Brief Notes in Advanced DSP : Fourier Analysis with MATLAB

Based on the authors’ research in Fourier analysis, Brief Notes in Advanced DSP: Fourier Analysis with MATLAB® addresses many concepts and applications of digital signal processing (DSP). The included MATLAB® codes illustrate how to apply the ideas in practice. The book begins with the basic concept of the discrete Fourier transformation and its properties. It then describes lifting schemes, integer transformations, the discrete cosine transform, and the paired transform method for calculating the discrete Hadamard transform. The text also examines the decomposition of the 1-D signal by so-called section basis signals as well as new forms of 2-D signal/image representation and decomposition by direction signals/images. Focusing on Fourier transform wavelets and Givens–Haar transforms, the last chapter discusses the problem of signal multiresolution. This book presents numerous interesting problems and concepts of unitary transformations, such as the Fourier, Hadamard, Hartley, Haar, paired, cosine, and new signal-induced transformations. It aids readers in using new forms and methods of signals and images in the frequency and frequency-and-time domains. Cover Page ......Page 1 Brief Notes in Advanced DSP: Fourier Analysis with MATLAB®......Page 3 Contents......Page 5 Biography......Page 8 Preface......Page 9 References......Page 0 1.1 Properties of the discrete Fourier transform......Page 12 1.2 Fourier transform splitting......Page 17 1.3 Fast Fourier transform......Page 23 1.3.1 Unitary paired transform......Page 25 1.3.2 Fast 8-point DFT......Page 28 1.3.3 Fast 16-point DFT......Page 30 1.4 Codes for the paired FFT......Page 36 1.5 Paired and Haar transforms......Page 39 1.5.1 Haar functions......Page 40 1.5.2 Codes for the Haar transform......Page 44 1.5.3 Comparison with the paired transform......Page 45 Problems......Page 53 2.1.1 Lifting scheme implementation......Page 56 2.2 Lifting schemes for DFT......Page 60 2.3 One-point integer transform......Page 67 2.3.1 The eight-point integer Fourier transform......Page 70 2.3.2 Eight-point inverse integer DFT......Page 74 2.3.4 16-point IDFT with 8 and 12 control bits......Page 77 2.3.5 Inverse 16-point integer DFT......Page 78 2.3.6 Codes for the forward 16-point integer FFT......Page 89 2.3.6.1 General case = 2r, ≥ 4.......Page 92 2.4 DFT in vector form......Page 95 2.4.1 DFT in real space......Page 96 2.4.2.1 The = 6 case......Page 101 2.4.2.2 The = 3 case......Page 107 2.4.2.3 The = 4 case......Page 110 2.5 Roots of the unit......Page 112 2.5.1 Elliptic DFT......Page 116 2.6 Codes for the block DFT......Page 128 2.7 General elliptic Fourier transforms......Page 131 2.7.1 N-block GEFT......Page 133 Problems......Page 136 3.1 Partitioning the DCT......Page 139 3.1.1 4-point DCT of type IV......Page 150 3.1.2 IV DCTFastfour-pointtype......Page 152 3.1.3 8-point DCT of type IV......Page 155 3.2 Paired algorithm for the N-point DCT......Page 161 3.2.1 Paired functions......Page 162 3.2.2 Complexity of the calculation......Page 163 3.4 Reversible integer DCT......Page 165 3.4.1 Integer four-point DCTs......Page 166 3.4.2 Integer eight-point DCT......Page 169 3.5 Method of nonlinear equations......Page 170 3.5.1 Calculation of coefficients......Page 172 3.5.2 Error of approximation......Page 174 3.6.1 Reversible two-point transforms......Page 178 3.6.2 Reversible two-point DCT of type II......Page 180 3.6.3 Kernel transform......Page 181 3.6.4 Reversible two-point IDCT of type IV......Page 184 3.6.5 Parameterized two-point IDCT......Page 187 3.6.6 Codes for the integer 2-point DCT......Page 188 3.6.7 Four- and eight-point IDCTs......Page 190 Problems......Page 191 4.1 The Walsh and Hadamard transform......Page 194 4.1.1 Codes for the paired DHdT......Page 200 4.2 Mixed Hadamard