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Preface xi <p>List of Abbreviations xiii</p> <p><b>1 Introduction 1</b></p> <p>1.1 The Organization of This Book 2</p> <p><b>2 Signals 5</b></p> <p>2.1 Signal Classifications 5</p> <p>2.1.1 Periodic and Aperiodic Signals 5</p> <p>2.1.2 Even and Odd Signals 6</p> <p>2.1.3 Energy Signals 7</p> <p>2.1.4 Causal and Noncausal Signals 9</p> <p>2.2 Basic Signals 9</p> <p>2.2.1 Unit-Impulse Signal 9</p> <p>2.2.2 Unit-Step Signal 10</p> <p>2.2.3 The Sinusoid 10</p> <p>2.3 The Sampling Theorem and the Aliasing Effect 12</p> <p>2.4 Signal Operations 13</p> <p>2.4.1 Time Shifting 13</p> <p>2.4.2 Time Reversal 14</p> <p>2.4.3 Time Scaling 14</p> <p>2.5 Summary 17</p> <p>Exercises 17</p> <p><b>3 Convolution and Correlation 21</b></p> <p>3.1 Convolution 21</p> <p>3.1.1 The Linear Convolution 21</p> <p>3.1.2 Properties of Convolution 24</p> <p>3.1.3 The Periodic Convolution 25</p> <p>3.1.4 The Border Problem 25</p> <p>3.1.5 Convolution in the DWT 26</p> <p>3.2 Correlation 28</p> <p>3.2.1 The Linear Correlation 28</p> <p>3.2.2 Correlation and Fourier Analysis 29</p> <p>3.2.3 Correlation in the DWT 30</p> <p>3.3 Summary 31</p> <p>Exercises 31</p> <p><b>4 Fourier Analysis of Discrete Signals 37</b></p> <p>4.1 Transform Analysis 37</p> <p>4.2 The Discrete Fourier Transform 38</p> <p>4.2.1 Parseval’s Theorem 43</p> <p>4.3 The Discrete-Time Fourier Transform 44</p> <p>4.3.1 Convolution 48</p> <p>4.3.2 Convolution in the DWT 48</p> <p>4.3.3 Correlation 50</p> <p>4.3.4 Correlation in the DWT 50</p> <p>4.3.5 Time Expansion 52</p> <p>4.3.6 Sampling Theorem 52</p> <p>4.3.7 Parseval’s Theorem 54</p> <p>4.4 Approximation of the DTFT 55</p> <p>4.5 The Fourier Transform 56</p> <p>4.6 Summary 56</p> <p>Exercises 57</p> <p><b>5 Thez-Transform 59</b></p> <p>5.1 The z-Transform 59</p> <p>5.2 Properties of the z-Transform 60</p> <p>5.2.1 Linearity 60</p> <p>5.2.2 Time Shift of a Sequence 61</p> <p>5.2.3 Convolution 61</p> <p>5.3 Summary 62</p> <p>Exercises 62</p> <p><b>6 Finite Impulse Response Filters 63</b></p> <p>6.1 Characterization 63</p> <p>6.1.1 Ideal Lowpass Filters 64</p> <p>6.1.2 Ideal Highpass Filters 65</p> <p>6.1.3 Ideal Bandpass Filters 66</p> <p>6.2 Linear Phase Response 66</p> <p>6.2.1 Even-Symmetric FIR Filters with Odd Number of Coefficients 67</p> <p>6.2.2 Even-Symmetric FIR Filters with Even Number of Coefficients 68</p> <p>6.3 Summary 69</p> <p>Exercises 69</p> <p><b>7 Multirate Digital Signal Processing 71</b></p> <p>7.1 Decimation 72</p> <p>7.1.1 Downsampling in the Frequency-Domain 72</p> <p>7.1.2 Downsampling Followed by Filtering 75</p> <p>7.2 Interpolation 77</p> <p>7.2.1 Upsampling in the Frequency-Domain 77</p> <p>7.2.2 Filtering Followed by Upsampling 78</p> <p>7.3 Two-Channel Filter Bank 79</p> <p>7.3.1 Perfect Reconstruction Conditions 81</p> <p>7.4 Polyphase Form of the Two-Channel Filter Bank 84</p> <p>7.4.1 Decimation 84</p> <p>7.4.2 Interpolation 87</p> <p>7.4.3 Polyphase Form of the Filter Bank 91</p> <p>7.5 