Mathematics of Signal Processing: A First Course

410 Pages·1.898 MB·English

by  Byrne C.L.

Mathematics of Signal Processing:
A First Course
Charles L. Byrne
Department of Mathematical Sciences
University of Massachusetts Lowell
Lowell, MA 01854
March 31, 2013
(Text for 92.548 Mathematics of Signal Processing)
(The most recent version is available as a pdf file at
http://faculty.uml.edu/cbyrne/cbyrne.html)
2
Contents
I Introduction xiii
1 Preface 1
1.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Course Aims and Topics . . . . . . . . . . . . . . . . . . . . 1
1.2.1 Some Examples of Remote Sensing . . . . . . . . . . 2
1.2.2 A Role for Mathematics . . . . . . . . . . . . . . . . 4
1.2.3 Limited Data . . . . . . . . . . . . . . . . . . . . . . 4
1.2.4 Course Emphasis . . . . . . . . . . . . . . . . . . . . 4
1.2.5 Course Topics . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Applications of Interest . . . . . . . . . . . . . . . . . . . . 5
1.4 Sensing Modalities . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.1 Active and Passive Sensing . . . . . . . . . . . . . . 5
1.4.2 A Variety of Modalities . . . . . . . . . . . . . . . . 6
1.5 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Using Prior Knowledge . . . . . . . . . . . . . . . . . . . . . 9
2 Urn Models in Remote Sensing 13
2.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The Urn Model . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Some Mathematical Notation . . . . . . . . . . . . . . . . . 14
2.4 An Application to SPECT Imaging . . . . . . . . . . . . . . 15
2.5 Hidden Markov Models . . . . . . . . . . . . . . . . . . . . 16
II Fundamental Examples 19
3 Transmission and Remote Sensing- I 21
3.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Fourier Series and Fourier Coefficients . . . . . . . . . . . . 21
3.3 The Unknown Strength Problem . . . . . . . . . . . . . . . 22
3.3.1 Measurement in the Far-Field . . . . . . . . . . . . . 23
3.3.2 Limited Data . . . . . . . . . . . . . . . . . . . . . . 24
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3.3.3 Can We Get More Data? . . . . . . . . . . . . . . . 25
3.3.4 The Fourier Cosine and Sine Transforms . . . . . . . 25
3.3.5 Over-Sampling . . . . . . . . . . . . . . . . . . . . . 26
3.3.6 Other Forms of Prior Knowledge . . . . . . . . . . . 27
3.4 Estimating the Size of Distant Objects . . . . . . . . . . . . 28
3.5 The Transmission Problem . . . . . . . . . . . . . . . . . . 30
3.5.1 Directionality . . . . . . . . . . . . . . . . . . . . . . 30
3.5.2 The Case of Uniform Strength . . . . . . . . . . . . 30
3.6 Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.7 One-Dimensional Arrays . . . . . . . . . . . . . . . . . . . . 32
3.7.1 Measuring Fourier Coefficients . . . . . . . . . . . . 32
3.7.2 Over-sampling . . . . . . . . . . . . . . . . . . . . . 34
3.7.3 Under-sampling . . . . . . . . . . . . . . . . . . . . . 35
III Signal Models 41
4 Undetermined-Parameter Models 43
4.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Fundamental Calculations . . . . . . . . . . . . . . . . . . . 43
4.2.1 Evaluating a Trigonometric Polynomial . . . . . . . 44
4.2.2 Determining the Coefficients . . . . . . . . . . . . . 44
4.3 Two Examples . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3.1 The Unknown Strength Problem . . . . . . . . . . . 45
4.3.2 Sampling in Time . . . . . . . . . . . . . . . . . . . 46
4.3.3 The Issue of Units . . . . . . . . . . . . . . . . . . . 46
4.4 Estimation and Models. . . . . . . . . . . . . . . . . . . . . 47
4.5 A Polynomial Model . . . . . . . . . . . . . . . . . . . . . . 47
4.6 Linear Trigonometric Models . . . . . . . . . . . . . . . . . 48
4.6.1 Equi-Spaced Frequencies . . . . . . . . . . . . . . . . 49
4.6.2 Equi-Spaced Sampling . . . . . . . . . . . . . . . . . 49
4.7 Recalling Fourier Series . . . . . . . . . . . . . . . . . . . . 50
4.7.1 Fourier Coefficients . . . . . . . . . . . . . . . . . . . 50
