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  • Research Article
  • Open Access

Fourier Transforms of Finite Chirps

  • 1 and
  • 2
EURASIP Journal on Advances in Signal Processing20062006:070204

  • Received: 15 October 2004
  • Accepted: 5 April 2005
  • Published:


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


Authors’ Affiliations

Department of Mathematics, University of Missouri, Columbia, USA
Department of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson AFB, USA


  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. 10.1109/78.875469MathSciNetView ArticleMATHGoogle Scholar
  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. Berndt BC, Evans RJ: The determination of Gauss sums. Bulletin of the American Mathematical Society (New Series) 1981, 5(2):107-129. 10.1090/S0273-0979-1981-14930-2MathSciNetView ArticleMATHGoogle Scholar
  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-XMathSciNetView ArticleMATHGoogle Scholar
  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.1972.1162342MathSciNetView ArticleGoogle Scholar
  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. 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-XMathSciNetView ArticleMATHGoogle Scholar
  8. Fickus M: An elementary proof of a generalized Schaar identity. preprint.Google Scholar


© Casazza and Fickus 2006