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2012 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Fractional Processes and Fractional-Order Signal Processing

Techniques and Applications

verfasst von: Hu Sheng, YangQuan Chen, TianShuang Qiu

Verlag: Springer London

Buchreihe : Signals and Communication Technology

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Fractional processes are widely found in science, technology and engineering systems. In Fractional Processes and Fractional-order Signal Processing, some complex random signals, characterized by the presence of a heavy-tailed distribution or non-negligible dependence between distant observations (local and long memory), are introduced and examined from the ‘fractional’ perspective using simulation, fractional-order modeling and filtering and realization of fractional-order systems. These fractional-order signal processing (FOSP) techniques are based on fractional calculus, the fractional Fourier transform and fractional lower-order moments. Fractional Processes and Fractional-order Signal Processing: presents fractional processes of fixed, variable and distributed order studied as the output of fractional-order differential systems; introduces FOSP techniques and the fractional signals and fractional systems point of view; details real-world-application examples of FOSP techniques to demonstrate their utility; and provides important background material on Mittag–Leffler functions, the use of numerical inverse Laplace transform algorithms and supporting MATLAB® codes together with a helpful survey of relevant webpages. Readers will be able to use the techniques presented to re-examine their signals and signal-processing methods. This text offers an extended toolbox for complex signals from diverse fields in science and engineering. It will give academic researchers and practitioners a novel insight into the complex random signals characterized by fractional properties, and some powerful tools to analyze those signals.

Inhaltsverzeichnis

Frontmatter

Overview of Fractional Processes and Fractional-Order Signal Processing Techniques

Frontmatter
Chapter 1. Introduction
Abstract
Chapter 1 briefly introduces the research motivation of the monograph, and the basic theories of fractionalorder signal processing techniques. Fractional processes and the fractional-order signal processing techniques are extended from conventional integer-order processes and integer-order signal processing techniques, respectively. Fractional processes are characterized by heavy-tailed distribution, power-law decay of autocorrelation, or local memory. In order to effectively study these fractional processes, many fractionalorder signal processing techniques were provided, such as fractional filter, simulation of fractional processes, and fractional modeling. Fractional-order signal processing techniques are based on the basic theories of α stable distribution, fractional calculus and fractional Fourier transform. Besides, the contributions of the monograph are briefly summarized at the end of the chapter.
Hu Sheng, YangQuan Chen, TianShuang Qiu
Chapter 2. An Overview of Fractional Processes and Fractional-Order Signal Processing Techniques
Abstract
Chapter 2 provides an overview of basic concepts of fractional processes and fractional-order signal processing techniques from the perspective of fractional signals and fractional-order systems. Based on the fractional calculus, fractional-order systems are classified into three categories: constant-order fractional systems, variable-order fractional systems, and distributed-order fractional systems. Fractional processes, which can be considered as outputs of the fractional-order systems, have significant and complex long-memory properties. In order to best understand the fractional-order systems and extract valuable information from the fractional-order signals, fractional-order signal processing techniques are put forward for different kinds of fractional signals. All discussions on fractional-order signal processing techniques are centered around fractional calculus, fractional Fourier transform and α-stable distribution.
Hu Sheng, YangQuan Chen, TianShuang Qiu

Fractional Processes

Frontmatter
Chapter 3. Constant-Order Fractional Processes
Abstract
Chapter 3 deals with the constant-order fractional processes and the Hurst parameter estimators evaluation. A fractional process with a constant long memory parameter can be regarded as the output signal of a fractional-order system driven by white Gaussian noise. Typical constant-order fractional processes including fractional Brownian motion, fractional Gaussian noise, fractional stable motion, and fractional stable noise. A constant-order fractional process can be characterized by its long memory parameter H, the Hurst parameter or Hurst exponent. In this chapter, long-range dependent processes and Hurst parameter estimators are introduced. Furthermore, the robustness and the accuracy of twelve Hurst parameter estimators are extensively studied.
Hu Sheng, YangQuan Chen, TianShuang Qiu
Chapter 4. Multifractional Processes
Abstract
Chapter 4 deals with multifractional processes with a time varying local Hölder parameter, and the evaluation of various local Hölder estimators. The multifractional processes are the extension of fractional processes by generalizing the constant Hurst parameter H to the case where H is indexed by a time-dependent local Hölder exponent H(t). Multifractional Gaussian noise and multifractional Brownian motion are typical examples of multifractional processes. Multifractional stable noise and the multifractional stable motion are typical examples of multifractional processes with infinite second-order statistics. To better take advantage of multifractional processes, this chapter investigates both the tracking performance and the robustness issue of the twelve sliding-windowed Hurst estimators for noisy multifractional processes and for multifractional processes with infinite second-order statistics.
Hu Sheng, YangQuan Chen, TianShuang Qiu

