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Cao, Yu-Jing; Li, Bing-Zhao; Li, Yong-Gang; Chen, Yi-Hong

Logarithmic uncertainty relations for odd or even signals associate with Wigner-Ville distribution. (English) Zbl 1346.94018

Circuits Syst. Signal Process. 35, No. 7, 2471-2486 (2016).
Summary: Heisenberg’s uncertainty relation is a basic principle in the applied mathematics and signal processing community. The logarithmic uncertainty relation, which is a more general form of Heisenberg’s uncertainty relation, describes the relationship between a function and its Fourier transform. In this paper, we consider several logarithmic uncertainty relations for a odd or even signal \(f(t)\) related to the Wigner-Ville distribution and the linear canonical transform. First, the logarithmic uncertainty relations associated with the Wigner-Ville distribution of a signal \(f(t)\) based on the Fourier transform are obtained. We then generalize the logarithmic uncertainty relation to the linear canonical transform domain and derive a number of theorems relating to the Wigner-Ville distribution and the ambiguity function; finally, the logarithmic uncertainty relations are obtained for the Wigner-Ville distribution associated with the linear canonical transform. We present an example in which the theorems derived in this paper can be used to provide an estimation for a practical signal.

Cited in 3 Documents

MSC:

94A12 Signal theory (characterization, reconstruction, filtering, etc.)
94A11 Application of orthogonal and other special functions

Keywords:

logarithmic uncertainty relation; Wigner-Ville distribution; linear canonical transform; Fourier transform
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References:

[1] R.F. Bai, B.Z. Li, Q.Y. Cheng, Wigner-Ville distribution associated with the linear canonical transform. J. Appl. Math. 2012, 1-14 (2012) · Zbl 1251.94012
[2] W. Beckner, Pitt’s inequality and the uncertainty principle. Proc. Am. Math. Soc. 123, 1897-1905 (1995) · Zbl 0842.58077
[3] B. Boashash, Time Frequency Analysis (Elsevier, Amsterdam, 2003) · Zbl 1041.94510
[4] W.T. Che, B.Z. Li, T.Z. Xu, The ambiguity function associated with the liner canonical transform. EURASIP J. Adv. Signal Process 2012(1), 138 (2012)
[5] G. Chen, J. Chen, G.M. Dong, Chirplet Wigner-Ville distribution for time-frequency representation and its application. Mech. Syst. Signal Process. 41, 1-13 (2013) · doi:10.1016/j.ymssp.2013.08.010
[6] L. Cohen, Time-Frequency Analysis (Prentice-Hall, New York, 1995)
[7] L. Debnath, Recent developments in the Wigner-Ville distribution and time-frequency signal analysis. Proc. Indian Natl. Sci. Acad. Phys. Sci. 68, 35-56 (2002) · Zbl 1229.94016
[8] J.J. Ding, S.C. Pei, Heisenberg’s uncertainty principles for the 2-D nonseparable linear canonical transforms. Signal Process. 93, 1027-1043 (2013) · doi:10.1016/j.sigpro.2012.11.023
[9] D. Ding, Z. Wang, B. Shen, H. Dong, Envelope-constrained H filtering with fading measurements and randomly occurring nonlinearities: the finite horizon case. Automatica 55, 37-45 (2015) · Zbl 1378.93125 · doi:10.1016/j.automatica.2015.02.024
[10] G.B. Folland, A. Sitaram, The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3, 207-238 (1997) · Zbl 0885.42006 · doi:10.1007/BF02649110
[11] R.C. Harney, A method for obtaining force law information by applying the Heisenberg uncertainty principle. Am. J. Phys. 41, 67-70 (1973) · doi:10.1119/1.1987122
[12] A. K. Khitrin, A quantitative aspect of the Heisenberg uncertainty (2012, preprint). arXiv:1206.1576
[13] Y.G Li, B.Z Li, H.F. Sun, Uncertainty principles for Wigner-Ville distribution associated with the linear canonical transforms. Abstr. Appl. Anal. 2014, 1-9 (2014) · Zbl 1472.42009
[14] C.P. Li, B.Z. Li, T.Z. Xu, Approximating bandlimited signals associated with the LCT domain from nonuniform samples at unknown locations. Signal Process. 92, 1658-1664 (2012) · doi:10.1016/j.sigpro.2011.12.024
[15] A. Ouakasse, G. Melard, On-line estimation of ARMA models using Fisher-scoring. Syst. Sci. Control Eng. Open Access J. 2(1), 406-432 (2014) · doi:10.1080/21642583.2014.912572
[16] S.C. Pei, J.J. Ding, Relations between fractional operations and time-frequency distributions, and their applications. IEEE Trans. Signal Process. 49, 1638-1655 (2001) · Zbl 1369.94258 · doi:10.1109/78.934134
[17] B. Shen, Z. Wang, H. Shu, G. Wei, Robust H finite-horizon filtering with randomly occurred nonlinearities and quantization effects. Automatica 46(11), 1743-1751 (2010) · Zbl 1218.93103 · doi:10.1016/j.automatica.2010.06.041
[18] K.K. Sharma, S.D. Joshi, Uncertainty principles for real signals in linear canonical transform domains. IEEE Trans. Signal Process. 56, 2677-2683 (2008) · Zbl 1390.94407 · doi:10.1109/TSP.2008.917384
[19] E.K. Stelzer, S. Grill, The uncertainty principle applied to estimate focal spot dimensions. Opt. Commun. 173, 51-56 (2000) · doi:10.1016/S0030-4018(99)00644-6
[20] K.B. Wolf, Integral Transforms in Science and Engineering (Plenum Press, New York, 1979) · Zbl 0409.44001 · doi:10.1007/978-1-4757-0872-1
[21] L. Weiss, Wavelets and wideband correlation processing. IEEE Signal Process. Mag. 11(1), 13-32 (1994) · doi:10.1109/79.252866
[22] D. Wei, Q. Ran, Y. Li, Reconstruction of band-limited signals from multichannel and periodic nonuniform samples in the linear canonical transform domain. Opt. Commun. 284, 4307-4315 (2011) · doi:10.1016/j.optcom.2011.05.010
[23] T.Z. Xu, B.Z. Li, Linear Canonical Transform and Its Applications (Science Press, Beijing, 2013)
[24] G.L. Xu, X.T. Wang, X.G. Xu, Generalized Hilbert transform and its properties in 2D LCT domain. Signal Process. 89, 1395-1402 (2009) · Zbl 1178.94039 · doi:10.1016/j.sigpro.2009.04.038
[25] G.L. Xu, X.T. Wang, X.G. Xu, Three uncertainty relations for real signals associated with linear canonical transform. IET Signal Proc. 3, 85-92 (2009) · doi:10.1049/iet-spr:20080019
[26] G.L. Xu, X.T. Wang, X.G. Xu, Uncertainty inequalities for linear canonical transform. IET Signal Proc. 3, 392-402 (2009) · doi:10.1049/iet-spr.2008.0102
[27] J. Zhao, R. Tao, Y. Wang, On signal moments and uncertainty relations associated with linear canonical transform. Signal Process. 90, 2686-2689 (2010) · Zbl 1194.94162 · doi:10.1016/j.sigpro.2010.03.017
[28] J. Zhao, R. Tao, Y.L. Li, Y. Wang, Uncertainty principles for linear canonical transform. IEEE Trans. Signal Process. 57, 2856-2858 (2009) · Zbl 1391.44002 · doi:10.1109/TSP.2009.2020039
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