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Multi-dimensional linear canonical transform with applications to sampling and multiplicative filtering

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Abstract

This paper presents a novel and elegant convolution structure for the multi-dimensional linear canonical transform involving a pure multi-dimensional kernel obtained via a general \(2n\times 2n\) real, symplectic matrix M with \(n(2n+1)\) independent parameters. The primary intention is to develop the convolution theorem associated with the novel linear canonical convolution. The convolution structure is subsequently invoked to establish the sampling theorem for the band-limited signals in the multi-dimensional linear canonical domain. The validity and efficiency of the sampling procedure are demonstrated via a lucid example. Besides, the Heisenberg’s and Beckner’s uncertainty principles associated with the multi-dimensional linear canonical transform are also studied in detail. Finally, we study and design the multiplicative filter in the multi-dimensional linear canonical domain by utilizing the proposed multi-dimensional convolution structure.

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References

  • Akay, O., & Boudreaux-Bartels, G. F. (2001). Fractional convolution and correlation via operator methods and application to detection of linear FM signals. IEEE Transactions on Signal Processing, 49, 979–993.

    Google Scholar 

  • Bahri, M., Shah, F. A., & Tantary, A. Y. (2020). Uncertainty principles for the continuous shearlet transforms in aArbitrary space dimensions. Integral Transforms and Special Functions, 31(7), 538–555.

    MathSciNet  MATH  Google Scholar 

  • Bahri, M., Zulfajar, Z., & Ashino, R. (2014). Convolution and correlation theorem for linear canonical transform and properties. Information An International Interdisciplinary Journal, 17(6), 2509–2521.

    MathSciNet  MATH  Google Scholar 

  • Barshan, B., Kutay, M. A., & Ozaktas, H. M. (1997). Optimal filtering with linear canonical transformations. Optics Communication, 135, 32–36.

    Google Scholar 

  • Beckner, W. (1995). Pitt’s inequality and the uncertainty principle. Proceedings of the American Mathematical Society, 123, 1897–1905.

    MathSciNet  MATH  Google Scholar 

  • Chen, T., & Vaidyanathan, P. (1993). Recent developments in multidimensional multirate systems. IEEE Transaction on Circuits and System, 3(2), 116–137.

    Google Scholar 

  • Collins, S. A., Jr. (1970). Lens-system diffraction integral written in terms of matrix optics. Journal of the Optical Society of America, 60, 1772–1780.

    Google Scholar 

  • Debnath, L., & Shah, F. A. (2015). Wavelet Transforms and Their Applications. Boston: Birkhäuser.

    MATH  Google Scholar 

  • Debnath, L., & Shah, F. A. (2017). Lectuer Notes on Wavelet Transforms. Boston: Birkhäuser.

    MATH  Google Scholar 

  • Deng, B., Tao, R., & Wang, Y. (2006). Convolution theorems for the linear canonical transform and their applications. Science in China Series F: Information Sciences, 49, 592–603.

    MathSciNet  Google Scholar 

  • Dubois, E. (1985). The Sampling and reconstruction of time-varying imagery with application in video systems. Proceedings of the IEEE, 73(4), 502–522.

    Google Scholar 

  • Dubois, E. (1992). Motion-compensated filtering of time-varying images. Multidimensional System Signal Processing, 3(2), 211–239.

    Google Scholar 

  • Feng, Q., & Li, B. Z. (2016). Convolution and correlation theorems for the two-dimensional linear canonical transform and its applications. IET Signal Processing, 10(2), 125–132.

    Google Scholar 

  • Folland, G. B., & Sitaram, A. (1997). The uncertainty principle: A mathematical survey. Journal of Fourier Analysis and Applications, 3, 207–238.

    MathSciNet  MATH  Google Scholar 

  • Gopinath, R. A., & Burrus, C. S. (1994). On upsampling, downsampling, and rational sampling rate filter banks. IEEE Transactions on Signal Processing, 42(4), 812–824.

    Google Scholar 

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

    MATH  Google Scholar 

  • Healy, J. J., & Sheridan, J. T. (2009). Sampling and discretization of the linear canonical transform. Signal Processing, 89, 641–648.

    MATH  Google Scholar 

  • Hennelly, B. M., & Sheridan, J. T. (2005). Fast numerical algorithm for the linear canonical transform. Journal of the Optical Society of America. A, 22, 928–937.

    MathSciNet  Google Scholar 

  • Kutay, M. A., & Ozaktas, H. M. (1998). Optimal image restoration with the fractional Fourier transform. Journal of the Optical Society of America A. Optics and Image Science, 15(4), 825–833.

    Google Scholar 

  • Lee, Q. Y., Lia, B. Z., & Cheng, Q. Y. (2012). Discrete linear canonical transform of finite chirps. Engineering, 29, 3663–3667.

    Google Scholar 

  • Li, B. Z., Tao, R., & Wang, Y. (2007). New sampling formulae related to linear canonical transform. Signal Processing, 87, 983–990.

    MATH  Google Scholar 

  • Li, B. Z., & Xu, T. Z. (2012). Sampling in the linear canonical transform domain. Mathematical Problems in Engineering. https://doi.org/10.1155/2012/504580.

    Article  MathSciNet  MATH  Google Scholar 

  • Lu, Y. M., Do, M. N., & Laugesen, R. S. (2009). A computable Fourier condition generating alias-free sampling lattices. IEEE Transactions on Signal Processing, 57(5), 1768–1782.

    MathSciNet  MATH  Google Scholar 

  • Moshinsky, M., & Quesne, C. (1971). Linear canonical transformations and their unitary representations. Journal of Mathematics and Physics, 12(8), 1772–1780.

