Skip to main content
Log in

Operator theory-based discrete fractional Fourier transform

  • Original Paper
  • Published:
Signal, Image and Video Processing Aims and scope Submit manuscript

Abstract

The fractional Fourier transform is of importance in several areas of signal processing with many applications including optical signal processing. Deploying it in practical applications requires discrete implementations, and therefore defining a discrete fractional Fourier transform (DFRT) is of considerable interest. We propose an operator theory-based approach to defining the DFRT. By deploying hyperdifferential operators, a DFRT matrix can be defined compatible with the theory of the discrete Fourier transform. The proposed DFRT only uses the ordinary Fourier transform and the coordinate multiplication and differentiation operations. We also propose and compare several alternative discrete definitions of coordinate multiplication and differentiation operations, each of which leads to an alternative DFRT definition. Unitarity and approximation to the continuous transform properties are also investigated in detail. The proposed DFRT is highly accurate in approximating the continuous transform.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Ozaktas, H.M., Mendlovic, D.: Fourier transforms of fractional order and their optical interpretation. Opt. Commun. 101(3–4), 163–169 (1993)

    Article  Google Scholar 

  2. Mendlovic, D., Ozaktas, H.M.: Fractional Fourier transforms and their optical implementation: I. J. Opt. Soc. Am. A 10, 1875–1881 (1993)

    Article  Google Scholar 

  3. Ozaktas, H.M., Mendlovic, D.: Fractional Fourier transforms and their optical implementation: II. J. Opt. Soc. Am. A 10, 2522–2531 (1993)

    Article  Google Scholar 

  4. Ozaktas, H.M., Zalevsky, Z., Kutay, M.A.: The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, New York (2001)

    Google Scholar 

  5. Almeida, L.B.: The fractional Fourier transform and time–frequency representations. IEEE Trans. Signal Process. 42, 3084–3091 (1994)

    Article  Google Scholar 

  6. Kutay, M.A., Ozaktas, H.M., Arikan, O., Onural, L.: Optimal filtering in fractional Fourier domains. IEEE Trans. Signal Process. 45, 1129–1143 (1997)

    Article  Google Scholar 

  7. Zayed, A.I.: A convolution and product theorem for the fractional Fourier transform. IEEE Signal Process. Lett. 5(4), 101–103 (1998)

    Article  Google Scholar 

  8. Ozaktas, H.M., Barshan, B., Mendlovic, D., Onural, L.: Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms. J. Opt. Soc. Am. A 11(2), 547–559 (1994)

    Article  MathSciNet  Google Scholar 

  9. Pei, S.C., Ding, J.J.: Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing. IEEE Trans. Signal Process. 55(10), 4839–4850 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sejdic, E., Djurovic, I., Stankovic, J.: Fractional Fourier transform as a signal processing tool: an overview of recent developments. Signal Process. 91(6), 1351–1369 (2011)

    Article  MATH  Google Scholar 

  11. Tao, R., Li, B.Z., Wang, Y.: Spectral analysis and reconstruction for periodic nonuniformly sampled signals in fractional Fourier domain. IEEE Trans. Signal Process. 55(7), 3541–3547 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mohindru, P., Khanna, R., Bhatia, S.S.: New tuning model for rectangular windowed FIR filter using fractional Fourier transform. Signal Image Video Process. 9(4), 761–767 (2015)

    Article  Google Scholar 

  13. Anh, P.K., Castro, L.P., Thao, P.T., Tuan, N.M.: New sampling theorem and multiplicative filtering in the FRFT domain. Signal Image Video Process. 13(5), 951–958 (2019)

    Article  Google Scholar 

  14. Pei, S.C., Ding, J.J.: Relations between the fractional operations and time–frequency distributions, and their applications. IEEE Trans. Signal Process. 49(8), 1638–1655 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pei, S.C., Ding, J.J.: Relations between the fractional operations and the Wigner distribution, ambiguity function. IEEE Trans. Signal Process. 49(8), 209–219 (2001)

    Google Scholar 

  16. Shi, J., Han, M., Zhang, N.: Uncertainty principles for discrete signals associated with the fractional Fourier and linear canonical transforms. Signal Image Video Process. 10(8), 1519–1525 (2016)

