Introduction to wavelet transforms:
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| Format: | Book |
| Language: | English |
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Boca Raton ; London ; New York
CRC Press
2021
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| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xxviii, 455 Seiten Illustrationen 24 cm |
| ISBN: | 9781032174839 1032174838 |
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| 245 | 1 | 0 | |a Introduction to wavelet transforms |c by Nirdosh Bhatnagar |
| 264 | 1 | |a Boca Raton ; London ; New York |b CRC Press |c 2021 | |
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| adam_text | Contents Preface.................................................................................................................. xv List of Symbols................................................................................................... xix Greek Symbols ................................................................................................... xxvii Part I. Basics of Wavelet Transforms 1. Introduction to Wavelets .......................................................................... 1.1 Introduction........................................................................................ 1.2 Representation of Functions.............................................................. 1.2.1 Basis Representation.......................................................... 1.2.2 Representation via Frames................................................ 1.2.3 Riesz Basis Representation................................................ 1.2.4 Multiscale Representation.................................................. 1.2.5 Representation via Dictionaries........................................ 1.2.6 Redundancy in Representation........................................... 1.3 Fourier Analysis ................................................................................ 1.3.1 Fourier Series....................................................................... 1.3.2 Fourier Transform and Spectral Analysis........................ 1.4 Wavelet Analysis................................................................................ 1.5 Why Use
Wavelets?............................................................................ 1.6 Story of Wavelets .............................................................................. 1.7 Applications........................................................................................ Problems........................................................................................................ 3 3 4 4 5 6 6 6 7 7 8 8 9 11 12 13 14
viii Contents 2. Continuous Wavelet Transform............................................................... 2.1 Introduction........................................................................................ 2.2 Basics of Continuous Wavelet Transform........................................ 2.3 Properties of Continuous Wavelet Transform.................................. 2.4 Examples............................................................................................ 2.4.1 Wavelets.............................................................................. 2.4.2 Continuous Wavelet Transforms ...................................... 2.5 Regularity of Wavelets...................................................................... Problems........................................................................................................ 15 15 15 17 19 19 21 21 22 3. Discrete Wavelet Transform...................................................................... 3.1 Introduction........................................................................................ 3.2 Basics of Discrete Wavelet Transform ............................................ 3.3 Multiresolution Analysis .................................................................. 3.4 Scaling Function................................................................................ 3.5 Characterization of the W) Spaces.................................................. 3.6 Expansions and Transformations...................................................... 3.6.1 Coefficient Relationships between
Different Scales......... 3.6.2 Pyramid Algorithm............................................................ 3.7 Digital Filter Interpretation .............................................................. 3.8 Computation of the Scaling Function.............................................. 3.9 An Alternate Multiresolution Analysis............................................ Problems........................................................................................................ 25 25 25 26 29 31 35 36 38 39 40 41 42 4. Daubechies Wavelets.................................................................................. 4.1 Introduction........................................................................................ 4.2 Regularity and Moments.................................................................... 4.2.1 Regularity............................................................................ 4.2.2 Moments.............................................................................. 4.3 Compactness...................................................................................... 4.4 Construction of Daubechies Scaling Coefficients.......................... 4.5 Computation of Scaling and Mother Wavelet Functions................ Problems........................................................................................................ 55 55 55 56 56 59 61 72 72 5. Some Examples of Wavelets...................................................................... 5.1
Introduction........................................................................................ 5.2 Shannon Wavelets.............................................................................. 5.3 Meyer Wavelets.................................................................................. 5.4 Splines ................................................................................................ 5.4.1 Properties of B-Splines...................................................... 5.4.2 Examples of B-Splines...................................................... 5.4.3 Orthogonalization of B-Splines........................................ Problems........................................................................................................ 83 83 83 85 87 89 91 92 95
