Wigner-Ville distribution and ambiguity function of QPFT signals

Mohammad Younus Bhat, Aamir Hamid Dar

Abstract


The quadratic phase Fourier transform(QPFT) has received my attention in recent years because of its applications in signal processing. At the same time the applica- tions of Wigner-Ville distribution (WVD) and ambiguity function (AF) in signal analysis and image processing can not be excluded. In this paper we investigated the Wigner- Ville Distribution (WVD) and ambiguity function (AF) associated with quadratic phase Fourier transform (WVD-QPFT/AF-QPFT). Firstly, we propose the definition of the WVD-QPFT, and then several important properties of newly defined WVD-QPFT, such as nonlinearity, boundedness, reconstruction formula, orthogonality relation and Plancherel formula are derived. Secondly, we propose the definition of the AF-QPFT, and its with classical AF, then several important properties of newly defined AF-QPFT, such as non-linearity, the reconstruction formula, the time-delay marginal property, the quadratic-phase marginal property and orthogonal relation are studied. Further, a novel quadratic convolution operator and a related correlation operator for WVD-QPFT are proposed. Based on the proposed operators, the corresponding generalized convolution, correlation theorems are studied. Finally, a novel algorithm for the detection of linear frequency-modulated(LFM) signal is presented by using the proposed WVD-QPFT and AF-QPFT.


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References


L.P. Castro, M.R. Haque, M.M. Murshed, S. Saitoh, N.M. Tuan, Quadratic Fourier transforms, Ann. Funct. Anal. AFA 5 (2014), no. 1, 10–23.

L.P. Castro, L.T. Minh, N.M. Tuan, New convolutions for quadratic-phase Fourier integral operators and their applications, Mediterr J Math. 15 (2018), no. 13, 1–17. DOI:10.1007/s00009- 017-1063-y

L.P. Castro, L.T. Minh, N.M. Tuan, New convolutions for quadratic-phase Fourier integral operators and their applications, Mediterr. J. Math. 15 (2018), 1–17.

M.Y. Bhat, A.H. Dar, Towards Quaternion Quadratic-phase Fourier Transform, Math. Methods Appl. Sci. (2023). DOI:10.1002/mma.9126

M.Y. Bhat, A H. Dar, k-Ambiguity function in the framework of Offset Linear Canonical Trans- form, Int. J. Wavelets, Multires. Inform. Process (2023). DOI:10.1142/s0219691322500357

A. Prasad, P.B. Sharma, The quadratic-phase Fourier wavelet transform, Math. Meth. Appl. Sci. 43 (2020), no. 3, 1953–1969. DOI:10.1002/mma.6018

M.Y. Bhat, A.H. Dar, D. Urynbassarova, A. Urynbassarova, Quadratic-phase wave packet trans- form, Optik 261 (2022), 169120. DOI:10.1016/j.ijleo.2022.169120

M. Bahri, R. Ashino, R. Vaillancourt, Convolution theorems for quaternion Fourier transform: properties and applications, Abst. Appl Anal. 2013 (2013), Article ID 162769.

M. Bahri, E.S.M. Hitzer, A. Hayashi, R. Ashino, An uncertainty principle for quaternion Fourier transform, Comps. Maths with Appl. 56 (2008), no. 9, 2398–2410.

M. Bahri, Correlation theorem for Wigner-Ville distribution, Far East J. Math. Sci. 80 (2013), no. 1, 123–133.

R.F. Bai, B.Z. Li, Q.Y. Cheng, Wigner-Ville distribution associated with the linear canonical transform, J. Appl. Maths. 2012 (2012), Article ID 740161.

L. Debnath, B.V. Shankara, N. Rao, On new two- dimensional Wigner-Ville nonlinear integral transforms and their basic properties, Int. Trans. Sp. Funct. 21 (2010), no. 3, 165–174.

W.B. Gao, B.Z. Li, Convolution and correlation theorems for the windowed offset linear canon- ical transform, arxiv: 1905.01835v2 [math.GM](2019).

E.M.S. Hitzer, Quaternion Fourier transform on quaternion fields and generalizations, Adv. Appl. Clifford Algs. 17 (2007), no. 3, 497–517.

