XFT: A Discrete Fourier Transform

  • Rafael G. Campos
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


We present in this chapter a procedure to obtain a novel discrete Fourier transform. The discrete Fourier transform obtained in this way is called XFT (from eXtended Fourier Transform) in order to distinguish it from the usual discrete Fourier transform DFT, which was studied in Chap.  2. It is shown in this chapter that the XFT appears as a quadrature of the fractional Fourier transform and that the XFT can be applied as a discrete Fourier transform to solve some simple examples. A discrete scheme for the Hermite functions, the Schrödinger equation, the Fourier cosine/sine transform, and for partial fractional differentiation is also given. The aliasing problem is discussed.


Extended discrete Fourier transform Discrete Hermite functions Square-integrable functions Quadrature Sine/cosine transform Differentiation matrices Fast algorithms Signal processing Sampling Aliasing Partial differentiation matrices 


  1. 4.
    I. Amidror. Mastering the Discrete Fourier Transform in One, Two or Several Dimensions. Pitfalls and Artifacts. Springer-Verlag, London, 2013.Google Scholar
  2. 6.
    N.M. Atakishiyev, L.E. Vicent, and K.B Wolf. Continuous vs. discrete fractional Fourier transforms. J. Comput. Appl. Math., 107:73–95, 1999.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 7.
    L. Auslander and F.A. Grünbaum. The Fourier transform and the discrete Fourier transform. Inverse Problems, 5:149–164, 1989.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 9.
    L. Barker, C. Candan, T. Hakioglu, M.A. Kutay, and H.M. Ozaktas. The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform. J. Phys. A: Math. Gen., 33:2209–2222, 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 17.
    M. Bruschi and F. Calogero. Finite-dimensional matrix representations of the operator of differentiation through the algebra of raising and lowering operators: General properties and explicit examples. Nuovo Cimento B, 62:337–351, 1981.MathSciNetCrossRefGoogle Scholar
  6. 23.
    F. Calogero. Matrices, differential operators and polynomials. J. Math. Phys., 22:919–934, 1981.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 24.
    F. Calogero. Lagrangian interpolation and differentiation. Lett. Nuovo Cimento, 35:273–278, 1983.MathSciNetCrossRefGoogle Scholar
  8. 25.
    F. Calogero. Interpolation, differentiation and solution of eigenvalue problems in more than one dimension. Lett. Nuovo Cimento, 39:305–311, 1984.MathSciNetCrossRefGoogle Scholar
  9. 26.
    F. Calogero. Classical Many-Body Problems Amenable to Exact Treatments, volume 66 of Lecture Notes in Physics Monographs. Springer-Verlag, Berlin, 2001.CrossRefzbMATHGoogle Scholar
  10. 27.
    F. Calogero and E. Franco. Numerical tests of a novel technique to compute the eigenvalues of differential operators. Nuovo Cimento B, 89:161–208, 1985.MathSciNetCrossRefGoogle Scholar
  11. 28.
    W.L. Cameron. Precise expression relating the Fourier transform of a continuous signal to the fast Fourier transform of signal samples. IEEE Trans. Signal Process., 43:2811–2821, 1995.CrossRefGoogle Scholar
  12. 29.
    R.G. Campos. Time-reversal breaking in fractional differential problems. Eur. Phys. J. Plus, 130:121–125, 2015.CrossRefGoogle Scholar
  13. 32.
    R.G. Campos, J.Rico-Melgoza, and E. Chávez. A new formulation of the fast fractional Fourier fransform. SIAM J. Sci. Comput., 34(2):A1110–A1125, 2012.CrossRefzbMATHGoogle Scholar
  14. 33.
    R.G. Campos and L.Z. Juárez. A discretization of the continuous Fourier transform. Nuovo Cimento B, 107:703–711, 1992.MathSciNetCrossRefGoogle Scholar
  15. 34.
    R.G. Campos and F. Marcellán. Quadratures and integral transforms arising from generating functions. Appl. Math. Comput., 297:8–18, 2017.MathSciNetzbMATHGoogle Scholar
  16. 35.