Contents 6. Applications................................................................................................. 6.1 Introduction........................................................................................ 6.2 Signal Denoising via Wavelets.......................................................... 6.3 Image Compression............................................................................ 6.4 Wavelet Neural Networks.................................................................. 6.4.1 Artificial Neural Network.................................................. 6.4.2 Gradient Descent................................................................ 6.4.3 Wavelets and NeuralNetworks ......................................... 6.4.4 Learning Algorithm............................................................ 6.4.5 Wavelons with Vector Inputs ............................................ Problems........................................................................................................ ix 107 107 107 110 114 115 117 120 121 123 127 Part II. Intermediate Topics 7. 8. Periodic Wavelet Transform .................................................................... 7.1 Introduction........................................................................................ 7.2 Periodization of a Function .............................................................. 7.3 Periodization of Scaling and Wavelet Functions............................ 7.4 Periodic Multiresolution Analysis.................................................... 7.5 Periodic Series
Expansions .............................................................. 7.6 Fast Periodic Wavelet Transform...................................................... 7.6.1 Computational Complexity......................................... 140 7.6.2 A Matrix Formulation................................................. 140 Problems........................................................................................................ 131 131 131 132 134 135 137 145 Biorthogonal Wavelet Transform........................................................... 151 8.1 Introduction........................................................................................ 151 8.2 Biorthogonal Representations of a Function .................................. 151 8.3 Biorthogonal Wavelets...................................................................... 153 8.3.1 Motivation for the Use of Biorthogonal Wavelet Bases.. 154 8.3.2 Biorthogonal Spaces................................................... 155 8.3.3 Biorthogonal Space Bases ................................................ 156 8.3.4 Biorthogonal Scaling Functions and Dual Wavelets.... 157 8.3.5 Biorthogonal Relationships in the Frequency Domain .. 158 8.3.6 Relationships between Scaling Coefficients............. 161 8.3.7 Support Values............................................................. 162 8.4 Decomposition and Reconstruction of Functions .......................... 163 8.4.1 Basics........................................................................... 163 8.4.2 Digital Filter
1ոէ6րրր6էՅձւօո......................................... 165 8.4.3 Symmetric h(n)’s and fi(n)’s..................................... 165 8.4.4 Moments....................................................................... 166 8.5 Construction of Biorthogonal Scaling Coefficients........................ 168
ř x Contents f 8.6 B-Spline-Based Biorthogonal Wavelets............................................ 8.7 Semi-Orthogonal Wavelets.................................................................. Problems.......................................................................................................... 9. 172j 176 177f Coiflets............................................................................................................ 179ļ 9.1 Introduction.......................................................................................... 9.2 Preliminaries........................................................................................ 9.3 Construction of Coiflets...................................................................... Problems.......................................................................................................... 179ī 179; 181; 186; 10. The Lifting Technique ................................................................................. 10.1 Introduction.......................................................................................... 10.2 Laurent Polynomials............................................................................ 10.3 Greatest Common Divisor of Two Laurent Polynomials ................. 10.4 Biorthogonal Wavelet Transform........................................................ 10.4.1 Perfect Deconstruction and Reconstruction...................... 10.4.2 Single-Stage Deconstruction and Reconstruction ............ 10.5 The Lifting
Technique.......................................................................... 10.5.1 Lifting Technique via Polyphase Matrix............................ 10.5.2 Polyphase Matrix Factorization.......................................... 10.5.3 Examples .............................................................................. 10.6 Second-Generation Wavelets.............................................................. Problems.......................................................................................................... 191 191[ 191 ֊ 193j 196į 197; 200 I 202 202 205 208 213 215 11. Wavelet Packets.............................................................................................. 11.1 Introduction.......................................................................................... 11.2 Elements of Graph Theory................................................................. 11.3 Elementary Properties of Wavelet Packets........................................ 11.3.1 Basic Wavelet Packets.......................................................... 11.3.2 General Wavelet Packets...................................................... 11.4 Wavelet Packet Transformation.......................................................... 11.5 Best Basis Selection Algorithm.......................................................... 11.5.1 Cost Function and Measures................................................ 11.5.2 Characteristics of Wavelet Packet Trees............................ 11.5.3 Algorithm for Selectionof Best Basis.................................
