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Birkhäuser

Time-Frequency Representations

  • Book
  • © 1998

Overview

Authors:
  1. Richard Tolimieri

    1. Department of Electrical Engineering, City College of New York, New York, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Myoung An

    1. A. J. Devaney Associates, Boston, USA

    View author publications

    You can also search for this author in PubMed   Google Scholar

Part of the book series: Applied and Numerical Harmonic Analysis (ANHA)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xiii
    Download chapter PDF

  2. Review of algebra

    • Richard Tolimieri, Myoung An
    Pages 1-18

  3. Linear algebra and abelian groups

    • Richard Tolimieri, Myoung An
    Pages 19-24

  4. Fourier transform over A

    • Richard Tolimieri, Myoung An
    Pages 25-46

  5. Poisson summation formula

    • Richard Tolimieri, Myoung An
    Pages 47-56

  6. Zak transform

    • Richard Tolimieri, Myoung An
    Pages 57-75

  7. Weyl-Heisenberg systems

    • Richard Tolimieri, Myoung An
    Pages 77-92

  8. Zak transform and W-H systems

    • Richard Tolimieri, Myoung An
    Pages 93-116

  9. Algorithms for W-H systems

    • Richard Tolimieri, Myoung An
    Pages 117-133

  10. Orthogonal projection theorem

    • Richard Tolimieri, Myoung An
    Pages 135-139

  11. Cross-ambiguity function

    • Richard Tolimieri, Myoung An
    Pages 141-150

  12. Ambiguity surfaces

    • Richard Tolimieri, Myoung An
    Pages 151-153

  13. Orthonormal W-H systems

    • Richard Tolimieri, Myoung An
    Pages 155-168

  14. Duality

    • Richard Tolimieri, Myoung An
    Pages 169-185

  15. Frames

    • Richard Tolimieri, Myoung An
    Pages 187-197

  16. Implementation

    • Richard Tolimieri, Myoung An
    Pages 199-218

  17. Algebra of multirate structures

    • Richard Tolimieri, Myoung An
    Pages 219-238

  18. Multirate structures

    • Richard Tolimieri, Myoung An
    Pages 239-259

  19. A Time-frequency search for stock market anomalies

    • Sudeshna Adak, Abhinanda Sarkar
    Pages 261-273

  20. Back Matter

    Pages 275-284
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Keywords

About this book

The aim of this work is to present several topics in time-frequency analysis as subjects in abelian group theory. The algebraic point of view pre­ dominates as questions of convergence are not considered. Our approach emphasizes the unifying role played by group structures on the development of theory and algorithms. This book consists of two main parts. The first treats Weyl-Heisenberg representations over finite abelian groups and the second deals with mul­ tirate filter structures over free abelian groups of finite rank. In both, the methods are dimensionless and coordinate-free and apply to one and multidimensional problems. The selection of topics is not motivated by mathematical necessity but rather by simplicity. We could have developed Weyl-Heisenberg theory over free abelian groups of finite rank or more generally developed both topics over locally compact abelian groups. However, except for having to dis­ cuss conditions for convergence, Haar measures, and other standard topics from analysis the underlying structures would essentially be the same. A re­ cent collection of papers [17] provides an excellent review of time-frequency analysis over locally compact abelian groups. A further reason for limiting the scope of generality is that our results can be immediately applied to the design of algorithms and codes for time­ frequency processing.

Reviews

"On the one side, the book seems to be dedicated to theory-interested specialists in the area of harmonic analysis, on the other side the last chapter shows that highly theoretical constructs can efficiently be used in modern applications." —Simulation News Europe

"[The book], devoted to the subject of time-frequency representation, shows that there are still many interesting and useful aspects of Gabor analysis, in particular, in the discrete setting." —Zentralblatt Math

 

Authors and Affiliations

  • Department of Electrical Engineering, City College of New York, New York, USA

    Richard Tolimieri

  • A. J. Devaney Associates, Boston, USA

    Myoung An

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