Citation:  SHI Xiya, WU Anyang, SUN Yun, et al., “Unique Parameters Selection Strategy of Linear Canonical Wigner Distribution via Multiobjective Optimization Modeling,” Chinese Journal of Electronics, vol. 32, no. 3, pp. 453464, 2023, doi: 10.23919/cje.2021.00.338 
[1] 
T. Z. Xu and B. Z. Li, Linear Canonical Transform and Its Applications. Science Press, Beijing, 2013. (in Chinese)

[2] 
J. J. Healy, M. A. Kutay, H. M. Ozaktas, et al., Linear Canonical Transforms: Theory and Applications. Springer, New York, NY, USA, 2016.

[3] 
T. Alieva and M. J. Bastiaans, “Alternative representation of the linear canonical integral transform,” Optics Letters, vol.30, no.24, pp.3302–3304, 2005. doi: 10.1364/OL.30.003302

[4] 
A. Bhandari and A. I. Zayed, “Shiftinvariant and sampling spaces associated with the special affine Fourier transform,” Applied and Computational Harmonic Analysis, vol.47, no.1, pp.30–52, 2019. doi: 10.1016/j.acha.2017.07.002

[5] 
M. J. Bastiaans and T. Alieva, “Classification of lossless firstorder optical systems and the linear canonical transformation,” Journal of the Optical Society of America A, vol.24, no.4, pp.1053–1062, 2007. doi: 10.1364/JOSAA.24.001053

[6] 
S. A. Collins, “Lenssystem diffraction integral written in terms of matrix optics,” Journal of the Optical Society of America, vol.60, no.9, pp.1168–1177, 1970. doi: 10.1364/JOSA.60.001168

[7] 
M. Rahman, Applications of Fourier Transforms to Generalized Functions. WIT Press, Boston, MA, USA, 2011.

[8] 
J. Shi, Y. N. Zhao, W. Xiang, et al., “Deep scattering network with fractional wavelet transform,” IEEE Transactions on Signal Processing, vol.69, pp.4740–4757, 2021. doi: 10.1109/TSP.2021.3098936

[9] 
J. Shi, X. P. Liu, W. Xiang, et al., “Novel fractional wavelet packet transform: Theory, implementation, and applications,” IEEE Transactions on Signal Processing, vol.68, pp.4041–4054, 2020. doi: 10.1109/TSP.2020.3006742

[10] 
J. Shi, J. B. Zheng, X. P. Liu, et al., “Novel shorttime fractional Fourier transform: Theory, implementation, and applications,” IEEE Transactions on Signal Processing, vol.68, pp.3280–3295, 2020. doi: 10.1109/TSP.2020.2992865

[11] 
J. M. Ma, R. Tao, Y. Z. Li, et al., “Fractional spectrum analysis for nonuniform sampling in the presence of clock jitter and timing offset,” IEEE Transactions on Signal Processing, vol.68, pp.4148–4162, 2020. doi: 10.1109/TSP.2020.3007360

[12] 
D. Y. Wei and Y. J. Zhang, “Fractional Stockwell transform: Theory and applications,” Digital Signal Processing, vol.115, article no.103090, 2021.

[13] 
Y. Guo, B. Z. Li, and L. D. Yang, “Novel fractional wavelet transform: Principles, MRA and application,” Digital Signal Processing, vol.110, article no.102937, 2021.

[14] 
D. P. Kelly, “Numerical calculation of the Fresnel transform,” Journal of the Optical Society of America A, vol.31, no.4, pp.755–764, 2014. doi: 10.1364/JOSAA.31.000755

[15] 
R. T. Ranaivoson, R. Andriambololona, H. Rakotoson, et al., “Linear canonical transformations in relativistic quantum physics,” Physica Scripta, vol.96, no.6, article no.065204, 2021.

[16] 
J. Shi, X. P. Liu, Y. N. Zhao, et al., “Filter design for constrained signal reconstruction in linear canonical transform domain,” IEEE Transactions on Signal Processing, vol.66, no.24, pp.6534–6548, 2018. doi: 10.1109/TSP.2018.2878549

[17] 
J. T. Wang, Y. Wang, W. J. Wang, et al., “Discrete linear canonical wavelet transform and its applications,” EURASIP Journal on Advances in Signal Processing, vol.2018, no.1, article no.articleno.29, 2018. doi: 10.1186/s136340180550z

[18] 
S. C. Pei and J. J. Ding, “Relations between fractional operations and timefrequency distributions, and their applications,” IEEE Transactions on Signal Processing, vol.49, no.8, pp.1638–1655, 2001. doi: 10.1109/78.934134

[19] 
L. Cohen, TimeFrequency Analysis. PrenticeHall, New York, NY, USA, 1995.

