Variance-SNR Based Noise Suppression on Linear Canonical Choi-Williams Distribution of LFM Signals

ZHANG Zhichao

doi:10.1049/cje.2020.00.367

Volume 31 Issue 5

Sep. 2022

Turn off MathJax

Article Contents

Article Navigation > Chinese Journal of Electronics > 2022 > 31(5): 804-820

ZHANG Zhichao, “Variance-SNR Based Noise Suppression on Linear Canonical Choi-Williams Distribution of LFM Signals,” Chinese Journal of Electronics, vol. 31, no. 5, pp. 804-820, 2022, doi: 10.1049/cje.2020.00.367

Citation:

ZHANG Zhichao, “Variance-SNR Based Noise Suppression on Linear Canonical Choi-Williams Distribution of LFM Signals,” Chinese Journal of Electronics, vol. 31, no. 5, pp. 804-820, 2022, doi: 10.1049/cje.2020.00.367

Citation:

PDF( 4864 KB)

Variance-SNR Based Noise Suppression on Linear Canonical Choi-Williams Distribution of LFM Signals

doi: 10.1049/cje.2020.00.367

ZHANG Zhichao^{1, 2
,}

1.
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2.
School of Computer Science and Engineering, Macau University of Science and Technology, Macau 999078, China

Funds: This work was supported by the National Natural Science Foundation of China (61901223), the Natural Science Foundation of Jiangsu Province (BK20190769), the Jiangsu Planned Projects for Postdoctoral Research Funds (2021K205B), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (19KJB510041), the Jiangsu Province High-Level Innovative and Entrepreneurial Talent Introduction Program (R2020SCB55), the Macau Young Scholars Program (AM2020015), and the Startup Foundation for Introducing Talent of NUIST (2019r024)

More Information

Author Bio:
(corresponding author) was born in Jingdezhen City, Jiangxi Province, China, in 1991. He received the B.S. degree in mathematics and applied mathematics from Gannan Normal University, Ganzhou, Jiangxi, China, in 2012, and the Ph.D. degree in Mathematics of Uncertainty Processing from Sichuan University, Chengdu, Sichuan, China, in 2018. From September 2017 to September 2018, he was awarded a grant from the China Scholarship Council to study as a visiting student researcher with the Department of Electrical and Computer Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA. Since 2019, he has been with the School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, China, where he is currently a Full Professor and Doctoral Supervisor. He is currently working as a Macau Young Scholars Postdoctoral Fellow in Information and Communication Engineering with the School of Computer Science and Engineering, Macau University of Science and Technology, Macau, China. His research interests cover the mathematical theories, methods and applications in signal and information processing, including fundamental theories such as Fourier analysis, functional analysis and harmonic analysis, applied theories such as signal representation, sampling, reconstruction, filter, separation, detection and estimation, and engineering technologies such as satellite communications, radar detection and electronic countermeasures. (Email: zzc910731@163.com)
Received Date: 2020-11-01
Accepted Date: 2022-02-28

Available Online: 2022-03-17

Publish Date: 2022-09-05

Abstract

Abstract

By solving the existing expectation-signal-to-noise ratio (expectation-SNR) based inequality model of the closed-form instantaneous cross-correlation function type of Choi-Williams distribution (CICFCWD), the linear canonical transform (LCT) free parameters selection strategies obtained are usually unsatisfactory. Since the second-order moment variance outperforms the first-order moment expectation in accurately characterizing output SNRs, this paper uses the variance analysis technique to improve parameters selection strategies. The CICFCWD’s average variance of deterministic signals embedded in additive zero-mean stationary circular Gaussian noise processes is first obtained. Then the so-called variance-SNRs are defined and applied to model a variance-SNR based inequality. A stronger inequalities system is also formulated by integrating expectation-SNR and variance-SNR based inequality models. Finally, a direct application of the system in noisy one-component and bi-component linear frequency-modulated (LFM) signals detection is studied. Analytical algebraic constraints on LCT free parameters newly derived seem more accurate than the existing ones, achieving better noise suppression effects. Our methods have potential applications in optical, radar, communication and medical signal processing.
- Average variance,
- Choi-Williams distribution,
- Linear canonical transform,
- Linear frequency-modulated signal

FullText(HTML)

References(118)

