http://iet.metastore.ingenta.com
1887

Algebraic representation for fractional Fourier transform on one-dimensional discrete signal models

Algebraic representation for fractional Fourier transform on one-dimensional discrete signal models

For access to this article, please select a purchase option:

Buy article PDF
£12.50
(plus tax if applicable)
Buy Knowledge Pack
10 articles for £75.00
(plus taxes if applicable)

IET members benefit from discounts to all IET publications and free access to E&T Magazine. If you are an IET member, log in to your account and the discounts will automatically be applied.

Learn more about IET membership 

Recommend Title Publication to library

You must fill out fields marked with: *

Librarian details
Name:*
Email:*
Your details
Name:*
Email:*
Department:*
Why are you recommending this title?
Select reason:
 
 
 
 
 
IET Signal Processing — Recommend this title to your library

Thank you

Your recommendation has been sent to your librarian.

Algebraic signal processing provides a general framework for studying theoretical problems (sampling, transform domain analysis etc.) in the classical signal processing. In this study, the authors extend algebraic representation for the conventional Fourier transform (FT) to the fractional FT (FRFT) domain, from which the algebraic structures for the FRFT on infinite and finite one-dimensional signal models are obtained. They show that FRFTs on the infinite and finite discrete-time (DT) signal models, respectively, are none other than the DTFRFT and the closed-form discrete FRFT. They also derive FRFTs on the infinite and finite discrete-nearest neighbour signal models, and finally they discuss their applications in optical and time–frequency signal processing.