    Article  Google Scholar 

  17. Lohmann, A.W.: Image rotation, Wigner rotation, and the fractional Fourier transform. J. Opt. Soc. Am. A 10(10), 2181–2186 (1993)

    Article  Google Scholar 

  18. Koç, A., Bartan, B., Gundogdu, E., Çukur, T., Ozaktas, H.M.: Sparse representation of two- and three-dimensional images with fractional Fourier, Hartley, linear canonical, and Haar wavelet transforms. Expert Syst. Appl. 77, 247–255 (2017)

    Article  Google Scholar 

  19. Zhang, X., Ling, B.W., Tao, R., Yang, Z., Woo, W., Sanei, S., Teo, K.L.: Optimal design of orders of DFrFTs for sparse representations. IET Signal Process. 12(8), 1023–1033 (2018)

    Article  Google Scholar 

  20. Sharma, K.K., Sharma, M.: Image fusion based on image decomposition using self-fractional Fourier functions. Signal Image Video Process. 8(7), 1335–1344 (2014)

    Article  Google Scholar 

  21. Jindal, N., Singh, K.: Image and video processing using discrete fractional transforms. Signal Image Video Process. 8(8), 1543–1553 (2014)

    Article  Google Scholar 

  22. Tian, N.-L., Zhang, X.-Z., Ling, B.W.-K., Yang, Z.-J.: Two-dimensional discrete fractional Fourier transform-based content removal algorithm. Signal Image Video Process. 10(7), 1311–1318 (2016)

    Article  Google Scholar 

  23. Mendlovic, D., Zalevsky, Z., Ozaktas, H.M.: The applications of the fractional Fourier transform to optical pattern recognition. In: Yu, F.T.S., Jutamulia, S. (eds.) Optical Pattern Recognition. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  24. Jacob, R., Thomas, T., Unnikrishnan, A.: Applications of fractional Fourier transform in sonar signal processing. IETE J. Res. 55(1), 16–27 (2009)

    Article  Google Scholar 

  25. Zhao, Z., Tao, R., Li, G., Wang, Y.: Fractional sparse energy representation method for ISAR imaging. IET Radar Sonar Navig. 12(9), 988–997 (2018)

    Article  Google Scholar 

  26. Ahmad, M.I., Sardar, M.U., Ahmad, I.: Blind beamforming using fractional Fourier transform domain cyclostationarity. Signal Image Video Process. 12(2), 379–383 (2018)

    Article  Google Scholar 

  27. Ahmad, M.I.: Optimum FRFT domain cyclostationarity based adaptive beamforming. Signal Image Video Process. 13(3), 551–556 (2019)

    Article  Google Scholar 

  28. Ozaktas, H.M., Mendlovic, D.: Fractional Fourier optics. J. Opt. Soc. Am. A 12, 743–751 (1995)

    Article  Google Scholar 

  29. Ozaktas, H.M., Erden, M.F.: Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems. Opt. Commun. 143, 75–86 (1997)

    Article  Google Scholar 

  30. Healy, J.J., Kutay, M.A., Ozaktas, H.M., Sheridan, J.T. (eds.): Linear Canonical Transforms: Theory and Applications. Springer, New York (2016)

    MATH  Google Scholar 

  31. Chen, X., Guan, J., Huang, Y., Liu, N., He, Y.: Radon-linear canonical ambiguity function-based detection and estimation method for marine target with micromotion. IEEE Trans. Geosci. Remote Sens. 53(4), 2225–2240 (2015)

    Article  Google Scholar 

  32. Qiu, W., Li, B., Li, X.: Speech recovery based on the linear canonical transform. Speech Commun. 55(1), 40–50 (2013)

    Article  Google Scholar 

  33. Singh, N., Sinha, A.: Chaos based multiple image encryption using multiple canonical transforms. Opt. Laser Technol. 42, 724–731 (2010)

    Article  Google Scholar 

  34. Zhu, B.H., Liu, S.T., Ran, Q.W.: Optical image encryption based on multifractional Fourier transforms. Opt. Lett. 25(16), 1159–1161 (2000)

    Article  Google Scholar 

  35. Li, B., Shi, Y.: Image watermarking in the linear canonical transform domain. Math. Probl. Eng. (2014). https://doi.org/10.1155/2014/645059

