# Discrete Wavelet Transform: A Signal Processing Approach

## Evaluation Copy

Request an Evaluation Copy

# Discrete Wavelet Transform: A Signal Processing Approach

ISBN: 978-1-119-11311-9 December 2015 344 Pages

## Description

Provides easy learning and understanding of DWT from a signal processing point of view

• Presents DWT from a digital signal processing point of view, in contrast to the usual mathematical approach, making it highly accessible
• Offers a comprehensive coverage of related topics, including convolution and correlation, Fourier transform, FIR filter, orthogonal and biorthogonal filters
• Organized systematically, starting from the fundamentals of signal processing to the more advanced topics of DWT and Discrete Wavelet Packet Transform.
• Written in a clear and concise manner with abundant examples, figures and detailed explanations
• Features a companion website that has several MATLAB programs for the implementation of the DWT with commonly used filters

“This well-written textbook is an introduction to the theory of discrete wavelet transform (DWT) and its applications in digital signal and image processing.”
-- Prof. Dr. Manfred Tasche - Institut für Mathematik, Uni Rostock

Full review at https://zbmath.org/?q=an:06492561

## Related Resources

##### Instructor

View Instructor Companion Site

Preface xi

List of Abbreviations xiii

1 Introduction 1

1.1 The Organization of This Book 2

2 Signals 5

2.1 Signal Classifications 5

2.1.1 Periodic and Aperiodic Signals 5

2.1.2 Even and Odd Signals 6

2.1.3 Energy Signals 7

2.1.4 Causal and Noncausal Signals 9

2.2 Basic Signals 9

2.2.1 Unit-Impulse Signal 9

2.2.2 Unit-Step Signal 10

2.2.3 The Sinusoid 10

2.3 The Sampling Theorem and the Aliasing Effect 12

2.4 Signal Operations 13

2.4.1 Time Shifting 13

2.4.2 Time Reversal 14

2.4.3 Time Scaling 14

2.5 Summary 17

Exercises 17

3 Convolution and Correlation 21

3.1 Convolution 21

3.1.1 The Linear Convolution 21

3.1.2 Properties of Convolution 24

3.1.3 The Periodic Convolution 25

3.1.4 The Border Problem 25

3.1.5 Convolution in the DWT 26

3.2 Correlation 28

3.2.1 The Linear Correlation 28

3.2.2 Correlation and Fourier Analysis 29

3.2.3 Correlation in the DWT 30

3.3 Summary 31

Exercises 31

4 Fourier Analysis of Discrete Signals 37

4.1 Transform Analysis 37

4.2 The Discrete Fourier Transform 38

4.2.1 Parseval’s Theorem 43

4.3 The Discrete-Time Fourier Transform 44

4.3.1 Convolution 48

4.3.2 Convolution in the DWT 48

4.3.3 Correlation 50

4.3.4 Correlation in the DWT 50

4.3.5 Time Expansion 52

4.3.6 Sampling Theorem 52

4.3.7 Parseval’s Theorem 54

4.4 Approximation of the DTFT 55

4.5 The Fourier Transform 56

4.6 Summary 56

Exercises 57

5 Thez-Transform 59

5.1 The z-Transform 59

5.2 Properties of the z-Transform 60

5.2.1 Linearity 60

5.2.2 Time Shift of a Sequence 61

5.2.3 Convolution 61

5.3 Summary 62

Exercises 62

6 Finite Impulse Response Filters 63

6.1 Characterization 63

6.1.1 Ideal Lowpass Filters 64

6.1.2 Ideal Highpass Filters 65

6.1.3 Ideal Bandpass Filters 66

6.2 Linear Phase Response 66

6.2.1 Even-Symmetric FIR Filters with Odd Number of Coefficients 67

6.2.2 Even-Symmetric FIR Filters with Even Number of Coefficients 68

6.3 Summary 69

Exercises 69

7 Multirate Digital Signal Processing 71

7.1 Decimation 72

7.1.1 Downsampling in the Frequency-Domain 72

7.1.2 Downsampling Followed by Filtering 75

