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Foundations of Signal Processing.pdf

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    Foundations of Signal Processing Martin Vetterli ´Ecole Polytechnique F´ed´erale de Lausanne Jelena Kovaˇcevi´c Carnegie Mellon University Vivek K Goyal Massachusetts Institute of Technology & Boston University May 31, 2014 Copyright (c) 2014 Martin Vetterli, Jelena Kovaˇcevi´c, and Vivek K Goyal. These materials are protected by copyright under the Attribution-NonCommercial-NoDerivs 3.0 Unported License from Creative Commons. FoundationsofSignalProcessingCopyright2014M.Vetterli,J.Kovaˇcevi´c,andV.K.GoyalCambridgeUniversityPress(ISBN110703860X)v1.1[May2014][freeversion]Commentstobook-errata@FourierAndWavelets.org
    To Marie-Laure, for her ∞ patience and many other qualities, Thomas and No´emie, whom I might still convince of the beauty of this material, and my parents, who gave me all the opportunities one can wish for. — MV To Danica and Giovanni, who make life beautiful. To my parents, who made me who I am. — JK To Allie, Sundeep, and my family, who encourage me unceasingly, and to the educators who made me want to be one of them. — VKG FoundationsofSignalProcessingCopyright2014M.Vetterli,J.Kovaˇcevi´c,andV.K.GoyalCambridgeUniversityPress(ISBN110703860X)v1.1[May2014][freeversion]Commentstobook-errata@FourierAndWavelets.org
    The cover illustration captures an experiment first described by Isaac Newton in Opticks in 1730, showing that white light can be split into its color components and then synthesized back into white light. It is a physical implementation of a decom- position of white light into its Fourier components – the colors of the rainbow – followed by a synthesis to recover the original. FoundationsofSignalProcessingCopyright2014M.Vetterli,J.Kovaˇcevi´c,andV.K.GoyalCambridgeUniversityPress(ISBN110703860X)v1.1[May2014][freeversion]Commentstobook-errata@FourierAndWavelets.org
    Contents Quick reference Acknowledgments Preface 1 On rainbows and spectra 2 From Euclid to Hilbert Introduction 2.1 2.2 Vector spaces Inner product 2.2.1 Definition and properties 2.2.2 2.2.3 Norm 2.2.4 Standard spaces 2.3 Hilbert spaces 2.3.1 Convergence 2.3.2 Completeness 2.3.3 Linear operators 2.4 Approximations, projections, and decompositions 2.4.1 Projection theorem 2.4.2 Projection operators 2.4.3 Direct sums and subspace decompositions 2.4.4 Minimum mean-squared error estimation 2.5 Bases and frames 2.5.1 Bases and Riesz bases 2.5.2 Orthonormal bases 2.5.3 Biorthogonal pairs of bases 2.5.4 Frames 2.5.5 Matrix representations of vectors and linear operators 2.6 Computational aspects 2.6.1 Cost, complexity, and asymptotic notations 2.6.2 Precision 2.6.3 Conditioning 2.6.4 Solving systems of linear equations 2.A Elements of analysis and topology 2.A.1 Basic definitions ix page xv xxi xxiii 1 9 10 18 18 23 27 30 35 36 37 40 50 51 54 60 63 69 69 76 86 101 109 119 120 123 126 129 135 135 FoundationsofSignalProcessingCopyright2014M.Vetterli,J.Kovaˇcevi´c,andV.K.GoyalCambridgeUniversityPress(ISBN110703860X)v1.1[May2014][freeversion]Commentstobook-errata@FourierAndWavelets.org
