线性正则域的Hartley变换及其不确定性原理
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摘要
Hartley变换是傅里叶变换的推广,它的一个非常好的性质就是把实信号变换成实信号,从而减少计算量。近些年,随着分数阶傅里叶变换在信号处理中被广泛的应用,线性正则变换也逐渐被应用到信号处理,所以把Hartley变换推广到正则域是一个有研究价值的问题。本文首先通过变化傅里叶变换域Hartley变换的核函数,得到了一个具有共轭性的核函数,之后,通过把该核函数替换成线性正则变换的核函数,从而得到了正则域的Hartley变换,在这个定义的基础上,得到了正则域Hartley变换满足实数性质和奇偶不变性,之后再利用线性正则变换的Heisenberg不确定性原理,得到了正则域Hartley变换的Heisenberg不确定性原理。
Hartley transform is a generalization of Fourier transform and it transforms the real signal into real signal thereby reducing the amount of computation. In recent years, with the wide applications of fractional Fourier transform in signal processing, linear canonical transform has gradually been applied to signal processing. Hence, it is a valuable problem to generalize Hartley transform in linear canonical transform domain. In this paper, a kernel function with conjugate property is obtained by changing kernel function of Hartley transform in Fourier transform domain. After that, we obtain Hartley transform in linear canonical transform domain by using kernel function of linear canonical transform. Then, Hartley transform in linear canonical transform domain has the properties of real number and odd-even invariance. Finally, by using Heisenberg uncertainty principle in linear canonical transform domain, we obtain Heisenberg uncertainty principle of Hartley transform in linear canonical transform domain.
Hartley transform is a generalization of Fourier transform and it transforms the real signal into real signal thereby reducing the amount of computation. In recent years, with the wide applications of fractional Fourier transform in signal processing, linear canonical transform has gradually been applied to signal processing. Hence, it is a valuable problem to generalize Hartley transform in linear canonical transform domain. In this paper, a kernel function with conjugate property is obtained by changing kernel function of Hartley transform in Fourier transform domain. After that, we obtain Hartley transform in linear canonical transform domain by using kernel function of linear canonical transform. Then, Hartley transform in linear canonical transform domain has the properties of real number and odd-even invariance. Finally, by using Heisenberg uncertainty principle in linear canonical transform domain, we obtain Heisenberg uncertainty principle of Hartley transform in linear canonical transform domain.
引文
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[5]Wolf K B.Integral Transforms in Science and Engineering[M].New York:Plenum Press,1979.
[6]Liu Y L,Kou K I,Ho I T,et al.New sampling formulae for non-bandlimited signals associated with linear canonical transform and nonlinear Fourier atoms[J].Signal Processing,2010,90(3):933–945.
[7]许天周,李炳照.线性正则变换及其应用[M].北京:科学出版社,2013.
[8]李炳照,陶然,王越.线性正则变换域的框架理论研究[J].电子学报,2007,35(7):1387–1390.
[9]Bai R F,Li B Z,Cheng Q Y.Wigner-Ville distribution associated with the linear canonical transform[J].Journal of Applied Mathematics,2012,2012:740161
[10]Xu G L,Wang X T,Xu X G.Three uncertainty relations for real signals associated with linear canonical transform[J].IET Signal Processing,2009,3(1):85–92.
[11]Folland G B,Sitaram A.The uncertainty principle:a mathematical survey[J].Journal of Fourier Analysis and Applications,1997,3(3):207–238.
[12]Shannon C E.Communication theory of secrecy systems[J].The Bell System Technical Journal,1949,28(4):656–715.
[13]Leipnik R.Entropy and the uncertainty principle[J].Information and Control,1959,2(1):64–79.
[14]Xu G L,Wang X T,Xu X G.Uncertainty inequalities for linear canonical transform[J].IET Signal Processing,2009,3(5):392–402.
[15]Moshinsky M,Quesne C.Linear canonical transformations and their unitary representations[J].Journal of Mathematical Physics,1971,12(8):1772–1780.
[16]Collins S A.Lens-system diffraction integral written in terms of matrix optics[J].Journal of the Optical Society of America,1970,60(9):1168–1177.
[17]Zhang Z C.New convolution and product theorem for the linear canonical transform and its applications[J].Optik-International Journal for Light and Electron Optics,2016,127(11):4894–4902.
[18]Azoug S E,Bouguezel S.A non-linear preprocessing for opto-digital image encryption using multiple-parameter discrete fractional Fourier transform[J].Optics Communications,2016,359:85–94.
[2]Bailey D H,Swarztrauber P N.A fast method for the numerical evaluation of continuous Fourier and Laplace transforms[J].SIAM Journal on Scientific Computing,1994,15(5):1105–1110.
[3]Duffieux P M.The Fourier Transform and Its Applications to Optics[M].2nd ed.New York,USA:John Wiley&Sons,1983.
[4]Ozaktas H M,Kutay M A,Zalevsky Z.The Fractional Fourier Transform with Applications in Optics and Signal Processing[M].New York:Wiley,2000.
[5]Wolf K B.Integral Transforms in Science and Engineering[M].New York:Plenum Press,1979.
[6]Liu Y L,Kou K I,Ho I T,et al.New sampling formulae for non-bandlimited signals associated with linear canonical transform and nonlinear Fourier atoms[J].Signal Processing,2010,90(3):933–945.
[7]许天周,李炳照.线性正则变换及其应用[M].北京:科学出版社,2013.
[8]李炳照,陶然,王越.线性正则变换域的框架理论研究[J].电子学报,2007,35(7):1387–1390.
[9]Bai R F,Li B Z,Cheng Q Y.Wigner-Ville distribution associated with the linear canonical transform[J].Journal of Applied Mathematics,2012,2012:740161
[10]Xu G L,Wang X T,Xu X G.Three uncertainty relations for real signals associated with linear canonical transform[J].IET Signal Processing,2009,3(1):85–92.
[11]Folland G B,Sitaram A.The uncertainty principle:a mathematical survey[J].Journal of Fourier Analysis and Applications,1997,3(3):207–238.
[12]Shannon C E.Communication theory of secrecy systems[J].The Bell System Technical Journal,1949,28(4):656–715.
[13]Leipnik R.Entropy and the uncertainty principle[J].Information and Control,1959,2(1):64–79.
[14]Xu G L,Wang X T,Xu X G.Uncertainty inequalities for linear canonical transform[J].IET Signal Processing,2009,3(5):392–402.
[15]Moshinsky M,Quesne C.Linear canonical transformations and their unitary representations[J].Journal of Mathematical Physics,1971,12(8):1772–1780.
[16]Collins S A.Lens-system diffraction integral written in terms of matrix optics[J].Journal of the Optical Society of America,1970,60(9):1168–1177.
[17]Zhang Z C.New convolution and product theorem for the linear canonical transform and its applications[J].Optik-International Journal for Light and Electron Optics,2016,127(11):4894–4902.
[18]Azoug S E,Bouguezel S.A non-linear preprocessing for opto-digital image encryption using multiple-parameter discrete fractional Fourier transform[J].Optics Communications,2016,359:85–94.