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E-raamat: Brief Notes in Advanced DSP: Fourier Analysis with MATLAB

(University of Texas, San Antonio, USA),
  • Formaat: 354 pages
  • Ilmumisaeg: 03-Oct-2018
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781439801383
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Brief Notes in Advanced DSP: Fourier Analysis with MATLABSuurem pilt
  • Formaat: 354 pages
  • Ilmumisaeg: 03-Oct-2018
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781439801383
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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 GivensHaar 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.
Biography ix
Preface xi
1 Discrete Fourier Transform 1
1.1 Properties of the discrete Fourier transform
1
1.2 Fourier transform splitting
6
1.3 Fast Fourier transform
12
1.3.1 Unitary paired transform
14
1.3.2 Fast 8-point DFT
17
1.3.3 Fast 16-point DFT
19
1.4 Codes for the paired FFT
25
1.5 Paired and Haar transforms
28
1.5.1 Haar functions
29
1.5.2 Codes for the Haar transform
33
1.5.3 Comparison with the paired transform
34
2 Integer Fourier Transform 45
2.1 Reversible integer Fourier transform
45
2.1.1 Lifting scheme implementation
45
2.2 Lifting schemes for DFT
49
2.3 One-point integer transform
56
2.3.1 The eight-point integer Fourier transform
59
2.3.2 Eight-point inverse integer DFT
63
2.3.3 General method of control bits
66
2.3.4 16-point IDFT with 8 and 12 control bits
66
2.3.5 Inverse 16-point integer DFT
67
2.3.6 Codes for the forward 16-point integer ITT
78
2.4 DFT in vector form
84
2.4.1 DFT in real space
85
2.4.2 Integer representation of the DFT
90
2.5 Roots of the unit
101
2.5.1 Elliptic DFT
105
2.6 Codes for the block DFT
117
2.7 General elliptic Fourier transforms
120
2.7.1 N-block GEFT
122
3 Cosine Transform 129
3.1 Partitioning the DCT
129
3.1.1 4-point DCT of type IV
140
3.1.2 Fast four-point type IV DCT
142
3.1.3 8-point DCT of type IV
145
3.2 Paired algorithm for the N-point DCT
151
3.2.1 Paired functions
152
3.2.2 Complexity of the calculation
153
3.3 Codes for the paired transform
155
3.4 Reversible integer DCT
155
3.4.1 Integer four-point DCTs
156
3.4.2 Integer eight-point DCT
159
3.5 Method of nonlinear equations
160
3.5.1 Calculation of coefficients
162
3.5.2 Error of approximation
164
3.6 Canonical representation of the integer DCT
168
3.6.1 Reversible two-point transforms
168
3.6.2 Reversible two-point DCT of type II
170
3.6.3 Kernel transform
171
3.6.4 Reversible two-point IDCT of type IV
174
3.6.5 Parameterized two-point IDCT
177
3.6.6 Codes for the integer 2-point DCT
178
3.6.7 Four- and eight-point IDCTs
180
4 Hadamard Transform 185
4.1 The Walsh and Hadamard transform
185
4.1.1 Codes for the paired DHdT
191
4.2 Mixed Hadamard transformation.
193
4.2.1 Square roots of mixed transformations
196
4.2.2 High degree roots of the DHdT
199
4.2.3 S-x transformation
201
4.3 Generalized bit-and transformations
203
4.3.1 Projection operators
211
4.4 T-decomposition of Hadamard matrices
212
4.4.1 Square roots of the Hadamard transfomnation
214
4.4.2 Square roots of the identity transformation
215
4.4.3 The 4th degree roots of the identity transformation
221
4.5 Mixed Fourier transformations
224
4.5.1 Square roots of the Fourier transformation
225
4.5.2 Series of Fourier transforms
229
4.6 Mixed transformations: Continuous case
234
4.6.1 Linear convolution
238
5 Paired Transform-Based Decomposition 243
5.1 Decomposition of 1-D signals
243
5.1.1 Section basis signals
249
5.2 2-D paired representation
251
5.2.1 Set-frequency characteristics
254
5.2.2 Image reconstruction by projections
257
5.2.3 Series images
263
5.2.4 Resolution map
265
5.2.5 A-series linear transformation
267
5.2.6 Method of splitting-signals for image enhancement
268
5.2.7 Fast methods of a-rooting
271
5.2.8 Method of series images
283
6 Fourier Transform and Multiresolution 285
6.1 Fourier transform
285
6.1.1 Powers of the Fourier transform
289
6.2 Representation by frequency-time wavelets
292
6.2.1 Wavelet transforms
292
6.2.2 Fourier transform wavelet
293
6.2.3 Cosine- and sine-wavelet transforms
298
6.2.4 B-wavelet transforms
302
6.2.5 Hartley transform representation
304
6.3 Time-frequency correlation analysis
306
6.3.1 Wavelet transform and -resolution
309
6.3.2 Cosine and sine correlation-type transforms
311
6.3.3 Paired transform and Fourier function
313
6.4 Givens-Haar transformations
315
6.4.1 Fast transforms with Haar path
320
6.4.2 Experimental results
324
6.4.3 Characteristics of basic waves
326
6.4.4 Givens-Haar transforms of any order
330
References 339
Index 347
Artyom M. Grigoryan, Merughan Grigoryan
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