Ebook: Time-Frequency Representations
- Tags: Computational Mathematics and Numerical Analysis, Signal Image and Speech Processing, Computational Science and Engineering, Applications of Mathematics
- Series: Applied and Numerical Harmonic Analysis
- Year: 1996
- Publisher: Birkhäuser Basel
- Edition: 1
- Language: English
- pdf
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.
Content:
Front Matter....Pages i-xiii
Review of algebra....Pages 1-18
Linear algebra and abelian groups....Pages 19-24
Fourier transform over A ....Pages 25-46
Poisson summation formula....Pages 47-56
Zak transform....Pages 57-75
Weyl-Heisenberg systems....Pages 77-92
Zak transform and W-H systems....Pages 93-116
Algorithms for W-H systems....Pages 117-133
Orthogonal projection theorem....Pages 135-139
Cross-ambiguity function....Pages 141-150
Ambiguity surfaces....Pages 151-153
Orthonormal W-H systems....Pages 155-168
Duality....Pages 169-185
Frames....Pages 187-197
Implementation....Pages 199-218
Algebra of multirate structures....Pages 219-238
Multirate structures....Pages 239-259
A Time-frequency search for stock market anomalies....Pages 261-273
Back Matter....Pages 275-284
Content:
Front Matter....Pages i-xiii
Review of algebra....Pages 1-18
Linear algebra and abelian groups....Pages 19-24
Fourier transform over A ....Pages 25-46
Poisson summation formula....Pages 47-56
Zak transform....Pages 57-75
Weyl-Heisenberg systems....Pages 77-92
Zak transform and W-H systems....Pages 93-116
Algorithms for W-H systems....Pages 117-133
Orthogonal projection theorem....Pages 135-139
Cross-ambiguity function....Pages 141-150
Ambiguity surfaces....Pages 151-153
Orthonormal W-H systems....Pages 155-168
Duality....Pages 169-185
Frames....Pages 187-197
Implementation....Pages 199-218
Algebra of multirate structures....Pages 219-238
Multirate structures....Pages 239-259
A Time-frequency search for stock market anomalies....Pages 261-273
Back Matter....Pages 275-284
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