H.Y. Huo, W.C. Sun, L. Xiao, Uncertainty principles associated with the offset linear canonical transform Mathl. Methods Appl. Scis. 42 (2019), no. 2, 466–474.

K.I. Kou, J.-Yu Ou, J. Morais, On uncertainty principle for quaternionic linear canonical trans- form, Abstr. Appl. Anal. 2013 (2013), Article ID 725952.

K.I. Kou, J. Morais, Y. Zhang, Generalized prolate spheroidal wave functions for offset linear canonical transform in clifford analysis, Math. Methods Appl. Sci. 36 (2013), no. 9, 1028–1041.

Y.G. Li, B.Z. Li, H.F. Sun, Uncertainty principle for Wigner-Ville distribution associated with the linear canonical transform, Abstr. Appl. Anal. 2014 (2014), Article ID 470459.

Y.E. Song, X.Y. Zhang, C.H. Shang, H.X. Bu, X.Y. Wang, The Wigner-Ville distribution based on the linear canonical transform and its applications for QFM signal parameters estimation, J. App. Maths. 2014 (2014), Article ID 516457. DOI: 10.1155/2014/516457

D. Urynbassarova, B.Z. Li, R. Tao, The Wigner-Ville distribution in the linear canonical trans- form domain, IAENG Int. J. Appl. Maths. 46 (2016), no. 4, 559–563.

D. Urynbassarova, B.Zhao, R.Tao, Convolution and Correlation Theorems for Wigner-Ville Distribution Associated with the Offset Linear Canonical Transform, Optik 157 (2018), 455– 466. DOI:10.1016/j.ijleo.2017.08.099

D. Wei, Q. Ran, Y. Li, A convolution and correlation theorem for the linear canonical transform and its application, Circuits Syst. Signal Process. 31 (2012), no. 1, 301–312.

M.Y. Bhat, A.H. Dar, Scaled Wigner Distribution in the Offset Linear Canonical Domain, Optik 262(2022), 169286. DOI:10.1016/j.ijleo.2022.169286

D. Wei, Q. Ran, Y. Li, New convolution theorem for the linear canonical transform and its translation invariance property, Optik 123 (2012), no. 16, 1478–1481.

Z.C. Zhang, Novel Wigner distribution and ambiguity function associated with the linear canon- ical transform, Optik 127 (2015), 995–5012.

Z.C. Zhang, Unified Wigner–Ville distribution and ambiguity function in thelinear canonical transform domain, Optik 114 (2015), 45–60.

J.A. Johnston, Wigner distribution and FM radar signal design, Proc. Inst. Electr. Eng. F Radar Signal Process 136 (1989), 81–88.

M.Y. Bhat, A.H. Dar, Convolution and correlation theorems for Wigner-Ville distribution asso- ciated with the quaternion offset linear canonical transform, Signal, Image and Video Processing 16 (2022), 1235–1242.

M.Y. Bhat, A.H. Dar, Quadratic-phase scaled Wigner distribution: convolution and correlation, Signal Image and Video Processing 17 (2023), 2779–2788. DOI:10.1007/s11760-023-02495-1

M.Y. Bhat, I.B. Almanjahie, A.H. Dar, J.G. Dar, Wigner-Ville Distribution and Ambiguity Function Associated with the Quaternion Offset Linear Canonical Transform, Demonstratio Mathematica, 55 (2022), no. 1. DOI:10.1515/dema-2022-0175

A.H. Dar, M.Y. Bhat, Scaled Ambiguity function and Scaled Wigner Distribution for LCT Signals, Optik 267 (2022), 169678. DOI:10.1016/j.ijleo.2022.169678

M.Y. Bhat, A.H. Dar, The 2-D Hyper-complex Gabor Quadratic-phase Fourier Transform and Uncertainty Principles, The Journal of Analysis 31 (2023), 243–260. DOI:10.1007/s41478-022- 00445-7

M.Y. Bhat, A.H. Dar, A.A. Bhat, D.K. Jain, Scaled Ambiguity Function Associ- ated with Quadratic-Phase Fourier Transform, In: (M.Y. Bhat (Ed.)) Time Frequency Analysis of Some Generalized Fourier Transforms, IntechOpen (2023). Available at https://www.intechopen.com/chapters/84949. DOI:10.5772/intechopen.108668




DOI: https://doi.org/10.52846/ami.v50i2.1640