The purpose of this book is twofold. Our starting point is the design of layered media with a prescribed reflection coefficient. In the first part of this book we show that the space of physically realizable reflection coefficients is rather restricted by a number of properties. In the second part we consider a constrained approximation problem in Hardy spaces. This can be viewed as an optimization problem for the frequency response of a causal LTI system with limited gain.
Author(s): Arne Schneck
Year: 2009
Language: English
Pages: 148
Cover......Page 1
Bounds for Optimization of the Reflection Coefficient by Constrained Optimization in Hardy Spaces......Page 5
9783866443822......Page 6
Contents......Page 11
0.1 Motivation......Page 15
0.2 Goals......Page 19
0.3 Overview......Page 20
0.4 Acknowledgements......Page 22
1.1 The Helmholtz equation......Page 25
1.2 Pulses......Page 28
1.3 Dispersion......Page 30
2 Hardy Spaces, LTI Systems and the Paley-Wiener Theorem......Page 35
2.1.1 Hardy spaces on the disk: H^p(\mathbb{D})......Page 36
2.1.2 Hardy spaces on the half-plane: H^p(\mathbb{C}^+)......Page 39
2.2 LTI systems......Page 41
2.3 The Paley-Wiener Theorem......Page 43
3 Scattering Theory for the 1D Helmholtz Equation......Page 47
3.1.1 Jost solutions and an integral formulation......Page 48
3.1.2 Estimates for Jost solutions......Page 49
3.1.3 Reflection and transmission coefficient R and T......Page 52
3.1.4 Further estimates......Page 58
3.2.1 Definition of R and T via an initial value problem......Page 60
3.2.2 Definition of R and T via a boundary value problem......Page 61
3.2.3 A weak formulation......Page 62
3.2.4 Continuity in the weak* topology of L^\infty......Page 64
3.3.1 Changing the surrounding medium......Page 67
3.3.2 Shifting n......Page 70
3.3.3 Hardy space properties of R and T......Page 71
3.4 An optimization problem for the reflection coefficient......Page 75
3.5 Further remarks......Page 77
4 Constrained Optimization in Hardy Spaces: Theory......Page 79
4.1 Existence (1 <= p <= \infty) and uniqueness (1 < p < \infty)......Page 81
4.2 Extremal properties and uniqueness (1 <= p <= \infty)......Page 82
4.4 Approximation by smooth functions, 1 <= p < \infty......Page 90
4.5 Approximation by smooth functions, p = \infty......Page 94
5.1 Discretization......Page 107
5.1.1 Assumptions and notation......Page 108
5.1.2 Semi-discrete problem......Page 109
5.1.3 Fully discrete problem......Page 111
5.2.2 p = 2, exact quadrature......Page 119
5.3 QCQP formulation of the discrete problems......Page 120
5.3.1 p = 2, rectangle rule......Page 121
5.3.2 p = 2, exact quadrature......Page 122
5.3.5 Summary......Page 123
5.4 Second-order cone programs (SOCPs)......Page 124
5.5.1 General strategy to rewrite QCQPs as SOCPs......Page 126
5.5.2 p = 2, rectangle rule......Page 128
5.5.4 p = \infty......Page 129
5.6 Numerical experiments......Page 130
5.6.1 Example 1: Artificial example......Page 132
5.6.2 Example 2: Wideband dispersion compensating mirror......Page 134
5.6.3 Example 3: DCM with pump window......Page 139
Bibliography......Page 145
Back Cover......Page 152
