The Wigner distribution for classical systems. (English) Zbl 0998.70012
Summary: We present an explicit procedure for obtaining the equation of motion for the Wigner distribution when the underlying governing equation is a linear ordinary or partial differential equation. The cases of constant and variable coefficients are considered.
MSC:
| 70G10 | Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics |
Keywords:
equation of motion; linear ordinary differential equation; linear partial differential equationReferences:
| [1] | Schleich, W. P., Quantum Optics in Phase Space (2001), Wiley: Wiley England · Zbl 0961.81136 |
| [2] | Scully, M. O.; Zubairy, M. S., Quantum Optics (1997), Cambridge University Press: Cambridge University Press Cambridge, England |
| [3] | Wigner, E. P., Phys. Rev., 40, 749 (1932) · Zbl 0004.38201 |
| [4] | Kirkwood, J. G., Phys. Rev., 44, 31 (1933) · Zbl 0007.23501 |
| [5] | Cohen, L., Proc. of the IEEE, 77, 941 (1989) |
| [6] | Cohen, L., Time-Frequency Analysis (1995), Prentice-Hall |
| [7] | Koenig, W.; Dunn, H.; Lacy, L., J. Acoust. Soc. Am., 18, 1, 19 (1946) |
| [8] | Gabor, D., Theory of communication, IEEE J. Comm. Engr., 93, 429 (1946) |
| [9] | Ville, J., Cables et Transmissions, 2A, 61 (1948) |
| [10] | Cohen, L., J. Math. Phys., 7, 781 (1966) |
| [11] | Choi, H. I.; Williams, W. J., IEEE Trans. on Acoust., Speech, Signal Processing, 37, 862 (1989) |
| [12] | Zhao, Y.; Atlas, L. E.; Marks, R. J., IEEE Trans. Acoust., Speech, Signal Processing, 38, 1084 (1990) |
| [13] | Jeong, J.; Williams, W., IEEE Trans. Sig. Proc., 40, 402 (1992) |
| [14] | Sala, R.; Palao, J. P.; Muga, J. G., Phys. Lett. A, 231, 305 (1997) |
| [15] | Sala, R.; Muga, J. G., Phys. Lett. A, 231, 180 (1994) |
| [16] | Amin, M.; Williams, W., IEEE Trans. Signal Processing, 46, 10, 2796 (1998) · Zbl 0978.93080 |
| [17] | Loughlin, P.; Pitton, J.; Atlas, L. E., IEEE Trans. Signal Processing, 41, 750 (1993) · Zbl 0825.94085 |
| [18] | Gaunaurd, G. C.; Strifors, H. C., IEEE Trans. Ultrasonics, Ferroelectrics, and Frequency Control, 4 (1993) |
| [19] | Loughlin, P.; Groutage, D.; Rohrbaugh, R., Proc. IEEE ICASSP’97, 3, 2125 (1997) |
| [20] | Pitton, J.; Atlas, L., IEEE Trans. on Speech and Audio Proc., 2, 4, 554 (1994) |
| [21] | Pielemeier, W. J.; Wakefield, G. H., JASA, 99, 4, 2382 (1996) |
| [22] | Samimi, B.; Rizzoni, G., Proceedings of the IEEE, 84, 1330 (1996) |
| [23] | Atlas, L.; Bernard, G. D.; Narayanan, S. B., Proceedings of the IEEE, 84, 9, 1319 (1996) |
| [24] | Amin, M. G., Proceedings of the IEEE, 74, 1800 (1986) |
| [25] | Pitton, J., J. Franklin Institute, 337, 379 (2000) · Zbl 0988.93081 |
| [26] | Williams, W., Biological applications and interpretations of time-frequency signal analysis, (Time-Frequency and Wavelets in Biomedical Signal Processing (1999), IEEE Press) · Zbl 0951.92016 |
| [27] | Wood, J. C.; Barry, D. T., IEEE Trans. Signal Processing, 42, 8, 2094 (1994) |
| [28] | Gaunaurd, G. C.; Strifors, H. C., Proceedings of the IEEE (Special Issue on Time-Frequency Analysis), 84, 1231 (1996) |
| [29] | Galleani, L.; Lo Presti, L., J. Modern Optics, 49, 3/4, 561 (2002) |
| [30] | Loughlin, P., Special Issue on Applications of Time-Frequency Analysis. Special Issue on Applications of Time-Frequency Analysis, Proc. of the IEEE, 84, 9 (1996) |
| [31] | Moyal, J. E., Proc. Cambridge Philos. Soc., 45, 99 (1949) · Zbl 0031.33601 |
| [32] | L. Galleani, L. Cohen, A detailed analysis was presented at the 31st Winter Colloquium on The Physics of Quantum Electronics, Snowbird, Utah, USA, January 7-11, 2001; L. Galleani, L. Cohen, A detailed analysis was presented at the 31st Winter Colloquium on The Physics of Quantum Electronics, Snowbird, Utah, USA, January 7-11, 2001 |
| [33] | Martin, W.; Flandrin, P., IEEE Trans. Acoust., Speech, Signal Processing, 33, 1461 (1985) |
| [34] | L. Galleani, L. Cohen, Direct time-frequency approach to dynamical systems, IEEE-EURASIP NSIP 2001, June 3-6 2001, Baltimore, MD, USA; L. Galleani, L. Cohen, Direct time-frequency approach to dynamical systems, IEEE-EURASIP NSIP 2001, June 3-6 2001, Baltimore, MD, USA |
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