Problems.......................................................................................................... 219 219 219 221 222 226 228 230 231 232 233 234 12. Lapped Orthogonal Transform.................................................................. 12.1 Introduction.......................................................................................... 12.2 Orthogonal Transforms........................................................................ 12.3 Transform Efficiency............................................................................ 12.3.1 Covariance Matrices............................................................ 12.3.2 Transform Metrics................................................................ 12.4 AR(1) Process...................................................................................... 239 239 240 242 242 244 245
Contents 12.5 Karhunen-Loéve Transform ............................................................ 12.5.1 KIT Matrix ........................................................................ 12.5.2 Properties of the KIT Matrix............................................ 12.5.3 Karhunen-Loéve Transform of Vector x.......................... 12.6 Discrete Cosine Transform................................................................ 12.6.1 Basics of the DCT.............................................................. 12.6.2 Computation of the DCT .................................................. 12.6.3 DCT Basis Vectors as Eigenvectors of Special Matrices . 12.7 Lapped Transform............................................................................. Problems........................................................................................................ 247 248 249 249 251 252 253 257 257 262 Part III. Signal Processing 13. Discrete Fourier Transform...................................................................... 13.1 Introduction........................................................................................ 13.2 Elements of the DFT.......................................................................... 13.2.1 Properties of the DFT........................................................ 13.2.2 Computation of the DFT.................................................... 13.3 DFT Computation for Ramanujan Numbers .................................. 13.3.1 Ramanujan Numbers.......................................................... 13.3.2 Recursive
Computations.................................................... 13.3.3 Discrete Fourier Transform Computation........................ Problems ........................................................................................................ 279 279 279 280 281 285 286 288 290 291 14. The z-Transform and Discrete-Time Fourier Transform................... 14.1 Introduction........................................................................................ 14.2 ^֊Transform........................................................................................ 14.2.1 Properties............................................................................ 14.2.2 Down-Sampled and Up-Sampled Sequences.................. 14.2.3 Inversion.............................................................................. 14.3 Discrete-Time Fourier Transform.................................................... Problems........................................................................................................ 293 293 293 294 296 296 297 299 15. Elements of Continuous-Time Signal Processing.................................. 15.1 Introduction........................................................................................ 15.2 Continuous-Time Signal Processing................................................ Problems........................................................................................................ 301 301 301 305 16. Elements of Discrete-Time Signal Processing........................................ 16.1
Introduction........................................................................................ 16.2 Discrete-Time Signal Processing...................................................... 16.3 2֊Transform Analysis of a Discrete-Time Linear System.............. 16.4 Special Filters .................................................................................... 307 307 307 310 313
Contents xii 16.4.1 Linear Phase Filter.................................................................. 16.4.2 All-Pass Filter ....................................................................... 16.4.3 Minimum-Phase Filter ........................................................ 16.4.4 Subband Coding..................................................................... Problems............................................................................................................ 314 314 315 317 319 Part IV. Mathematical Concepts 17. Set-Theoretic Concepts and Number Theory.......................................... 17.1 17.2 Introduction............................................................................................ Sets.......................................................................................................... 17.2.1 Set Operations....................................................................... 17.2.2 Interval Notation................................................................... 17.3 Functions and Sequences..................................................................... 17.3.1 Sequences............................................................................... 17.4 Elementary Number-Theoretic Concepts.......................................... 17.4.1 Countability........................................................................... 17.4.2 Divisibility............................................................................. 17.4.3 Prime Numbers
..................................................................... 17.4.4 Greatest Common Divisor .................................................. 17.4.5 Polynomials........................................................................... 17.5 Congruence Arithmetic......................................................................... Problems............................................................................................................. 18. Matrices and Determinants......................................................................... 18.1 18.2 Introduction............................................................................................ Elements of Matrix Theory ................................................................. 18.2.1 Basic Matrix Operations...................................................... 18.2.2 Different Types of Matrices................................................ 18.2.3 Matrix Norm ......................................................................... 18.3 Determinants.......................................................................................... 18.4 More Matrix Theory............................................................................. 18.4.1 Rank of a Matrix................................................................... 18.4.2 Matrices as Linear Transformations .................................. 18.5 Spectral Analysis of Matrices .............................................................