References

[1]	J. J. Healy, M. A. Kutay, H. M. Ozaktas, et al. Eds., Linear Canonical Transforms: Theory and Applications, Springer, New York, USA, 2016.
[2]	S. A. Collins, “Lens-system 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
[3]	M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” Journal of Mathematical Physics, vol.12, no.8, pp.1772–1780, 1971. doi: 10.1063/1.1665805
[4]	T. Xu and B. Li, Linear Canonical Transform and its Applications, Science Press, Beijing, China, 2013.
[5]	Z. Zhang, “Convolution theorems for two-dimensional LCT of angularly periodic functions in polar coordinates,” IEEE Signal Processing Letters, vol.26, no.8, pp.1242–1246, 2019. doi: 10.1109/LSP.2019.2926829
[6]	R. Bracewell, “The Fourier transform and its application,” American Journal of Physics, vol.34, article no.712, 1966. doi: 10.1119/1.1973431
[7]	J. F. James, A Student’s Guide to Fourier Transforms: With Applications in Physics and Engineering, Cambridge: Cambridge University Press, 2011.
[8]	H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, Wiley, New York, USA, 2001.
[9]	R. Tao, B. Deng, and Y. Wang, Fractional Fourier Transform and Its Applications, Tsinghua Univ. Press, Beijing, China, 2009.
[10]	J. Shi, X. Sha, and Q. Zhang, Fractional Order Signal Processing Theory and Methods, Harbin Inst. Tech. Press, Harbin, China, 2018.
[11]	D. Wei and Y. Li, “Generalized sampling expansions with multiple sampling rates for lowpass and bandpass signals in the fractional Fourier transform domain,” IEEE Transactions on Signal Processing, vol.64, no.18, pp.4861–4874, 2016. doi: 10.1109/TSP.2016.2560148
[12]	J. Shi, X. Liu, X. Sha, et al., “A sampling theorem for fractional wavelet transform with error estimates,” IEEE Transactions on Signal Processing, vol.65, no.18, pp.4797–4811, 2017. doi: 10.1109/TSP.2017.2715009
[13]	L. Xu, F. Zhang, and R. Tao, “Fractional spectral analysis of randomly sampled signals and applications,” IEEE Transactions on Instrumentation and Measurement, vol.66, no.11, pp.2869–2881, 2017. doi: 10.1109/TIM.2017.2728438
[14]	H. Miao, F. Zhang, and R. Tao, “Fractional Fourier analysis using the möbius inversion formula,” IEEE Transactions on Signal Processing, vol.67, no.12, pp.3181–3196, 2019. doi: 10.1109/TSP.2019.2912878
[15]	J. Ma, R. Tao, Y. Li, et al., “Fractional power spectrum and fractional correlation estimations for nonuniform sampling,” IEEE Signal Processing Letters, vol.27, pp.930–934, 2020. doi: 10.1109/LSP.2020.2997561
[16]	J. Shi, J. Zheng, X. Liu, et al., “Novel short-time fractional Fourier transform: Theory, implementation, and applications,” IEEE Transactions on Signal Processing, vol.68, pp.3280–3295, 2020. doi: 10.1109/TSP.2020.2992865
[17]	J. Ma, R. Tao, Y. 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
[18]	J. Shi, X. 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
[19]	Y. Liu, F. Zhang, H. Miao, et al., “The hopping discrete fractional Fourier transform,” Signal Processing, vol.178, article no.107763, 2021. doi: 10.1016/j.sigpro.2020.107763
[20]	J. Shi, Y. 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
[21]	H. I. Choi and W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.37, no.6, pp.862–871, 1989. doi: 10.1109/ASSP.1989.28057
[22]	Z. Zhang, “Choi-Williams distribution in linear canonical domains and its application in noisy LFM signals detection,” Communications in Nonlinear Science and Numerical Simulation, vol.82, article no.105025, 2020. doi: 10.1016/j.cnsns.2019.105025
[23]	Z. 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
[24]	Z. Zhang, “The optimal linear canonical Wigner distribution of noisy linear frequency-modulated signals,” IEEE Signal Processing Letters, vol.26, no.8, pp.1127–1131, 2019. doi: 10.1109/LSP.2019.2922510