4.7.2 Riemann Sums . . . . . . . . . . . . . . . . . . . . . 50
4.8 Simplifying the Calculations . . . . . . . . . . . . . . . . . . 51
4.8.1 The Main Theorem. . . . . . . . . . . . . . . . . . . 51
4.8.2 The Proofs as Exercises . . . . . . . . . . . . . . . . 53
4.8.3 More Computational Issues . . . . . . . . . . . . . . 55
4.9 Approximation, Models, or Truth? . . . . . . . . . . . . . . 55
4.9.1 Approximating the Truth . . . . . . . . . . . . . . . 55
4.9.2 Modeling the Data . . . . . . . . . . . . . . . . . . . 55
4.10 From Real to Complex . . . . . . . . . . . . . . . . . . . . . 57
CONTENTS iii
5 Complex Numbers 59
5.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Definition and Basics . . . . . . . . . . . . . . . . . . . . . . 59
5.3 Complex Numbers as Matrices . . . . . . . . . . . . . . . . 61
6 Complex Exponential Functions 63
6.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 63
6.2 The Complex Exponential Function . . . . . . . . . . . . . 63
6.2.1 Real Exponential Functions . . . . . . . . . . . . . . 64
6.2.2 Why is h(x) an Exponential Function? . . . . . . . . 64
6.2.3 What is ez, for z complex? . . . . . . . . . . . . . . 65
6.3 Complex Exponential Signal Models . . . . . . . . . . . . . 66
6.4 Coherent and Incoherent Summation . . . . . . . . . . . . . 67
6.5 Uses in Quantum Electrodynamics . . . . . . . . . . . . . . 67
6.6 Using Coherence and Incoherence . . . . . . . . . . . . . . . 68
6.6.1 The Discrete Fourier Transform . . . . . . . . . . . . 68
6.7 Some Exercises on Coherent Summation . . . . . . . . . . . 69
6.8 Complications . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.8.1 Multiple Signal Components . . . . . . . . . . . . . 71
6.8.2 Resolution. . . . . . . . . . . . . . . . . . . . . . . . 72
6.8.3 Unequal Amplitudes and Complex Amplitudes . . . 72
6.8.4 Phase Errors . . . . . . . . . . . . . . . . . . . . . . 72
6.9 Undetermined Exponential Models . . . . . . . . . . . . . . 72
6.9.1 Prony’s Problem . . . . . . . . . . . . . . . . . . . . 73
6.9.2 Prony’s Method . . . . . . . . . . . . . . . . . . . . 73
7 Transmission and Remote Sensing- II 77
7.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 77
7.2 Directional Transmission . . . . . . . . . . . . . . . . . . . . 77
7.3 Multiple-Antenna Arrays . . . . . . . . . . . . . . . . . . . 78
7.3.1 The Array of Equi-Spaced Antennas . . . . . . . . . 78
7.3.2 The Far-Field Strength Pattern . . . . . . . . . . . . 78
7.3.3 Can the Strength be Zero? . . . . . . . . . . . . . . 79
7.3.4 Diffraction Gratings . . . . . . . . . . . . . . . . . . 80
7.4 Phase and Amplitude Modulation . . . . . . . . . . . . . . 81
7.5 Steering the Array . . . . . . . . . . . . . . . . . . . . . . . 81
7.6 Maximal Concentration in a Sector . . . . . . . . . . . . . . 82
7.7 Higher Dimensional Arrays . . . . . . . . . . . . . . . . . . 83
7.7.1 The Wave Equation . . . . . . . . . . . . . . . . . . 83
7.7.2 Planewave Solutions . . . . . . . . . . . . . . . . . . 84
7.7.3 Superposition and the Fourier Transform . . . . . . 85
7.7.4 The Spherical Model . . . . . . . . . . . . . . . . . . 85
7.7.5 The Two-Dimensional Array . . . . . . . . . . . . . 85
7.7.6 The One-Dimensional Array. . . . . . . . . . . . . . 86
iv CONTENTS
7.7.7 Limited Aperture . . . . . . . . . . . . . . . . . . . . 87
7.7.8 Other Limitations on Resolution . . . . . . . . . . . 87
7.8 An Example: The Solar-Emission Problem. . . . . . . . . . 88
7.9 Another Example: Scattering in Crystallography . . . . . . 88
IV Fourier Methods 95
8 Fourier Analysis 97
8.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 97
8.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . 97
8.3 The Unknown Strength Problem Again . . . . . . . . . . . 98
8.4 Two-Dimensional Fourier Transforms . . . . . . . . . . . . . 100
8.4.1 Two-Dimensional Fourier Inversion . . . . . . . . . . 101
8.5 Fourier Series and Fourier Transforms . . . . . . . . . . . . 101
8.5.1 Support-Limited F(ω) . . . . . . . . . . . . . . . . . 101
