Signal analysis gives an insight into the properties of signals and stochastic processes by methodology. Linear transforms are integral to the continuing growth of signal processes as they characterize and classify signals. In particular, those transforms that provide time-frequency signal analysis are attracting greater numbers of researchers and are becoming an area of considerable importance. The key characteristic of these transforms, along with a certain time-frequency localization called the wavelet transform and various types of multirate filter banks, is their high computational efficiency. It is this computational efficiently which accounts for their increased application. This book provides a complete overview and introduction to signal analysis. It presents classical and modern signal analysis methods in a sequential structure starting with the background to signal theory. Progressing through the book the author introduces more advanced topics in an easy to understand style. Including recent and emerging topics such as filter banks with perfect reconstruction, time frequency and wavelets. With great accuracy and technical merit, this book makes a useful and original contribution to the current literature.
Author(s): Alfred Mertins
Edition: Revised
Publisher: Wiley
Year: 1999
Language: English
Pages: 328
Signal Analysis: Wavelets, Filter Banks, Time-Frequency Transforms & Applications......Page 1
Copyright......Page 4
Contents......Page 5
Preface......Page 10
1.1.1 Energy & Power Signals......Page 12
1.1.2 Normed Spaces......Page 13
1.1.3 Metric Spaces......Page 14
1.1.4 Inner Product Spaces......Page 15
1.2.1 Continuous-Time Signals......Page 19
1.2.2 Discrete-Time Signals......Page 20
1.3 Random Signals......Page 21
1.3.1 Properties of Random Variables......Page 22
1.3.2 Random Processes......Page 24
1.3.3 Transmission of Stochastic Processes through Linear Systems......Page 31
2.1 Integral Transforms......Page 33
2.2 Fourier Transform......Page 37
2.3 Hartley Transform......Page 40
2.4.1 Definition......Page 45
2.5 Representation of Bandpass Signals......Page 46
2.5.1 Analytic Signal & Complex Envelope......Page 47
2.5.2 Stationary Bandpass Processes......Page 54
3.1 Introduction......Page 58
3.2.1 Calculation of Coefficients......Page 60
3.2.2 Orthogonal Projection......Page 61
3.2.4 Parseval's Relation......Page 62
3.2.5 Complete Orthonormal Sets......Page 63
3.2.6 Examples of Complete Orthonormal Sets......Page 64
3.3 General Series Expansions......Page 67
3.3.1 Calculating Representation......Page 68
3.3.2 Orthogonal Projection......Page 71
3.3.3 Orthogonal Projection of n-Tuples......Page 73
3.4.1 QR Decomposition......Page 75
3.4.2 Moore-Penrose Pseudoinverse......Page 77
3.4.3 Nullspace......Page 79
3.4.4 Householder Transform......Page 80
3.4.5 Givens Rotations......Page 84
4.1 z-Transform......Page 86
4.2 Discrete-Time Fourier Transform......Page 91
4.3 Discrete Fourier Transform (DFT)......Page 93
4.4.1 Radix-2 Decimation-in-Time FFT......Page 96
4.4.2 Radix-2 Decimation-in-Frequency FFT......Page 99
4.4.3 Radix-4 FFT......Page 101
4.4.4 Split-Radix FFT......Page 102
4.4.5 Further FFT Algorithms......Page 103
4.5 Discrete Cosine Transforms......Page 104
4.6 Discrete Sine Transforms......Page 107
4.7 Discrete Hartley Transform......Page 108
4.8 Hadamard & Walsh-Hadamard Transforms......Page 111
5.1 Continuous-Time Karhunen-Lo'eve Transform......Page 112
5.2 Discrete Karhunen-Lohe Transform......Page 114
5.3 KLT of Real-Valued AR(1) Processes......Page 120
5.4 Whitening Transforms......Page 122
5.5.1 Least-Squares Estimation......Page 124
5.5.2 Best Linear Unbiased Estimator (BLUE)......Page 125
5.5.3 Minimum Mean Square Error Estimation......Page 127