    Article  Google Scholar 

  36. Elhoseny, H.M., Ahmed, H.E.H., Abbas, A.M., Kazemian, H.B., Faragallah, O.S., El-Rabaie, S.M., Abd El-Samie, F.E.: Chaotic encryption of images in the fractional Fourier transform domain using different modes of operation. Signal Image Video Process. 9(3), 611–622 (2015)

    Article  Google Scholar 

  37. Elshazly, E.H., Faragallah, O.S., Abbas, A.M., Ashour, M.A., El-Rabaie, E.-S.M., Kazemian, H., Alshebeili, S.A., Abd El-Samie, F.E., El-sayed, H.S.: Robust and secure fractional wavelet image watermarking. Signal Image Video Process. 9(1), 89–98 (2015)

    Article  Google Scholar 

  38. Ozaktas, H.M., Arıkan, O., Kutay, M.A., Bozdağı, G.: Digital computation of the fractional Fourier transform. IEEE Trans. Signal Process. 44, 2141–2150 (1996)

    Article  Google Scholar 

  39. Pei, S.C., Ding, J.J.: Closed-form discrete fractional and affine Fourier transforms. IEEE Trans. Signal Process. 48, 1338–1353 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  40. Pei, S.C., Yeh, M.H.: Improved discrete fractional Fourier transform. Opt. Lett. 22(14), 1047–1049 (1997)

    Article  Google Scholar 

  41. Pei, S.C., Yeh, M.H., Tseng, C.C.: Discrete fractional Fourier transform based on orthogonal projections. IEEE Trans. Signal Process. 47(5), 1335–1348 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  42. Candan, C., Kutay, M.A., Ozaktas, H.M.: The discrete fractional Fourier transform. IEEE Trans. Signal Process. 48(5), 1329–1337 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  43. Pei, S.C., Hsue, W., Ding, J.: Discrete fractional Fourier transform based on new nearly tridiagonal commuting matrices. IEEE Trans. Signal Process. 54(10), 3815–3828 (2006)

    Article  MATH  Google Scholar 

  44. Hanna, M.T.: Direct batch evaluation of optimal orthonormal eigenvectors of the dft matrix. IEEE Trans. Signal Process. 56(5), 2138–2143 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  45. de Oliveira Neto, J.R., Lima, J.B.: Discrete fractional Fourier transforms based on closed-form Hermite–Gaussian-like DFT eigenvectors. IEEE Trans. Signal Process. 65(23), 6171–6184 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  46. Serbes, A., Durak-Ata, L.: Efficient computation of DFT commuting matrices by a closed-form infinite order approximation to the second differentiation matrix. Signal Process. 91(3), 582–589 (2011)

    Article  MATH  Google Scholar 

  47. Barker, L., Candan, C., Hakioglu, T., Kutay, M.A., Ozaktas, H.M.: The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform. J. Phys. A Math. Gen. 33(11), 2209 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  48. Su, X., Tao, R., Kang, X.: Analysis and comparison of discrete fractional Fourier transforms. Signal Process. 160, 284–298 (2019)

    Article  Google Scholar 

  49. Koç, A., Bartan, B., Ozaktas, H.M.: Discrete linear canonical transform based on hyperdifferential operators. IEEE Trans. Signal Process. 67(9), 2237–2248 (2019)

    Article  MathSciNet  Google Scholar 

  50. Wolf, K.B.: Construction and properties of canonical transforms. In: Integral Transforms in Science and Engineering, Chapter 9. Plenum Press, New York (1979)

    Chapter  Google Scholar 

  51. Gottlieb, D., Hussaini, M.Y., Orszag, S.A.: Introduction—theory and applications of spectral methods. In: Voigt, R.G., Gottlieb, D., Hussaini, M.Y. (eds.) Spectral Methods for Partial Differential Equations, Chapter 1, pp. 1–54. SIAM, Philadelphia (1984)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Aykut Koç.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Koç, A. Operator theory-based discrete fractional Fourier transform. SIViP 13, 1461–1468 (2019). https://doi.org/10.1007/s11760-019-01553-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11760-019-01553-x

Keywords

Navigation