[25]	Y. Wu, B. Li, and Q. Cheng, “A quantitative SNR analysis of LFM signals in the linear canonical transform domain with Gaussian windows,” in Proceedings 2013 International Conference on Mechatronic Sciences, Electric Engineering and Computer (MEC), pp.1426–1430, 2013.
[26]	B. M. Hamschin, J. D. Ferguson, and M. T. Grabbe, “Interception of multiple low-power linear frequency modulated continuous wave signals,” IEEE Transactions on Aerospace and Electronic Systems, vol.53, no.2, pp.789–804, 2017. doi: 10.1109/TAES.2017.2665140
[27]	J. Zheng, H. Liu, and Q. Liu, “Parameterized centroid frequency-chirp rate distribution for LFM signal analysis and mechanisms of constant delay introduction,” IEEE Transactions on Signal Processing, vol.65, no.24, pp.6435–6447, 2017. doi: 10.1109/TSP.2017.2755604
[28]	A. Serbes, “On the estimation of LFM signal parameters: Analytical formulation,” IEEE Transactions on Aerospace and Electronic Systems, vol.54, no.2, pp.848–860, 2018. doi: 10.1109/TAES.2017.2767978
[29]	G. Bai, Y. Cheng, W. Tang, et al., “Chirp rate estimation for LFM signal by multiple DPT and weighted combination,” IEEE Signal Processing Letters, vol.26, no.1, pp.149–153, 2019. doi: 10.1109/LSP.2018.2882300
[30]	A. A. Dezfuli, A. Shokouhmand, A. H. Oveis, et al., “Reduced complexity and near optimum detector for linear‐frequency‐modulated and phase‐modulated LPI radar signals,” IET Radar, Sonar & Navigation, vol.13, no.4, pp.593–600, 2019.
[31]	S. Li, M. Xue, T. Qing, et al., “Ultrafast and ultrahigh-resolution optical vector analysis using linearly frequency-modulated waveform and dechirp processing,” Optics Letters, vol.44, no.13, pp.3322–3325, 2019. doi: 10.1364/OL.44.003322
[32]	X. Fu, B. Wang, M. Xiang, et al., “Residual RCM correction for LFM-CW mini-SAR system based on fast-time split-band signal interferometry,” IEEE Transactions on Geoscience and Remote Sensing, vol.57, no.7, pp.4375–4387, 2019. doi: 10.1109/TGRS.2019.2890978
[33]	Y. Luo, L. Zhang and H. Ruan, “An acquisition algorithm based on FRFT for weak GNSS signals in A dynamic environment,” IEEE Communications Letters, vol.22, no.6, pp.1212–1215, 2018. doi: 10.1109/LCOMM.2018.2828834
[34]	T. Misaridis and J. A. Jensen, “Use of modulated excitation signals in medical ultrasound. Part II: Design and performance for medical imaging applications,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol.52, no.2, pp.192–207, 2005. doi: 10.1109/TUFFC.2005.1406546
[35]	S. Qiang, X. Jiang, P. Han, et al., “Instantaneous cross-correlation function type of WD based LFM signals analysis via output SNR inequality modeling,” EURASIP Journal on Advances in Signal Processing, vol.2021, no.1, pp.1–24, 2021. doi: 10.1186/s13634-020-00710-6
[36]	A. Wu, X. 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.
[37]	Z. Zhang, “Variance analysis of linear canonical Wigner distribution,” Optik, vol.212, article no.164633, 2020. doi: 10.1016/j.ijleo.2020.164633
[38]	I. Djurović and V. N. Ivanović, “Analysis of closed-form Wigner distribution in linear canonical domains for continuous-time noisy signals,” 2020 9th Mediterranean Conference on Embedded Computing (MECO), Budva, Montenegro, pp.1–4, 2020.
[39]	Z. Zhang, “Variance analysis of noisy LFM signal in linear canonical Cohen’s class,” Optik, vol.216, article no.164610, 2020. doi: 10.1016/j.ijleo.2020.164610
[40]	Z. Zhang, D. Li, Y. Chen, et al., “Linear canonical Wigner distribution of noisy LFM signals via multiobjective optimization analysis involving variance-SNR,” IEEE Communications Letters, vol.25, no.2, pp.546–550, 2021. doi: 10.1109/LCOMM.2020.3031982
[41]	Z. Zhang, S. Qiang, X. Jiang, et al., “Linear canonical Wigner distribution of noisy LFM signals via variance-SNR based inequalities system analysis,” Optik, vol.237, article no.166712, 2021. doi: 10.1016/j.ijleo.2021.166712
