Positive trigonometric polynomials and signal processing applications

CONTENTS......Page 6
Preface......Page 10
1.1 Types of polynomials......Page 14
1.2 Positive polynomials......Page 16
1.3 Toeplitz positivity conditions......Page 21
1.4 Positivity on an interval......Page 23
1.5.2 Positive polynomials in R[t] as sum-of-squares......Page 27
1.5.3 Proof of Theorem 1.11......Page 28
1.5.4 Proof of Theorem 1.13......Page 29
1.5.5 Proof of Theorem 1.15......Page 30
1.5.6 Proof of Theorem 1.17......Page 31
1.6 Bibliographical and historical notes......Page 32
2.1 Parameterization of trigonometric polynomials......Page 34
2.2 Optimization using the trace parameterization......Page 39
2.3 Toeplitz quadratic optimization......Page 43
2.4 Duality......Page 45
2.5 Kalman-Yakubovich-Popov lemma......Page 46
2.6.1 SDP computation of a rank-1 Gram matrix......Page 48
2.6.2 Spectral factorization using a Riccati equation......Page 50
2.7 Parameterization of real polynomials......Page 52
2.8.1 Basis of trigonometric polynomials......Page 55
2.8.2 Transformation to real polynomials......Page 59
2.8.3 Gram pair matrix parameterization......Page 60
2.9 Interpolation representations......Page 64
2.10 Mixed representations......Page 66
2.10.1 Complex polynomials and the DFT......Page 67
2.10.2 Cosine polynomials and the DCT......Page 68
2.11 Fast algorithms......Page 69
2.12.1 A SeDuMi program......Page 70
2.12.2 Proof of Theorem 2.16......Page 71
2.12.4 Proof of Theorem 2.21......Page 73
2.13 Bibliographical and historical notes......Page 74
3. MULTIVARIATE POLYNOMIALS......Page 78
3.1 Multivariate polynomials......Page 79
3.2 Sum-of-squares multivariate polynomials......Page 81
3.3 Sum-of-squares of real polynomials......Page 84
3.4 Gram matrix parameterization of multivariate trigonometric polynomials......Page 86
3.5.1 Relaxation principle......Page 90
3.5.2 A case study......Page 91
3.5.3 Optimality certificate......Page 94
3.6.1 Sparse polynomials and Gram representation......Page 97
3.6.2 Relaxations......Page 100
3.7.1 Gram parameterization......Page 101
3.7.2 Sum-of-squares relaxations......Page 102
3.7.3 Sparseness treatment......Page 103
3.8 Pairs of relaxations......Page 106
3.9.1 Basic Gram pair parameterization......Page 107
3.9.2 Parity discussion......Page 109
3.9.3 LMI form......Page 110
3.10 Polynomials with matrix coefficients......Page 114
3.11.1 Transformation between trigonometric and real nonnegative polynomials......Page 118
3.11.2 A program using the Gram pair parameterization......Page 120
3.12 Bibliographical and historical notes......Page 122
4.1 Real polynomials positive on compact domains......Page 127
4.2 Trigonometric polynomials positive on frequency domains......Page 131
4.2.1 Gram set parameterization......Page 133
4.2.2 Gram-pair set parameterization......Page 136
4.3 Bounded Real Lemma......Page 137
4.3.1 Gram set BRL......Page 138
4.3.2 Gram-pair set BRL......Page 141
4.4 Positivstellensatz for trigonometric polynomials......Page 142
4.5 Proof of Theorem 4.11......Page 145
4.6 Bibliographical and historical notes......Page 147
5.1 Design of FIR filters......Page 149
5.1.1 Optimization of linear-phase FIR filters......Page 151
5.1.2 Magnitude optimization......Page 153
5.1.3 Approximate linear phase FIR filters......Page 154
5.2 Design of 2-D FIR filters......Page 157
5.2.1 2-D frequency domains......Page 158
5.2.2 Linear phase designs......Page 159
5.2.3 Approximate linear phase designs......Page 164
5.3 FIR deconvolution......Page 166
5.3.1 Basic optimization problem......Page 168
5.3.2 Deconvolution of periodic FIR filters......Page 169
5.3.3 Robust H∞ deconvolution......Page 171
5.3.4 2-D H∞ deconvolution......Page 172
5.4 Bibliographical and historical notes......Page 174
6.1 Two-channel filterbanks......Page 178
6.1.1 Orthogonal FB design......Page 180
6.1.2 Towards signal-adapted wavelets......Page 183
6.1.3 Design of symmetric complex-valued FBs......Page 187
6.2 GDFT modulated filterbanks......Page 191
6.2.1 GDFT modulation: definitions and properties......Page 192
6.2.2 Design of near-orthogonal GDFT modulated FBs......Page 196
6.3 Bibliographical and historical notes......Page 198
7.1 Multidimensional stability tests......Page 202
7.1.1 Stability test via positivity......Page 203
7.1.2 Stability of Fornasini-Marchesini model......Page 205
7.1.3 Positivstellensatz for testing stability......Page 207
7.2 Robust stability......Page 209
7.2.1 Real polynomials test......Page 210
7.2.2 Trigonometric polynomials test......Page 212
7.3.1 Positive realness stability domain......Page 214
7.3.3 Proof of Theorem 7.16......Page 218
7.4 Bibliographical and historical notes......Page 220
8.1 Magnitude design of IIR filters......Page 222
8.2 Approximate linear-phase designs......Page 224
8.2.1 Optimization with fixed denominator......Page 226
8.2.2 IIR filter design using convex stability domains......Page 228
8.3 2-D IIR filter design......Page 231
8.4 Bibliographical and historical notes......Page 233
Appendix A: semidefinite programming......Page 236
B.1 Root finding......Page 238
B.2 Newton-Raphson algorithm......Page 239
B.4 Hilbert transform method......Page 240
B.5 Polynomials with matrix coefficients......Page 241
References......Page 242
E......Page 250
S......Page 251
Z......Page 252