can be implemented efficiently allowing very large data set. We used fractional Gaussian noises (FGN) which typically possesses long-range dependence with known Hurst parameters to test the accuracy of the proposed Hurst parameter estimator. For justifying the advantage of the proposed estimator, some other existing Hurst parameter estimation methods, such as wavelet-based method and a global estimator based on dispersional analysis, are compared. The proposed estimator can process the very long experimental time

The long range dependence (LRD) of stationary process is characterized by the Hurst parameter. In practice, previous methods for estimation of the Hurst parameter might have poor performance when processing the non-stationary time series or trying to distinguish the slight difference between very long stochastic processes. This paper explores the use of fractional Fourier transform (FrFT) for estimating the Hurst parameter. The time series was processed locally to achieve a reliable local estimation of the Hurst parameter. The biocorrosion signal which is very popular in biological engineering was studied as an example to show the long range dependence properties. After comparing with the commonly used wavelet based method and another method based on Matlab's polyfit, the new Hurst parameter estimator proposed in this paper is proved to be more robust for non-stationarity and can show the slight difference clearly between those very long sets of biocorrosion data.Index Terms-biocorrosion signal, fractional Fourier transform, Hurst parameter, long range dependence, parameter estimation

A fractional Fourier transform (FrFT) based estimation method is introduced in this paper to analyze the long range dependence (LRD) in time series. The degree of LRD can be characterized by the Hurst parameter. The FrFT-based estimation of Hurst parameter proposed in this paper can be implemented efficiently allowing very large data set. We used fractional Gaussian noises (FGN) which typically possesses long-range dependence with known Hurst parameters to test the accuracy of the proposed Hurst parameter estimator. For justifying the advantage of the proposed estimator, some other existing Hurst parameter estimation methods, such as wavelet-based method and a global estimator based on dispersional analysis, are compared. The proposed estimator can process the very long experimental time series locally to achieve a reliable estimation of the Hurst parameter.