Fourier Transforms of Finite Chirps

  • Peter G. Casazza
  • Matthew Fickus
Open Access
Research Article
Part of the following topical collections:
  1. Frames and Overcomplete Representations in Signal Processing, Communications, and Information Theory


Chirps arise in many signal processing applications. While chirps have been extensively studied as functions over both the real line and the integers, less attention has been paid to the study of chirps over finite groups. We study the existence and properties of chirps over finite cyclic groups of integers. In particular, we introduce a new definition of a finite chirp which is slightly more general than those that have been previously used. We explicitly compute the discrete Fourier transforms of these chirps, yielding results that are number-theoretic in nature. As a consequence of these results, we determine the degree to which the elements of certain finite tight frames are well distributed.


Fourier Fourier Transform Information Technology Signal Processing Quantum Information 


  1. 1.
    Xia X-G: Discrete chirp-Fourier transform and its application to chirp rate estimation. IEEE Transactions on Signal Processing 2000, 48(11):3122–3133. 10.1109/78.875469MathSciNetCrossRefGoogle Scholar
  2. 2.
    Mann S, Haykin S: The chirplet transform: A generalization of Gabor's logon transform. Proceedings of Vision Interface, June 1991, Calgary, Alberta, Canada 205–212.Google Scholar
  3. 3.
    Berndt BC, Evans RJ: The determination of Gauss sums. Bulletin of the American Mathematical Society (New Series) 1981, 5(2):107–129. 10.1090/S0273-0979-1981-14930-2MathSciNetCrossRefGoogle Scholar
  4. 4.
    Auslander L, Tolimieri R: Is computing with the finite Fourier transform pure or applied mathematics? Bulletin of the American Mathematical Society (New Series) 1979, 1(6):847–897. 10.1090/S0273-0979-1979-14686-XMathSciNetCrossRefGoogle Scholar
  5. 5.
    McClellan JH, Parks TW: Eigenvalue and eigenvector decomposition of the discrete Fourier transform. IEEE Transactions on Audio and Electroacoustics 1972, 20(1):66–74. 10.1109/TAU.1972.1162342MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kaiblinger N: Metaplectic representation, eigenfunctions of phase space shifts, and Gelfand-Shilov spaces for lca groups, M.S. thesis. University of Vienna, Vienna, Austria; 1999.Google Scholar
  7. 7.
    Strohmer T, Heath RW Jr.: Grassmannian frames with applications to coding and communications. Applied and Computational Harmonic Analysis 2003, 14(3):257–275. 10.1016/S1063-5203(03)00023-XMathSciNetCrossRefGoogle Scholar
  8. 8.
    Fickus M: An elementary proof of a generalized Schaar identity. preprint.Google Scholar

Copyright information

© Casazza and Fickus 2006

Authors and Affiliations

  • Peter G. Casazza
    • 1
  • Matthew Fickus
    • 2
  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Department of Mathematics and StatisticsAir Force Institute of TechnologyWright-Patterson AFBUSA

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