[20] 
M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution: Theory and Applications in Signal Processing, W. Mecklenbräuker and F. Hlawatsch, Eds., Elsevier, Amsterdam, The Netherlands, pp.375–426, 1997.

[21] 
M. J. Bastiaans, “Wigner distribution function and its application to firstorder optics,” Journal of the Optical Society of America, vol.69, no.12, pp.1710–1716, 1979. doi: 10.1364/JOSA.69.001710

[22] 
J. Z. Jiao, B. Wang, and H. Liu, “Wigner distribution function and optical geometrical transformation,” Applied Optics, vol.23, no.8, pp.1249–1254, 1984. doi: 10.1364/AO.23.001249

[23] 
Z. Y. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in Proceedings of the 2009 IEEE International Conference on Computational Photography, San Francisco, CA, USA, pp.1–10, 2010.

[24] 
M. Mout, M. Wick, F. Bociort, et al., “Ray tracing the Wigner distribution function for optical simulations,” Optical Engineering, vol.57, no.1, article no.014106, 2018.

[25] 
R. OrtegaMartínez, C. J. RománMoreno, and A. L. Rivera, “The Wigner function in paraxial optics I. Matrix methods in Fourier optics,” Revista Mexicana de Física, vol.48, no.6, pp.565–574, 2002.

[26] 
M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” Journal of the Optical Society of America A, vol.3, no.8, pp.1227–1238, 1986. doi: 10.1364/JOSAA.3.001227

[27] 
P. Tovar, L. E. Y. Herrera, R. M. Ribeiro, et al., “Photonic generation of NLFM microwave pulses from DFBlaser chirp,” IEEE Photonics Technology Letters, vol.31, no.17, pp.1417–1420, 2019. doi: 10.1109/LPT.2019.2929709

[28] 
M. Alshaya, M. Yaghoobi, and B. Mulgrew, “Highresolution wideswath IRCIfree MIMO SAR,” IEEE Transactions on Geoscience and Remote Sensing, vol.58, no.1, pp.713–725, 2020. doi: 10.1109/TGRS.2019.2940075

[29] 
Z. N. Liang, Q. H. Liu, and T. Long, “A novel subarray digital modulation technique for wideband phased array radar,” IEEE Transactions on Instrumentation and Measurement, vol.69, no.10, pp.7365–7376, 2020. doi: 10.1109/TIM.2020.2984417

[30] 
P. R. Atkins, T. Collins, and K. G. Foote, “Transmitsignal design and processing strategies for sonar target phase measurement,” IEEE Journal of Selected Topics in Signal Processing, vol.1, no.1, pp.91–104, 2007. doi: 10.1109/JSTSP.2007.897051

[31] 
K. L. Gammelmark and J. A. Jensen, “Multielement synthetic transmit aperture imaging using temporal encoding,” IEEE Transactions on Medical Imaging, vol.22, no.4, pp.552–563, 2003. doi: 10.1109/TMI.2003.809088

[32] 
R. F. Bai, B. Z. Li, and Q. Y. Cheng, “WignerVille distribution associated with the linear canonical transform,” Journal of Applied Mathematics, vol.2012, article no. 740161, 2012.

[33] 
Z. C. Zhang, “New Wigner distribution and ambiguity function based on the generalized translation in the linear canonical transform domain,” Signal Processing, vol.118, pp.51–61, 2016. doi: 10.1016/j.sigpro.2015.06.010

[34] 
Z. C. Zhang, “Unified WignerVille distribution and ambiguity function in the linear canonical transform domain,” Signal Processing, vol.114, pp.45–60, 2015. doi: 10.1016/j.sigpro.2015.02.016

[35] 
A. Y. Wu, X. Y. Shi, Y. Sun, et al., “A computationally efficient optimal Wigner distribution in LCT domains for detecting noisy LFM signals,” Mathematical Problems in Engineering, vol.2022, article no.2036285, 2022.

