## Time-Frequency RepresentationsThe 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. |

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### Contents

Review of algebra | 1 |

Linear algebra and abelian groups | 19 |

Fourier transform over A | 25 |

Poisson summation formula | 47 |

Zak transform | 57 |

WeylHeisenberg systems | 77 |

Zak transform and WH systems | 93 |

Algorithms for WH systems | 117 |

Ambiguity surfaces | 151 |

Orthonormal WH systems | 155 |

Duality | 169 |

Frames | 187 |

Implementation | 199 |

Algebra of multirate structures | 219 |

Multirate structures | 239 |

A Timefrequency search for stock market anomalies | 261 |

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### Common terms and phrases

algebra algorithm Ao-coset representatives Ao-periodic function array biorthogonal to g called Chapter character group complete system completing the proof Compute Continuing Example Corollary coset critical sampling subgroup cross-ambiguity function cyclic group Denote described direct sum duality f over g finite abelian group following result Fourier coefficient set Fourier expansion Fourier transform free abelian group function in L(A functional equation G L(g given homomorphism implies injection inner product integer over-sampling subgroup integer sampling rate isomorphism ker(Y Lemma linear LZ/N mapping multiplication multirate orthogonal projection orthonormal W-H system permutation permutation matrix Poisson summation formula quotient group rate filter bank sampling rate sampling rate filter satisfying short exact sequence spectra Spectrum of Segment standard presentation Suppose g surjection system of Ao-coset tensor product theory time-frequency transform of f vanishes vector W-H coefficient set W-H operators Weyl-Heisenberg Z/Ni Z/Nr Zak space Zak transform zero set

### References to this book

Modern Sampling Theory: Mathematics and Applications John J. Benedetto,Paulo J.S.G. Ferreira Limited preview - 2001 |

Applications in Time-Frequency Signal Processing Antonia Papandreou-Suppappola No preview available - 2002 |