Fractional-Order Signal Processing

Frontmatter
Chapter 5. Constant-Order Fractional Signal Processing
Abstract
Chapter 5 introduces the constant-order fractional signal processing techniques. The constant-order fractional signal processing techniques includes the simulation of constant-order fractional processes, constant-order fractional system modeling, fractional-order filter, and analogue realization of constant-order fractional systems. The relationship between constant-order fractional processes and constant-order fractional systems is investigated. Based on this relationship, the fractional Gaussian noise and fractional stable noise can both be simulated using the constant-order fractional integrator. In order to capture the long-range dependent property of the constant-order fractional processes, some constant-order fractional models, including FARIMA, FIGARCH and FARIMA with stable innovations were introduced. In addition, a fractional second-order filter G(s)=(s 2+as+b)−γ and its asymptotic properties were studied. At the end of the chapter, the analogue realization of the constant-order fractional integrator and differentiator was provided to meet the needs of practical applications.
Hu Sheng, YangQuan Chen, TianShuang Qiu
Chapter 6. Variable-Order Fractional Signal Processing
Abstract
Chapter 6 introduces variable-order fractional signal processing techniques. The simulation of multifractional processes was realized by replacing the constant-order fractional integrator with a variable-order integrator. So, the generated multifractional processes exhibit the local memory property. Similarly, variable-order fractional system models were built by replacing the constant-order long memory parameter d with a variable-order local memory parameter d t . The variable-order fractional system models can characterize the local memory of the fractional processes. A physical experimental study of the temperature-dependent variable-order fractional integrator and differentiator was introduced at the end of this chapter. Some potential applications of the variable-order fractional integrator and differentiator are briefly discussed.
Hu Sheng, YangQuan Chen, TianShuang Qiu
Chapter 7. Distributed-Order Fractional Signal Processing
Abstract
Chapter 7 studies the impulse response functions of the distributed-order integrator/differentiator, fractional-order distributed low-pass filter, and the fractional-order distributed parameter low-pass filter from the complex path integral expressed in the definite integral form. Based on these results, we obtained some asymptotic properties, and we can accurately compute the integrals on the whole time domain. Moreover, for practical applications, we presented a technique known as “impulse-response-invariant discretization” to perform the discretization of the above three distributed-order filters. Lastly, it was shown that the distributed-order fractional filters had some unique features compared with the classical integer-order or constant-order fractional filters.
Hu Sheng, YangQuan Chen, TianShuang Qiu

