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The Propagation of Noise Fields in a Dispersive Medium

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Classical, Semi-classical and Quantum Noise

Abstract

Our aim is to show how a noise field propagates in a medium that has dispersion and attenuation. For example, suppose noise with known statistical properties is generated at a particular spatial point and we ask for its statistical properties at other spatial points at a later time. We show that phase space methods are particularly effective and intuitive to study such problems. We deal with fields and stochastic processes that are not necessarily stationary either locally or globally. In the next section, we review deterministic wave propagation and we also introduce a new method for studying propagation by way of the Wigner distribution. Subsequently, we discuss the issue as to how to handle nonstationary stochastic processes using the Wigner spectrum. After these issues we address the propagation of noise fields in a dispersive deterministic medium. By a deterministic medium we mean that the medium has no random aspects and that the random aspects come in only in the initial generation of the noise field.

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Notes

  1. 1.

    All integrals go from − ∞ to ∞ unless otherwise noted.

References

  1. Cohen L (1989) Time-frequency distributions – A review. Proc IEEE 77:941–981

    Article  Google Scholar 

  2. Cohen L (1995) Time-frequency analysis. Prentice-Hall, NJ

    Google Scholar 

  3. Cohen L (2001) Wigner distribution and pulse propagation. Proc SPIE 4474:20–24

    Article  Google Scholar 

  4. Cohen L, Loughlin P (2007) Dispersion its effects, and compensation. In: Sadjadi F (ed) Physics of automatic target recognition. Springer, Berlin

    Google Scholar 

  5. Cohen L, Loughlin P, Okopal G (2008) Exact and approximate moments of a propagating pulse. J Mod Optic 55:3349–3358

    Article  Google Scholar 

  6. Galleani L, Lo Presti L, Fosson S (2003) The Wigner spectrum and its application to stochastic processes. IEEE-EURASIP NSIP 03, 8–11 June 2003, Grado, Italy

    Google Scholar 

  7. Galleani L, Cohen L (2002) The Wigner distribution for classical systems. Phys Lett A 302:149–155

    Article  MathSciNet  Google Scholar 

  8. Galleani L, Cohen L (2003) Time-frequency Wigner distribution approach to differential equations. In: Barner K, Arce G (eds.) Nonlinear signal and image processing: theory, methods, and applications. CRC Press, Boca Raton

    Google Scholar 

  9. Galleani L, Cohen L, Suter B (2006) Locally stationary noise and random processes. In: Gabriel Cristóbal (ed.) Information optics. AIP, New York, pp 514–519

    Google Scholar 

  10. Galleani L, Cohen L, Suter B (2007) Local stationarity and time-frequency distributions. Proc SPIE 6313:63130Z-1-5

    Google Scholar 

  11. Jackson JD (1992) Classical electrodynamics. Wiley, New York

    MATH  Google Scholar 

  12. Lighthill J (1978) Waves in fluids. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  13. Morse PH and Ingard KU (1968) Theoretical acoustics. McGraw-Hill, New York

    Google Scholar 

  14. Tolstoy I, Clay CS (1966) Ocean acoustics: theory and experiment in underwater sound. Acoustical Society of America, New York

    Google Scholar 

  15. Whitham GB (1974) Linear and nonlinear waves. Wiley, New York

    MATH  Google Scholar 

  16. Graff K (1975) Wave motion in elastic solids. Oxford University Press, Oxford

    MATH  Google Scholar 

  17. Mark WD (1970) Spectral analysis of the convolution and filtering of non-stationary stochastic processes. J Sound Vib 11:19–63

    Article  Google Scholar 

  18. Martin W (1982) Time-frequency analysis of random signals. In: Proc ICASSP ’82, pp 1325—1328

    Google Scholar 

  19. Martin W, Flandrin P (1985) Wigner–Ville spectral analysis of nonstationary processes. IEEE Trans Acoust Speech Signal Process 33:1461–1470

    Article  Google Scholar 

  20. Loughlin P, Cohen L (2005) A Wigner approximation method for wave propagation. J Acoust Soc Am 118(3):1268–1271

    Article  Google Scholar 

  21. Loughlin P, Cohen L (2004) Phase-space approach to wave propagation with dispersion and damping. Proc SPIE 5559:221–231

    Article  Google Scholar 

  22. Silverman RA (1957) Locally stationary random processes. IRE Trans Inf Theory 3:182–187

    Article  Google Scholar 

  23. Wigner EP (1932) On the quantum correction for thermodynamic equilibrium. Phys Rev 40:749–759

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Physics, Hunter College of the City University of New York, New York City, USA

    Leon Cohen

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  1. Leon Cohen
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Correspondence to Leon Cohen .

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Editors and Affiliations

  1. Hunter College & Graduate Center, City University of New York, Park Ave. 695, New York, 10021, New York, USA

    Leon Cohen

  2. Dept. Electrical Engineering, Princeton University, Olden Street, Princeton, 08544, New Jersey, USA

    H. Vincent Poor

  3. Mechanical & Aerospace, Engineering Dept., Princeton University, Olden Street, Princeton, 08544, New Jersey, USA

    Marlan O. Scully

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Cohen, L. (2012). The Propagation of Noise Fields in a Dispersive Medium. In: Cohen, L., Poor, H., Scully, M. (eds) Classical, Semi-classical and Quantum Noise. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6624-7_3

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  • DOI: https://doi.org/10.1007/978-1-4419-6624-7_3

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  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-6623-0

  • Online ISBN: 978-1-4419-6624-7

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