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Foundations of Signal Processing

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Our main goals in this book and its companion volume, Fourier and Wavelet Signal Processing (FWSP) [57], are to enable an understanding of state-of-the-art signal processing methods and techniques, as well as to provide a solid foundation for those hoping to advance the theory and practice of signal processing.

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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

These materials are protected by copyright under the

Attri bution-NonCommercial-NoDerivs 3.0 Unported License

from Creative Commons.

Cambridge University Press (ISBN 110703860X) v1.1 [May 2014] [free version] Comments to b ook-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 e nc ourage me unceasingly, and

to the educators who made me want to be one of them.

— VKG

Cambridge University Press (ISBN 110703860X) v1.1 [May 2014] [free version] Comments to b ook-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 co lor components and

then synthesized back into white light. It is a physical implementation of a decom-

position of white light into its Fourier co mponents – the color s of the rainbow –

followed by a synthesis to recover the o riginal.

Cambridge University Press (ISBN 110703860X) v1.1 [May 2014] [free version] Comments to b ook-errata@FourierAndWavelets.org

Contents

Quick reference page xv

Acknowledgments xxi

Preface xxiii

1 On rainbows and s pectra 1

2 From Euclid to Hilbert 9

2.1 Introduction 10

2.2 Vector spaces 18

2.2.1 Deﬁnition and properties 18

2.2.2 Inner product 23

2.2.3 Norm 27

2.2.4 Standard spaces 30

2.3 Hilbert spaces 35

2.3.1 Convergence 36

2.3.2 Completeness 37

2.3.3 Linear operators 40

2.4 Approximations, projections, and decompositions 50

2.4.1 Projection theorem 51

2.4.2 Projection operators 54

2.4.3 Direct sums and subspace decompositions 60

2.4.4 Minimum mean-squared error estimation 63

2.5 Bases and frames 69

2.5.1 Bases and Riesz bases 69

2.5.2 Orthonormal b ases 76

2.5.3 Biorthogonal pairs of bases 86

2.5.4 Frames 101

2.5.5 Matrix representations of vectors and linear operators 109

2.6 Computational aspects 119

2.6.1 Cost, complexity, and asymptotic notations 120

2.6.2 Precision 123

2.6.3 Conditioning 126

2.6.4 Solving sy stems of linear equations 129

2.A Elements of analysis and topology 135

2.A.1 Basic deﬁ nitions 135

Cambridge University Press (ISBN 110703860X) v1.1 [May 2014] [free version] Comments to b ook-errata@FourierAndWavelets.org

x Contents

2.A.2 Convergence 136

2.A.3 Interchange theorems 138

2.A.4 Inequalities 139

2.A.5 Integration by parts 140

2.B Elements of linear algebra 141

2.B.1 Basic deﬁnitions and properties 141

2.B.2 Special matrices 147

2.C Elements of probability 151

2.C.1 Basic deﬁnitions 151

2.C.2 Standard distributions 154

2.C.3 Estimation 155

2.D Basis concepts 159

Chapter at a glance 161

Historical remarks 162

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