References

    1. 1)
      • 1. Püschel, M., Moura, J.M.F.: ‘Algebraic signal processing theory’. Available at http://www.arxiv.org/abs/cs.IT/0612077, accessed date 15 December 2006.
    2. 2)
      • 2. Ip, E.M., Kahn, J.M.: ‘Fiber impairment compensation using coherent detection and digital signal processing’, J. Lightwave Technol., 2010, 28, (4), pp. 502519.
    3. 3)
      • 3. James, J.F.: ‘A student's guide to Fourier transforms: with applications in physics and engineering’ (Cambridge University Press, Cambridge, 2011).
    4. 4)
      • 4. Püschel, M., Moura, J.M.F.: ‘Algebraic signal processing theory: foundation and 1-D time’, IEEE Trans. Signal Process., 2008, 56, (8), pp. 35723585.
    5. 5)
      • 5. Püschel, M., Moura, J.M.F.: ‘Algebraic signal processing theory: 1-D space’, IEEE Trans. Signal Process., 2008, 56, (8), pp. 35863599.
    6. 6)
      • 6. Sandryhaila, A., Kovačević, J., Püschel, M.: ‘Algebraic signal processing theory: 1-D nearest neighbor models’, IEEE Trans. Signal Process., 2012, 60, (5), pp. 22472259.
    7. 7)
      • 7. Namias, V.: ‘The fractional order Fourier transform and its application to quantum mechanics’, J. Inst. Math. Appl., 1980, 25, (3), pp. 241265.
    8. 8)
      • 8. Pei, S.C., Ding, J.J.: ‘Closed-form discrete fractional and affine Fourier transforms’, IEEE Trans. Signal Process., 2000, 48, (5), pp. 13381353.
    9. 9)
      • 9. Ozaktas, H.M., Barshan, B., Mendlovic, D., et al: ‘Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms’, J. Opt. Soc. Am. A, 1994, 11, (2), pp. 547559.
    10. 10)
      • 10. Wei, D.Y.: ‘Novel convolution and correlation theorems for the fractional Fourier transform’, Optik, 2016, 127, (7), pp. 36693675.
    11. 11)
      • 11. Zhang, Z.C.: ‘New convolution and product theorem for the linear canonical transform and its applications’, Optik, 2016, 127, (11), pp. 48944902.
    12. 12)
      • 12. Oktem, F.S., Ozaktas, H.M.: ‘Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space-bandwidth product’, J. Opt. Soc. Am. A, 2010, 27, (8), pp. 18851895.
    13. 13)
      • 13. Ozaktas, H.M., Ark, S.Ö., Coşkun, T.: ‘Fundamental structure of Fresnel diffraction: natural sampling grid and the fractional Fourier transform’, Opt. Lett., 2011, 36, (13), pp. 25242526.
    14. 14)
      • 14. Shinde, S., Gadre, V.M.: ‘An uncertainty principle for real signals in the fractional Fourier transform domain’, IEEE Trans. Signal Process., 2001, 49, (11), pp. 25452548.
    15. 15)
      • 15. Shi, J., Liu, X.P., Zhang, N.T.: ‘On uncertainty principle for signal concentrations with fractional Fourier transform’, Signal Process., 2012, 92, (12), pp. 28302836.
    16. 16)
      • 16. Zhou, H.B., Li, X., Tang, M., et al: ‘Joint timing/frequency offset estimation and correction based on FrFT encoded training symbols for PDM CO-OFDM systems’, Opt. Express, 2016, 24, (25), pp. 2825628269.
    17. 17)
      • 17. Zhang, Y., Dong, B.Z., Gu, B.Y., et al: ‘Beam shaping in the fractional Fourier transform domain’, J. Opt. Soc. Am. A, 1998, 15, (5), pp. 11141120.
    18. 18)
      • 18. Pei, S.C., Ding, J.J.: ‘Simplified fractional Fourier transforms’, J. Opt. Soc. Am. A, 2000, 17, (12), pp. 23552367.
    19. 19)
      • 19. Guo, C., Tan, J.B., Liu, Z.J.: ‘Precision influence of a phase retrieval algorithm in fractional Fourier domains from position measurement error’, Appl. Opt., 2015, 54, (22), pp. 69406947.
    20. 20)
      • 20. Kong, D.Z., Shen, X.J., Cao, L.C., et al: ‘Phase retrieval for attacking fractional Fourier transform encryption’, Appl. Opt., 2017, 56, (12), pp. 34493456.
    21. 21)
      • 21. Lohmann, A.W., Zalevsky, Z., Mendlovic, D.: ‘Synthesis of pattern recognition filters for fractional Fourier processing’, Opt. Commun., 1996, 128, (4–6), pp. 199204.
    22. 22)
      • 22. Ran, Q.W., Zhang, H.Y., Zhang, J., et al: ‘Deficiencies of the cryptography based on multiple-parameter fractional Fourier transform’, Opt. Lett., 2009, 34, (11), pp. 17291731.
    23. 23)
      • 23. Liu, X.B., Mei, W.B., Du, H.Q.: ‘Optical image encryption based on compressive sensing and chaos in the fractional Fourier domain’, J. Mod. Opt., 2014, 61, (19), pp. 15701577.
    24. 24)
      • 24. Alieva, T., Lopez, V., Agullo-Lopez, F., et al: ‘The fractional Fourier transform in optical propagation problems’, J. Mod. Opt., 1994, 41, (5), pp. 10371044.
    25. 25)
      • 25. Cowell, D.M., Freear, S.: ‘Separation of overlapping linear frequency modulated (LFM) signals using the fractional Fourier transform’, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 2010, 57, (10), pp. 23242333.
    26. 26)
      • 26. Ozaktas, H.M., Zalevsky, Z., Kutay, M.A.: ‘The fractional Fourier transform with applications in optics and signal processing’ (Wiley, New York, 2001).
    27. 27)
      • 27. Tao, R., Deng, B., Wang, Y.: ‘Fractional Fourier transform and its applications’ (Tsinghua University Press, Beijing, 2009).
    28. 28)
      • 28. Kraniauskas, P., Cariolaro, G., Erseghe, T.: ‘Method for defining a class of fractional operations’, IEEE Trans. Signal Process., 1998, 46, (10), pp. 28042807.
    29. 29)
      • 29. Erseghe, T., Kraniauskas, P., Cariolaro, G.: ‘Unified fractional Fourier transform and sampling theorem’, IEEE Trans. Signal Process., 1999, 47, (12), pp. 34193423.
    30. 30)
      • 30. Ozaktas, H.M., Arkan, O., Kutay, M.A., et al: ‘Digital computation of the fractional Fourier transform’, IEEE Trans. Signal Process., 1996, 44, (9), pp. 21412150.
    31. 31)
      • 31. Santhanam, B., McClellan, J.H.: ‘The discrete rotational Fourier transform’, IEEE Trans. Signal Process., 1996, 44, (4), pp. 994998.
    32. 32)
      • 32. Pei, S.C., Yeh, M.H.: ‘Improved discrete fractional Fourier transform’, Opt. Lett., 1997, 22, (14), pp. 10471049.
    33. 33)
      • 33. Arkan, O., Kutay, M.A., Ozaktas, H.M., et al: ‘The discrete fractional Fourier transformation’. IEEE Int. Symp. TFTS, 1996, vol. 4, pp. 205207.
    34. 34)
      • 34. Richman, M.S., Parks, T.W.: ‘Understanding discrete rotations’. Proc. ICASSP 1997, vol. 3, pp. 20572060.
    35. 35)
      • 35. Wolf, K.B.: ‘Discrete and finite fractional Fourier transforms’. Proc. Workshop on Group Theory and Numerical Methods (Université de Montréal, 26–31 May 2003), CRM Proc. and Lecture Series, 2004, vol. 39, pp. 267276.
    36. 36)
      • 36. Weimann, S., Perez-Leija, A., Lebugle, M., et al: ‘Implementation of quantum and classical discrete fractional Fourier transforms’, Nat. Commun., 2016, 7, p. 11027.
    37. 37)
      • 37. Tao, R., Li, B.Z., Sun, H.F.: ‘Research progress of the algebraic and geometric signal processing’, Def. Technol., 2013, 9, (1), pp. 4047.
    38. 38)
      • 38. Stern, A.: ‘Uncertainty principles in linear canonical transform domains and some of their implications in optics’, J. Opt. Soc. Am. A, 2008, 25, (3), pp. 647652.
    39. 39)
      • 39. Zhang, Z.C.: ‘Tighter uncertainty principles for linear canonical transform in terms of matrix decomposition’, Dig. Signal Process., 2017, 69, pp. 7085.
    40. 40)
      • 40. Potts, D., Steidl, G., Tasche, M.: ‘Fast algorithms for discrete polynomial transforms’, Math. Comput., 1998, 67, (224), pp. 15771590.
    41. 41)
      • 41. Driscoll, J.R., Healy, D.M.Jr., Rockmore, D.N.: ‘Fast discrete polynomial transforms with applications to data analysis for distance transitive graphs’, SIAM J. Comput., 1997, 26, (4), pp. 10661099.
    42. 42)
      • 42. Zhang, Z.C.: ‘Sampling theorem for the short-time linear canonical transform and its applications’, Signal Process., 2015, 113, (8), pp. 138146.
    43. 43)
      • 43. Wei, D.Y., Li, Y.M., Wang, R.K.: ‘Time–frequency analysis method based on affine Fourier transform and Gabor transform’, IET Signal Process., 2017, 11, (3), pp. 213220.
    44. 44)
      • 44. Zhang, Z.C., Yu, T., Luo, M.K., et al: ‘Multichannel sampling expansions in the linear canonical transform domain associated with explicit system functions and finite samples’, IET Signal Process., 2017, 11, (7), pp. 814824.
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-spr.2017.0217
Loading

Related content

content/journals/10.1049/iet-spr.2017.0217
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading
This is a required field
Please enter a valid email address