    x 3 2.A.2 Convergence 2.A.3 Interchange theorems 2.A.4 Inequalities 2.A.5 Integration by parts 2.B Elements of linear algebra 2.B.1 Basic definitions and properties 2.B.2 Special matrices 2.C Elements of probability 2.C.1 Basic definitions 2.C.2 Standard distributions 2.C.3 Estimation 2.D Basis concepts Chapter at a glance Historical remarks Further reading Exercises with solutions Exercises Sequences and discrete-time systems 3.1 3.2 Infinite-length sequences Introduction Sequences 3.2.1 3.2.2 Finite-length sequences 3.2.3 Two-dimensional sequences Systems 3.3.1 Discrete-time systems and their properties 3.3.2 Difference equations 3.3.3 Linear shift-invariant systems 3.3 3.4 Discrete-time Fourier transform 3.4.1 Definition of the DTFT 3.4.2 Existence and convergence of the DTFT 3.4.3 Properties of the DTFT 3.4.4 Frequency response of filters z-transform 3.5.1 Definition of the z-transform 3.5.2 Existence and convergence of the z-transform 3.5.3 Properties of the z-transform 3.5.4 z-transform of filters 3.5 3.6 Discrete Fourier transform 3.6.1 Definition of the DFT 3.6.2 Properties of the DFT 3.6.3 Frequency response of filters 3.7 Multirate sequences and systems 3.7.1 Downsampling 3.7.2 Upsampling 3.7.3 Combinations of downsampling and upsampling 3.7.4 Combinations of downsampling, upsampling, and filtering 3.7.5 Polyphase representation Stochastic processes and systems 3.8.1 Stochastic processes 3.8 Contents 136 138 139 140 141 141 147 151 151 154 155 159 161 162 162 163 169 181 182 185 185 192 193 195 195 202 205 216 216 218 221 227 233 234 235 240 249 252 252 255 259 264 265 268 270 272 278 285 285 FoundationsofSignalProcessingCopyright2014M.Vetterli,J.Kovaˇcevi´c,andV.K.GoyalCambridgeUniversityPress(ISBN110703860X)v1.1[May2014][freeversion]Commentstobook-errata@FourierAndWavelets.org
    Contents 3.8.2 Systems 3.8.3 Discrete-time Fourier transform 3.8.4 Multirate sequences and systems 3.8.5 Minimum mean-squared error estimation 3.9 Computational aspects 3.9.1 Fast Fourier transforms 3.9.2 Convolution 3.9.3 Multirate operations 3.A Elements of analysis 3.A.1 Complex numbers 3.A.2 Difference equations 3.A.3 Convergence of the convolution sum 3.A.4 Dirac delta function 3.B Elements of algebra 3.B.1 Polynomials 3.B.2 Vectors and matrices of polynomials 3.B.3 Kronecker product Chapter at a glance Historical remarks Further reading Exercises with solutions Exercises 4 Functions and continuous-time systems Introduction 4.1 4.2 Functions 4.3 4.2.1 Functions on the real line 4.2.2 Periodic functions Systems 4.3.1 Continuous-time systems and their properties 4.3.2 Differential equations 4.3.3 Linear shift-invariant systems 4.4 Fourier transform 4.4.1 Definition of the Fourier transform 4.4.2 Existence and inversion of the Fourier transform 4.4.3 Properties of the Fourier transform 4.4.4 Frequency response of filters 4.4.5 Regularity and spectral decay 4.4.6 Laplace transform 4.5 Fourier series 4.5.1 Definition of the Fourier series 4.5.2 Existence and convergence of the Fourier series 4.5.3 Properties of the Fourier series 4.5.4 Frequency response of filters Stochastic processes and systems 4.6.1 Stochastic processes 4.6.2 Systems 4.6.3 Fourier transform 4.6 Chapter at a glance Historical remarks xi 288 292 294 300 303 303 307 311 313 313 315 316 316 318 318 321 324 325 328 328 329 336 343 344 345 345 351 351 352 355 355 359 359 360 365 373 374 379 380 381 383 385 394 395 395 397 399 401 403 FoundationsofSignalProcessingCopyright2014M.Vetterli,J.Kovaˇcevi´c,andV.K.GoyalCambridgeUniversityPress(ISBN110703860X)v1.1[May2014][freeversion]Commentstobook-errata@FourierAndWavelets.org
