Sheng H., Chen Y., Qiu T. Fractional Processes and Fractional-Order Signal Processing. Techniques and Applications

Springer, 2012. — 322 p.In this monograph, we will introduce some complex random signals which are characterized by the presence of heavy-tailed distribution or non-negligible dependence between distant observations, from the ‘fractional’ point of view. Furthermore, the analysis techniques for these fractional processes are investigated using ‘fractional thinking’. The term ‘fractional process’ in this monograph refers to some random signals which manifest themselves by heavy-tailed distribution, long range dependence (LRD)/long memory, or local memory. Fractional processes are widely found in science, technology and engineering systems. Typical heavy-tailed distributed signals include underwater acoustic signals, low-frequency atmospheric noises, many types of man-made noises, and so on. Typical LRD/long memory processes and local memory processes can be observed in financial data, communications networks data and biological data. These properties, i.e., heavy-tailed distribution, LRD/long memory, and local memory, always lead to difficulty in correctly obtaining the statistical characteristics and extracting the desired information from these fractional processes. These properties cannot be neglected in time series analysis, because the tail thickness of the distribution, LRD, or local memory properties of the time series are critical in characterizing the essence of the resulting natural or man-made phenomena of the signals. Therefore, some valuable fractional-order signal processing (FOSP) techniques were provided to analyze these fractional processes. FOSP techniques, which are based on the fractional calculus, FLOM and FrFT, include simulation of fractional processes, fractional-order system modeling, fractional-order filtering, realization of fractional systems, etc. So, random signals which exhibit evident ‘fractional’ properties should be investigated using FOSP techniques to obtain better analysis results.
This monograph includes four parts. The first part is the overview of fractional processes and FOSP techniques. The second part presents fractional processes, which are studied as the output of the fractional order differential systems, including constant-order fractional processes and variable-order fractional processes. The third part introduces the FOSP techniques from the ‘fractional signals and fractional systems’ point of view. In the last part of the monograph, some application examples of FOSP techniques are presented to help readers understand and appreciate the fractional processes and fractional techniques. We sincerely wish that this monograph will give our readers a novel insight into the complex random signals characterized by ‘fractional’ properties, and some powerful tools to characterize those signals.Overview of Fractional Processes and Fractional-Order Signal Processing Techniques.An Overview of Fractional Processes and Fractional-Order Signal Processing Techniques.
Fractional Processes.
Constant-Order Fractional Processes.
Multifractional Processes.
Fractional-Order Signal Processing.
Constant-Order Fractional Signal Processing.
Variable-Order Fractional Signal Processing.
Distributed-Order Fractional Signal Processing.
Applications of Fractional-Order Signal Processing Techniques.
Fractional Autoregressive Integrated Moving Average with Stable Innovations Model of Great Salt Lake Elevation Time Series.
Analysis of Biocorrosion Electrochemical Noise Using Fractional Order Signal Processing Techniques.
Optimal Fractional-Order Damping Strategies.
Heavy-Tailed Distribution and Local Memory in Time Series of Molecular Motion on the Cell Membrane.
Non-linear Transform Based Robust Adaptive Latency Change Estimation of Evoked Potentials.
Multifractional Property Analysis of Human Sleep Electroencephalogram Signals.
Conclusions.
A Mittag-Leffler Function.
B Application of Numerical Inverse Laplace Transform Algorithms in Fractional-Order Signal Processing.
C Some UsefulWebpages.
D MatLAB Codes of Impulse Response Invariant Discretization of Fractional-Order Filters.