Convolution Theorem Associated with the QWFRFT

MEI Yinyin; FENG Qiang; GAO Xiuxiu; ZHAO Yanbo

doi:10.23919/cje.2021.00.225

Volume 32 Issue 3

May 2023

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Article Navigation > Chinese Journal of Electronics > 2023 > 32(3): 485-492

MEI Yinyin, FENG Qiang, GAO Xiuxiu, et al. “Convolution Theorem Associated with the QWFRFT”. Chinese Journal of Electronics, vol. 32 no. 3. doi: 10.23919/cje.2021.00.225

Citation:

MEI Yinyin, FENG Qiang, GAO Xiuxiu, et al. “Convolution Theorem Associated with the QWFRFT”. Chinese Journal of Electronics, vol. 32 no. 3. doi: 10.23919/cje.2021.00.225

MEI Yinyin, FENG Qiang, GAO Xiuxiu, et al. “Convolution Theorem Associated with the QWFRFT”. Chinese Journal of Electronics, vol. 32 no. 3. doi: 10.23919/cje.2021.00.225

Citation:

MEI Yinyin, FENG Qiang, GAO Xiuxiu, et al. “Convolution Theorem Associated with the QWFRFT”. Chinese Journal of Electronics, vol. 32 no. 3. doi: 10.23919/cje.2021.00.225

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Convolution Theorem Associated with the QWFRFT

doi: 10.23919/cje.2021.00.225

1.
School of Mathematics and Computer Science, Yan’an University, Yan’an 716000, China

Funds: This work was supported by the National Natural Science Foundation of China (61861044, 11961072, 62001193 ), the Natural Science Foundation of Shaanxi Province (2022JM-400, 2020JM-547), and the Graduate Education Innovation Program of Yan’an University (YCX2021047)

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Author Bio:
Yinyin MEI was born in Shaanxi Province, China, in 1995. She received the B.S. degree from Yan’an University. She is an M.S. candidate in the School of Mathematics and Computer Science of Yan’an University. Her research interests include fractional convolution theory and fast algorithm. (Email: 2392471433@qq.com)

Qiang FENG (corresponding author) was born in Shaanxi Province, China, in 1975. He received the B.S. and M.S. degrees from Yan’an University, Shaanxi, China, in 1998 and 2006, respectively, and the Ph.D. degree from Beijing Institute of Technology, Beijing, China, in 2018. He is currently an Associate Professor with the School of Mathematics and Computer Science in Yan’an University. His research interests include fractional Fourier transform, linear canonical transform, and mathematical methods in signal processing. (Email: yadxfq@yau.edu.cn)
Received Date: 2021-07-06
Accepted Date: 2022-08-14

Available Online: 2022-04-14

Publish Date: 2023-05-05

Abstract

Abstract

The quaternion windowed fractional Fourier transform (QWFRFT) is a generalized form of the quaternion fractional Fourier transform (QFRFT), it plays a crucial role in signal processing for the analysis of multidimensional signals. In this paper, we first give the definition of the two-sided QWFRFT and some fundamental properties. Then, the quaternion convolution is proposed, and the relation between the quaternion convolution and the classical convolution is also given. Based on the quaternion convolution of the QWFRFT, relevant convolution theorems for the QWFRFT are studied. Moreover, the fast algorithm for QWFRFT is discussed. Finally, the complexity of QWFRFT and the quaternion windowed fractional convolution are given.
- Windowed fractional Fourier transform,
- Convolution theorem,
- Quaternion algebra

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References(36)

