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Spectral Analysis of Sampled Band-Limited Signals in the Offset Linear Canonical Transform Domain

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Abstract

The spectral analysis of sampled signals is one of the fundamental topics in the signal processing community. The properties and applications of uniformly and periodic nonuniformly sampled one- or two-dimensional signals in the traditional Fourier domain have been extensively studied. As the offset linear canonical transform (OLCT) has been shown to be a powerful tool for signal processing and optics, it is therefore worthwhile and interesting to consider the spectral analysis of sampled signals in the OLCT domain. In this paper, we investigate the spectrum of uniformly and periodic nonuniformly sampled one- and two-dimensional signals in the OLCT domain. First, the general spectral representation of uniformly sampled one-dimensional signals has been obtained. The reconstruction formula for uniform sampling from one-dimensional signal also has been performed. Based on the results, these theories are all extended to the two-dimensional case. Second, the digital spectra of periodic nonuniformly sampled one- and two-dimensional signals have been derived analytically. Finally, exhaustive analysis of sampled chirp signals in the OLCT domain has been carried out, and the simulations are presented to verify the correctness of the the results.

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References

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

    Article  Google Scholar 

  2. O. Besson, M. Ghogho, A. Swami, Parameter estimation for random amplitude chirp signals. IEEE Trans. Signal Process. 47, 3208–3219 (1999)

    Article  Google Scholar 

  3. Y.C. Eldar, A.V. Oppenheim, Filterbank reconstruction of bandlimited signals from nonuniform and generalized samples. IEEE Trans. Signal Process. 48, 2864–2875 (2000)

    Article  MathSciNet  Google Scholar 

  4. A. Feuer, G.C. Goodwin, Reconstruction of multidimensional bandlimited signals from nonuniform and generalized samples. IEEE Trans. Signal Process. 53, 4273–4282 (2005)

    Article  MathSciNet  Google Scholar 

  5. D.F.V. James, G.S. Agarwal, The generalized Fresnel transform and its applications to optics. Opt. Commun. 126, 207–212 (1996)

    Article  Google Scholar 

  6. Y.C. Jenq, Digital spectra of nonuniformly sampled signals: fundamentals and high-speed waveform digitizers. IEEE Trans. Instrum. Meas. 37, 245–251 (1988)

    Article  Google Scholar 

  7. Y.C. Jenq, L. Cheng, Digital spectrum of a nonuniformly sampled two-dimensional signal and its reconstruction. IEEE Trans. Instrum. Meas. 54, 1180–1187 (2005)

    Article  Google Scholar 

  8. A.J. Jerri, The Shannon sampling theorem—its various extensions and applications: a tutorial review. Proc. IEEE. 65, 1565–1598 (1977)

    Article  MATH  Google Scholar 

  9. B.Z. Li, T.Z. Xu, Spectral analysis of sampled signals in the linear canonical transform domain. Math. Prob. Eng. (2012). doi:10.1155/2012/536464

  10. F. Marvasti, Nonuniform Sampling Theory and Practice (Kluwer Academic, New York, 2000)

    Google Scholar 

  11. M. Moshinsky, C. Quesne, Linear canonical transformations and their unitary representations. J. Math. Phys. 12, 1772–1783 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Papoulis, Generalized sampling expansion. IEEE Trans. Circuit Syst. 24, 652–654 (1977)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. S.C. Pei, J.J. Ding, Eigenfunctions of the offset Fourier, fractional Fourier, and linear canonical transforms. J. Opt. Soc. Am. A 20, 522–532 (2003)

    Article  MathSciNet  Google Scholar 

  15. S.C. Pei, J.J. Ding, Eigenfunctions of Fourier and fractional Fourier transforms with complex offsets and parameters. IEEE Trans. Circuits Syst. I (54), 1599–1611 (2007)

    Article  MathSciNet  Google Scholar 

  16. A. Sahin, H.M. Ozaktas, D. Mendlovic, Optical implementation of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters. Appl. Opt. 37, 2130–2141 (1998)

    Article  Google Scholar 

  17. K.K. Sharma, S.D. Joshi, Signal separation using linear canonical and fractional Fourier transforms. Opt. Commun. 265, 454–460 (2006)

    Article  Google Scholar 

  18. M.I. Skolnik, Radar Handbook (McGraw-Hill, New York, 1990)

    Google Scholar 

  19. A. Stern, Sampling of linear canonical transformed signals. Signal Process. 86, 1421–1425 (2006)

    Article  MATH  Google Scholar 

  20. A. Stern, Sampling of compact signals in the offset linear canonical domain. Signal Image Video Process. 1, 359–367 (2007)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. M. Unser, Sampling-50 years after Shannon. Proc. IEEE. 88, 569–587 (2000)

    Article  Google Scholar 

  23. D.Y. Wei, Q.W. Ran, Y.M. Li, A convolution and correlation theorem for the linear canonical transform and its application. Circuits Syst. Signal Process. 31, 301–312 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  24. D.Y. Wei, Y.M. Li, Sampling reconstruction of N-dimensional bandlimited images after multilinear filtering in fractional Fourier domain. Opt. Commun. 295, 26–35 (2013)

    Article  Google Scholar 

  25. D.Y. Wei, Y.M. Li, Reconstruction of multidimensional bandlimited signals from multichannel samples in the linear canonical transform domain. IET Signal Process. 8, 647–657 (2014)

    Article  Google Scholar 

  26. G.L. Xu, X.T. Wang, X.G. Xu, Generalized Hilbert transform and its properties in 2D LCT domain. Signal Process. 89, 1395–1402 (2009)

    Article  MATH  Google Scholar 

  27. Q. Xiang, K.Y. Qin, Convolution, correlation, and sampling theorems for the offset linear canonical transform. Signal Image Video Process. (2012). doi:10.1007/s11760-012-0342-0

  28. Q. Xiang, K.Y. Qin, Q.Z. Huang, Multichannel sampling of signals band-limited in offset linear canonical transform domains. Circuits Syst. Signal Process. 32, 2385–2406 (2013)

    Article  MathSciNet  Google Scholar 

  29. T.Z. Xu, B.Z. Li, The Linear Canonical Transform and its Applications (Science Press, Beijing, 2013)

    Google Scholar 

  30. J.L. Yen, On nonuniform sampling of bandwidth-limited signals. IEEE Trans. Circuit Theor. 3, 251–257 (1956)

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions that improved the clarity and quality of this manuscript. This work was supported by the National Natural Science Foundation of China (61374135) and China Central Universities Foundation (CDJZR11170010).

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Authors and Affiliations

  1. State Key Laboratory of Power Transmission Equipment and System Security and New Technology, College of Automation, Chongqing University, Chongqing, 400044, China

    Shuiqing Xu & Yi Chai

  2. College of Automation, Chongqing University, Chongqing, 400044, China

    Youqiang Hu

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  1. Shuiqing Xu
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  2. Yi Chai
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  3. Youqiang Hu
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Correspondence to Shuiqing Xu.

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Xu, S., Chai, Y. & Hu, Y. Spectral Analysis of Sampled Band-Limited Signals in the Offset Linear Canonical Transform Domain. Circuits Syst Signal Process 34, 3979–3997 (2015). https://doi.org/10.1007/s00034-015-0053-1

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  • DOI: https://doi.org/10.1007/s00034-015-0053-1

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