Sheng / Chen / Qiu Fractional Processes and Fractional-Order Signal Processing

Techniques and Applications
1. Auflage 2011
ISBN: 978-1-4471-2233-3
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 310 Seiten

Reihe: Signals and Communication Technology

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.

Doctor YangQuan Chen has authored over 200 academic papers plus numerous technical reports. He co-authored two textbooks: 'System Simulation Techniques with MATLAB®/Simulink' (with Dingyu Xue . Tsinghua University Press, April 2002, ISBN 7-302-05341-3/TP3137, in Chinese) and 'Solving Advanced Applied Mathematical Problems Using Matlab' (with Dingyu Xue. Tsinghua University Press. August 2004. 419 pages in Chinese, ISBN 7-302-09311-3/O.392); and six research monographs: 'Plastic Belt for Projectiles' (with Y. Shi. Shaanxi Science and Technology Press, Jan. 1995, ISBN 7-5369-2277-9/TJ.1, in Chinese), 'Iterative Learning Control ' (with C. Wen . Lecture Notes in Control and Information Sciences, Springer-Verlag, Nov. 1999, ISBN: 978-1-85233-190-0), 'Iterative Learning Control' (with Hyo-Sung Ahn and Kevin L. Moore. Springer, July 2007, ISBN: 978-1-84628-846-3), 'Optimal Observation for Cyber-physical Systems'(with Zhen Song, Chellury Sastry and Nazif Tas. Springer, July 2009, ISBN: 978-1-84882-655-7), 'Fractional-order Systems and Controls' (with Concepción A. Monje, Blas M. Vinagre, Dingyu Xue and Vicente Feliu, ISBN: 978-1-84996-334-3), and 'Optimal Mobile Sensing and Actuation Strategies in Cyber-physical Systems' (with Christophe Tricaud). His current research interests include autonomous navigation and intelligent control of a team of unmanned ground vehicles, machine vision for control and automation, distributed control systems (MAS-net: mobile actuator-sensor networks), fractional-order control, interval computation, and iterative/repetitive/adaptive learning control. Currently, he serves as an Associate Editor for IEEE Control Systems Society, Conference Editorial Board (CSSCEB ). He was also an Associate Editor of ISA Review Board for AACC 's American Control Conference ( ACC2005 ). He has been the Co-Organizer and Instructor of the Tutorial Workshops on 'Fractional-order Calculus in Control and Robotics' at IEEE 2002 Conference on Decision and Control (CDC'02), and 'Applied Fractional Calculus in Controls and Signal Processing' at CDC'10 and a founding member of the ASME subcommittee on 'Fractional Dynamics'.

Autoren/Hrsg.

