Discrete Fourier and Wavelet Transforms An Introduction through Algebra with Applications to Signal Processing 1st edition by Roe W Goodman ISBN 9814725767 9789814725767

1. Linear Algebra and Signal Processing

1.1 Overview

1.2 Sampling and Quantization

1.3 Vector Spaces

1.4 Bases and Dual Bases

1.5 Linear Transformations and Matrices

1.5.1 Matrix form of a linear transformation

1.5.2 Direct sums of vector spaces

1.5.3 Partitioned matrices and block multiplication

1.6 Vector Graphics and Animation

1.6.1 Geometric transformations of images

1.6.2 Affine transformations

1.7 Inner Products, Orthogonal Projections, and Unitary Matrices

1.8 Fourier Series

1.9 Computer Explorations

1.9.1 Sampling and quantizing an audio signal

1.9.2 Vector graphics

1.10 Exercises

2. Discrete Fourier Transform

2.1 Overview

2.2 Sampling and Aliasing

2.3 Discrete Fourier Transform and Fourier Matrix

2.4 Shift-Invariant Transformations and Circulant Matrices

2.4.1 Moving averages and shift operator

2.4.2 Shift-invariant transformations

2.4.3 Eigenvectors and eigenvalues of circulant matrices

2.5 Circular Convolution and Filters

2.6 Downsampling and Fast Fourier Transform

2.7 Computer Explorations

2.7.1 Fourier matrix and sampling

2.7.2 Applications of the discrete Fourier transform

2.7.3 Circulant matrices and circular convolution

2.7.4 Fast Fourier transform

2.8 Exercises

3. Discrete Wavelet Transforms

3.1 Overview

3.2 Haar Wavelet Transform for Digital Signals

3.2.1 Basic example

3.2.2 Prediction and update transformations

3.3 Multiple Scale Haar Wavelet Transform

3.3.1 Matrix description of multiresolution representation

3.3.2 Signal processing using the multiresolution representation

3.4 Wavelet Transforms for Periodic Signals by Lifting

3.4.1 CDF(2, 2) transform

3.4.2 Daub4 transform

3.5 Wavelet Bases for Periodic Signals

3.5.1 Lifting steps and polyphase matrices

3.5.2 One-scale wavelet matrices

3.5.3 Trend and detail subspaces

3.5.4 Multiscale wavelet matrices

3.6 Two-Dimensional Wavelet Transforms

3.6.1 Images as matrices

3.6.2 One-scale 2D wavelet transform

3.6.3 Multiscale 2D wavelet transform

3.6.4 Image compression using wavelet transforms

3.7 Computer Explorations

3.7.1 Haar transform

3.7.2 CDF(2, 2) wavelet transform

3.7.3 Daub4 wavelet transform

3.7.4 Fast multiscale Haar transform

3.7.5 Fast multiscale Daub4 transform

3.7.6 Signal processing with the multiscale Haar transform

3.8 Exercises

4. Wavelet Transforms from Filter Banks

4.1 Overview

4.2 Filtering, Downsampling, and Upsampling

4.2.1 Signals and z-transforms

4.2.2 Convolution

4.2.3 Linear shift-invariant filters

4.2.4 Downsampling and upsampling

4.2.5 Periodic signals

4.2.6 Filtering and downsampling of periodic signals

4.2.7 Discrete Fourier transform and z-transform

4.3 Filter Banks and Polyphase Matrices

4.3.1 Lazy filter bank

4.3.2 Filter banks from lifting

4.4 Filter Banks and Modulation Matrices

4.4.1 Lowpass and highpass filters

4.4.2 Filter banks from filter pairs

4.4.3 Perfect reconstruction from analysis filters

4.5 Perfect Reconstruction Filter Pairs

4.5.1 Perfect reconstruction from lowpass filters

4.5.2 Lowpass filters and the Bezout polynomials

4.5.3 CDF(p, q) filters

4.6 Comparing Polyphase and Modulation Matrices

4.7 Lifting Step Factorization of Polyphase Matrices

4.8 Biorthogonal Wavelet Bases

4.9 Orthogonal Filter Banks

4.10 Daubechies Wavelet Transforms

4.10.1 Power spectral response function

4.10.2 Construction of the Daub4 filters

4.10.3 Construction of the Daub2K filters

4.11 Computer Explorations

4.11.1 Signal processing with the CDF(2, 2) transform

4.11.2 Two-dimensional discrete wavelet transforms

4.11.3 Image compression and multiscale analysis

4.11.4 Fast two-dimensional wavelet transforms

4.11.5 Denoising and compressing images

4.12 Exercises

5. Wavelet Transforms for Analog Signals

5.1 Overview

5.2 Linear Transformations of Analog Signals

5.2.1 Finite-energy analog signals

5.2.2 Orthogonal projections

5.2.3 Shift and dilation operators

5.3 Haar Wavelet Transform for Analog Signals

5.3.1 Haar scaling function

5.3.2 Haar multiresolution analysis

5.3.3 Haar wavelet and wavelet transform

5.4 Scaling and Wavelet Functions from Orthogonal Filter Banks

5.4.1 Cascade algorithm

5.4.2 Orthogonality relations

5.5 Multiresolution Analysis of Analog Signals

5.5.1 Multiresolution spaces

5.5.2 Trend and detail projections

5.5.3 Fast multiscale wavelet transform

5.5.4 Vanishing moments for wavelet functions

5.5.5 Guides to wavelet theory and applications

5.6 Computer Explorations

5.6.1 Generating scaling and wavelet functions

5.6.2 Using wavelet transforms to find singularities

5.7 Exercises

Appendix A Some Mathematical and Software Tools

A.1 Complex Numbers and Roots of Polynomials

A.2 Exponential Function and Roots of Unity

A.3 Computations in MATLAB and UVI_WAVE

 A.3.1 Introduction to MATLAB

 A.3.2 UVI_WAVE software

Appendix B Solutions to Exercises

B.1 Solutions to Exercises 1.10

B.2 Solutions to Exercises 2.8

B.3 Solutions to Exercises 3.8

B.4 Solutions to Exercises 4.12

B.5 Solutions to Exercises 5.7

Bibliography

Index