Applications of Fractional-Order Signal Processing Techniques

Frontmatter
Chapter 8. Fractional Autoregressive Integrated Moving Average with Stable Innovations Model of Great Salt Lake Elevation Time Series
Abstract
Chapter 8 presents an application example of fractional-order signal processing techniques in hydrology. The fractional-order signal processing techniques presented in Chap. 5 are used to study the north part of Great Salt Lake water-surface elevation time series, which possess long-range dependence and infinite variance properties. In this application example we show that FARIMA with stable innovations model can successfully characterize the Great Salt Lake historical water levels and predict its future rise and fall with much better accuracy. Therefore, we can observe that fractional-order signal processing techniques provide more powerful tools for forecasting the Great Salt Lake elevation time series with long-range dependent and infinite variance properties.
Hu Sheng, YangQuan Chen, TianShuang Qiu
Chapter 9. Analysis of Biocorrosion Electrochemical Noise Using Fractional Order Signal Processing Techniques
Abstract
Chapter 9 provides an application example of fractional-order signal processing techniques in biomedical signals. The electrochemical noise data of the bare Ti-6Al-4V bioimplant and the TiO2 nanoparticle coated Ti-6Al-4V bioimplant in three simulated biofluid solutions are analyzed. To draw a comparison between conventional analysis methods and fractional-order signal processing techniques, we first characterized the biocorrosion behavior using time domain statistic parameters, magnitudes of the Fourier transform, and spectral noise impedance. Although these techniques provided some valuable biocorrosion information, it is difficult to assess biocorrosion rate in long term electrochemical noise measurement. Compared with the conventional time or frequency domain based analysis techniques, fractional Fourier transform, fractional power spectrum, and the local Hurst parameter analysis of electrochemical noise data all provided improved results as observed from the signal processing figures.
Hu Sheng, YangQuan Chen, TianShuang Qiu
Chapter 10. Optimal Fractional-Order Damping Strategies
Abstract
Chapter 10 studies the optimal non-delayed fractional-order damping, time-delayed fractional-order damping, and optimal distributed order fractional damping based on ISE, ITSE, IAE and ITAE performance criteria. The comparisons of the step responses of the integer-order and the three types of fractional-order damping systems indicate that the optimal fractional-order damping systems achieve much better step responses than optimal integer-order systems in some instances, but sometimes the integer-order damping systems performs as well as fractional-order ones. Furthermore, time delay can sometimes be used to gain benefit in control systems, and, especially, the fractional-order damping plus properly chosen delay can bring outstanding performance. Time-delayed fractional-order damping systems can produce a faster rise time and less overshoot than others.
Hu Sheng, YangQuan Chen, TianShuang Qiu
Chapter 11. Heavy-Tailed Distribution and Local Memory in Time Series of Molecular Motion on the Cell Membrane
Abstract
Chapter 11 analyzes the heavy-tailed distribution and local memory characteristics of ten Class I major histocompatibility complex molecular jump time series. The histograms of ten jump time series are fitted by α-stable distributions, and the tail thickness of the distributions is quantified using the characteristic exponent parameter α. The long memory and local memory characteristics are tested using the Diffusion Entropy method and the sliding-windowed Koutsoyiannis’ method, respectively. The levels of long memory and local memory are quantified by Hurst parameter H and local Hölder exponent H(t). The analysis results show that the Class I major histocompatibility complex molecular jump time series obviously have heavy-tailed distribution and local memory characteristics. The local Hölder exponent can reflect the essential changes of these Class I major histocompatibility complex molecular motions.
Hu Sheng, YangQuan Chen, TianShuang Qiu
Chapter 12. Non-linear Transform Based Robust Adaptive Latency Change Estimation of Evoked Potentials
Abstract
Chapter 12 provides an application example of fractional-order signal processing techniques in evoked potentials signals. To improve the latency change estimation of evoked potentials under the lower order α-stable noise conditions by proposing, a new adaptive evoked potentials latency change estimation algorithm based on the fractional lower order moment and the nonlinear transform of the error function is proposed. The computer simulation shows that this new algorithm is robust under the lower order α-stable noise conditions, and it also achieves a better performance than the direct least mean square, direct least mean p-norm and signed adaptive algorithms without the need to estimate the α value of the evoked potentials signals and noises.
Hu Sheng, YangQuan Chen, TianShuang Qiu
Chapter 13. Multifractional Property Analysis of Human Sleep Electroencephalogram Signals
Abstract
In Chap. 13, different human sleep stages are investigated by studying the fractional and multifractional properties of sleep EEG signals. From analyzing the results for the fractional property of short term sleep EEG signals in different sleep stages, we can conclude that the average Hurst parameter H is different during different sleep stages. In comparison, the analysis results of multifractional characteristics for long term sleep EEG signals provided more detailed and more valuable information on various sleep stages. In different sleep stages, the fluctuations of local Hölder exponent H(t) exhibit distinctive properties, which are closely related to the distinct characteristics in a specific sleep stage. The emphasis of this study is to provide a novel and more effective analysis technique for dynamic sleep EEG signals.
Hu Sheng, YangQuan Chen, TianShuang Qiu
Chapter 14. Conclusions
Abstract
Chapter 14 concludes the monograph and proposes specific research problems to be solved by taking the advantages of fractional-order signal processing techniques. The monograph presents fractional processes and fractional-order signal processing techniques from the perspective of fractional signals and fractional-order systems. Fractional processes, which can be considered as outputs of the fractional-order systems, are categorized as constant-order fractional processes, variable-order fractional processes and distributed-order fractional processes. In order to best understand the fractional-order systems and extract valuable information from the fractional-order signals, fractional-order signal processing techniques are put forward for different kinds of fractional signals. Fractional-order signal processing techniques have been extensively used in econometrics, communication, biomedicine, hydrology, linguistics, and so on. we hope that the readers will use fractional thinking to understand natural or man-made phenomena, and use fractional techniques to solve the problems and gain additional insights after reading the monograph.
Hu Sheng, YangQuan Chen, TianShuang Qiu
Backmatter
Metadaten
Titel
Fractional Processes and Fractional-Order Signal Processing
verfasst von
Hu Sheng
YangQuan Chen
TianShuang Qiu
Copyright-Jahr
2012
Verlag
Springer London
Electronic ISBN
978-1-4471-2233-3
Print ISBN
978-1-4471-2232-6
DOI
https://doi.org/10.1007/978-1-4471-2233-3

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