Sheng, Hu

Chen, Yangquan

Qiu, Tianshuang

Inhaltsverzeichnis

1;Fractional Processes and Fractional-Order Signal Processing;3
1.1;Foreword;6
1.2;Preface;9
1.3;Acknowledgements;12
1.4;Contents;16
1.5;Acronyms;22
2;Part I: Overview of Fractional Processes and Fractional-Order Signal Processing Techniques;24
2.1;Chapter 1: Introduction;25
2.1.1;1.1 An Introduction to Fractional Processes and Analysis Methods;25
2.1.2;1.2 Basis of Stochastic Processes;28
2.1.2.1;1.2.1 Statistics of Stochastic Processes;28
2.1.2.2;1.2.2 Properties of Stochastic Processes;29
2.1.2.2.1;Mean Function;29
2.1.2.2.2;Variance Function;29
2.1.2.2.3;Correlation Function;29
2.1.2.2.4;Autocovariance Function;30
2.1.2.2.5;Cross-Correlation Function;30
2.1.2.2.6;Cross-Covariance Function;30
2.1.2.2.7;Moments;30
2.1.2.3;1.2.3 Gaussian Distribution and Gaussian Processes;31
2.1.2.4;1.2.4 Stationary Processes;32
2.1.3;1.3 Analysis of Random Signals;32
2.1.3.1;1.3.1 Estimation of Properties for Stochastic Signals;32
2.1.3.1.1;Estimation of the Mean Value;33
2.1.3.1.2;Estimation of the Variance;33
2.1.3.1.3;Estimation of the Covariance Function;33
2.1.3.1.4;Estimation of the Correlation Function;33
2.1.3.1.5;Estimation of the Cross-Covariance Function;33
2.1.3.1.6;Estimation of the Cross-Correlation;34
2.1.3.1.7;Estimation of the Moments;34
2.1.3.2;1.3.2 Simulation of Random Signals;34
2.1.3.3;1.3.3 Signal Filtering;35
2.1.3.3.1;Analogue Filters;35
2.1.3.3.2;Digital Filter;36
2.1.3.4;1.3.4 Modeling Random Processes;37
2.1.3.5;1.3.5 Transform Domain Analysis;38
2.1.3.5.1;Fourier Transform;38
2.1.3.5.2;Laplace Transform;38
2.1.3.5.3;Z-Transform;39
2.1.3.5.4;Wavelet Transform;39
2.1.3.5.5;Hilbert Transform;40
2.1.3.5.6;Mellin Transform;40
2.1.3.6;1.3.6 Other Analysis Methods;41
2.1.4;1.4 Research Motivation;41
2.1.4.1;1.4.1 Heavy Tailed Distributions;41
2.1.4.2;1.4.2 Long Range Dependence;42
2.1.4.3;1.4.3 Local Memory;44
2.1.5;1.5 Basics of Fractional-Order Signal Processing;45
2.1.5.1;1.5.1 Fractional Calculus;45
2.1.5.1.1;Constant-Order Fractional Calculus;45
2.1.5.1.2;Distributed-Order Fractional Calculus;46
2.1.5.1.3;Variable-Order Fractional Calculus;46
2.1.5.2;1.5.2 alpha-Stable Distribution;47
2.1.5.3;1.5.3 Fractional Fourier Transform;48
2.1.6;1.6 Brief Summary of Contributions of the Monograph;50
2.1.7;1.7 Structure of the Monograph;50
2.2;Chapter 2: An Overview of Fractional Processes and Fractional-Order Signal Processing Techniques;52
2.2.1;2.1 Fractional Processes;52
2.2.1.1;2.1.1 Fractional Processes and Fractional-Order Systems;53
2.2.1.1.1;Review of Conventional Random Processes and Integer-Order Systems;53
2.2.1.1.2;Constant-Order Fractional Processes and Constant-Order Fractional Systems;54
2.2.1.2;2.1.2 Stable Processes;56
2.2.1.3;2.1.3 Fractional Brownian Motion;57
2.2.1.4;2.1.4 Fractional Gaussian Noise;58
2.2.1.5;2.1.5 Fractional Stable Motion;58
2.2.1.6;2.1.6 Fractional Stable Noise;59
2.2.1.7;2.1.7 Multifractional Brownian Motion;59
2.2.1.8;2.1.8 Multifractional Gaussian Noise;59
2.2.1.9;2.1.9 Multifractional Stable Motion;60
2.2.1.10;2.1.10 Multifractional Stable Noise;60
2.2.2;2.2 Fractional-Order Signal Processing Techniques;60
2.2.2.1;2.2.1 Simulation of Fractional Random Processes;60
2.2.2.2;2.2.2 Fractional Filter;61
2.2.2.3;2.2.3 Fractional-Order Systems Modeling;62
2.2.2.4;2.2.4 Realization of Fractional Systems;62
2.2.2.4.1;Analogue Realization of Fractional Systems;62
2.2.2.4.2;Digital Realization of Fractional Systems;64
