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

Wang Rongbo; Feng Qiang

doi:10.12086/oee.2018.170722

Article navigation > Opto-Electronic Engineering > 2018 Vol. 45 > No. 6 > 170722

Next Article Previous Article

Wang Rongbo, Feng Qiang. Convolution theorems for the linear canonical sine and cosine transform and its application[J]. Opto-Electronic Engineering, 2018, 45(6): 170722. doi: 10.12086/oee.2018.170722

Citation:

Wang Rongbo, Feng Qiang. Convolution theorems for the linear canonical sine and cosine transform and its application[J]. Opto-Electronic Engineering, 2018, 45(6): 170722. doi: 10.12086/oee.2018.170722

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

Wang Rongbo,
Feng Qiang^,

School of Mathematics and Computer Science, Yan'an University, Yan'an, Shaanxi 716000, China

Fund Project: Supported by National Natural Science Foundation of China (61671063) and the Science Foundation of Yan'an University (YDY2017-05)

More Information

Corresponding author: Feng Qiang, E-mail:yadxfq@yau.edu.cn

Received Date 30 December 2017

Revised Date 15 March 2018

Published Date 01 June 2018

Abstract

Abstract

For the denoising problem of odd and even signals, a multiplicative filter design method based on the convolution theorem of the linear canonical sine and cosine transform is proposed. Two kinds of convolution theorems associated with the linear canonical sine and cosine transform based on the existing linear canonical transform domain convolution theory are derived. Using this two convolution theorems, two kinds of the multiplicative filtering models of the band-limited signal are designed by choosing an appropriate filter function in linear canonical sine and cosine transform domain. And the complexity of these schemes is analyzed. The results indicate that these filtering models are particularly suitable for handling odd and even signals, and can effectively improve computational efficiency by reducing computational complexity.
- linear canonical transform /
- linear canonical sine and cosine transform /
- convolution theorem /
- filter

FullText(HTML)

References

[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. doi: 10.1109/78.917802 CrossRef Google Scholar
[2]	Mustard D. Uncertainty principles invariant under the fractional Fourier transform[J]. The ANZIAM Journal, 1991, 33(2): 180-191. Google Scholar
[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. doi: 10.1109/78.960402 CrossRef Google Scholar
[4]	李红, 李明伟.编码曝光相机设计与实现[J].光电工程, 2016, 43(9): 72-77. Google Scholar Li H, Li M W. Design and implementation of coded exposure camera[J]. Opto-Electronic Engineering, 2016, 43(9): 72-77. Google Scholar
[5]	葛明锋, 亓洪兴, 王雨曦, 等.高分辨力成像光谱仪光谱定标研究[J].光电工程, 2015, 42(12): 14-19. doi: 10.3969/j.issn.1003-501X.2015.12.003 CrossRef Google Scholar 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. doi: 10.3969/j.issn.1003-501X.2015.12.003 CrossRef Google Scholar
[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. doi: 10.1090/S0025-5718-1965-0178586-1 CrossRef Google Scholar
[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. doi: 10.1093/imamat/25.3.241 CrossRef Google Scholar
[8]	陶然, 齐林, 王越.分数阶傅里叶变换及其应用[M].北京:清华大学出版社, 2004. Google Scholar Tao R, Qi L, Wang Y. Theory Applications of the Fractional Fourier transform[M]. Tsinghua University Press: China, 2004. Google Scholar
[9]	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. Google Scholar Xu T Z, Li B Z. Linear Canonical Transform and Its Applications[M]. Beijing: Science Press, 2013. Google Scholar
[10]	Sneddon I N. Fourier Transforms[M]. New York: McGraw-Hill, 1951. Google Scholar
[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. doi: 10.1007/s11432-006-2016-4 CrossRef Google Scholar
[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. doi: 10.1007/s00034-011-9319-4 CrossRef Google Scholar
[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. doi: 10.1016/j.ijleo.2011.08.054 CrossRef Google Scholar
[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. doi: 10.1109/LSP.2009.2026107 CrossRef Google Scholar
[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. doi: 10.1007/s11760-012-0348-7 CrossRef Google Scholar
[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. doi: 10.1049/iet-spr.2015.0028 CrossRef Google Scholar
[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. doi: 10.1016/j.ijleo.2016.03.025 CrossRef Google Scholar
[18]	Stern A. Sampling of linear canonical transformed signals[J]. Signal Processing, 2006, 86(7): 1421-1425. doi: 10.1016/j.sigpro.2005.07.031 CrossRef Google Scholar
[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. doi: 10.1109/TSP.2007.893931 CrossRef Google Scholar
[20]	Li B Z, Tao R, Wang Y. New sampling formulae related to linear canonical transform[J]. Signal Processing, 2007, 87(5): 983-990. doi: 10.1016/j.sigpro.2006.09.008 CrossRef Google Scholar
[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. doi: 10.1109/TSP.2011.2176338 CrossRef Google Scholar
[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. doi: 10.1017/S0956792514000357 CrossRef Google Scholar
[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. doi: 10.1016/j.ijleo.2015.09.226 CrossRef Google Scholar
[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. doi: 10.1364/JOSAA.25.000647 CrossRef Google Scholar
[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. doi: 10.1109/TSP.2008.917384 CrossRef Google Scholar
[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. doi: 10.1109/TSP.2009.2020039 CrossRef Google Scholar
[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. doi: 10.1109/TSP.2010.2050201 CrossRef Google Scholar
[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. doi: 10.1109/TSP.2013.2273440 CrossRef Google Scholar
[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. doi: 10.1007/s11760-016-0965-7 CrossRef Google Scholar
[30]	Pei S C, Ding J J. Fractional cosine, sine, and Hartley transforms[J]. IEEE Transactions on Signal Processing, 2002, 50(7): 1661-1680. doi: 10.1109/TSP.2002.1011207 CrossRef Google Scholar
[31]	Paley R C, Wiener N. Fourier Transforms in the Complex Domain[M]. New York: American Mathematical Society, 1934. Google Scholar
[32]	Churchill R V. Fourier Series and Boundary Value Problems[M]. New York: McGraw-Hill, 1941. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. doi: 10.1080/10652460500432071 CrossRef Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	Kakichev V A. On the convolution for integral transforms (in Russian)[J]. Vestsi Akademii Navuk BSSR, Seriya Fizika-Mathematics, 1967, 2: 48-57. Google Scholar
[39]	Thao N X, Hai N T. Convolution for Integral Transforms and Their Applications[M]. Moscow: Russian Academy, 1997. Google Scholar
[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. doi: 10.4134/CKMS.c150244 CrossRef Google Scholar
[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. doi: 10.1002/mma.v40.10 CrossRef Google Scholar
[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. doi: 10.1109/TASSP.1984.1164443 CrossRef Google Scholar