[36] 
Z. C. Zhang and M. K. Luo, “New integral transforms for generalizing the Wigner distribution and ambiguity function,” IEEE Signal Processing Letters, vol.22, no.4, pp.460–464, 2015. doi: 10.1109/LSP.2014.2362616

[37] 
Z. C. Zhang, “The optimal linear canonical Wigner distribution of noisy linear frequencymodulated signals,” IEEE Signal Processing Letters, vol.26, no.8, pp.1127–1131, 2019. doi: 10.1109/LSP.2019.2922510

[38] 
Z. C. Zhang, “Linear canonical Wigner distribution based noisy LFM signals detection through the output SNR improvement analysis,” IEEE Transactions on Signal Processing, vol.67, no.21, pp.5527–5542, 2019. doi: 10.1109/TSP.2019.2941071

[39] 
Z. C. Zhang, “Variance analysis of linear canonical Wigner distribution,” Optik, vol.212, article no.164633, 2020.

[40] 
Z. C. Zhang, D. Li, Y. J. Chen, et al., “Linear canonical Wigner distribution of noisy LFM signals via multiobjective optimization analysis involving varianceSNR,” IEEE Communications Letters, vol.25, no.2, pp.546–550, 2021. doi: 10.1109/LCOMM.2020.3031982

[41] 
Z. C. Zhang, S. Z. Qiang, X. Jiang, et al., “Linear canonical Wigner distribution of noisy LFM signals via varianceSNR based inequalities system analysis,” Optik, vol.237, article no.166712, 2021.

[42] 
J. Shi, X. P. Liu, F. G. Yan, et al., “Error analysis of reconstruction from linear canonical transform based sampling,” IEEE Transactions on Signal Processing, vol.66, no.7, pp.1748–1760, 2018.

[43] 
Q. Feng, B. Z. Li, and J. M. Rassias, “Weighted HeisenbergPauliWeyl uncertainty principles for the linear canonical transform,” Signal Processing, vol.165, pp.209–221, 2019. doi: 10.1016/j.sigpro.2019.07.008

[44] 
D. Y. Wei and Y. M. Li, “Convolution and multichannel sampling for the offset linear canonical transform and their applications,” IEEE Transactions on Signal Processing, vol.67, no.23, pp.6009–6024, 2019. doi: 10.1109/TSP.2019.2951191

[45] 
W. B. Gao and B. Z. Li, “Uncertainty principles for the shorttime linear canonical transform of complex signals,” Digital Signal Processing, vol.111, article no.102953, 2021.

[46] 
L. Stanković, S. Stanković, and M. Daković, “From the STFT to the Wigner distribution [lecture notes],” IEEE Signal Processing Magazine, vol.31, no.3, pp.163–174, 2014. doi: 10.1109/MSP.2014.2301791

[47] 
A. Mahmood and R. Jäntti, “Packet error rate analysis of uncoded schemes in blockfading channels using extreme value theory,” IEEE Communications Letters, vol.21, no.1, pp.208–211, 2017. doi: 10.1109/LCOMM.2016.2615300

[48] 
L. H. Jia, K. B. Jia, and X. P. Fan, “Adaptive Lagrangian multiplier for quantization parameter cascading in HEVC hierarchical coding,” IEEE Signal Processing Letters, vol.27, pp.1220–1224, 2020. doi: 10.1109/LSP.2020.3005821

[49] 
G. K. Palshikar, “Simple algorithms for peak detection in timeseries,” in Proceedings of the 1st International Conference Advanced Data Analysis, Ahmedabad, India, 2009.

[50] 
K. T. Fang, D. K. J. Lin, P. Winker, et al., “Uniform design: Theory and application,” Technometrics, vol.42, no.3, pp.237–248, 2000. doi: 10.1080/00401706.2000.10486045

[51] 
N. R. Costa and J. A. Lourenço, “Exploring Pareto frontiers in the response surface methodology,” in Transactions on Engineering Technologies: World Congress on Engineering 2014, G. C. Yang, S. I. Ao, and L. Gelman, Eds., Springer, Berlin Heidelberg, pp.399–412, 2015.

[52] 
Z. Beheshti and S. M. H. Shamsuddin, “A review of populationbased metaheuristic algorithm,” International Journal of Advances in Soft Computing and its Applications, vol.5, no.1, pp.1–35, 2013.

[53] 
L. Debnath and D. Bhatta, Integral Transforms and Their Applications, 3rd ed., CRC Press, London, 2015.