[42]	X. Shi, A. Wu, Y. Sun, et al., “Unique parameters selection strategy of linear canonical Wigner distribution via multiobjective optimization modeling, ” Chinese Journal of Electronics, in press, DOI: 10.1049/cje.2021.00.338, 2022.
[43]	B. M. Hennelly and J. T. Sheridan, “Fast numerical algorithm for the linear canonical transform,” Journal of the Optical Society of America. A, Optics, Image Science, and Vision, vol.22, no.5, pp.928–937, 2005. doi: 10.1364/JOSAA.22.000928
[44]	A. Koc, H. M. Ozaktas, C. Candan, et al., “Digital computation of linear canonical transforms,” IEEE Transactions on Signal Processing, vol.56, no.6, pp.2383–2394, 2008. doi: 10.1109/TSP.2007.912890
[45]	A. Koç, B. Bartan, and H. M. Ozaktas, “Discrete linear canonical transform based on hyperdifferential operators,” IEEE Transactions on Signal Processing, vol.67, no.9, pp.2237–2248, 2019. doi: 10.1109/TSP.2019.2903031
[46]	J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Processing, vol.89, no.4, pp.641–648, 2009. doi: 10.1016/j.sigpro.2008.10.011
[47]	A. Ko, F. S. Oktem, H. M. Ozaktas, et al., Fast Algorithms For Digital Computation of Linear Canonical Transforms, Linear Canonical Transforms, New York, NY: Springer New York, pp.293–327, 2016.
[48]	S. C. Pei and Y. C. Lai, “Discrete linear canonical transforms based on dilated Hermite functions,” Journal of the Optical Society of America A, vol.28, no.8, pp.1695–1708, 2011. doi: 10.1364/JOSAA.28.001695
[49]	S. C. Pei and S. G. Huang, “Fast discrete linear canonical transform based on CM-CC-CM decomposition and FFT,” IEEE Transactions on Signal Processing, vol.64, no.4, pp.855–866, 2016. doi: 10.1109/TSP.2015.2491891
[50]	D. Wei, R. Wang, and Y. Li, “Random discrete linear canonical transform,” Journal of the Optical Society of America A, vol.33, no.12, pp.2470–2476, 2016. doi: 10.1364/JOSAA.33.002470
[51]	Y. N. Sun and B. Z. Li, “Segmented fast linear canonical transform,” Journal of the Optical Society of America A, vol.35, no.8, article no.1346, 2018. doi: 10.1364/JOSAA.35.001346
[52]	Bao Yiping and Li. Bingzhao, “Modelling the noise influence associated with the discrete linear canonical transform,” IET Signal Processing, vol.12, no.6, pp.756–760, 2018. doi: 10.1049/iet-spr.2017.0319
[53]	Y. Sun and B. Li, “Sliding discrete linear canonical transform,” IEEE Transactions on Signal Processing, vol.66, no.17, pp.4553–4563, 2018. doi: 10.1109/TSP.2018.2855658
[54]	Y. Sun, B. Li, and R. Tao, “Research progress on discretization of linear canonical transform,” Opto-Electronic Engineering, vol.45, no.6, pp.29–49, 2018.
[55]	Y. Sun and B. Li, “Digital computation of linear canonical transform for local spectra with flexible resolution ability,” Science China Information Sciences, vol.62, no.4, article no.49301, 2019. doi: 10.1007/s11432-018-9585-1
[56]	D. Wei and H. Hu, “Sparse discrete linear canonical transform and its applications,” Signal Processing, no.183, article no.108046, 2021.
[57]	Y. Sun, B. Li, and R. Tao, “Research progress on discretization of linear canonical transform,” Opto-Electronic Engineering, vol.45, no.6, pp.205–216, 2018.
[58]	D. Bing, T. Ran, and W. Yue, “Convolution theorems for the linear canonical transform and their applications,” Science in China Series F: Information Sciences, vol.49, no.5, pp.592–603, 2006. doi: 10.1007/s11432-006-2016-4
[59]	D. Wei, Q. Ran, Y. Li, et al., “A convolution and product theorem for the linear canonical transform,” IEEE Signal Processing Letters, vol.16, no.10, pp.853–856, 2009. doi: 10.1109/LSP.2009.2026107
