Pohl / Boche Advanced Topics in System and Signal Theory

A Mathematical Approach
1. Auflage 2009
ISBN: 978-3-642-03639-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 4, 241 Seiten

Reihe: Foundations in Signal Processing, Communications and Networking

The requirement of causality in system theory is inevitably accompanied by the appearance of certain mathematical operations, namely the Riesz proj- tion,theHilberttransform,andthespectralfactorizationmapping.Aclassical exampleillustratingthisisthedeterminationoftheso-calledWiener?lter(the linear, minimum means square error estimation ?lter for stationary stochastic sequences [88]). If the ?lter is not required to be causal, the transfer function of the Wiener ?lter is simply given by H(?)=? (?)/? (?),where ? (?) xy xx xx and ? (?) are certain given functions. However, if one requires that the - xy timation ?lter is causal, the transfer function of the optimal ?lter is given by 1 ? (?) xy H(?)= P ,?? (??,?] . + [? ] (?) [? ] (?) xx + xx? Here [? ] and [? ] represent the so called spectral factors of ? ,and xx + xx? xx P is the so called Riesz projection. Thus, compared to the non-causal ?lter, + two additional operations are necessary for the determination of the causal ?lter, namely the spectral factorization mapping ? ? ([? ] ,[? ] ),and xx xx + xx? the Riesz projection P .

Autoren/Hrsg.

Pohl, Volker

Boche, Holger

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;Part I Mathematical Preliminaries;10
3.1;Function Spaces and Operators;11
3.1.1;Banach and Hilbert spaces;11
3.1.2;Operators on Banach spaces;15
3.1.3;Spaces of Smooth Functions;19
3.2;Fourier Analysis and Analytic Functions;23
3.2.1;Trigonometric Series;23
3.2.2;Hardy Spaces on the Unit Disk;37
3.2.3;Vector-valued Hardy Spaces;50
3.2.4;Operator-valued Analytic Functions;54
3.3;Banach Algebras;59
3.3.1;The Invertible Elements;62
3.3.2;Complex Homomorphisms and Ideals;68
3.3.3;Involutions;73
3.4;Signal Models and Linear Systems;75
3.4.1;Signal Models;75
3.4.2;Linear Systems -- Properties and Representation;78
4;Part II Fundamental Operators;87
4.1;Poisson Integral and Hilbert Transformation;88
4.1.1;Definitions;88
4.1.2;The Poisson Integral;90
4.1.3;The Conjugate Poisson Integral;96
4.2;Causal Projections;106
4.2.1;Complemented Subspaces and Projections;107
4.2.2;Projections from Lp to Hp;110
4.2.3;Projections in Spaces of Smooth Functions;113
4.2.4;Inner-Outer Factorization on Subspaces of H;118
5;Part III Causality Aspects in Signal and System Theory;126
5.1;Disk Algebra Bases;127
5.1.1;On the Existence of Disk Algebra Bases;128
5.1.2;Robust Approximation in Disk Algebra Bases;130
5.1.3;Bases in Spaces of Smooth Functions;135
5.1.4;Uniformly Stable Basis Representations;140
5.2;Causal Approximations;143
5.2.1;Non-linear and Causal Approximations;145
5.2.2;Non-causal, Linear Approximations;146
5.2.3;Behavior of Causal Approximations;147
5.2.4;Causal Approximations for Smooth Functions;156
5.3;On Algorithms for Calculating the Hilbert Transform;158
5.4;Spectral Factorization;168
5.4.1;Regularity of Stochastic Sequences;168
5.4.2;Definition and Basic Properties;173
5.4.3;Factorization on Algebras of Continuous Functions;176
5.4.4;Error Bounds for Polynomial Data;196
5.4.5;Approximation of Spectral Densities;213
5.4.6;Spectral Factorization of Approximated Spectra;222
6;List of Symbols;238
7;References;241