Fractional order signal processing : introductory concepts and applications

✍ Scribed by Das S., Pan I.

• Publisher: Springer
• Year: 2012
• Tongue: English
• Leaves: 110
• Series: SpringerBriefs in applied sciences and technology
• Category: Library

✦ Synopsis

Introduction.- Basics of Fractional Order Signals and Systems.- Long Range Dependence, Stable Distributions and Self-similarity.- Fractional Order Integral Transforms.- Fractional Order System Identification.- Fractional Order Statistical Signal Processing.- MATLAB Based Simulation Tools

✦ Table of Contents

Cover......Page 1
Fractional Order Signal
Processing......Page 3
Preface......Page 6
Contents......Page 9
7 MATLAB Based Simulation Tools......Page 12
1.1.2 Definitions in Fractional Calculus......Page 14
1.3 Some Visualizations of Fractional Calculus......Page 16
1.4 Geometric and Physical Interpretation of Fractional Order Differentiation and Integration......Page 18
1.5 Application of Fractional Calculus in Science and Engineering......Page 21
References......Page 22
2.1.1 Transfer Function Representation......Page 25
2.1.2 State-Space Representation......Page 26
2.2.1 Continuous Time Realization......Page 28
2.2.2 Discrete Time Realization......Page 30
2.3.1 ARFIMA Models......Page 34
2.4 Fractional Order Convolution and Correlation......Page 35
2.5.1 H2 Norm of FO Systems......Page 37
2.5.2 H infty Norm of FO Systems......Page 38
References......Page 39
3.1 Introduction......Page 42
3.2 Rational Transfer Function Models of Signals and its Extension to ARFIMA Models......Page 43
3.3 Few Characteristics of Fractional Pole Zero Models......Page 45
3.4 Stable Distributions......Page 46
3.5 Statistical Self-Similarity of Stochastic Processes......Page 47
3.5.1 Self-Similar Processes with Stationary Increments......Page 48
3.5.3 Persistence and Anti-Persistence......Page 49
3.5.4 Fractional Gaussian Noise......Page 50
3.5.5 Simulation of Fractional Brownian Motion and Fractional Gaussian Noise......Page 51
3.6 Multi-Fractional Gaussian Noise......Page 54
3.6.1 Concept of Variable Order Integrator......Page 55
3.8 Hurst Parameter Estimators......Page 56
3.8.1 Estimation of the Constant Hurst Index H in the Interval (0.5, 1)......Page 57
3.8.2 Estimation of the Time Varying Hurst Parameter for a Locally Self Similar Process......Page 58
References......Page 60
4.1 Fractional Fourier Transform as a Generalization of the Standard Integer Order Fourier Transform......Page 62
4.1.3 Relation of the FrFT with Other Transforms......Page 64
4.1.4 Two Dimensional Transform......Page 66
4.1.6 Filtering and Noise Removal in Fractional Domain......Page 67
4.1.7 FrFT Applications in Other Areas of Signal Processing......Page 69
4.2 Fractional Sine, Cosine and Hartley Transforms......Page 71
4.3.1 The B-spline and the Fractional B-spline......Page 72
4.3.2 Fractional B-spline Wavelets......Page 73
References......Page 74
5.1 Time Domain Identification Methods......Page 78
5.1.1 Equation Error Model......Page 79
5.1.2 Output Error Model......Page 82
5.2.1 Levy's Identification Method for Fractional Order Systems......Page 85
5.2.2 Managing Multiple Frequencies......Page 88
5.2.3 Adaptation of Levy's Algorithm Using Weights......Page 89
5.3 Other Identification Techniques......Page 90
References......Page 91
6.1.1 Brief Overview of the Kalman Filter......Page 92
6.1.2 The Fractional Kalman Filter......Page 93
6.1.3 The Extended Fractional Kalman Filter for Non-linear Systems......Page 95
6.1.4 Improved Fractional Kalman Filter......Page 96
6.2.1 Importance of Non-Gaussian Statistical Signal Processing......Page 98
6.2.2 Characteristic Functions, FLOM and Covariations......Page 99
6.2.3 Parameter Estimation for Symmetric Stable Distributions......Page 101
6.2.4 Estimation of Covariation......Page 103
References......Page 104
7.1 Numerical Tools for Fractional Calculus......Page 106
7.3 Rational Approximation of FO Differ-integrators......Page 107
7.7 Hurst Parameter Estimators......Page 108
References......Page 109