[60]	D. Wei, Q. Ran, and Y. Li, “Multichannel sampling expansion in the linear canonical transform domain and its application to superresolution,” Optics Communications, vol.284, no.23, pp.5424–5429, 2011. doi: 10.1016/j.optcom.2011.08.015
[61]	J. Shi, X. Sha, Q. Zhang, et al., “Extrapolation of bandlimited signals in linear canonical transform domain,” IEEE Transactions on Signal Processing, vol.60, no.3, pp.1502–1508, 2012. doi: 10.1109/TSP.2011.2176338
[62]	X. Wang, Q. Zhang, Y. Zhou, et al., “Linear canonical transform related operators and their applications to signal analysis—Part I: Fundamentals,” Chinese Journal of Electronics, vol.24, no.1, pp.102–109, 2015. doi: 10.1049/cje.2015.01.017
[63]	X. Wang, Q. Zhang, Y. Zhou, et al., “Linear canonical transform related operators and their applications to signal analysis—Part II: Applications,” Chinese Journal of Electronics, vol.24, no.2, pp.312–316, 2015. doi: 10.1049/cje.2015.04.014
[64]	Q. Feng and Bingzhao Li, “Convolution and correlation theorems for the two‐dimensional linear canonical transform and its applications,” IET Signal Processing, vol.10, no.2, pp.125–132, 2016. doi: 10.1049/iet-spr.2015.0028
[65]	A. Stern, “Sampling of linear canonical transformed signals,” Signal Processing, vol.86, no.7, pp.1421–1425, 2006. doi: 10.1016/j.sigpro.2005.07.031
[66]	B. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Processing, vol.87, no.5, pp.983–990, 2007. doi: 10.1016/j.sigpro.2006.09.008
[67]	R. Tao, B. Li, Y. Wang, et al., “On sampling of band-limited signals associated with the linear canonical transform,” IEEE Transactions on Signal Processing, vol.56, no.11, pp.5454–5464, 2008. doi: 10.1109/TSP.2008.929333
[68]	J. Shi, X. Liu, X. Sha, et al., “Sampling and reconstruction of signals in function spaces associated with the linear canonical transform,” IEEE Transactions on Signal Processing, vol.60, no.11, pp.6041–6047, 2012. doi: 10.1109/TSP.2012.2210887
[69]	J. Shi, X. Liu, L. He, et al., “Sampling and reconstruction in arbitrary measurement and approximation spaces associated with linear canonical transform,” IEEE Transactions on Signal Processing, vol.64, no.24, pp.6379–6391, 2016. doi: 10.1109/TSP.2016.2602808
[70]	L. Xu, R. Tao, and F. Zhang, “Multichannel consistent sampling and reconstruction associated with linear canonical transform,” IEEE Signal Processing Letters, vol.24, no.5, pp.658–662, 2017. doi: 10.1109/LSP.2017.2683535
[71]	J. Wang, S. Ren, Z. Chen, et al., “Periodically nonuniform sampling and reconstruction of signals in function spaces associated with the linear canonical transform,” IEEE Communications Letters, vol.22, no.4, pp.756–759, 2018. doi: 10.1109/LCOMM.2018.2801871
[72]	J. Shi, X. Liu, F. 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.
[73]	J. Shi, X. Liu, Y. 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
[74]	A. I. Zayed, “Sampling of signals bandlimited to a disc in the linear canonical transform domain,” IEEE Signal Processing Letters, vol.25, no.12, pp.1765–1769, 2018. doi: 10.1109/LSP.2018.2875341
[75]	A. Bhandari and A. I. Zayed, “Shift-invariant 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
[76]	D. Wei, W. Yang, and Y. Li, “Lattices sampling and sampling rate conversion of multidimensional bandlimited signals in the linear canonical transform domain,” Journal of the Franklin Institute, vol.356, no.13, pp.7571–7607, 2019. doi: 10.1016/j.jfranklin.2019.06.031
[77]	D. Wei and Y. 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
[78]	Z. Zhang, “Jittered sampling in linear canonical domain,” IEEE Communications Letters, vol.24, no.7, pp.1529–1533, 2020. doi: 10.1109/LCOMM.2020.2988947
[79]	H. Xin, B. Li, and X. Bai, “A novel sub-nyquist FRI sampling and reconstruction method in linear canonical transform domain,” Circuits, Systems, and Signal Processing, vol.40, no.12, pp.6173–6192, 2021. doi: 10.1007/s00034-021-01759-w