Summary 94</p> <p>Exercises 94</p> <p><b>8 The Haar Discrete Wavelet Transform 97</b></p> <p>8.1 Introduction 97</p> <p>8.1.1 Signal Representation 97</p> <p>8.1.2 The Wavelet Transform Concept 98</p> <p>8.1.3 Fourier and Wavelet Transform Analyses 98</p> <p>8.1.4 Time-Frequency Domain 99</p> <p>8.2 The Haar Discrete Wavelet Transform 100</p> <p>8.2.1 The Haar DWT and the 2-Point DFT 102</p> <p>8.2.2 The Haar Transform Matrix 103</p> <p>8.3 The Time-Frequency Plane 107</p> <p>8.4 Wavelets from the Filter Coefficients 111</p> <p>8.4.1 Two Scale Relations 116</p> <p>8.5 The 2-D Haar Discrete Wavelet Transform 118</p> <p>8.6 Discontinuity Detection 126</p> <p>8.7 Summary 127</p> <p>Exercises 128</p> <p><b>9 Orthogonal Filter Banks 131</b></p> <p>9.1 Haar Filter 132</p> <p>9.2 Daubechies Filter 135</p> <p>9.3 Orthogonality Conditions 146</p> <p>9.3.1 Characteristics of Daubechies Lowpass Filters 149</p> <p>9.4 Coiflet Filter 150</p> <p>9.5 Summary 154</p> <p>Exercises 155</p> <p><b>10 Biorthogonal Filter Banks 159</b></p> <p>10.1 Biorthogonal Filters 159</p> <p>10.2 5/3 Spline Filter 163</p> <p>10.2.1 Daubechies Formulation 170</p> <p>10.3 4/4 Spline Filter 170</p> <p>10.3.1 Daubechies Formulation 177</p> <p>10.4 CDF 9/7 Filter 178</p> <p>10.5 Summary 183</p> <p>Exercises 184</p> <p><b>11 Implementation of the Discrete Wavelet Transform 189</b></p> <p>11.1 Implementation of the DWT with Haar Filters 190</p> <p>11.1.1 1-Level Haar DWT 190</p> <p>11.1.2 2-Level Haar DWT 191</p> <p>11.1.3 1-Level Haar 2-D DWT 193</p> <p>11.1.4 The Signal-Flow Graph of the Fast Haar DWT Algorithms 194</p> <p>11.1.5 Haar DWT in Place 196</p> <p>11.2 Symmetrical Extension of the Data 198</p> <p>11.3 Implementation of the DWT with the D4 Filter 200</p> <p>11.4 Implementation of the DWT with Symmetrical Filters 203</p> <p>11.4.1 5/3 Spline Filter 203</p> <p>11.4.2 CDF 9/7 Filter 205</p> <p>11.4.3 4/4 Spline Filter 208</p> <p>11.5 Implementation of the DWT using Factorized Polyphase Matrix 210</p> <p>11.5.1 Haar Filter 211</p> <p>11.5.2 D4 Filter 213</p> <p>11.5.3 5/3 Spline Filter 216</p> <p>11.6 Summary 219</p> <p>Exercises 219</p> <p><b>12 The Discrete Wavelet Packet Transform 223</b></p> <p>12.1 The Discrete Wavelet Packet Transform 223</p> <p>12.1.1 Number of Representations 226</p> <p>12.2 Best Representation 227</p> <p>12.2.1 Cost Functions 230</p> <p>12.3 Summary 233</p> <p>Exercises 233</p> <p><b>13 The Discrete Stationary Wavelet Transform 235</b></p> <p>13.1 The Discrete Stationary Wavelet Transform 235</p> <p>13.1.1 The SWT 235</p> <p>13.1.2 The ISWT 236</p> <p>13.1.3 Algorithms for Computing the SWT and the ISWT 238</p> <p>13.1.4 2-D SWT 243</p> <p>13.2 Summary 244</p> <p>Exercises 244</p> <p><b>14 The Dual-Tree Discrete Wavelet Transform 247</b></p> <p>14.1 The Dual-Tree Discrete Wavelet Transform 248</p> <p>14.1.1 Parseval’s Theorem 248</p> <p>14.2 The Scaling and Wavelet Functions 252</p> <p>14.3 Computation of the DTDWT 253</p> <p>14.4 Summary 262</p> <p>Exercises 