Problems............................................................................................................ 19. Applied Analysis............................................................................................. 19.1 19.2 Introduction............................................................................................ Basic Concepts ...................................................................................... 19.2.1 Point Sets............................................................................... 19.2.2 Limits, Continuity, Derivatives, and Monotonicity ........ 327 327 327 328 330 330 331 332 332 332 333 333 335 336 340 343 343 343 344 345 348 349 350 350 351 351 353 355 355 355 355 356
Contents xut 19.2.3 Partial Derivatives.............................................................. 19.2.4 Singularity and Related Topics......................................... 19.3 Complex Analysis.............................................................................. 19.3.1 De Moivre and Euler Identities ........................................ 19.3.2 Limits, Continuity, Derivatives, and Analyticity............ 19.3.3 Contours or Curves............................................................ 19.3.4 Integration .......................................................................... 19.3.5 Infinite Series...................................................................... 19.4 Asymptotics........................................................................................ 19.5 Fields .................................................................................................. 19.6 Vector Spaces over Fields.................................................................. 19.7 Linear Mappings................................................................................ 19.8 Tensor Products.................................................................................. 19.9 Vector Algebra.................................................................................... 19.10 Vector Spaces Revisited.................................................................... 19.10.1 Normed Vector Space........................................................ 19.10.2 Complete Vector Space and Compactness ...................... 19.10.3 Inner Product
Space........................................................... 19.10.4 Orthogonality...................................................................... 19.10.5 Gram֊Schmidt Orthogonalization Process...................... 19.11 More Hilbert Spaces.......................................................................... 19.11.1 Non-Orthogonal Expansion.............................................. 19.11.2 Biorthogonal Bases............................................................ Problems........................................................................................................ 36! 363 363 365 366 367 368 368 369 372 373 378 379 382 384 384 385 386 388 389 390 391 392 393 20. Fourier Theory ........................................................................................... 20.1 Introduction........................................................................................ 20.2 Fourier Series...................................................................................... 20.2.1 Generalized Functions........................................................ 20.2.2 Conditions for the Existence of Fourier Series................ 20.2.3 Complex Fourier Series...................................................... 20.2.4 Trigonometric Fourier Series............................................ 20.2.5 Generalized Fourier Series................................................ 20.3 Transform Techniques........................................................................ 20.3.1 Fourier
Transform.............................................................. 20.3.2 Short-Time Fourier Transform.......................................... 20.3.3 Wigner-Ville Transform.................................................... Problems........................................................................................................ 397 397 397 397 399 400 401 402 403 403 411 412 413 21. Probability Theory and Stochastic Processes........................................ 21.1 Introduction........................................................................................ 21.2 Postulates of Probability Theory...................................................... 21.3 Random Variables.............................................................................. 421 421 421 423