[80]	K. K. Sharma and S. D. Joshi, “Uncertainty principle for real signals in the linear canonical transform domains,” IEEE Transactions on Signal Processing, vol.56, no.7, pp.2677–2683, 2008. doi: 10.1109/TSP.2008.917384
[81]	J. Zhao, R. Tao, Y. Li, et al., “Uncertainty principles for linear canonical transform,” IEEE Transactions on Signal Processing, vol.57, no.7, pp.2856–2858, 2009. doi: 10.1109/TSP.2009.2020039
[82]	G. Xu, X. Wang, and X. Xu, “On uncertainty principle for the linear canonical transform of complex signals,” IEEE Transactions on Signal Processing, vol.58, no.9, pp.4916–4918, 2010. doi: 10.1109/TSP.2010.2050201
[83]	J. Zhao, R. Tao, and Y. Wang, “On signal moments and uncertainty relations associated with linear canonical transform,” Signal Processing, vol.90, no.9, pp.2686–2689, 2010. doi: 10.1016/j.sigpro.2010.03.017
[84]	P. Dang, G. Deng, and T. Qian, “A tighter uncertainty principle for linear canonical transform in terms of phase derivative,” IEEE Transactions on Signal Processing, vol.61, no.21, pp.5153–5164, 2013. doi: 10.1109/TSP.2013.2273440
[85]	Y. Yang and K. I. Kou, “Uncertainty principles for hyper complex signals in the linear canonical transform domains,” Signal Processing, vol.95, pp.67–75, 2014. doi: 10.1016/j.sigpro.2013.08.008
[86]	Z. Zhang, “Tighter uncertainty principles for linear canonical transform in terms of matrix decomposition,” Digital Signal Processing, vol.69, pp.70–85, 2017. doi: 10.1016/j.dsp.2017.06.016
[87]	Z. Zhang, “Uncertainty principle for linear canonical transform using matrix decomposition of absolute spread matrix,” Digital Signal Processing, vol.89, pp.145–154, 2019. doi: 10.1016/j.dsp.2019.03.015
[88]	Q. Feng, B. Li, and J. M. Rassias, “Weighted Heisenberg-Pauli-Weyl uncertainty principles for the linear canonical transform,” Signal Processing, vol.165, pp.209–221, 2019. doi: 10.1016/j.sigpro.2019.07.008
[89]	J. J. Ding and S. C. Pei, “Heisenberg's uncertainty principles for the 2-D nonseparable linear canonical transforms,” Signal Processing, vol.93, no.5, pp.1027–1043, 2013. doi: 10.1016/j.sigpro.2012.11.023
[90]	Z. Zhang, “Uncertainty principle for real functions in free metaplectic transformation domains,” Journal of Fourier Analysis and Applications, vol.25, no.6, pp.2899–2922, 2019. doi: 10.1007/s00041-019-09686-w
[91]	Z. Zhang, “Generalized balian-low theorem associated with the linear canonical transform,” Results in Mathematics, vol.75, no.3, article no.129, 2020. doi: 10.1007/s00025-020-01255-8
[92]	S. C. Pei and J. J. Ding, “Relations between fractional operations and time-frequency distributions, and their applications,” IEEE Transactions on Signal Processing, vol.49, no.8, pp.1638–1655, 2001. doi: 10.1109/78.934134
[93]	W. Zhang, R. Tao, and Y. Wang, “Linear canonical S transform,” Chinese Journal of Electronics, vol.21, no.1, pp.63–66, 2011.
[94]	Q. Xiang and K. Qin, “A time-frequency analysis method based on linear canonical transform and short-time Fourier transform,” Acta Electronica Sinica, vol.39, no.7, pp.1508–1513, 2011.
[95]	L. Liu, M. Luo, and L. Lai, “Instantaneous frequency estimation based on the Wigner-ville distribution associated with linear canonical transform (WDL),” Chinese Journal of Electronics, vol.27, no.1, pp.123–127, 2018. doi: 10.1049/cje.2017.07.009
[96]	Y. Zhang and B. Li, “ϕ -linear canonical analytic signals,” Signal Processing, vol.143, pp.181–190, 2018. doi: 10.1016/j.sigpro.2017.09.008
[97]	J. Shi, X. Liu, X. Fang, et al., “Linear canonical matched filter: Theory, design, and applications,” IEEE Transactions on Signal Processing, vol.66, no.24, pp.6404–6417, 2018. doi: 10.1109/TSP.2018.2877193