263</p> <p><b>15 Image Compression 265</b></p> <p>15.1 Lossy Image Compression 266</p> <p>15.1.1 Transformation 266</p> <p>15.1.2 Quantization 268</p> <p>15.1.3 Coding 270</p> <p>15.1.4 Compression Algorithm 273</p> <p>15.1.5 Image Reconstruction 277</p> <p>15.2 Lossless Image Compression 284</p> <p>15.3 Recent Trends in Image Compression 289</p> <p>15.3.1 The JPEG2000 Image Compression Standard 290</p> <p>15.4 Summary 290</p> <p>Exercises 291</p> <p><b>16 Denoising 295</b></p> <p>16.1 Denoising 295</p> <p>16.1.1 Soft Thresholding 296</p> <p>16.1.2 Statistical Measures 297</p> <p>16.2 VisuShrink Denoising Algorithm 298</p> <p>16.3 Summary 303</p> <p>Exercises 303</p> <p>Bibliography 305</p> <p>Answers to Selected Exercises 307</p> <p>Index 319</p>

<p>"Doubtless, this nice book will stimulate the practical education in the theory of DWT and its applications." (<i>Zentralblatt MATH</i>, 2016)</p>

<b>Dr. D. Sundararajan</b>, Department Head of Electrical and Electronics Engineering, Adhiyamaan College of Engineering, India.<br />Dr. Sundararajan obtained his PhD in Electrical Engineering at Concordia University, Montreal, Canada in 1988. As the principle inventor of the latest family of DFT algorithms, he has written three books, three Patents (which have been granted by US, Canada and Britain), and several papers in IEEE Transactions and in the Proceedings of IEEE Conference.

<p>This easily accessible text makes the learning of the discrete wavelet transform (DWT) easy to understand. Relatively new, DWT is fast becoming a widely used technique in signal and image processing applications, and is essential to know for all signal processing specialists. To facilitate learning for students and professionals with general engineering backgrounds, the author presents DWT using a unique signal processing approach instead of the usual mathematical approaches. The book also includes a large number of examples and figures that illustrate various concepts.</p> <ul> <li>Presents DWT from a digital signal processing point of view, in contrast to the usual mathematical approach, making it highly accessible</li> <li>Offers a comprehensive coverage of related topics, including convolution and correlation, Fourier transform, FIR filter, orthogonal and biorthogonal filters</li> <li>Organized systematically, starting from the fundamentals of signal processing to the more advanced topics of DWT and Discrete Wavelet Packet Transform</li> <li>Written in a clear and concise manner with abundant examples, figures and detailed explanations</li> <li>Features a companion website that has several MATLAB programs for the implementation of the DWT with commonly used filters</li> </ul> <p><i>Discrete Wavelet Transform: A Signal Processing Approach</i> with its clarity and concision, as well as numerous examples, is written with graduate and advanced signal processing students in mind. Industry researchers and professionals will also find it an accessible and comprehensive refresher guide.</p>

SG