2.2.2.5;2.2.5 Other Fractional Tools;64
2.2.2.5.1;Fractional Hilbert Transform;64
2.2.2.5.2;Fractional Power Spectrum Density;65
2.2.2.5.3;Fractional Splines;66
2.2.3;2.3 Chapter Summary;67
3;Part II: Fractional Processes;68
3.1;Chapter 3: Constant-Order Fractional Processes;69
3.1.1;3.1 Introduction of Constant-Order Fractional Processes;69
3.1.1.1;3.1.1 Long-Range Dependent Processes;69
3.1.1.2;3.1.2 Fractional Brownian Motion and Fractional Gaussian Noise;71
3.1.1.2.1;Fractional Brownian Motion;71
3.1.1.2.2;Fractional Gaussian Noise;72
3.1.1.3;3.1.3 Linear Fractional Stable Motion and Fractional Stable Noise;73
3.1.1.3.1;Linear Fractional Stable Motion (LFSM);73
3.1.1.3.2;Fractional Stable Noise;74
3.1.2;3.2 Hurst Estimators: A Brief Summary;76
3.1.2.1;3.2.1 R/S Method;76
3.1.2.2;3.2.2 Aggregated Variance Method;76
3.1.2.3;3.2.3 Absolute Value Method;77
3.1.2.4;3.2.4 Variance of Residuals Method;77
3.1.2.5;3.2.5 Periodogram Method and the Modi?ed Periodogram Method;77
3.1.2.6;3.2.6 Whittle Estimator;78
3.1.2.7;3.2.7 Diffusion Entropy Method;78
3.1.2.8;3.2.8 Kettani and Gubner's Method;79
3.1.2.9;3.2.9 Abry and Veitch's Method;79
3.1.2.10;3.2.10 Koutsoyiannis' Method;79
3.1.2.11;3.2.11 Higuchi's Method;80
3.1.3;3.3 Robustness of Hurst Estimators;80
3.1.3.1;3.3.1 Test Signal Generation and Estimation Procedures;81
3.1.3.2;3.3.2 Comparative Results and Robustness Assessment;82
3.1.3.2.1;Results of R/S method;82
3.1.3.2.2;Results of Aggregated Variance Method;83
3.1.3.2.3;Results of Absolute Value Method;84
3.1.3.2.4;Results of Variance of Residuals Method;85
3.1.3.2.5;Results of Periodogram Method;86
3.1.3.2.6;Results of Modi?ed Periodogram Method;87
3.1.3.2.7;Results of Whittle Estimator;88
3.1.3.2.8;Results of Diffusion Entropy Method;89
3.1.3.2.9;Results of Kettani and Gubner's Method;90
3.1.3.2.10;Results of Abry and Veitch's Method;91
3.1.3.2.11;Results of Koutsoyiannis' Method;92
3.1.3.2.12;Results of Higuchi's Method;93
3.1.3.3;3.3.3 Quantitative Robustness Comparison and Guideline for Selection Estimator;94
3.1.4;3.4 Chapter Summary;96
3.2;Chapter 4: Multifractional Processes;97
3.2.1;4.1 Multifractional Processes;98
3.2.1.1;4.1.1 Multifractional Brownian Motion and Multifractional Gaussian Noise;98
3.2.1.2;4.1.2 Linear Multifractional Stable Motion and Multifractional Stable Noise;99
3.2.2;4.2 Tracking Performance and Robustness of Local Hölder Exponent Estimator;99
3.2.2.1;4.2.1 Test Signal Generation and Estimation Procedures;100
3.2.2.2;4.2.2 Estimation Results;102
3.2.2.3;4.2.3 Guideline for Estimator Selection;111
3.2.3;4.3 Chapter Summary;112
4;Part III: Fractional-Order Signal Processing;113
4.1;Chapter 5: Constant-Order Fractional Signal Processing;114
4.1.1;5.1 Fractional-Order Differentiator/Integrator and Fractional Order Filters;114
4.1.1.1;5.1.1 Continuous-Time Implementations of Fractional-Order Operators;115
4.1.1.1.1;Continued Fraction Approximations;115
4.1.1.1.2;Oustaloup Recursive Approximations;117
4.1.1.1.3;Modi?ed Oustaloup Filter;118
4.1.1.2;5.1.2 Discrete-Time Implementation of Fractional-Order Operators;120
4.1.1.2.1;FIR Filter Approximation: Grünwald-Letnikov de?nition;121
4.1.1.2.2;FIR Filter Approximation: Power Series Expansion;124
4.1.1.2.3;IIR Filter Approximation: Tustin Method with Prewarping;125
4.1.1.2.4;Direct Discretization: First-Order IIR Generating Functions;126
4.1.1.2.4.1;CFE Tustin Operator;127
4.1.1.2.4.2;Al-Alaoui Operator;129
4.1.1.2.5;Direct Discretization: Second-Order IIR Generating Function Method;131