xiv Contents 21.4 Average Measures.............................................................................. 21.4.1 Expectation.......................................................................... 21.4.2 Second-Order Expectations .............................................. 21.5 Independent Random Variables........................................................ 21.6 Moment-Generating Function.......................................................... 21.7 Examples of Some Distributions...................................................... 21.7.1 Discrete Distributions........................................................ 21.7.2 Continuous Distributions .................................................. 21.7.3 Multivariate Gaussian Distribution.................................. 21.8 Stochastic Processes.......................................................................... Problems........................................................................................................ 425 426 427 427 428 429 429 429 431 432 434 References............................................................................................................ 439 Index 449
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Contents Preface. xv List of Symbols. xix Greek Symbols . xxvii Part I. Basics of Wavelet Transforms 1. Introduction to Wavelets . 1.1 Introduction. 1.2 Representation of Functions. 1.2.1 Basis Representation. 1.2.2 Representation via Frames. 1.2.3 Riesz Basis Representation. 1.2.4 Multiscale Representation. 1.2.5 Representation via Dictionaries. 1.2.6 Redundancy in Representation. 1.3 Fourier Analysis . 1.3.1 Fourier Series. 1.3.2 Fourier Transform and Spectral Analysis. 1.4 Wavelet Analysis. 1.5 Why Use
Wavelets?. 1.6 Story of Wavelets . 1.7 Applications. Problems. 3 3 4 4 5 6 6 6 7 7 8 8 9 11 12 13 14
viii Contents 2. Continuous Wavelet Transform. 2.1 Introduction. 2.2 Basics of Continuous Wavelet Transform. 2.3 Properties of Continuous Wavelet Transform. 2.4 Examples. 2.4.1 Wavelets. 2.4.2 Continuous Wavelet Transforms . 2.5 Regularity of Wavelets. Problems. 15 15 15 17 19 19 21 21 22 3. Discrete Wavelet Transform. 3.1 Introduction. 3.2 Basics of Discrete Wavelet Transform . 3.3 Multiresolution Analysis . 3.4 Scaling Function. 3.5 Characterization of the W) Spaces. 3.6 Expansions and Transformations. 3.6.1 Coefficient Relationships between
Different Scales. 3.6.2 Pyramid Algorithm. 3.7 Digital Filter Interpretation . 3.8 Computation of the Scaling Function. 3.9 An Alternate Multiresolution Analysis. Problems. 25 25 25 26 29 31 35 36 38 39 40 41 42 4. Daubechies Wavelets. 4.1 Introduction. 4.2 Regularity and Moments. 4.2.1 Regularity. 4.2.2 Moments. 4.3 Compactness. 4.4 Construction of Daubechies Scaling Coefficients. 4.5 Computation of Scaling and Mother Wavelet Functions. Problems. 55 55 55 56 56 59 61 72 72 5. Some Examples of Wavelets. 5.1
Introduction. 5.2 Shannon Wavelets. 5.3 Meyer Wavelets. 5.4 Splines . 5.4.1 Properties of B-Splines. 5.4.2 Examples of B-Splines. 5.4.3 Orthogonalization of B-Splines. Problems. 83 83 83 85 87 89 91 92 95
Contents 6. Applications. 6.1 Introduction. 6.2 Signal Denoising via Wavelets. 6.3 Image Compression. 6.4 Wavelet Neural Networks. 6.4.1 Artificial Neural Network. 6.4.2 Gradient Descent. 6.4.3 Wavelets and NeuralNetworks . 6.4.4 Learning Algorithm. 6.4.5 Wavelons with Vector Inputs . Problems. ix 107 107 107 110 114 115 117 120 121 123 127 Part II. Intermediate Topics 7. 8. Periodic Wavelet Transform . 7.1 Introduction. 7.2 Periodization of a Function . 7.3 Periodization of Scaling and Wavelet Functions. 7.4 Periodic Multiresolution Analysis. 7.5 Periodic Series
Expansions . 7.6 Fast Periodic Wavelet Transform. 7.6.1 Computational Complexity. 140 7.6.2 A Matrix Formulation. 140 Problems. 131 131 131 132 134 135 137 145 Biorthogonal Wavelet Transform. 151 8.1 Introduction. 151 8.2 Biorthogonal Representations of a Function . 151 8.3 Biorthogonal Wavelets. 153 8.3.1 Motivation for the Use of Biorthogonal Wavelet Bases. 154 8.3.2 Biorthogonal Spaces. 155 8.3.3 Biorthogonal Space Bases . 156 8.3.4 Biorthogonal Scaling Functions and Dual Wavelets. 157 8.3.5 Biorthogonal Relationships in the Frequency Domain . 158 8.3.6 Relationships between Scaling Coefficients. 161 8.3.7 Support Values. 162 8.4 Decomposition and Reconstruction of Functions . 163 8.4.1 Basics. 163 8.4.2 Digital Filter
1ոէ6րրր6էՅձւօո. 165 8.4.3 Symmetric h(n)’s and fi(n)’s. 165 8.4.4 Moments. 166 8.5 Construction of Biorthogonal Scaling Coefficients. 168
ř x Contents f 8.6 B-Spline-Based Biorthogonal Wavelets. 8.7 Semi-Orthogonal Wavelets. Problems. 9. 172j 176 177f Coiflets. 179ļ 9.1 Introduction. 9.2 Preliminaries. 9.3 Construction of Coiflets. Problems. 179ī 179; 181; 186; 10. The Lifting Technique . 10.1 Introduction. 10.2 Laurent Polynomials. 10.3 Greatest Common Divisor of Two Laurent Polynomials . 10.4 Biorthogonal Wavelet Transform. 10.4.1 Perfect Deconstruction and Reconstruction. 10.4.2 Single-Stage Deconstruction and Reconstruction . 10.5 The Lifting
Technique. 10.5.1 Lifting Technique via Polyphase Matrix. 10.5.2 Polyphase Matrix Factorization. 10.5.3 Examples . 10.6 Second-Generation Wavelets. Problems. 191 191[ 191 ֊ 193j 196į 197; 200 I 202 202 205 208 213 215 11. Wavelet Packets. 11.1 Introduction. 11.2 Elements of Graph Theory. 11.3 Elementary Properties of Wavelet Packets. 11.3.1 Basic Wavelet Packets. 11.3.2 General Wavelet Packets. 11.4 Wavelet Packet Transformation. 11.5 Best Basis Selection Algorithm. 11.5.1 Cost Function and Measures. 11.5.2 Characteristics of Wavelet Packet Trees. 11.5.3 Algorithm for Selectionof Best Basis.
