Fractional Processes and Fractional-Order Signal Processing (eBook)

Techniques and Applications

YangQuan Chen, TianShuang Qiu, Hu Sheng (Autoren)

eBook Download: PDF

2011 | 2012
XXVI, 295 Seiten
Springer London (Verlag)
978-1-4471-2233-3 (ISBN)

Infos

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'.

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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(R) 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.

Inhalt

Fractional Processes and Fractional-Order Signal Processing 3
Foreword 6
Preface 9
Acknowledgements 12
Contents 16
Acronyms 22
Part I: Overview of Fractional Processes and Fractional-Order Signal Processing Techniques 24
Chapter 1: Introduction 25
1.1 An Introduction to Fractional Processes and Analysis Methods 25
1.2 Basis of Stochastic Processes 28
1.2.1 Statistics of Stochastic Processes 28
1.2.2 Properties of Stochastic Processes 29
Mean Function 29
Variance Function 29
Correlation Function 29
Autocovariance Function 30
Cross-Correlation Function 30
Cross-Covariance Function 30
Moments 30
1.2.3 Gaussian Distribution and Gaussian Processes 31
1.2.4 Stationary Processes 32
1.3 Analysis of Random Signals 32
1.3.1 Estimation of Properties for Stochastic Signals 32
Estimation of the Mean Value 33
Estimation of the Variance 33
Estimation of the Covariance Function 33
Estimation of the Correlation Function 33
Estimation of the Cross-Covariance Function 33
Estimation of the Cross-Correlation 34
Estimation of the Moments 34
1.3.2 Simulation of Random Signals 34
1.3.3 Signal Filtering 35
Analogue Filters 35
Digital Filter 36
1.3.4 Modeling Random Processes 37
1.3.5 Transform Domain Analysis 38
Fourier Transform 38
Laplace Transform 38
Z-Transform 39
Wavelet Transform 39
Hilbert Transform 40
Mellin Transform 40
1.3.6 Other Analysis Methods 41
1.4 Research Motivation 41
1.4.1 Heavy Tailed Distributions 41
1.4.2 Long Range Dependence 42
1.4.3 Local Memory 44
1.5 Basics of Fractional-Order Signal Processing 45
1.5.1 Fractional Calculus 45
Constant-Order Fractional Calculus 45
Distributed-Order Fractional Calculus 46
Variable-Order Fractional Calculus 46
1.5.2 alpha-Stable Distribution 47
1.5.3 Fractional Fourier Transform 48
1.6 Brief Summary of Contributions of the Monograph 50
1.7 Structure of the Monograph 50
Chapter 2: An Overview of Fractional Processes and Fractional-Order Signal Processing Techniques 52
2.1 Fractional Processes 52
2.1.1 Fractional Processes and Fractional-Order Systems 53
Review of Conventional Random Processes and Integer-Order Systems 53
Constant-Order Fractional Processes and Constant-Order Fractional Systems 54
2.1.2 Stable Processes 56
2.1.3 Fractional Brownian Motion 57
2.1.4 Fractional Gaussian Noise 58
2.1.5 Fractional Stable Motion 58
2.1.6 Fractional Stable Noise 59
2.1.7 Multifractional Brownian Motion 59
2.1.8 Multifractional Gaussian Noise 59
2.1.9 Multifractional Stable Motion 60
2.1.10 Multifractional Stable Noise 60
2.2 Fractional-Order Signal Processing Techniques 60
2.2.1 Simulation of Fractional Random Processes 60
2.2.2 Fractional Filter 61
2.2.3 Fractional-Order Systems Modeling 62
2.2.4 Realization of Fractional Systems 62
Analogue Realization of Fractional Systems 62
Digital Realization of Fractional Systems 64
2.2.5 Other Fractional Tools 64
Fractional Hilbert Transform 64
Fractional Power Spectrum Density 65
Fractional Splines 66
2.3 Chapter Summary 67
Part II: Fractional Processes 68
Chapter 3: Constant-Order Fractional Processes 69
3.1 Introduction of Constant-Order Fractional Processes 69
3.1.1 Long-Range Dependent Processes 69
3.1.2 Fractional Brownian Motion and Fractional Gaussian Noise 71
Fractional Brownian Motion 71
Fractional Gaussian Noise 72
3.1.3 Linear Fractional Stable Motion and Fractional Stable Noise 73
Linear Fractional Stable Motion (LFSM) 73
Fractional Stable Noise 74
3.2 Hurst Estimators: A Brief Summary 76
3.2.1 R/S Method 76
3.2.2 Aggregated Variance Method 76
3.2.3 Absolute Value Method 77
3.2.4 Variance of Residuals Method 77
3.2.5 Periodogram Method and the Modi?ed Periodogram Method 77
3.2.6 Whittle Estimator 78
3.2.7 Diffusion Entropy Method 78
3.2.8 Kettani and Gubner's Method 79
3.2.9 Abry and Veitch's Method 79
3.2.10 Koutsoyiannis' Method 79
3.2.11 Higuchi's Method 80
3.3 Robustness of Hurst Estimators 80
3.3.1 Test Signal Generation and Estimation Procedures 81