4.1.1.2.6;Direct Discretization: Step or Impulse Response Invariant Method;137
4.1.1.3;5.1.3 Frequency Response Fitting of Fractional-Order Filters;139
4.1.1.3.1;Continuous-Time Approximation;139
4.1.1.3.2;Discrete-Time Approximation;140
4.1.1.4;5.1.4 Transfer Function Approximations to Complicated Fractional-Order Filters;142
4.1.1.5;5.1.5 Sub-optimal Approximation of Fractional-Order Transfer Functions;144
4.1.2;5.2 Synthesis of Constant-Order Fractional Processes;148
4.1.2.1;5.2.1 Synthesis of Fractional Gaussian Noise;148
4.1.2.2;5.2.2 Synthesis of Fractional Stable Noise;150
4.1.3;5.3 Constant-Order Fractional System Modeling;150
4.1.3.1;5.3.1 Fractional Autoregressive Integrated Moving Average Model;151
4.1.3.2;5.3.2 Gegenbauer Autoregressive Moving Average Model;152
4.1.3.3;5.3.3 Fractional Autoregressive Conditional Heteroscedasticity Model;153
4.1.3.4;5.3.4 Fractional Autoregressive Integrated Moving Average with Stable Innovations Model;153
4.1.4;5.4 A Fractional Second-Order Filter;155
4.1.4.1;5.4.1 Derivation of the Analytical Impulse Response of (s2+as+b)-gamma;155
4.1.4.2;5.4.2 Impulse Response Invariant Discretization of (s2+as+b)-gamma;159
4.1.5;5.5 Analogue Realization of Constant-Order Fractional Systems;164
4.1.5.1;5.5.1 Introduction of Fractional-Order Component;164
4.1.5.2;5.5.2 Analogue Realization of Fractional-Order Integrator and Differentiator;165
4.1.6;5.6 Chapter Summary;167
4.2;Chapter 6: Variable-Order Fractional Signal Processing;168
4.2.1;6.1 Synthesis of Multifractional Processes;168
4.2.1.1;6.1.1 Synthesis of mGn;168
4.2.1.2;6.1.2 Examples of the Synthesized mGns;170
4.2.2;6.2 Variable-Order Fractional System Modeling;171
4.2.2.1;6.2.1 Locally Stationary Long Memory FARIMA(p,dt,q) Model;171
4.2.2.2;6.2.2 Locally Stationary Long Memory FARIMA(p,dt,q) with Stable Innovations Model;173
4.2.2.3;6.2.3 Variable Parameter FIGARCH Model;173
4.2.3;6.3 Analogue Realization of Variable-Order Fractional Systems;173
4.2.3.1;6.3.1 Physical Experimental Study of Temperature-Dependent Variable-Order Fractional Integrator and Differentiator;173
4.2.3.2;6.3.2 Application Examples of Analogue Variable-Order Fractional Systems;177
4.2.4;6.4 Chapter Summary;178
4.3;Chapter 7: Distributed-Order Fractional Signal Processing;180
4.3.1;7.1 Distributed-Order Integrator/Differentiator;181
4.3.1.1;7.1.1 Impulse Response of the Distributed-Order Integrator/Differentiator;182
4.3.1.2;7.1.2 Impulse Response Invariant Discretization of DOI/DOD;184
4.3.2;7.2 Distributed-Order Low-Pass Filter;186
4.3.2.1;7.2.1 Impulse Response of the Distributed-Order Low-Pass Filter;187
4.3.2.2;7.2.2 Impulse Response Invariant Discretization of DO-LPF;188
4.3.3;7.3 Distributed Parameter Low-Pass Filter;190
4.3.3.1;7.3.1 Derivation of the Analytical Impulse Response of the Fractional-Order Distributed Parameter Low-Pass Filter;191
4.3.3.2;7.3.2 Impulse Response Invariant Discretization of FO-DP-LPF;193
4.3.4;7.4 Chapter Summary;194
5;Part IV: Applications of Fractional-Order Signal Processing Techniques;196
5.1;Chapter 8: Fractional Autoregressive Integrated Moving Average with Stable Innovations Model of Great Salt Lake Elevation Time Series;197
5.1.1;8.1 Introduction;197
5.1.2;8.2 Great Salt Lake Elevation Data Analysis;198
5.1.3;8.3 FARIMA and FIGARCH Models of Great Salt Lake Elevation Time Series;202
5.1.4;8.4 FARIMA with Stable Innovations Model of Great Salt Lake Elevation Time Series;203