8.5.2 Shannon’s Sampling Theorem . . . . . . . . . . . . . 102
8.5.3 Sampling Terminology . . . . . . . . . . . . . . . . . 102
8.5.4 What Shannon Does Not Say . . . . . . . . . . . . . 103
8.5.5 Sampling from a Limited Interval . . . . . . . . . . . 103
8.6 The Problem of Finite Data . . . . . . . . . . . . . . . . . . 104
8.7 Best Approximation . . . . . . . . . . . . . . . . . . . . . . 104
8.7.1 The Orthogonality Principle. . . . . . . . . . . . . . 104
8.7.2 An Example . . . . . . . . . . . . . . . . . . . . . . 106
8.7.3 The DFT as Best Approximation . . . . . . . . . . . 106
8.7.4 The Modified DFT (MDFT) . . . . . . . . . . . . . 106
8.7.5 The PDFT . . . . . . . . . . . . . . . . . . . . . . . 107
8.8 The Vector DFT . . . . . . . . . . . . . . . . . . . . . . . . 108
8.9 Using the Vector DFT . . . . . . . . . . . . . . . . . . . . . 109
8.10 A Special Case of the Vector DFT . . . . . . . . . . . . . . 110
8.11 Plotting the DFT . . . . . . . . . . . . . . . . . . . . . . . . 111
8.12 The Vector DFT in Two Dimensions . . . . . . . . . . . . . 112
9 Properties of the Fourier Transform 115
9.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 115
9.2 Fourier-Transform Pairs . . . . . . . . . . . . . . . . . . . . 115
9.2.1 Decomposing f(x) . . . . . . . . . . . . . . . . . . . 116
9.3 Basic Properties of the Fourier Transform . . . . . . . . . . 116
9.4 Some Fourier-Transform Pairs . . . . . . . . . . . . . . . . . 117
9.5 Dirac Deltas . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.6 More Properties of the Fourier Transform . . . . . . . . . . 120
9.7 Convolution Filters . . . . . . . . . . . . . . . . . . . . . . . 121
9.7.1 Blurring and Convolution Filtering . . . . . . . . . . 121
9.7.2 Low-Pass Filtering . . . . . . . . . . . . . . . . . . . 123
CONTENTS v
9.8 Functions in the Schwartz Class . . . . . . . . . . . . . . . . 123
9.8.1 The Schwartz Class . . . . . . . . . . . . . . . . . . 124
9.8.2 A Discontinuous Function . . . . . . . . . . . . . . . 125
10 The Fourier Transform and Convolution Filtering 127
10.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 127
10.2 Linear Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 127
10.3 Shift-Invariant Filters . . . . . . . . . . . . . . . . . . . . . 127
10.4 Some Properties of a SILO . . . . . . . . . . . . . . . . . . 128
10.5 The Dirac Delta . . . . . . . . . . . . . . . . . . . . . . . . 129
10.6 The Impulse Response Function. . . . . . . . . . . . . . . . 129
10.7 Using the Impulse-Response Function . . . . . . . . . . . . 130
10.8 The Filter Transfer Function . . . . . . . . . . . . . . . . . 130
10.9 The Multiplication Theorem for Convolution . . . . . . . . 130
10.10Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . 131
10.11A Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
10.12Band-Limiting . . . . . . . . . . . . . . . . . . . . . . . . . 132
11 Infinite Sequences and Discrete Filters 133
11.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 133
11.2 Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
11.3 Shift-Invariant Discrete Linear Systems . . . . . . . . . . . 133
11.4 The Delta Sequence . . . . . . . . . . . . . . . . . . . . . . 134
11.5 The Discrete Impulse Response . . . . . . . . . . . . . . . . 134
11.6 The Discrete Transfer Function . . . . . . . . . . . . . . . . 134
11.7 Using Fourier Series . . . . . . . . . . . . . . . . . . . . . . 135
11.8 The Multiplication Theorem for Convolution . . . . . . . . 136
11.9 The Three-Point Moving Average . . . . . . . . . . . . . . . 136
11.10Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . 137
11.11Stable Systems . . . . . . . . . . . . . . . . . . . . . . . . . 138
11.12Causal Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 139
12 Convolution and the Vector DFT 141
12.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 141
12.2 Non-periodic Convolution . . . . . . . . . . . . . . . . . . . 141
12.3 The DFT as a Polynomial . . . . . . . . . . . . . . . . . . . 142
