Advanced Topics in System and Signal Theory: A Mathematical Approach (Foundations in Signal Processing, Communications and Networking, 4)

"This book provides an in-depth analysis of selected methods in signal and system theory with applications to problems in communications, stochastic processes and optimal filter theory. The authors take a consistent functional analysis and operator theoretic approach to linear system theory, using Banach algebra and Hardy space techniques. The themes connecting all the chapters are questions concerning the consequences of the causality constraint, which is necessary in all realizable systems, and the question of robustness of linear systems with respect to errors in the data. The first part of the book contains basic background on the necessary mathematical tools and provides a basic foundation of signal and system theory. Emphasis is given to the close relation between properties of linear systems such as causality, time-invariance, and robustness on the one hand and the algebraic structures and analytic properties of the mathematical objects, such as Banach algebras or Hardy spaces, on the other hand. The requirement of causality in system theory is inevitably accompanied by the appearance of certain mathematical operations, namely the Riesz projection and the Hilbert transform. These operations are studied in detail in part two. Part three relates the mathematical techniques that are developed in the first two parts to the behaviour of linear systems that are of interest from an engineering perspective, such as expansions of transfer functions in orthonormal bases, the approximation from measured data and the numerical calculation of the Hilbert transform, as well as spectral factorization."--Publisher's website 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 . Front Matter....Pages i-viii Front Matter....Pages 1-1 Function Spaces and Operators....Pages 3-14 Fourier Analysis and Analytic Functions....Pages 15-50 Banach Algebras....Pages 51-66 Signal Models and Linear Systems....Pages 67-78 Front Matter....Pages 79-79 Poisson Integral and Hilbert Transformation....Pages 81-98 Causal Projections....Pages 99-118 Front Matter....Pages 119-119 Disk Algebra Bases....Pages 121-136 Causal Approximations....Pages 137-151 On Algorithms for Calculating the Hilbert Transform....Pages 153-162 Spectral Factorization....Pages 163-232 Back Matter....Pages 1-8