5.1.5;8.5 Chapter Summary;205
5.2;Chapter 9: Analysis of Biocorrosion Electrochemical Noise Using Fractional Order Signal Processing Techniques;207
5.2.1;9.1 Introduction;207
5.2.2;9.2 Experimental Approach and Data Acquisition;208
5.2.3;9.3 Conventional Analysis Techniques;208
5.2.3.1;9.3.1 Conventional Time Domain Analysis of ECN Signals;208
5.2.3.2;9.3.2 Conventional Frequency Domain Analysis;210
5.2.4;9.4 Fractional-Orders Signal Processing Techniques;214
5.2.4.1;9.4.1 Fractional Fourier Transform Technique;214
5.2.4.2;9.4.2 Fractional Power Spectrum Density;215
5.2.4.3;9.4.3 Self-similarity Analysis;217
5.2.4.4;9.4.4 Local Self-similarity Analysis;219
5.2.5;9.5 Chapter Summary;219
5.3;Chapter 10: Optimal Fractional-Order Damping Strategies;221
5.3.1;10.1 Introduction;221
5.3.2;10.2 Distributed-Order Fractional Mass-Spring Viscoelastic Damper System;222
5.3.3;10.3 Frequency-Domain Method Based Optimal Fractional-Order Damping Systems;224
5.3.4;10.4 Time-Domain Method Based Optimal Fractional-Order Damping Systems;227
5.3.5;10.5 Chapter Summary;232
5.4;Chapter 11: Heavy-Tailed Distribution and Local Memory in Time Series of Molecular Motion on the Cell Membrane;234
5.4.1;11.1 Introduction;234
5.4.2;11.2 Heavy-Tailed Distribution;235
5.4.3;11.3 Time Series of Molecular Motion;236
5.4.4;11.4 In?nite Second-Order and Heavy-Tailed Distribution in Jump Time Series;237
5.4.5;11.5 Long Memory and Local Memory in Jump Time Series;240
5.4.6;11.6 Chapter Summary;243
5.5;Chapter 12: Non-linear Transform Based Robust Adaptive Latency Change Estimation of Evoked Potentials;249
5.5.1;12.1 Introduction;249
5.5.2;12.2 DLMS and DLMP Algorithms;250
5.5.2.1;12.2.1 Signal and Noise Model;250
5.5.2.2;12.2.2 DLMS and Its Degradation;250
5.5.2.3;12.2.3 DLMP and Its Improvement;251
5.5.3;12.3 NLST Algorithm;252
5.5.3.1;12.3.1 NLST Algorithm;252
5.5.3.2;12.3.2 Robustness Analysis of the NLST;252
5.5.4;12.4 Simulation Results and Discussion;255
5.5.5;12.5 Chapter Summary;258
5.6;Chapter 13: Multifractional Property Analysis of Human Sleep Electroencephalogram Signals;259
5.6.1;13.1 Introduction;259
5.6.2;13.2 Data Description and Methods;260
5.6.2.1;13.2.1 Data Description;260
5.6.2.2;13.2.2 Methods;261
5.6.3;13.3 Fractional Property of Sleep EEG Signals;261
5.6.4;13.4 Multifractional Property of Sleep EEG Signals;264
5.6.5;13.5 Chapter Summary;266
5.7;Chapter 14: Conclusions;267
6;Appendix A: Mittag-Lef?er Function;269
7;Appendix B: Application of Numerical Inverse Laplace Transform Algorithms in Fractional-Order Signal Processing;273
7.1;B.1 Introduction;273
7.2;B.2 Numerical Inverse Laplace Transform Algorithms;274
7.3;B.3 Some Application Examples of Numerical Inverse Laplace Transform Algorithms in Fractional Order Signal Processing;275
7.3.1;B.3.1 Example A;275
7.3.2;B.3.2 Example B;276
7.3.2.1;When a2-4b=0;276
7.3.2.2;When a2-4b>0;277
7.3.2.3;When a2-4b<0;277
7.3.3;B.3.3 Example C;277
7.3.4;B.3.4 Example D;279
7.3.5;B.3.5 Example E;279
7.4;B.4 Conclusion;282
8;Appendix C: Some Useful Webpages;283
8.1;C.1 Useful Homepages;283
8.2;C.2 Useful Codes;283
9;Appendix D: MATLAB Codes of Impulse Response Invariant Discretization of Fractional-Order Filters;285
9.1;D.1 Impulse Response Invariant Discretization of Distributed-Order Integrator;285
9.2;D.2 Impulse Response Invariant Discretization of Fractional Second-Order Filter;288
9.3;D.3 Impulse Response Invariant Discretization of Distributed-Order Low-Pass Filter;291
10;References;294
11;Index;308