    xii Contents Further reading Exercises with solutions Exercises 5 Sampling and interpolation 5.1 5.2 Finite-dimensional vectors Introduction 5.3 5.2.1 Sampling and interpolation with orthonormal vectors 5.2.2 Sampling and interpolation with nonorthogonal vectors Sequences 5.3.1 Sampling and interpolation with orthonormal sequences 5.3.2 Sampling and interpolation for bandlimited sequences 5.3.3 Sampling and interpolation with nonorthogonal sequences 5.4 Functions 5.4.1 Sampling and interpolation with orthonormal functions 5.4.2 Sampling and interpolation for bandlimited functions 5.4.3 Sampling and interpolation with nonorthogonal functions 5.5 Periodic functions 5.5.1 Sampling and interpolation with orthonormal periodic functions 5.5.2 Sampling and interpolation for bandlimited periodic functions 5.6 Computational aspects 5.6.1 Projection onto convex sets Chapter at a glance Historical remarks Further reading Exercises with solutions Exercises 6 Approximation and compression Introduction 6.1 6.2 Approximation of functions on finite intervals by polynomials 6.2.1 Least-squares approximation 6.2.2 Lagrange interpolation: Matching points 6.2.3 Taylor series expansion: Matching derivatives 6.2.4 Hermite interpolation: Matching points and derivatives 6.2.5 Minimax polynomial approximation 6.2.6 Filter design 6.3 Approximation of functions by splines 403 404 406 411 412 420 421 425 429 430 437 442 447 449 452 470 477 477 481 489 491 496 498 498 500 504 507 508 513 514 517 520 522 523 529 537 538 541 548 6.3.1 Splines and spline spaces 6.3.2 Bases for uniform spline spaces 6.3.3 Strang–Fix condition for polynomial representation 6.3.4 Continuous-time operators in spline spaces implemented with discrete- time processing 6.4 Approximation of functions and sequences by series truncation 6.4.1 Linear and nonlinear approximations 6.4.2 Linear approximation of random vectors and stochastic processes 6.4.3 Linear and nonlinear diagonal estimators 6.5 Compression 6.5.1 Lossless compression 6.5.2 Scalar quantization 554 560 560 566 571 576 577 579 FoundationsofSignalProcessingCopyright2014M.Vetterli,J.Kovaˇcevi´c,andV.K.GoyalCambridgeUniversityPress(ISBN110703860X)v1.1[May2014][freeversion]Commentstobook-errata@FourierAndWavelets.org
    Contents 6.5.3 Transform coding 6.6 Computational aspects 6.6.1 Huffman algorithm for lossless code design 6.6.2 6.6.3 Estimating from quantized samples Iterative design of quantizers Chapter at a glance Historical remarks Further reading Exercises with solutions Exercises 7 Localization and uncertainty 7.1 7.2 Localization for functions Introduction 7.2.1 Localization in time 7.2.2 Localization in frequency 7.2.3 Uncertainty principle for functions 7.3 Localization for sequences 7.3.1 Localization in time 7.3.2 Localization in frequency 7.3.3 Uncertainty principle for sequences 7.3.4 Uncertainty principle for finite-length sequences 7.4 Tiling the time–frequency plane 7.4.1 Localization for structured sets of functions 7.4.2 Localization for structured sets of sequences 7.5 Examples of local Fourier and wavelet bases 7.5.1 Local Fourier and wavelet bases for functions 7.5.2 Local Fourier and wavelet bases for sequences 7.6 Recap and a glimpse forward 7.6.1 Tools 7.6.2 Adapting tools to real-world problems 7.6.3 Music analysis, communications, and compression Chapter at a glance Historical remarks Further reading Exercises with solutions Exercises Image and quote attribution References Index xiii 584 591 591 593 594 597 598 599 601 607 615 616 619 620 622 624 627 629 630 633 637 638 638 640 645 645 651 656 657 658 659 666 667 667 669 671 673 675 681 FoundationsofSignalProcessingCopyright2014M.Vetterli,J.Kovaˇcevi´c,andV.K.GoyalCambridgeUniversityPress(ISBN110703860X)v1.1[May2014][freeversion]Commentstobook-errata@FourierAndWavelets.org
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