3.3.2 Comparative Results and Robustness Assessment 82
Results of R/S method 82
Results of Aggregated Variance Method 83
Results of Absolute Value Method 84
Results of Variance of Residuals Method 85
Results of Periodogram Method 86
Results of Modi?ed Periodogram Method 87
Results of Whittle Estimator 88
Results of Diffusion Entropy Method 89
Results of Kettani and Gubner's Method 90
Results of Abry and Veitch's Method 91
Results of Koutsoyiannis' Method 92
Results of Higuchi's Method 93
3.3.3 Quantitative Robustness Comparison and Guideline for Selection Estimator 94
3.4 Chapter Summary 96
Chapter 4: Multifractional Processes 97
4.1 Multifractional Processes 98
4.1.1 Multifractional Brownian Motion and Multifractional Gaussian Noise 98
4.1.2 Linear Multifractional Stable Motion and Multifractional Stable Noise 99
4.2 Tracking Performance and Robustness of Local Hölder Exponent Estimator 99
4.2.1 Test Signal Generation and Estimation Procedures 100
4.2.2 Estimation Results 102
4.2.3 Guideline for Estimator Selection 111
4.3 Chapter Summary 112
Part III: Fractional-Order Signal Processing 113
Chapter 5: Constant-Order Fractional Signal Processing 114
5.1 Fractional-Order Differentiator/Integrator and Fractional Order Filters 114
5.1.1 Continuous-Time Implementations of Fractional-Order Operators 115
Continued Fraction Approximations 115
Oustaloup Recursive Approximations 117
Modi?ed Oustaloup Filter 118
5.1.2 Discrete-Time Implementation of Fractional-Order Operators 120
FIR Filter Approximation: Grünwald-Letnikov de?nition 121
FIR Filter Approximation: Power Series Expansion 124
IIR Filter Approximation: Tustin Method with Prewarping 125
Direct Discretization: First-Order IIR Generating Functions 126
CFE Tustin Operator 127
Al-Alaoui Operator 129
Direct Discretization: Second-Order IIR Generating Function Method 131
Direct Discretization: Step or Impulse Response Invariant Method 137
5.1.3 Frequency Response Fitting of Fractional-Order Filters 139
Continuous-Time Approximation 139
Discrete-Time Approximation 140
5.1.4 Transfer Function Approximations to Complicated Fractional-Order Filters 142
5.1.5 Sub-optimal Approximation of Fractional-Order Transfer Functions 144
5.2 Synthesis of Constant-Order Fractional Processes 148
5.2.1 Synthesis of Fractional Gaussian Noise 148
5.2.2 Synthesis of Fractional Stable Noise 150
5.3 Constant-Order Fractional System Modeling 150
5.3.1 Fractional Autoregressive Integrated Moving Average Model 151
5.3.2 Gegenbauer Autoregressive Moving Average Model 152
5.3.3 Fractional Autoregressive Conditional Heteroscedasticity Model 153
5.3.4 Fractional Autoregressive Integrated Moving Average with Stable Innovations Model 153
5.4 A Fractional Second-Order Filter 155
5.4.1 Derivation of the Analytical Impulse Response of (s2+as+b)-gamma 155
5.4.2 Impulse Response Invariant Discretization of (s2+as+b)-gamma 159
5.5 Analogue Realization of Constant-Order Fractional Systems 164
5.5.1 Introduction of Fractional-Order Component 164
5.5.2 Analogue Realization of Fractional-Order Integrator and Differentiator 165
5.6 Chapter Summary 167
Chapter 6: Variable-Order Fractional Signal Processing 168
6.1 Synthesis of Multifractional Processes 168
6.1.1 Synthesis of mGn 168
6.1.2 Examples of the Synthesized mGns 170
6.2 Variable-Order Fractional System Modeling 171
6.2.1 Locally Stationary Long Memory FARIMA(p,dt,q) Model 171
6.2.2 Locally Stationary Long Memory FARIMA(p,dt,q) with Stable Innovations Model 173
6.2.3 Variable Parameter FIGARCH Model 173
6.3 Analogue Realization of Variable-Order Fractional Systems 173
6.3.1 Physical Experimental Study of Temperature-Dependent Variable-Order Fractional Integrator and Differentiator 173
6.3.2 Application Examples of Analogue Variable-Order Fractional Systems 177
6.4 Chapter Summary 178
Chapter 7: Distributed-Order Fractional Signal Processing 180
7.1 Distributed-Order Integrator/Differentiator 181
7.1.1 Impulse Response of the Distributed-Order Integrator/Differentiator 182
7.1.2 Impulse Response Invariant Discretization of DOI/DOD 184
7.2 Distributed-Order Low-Pass Filter 186
7.2.1 Impulse Response of the Distributed-Order Low-Pass Filter 187
7.2.2 Impulse Response Invariant Discretization of DO-LPF 188
7.3 Distributed Parameter Low-Pass Filter 190
