Brief Notes in Advanced DSP: Fourier Analysis with MATLAB

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