WANG Xiaobo, ZHANG Qiliang, ZHOU You, et al., “Linear Canonical Transform Related Operators and Their Applications to Signal Analysis—— Part I: Fundamentals,” Chinese Journal of Electronics, vol. 24, no. 1, pp. 102-109, 2015,
Citation: WANG Xiaobo, ZHANG Qiliang, ZHOU You, et al., “Linear Canonical Transform Related Operators and Their Applications to Signal Analysis—— Part I: Fundamentals,” Chinese Journal of Electronics, vol. 24, no. 1, pp. 102-109, 2015,

Linear Canonical Transform Related Operators and Their Applications to Signal Analysis—— Part I: Fundamentals

Funds:  This work is supported by the National Natural Science Foundation of China (No.11301024, No.61071222).
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  • Corresponding author: ZOU Hongxing was born in 1966 in Jiangsu Province. He currently is a professor at the Department of Automation, Tsinghua University, Beijing, China. His research interests include signal processing, pattern recognition and matrix factorization. (Hongxing_Zou@tsinghua.edu.cn)
  • Received Date: 2014-04-01
  • Rev Recd Date: 2014-06-01
  • Publish Date: 2015-01-10
  • In recent years, the Linear canonical transform (LCT) has been recognized as a powerful tool in signal processing scenarios and optics. In this two-part paper, the issue of unitary and Hermitian operators and their duality concept associated with LCT are addressed. In Part I, based on the proposed operators, three LCT-related topics are derived. Firstly, the definitions of convolution and correlation operations in LCT domain are constructed. Secondly, a new transform which is named as Linear canonical mellin transform (LCMT) is introduced and then convolution and correlation operations in LCMT domain are given. Thirdly, the so-called joint canonical distributions which satisfy the canonical transform marginal are given. Part II concerns two applications of the theory derived in Part I for signal analysis.
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  • L. Cohen, Time-Frequency Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1995.
    R.G. Baraniuk and D.L. Jones, "Unitary equivalence: A new twist on signal processing", IEEE Trans. Signal Processing, Vol.43, No.10, pp.2269-2282, 1995.
    A.M. Sayeed and D.L. Jones, "Integral transforms covariant to unitary operators and their implications for joint signal representations", IEEE Trans. Signal Processing, Vol.44, No.6, pp.1365-1377, 1996.
    R.G. Baraniuk, "Beyond time-frequency analysis: Energy densities in one and many dimensions", IEEE Trans. Signal Processing, Vol.46, No.9, pp.2305-2314, 1998.
    O. Akay and G.F. Boudreaux-Bartels, "Unitary and Hermitian fractional operators and their relation to the fractional Fourier transform", IEEE Signal Processing Lett., Vol.5, No.12, pp.312- 314, 1998.
    O. Akay and G.F. Boudreaux-Bartels, "Fractional convolution and correlation via operator methods and an application to detection of linear FM signals", IEEE Trans. Signal Processing, Vol.49, No.5, pp.979-993, 2001.
    M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations", J. Math. Phys., Vol.12, No.8, pp.1772-1783, 1971.
    K.B. Wolf, Integral Transforms in Science and Engineering, ch. 9: Canonical transforms, New York: Plenum, 1979.
    W. Rudin, Functional Analysis, 2nd ed. New York: McGraw- Hill, 1991.
    G.B. Folland, Harmonic Analysis in Phase Space. NJ: Princeton University Press, 1989.
    J.J. Healy and J.T. Sheridan, "Fast linear canonical transforms", J. Opt. Soc. Am. A, Vol.27, No.1, pp.21-30, 2010.
    J.J. Healy and K.B. Wolf, "Discrete canonical transforms that are Hadamard matrices", J. Phys. A: Math. Theor., Vol.44, 265302(10pp), 2011.
    N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory. New York Interscience, 1958.
    B. Deng, R. Tao and Y. Wang, "Convolution theorems for the linear canonical transform and their applications", Sci. China Ser. F—Inform. Sci., Vol.49, No.5, pp.592-603, 2006.
    D.Y. Wei, Q.W. Ran, Y.M. Li, J. Ma and L.Y. Tan, "A convolution and product theorem for the linear canonical transform", IEEE Signal Processing Lett., Vol.16, No.10, pp.853-856, 2009.
    D.Y. Wei, Q.W. Ran and Y.M. Li, "A convolution and correlation theorem for the linear canonical transform and its application", Circuits Syst. Signal Process., Vol.31, No.1, pp.301-312, 2012.
    D.Y. Wei, Q.W. Ran and Y. Li, "New convolution theorem for the linear canonical transform and its translation invariance property", Optik-Int. J. Light Electron. Opt., Vol.123, No.16, pp.1478-1481, 2012.
    J. Shi, X.P. Liu and N.T. Zhang, "Generalized convolution and product theorems associated with linear canonical transform", Signal, Image and Video Process., pp.1-8, doi:10.1007/s11760- 012-0348-7, 2012.
    R.J. Marks.II, Advanced Topics in Shannon Sampling and Interpolation Theory. Springer-Verlag, 1993.
    A.I. Zayed, Function and Generalized Function Transformations. Boca Raton, FL: CRC, 1996.
    H.X. Zou, D.J. Wang, Q.H. Dai and Y.D. Li, "Real, discrete, and real discrete FMmlet transforms", Chinese Journal of Electronics, Vol.13, No.1, pp.8-11, 2004.
    J. Jeong and J. Williams, "Kernel design for reduced interference distributions", IEEE Trans. Signal Processing, Vol.40, No.2, pp.402-412, 1992.
    M.J. Bastiaans, T. Alieva and L. Stankovi?, "On rotated timefrequency kernels", IEEE Signal Processing Lett., Vol.9, No.11, pp.378-381, 2002.
    L. Stankovi?, T. Alieva and M.J. Bastiaans, "Time-frequency signal analysis based on the windowed fractional Fourier transform", Signal Process., Vol.83, No.11, pp.2459-2468, 2003.
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