Problems. 219 219 219 221 222 226 228 230 231 232 233 234 12. Lapped Orthogonal Transform. 12.1 Introduction. 12.2 Orthogonal Transforms. 12.3 Transform Efficiency. 12.3.1 Covariance Matrices. 12.3.2 Transform Metrics. 12.4 AR(1) Process. 239 239 240 242 242 244 245
Contents 12.5 Karhunen-Loéve Transform . 12.5.1 KIT Matrix . 12.5.2 Properties of the KIT Matrix. 12.5.3 Karhunen-Loéve Transform of Vector x. 12.6 Discrete Cosine Transform. 12.6.1 Basics of the DCT. 12.6.2 Computation of the DCT . 12.6.3 DCT Basis Vectors as Eigenvectors of Special Matrices . 12.7 Lapped Transform. Problems. 247 248 249 249 251 252 253 257 257 262 Part III. Signal Processing 13. Discrete Fourier Transform. 13.1 Introduction. 13.2 Elements of the DFT. 13.2.1 Properties of the DFT. 13.2.2 Computation of the DFT. 13.3 DFT Computation for Ramanujan Numbers . 13.3.1 Ramanujan Numbers. 13.3.2 Recursive
Computations. 13.3.3 Discrete Fourier Transform Computation. Problems . 279 279 279 280 281 285 286 288 290 291 14. The z-Transform and Discrete-Time Fourier Transform. 14.1 Introduction. 14.2 ^֊Transform. 14.2.1 Properties. 14.2.2 Down-Sampled and Up-Sampled Sequences. 14.2.3 Inversion. 14.3 Discrete-Time Fourier Transform. Problems. 293 293 293 294 296 296 297 299 15. Elements of Continuous-Time Signal Processing. 15.1 Introduction. 15.2 Continuous-Time Signal Processing. Problems. 301 301 301 305 16. Elements of Discrete-Time Signal Processing. 16.1
Introduction. 16.2 Discrete-Time Signal Processing. 16.3 2֊Transform Analysis of a Discrete-Time Linear System. 16.4 Special Filters . 307 307 307 310 313
Contents xii 16.4.1 Linear Phase Filter. 16.4.2 All-Pass Filter . 16.4.3 Minimum-Phase Filter . 16.4.4 Subband Coding. Problems. 314 314 315 317 319 Part IV. Mathematical Concepts 17. Set-Theoretic Concepts and Number Theory. 17.1 17.2 Introduction. Sets. 17.2.1 Set Operations. 17.2.2 Interval Notation. 17.3 Functions and Sequences. 17.3.1 Sequences. 17.4 Elementary Number-Theoretic Concepts. 17.4.1 Countability. 17.4.2 Divisibility. 17.4.3 Prime Numbers
. 17.4.4 Greatest Common Divisor . 17.4.5 Polynomials. 17.5 Congruence Arithmetic. Problems. 18. Matrices and Determinants. 18.1 18.2 Introduction. Elements of Matrix Theory . 18.2.1 Basic Matrix Operations. 18.2.2 Different Types of Matrices. 18.2.3 Matrix Norm . 18.3 Determinants. 18.4 More Matrix Theory. 18.4.1 Rank of a Matrix. 18.4.2 Matrices as Linear Transformations . 18.5 Spectral Analysis of Matrices .