    R.G. Campos and C. Meneses. Differentiation matrices for meromorphic functions. Bol. Soc. Mat. Mexicana, 12:121–132, 2006.MathSciNetzbMATHGoogle Scholar
  17. 36.
    R.G. Campos, F. D. Mota, and E. Coronado. Quadrature formulas for integrals transforms generated by orthogonal polynomials. IMA J. Numer. Anal., 31:1181–1193, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 38.
    R.G. Campos and L.O. Pimentel. A finite-dimensional representation of the quantum angular momentum operator. Nuovo Cimento B, 116:31–46, 2001.MathSciNetGoogle Scholar
  19. 41.
    C. Candan. On higher order approximations for Hermite-Gaussian functions and discrete fractional Fourier transforms. IEEE Signal Process. Lett., 14:699–702, 2001.CrossRefGoogle Scholar
  20. 44.
    J.W. Cooley, P.A.W. Lewis, and P.D. Welch. Application of the fast Fourier transform to computation of Fourier integrals, Fourier series, and convolution integrals. IEEE Trans. Audio Electroacoustics, pages 79–84, 1967.CrossRefGoogle Scholar
  21. 50.
    A. Erdélyi, editor. Higher Transcendental Functions, volume I, II. McGraw-Hill, New York, 1953.zbMATHGoogle Scholar
  22. 51.
    A. Erdélyi, editor. Tables of Integral Transforms, volume I. McGraw-Hill, New York, 1954.zbMATHGoogle Scholar
  23. 58.
    Sean A. Fulop. Speech Spectrum Analysis. Springer-Verlag, Berlin, 2011.CrossRefzbMATHGoogle Scholar
  24. 60.
    C. Gasquet and P. Witomski. Fourier Analysis and Applications. Filtering, Numerical Computation, Wavelets. Texts in Applied Mathematics. Springer-Verlag, New York, 1999.CrossRefzbMATHGoogle Scholar
  25. 61.
    W. Gautschi. Orthogonal Polynomials, Computation and Approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, 2004.zbMATHGoogle Scholar
  26. 71.
    F.A. Grünbaum. The eigenvectors of the discrete Fourier transform: A version of the Hermite functions. J. Math. Anal. Appl., 88:355–363, 1982.MathSciNetCrossRefGoogle Scholar
  27. 72.
    F.A. Grünbaum. Discrete models of the harmonic oscillator and a discrete analog of Gauss’ hypergeometric equation. Ramanujan J., 5:263–270, 2001.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 73.
    F.A. Grünbaum and E.H. Zarantonello. On the extension of uniformly continuous mappings. Michigan Math. J., 15:65–74, 1968.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 85.
    N. Kaiblinger. Approximation of the Fourier transform and the dual Gabor window. J. Fourier Anal. Appl., 11:25–42, 2005.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 93.
    H.J. Korsch and K. Rapedius. Computations in quantum mechanics made easy. Eur. J. Phys., 37:055410, 2016.CrossRefzbMATHGoogle Scholar
  31. 95.
    A. Kuznetsov and M. Kwaśnicki. Minimal Hermite-type eigenbasis of the discrete Fourier transform. J. Fourier Anal. Appl.,, 2018.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 96.
    B. P. Lathi and Roger A. Green. Essentials of Digital Signal Processing. Cambridge University Press, Cambridge, 2010.Google Scholar
  33. 108.
    E. Merzbacher. Quantum Mechanics. John Wiley and Sons, Inc., New York, 1998.zbMATHGoogle Scholar
  34. 111.
    B. Muckenhoupt. Equiconvergence and almost everywhere convergence of Hermite and Laguerre series. SIAM J. Math. Anal., 1:295–321, 1970.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 112.
    V. Namias. The fractional order Fourier transform and its application to quantum mechanics. J. Inst. Math. Appl., 25:241–265, 1980.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 120.
    S.C. Pei, M.H. Yeh, and C.C. Tseng. Discrete fractional Fourier transform based on orthogonal projections. IEEE Trans. Signal Process., 47:1335–1348, 1999.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 137.
    G. Szegö. Orthogonal Polynomials. Colloquium Publications. AMS, Rhode Island, 1975.zbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Rafael G. Campos
    • 1
  1. 1.Science DepartmentUniversity of Quintana RooChetumalMexico

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