transformation......Page 202 4.2.1 Square roots of mixed transformations......Page 205 4.2.2 High degree roots of the DHdT......Page 208 4.2.3 S-x transformation......Page 210 4.3 Generalized bit-and transformations......Page 212 4.3.1 Projection operators......Page 220 4.4 T-decomposition of Hadamard matrices......Page 221 4.4.1 Square roots of the Hadamard transformation......Page 223 4.4.2 Square roots of the identity transformation......Page 224 4.4.3 The 4th degree roots of the identity transformation......Page 230 4.5 Mixed Fourier transformations......Page 233 4.5.1 Fourier transformationSquarerootsofthe......Page 234 4.5.2 Series of Fourier transforms......Page 238 4.6 Mixed transformations: Continuous case......Page 243 Problems......Page 247 5.1 Decomposition of 1-D signals......Page 266 5.1.1 Section basis signals......Page 272 5.2 2-D paired representation......Page 274 5.2.1 Set-frequency characteristics......Page 277 5.2.2 Image reconstruction by projections......Page 280 5.2.3 Series images......Page 286 5.2.4 Resolution map......Page 288 5.2.5 A-series linear transformation......Page 290 5.2.6 Method of splitting-signals for image enhancement......Page 291 5.2.7 Fast methods of α-rooting......Page 294 5.2.7.1 Fast paired method of α-rooting......Page 298 5.2.7.2 Directional denoising......Page 300 5.2.8 Method of series images......Page 306 Problems......Page 307 6.1 Fourier transform......Page 308 6.1.1 Powers of the Fourier transform......Page 312 6.2.1 Wavelet transforms......Page 315 6.2.2 Fourier transform wavelet......Page 316 6.2.3 Cosine- and sine-wavelet transforms......Page 321 6.2.4 B-wavelet transforms......Page 325 6.2.5 Hartley transform representation......Page 327 6.3 Time-frequency correlation analysis......Page 329 6.3.1 Wavelet transform and ψ-resolution......Page 332 6.3.2 Cosine and sine correlation-type transforms......Page 334 6.3.3 Paired transform and Fourier function......Page 336 6.4 Givens-Haar transformations......Page 338 6.4.1 Fast transforms with Haar path......Page 343 6.4.2 Experimental results......Page 347 6.4.3 Characteristics of basic waves......Page 349 6.4.4 Givens-Haar transforms of any order......Page 353 Problems......Page 358 Cover Page 1 Brief Notes in Advanced DSP: Fourier Analysis with MATLAB® 3 Contents 5 Biography 8 Preface 9 References -1 Chapter 1: Discrete Fourier Transform 12 1.1 Properties of the discrete Fourier transform 12 1.2 Fourier transform splitting 17 1.3 Fast Fourier transform 23 1.3.1 Unitary paired transform 25 1.3.2 Fast 8-point DFT 28 1.3.3 Fast 16-point DFT 30 1.4 Codes for the paired FFT 36 1.5 Paired and Haar transforms 39 1.5.1 Haar functions 40 1.5.2 Codes for the Haar transform 44 1.5.3 Comparison with the paired transform 45 Problems 53 Chapter 2: Integer Fourier Transform 56 2.1 Reversible integer Fourier transform 56 2.1.1 Lifting scheme implementation 56 2.2 Lifting schemes for DFT 60 2.3 One-point integer transform 67 2.3.1 The eight-point integer Fourier transform 70 2.3.2 Eight-point inverse integer DFT 74 2.3.3 General method of control bits 77 2.3.4 16-point IDFT with 8 and 12 control bits 77 2.3.5 Inverse 16-point integer DFT 78 2.3.6 Codes for the forward 16-point integer FFT 89 2.3.6.1 General case = 2r, ≥ 4. 92 2.4 DFT in vector form 95 2.4.1 DFT in real space 96 2.4.2 Integer representation of the DFT 101 2.4.2.1 The = 6 case 101 2.4.2.2 The = 3 case 107 2.4.2.3 The = 4 case 110 2.5 Roots of the unit 112 2.5.1 Elliptic DFT 116 2.6 Codes for the block DFT 128 2.7 General elliptic Fourier transforms 131 2.7.1 N-block GEFT 133 Problems 136 Chapter 3: Cosine Transform 139 3.1 Partitioning the DCT 139 3.1.1 4-point DCT of type IV 150 3.1.2 IV DCTFastfour-pointtype 152 3.1.3 8-point DCT of type IV 155 3.2 Paired algorithm for the N-point DCT 161 3.2.1 Paired functions 162 3.2.2 Complexity of the calculation 