Wavelets (eBook)

Wavelets 3
Preface 5
Contents 11
Chapter 1: Signals and Signal Processing in Manufacturing 15
1.1 Classification of Signals 15
1.1.1 Deterministic Signal 15
1.1.1.1 Periodic Signal 15
1.1.1.2 Transient Signal 16
1.1.2 Nondeterministic Signal 17
1.1.2.1 Stationary Signal 18
1.1.2.2 Nonstationary Signal 18
1.2 Signals in Manufacturing 19
1.3 Role of Signal Processing for Manufacturing 25
1.4References 27
Chapter 2: From Fourier Transform to Wavelet Transform: A Historical Perspective 30
2.1 Fourier Transform 31
2.2 Short-Time Fourier Transform 34
2.3 Wavelet Transform 39
2.4References 44
Chapter 3: Continuous Wavelet Transform 46
3.1 Properties of Continuous Wavelet Transform 48
3.1.1 Superposition Property 48
3.1.2 Covariant Under Translation 49
3.1.3 Covariant Under Dilation 49
3.1.4 Moyal Principle 50
3.2 Inverse Continuous Wavelet Transform 51
3.3 Implementation of Continuous Wavelet Transform 52
3.4 Some Commonly Used Wavelets 54
3.4.1 Mexican Hat Wavelets 54
3.4.2 Morlet Wavelet 54
3.4.3 Gaussian Wavelet 55
3.4.4 Frequency B-Spline Wavelet 56
3.4.5 Shannon Wavelet 56
3.4.6 Harmonic Wavelet 57
3.5 CWT of Representative Signals 58
3.5.1 CWT of Sinusoidal Function 58
3.5.2 CWT of Gaussian Pulse Function 59
3.5.3 CWT of Chirp Function 59
3.6 Summary 60
3.7References 60
Chapter 4: Discrete Wavelet Transform 62
4.1 Discretization of Scale and Translation Parameters 62
4.2 Multiresolution Analysis and Orthogonal Wavelet Transform 66
4.2.1 Multiresolution Analysis 66
4.2.2 Orthogonal Wavelet Transform 68
4.3 Dual-Scale Equation and Multiresolution Filters 69
4.4 The Mallat Algorithm 71
4.5 Commonly Used Base Wavelets 73
4.5.1 Haar Wavelet 74
4.5.2 Daubechies Wavelet 74
4.5.3 Coiflet Wavelet 75
4.5.4 Symlet Wavelet 76
4.5.5 Biorthogonal and Reverse Biorthogonal Wavelets 76
4.5.6 Meyer Wavelet 78
4.6 Application of Discrete Wavelet Transform 78
4.7 Summary 81
4.8References 81
Chapter 5: Wavelet Packet Transform 82
5.1 Theoretical Basis of Wavelet Packet 82
5.1.1 Definition 82
5.1.2 Wavelet Packet Property 85
5.1.2.1 Shift Orthogonality 85
5.1.2.2 Orthogonal Relationship between $$/bi u_{2n}^{/bi(/,j)} (t)$$ and $$/bi u_{2n + 1}^{(/,j)} (t)$$ 86
5.2 Recursive Algorithm 86
5.3 FFT-Based Harmonic Wavelet Packet Transform 87
5.3.1 Harmonic Wavelet Transform 87
5.3.2 Harmonic Wavelet Packet Algorithm 88
5.4 Application of Wavelet Packet Transform 91
5.4.1 Time-Frequency Analysis 91
5.4.2 Wavelet Packet for Denoising 92
5.5 Summary 92
5.6References 93
Chapter 6: Wavelet-Based Multiscale Enveloping 95
6.1 Signal Enveloping Through Hilbert Transform 95
6.2 Multiscale Enveloping Using Complex-Valued Wavelet 98
6.3 Application of Multiscale Enveloping 99
6.3.1 Ultrasonic Pulse Differentiation for Pressure Measurement in Injection Molding 99
6.3.1.1 Simulation 102
6.3.1.2 Experimental Study 103
6.3.2 Bearing Defect Diagnosis in Rotary Machine 105
6.3.2.1 Numerical Simulation Using the MuSEnS Algorithm 107
