×

A convolution theorem related to quaternion linear canonical transform. (English) Zbl 1474.44006

Summary: We introduce the two-dimensional quaternion linear canonical transform (QLCT), which is a generalization of the classical linear canonical transform (LCT) in quaternion algebra setting. Based on the definition of quaternion convolution in the QLCT domain we derive the convolution theorem associated with the QLCT and obtain a few consequences.

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
30G35 Functions of hypercomplex variables and generalized variables
44A35 Convolution as an integral transform
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bahri, M.; Zulfajar; Ashino, R., Convolution and correlation theorem for linear canonical transform and properties, INFORMATION-An International Interdisciplinary Journal, 17, 6B, 2509-2521, (2014) · Zbl 1323.42009
[2] Healy, J. J.; Kutay, M. A.; Ozaktas, H. M.; Sheridon, J. J., Linear Canonical Transform: Theory and Application. Linear Canonical Transform: Theory and Application, Springer Series in Optical Sciences, 198, (2016)
[3] Guanlai, X.; Xiaotong, W.; Xiaogang, Y., Uncertainty principles for the linear canonical transform of complex signal, IEEE Transactions on Signal Processing, 58, 9, 4916-4918, (2010) · Zbl 1392.94231
[4] Urynbassarova, D.; Li, B. Z.; Tao, R., Convolution and correlation theorems for Wigner-Ville distribution associated with the offset linear canonical transform, Optik - International Journal for Light and Electron Optics, 157, 455-466, (2018)
[5] Wei, D.; Ran, Q.; Li, Y., New convolution theorem for the linear canonical transform and its translation invariance property, Optik - International Journal for Light and Electron Optics, 123, 16, 1478-1481, (2012)
[6] Wei, D.; Ran, Q.; Li, Y., A convolution and correlation theorem for the linear canonical transform and its application, Circuits, Systems and Signal Processing, 31, 1, 301-312, (2012) · Zbl 1252.94035
[7] Xiang, Q.; Qin, K.-Y., On the relationship between the linear canonical transform and the Fourier transform, Proceedings of the 4th International Congress on Image and Signal Processing (CISP ’11), IEEE
[8] Zhao, J.; Tao, R.; Li, Y.-L.; Wang, Y., Uncertainty principles for linear canonical transform, IEEE Transactions on Signal Processing, 57, 7, 2856-2858, (2009) · Zbl 1391.44002
[9] Bahri, M.; Ashino, R., A simplified proof of uncertainty principle for quaternion linear canonical transform, Abstract and Applied Analysis, 2016, (2016) · Zbl 1470.42013
[10] Bahri, M.; Resnawati; Musdalifah, S., A version of uncertainty principle for quaternion linear canonical transform, Abstract and Applied Analysis, 2018, (2018) · Zbl 1470.42016
[11] Bahri, M.; Ashino, R., Logarithmic uncertainty principle for quaternion linear canonical transform, Proceedings of the 2016 International Conference on Wavelet Analysis and Pattern Recognition
[12] Kou, K.; Morais, J.; Zhang, Y., Generalized prolate spheroidal wave functions for offset linear canonical transform in Clifford analysis, Mathematical Methods in the Applied Sciences, 36, 9, 1028-1041, (2013) · Zbl 1269.30053
[13] Kou, K. I.; Ou, J.-Y.; Morais, J., On uncertainty principle for quaternionic linear canonical transform, Abstract and Applied Analysis, 2013, (2013) · Zbl 1275.42012
[14] Zhang, Y.-N.; Li, B.-Z., Novel uncertainty principles for two-sided quaternion linear canonical transform, Advances in Applied Clifford Algebras (AACA), 28, 15, (2018) · Zbl 1392.81164
[15] Bahri, M.; Ashino, R.; Vaillancourt, R., Convolution theorems for quaternion fourier transform: properties and applications, Abstract and Applied Analysis, 2013, (2013) · Zbl 1297.42015
[16] Xu, G.; Wang, X.; Xu, X., Fractional quaternion Fourier transform, convolution and correlation, Signal Processing, 88, 10, 2511-2517, (2008) · Zbl 1151.94364
[17] Bahri, M.; Lawi, A.; Aris, N.; Saleh, A. F.; Nur, M., Relationships between convolution and correlation for fourier transform and quaternion fourier transform, International Journal of Mathematical Analysis, 7, 41-44, 2101-2109, (2013) · Zbl 1284.42007
[18] Bahri, M., A modified uncertainty principle for two-sided quaternion Fourier transform, Advances in Applied Clifford Algebras (AACA), 26, 2, 513-527, (2016) · Zbl 1342.42009
[19] Bahri, M., On two-dimensional quaternion wigner-ville distribution, Journal of Applied Mathematics, 2014, (2014) · Zbl 1406.94006
[20] Hitzer, E. M., Quaternion Fourier transform on quaternion fields and generalizations, Advances in Applied Clifford Algebras (AACA), 20, 3, 497-517, (2007) · Zbl 1143.42006
[21] Cheng, D.; Kou, K. I., Plancherel theorem and quaternion Fourier transform for square integrable functions, Complex Variables and Elliptic Equations. An International Journal, 64, 2, 223-242, (2019) · Zbl 1405.42008
[22] Lian, P., Uncertainty principles for quaternion Fourier transform, Journal of Mathematical Analysis and Applications, 467, 2, 1258-1269, (2018) · Zbl 1431.42020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.