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 1D signal by so-called section basis signals as well as new forms of 2D 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.

Author Notes

Artyom M. Grigoryan, Merughan Grigoryan

Biography	p. ix
Preface	p. xi
1 Discrete Fourier Transform	p. 1
1.1 Properties of the discrete Fourier transform	p. 1
1.2 Fourier transform splitting	p. 6
1.3 Fast Fourier transform	p. 12
1.3.1 Unitary paired transform	p. 14
1.3.2 Fast 8-point DFT	p. 17
1.3.3 Fast 16-point DFT	p. 19
1.4 Codes for the paired FFT	p. 25
1.5 Paired and Haar transforms	p. 28
1.5.1 Haar functions	p. 29
1.5.2 Codes for the Haar transform	p. 33
1.5.3 Comparison with the paired transform	p. 34
2 Integer Fourier Transform	p. 45
2.1 Reversible integer Fourier transform	p. 45
2.1.1 Lifting scheme implementation	p. 45
2.2 Lifting schemes for DFT	p. 49
2.3 One-point integer transform	p. 56
2.3.1 The eight-point integer Fourier transform	p. 59
2.3.2 Eight-point inverse integer DFT	p. 63
2.3.3 General method of control bits	p. 66
2.3.4 16-point IDFT with 8 and 12 control bits	p. 66
2.3.5 Inverse 16-point integer DFT	p. 67
2.3.6 Codes for the forward 16-point integer FFT	p. 78
2.4 DFT in vector form	p. 84
2.4.1 DFT in real space	p. 85
2.4.2 Integer representation of the DFT	p. 90
2.5 Roots of the unit	p. 101
2.5.1 Elliptic DFT	p. 105
2.6 Codes for the block DFT	p. 117
2.7 General elliptic Fourier transforms	p. 120
2.7.1 N-block GEFT	p. 122
3 Cosine Transform	p. 129
3.1 Partitioning the DCT	p. 129
3.1.1 4-point DCT of type IV	p. 140
3.1.2 Fast four-point type IV DCT	p. 142
3.1.3 8-point DCT of type IV	p. 145
3.2 Paired algorithm for the N-point DCT	p. 151
3.2.1 Paired functions	p. 152
3.2.2 Complexity of the calculation	p. 153
3.3 Codes for the paired transform	p. 155
3.4 Reversible integer DCT	p. 155
3.4.1 Integer four-point DCTs	p. 156
3.4.2 Integer eight-point DCT	p. 159
3.5 Method of nonlinear equations	p. 160
3.5.1 Calculation of coefficients	p. 162
3.5.2 Error of approximation	p. 164
3.6 Canonical representation of the integer DCT	p. 168
3.6.1 Reversible two-point transforms	p. 168
3.6.2 Reversible two-point DCT of type II	p. 170
3.6.3 Kernel transform	p. 171
3.6.4 Reversible two-point IDCT of type IV	p. 174
3.6.5 Parameterized two-point IDCT	p. 177
3.6.6 Codes for the integer 2-point DCT	p. 178
3.6.7 Four-and eight-point IDCTs	p. 180
4 Hadamard Transform	p. 185
4.1 The Walsh and Hadamard transform	p. 185
4.1.1 Codes for the paired DHdT	p. 191
4.2 Mixed Hadamard transformation	p. 193
4.2.1 Square roots of mixed transformations	p. 196
4.2.2 High degree roots of the DHdT	p. 199
4.2.3 S-x transformation	p. 201
4.3 Generalized bit-and transformations	p. 203
4.3.1 Projection operators	p. 211
4.4 T-decomposition of Hadamard matrices	p. 212
4.4.1 Square roots of the Hadamard transformation	p. 214
4.4.2 Square roots of the identity transformation	p. 215
4.4.3 The 4th degree roots of the identity transformation	p. 221
4.5 Mixed Fourier transformations	p. 224
4.5.1 Square roots of the Fourier transformation	p. 225
4.5.2 Series of Fourier transforms	p. 229
4.6 Mixed transformations: Continuous case	p. 234
4.6.1 Linear convolution	p. 238
5 Paired Transform-Based Decomposition	p. 243
5.1 Decomposition of 1-D signals	p. 243
5.1.1 Section basis signals	p. 249
5.2 2-D paired representation	p. 251
5.2.1 Set-frequency characteristics	p. 254
5.2.2 Image reconstruction by projections	p. 257
5.2.3 Series images	p. 263
5.2.4 Resolution map	p. 265
5.2.5 A-series linear transformation	p. 267
5.2.6 Method of splitting-signals for image enhancement	p. 268
5.2.7 Fast methods of ¿-rooting	p. 271
5.2.8 Method of series images	p. 283
6 Fourier Transform and Multiresolution	p. 285
6.1 Fourier transform	p. 285
6.1.1 Powers of the Fourier transform	p. 289
6.2 Representation by frequency-time wavelets	p. 292
6.2.1 Wavelet transforms	p. 292
6.2.2 Fourier transform wavelet	p. 293
6.2.3 Cosine- and sine-wavelet transforms	p. 298
6.2.4 B-wavelet transforms	p. 302
6.2.5 Hartley transform representation	p. 304
6.3 Time-frequency correlation analysis	p. 306
6.3.1 Wavelet transform and ¿-resolution	p. 309
6.3.2 Cosine and sine correlation-type transforms	p. 311
6.3.3 Paired transform and Fourier function	p. 313
6.4 Givens-Haar transformations	p. 315
6.4.1 Fast transforms with Haar path	p. 320
6.4.2 Experimental results	p. 324
6.4.3 Characteristics of basic waves	p. 326
6.4.4 Givens-Haar transforms of any order	p. 330
References	p. 339
Index	p. 347

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