Convolution theorems for the linear canonical sine and cosine transform and its application

Wang Rongbo; Feng Qiang

doi:10.12086/oee.2018.170722

[1] Akay O, Boudreaux-Bartels G F. Fractional convolution and correlation via operator methods and an application to detection of linear FM signals[J]. IEEE Transactions on Signal Processing, 2001, 49(5): 979–993.

[2] Mustard D. Uncertainty principles invariant under the fractional Fourier transform[J]. The ANZIAM Journal, 1991, 33(2): 180–191.

[3] Shinde S, Gadre V M. An uncertainty principle for real signals in the fractional Fourier transform domain[J]. IEEE Transactions on Signal Processing, 2001, 49(11): 2545–2548.

[4] Li H, Li M W. Design and implementation of coded exposure camera[J]. Opto-Electronic Engineering, 2016, 43(9): 72–77.

[5] Ge M F, Qi H X, Wang Y X, et al. Spectral calibration for the high spectral resolution imager[J]. Opto-Electronic Engineering, 2015, 42(12): 14–19.

[6] Cooley J W, Tukey J W. An algorithm for the machine computation of complex Fourier series[J]. Mathematics of Computation, 1965, 19(90): 297–301.

[7] Namias V. The fractional order Fourier transform and its application to quantum mechanics[J]. IMA Journal of Applied Mathematics, 1980, 25(3): 241–265.

[8] Tao R, Qi L, Wang Y. Theory Applications of the Fractional Fourier transform[M]. Tsinghua University Press: China, 2004.

[9] Xu T Z, Li B Z. Linear Canonical Transform and Its Applications[ M]. Beijing: Science Press, 2013.

[10] Sneddon I N. Fourier Transforms[M]. New York: McGraw-Hill, 1951.

[11] Deng B, Tao R, Wang Y. Convolution theorems for the linear canonical transform and their applications[J]. Science in China Series F: Information Sciences, 2006, 49(5): 592–603.

[12] Wei D Y, Ran Q W, Li Y. A convolution and correlation theorem for the linear canonical transform and its application[J]. Circuits, Systems, and Signal Processing, 2012, 31(1): 301–312.

[13] Wei D Y, Ran Q W, Li Y M. New convolution theorem for the linear canonical transform and its translation invariance property[ J]. Optik, 2012, 123(16): 1478–1481.

[14] Wei D Y, Ran Q W, Li Y M, et al. A convolution and product theorem for the linear canonical transform[J]. IEEE Signal Processing Letters, 2009, 16(10): 853–856.

[15] Shi J, Liu X P, Zhang N T. Generalized convolution and product theorems associated with linear canonical transform[J]. Signal, Image and Video Processing, 2014, 8(5): 967–974.

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

[17] Zhang Z C. New convolution structure for the linear canonical transform and its application in filter design[J]. Optik, 2016, 127(13): 5259–5263.

[18] Stern A. Sampling of linear canonical transformed signals[J]. Signal Processing, 2006, 86(7): 1421–1425.

[19] Tao R, Li B Z, Wang Y. Spectral analysis and reconstruction for periodic nonuniformly sampled signals in fractional Fourier domain[J]. IEEE Transactions on Signal Processing, 2007, 55(7): 3541–3547.

[20] Li B Z, Tao R, Wang Y. New sampling formulae related to linear canonical transform[J]. Signal Processing, 2007, 87(5): 983–990.

[21] Shi J, Sha X J, Zhang Q Y, et al. Extrapolation of bandlimited signals in linear canonical transform domain[J]. IEEE Transactions on Signal Processing, 2012, 60(3): 1502–1508.

[22] Li B Z, Ji Q H. Sampling analysis in the complex reproducing kernel Hilbert space[J]. European Journal of Applied Mathematics, 2015, 26(1): 109–120.

[23] Wei D Y, Li Y M. The dual extensions of sampling and series expansion theorems for the linear canonical transform[J]. Optik, 2015, 126(24): 5163–5167.

[24] Stern A. Uncertainty principles in linear canonical transform domains and some of their implications in optics[J]. Journal of the Optical Society of America A, 2008, 25(3): 647–652.

[25] Sharma K K, Joshi S D. Uncertainty principle for real signals in the linear canonical transform domains[J]. IEEE Transactions on Signal Processing, 2008, 56(7): 2677–2683.

[26] Zhao J, Tao R, Li Y L, et al. Uncertainty principles for linear canonical transform[J]. IEEE Transactions on Signal Processing, 2009, 57(7): 2856–2858.

[27] Xu G L, Wang X T, Xu X G. On uncertainty principle for the linear canonical transform of complex signals[J]. IEEE Transactions on Signal Processing, 2010, 58(9): 4916–4918.

[28] Dang P, Deng G T, Qian T. A tighter uncertainty principle for linear canonical transform in terms of phase derivative[J]. IEEE Transactions on Signal Processing, 2013, 61(21): 5153–5164.

[29] Shi J, Han M, Zhang N T. Uncertainty principles for discrete signals associated with the fractional Fourier and linear canonical transforms[J]. Signal, Image and Video Processing, 2016, 10(8): 1519–1525.

[30] Pei S C, Ding J J. Fractional cosine, sine, and Hartley transforms[ J]. IEEE Transactions on Signal Processing, 2002, 50(7): 1661–1680.

[31] Paley R C, Wiener N. Fourier Transforms in the Complex Domain[ M]. New York: American Mathematical Society, 1934.

[32] Churchill R V. Fourier Series and Boundary Value Problems[M]. New York: McGraw-Hill, 1941.

[33] Thao N X, Kakichev V A, Tuan V K. On the generalized convolutions for Fourier cosine and sine transforms[J]. East West Journal of Mathematics, 1998, 1(1): 85–90.

[34] Thao N X, Tuan V K, Hong N T. A Fourier generalized convolution transform and applications to integral equations[J]. Fractional Calculus and Applied Analysis, 2012, 15(3): 493–508.

[35] Thao N X, Khoa N M. On the generalized convolution with a weight function for the Fourier sine and cosine transforms[J]. Integral Transforms and Special Functions, 2006, 17(9): 673–685.

[36] Thao N X, Tuan V K, Khoa N M. A generalized convolution with a weight function for the Fourier cosine and sine transforms[J]. Fractional Calculus and Applied Analysis, 2004, 7(3): 323–337.

[37] Thao N X, Khoa N M. On the convolution with a weight-function for the cosine-Fourier integral transform[J]. Acta Mathematica Vietnamica, 2004, 29(2): 149–162.

[38] Kakichev V A. On the convolution for integral transforms (in Russian)[J]. Vestsi Akademii Navuk BSSR, Seriya Fizika- Mathematics, 1967, 2: 48–57.

[39] Thao N X, Hai N T. Convolution for Integral Transforms and Their Applications[M]. Moscow: Russian Academy, 1997.

[40] Ganesan C, Roopkumar R. Convolution theorems for fractional Fourier cosine and sine transforms and their extensions to Boehmians[J]. Communications of the Korean Mathematical Society, 2016, 31(4): 791–809.

[41] Feng Q, Li B Z. Convolution theorem for fractional cosine-sine transform and its application[J]. Mathematical Methods in the Applied Sciences, 2017, 40(10): 3651–3665.

[42] Lee B G. A new algorithm for computing the discrete cosine transform[J]. IEEE Transactions on Acoustics, Speech, and Signal Processing, 1984, 32(6): 1243–1245.