Problems. 19. Applied Analysis. 19.1 19.2 Introduction. Basic Concepts . 19.2.1 Point Sets. 19.2.2 Limits, Continuity, Derivatives, and Monotonicity . 327 327 327 328 330 330 331 332 332 332 333 333 335 336 340 343 343 343 344 345 348 349 350 350 351 351 353 355 355 355 355 356
Contents xut 19.2.3 Partial Derivatives. 19.2.4 Singularity and Related Topics. 19.3 Complex Analysis. 19.3.1 De Moivre and Euler Identities . 19.3.2 Limits, Continuity, Derivatives, and Analyticity. 19.3.3 Contours or Curves. 19.3.4 Integration . 19.3.5 Infinite Series. 19.4 Asymptotics. 19.5 Fields . 19.6 Vector Spaces over Fields. 19.7 Linear Mappings. 19.8 Tensor Products. 19.9 Vector Algebra. 19.10 Vector Spaces Revisited. 19.10.1 Normed Vector Space. 19.10.2 Complete Vector Space and Compactness . 19.10.3 Inner Product
Space. 19.10.4 Orthogonality. 19.10.5 Gram֊Schmidt Orthogonalization Process. 19.11 More Hilbert Spaces. 19.11.1 Non-Orthogonal Expansion. 19.11.2 Biorthogonal Bases. Problems. 36! 363 363 365 366 367 368 368 369 372 373 378 379 382 384 384 385 386 388 389 390 391 392 393 20. Fourier Theory . 20.1 Introduction. 20.2 Fourier Series. 20.2.1 Generalized Functions. 20.2.2 Conditions for the Existence of Fourier Series. 20.2.3 Complex Fourier Series. 20.2.4 Trigonometric Fourier Series. 20.2.5 Generalized Fourier Series. 20.3 Transform Techniques. 20.3.1 Fourier
Transform. 20.3.2 Short-Time Fourier Transform. 20.3.3 Wigner-Ville Transform. Problems. 397 397 397 397 399 400 401 402 403 403 411 412 413 21. Probability Theory and Stochastic Processes. 21.1 Introduction. 21.2 Postulates of Probability Theory. 21.3 Random Variables. 421 421 421 423
xiv Contents 21.4 Average Measures. 21.4.1 Expectation. 21.4.2 Second-Order Expectations . 21.5 Independent Random Variables. 21.6 Moment-Generating Function. 21.7 Examples of Some Distributions. 21.7.1 Discrete Distributions. 21.7.2 Continuous Distributions . 21.7.3 Multivariate Gaussian Distribution. 21.8 Stochastic Processes. Problems. 425 426 427 427 428 429 429 429 431 432 434 References. 439 Index 449 |
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| id | DE-604.BV047509961 |
| illustrated | Illustrated |
| index_date | 2024-07-03T18:21:33Z |
| indexdate | 2024-07-10T09:14:04Z |
| institution | BVB |
| isbn | 9781032174839 1032174838 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-032910861 |
| oclc_num | 1284783907 |
| open_access_boolean | |
| owner | DE-706 DE-739 |
| owner_facet | DE-706 DE-739 |
| physical | xxviii, 455 Seiten Illustrationen 24 cm |
| publishDate | 2021 |
| publishDateSearch | 2021 |
| publishDateSort | 2021 |
| publisher | CRC Press |
| record_format | marc |
| spelling | Bhatnagar, Nirdosh Verfasser (DE-588)117615835X aut Introduction to wavelet transforms by Nirdosh Bhatnagar Boca Raton ; London ; New York CRC Press 2021 xxviii, 455 Seiten Illustrationen 24 cm txt rdacontent n rdamedia nc rdacarrier Wavelet (DE-588)4215427-3 gnd rswk-swf Wavelet-Transformation (DE-588)4814181-1 gnd rswk-swf Wavelets (Mathematics) Transformations (Mathematics) Wavelet (DE-588)4215427-3 s Wavelet-Transformation (DE-588)4814181-1 s DE-604 Äquivalent Druck-Ausgabe, Hardcover 978-0-367-43879-1 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032910861&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bhatnagar, Nirdosh Introduction to wavelet transforms Wavelet (DE-588)4215427-3 gnd Wavelet-Transformation (DE-588)4814181-1 gnd |
| subject_GND | (DE-588)4215427-3 (DE-588)4814181-1 |
| title | Introduction to wavelet transforms |
| title_auth | Introduction to wavelet transforms |
| title_exact_search | Introduction to wavelet transforms |
| title_exact_search_txtP | Introduction to wavelet transforms |
| title_full | Introduction to wavelet transforms by Nirdosh Bhatnagar |
| title_fullStr | Introduction to wavelet transforms by Nirdosh Bhatnagar |
| title_full_unstemmed | Introduction to wavelet transforms by Nirdosh Bhatnagar |
| title_short | Introduction to wavelet transforms |
| title_sort | introduction to wavelet transforms |
| topic | Wavelet (DE-588)4215427-3 gnd Wavelet-Transformation (DE-588)4814181-1 gnd |
| topic_facet | Wavelet Wavelet-Transformation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032910861&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT bhatnagarnirdosh introductiontowavelettransforms |