6.3.2.2 Case Study 109
6.4 Summary 111
6.5References 112
Chapter 7: Wavelet Integrated with Fourier Transform: A Unified Technique 114
7.1 Generalized Signal Transformation Frame 114
7.1.1 Fourier Transform in the Generalized Frame 117
7.1.2 Wavelet Transform in the Generalized Frame 118
7.2 Wavelet Transform with Spectral Postprocessing 120
7.2.1 Fourier Transform of the Measure Function 121
7.2.2 Fourier Transform of Wavelet-Extracted Data Set 123
7.3 Application to Bearing Defect Diagnosis 124
7.3.1 Effectiveness in Defect Feature Extraction 126
7.3.2 Selection of Decomposition Level 129
7.3.3 Effect of Bearing Operation Conditions 131
7.3.3.1 Variation of Radial Load 131
7.3.3.2 Variation of Axial Load 131
7.3.3.3 Variation of Rotational Speed 131
7.4 Summary 135
7.5References 135
Chapter 8: Wavelet Packet-Transform for Defect Severity Classification 136
8.1 Subband Feature Extraction 136
8.1.1 Energy Feature 137
8.1.2 Kurtosis 138
8.2 Key Feature Selection 139
8.2.1 Fisher Linear Discriminant Analysis 140
8.2.2 Principal Component Analysis 142
8.3 Neural-Network Classifier 145
8.4 Formulation of WPT-Based Defect Severity Classification 147
8.5 Case Studies 148
8.5.1 Case Study I: Roller Bearing Defect Severity Evaluation 148
8.5.2 Case Study II: Ball Bearing Defect Severity Evaluation 153
8.6 Summary 157
8.7References 157
Chapter 9: Local Discriminant Bases for Signal Classification 159
9.1 Dissimilarity Measures 159
9.1.1 Relative Entropy 160
9.1.2 Energy Difference 161
9.1.3 Correlation Index 161
9.1.4 Nonstationarity 162
9.2 Local Discriminant Bases 163
9.3 Case Study 165
9.4 Application to Gearbox Defect Classification 168
9.5 Summary 172
9.6References 172
Chapter 10: Selection of Base Wavelet 174
10.1 Overview of Base Wavelet Selection 174
10.1.1 Qualitative Measure 175
10.1.2 Quantitative Measure 177
10.2 Wavelet Selection Criteria 178
10.2.1 Energy and Shannon Entropy 179
10.2.2 Information Theoretic Measure 181
10.2.2.1 Joint Entropy 181
10.2.2.2 Conditional Entropy 181
10.2.2.3 Mutual Information 182
10.2.2.4 Relative Entropy 184
10.3 Numerical Study on Base Wavelet Selection 185
10.3.1 Evaluation Using Real-Valued Wavelets 185
10.3.2 Evaluation Using Complex-Valued Wavelets 188
10.4 Base Wavelet Selection for Bearing Vibration Signal 192
10.5 Summary 194
10.6References 195
Chapter 11: Designing Your Own Wavelet 197
11.1 Overview of Wavelet Design 197
11.2 Construction of an Impulse Wavelet 198
11.3 Impulse Wavelet Application 206
11.4 Summary 210
11.5References 211
Chapter 12: Beyond Wavelets 212
12.1 Second Generation Wavelet Transform 212
12.1.1 Theoretical Basis of SGWT 213
12.1.2 Illustration of SGWT in Signal Processing 215
12.2 Ridgelet Transform 217
12.2.1 Theoretical Basis of Ridgelet Transform 217
12.2.2 Application of the Ridgelet Transform 219
12.3 Curvelet Transform 221
12.3.1 Curvelet Transform 221
12.3.2 Application of the Curvelet Transform 224
12.4 Summary 225
12.5References 226
Index 228