7.3.1 Derivation of the Analytical Impulse Response of the Fractional-Order Distributed Parameter Low-Pass Filter 191
7.3.2 Impulse Response Invariant Discretization of FO-DP-LPF 193
7.4 Chapter Summary 194
Part IV: Applications of Fractional-Order Signal Processing Techniques 196
Chapter 8: Fractional Autoregressive Integrated Moving Average with Stable Innovations Model of Great Salt Lake Elevation Time Series 197
8.1 Introduction 197
8.2 Great Salt Lake Elevation Data Analysis 198
8.3 FARIMA and FIGARCH Models of Great Salt Lake Elevation Time Series 202
8.4 FARIMA with Stable Innovations Model of Great Salt Lake Elevation Time Series 203
8.5 Chapter Summary 205
Chapter 9: Analysis of Biocorrosion Electrochemical Noise Using Fractional Order Signal Processing Techniques 207
9.1 Introduction 207
9.2 Experimental Approach and Data Acquisition 208
9.3 Conventional Analysis Techniques 208
9.3.1 Conventional Time Domain Analysis of ECN Signals 208
9.3.2 Conventional Frequency Domain Analysis 210
9.4 Fractional-Orders Signal Processing Techniques 214
9.4.1 Fractional Fourier Transform Technique 214
9.4.2 Fractional Power Spectrum Density 215
9.4.3 Self-similarity Analysis 217
9.4.4 Local Self-similarity Analysis 219
9.5 Chapter Summary 219
Chapter 10: Optimal Fractional-Order Damping Strategies 221
10.1 Introduction 221
10.2 Distributed-Order Fractional Mass-Spring Viscoelastic Damper System 222
10.3 Frequency-Domain Method Based Optimal Fractional-Order Damping Systems 224
10.4 Time-Domain Method Based Optimal Fractional-Order Damping Systems 227
10.5 Chapter Summary 232
Chapter 11: Heavy-Tailed Distribution and Local Memory in Time Series of Molecular Motion on the Cell Membrane 234
11.1 Introduction 234
11.2 Heavy-Tailed Distribution 235
11.3 Time Series of Molecular Motion 236
11.4 In?nite Second-Order and Heavy-Tailed Distribution in Jump Time Series 237
11.5 Long Memory and Local Memory in Jump Time Series 240
11.6 Chapter Summary 243
Chapter 12: Non-linear Transform Based Robust Adaptive Latency Change Estimation of Evoked Potentials 249
12.1 Introduction 249
12.2 DLMS and DLMP Algorithms 250
12.2.1 Signal and Noise Model 250
12.2.2 DLMS and Its Degradation 250
12.2.3 DLMP and Its Improvement 251
12.3 NLST Algorithm 252
12.3.1 NLST Algorithm 252
12.3.2 Robustness Analysis of the NLST 252
12.4 Simulation Results and Discussion 255
12.5 Chapter Summary 258
Chapter 13: Multifractional Property Analysis of Human Sleep Electroencephalogram Signals 259
13.1 Introduction 259
13.2 Data Description and Methods 260
13.2.1 Data Description 260
13.2.2 Methods 261
13.3 Fractional Property of Sleep EEG Signals 261
13.4 Multifractional Property of Sleep EEG Signals 264
13.5 Chapter Summary 266
Chapter 14: Conclusions 267
Appendix A: Mittag-Lef?er Function 269
Appendix B: Application of Numerical Inverse Laplace Transform Algorithms in Fractional-Order Signal Processing 273
B.1 Introduction 273
B.2 Numerical Inverse Laplace Transform Algorithms 274
B.3 Some Application Examples of Numerical Inverse Laplace Transform Algorithms in Fractional Order Signal Processing 275
B.3.1 Example A 275
B.3.2 Example B 276
When a2-4b=0 276
When a2-4b> 0
When a2-4b< 0
B.3.3 Example C 277
B.3.4 Example D 279
B.3.5 Example E 279
B.4 Conclusion 282
Appendix C: Some Useful Webpages 283
C.1 Useful Homepages 283
C.2 Useful Codes 283
Appendix D: MATLAB Codes of Impulse Response Invariant Discretization of Fractional-Order Filters 285
D.1 Impulse Response Invariant Discretization of Distributed-Order Integrator 285
D.2 Impulse Response Invariant Discretization of Fractional Second-Order Filter 288
D.3 Impulse Response Invariant Discretization of Distributed-Order Low-Pass Filter 291
References 294
Index 308

Produkt-Details

• Erscheint lt. Verlag: 20.10.2011
• Reihe/Serie: Signals and Communication Technology
• Zusatzinfo: XXVI, 295 p. 162 illus., 146 illus. in color.
• Verlagsort: London
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik ► Grafik / Design
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Themenwelt: Technik ► Nachrichtentechnik
• Schlagworte: Fractional Fourier Transform • Fractional-order Random Processes • Fractional-order Signal Processing • Information and Communication, Circuits • Long-range Dependence • Nonstationary Stochastic Processes • stable processes
• ISBN-10: 1-4471-2233-X / 144712233X
• ISBN-13: 978-1-4471-2233-3 / 9781447122333