By Albert Boggess

ISBN-10: 0130228095

ISBN-13: 9780130228093

ISBN-10: 0470431172

ISBN-13: 9780470431177

ISBN-10: 1118211154

ISBN-13: 9781118211151

ISBN-10: 1118626265

ISBN-13: 9781118626269

A finished, self-contained therapy of Fourier research and wavelets—now in a brand new edition

Through expansive assurance and easy-to-follow reasons, a primary direction in Wavelets with Fourier research, moment version presents a self-contained mathematical remedy of Fourier research and wavelets, whereas uniquely proposing sign research functions and difficulties. crucial and basic principles are offered so as to make the booklet available to a huge viewers, and, additionally, their functions to sign processing are saved at an straightforward level.

The publication starts off with an creation to vector areas, internal product areas, and different initial subject matters in research. next chapters feature:

The improvement of a Fourier sequence, Fourier remodel, and discrete Fourier analysis

Improved sections dedicated to non-stop wavelets and two-dimensional wavelets

The research of Haar, Shannon, and linear spline wavelets

The normal conception of multi-resolution analysis

Updated MATLAB code and multiplied purposes to sign processing

The development, smoothness, and computation of Daubechies' wavelets

Advanced themes similar to wavelets in greater dimensions, decomposition and reconstruction, and wavelet transform

Applications to sign processing are supplied during the ebook, such a lot concerning the filtering and compression of indications from audio or video. a few of these functions are offered first within the context of Fourier research and are later explored within the chapters on wavelets. New routines introduce extra purposes, and whole proofs accompany the dialogue of every provided idea. huge appendices define extra complicated proofs and partial suggestions to routines in addition to up to date MATLAB workouts that complement the provided examples.

A First direction in Wavelets with Fourier research, moment version is a wonderful e-book for classes in arithmetic and engineering on the upper-undergraduate and graduate degrees. it's also a worthwhile source for mathematicians, sign processing engineers, and scientists who desire to know about wavelet thought and Fourier research on an uncomplicated level.

Table of Contents

Preface and Overview.

0 internal Product Spaces.

0.1 Motivation.

0.2 Definition of internal Product.

0.3 The areas L2 and l2.

0.4 Schwarz and Triangle Inequalities.

0.5 Orthogonality.

0.6 Linear Operators and Their Adjoints.

0.7 Least Squares and Linear Predictive Coding.

Exercises.

1 Fourier Series.

1.1 Introduction.

1.2 Computation of Fourier Series.

1.3 Convergence Theorems for Fourier Series.

Exercises.

2 The Fourier Transform.

2.1 casual improvement of the Fourier Transform.

2.2 homes of the Fourier Transform.

2.3 Linear Filters.

2.4 The Sampling Theorem.

2.5 The Uncertainty Principle.

Exercises.

3 Discrete Fourier Analysis.

3.1 The Discrete Fourier Transform.

3.2 Discrete Signals.

3.3 Discrete indications & Matlab.

Exercises.

4 Haar Wavelet Analysis.

4.1 Why Wavelets?

4.2 Haar Wavelets.

4.3 Haar Decomposition and Reconstruction Algorithms.

4.4 Summary.

Exercises.

5 Multiresolution Analysis.

5.1 The Multiresolution Framework.

5.2 imposing Decomposition and Reconstruction.

5.3 Fourier remodel Criteria.

Exercises.

6 The Daubechies Wavelets.

6.1 Daubechies’ Construction.

6.2 category, Moments, and Smoothness.

6.3 Computational Issues.

6.4 The Scaling functionality at Dyadic Points.

Exercises.

7 different Wavelet Topics.

7.1 Computational Complexity.

7.2 Wavelets in better Dimensions.

7.3 pertaining to Decomposition and Reconstruction.

7.4 Wavelet Transform.

Appendix A: Technical Matters.

Appendix B: options to chose Exercises.

Appendix C: MATLAB® Routines.

Bibliography.

Index.

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**Extra info for A First Course in Wavelets with Fourier Analysis**

**Example text**

Then5). Eq. 5) reduces to (l/n) f:,:rr 1 dx 2. 6) follows byfollows subtracting Eqs. 10) andsin(kx) then integrating ask Equation before. Equation (1. 7). 7) to compute the We start with relations the equation FouriNower coefficients. 11) k=I >k cos(kx) + bk sin(kx). : 1, we multiply both sides by cos nx and integrate: -1 1-T{ f (x) cos nx dx 1 1-T{ (oo + I >k cos(kx) + bk sin(kx) cos nx dx. : 1. Similarly, by multiplyingT{Eq. : 1. T{ oo by integrating Eq. 11) to give As a special case, we compute -2n1 1 T{ f(x) dx 2n1 1 T{ (oo + L00 ok cos(kx) + bk sin(kx) dx.

10) Adding these two identities and integrating gives 1 / Jr (cos((n + k)x + cos((n - k)x)) dx. cosnx coskx dx = 2 1Jr -n -n = -n Proof. - 71: -71: 43 COMPUTATION OF FOURIER SERIES The right side can be easily integrated. = k, then l:rr cosnxcoskxdx -21 [ sin(nn ++kk)x + sin(nn -k- k)x ] 0. : 1, then I: cos2 nx dx I: (1/2)(1 + cos2nx) dx n. then5). Eq. 5) reduces to (l/n) f:,:rr 1 dx 2. 6) follows byfollows subtracting Eqs. 10) andsin(kx) then integrating ask Equation before. Equation (1. 7). 7) to compute the We start with relations the equation FouriNower coefficients.

3. x Odd function f (-x) = - a f (x). For if f is even and g is odd, then g(-x)f (-x) = -g(x)f (x) and so fgAnother isexample, odd. important property of even and odd functions is given in the next lemma. 7 • If F is an even function, then f-a F(x) dx 2 Jor F(x) dx. a If F is an odd function, then 1_: F(x) dx = 0. 3. If F is even, then the integral over the left hal f -interval OJ is the same as the integral over the right half interval [0, a]. Thus, the integral over [-a, aJ is twice the integral over [0, aJ.

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