References

[1]	H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, John Wiley & Sons, New York, NY, USA, pp.25–33, 2001.
[2]	L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Transactions on Signal Processing, vol.42, no.11, pp.3084–3091, 1994. doi: 10.1109/78.330368
[3]	V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” IMA Journal of Applied Mathematics, vol.25, no.3, pp.241–265, 1980. doi: 10.1093/imamat/25.3.241
[4]	L. Zhang, G. H. Wang, X. Y. Zhang, et al., “The research on LFM radar countering frequency-shift jamming methods,” Acta Electronica Sinica, vol.49, no.3, pp.510–517, 2021. (in Chinese) doi: 10.12263/DZXB.20191334
[5]	L. Stanković, T. Alieva, and M. J. Bastiaans, “Time-frequency signal analysis based on the windowed fractional Fourier transform,” Signal Processing, vol.83, no.11, pp.2459–2468, 2003. doi: 10.1016/S0165-1684(03)00197-X
[6]	N. Goel and J. Singh, “Analysis of kaiser and Gaussian window functions in the fractional Fourier transform domain and its application,” Iranian Journal of Science and Technology, Transactions of Electrical Engineering, vol.43, no.2, pp.181–188, 2019. doi: 10.1007/s40998-018-0100-6
[7]	Q. Y. Zhang, “Uniqueness guarantees for phase retrieval from discrete windowed fractional Fourier transform,” Optik, vol.158, pp.1491–1498, 2018. doi: 10.1016/j.ijleo.2018.01.052
[8]	P. Mohindru, R. Khanna, and S. S. Bhatia, “New tuning model for rectangular windowed FIR filter using fractional Fourier transform,” Signal, Image and Video Processing, vol.9, no.4, pp.761–767, 2015. doi: 10.1007/s11760-013-0515-5
[9]	F. J. Yan and B. Z. Li, “Windowed fractional Fourier transform on graphs: Properties and fast algorithm,” Digital Signal Processing, vol.118, article no.103210, 2021. doi: 10.1016/J.DSP.2021.103210
[10]	W. B. Gao and B. Z. Li, “Convolution theorem involving n-dimensional windowed fractional Fourier transform,” Science China Information Sciences, vol.64, no.6, article no.169302, 2021. doi: 10.1007/s11432-020-2909-5
[11]	M. Bahri and R. Ashino, “Some properties of windowed linear canonical transform and its logarithmic uncertainty principle,” International Journal of Wavelets, Multiresolution and Information Processing, vol.14, no.3, article no.1650015, 2016. doi: 10.1142/S0219691316500156
[12]	R. Tao, Y. L. Li, and Y. Wang, “Short-time fractional Fourier transform and its applications,” IEEE Transactions on Signal Processing, vol.58, no.5, pp.2568–2580, 2010. doi: 10.1109/TSP.2009.2028095
[13]	W. R. Hamilton, Elements of Quaternions, Longmans, London, UK, pp.199–202, 1901.
[14]	I. L. Kantor and A. S. Solodovnikov, Hypercomplex Numbers: An Elementary Introduction to Algebras. Springer, New York, NY, USA, pp.15–25, 1989.
[15]	B. J. Chen, X. W. Ju, Y. Gao, et al., “A quaternion two-stream R-CNN network for pixel-level color image splicing localization,” Chinese Journal of Electronics, vol.30, no.6, pp.1069–1079, 2021. doi: 10.1049/cje.2021.08.004
[16]	H. S. Ye, N. R. Zhou, and L. H. Gong, “Multi-image compression-encryption scheme based on quaternion discrete fractional Hartley transform and improved pixel adaptive diffusion,” Signal Processing, vol.175, article no.107652, 2020. doi: 10.1016/j.sigpro.2020.107652
[17]	M. Bahri, E. S. M. Hitzer, A. Hayashi, et al., “An uncertainty principle for quaternion Fourier transform,” Computers & Mathematics with Applications, vol.56, no.9, pp.2398–2410, 2008. doi: 10.1016/j.camwa.2008.05.032
[18]	R. Bujack, H. De Bie, N. De Schepper, et al., “Convolution products for hypercomplex Fourier transforms,” Journal of Mathematical Imaging and Vision, vol.48, no.3, pp.606–624, 2014. doi: 10.1007/s10851-013-0430-y