163 3.3 Codes for the paired transform 165 3.4 Reversible integer DCT 165 3.4.1 Integer four-point DCTs 166 3.4.2 Integer eight-point DCT 169 3.5 Method of nonlinear equations 170 3.5.1 Calculation of coefficients 172 3.5.2 Error of approximation 174 3.6 Canonical representation of the integer DCT 178 3.6.1 Reversible two-point transforms 178 3.6.2 Reversible two-point DCT of type II 180 3.6.3 Kernel transform 181 3.6.4 Reversible two-point IDCT of type IV 184 3.6.5 Parameterized two-point IDCT 187 3.6.6 Codes for the integer 2-point DCT 188 3.6.7 Four- and eight-point IDCTs 190 Problems 191 Chapter 4: Hadamard Transform 194 4.1 The Walsh and Hadamard transform 194 4.1.1 Codes for the paired DHdT 200 4.2 Mixed Hadamard transformation 202 4.2.1 Square roots of mixed transformations 205 4.2.2 High degree roots of the DHdT 208 4.2.3 S-x transformation 210 4.3 Generalized bit-and transformations 212 4.3.1 Projection operators 220 4.4 T-decomposition of Hadamard matrices 221 4.4.1 Square roots of the Hadamard transformation 223 4.4.2 Square roots of the identity transformation 224 4.4.3 The 4th degree roots of the identity transformation 230 4.5 Mixed Fourier transformations 233 4.5.1 Fourier transformationSquarerootsofthe 234 4.5.2 Series of Fourier transforms 238 4.6 Mixed transformations: Continuous case 243 4.6.1 Linear convolution 247 Problems 247 Chapter 5: Paired Transform-Based Decomposition 266 5.1 Decomposition of 1-D signals 266 5.1.1 Section basis signals 272 5.2 2-D paired representation 274 5.2.1 Set-frequency characteristics 277 5.2.2 Image reconstruction by projections 280 5.2.3 Series images 286 5.2.4 Resolution map 288 5.2.5 A-series linear transformation 290 5.2.6 Method of splitting-signals for image enhancement 291 5.2.7 Fast methods of α-rooting 294 5.2.7.1 Fast paired method of α-rooting 298 5.2.7.2 Directional denoising 300 5.2.8 Method of series images 306 Problems 307 Chapter 6: Fourier Transform and Multiresolution 308 6.1 Fourier transform 308 6.1.1 Powers of the Fourier transform 312 6.2 Representation by frequency-time wavelets 315 6.2.1 Wavelet transforms 315 6.2.2 Fourier transform wavelet 316 6.2.3 Cosine- and sine-wavelet transforms 321 6.2.4 B-wavelet transforms 325 6.2.5 Hartley transform representation 327 6.3 Time-frequency correlation analysis 329 6.3.1 Wavelet transform and ψ-resolution 332 6.3.2 Cosine and sine correlation-type transforms 334 6.3.3 Paired transform and Fourier function 336 6.4 Givens-Haar transformations 338 6.4.1 Fast transforms with Haar path 343 6.4.2 Experimental results 347 6.4.3 Characteristics of basic waves 349 6.4.4 Givens-Haar transforms of any order 353 Problems 358 References -1

based On The Authors’ Research In Fourier Analysis, Brief Notes In Advanced Dsp: Fourier Analysis With Matlab® Addresses Many Concepts And Applications Of Digital Signal Processing (dsp). The Included Matlab® Codes Illustrate How To Apply The Ideas In Practice.

the Book Begins With The Basic Concept Of The Discrete Fourier Transformation And Its Properties. It Then Describes Lifting Schemes, Integer Transformations, The Discrete Cosine Transform, And The Paired Transform Method For Calculating The Discrete Hadamard Transform. The Text Also Examines The Decomposition Of The 1-d Signal By So-called Section Basis Signals As Well As New Forms Of 2-d Signal/image Representation And Decomposition By Direction Signals/images. Focusing On Fourier Transform Wavelets And Givens—haar Transforms, The Last Chapter Discusses The Problem Of Signal Multiresolution.

this Book Presents Numerous Interesting Problems And Concepts Of Unitary Transformations, Such As The Fourier, Hadamard, Hartley, Haar, Paired, Cosine, And New Signal-induced Transformations. It Aids Readers In Using New Forms And Methods Of Signals And Images In The Frequency And Frequency-and-time Domains.