Overview

Overview

Overview: In the modern optical signal processing domain, the collected signals must be denoised before the signal is analyzed and processed. The multiplicative filtering is one of the effective denoising methods in signal processing field based on the convolution theorem. The classical convolution theorem shows that the convolution of the two signals in time domain is leads to simple multiplication of their Fourier transforms in the Fourier transform domain. But Fourier transform is a holistic transformation based on the time domain or frequency domain, which is not suitable for modern optical signal processing.

As generalization of the Fourier transform and the fractional Fourier transform, Linear canonical transform has become a one of the powerful tools for modern optical signal analysis and processing, and has achieved fruitful research results in recent years. In order to further reduce computation and improve computing efficiency, convolution theory and application based on linear canonical transform has become one of the hot topic research in modern optical signal processing. Therefore, this paper will mainly focus on the research of convolution theory and application based on canonical sine transform and canonical cosine transform which have very close relations with the linear canonical transform, and have important role in signal processing, optics and other fields. Because canonical sine transform has no even eigenfunction and canonical cosine transform have no odd eigenfunction, therefore, it is much more efficient to use the canonical sine transform to deal with the odd signal and use the canonical cosine transform to deal with the even signal. Moreover, the complexity of the canonical sine transform and canonical cosine transform is one half of the complexities of the linear canonical transform, then, it is more suitable for engineering applications.

Hence, for the denoising problem of odd and even signals, a multiplicative filter design method based on the convolution theorem of the canonical sine and cosine transform is proposed. Two kinds of the convolution theorems associated with the canonical sine and cosine transform based on the existing linear canonical transform domain convolution theory are derived. Using this two convolution theorems, a kind of the multiplicative filtering model of the band-limited signal is designed by choosing an appropriate filter function in canonical sine and cosine transform domain. And the complexity of this scheme is analyzed. The results indicate that this filtering model is particularly suitable for handling odd and even signals, and can effectively improve computational efficiency by reducing computational complexity.