12.4 The Vector DFT and Periodic Convolution . . . . . . . . . 143
12.4.1 The Vector DFT . . . . . . . . . . . . . . . . . . . . 143
12.4.2 Periodic Convolution . . . . . . . . . . . . . . . . . . 143
12.5 The vDFT of Sampled Data . . . . . . . . . . . . . . . . . . 144
12.5.1 Superposition of Sinusoids . . . . . . . . . . . . . . . 145
12.5.2 Rescaling . . . . . . . . . . . . . . . . . . . . . . . . 145
12.5.3 The Aliasing Problem . . . . . . . . . . . . . . . . . 146
12.5.4 The Discrete Fourier Transform . . . . . . . . . . . . 146
vi CONTENTS
12.5.5 Calculating Values of the DFT . . . . . . . . . . . . 146
12.5.6 Zero-Padding . . . . . . . . . . . . . . . . . . . . . . 147
12.5.7 What the vDFT Achieves . . . . . . . . . . . . . . . 147
12.5.8 Terminology . . . . . . . . . . . . . . . . . . . . . . 147
12.6 Understanding the Vector DFT . . . . . . . . . . . . . . . . 148
13 The Fast Fourier Transform (FFT) 151
13.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 151
13.2 Evaluating a Polynomial . . . . . . . . . . . . . . . . . . . . 151
13.3 The DFT and Vector DFT . . . . . . . . . . . . . . . . . . 152
13.4 Exploiting Redundancy . . . . . . . . . . . . . . . . . . . . 153
13.5 The Two-Dimensional Case . . . . . . . . . . . . . . . . . . 154
14 Plane-wave Propagation 155
14.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 155
14.2 The Bobbing Boats . . . . . . . . . . . . . . . . . . . . . . . 155
14.3 Transmission and Remote-Sensing . . . . . . . . . . . . . . 156
14.4 The Transmission Problem . . . . . . . . . . . . . . . . . . 157
14.5 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
14.6 Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . 158
14.7 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . 159
14.8 Planewave Solutions . . . . . . . . . . . . . . . . . . . . . . 160
14.9 Superposition and the Fourier Transform . . . . . . . . . . 160
14.9.1 The Spherical Model . . . . . . . . . . . . . . . . . . 160
14.10Sensor Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . 161
14.10.1The Two-Dimensional Array . . . . . . . . . . . . . 161
14.10.2The One-Dimensional Array. . . . . . . . . . . . . . 161
14.10.3Limited Aperture . . . . . . . . . . . . . . . . . . . . 162
14.11The Remote-Sensing Problem . . . . . . . . . . . . . . . . . 162
14.11.1The Solar-Emission Problem . . . . . . . . . . . . . 162
14.12Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
14.13The Limited-Aperture Problem . . . . . . . . . . . . . . . . 164
14.14Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
14.14.1The Solar-Emission Problem Revisited . . . . . . . . 166
14.15Discrete Data . . . . . . . . . . . . . . . . . . . . . . . . . . 167
14.15.1Reconstruction from Samples . . . . . . . . . . . . . 167
14.16The Finite-Data Problem . . . . . . . . . . . . . . . . . . . 168
14.17Functions of Several Variables . . . . . . . . . . . . . . . . . 168
14.17.1Two-Dimensional Farfield Object . . . . . . . . . . . 168
14.17.2Limited Apertures in Two Dimensions . . . . . . . . 169
14.18Broadband Signals . . . . . . . . . . . . . . . . . . . . . . . 169
CONTENTS vii
V Nonlinear Models 173
15 Random Sequences 175
15.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 175
15.2 What is a Random Variable? . . . . . . . . . . . . . . . . . 175
15.3 The Coin-Flip Random Sequence . . . . . . . . . . . . . . . 176
15.4 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
15.5 Filtering Random Sequences . . . . . . . . . . . . . . . . . . 178
15.6 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
15.7 Correlation Functions and Power Spectra . . . . . . . . . . 179
15.8 The Dirac Delta in Frequency Space . . . . . . . . . . . . . 181
15.9 Random Sinusoidal Sequences . . . . . . . . . . . . . . . . . 181
15.10Random Noise Sequences . . . . . . . . . . . . . . . . . . . 182
15.11Increasing the SNR . . . . . . . . . . . . . . . . . . . . . . . 183
15.12Colored Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 183