5.6.1 Wiener Filters......Page 135
5.6.2 One-Step Linear Prediction......Page 138
5.6.3 Filter Design on Basis of Finite Data Ensembles......Page 141
5.7.1 Estimation of Autocorrelation Sequences......Page 144
5.7.2 Non-Parametric Estimation of Power Spectral Densities......Page 145
5.7.3 Parametric Methods in Spectral Estimation......Page 152
Ch6 Filter Banks......Page 154
6.1.1 Decimation & Interpolation......Page 155
6.1.2 Polyphase Decomposition......Page 158
6.2.1 PR Condition......Page 159
6.2.2 Quadrature Mirror Filters......Page 160
6.2.3 General Perfect Reconstruction Two-Channel Filter Banks......Page 161
6.2.4 Matrix Representations......Page 162
6.2.5 Paraunitary Two-Channel Filter Banks......Page 166
6.2.6 Paraunitary Filter Banks in Lattice Structure......Page 169
6.2.7 Linear-Phase Filter Banks in Lattice Structure......Page 170
6.2.8 Lifting Structures......Page 171
6.3 Tree-Structured Filter Banks......Page 173
6.4.1 Input-Output Relations......Page 175
6.4.2 The Polyphase Representation......Page 177
6.4.4 Design of Critically Subsampled M-Channel FIR Filter Banks......Page 179
6.5 DFT Filter Banks......Page 181
6.6 Cosine-Modulated Filter Banks......Page 185
6.6.1 Critically Subsampled Case......Page 186
6.6.2 Paraunitary Case......Page 190
6.6.3 Oversampled Cosine-Modulated Filter Banks......Page 194
6.6.4 Pseudo-QMF Banks......Page 195
6.7 Lapped Orthogonal Transforms......Page 197
6.8 Subband Coding of Images......Page 199
6.9 Processing of Finite-Length Signals......Page 200
6.10 Transmultiplexers......Page 206
7.1.1 Definition......Page 207
7.1.2 Time-Frequency Resolution......Page 209
7.1.3 Uncertainty Principle......Page 211
7.1.4 Spectrogram......Page 212
7.1.5 Reconstruction......Page 213
7.1.6 Reconstruction via Series Expansion......Page 215
7.2 Discrete-Time Signals......Page 216
7.3 Spectral Subtraction based on STFT......Page 218
8.1 Continuous-Time Wavelet Transform......Page 221
8.2 Wavelets for Time-Scale Analysis......Page 225
8.3.1 Integral Reconstruction......Page 228
8.3.2 Semi-Discrete Dyadic Wavelets......Page 230
8.4.1 Dyadic Sampling......Page 234
8.4.2 Better Frequency Resolution--Decomposition of Octaves......Page 236
8.5.1 Multiresolution Analysis......Page 238
8.5.2 Wavelet Analysis by Multirate Filtering......Page 243
8.5.3 Wavelet Synthesis by Multirate Filtering......Page 244
8.5.4 Relationship between Filters & Wavelets......Page 245
8.6.1 General Procedure......Page 248
8.6.3 Partition of Unity......Page 252
8.6.4 Norm of Constructed Scaling Functions & Wavelets......Page 253
8.6.5 Moments......Page 254
8.6.6 Regularity......Page 255
8.6.7 Wavelets with Finite Support......Page 256
8.7.1 Design of Biorthogonal Linear-Phase Wavelets......Page 258
8.7.2 Orthonormal Daubechies Wavelets......Page 263
8.7.3 Coiflets......Page 264
8.8 Wavelet Transform of Discrete-Time Signals......Page 266
8.8.1 A Trous Algorithm......Page 267
8.8.2 Relationship between Mallat & A Trous Algorithms......Page 270
8.8.3 Discrete-Time Morlet Wavelet......Page 271
8.9 DWT-Based Image Compression......Page 272
8.10 Wavelet-Based Denoising......Page 274
9.1 Ambiguity Function......Page 276
9.2.1 Definition & Properties......Page 280
9.2.2 Examples......Page 285
9.2.3 Cross-Terms & Cross Wigner Distributions......Page 286
9.2.4 Linear Operations......Page 290
9.3 General Time-Frequency Distributions......Page 291
9.3.1 Shift-Invariant Time-Frequency Distributions......Page 292
9.3.2 Examples of Shift-Invariant Time-Frequency Distributions......Page 294
9.3.3 Affine-Invariant Time-Frequency Distributions......Page 300
9.3.4 Discrete-Time Calculation of Time-Frequency Distributions......Page 301
9.4 Wigner-Ville Spectrum......Page 303
Bibliography......Page 310
Index......Page 322