7.2 Interpolation 77

7.2.1 Upsampling in the Frequency-Domain 77

7.2.2 Filtering Followed by Upsampling 78

7.3 Two-Channel Filter Bank 79

7.3.1 Perfect Reconstruction Conditions 81

7.4 Polyphase Form of the Two-Channel Filter Bank 84

7.4.1 Decimation 84

7.4.2 Interpolation 87

7.4.3 Polyphase Form of the Filter Bank 91

7.5 Summary 94

Exercises 94

8 The Haar Discrete Wavelet Transform 97

8.1 Introduction 97

8.1.1 Signal Representation 97

8.1.2 The Wavelet Transform Concept 98

8.1.3 Fourier and Wavelet Transform Analyses 98

8.1.4 Time-Frequency Domain 99

8.2 The Haar Discrete Wavelet Transform 100

8.2.1 The Haar DWT and the 2-Point DFT 102

8.2.2 The Haar Transform Matrix 103

8.3 The Time-Frequency Plane 107

8.4 Wavelets from the Filter Coefficients 111

8.4.1 Two Scale Relations 116

8.5 The 2-D Haar Discrete Wavelet Transform 118

8.6 Discontinuity Detection 126

8.7 Summary 127

Exercises 128

9 Orthogonal Filter Banks 131

9.1 Haar Filter 132

9.2 Daubechies Filter 135

9.3 Orthogonality Conditions 146

9.3.1 Characteristics of Daubechies Lowpass Filters 149

9.4 Coiflet Filter 150

9.5 Summary 154

Exercises 155

10 Biorthogonal Filter Banks 159

10.1 Biorthogonal Filters 159

10.2 5/3 Spline Filter 163

10.2.1 Daubechies Formulation 170

10.3 4/4 Spline Filter 170

10.3.1 Daubechies Formulation 177

10.4 CDF 9/7 Filter 178

10.5 Summary 183

Exercises 184

11 Implementation of the Discrete Wavelet Transform 189

11.1 Implementation of the DWT with Haar Filters 190

11.1.1 1-Level Haar DWT 190

11.1.2 2-Level Haar DWT 191

11.1.3 1-Level Haar 2-D DWT 193

11.1.4 The Signal-Flow Graph of the Fast Haar DWT Algorithms 194

11.1.5 Haar DWT in Place 196

11.2 Symmetrical Extension of the Data 198

11.3 Implementation of the DWT with the D4 Filter 200

11.4 Implementation of the DWT with Symmetrical Filters 203

11.4.1 5/3 Spline Filter 203

11.4.2 CDF 9/7 Filter 205

11.4.3 4/4 Spline Filter 208

11.5 Implementation of the DWT using Factorized Polyphase Matrix 210

11.5.1 Haar Filter 211

11.5.2 D4 Filter 213

11.5.3 5/3 Spline Filter 216

11.6 Summary 219

Exercises 219

12 The Discrete Wavelet Packet Transform 223

12.1 The Discrete Wavelet Packet Transform 223

12.1.1 Number of Representations 226

12.2 Best Representation 227

12.2.1 Cost Functions 230

12.3 Summary 233

Exercises 233

13 The Discrete Stationary Wavelet Transform 235

13.1 The Discrete Stationary Wavelet Transform 235

13.1.1 The SWT 235

13.1.2 The ISWT 236

13.1.3 Algorithms for Computing the SWT and the ISWT 238

13.1.4 2-D SWT 243

13.2 Summary 244

Exercises 244

14 The Dual-Tree Discrete Wavelet Transform 247

14.1 The Dual-Tree Discrete Wavelet Transform 248

14.1.1 Parseval’s Theorem 248

14.2 The Scaling and Wavelet Functions 252

14.3 Computation of the DTDWT 253

14.4 Summary 262

Exercises 263

15 Image Compression 265

15.1 Lossy Image Compression 266

15.1.1 Transformation 266

15.1.2 Quantization 268

15.1.3 Coding 270

15.1.4 Compression Algorithm 273

15.1.5 Image Reconstruction 277

15.2 Lossless Image Compression 284

15.3 Recent Trends in Image Compression 289

15.3.1 The JPEG2000 Image Compression Standard 290

15.4 Summary 290

Exercises 291

16 Denoising 295

16.1 Denoising 295

16.1.1 Soft Thresholding 296

16.1.2 Statistical Measures 297

16.2 VisuShrink Denoising Algorithm 298

16.3 Summary 303

Exercises 303

Bibliography 305