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Advanced Topics in System and Signal Theory

A Mathematical Approach

  • Book
  • © 2010

Overview

Authors:
  1. Volker Pohl

    1. Nachrichtentechnik, Heinrich-Hertz-Inst., Fraunhofer-Institut für, Berlin, Germany

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  2. Holger Boche

    1. Nachrichtentechnik, Heinrich-Hertz-Inst., Fraunhofer-Institut für, Berlin, Germany

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  • Gives a concise picture of the mathematical foundations of network calculus
  • Basic treatment of Quality-of-Service Optimization
  • Includes supplementary material: sn.pub/extras

Part of the book series: Foundations in Signal Processing, Communications and Networking (SIGNAL, volume 4)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-viii
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  2. Mathematical Preliminaries

    1. Front Matter

      Pages 1-1
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  3. Part I Mathematical Preliminaries

    1. Function Spaces and Operators

      • Volker Pohl, Holger Boche
      Pages 3-14

    2. Fourier Analysis and Analytic Functions

      • Volker Pohl, Holger Boche
      Pages 15-50

    3. Banach Algebras

      • Volker Pohl, Holger Boche
      Pages 51-66

    4. Signal Models and Linear Systems

      • Volker Pohl, Holger Boche
      Pages 67-78

  4. Fundamental Operators

    1. Front Matter

      Pages 79-79
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  5. Part II Fundamental Operators

    1. Poisson Integral and Hilbert Transformation

      • Volker Pohl, Holger Boche
      Pages 81-98

    2. Causal Projections

      • Volker Pohl, Holger Boche
      Pages 99-118

  6. Causality Aspects in Signal and System Theory

    1. Front Matter

      Pages 119-119
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  7. Part III Causality Aspects in Signal and System Theory

    1. Disk Algebra Bases

      • Volker Pohl, Holger Boche
      Pages 121-136

    2. Causal Approximations

      • Volker Pohl, Holger Boche
      Pages 137-151

    3. On Algorithms for Calculating the Hilbert Transform

      • Volker Pohl, Holger Boche
      Pages 153-162

    4. Spectral Factorization

      • Volker Pohl, Holger Boche
      Pages 163-232

  8. Back Matter

    Pages 1-8
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Keywords

About this book

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 .

Reviews

From the reviews:

“It is intended primarily for engineers working in the area of robustness problems under a casualty constraint. At the beginning it presents the mathematical methods, necessary to approaching theses problems. Then, some related classical results concerning the boundedness and continuity of the Hilbert transform and Riesz projection are presented. Finally, these methods and results are applied to selected topics from signal processing.” (Tzvetan Semerdjiev, Zentralblatt MATH, Vol. 1179, 2010)

“The monograph gives an elaborate treatment of some tools used in signal and system theory, which have applications in communications, stochastic processes and optimal filter theory. … Appropriate references are provided for the reader interested in details. … The book is a valuable addition to the mathematical literature related to signal processing, and should be useful to students in engineering as well as to mathematicians interested in applications.”­­­ (Amol J. Sasane, Mathematical Reviews, Issue 2010 j)

Authors and Affiliations

  • Nachrichtentechnik, Heinrich-Hertz-Inst., Fraunhofer-Institut für, Berlin, Germany

    Volker Pohl, Holger Boche

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