[98]	W. Gao and B. Li, “The octonion linear canonical transform: Definition and properties,” Signal Processing, vol.188, article no.108233, 2021. doi: 10.1016/j.sigpro.2021.108233
[99]	D. Wei and H. Hu, “Theory and applications of short-time linear canonical transform,” Digital Signal Processing, vol.118, article no.103239, 2021. doi: 10.1016/j.dsp.2021.103239
[100]	H. Bu, X. Bai, and R. Tao, “Compressed sensing SAR imaging based on sparse representation in fractional Fourier domain,” Science China Information Sciences, vol.55, no.8, pp.1789–1800, 2012. doi: 10.1007/s11432-012-4607-6
[101]	L. Zhao, J. T. Sheridan, and J. J. Healy, “Unitary algorithm for nonseparable linear canonical transforms applied to iterative phase retrieval,” IEEE Signal Processing Letters, vol.24, no.6, pp.814–817, 2017. doi: 10.1109/LSP.2017.2684829
[102]	Z. Zhang and M. 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
[103]	Z. Zhang, “Unified Wigner-Ville 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
[104]	Z. 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
[105]	H. Xin and B. Li, “On a new Wigner-Ville distribution associated with linear canonical transform,” EURASIP Journal on Advances in Signal Processing, vol.2021, no.1, article no.56, 2021. doi: 10.1186/s13634-021-00753-3
[106]	Y. Guo and bing‐zhao Li, “Blind image watermarking method based on linear canonical wavelet transform and QR decomposition,” IET Image Processing, vol.10, no.10, pp.773–786, 2016. doi: 10.1049/iet-ipr.2015.0818
[107]	Y. G. Su, C. Tang, B. Y. Li, et al., “Optical colour image watermarking based on phase-truncated linear canonical transform and image decomposition,” Journal of Optics, vol.20, no.5, article no.055702, 2018. doi: 10.1088/2040-8986/aabbee
[108]	X. Chen, J. Guan, N. Liu, et al., “Detection of a low observable sea-surface target with micromotion via the radon-linear canonical transform,” IEEE Geoscience and Remote Sensing Letters, vol.11, no.7, pp.1225–1229, 2014. doi: 10.1109/LGRS.2013.2290024
[109]	X. Chen, J. Guan, Y. Huang, et al., “Radon-linear canonical ambiguity function-based detection and estimation method for marine target with micromotion,” IEEE Transactions on Geoscience and Remote Sensing, vol.53, no.4, pp.2225–2240, 2015. doi: 10.1109/TGRS.2014.2358456
[110]	M. G. Amin, “Minimum variance time-frequency distribution kernels for signals in additive noise,” IEEE Transactions on Signal Processing, vol.44, no.9, pp.2352–2356, 1996. doi: 10.1109/78.536695
[111]	S. B. Hearon and M. G. Amin, “Minimum-variance time-frequency distribution kernels,” IEEE Transactions on Signal Processing, vol.43, no.5, pp.1258–1262, 1995. doi: 10.1109/78.382412
[112]	L. Stankovic and V. Ivanovic, “Further results on the minimum variance time-frequency distribution kernels,” IEEE Transactions on Signal Processing, vol.45, no.6, pp.1650–1655, 1997. doi: 10.1109/78.600007
[113]	A. Mefleh, Contributions to Extreme Value Theory: Trend Detection for Heteroscedastic Extremes, Universite Bourgogne Franche-Comte, Besancon, France, 2018.
[114]	P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed., Academic Press, 1984.
[115]	S. Vadrevu and M. S. Manikandan, “A robust pulse onset and peak detection method for automated PPG signal analysis system,” IEEE Transactions on Instrumentation and Measurement, vol.68, no.3, pp.807–817, 2019. doi: 10.1109/TIM.2018.2857878
[116]	Y. Wang, B. Xu, G. Sun, et al., “A two-phase differential evolution for uniform designs in constrained experimental domains,” IEEE Transactions on Evolutionary Computation, vol.21, no.5, pp.665–680, 2017. doi: 10.1109/TEVC.2017.2669098
[117]	K. I. Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Cham: Springer International Publishing, 2018.
[118]	L. Stankovic, S. Stankovic, and M. Dakovic, “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