[19]	Y. El Haoui and S. Fahlaoui, “The uncertainty principle for the two-sided quaternion Fourier transform,” Mediterranean Journal of Mathematics, vol.14, no.6, article no.221, 2017. doi: 10.1007/s00009-017-1024-5
[20]	S. C. Pei, J. J. Ding, and J. H. Chang, “Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT,” IEEE Transactions on Signal Processing, vol.49, no.11, pp.2783–2797, 2001. doi: 10.1109/78.960426
[21]	P. Lian, “Quaternion and fractional Fourier transform in higher dimension,” Applied Mathematics and Computation, vol.389, article no.125585, 2021. doi: 10.1016/j.amc.2020.125585
[22]	G. L. Xu, X. T. Wang, and X. G. Xu, “Fractional quaternion Fourier transform, convolution and correlation,” Signal Processing, vol.88, no.10, pp.2511–2517, 2008. doi: 10.1016/j.sigpro.2008.04.012
[23]	Z. F. Li, H. P. Shi, and Y. Y. Qiao, “Two-sided fractional quaternion Fourier transform and its application,” Journal of Inequalities and Applications, vol.2021, no.1, article no.121, 2021. doi: 10.1186/s13660-021-02654-3
[24]	M. Bahri and R. Ashino, “Two-dimensional quaternion linear canonical transform: Properties, convolution, correlation, and uncertainty principle,” Journal of Mathematics, vol.2019, article no.1062979, 2019. doi: 10.1155/2019/1062979
[25]	Y. Yang and K. I. Kou, “Uncertainty principles for hypercomplex signals in the linear canonical transform domains,” Signal Processing, vol.95, pp.67–75, 2014. doi: 10.1016/j.sigpro.2013.08.008
[26]	M. Bahri, E. S. M. Hitzer, R. Ashino, et al., “Windowed Fourier transform of two-dimensional quaternionic signals,” Applied Mathematics and Computation, vol.216, no.8, pp.2366–2379, 2010. doi: 10.1016/j.amc.2010.03.082
[27]	Y. X. Fu, U. Kähler, and P. Cerejeiras, “The Balian-Low theorem for the windowed quaternionic Fourier transform,” Advances in Applied Clifford Algebras, vol.22, no.4, pp.1025–1040, 2012. doi: 10.1007/s00006-012-0324-x
[28]	L. Akila and R. Roopkumar, “Multidimensional quaternionic Gabor transforms,” Advances in Applied Clifford Algebras, vol.26, no.3, pp.985–1011, 2016. doi: 10.1007/s00006-015-0634-x
[29]	K. Brahim and E. Tefjeni, “Uncertainty principle for the two sided quaternion windowed Fourier transform,” Journal of Pseudo-Differential Operators and Applications, vol.11, no.1, pp.159–185, 2020. doi: 10.1007/s11868-019-00283-5
[30]	M. Bahri, “A generalized windowed Fourier transform for quaternions,” Far East Journal of Mathematical Sciences, vol.42, no.1, pp.35–47, 2010.
[31]	M. Bahri and R. Ashino, “Uncertainty principles related to quaternionic windowed Fourier transform,” International Journal of Wavelets, Multiresolution and Information Processing, vol.18, no.3, article no.2050015, 2020. doi: 10.1142/S0219691320500150
[32]	R. Rajakumar, “Quaternionic short-time fractional Fourier transform,” International Journal of Applied and Computational Mathematics, vol.7, article no.100, 2021. doi: 10.1007/s40819-021-01055-w
[33]	W. B. Gao and B. Z. Li, “Quaternion windowed linear canonical transform of two-dimensional signals,” Advances in Applied Clifford Algebras, vol.30, no.1, article no.16, 2020. doi: 10.1007/s00006-020-1042-4
[34]	W. B. Gao and B. Z. Li, “Uncertainty principle for the two-sided quaternion windowed linear canonical transform,” Circuits, Systems, and Signal Processing, vol.41, no.3, pp.1324–1348, 2022. doi: 10.1007/s00034-021-01841-3
[35]	H. J. Nussbaumer, Fast Fourier Transform and Convolution Algorithms, Springer, Berlin, Heidelberg, pp. 80–85, 1981, doi: 10.1007/978-3-662-00551-4.
[36]	J. H. Wang, Y. Guo, Z. H. Wang, et al., “Advancing graph convolution network with revised Laplacian matrix,” Chinese Journal of Electronics, vol.29, no.6, pp.1134–1140, 2020. doi: 10.1049/cje.2020.09.015