Brief Notes in Advanced DSP: Fourier Analysis with MATLAB

by Grigoryan; Artyom M.

• ISBN13: 9781439801376
• ISBN10: 1439801371
• Edition: 1st
• Format: Hardcover
• Copyright: 2009-02-20
• Publisher: CRC Press

Summary

In addressing new concepts and methods of advanced digital signal processing and their applications, this book looks at recent research in Fourier analysis. Instead of focusing on well-known fundamental properties of thetransformations found in many books, this work shows readers new approaches and new forms of signal and image representation that can be used effectively in signal and image processing. It includes new algorithms for integer approximation of the Fourier and cosine transforms, new methods of multi-resolution based on the Fourier analysis rather than wavelet-theory, and MATLAB ® -based codes for implementing fast transforms and many methods of signal and image processing.

Table of Contents

Biography	p. ix
Preface	p. xi
Discrete Fourier Transform	p. 1
Properties of the discrete Fourier transform	p. 1
Fourier transform splitting	p. 6
Fast Fourier transform	p. 12
Unitary paired transform	p. 14
Fast 8-point DFT	p. 17
Fast 16-point DFT	p. 19
Codes for the paired FFT	p. 25
Paired and Haar transforms	p. 28
Haar functions	p. 29
Codes for the Haar transform	p. 33
Comparison with the paired transform	p. 34
Integer Fourier Transform	p. 45
Reversible integer Fourier transform	p. 45
Lifting scheme implementation	p. 45
Lifting schemes for DFT	p. 49
One-point integer transform	p. 56
The eight-point integer Fourier transform	p. 59
Eight-point inverse integer DFT	p. 63
General method of control bits	p. 66
16-point IDFT with 8 and 12 control bits	p. 66
Inverse 16-point integer DFT	p. 67
Codes for the forward 16-point integer FFT	p. 78
DFT in vector form	p. 84
DFT in real space	p. 85
Integer representation of the DFT	p. 90
Roots of the unit	p. 101
Elliptic DFT	p. 105
Codes for the block DFT	p. 117
General elliptic Fourier transforms	p. 120
N-block GEFT	p. 122
Cosine Transform	p. 129
Partitioning the DCT	p. 129
4-point DCT of type IV	p. 140
Fast four-point type IV DCT	p. 142
8-point DCT of type IV	p. 145
Paired algorithm for the N-point DCT	p. 151
Paired functions	p. 152
Complexity of the calculation	p. 153
Codes for the paired transform	p. 155
Reversible integer DCT	p. 155
Integer four-point DCTs	p. 156
Integer eight-point DCT	p. 159
Method of nonlinear equations	p. 160
Calculation of coefficients	p. 162
Error of approximation	p. 164
Canonical representation of the integer DCT	p. 168
Reversible two-point transforms	p. 168
Reversible two-point DCT of type II	p. 170
Kernel transform	p. 171
Reversible two-point IDCT of type IV	p. 174
Parameterized two-point IDCT	p. 177
Codes for the integer 2-point DCT	p. 178
Four-and eight-point IDCTs	p. 180
Hadamard Transform	p. 185
The Walsh and Hadamard transform	p. 185
Codes for the paired DHdT	p. 191
Mixed Hadamard transformation	p. 193
Square roots of mixed transformations	p. 196
High degree roots of the DHdT	p. 199
S-x transformation	p. 201
Generalized bit-and transformations	p. 203
Projection operators	p. 211
T-decomposition of Hadamard matrices	p. 212
Square roots of the Hadamard transformation	p. 214
Square roots of the identity transformation	p. 215
The 4th degree roots of the identity transformation	p. 221
Mixed Fourier transformations	p. 224
Square roots of the Fourier transformation	p. 225
Series of Fourier transforms	p. 229
Mixed transformations: Continuous case	p. 234
Linear convolution	p. 238
Paired Transform-Based Decomposition	p. 243
Decomposition of 1-D signals	p. 243
Section basis signals	p. 249
2-D paired representation	p. 251
Set-frequency characteristics	p. 254
Image reconstruction by projections	p. 257
Series images	p. 263
Resolution map	p. 265
A-series linear transformation	p. 267
Method of splitting-signals for image enhancement	p. 268
Fast methods of ¿-rooting	p. 271
Method of series images	p. 283
Fourier Transform and Multiresolution	p. 285
Fourier transform	p. 285
Powers of the Fourier transform	p. 289
Representation by frequency-time wavelets	p. 292
Wavelet transforms	p. 292
Fourier transform wavelet	p. 293
Cosine- and sine-wavelet transforms	p. 298
B-wavelet transforms	p. 302
Hartley transform representation	p. 304
Time-frequency correlation analysis	p. 306
Wavelet transform and ¿-resolution	p. 309
Cosine and sine correlation-type transforms	p. 311
Paired transform and Fourier function	p. 313
Givens-Haar transformations	p. 315
Fast transforms with Haar path	p. 320
Experimental results	p. 324
Characteristics of basic waves	p. 326
Givens-Haar transforms of any order	p. 330
References	p. 339
Index	p. 347
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