15.13Spread-Spectrum Communication . . . . . . . . . . . . . . . 183
15.14Stochastic Difference Equations . . . . . . . . . . . . . . . . 184
15.15Random Vectors and Correlation Matrices . . . . . . . . . . 185
16 Classical and Modern Methods 187
16.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 187
16.2 The Classical Methods . . . . . . . . . . . . . . . . . . . . . 187
16.3 Modern Signal Processing and Entropy. . . . . . . . . . . . 187
16.4 Related Methods . . . . . . . . . . . . . . . . . . . . . . . . 188
17 Entropy Maximization 191
17.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 191
17.2 Estimating Non-Negative Functions . . . . . . . . . . . . . 191
17.3 Philosophical Issues . . . . . . . . . . . . . . . . . . . . . . 192
17.4 The Autocorrelation Sequence {r(n)} . . . . . . . . . . . . 193
17.5 Minimum-Phase Vectors . . . . . . . . . . . . . . . . . . . . 194
17.6 Burg’s MEM . . . . . . . . . . . . . . . . . . . . . . . . . . 195
17.6.1 The Minimum-Phase Property . . . . . . . . . . . . 196
17.6.2 Solving Ra=δ Using Levinson’s Algorithm . . . . . 197
17.7 A Sufficient Condition for Positive-definiteness . . . . . . . 198
18 Eigenvector Methods in Estimation 207
18.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 207
18.2 Some Eigenvector Methods . . . . . . . . . . . . . . . . . . 207
18.3 The Sinusoids-in-Noise Model . . . . . . . . . . . . . . . . . 207
18.4 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . 208
18.5 Determining the Frequencies . . . . . . . . . . . . . . . . . 209
18.6 The Case of Non-White Noise . . . . . . . . . . . . . . . . . 210
18.7 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
viii CONTENTS
19 The IPDFT 213
19.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 213
19.2 The Need for Prior Information in Non-Linear Estimation . 213
19.3 What Wiener Filtering Suggests . . . . . . . . . . . . . . . 214
19.4 Using a Prior Estimate . . . . . . . . . . . . . . . . . . . . . 215
19.5 Properties of the IPDFT . . . . . . . . . . . . . . . . . . . . 215
19.6 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
VI Wavelets 223
20 Analysis and Synthesis 225
20.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 225
20.2 The Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . 225
20.3 Polynomial Approximation . . . . . . . . . . . . . . . . . . 226
20.4 Signal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 226
20.5 Practical Considerations in Signal Analysis . . . . . . . . . 227
20.5.1 The Finite Data Problem . . . . . . . . . . . . . . . 229
20.6 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
20.7 Bases, Riesz Bases and Orthonormal Bases . . . . . . . . . 231
21 Ambiguity Functions 233
21.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 233
21.2 Radar Problems . . . . . . . . . . . . . . . . . . . . . . . . 233
21.3 The Wideband Cross-Ambiguity Function . . . . . . . . . . 234
21.4 The Narrowband Cross-Ambiguity Function . . . . . . . . . 236
21.5 Range Estimation . . . . . . . . . . . . . . . . . . . . . . . 237
22 Time-Frequency Analysis 239
22.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 239
22.2 Non-stationary Signals . . . . . . . . . . . . . . . . . . . . . 239
22.3 The Short-Time Fourier Transform . . . . . . . . . . . . . . 240
22.4 The Wigner-Ville Distribution. . . . . . . . . . . . . . . . . 241
23 Wavelets 243
23.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 243
23.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
23.3 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . 244
23.4 The Integral Wavelet Transform . . . . . . . . . . . . . . . 245
23.5 Wavelet Series Expansions . . . . . . . . . . . . . . . . . . . 246
23.6 Multiresolution Analysis . . . . . . . . . . . . . . . . . . . . 247
23.6.1 The Shannon Multiresolution Analysis . . . . . . . . 247
23.6.2 The Haar Multiresolution Analysis . . . . . . . . . . 248
23.6.3 Wavelets and Multiresolution Analysis . . . . . . . . 249