    MathSciNet  MATH  Google Scholar 

  • Robert, J., & Marks, I. I. (1993). Advanced Topics in Shannon Sampling and Interpolation Theory. Berlin: Springer.

    MATH  Google Scholar 

  • Shah, F. A., & Tantary, A. Y. (2018). Polar wavelet transform and the associated uncertainty principles. International Journal of Theoretical Physics, 57(6), 1774–1786.

    MathSciNet  MATH  Google Scholar 

  • Shah, F. A., & Tantary, A. Y. (2020). Linear canonical Stockwell transform. Journal of Mathematical Analysis and Applications, 443, 1–28.

    MathSciNet  MATH  Google Scholar 

  • Shah, F. A., & Tantary, A. Y. (2021). Lattice-based multi-channel sampling theorem for linear canonical transform. Digital Signal Processing. https://doi.org/10.1016/j.dsp.2021.103168.

    Article  Google Scholar 

  • Shi, J., Chi, Y., & Zhang, N. (2010). Multichannel sampling and reconstruction of bandlimited signals in fractional Fourier domain. IEEE Sig. Process. Letters., 17(11), 909–912.

    Google Scholar 

  • Shi, J., Liu, X., Sha, X., & Zhang, N. (2017). Sampling and reconstruction of signals in function spaces associated with the linear canonical transform. IEEE Signal Processing Letters, 60(11), 6041–6047.

    MathSciNet  MATH  Google Scholar 

  • Shi, J., Liu, X., & Zhang, N. (2014). Generalized convolution and product theorems associated with linear canonical transform. Signal, Image and Video Processing, 8, 967–974.

    Google Scholar 

  • Shinde, S., & Gadre, V. M. (2001). An uncertainty principle for real signals in the fractional Fourier transform domain. IEEE Transactions on Signal Processing, 49, 2545–2548.

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Stern, A. (2008). Uncertainty principles in linear canonical transform domains and some of their implications in optics. Journal of the Optical Society of America A. Optics and Image Science, 25(3), 647–652.

    MathSciNet  Google Scholar 

  • Unser, M. (2000). Sampling-50 years after Shannon. Proceedings of the IEEE, 88(4), 569–587.

    MATH  Google Scholar 

  • Wei, D., & Li, Y. M. (2014). Generalized wavelet transform based on the convolution operator in the linear canonical transform domain. Optik, 125, 4491–4496.

    Google Scholar 

  • Wei, D., & Li, Y. (2014). Reconstruction of multidimensional bandlimited signals from multichannel samples in linear canonical transform domain. IET Signal Processing, 8(6), 647–657.

    Google Scholar 

  • Wei, D. Y., Ran, Q. W., & Li, Y. M. (2011). A convolution and correlation theorem for the linear canonical transform and its application. Circuits System Signal Process, 31(1), 301–12.

    MathSciNet  MATH  Google Scholar 

  • Wei, D., Yang, W., & Li, Y. M. (2019). Lattices sampling and sampling rate conversion of multi-dimensional bandlimited signals in the linear canonical transform domain. Journal of the Franklin Institute, 356(13), 7571–7607.

    MathSciNet  MATH  Google Scholar 

  • Wilczok, E. (2000). New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform. Documenta Mathematica, 5, 201–226.

    MathSciNet  MATH  Google Scholar 

  • Woods, J. W. (2011). Multidimensional Signal, Image, and Video Processing and Coding. Cambridge: Academic press.

    Google Scholar 

  • Xiang, Q., & Qin, K. (2014). Convolution, correlation, and sampling theorems for the offset linear canonical transform. Signal, Image and Video Processing, 8(3), 433–442.

    Google Scholar 

  • Xua, S., Chenb, Z., Zhang, K., & He, Y. (2020). Aliased polyphase sampling theorem for the offset linear canonical transform. Optik. https://doi.org/10.1016/j.ijleo.2019.163410.

    Article  Google Scholar 

  • Xu, T. Z., & Li, B. Z. (2013). Linear Canonical Transform and Its Applications. Beijing: Science Press.

    Google Scholar 

  • Zayed, A. I. (1993). Advances in Shannon’s Sampling Theory. Boca Raton: CRC Press.

    MATH  Google Scholar 

  • Zayed, A. I. (1996). Function and Generalized Function Transformations. Boca Raton: CRC Press.

    MATH  Google Scholar 

  • Zayed, A. I. (2018). Sampling of signals bandlimited to a disc in the linear canonical transform domain. IEEE Signal Processing Letters, 25(12), 1765–1769.

    Google Scholar 

  • Zayed, A. I., & Garcia, A. G. (1999). New sampling formulae for the fractional Fourier transform. Signal Processing, 77, 111–114.

    MATH  Google Scholar 

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Acknowledgements

The first author is supported by SERB (DST), Government of India under Grant No. EMR/2016/007951.

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  1. Department of Mathematics, University of Kashmir, South Campus, Anantnag, Jammu and Kashmir, 192101, India

    Firdous A. Shah & Azhar Y. Tantary

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  1. Firdous A. Shah
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  2. Azhar Y. Tantary
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Correspondence to Firdous A. Shah.

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Shah, F.A., Tantary, A.Y. Multi-dimensional linear canonical transform with applications to sampling and multiplicative filtering. Multidim Syst Sign Process 33, 621–650 (2022). https://doi.org/10.1007/s11045-021-00816-6

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  • DOI: https://doi.org/10.1007/s11045-021-00816-6

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