# XFT: A Discrete Fourier Transform

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

## Abstract

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. . 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.

## Keywords

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

## References

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.
3. 7.
L. Auslander and F.A. Grünbaum. The Fourier transform and the discrete Fourier transform. Inverse Problems, 5:149–164, 1989.
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.
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.
6. 23.
F. Calogero. Matrices, differential operators and polynomials. J. Math. Phys., 22:919–934, 1981.
7. 24.
F. Calogero. Lagrangian interpolation and differentiation. Lett. Nuovo Cimento, 35:273–278, 1983.
8. 25.
F. Calogero. Interpolation, differentiation and solution of eigenvalue problems in more than one dimension. Lett. Nuovo Cimento, 39:305–311, 1984.
9. 26.
F. Calogero. Classical Many-Body Problems Amenable to Exact Treatments, volume 66 of Lecture Notes in Physics Monographs. Springer-Verlag, Berlin, 2001.
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.
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.
12. 29.
R.G. Campos. Time-reversal breaking in fractional differential problems. Eur. Phys. J. Plus, 130:121–125, 2015.
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.
14. 33.
R.G. Campos and L.Z. Juárez. A discretization of the continuous Fourier transform. Nuovo Cimento B, 107:703–711, 1992.
15. 34.
R.G. Campos and F. Marcellán. Quadratures and integral transforms arising from generating functions. Appl. Math. Comput., 297:8–18, 2017.
16. 35.
R.G. Campos and C. Meneses. Differentiation matrices for meromorphic functions. Bol. Soc. Mat. Mexicana, 12:121–132, 2006.
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.
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.
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.
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.
21. 50.
A. Erdélyi, editor. Higher Transcendental Functions, volume I, II. McGraw-Hill, New York, 1953.
22. 51.
A. Erdélyi, editor. Tables of Integral Transforms, volume I. McGraw-Hill, New York, 1954.
23. 58.
Sean A. Fulop. Speech Spectrum Analysis. Springer-Verlag, Berlin, 2011.
24. 60.
C. Gasquet and P. Witomski. Fourier Analysis and Applications. Filtering, Numerical Computation, Wavelets. Texts in Applied Mathematics. Springer-Verlag, New York, 1999.
25. 61.
W. Gautschi. Orthogonal Polynomials, Computation and Approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, 2004.
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.
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.
28. 73.
F.A. Grünbaum and E.H. Zarantonello. On the extension of uniformly continuous mappings. Michigan Math. J., 15:65–74, 1968.
29. 85.
N. Kaiblinger. Approximation of the Fourier transform and the dual Gabor window. J. Fourier Anal. Appl., 11:25–42, 2005.
30. 93.
H.J. Korsch and K. Rapedius. Computations in quantum mechanics made easy. Eur. J. Phys., 37:055410, 2016.
31. 95.
A. Kuznetsov and M. Kwaśnicki. Minimal Hermite-type eigenbasis of the discrete Fourier transform. J. Fourier Anal. Appl., https://doi.org/10.1007/s00041-018-9600-z, 2018.
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.
34. 111.
B. Muckenhoupt. Equiconvergence and almost everywhere convergence of Hermite and Laguerre series. SIAM J. Math. Anal., 1:295–321, 1970.
35. 112.
V. Namias. The fractional order Fourier transform and its application to quantum mechanics. J. Inst. Math. Appl., 25:241–265, 1980.
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.
37. 137.
G. Szegö. Orthogonal Polynomials. Colloquium Publications. AMS, Rhode Island, 1975.