Skip to main content

The Spread of a Noise Field in a Dispersive Medium


We discuss the production of induced noise by a pulse and the propagation of the noise in a dispersive medium. We present a simple model where the noise is the sum of pulses and where the mean of each pulse is random. We obtain explicit expressions for the standard deviation of the spatial noise as a function of time. We also formulate the problem in terms of a time-frequency phase space approach and in particular we use the Wigner distribution to define the spatial/spatial-frequency distribution.

1. Introduction

In many situations noise is induced by a pulse due to the scattering of the pulse from many sources. To take a specific example, consider a pulse that hits a school of fish; each fish scatters the wave and the acoustic field seen at an arbitrary point is the sum of the waves received from each fish; the sum of which is noise like. Another example is the creation of a set of bubbles by a propeller in a finite region in space. When each bubble explodes it produces a wave and the acoustic pressure seen is the sum of the waves produced by each bubble. Of course this is a simplified view since there could be many other effects such as multiple scattering. Now as time evolves the noise field is propagating and can be changing in a significant way if the medium has dispersion. Our aim here is to investigate how a noise field which is composed of a group of pulses behave as it is propagating and in particular we want to investigate the spreading of the field as a function of time. Suppose we consider a space-time signal composed of the sum of elementary signal,


where is a deterministic function and and are random parameters. We have put in an overall for normalization convenience. The production of noise by expressions like (1) are sometimes called FOM models of noise production [17]. Here, we consider the case where the only random variables are the means of each of the elementary signals and where all are the same. Hence we write


where are the means of the elementary signals, assuming that the mean of is zero. The approach we take is the following. At time we form an ensemble of signals


We then evolve into in a medium with dispersion giving (2) and calculate the appropriate ensemble averaged moments of

For the elementary signal we define the moments in the standard way


Now if there are random parameters as for example the means then use to signify ensemble averaging. In particular


Note that in general it is not the case that the ensemble averaging and ordinary averaging can be interchanged. Which is to be used depends on the quantity we are considering. In this paper we will assume that


As to the means we assume they are taken from a distribution and we define the ensemble mean and standard deviation by


2. Pulse Propagation

We briefly review some of our previous results regarding pulse propagation in a dispersive medium. We consider one mode and also assume that the dispersion relation is real, which means there is no attenuation. The solution for each mode is [813],


where is the initial spatial spectrum


The pulse is normalized so that


If we define


then and form Fourier transform pairs for all time and hence the spatial moments can be obtained by way of


where is the position operator in the representation


The first two moments and standard deviation are


These expressions have been explicitly obtained [14, 15] to give




is the group velocity and where


2.1. An Exactly Solvable Case

An interesting and exactly solvable example that we will use is where the dispersion relation is


and the initial pulse is taken to be


At the means and standard deviations of and are


Using (17)–(21), one obtains that


and these yield


3. Spread of the Wave Group

For convenience we repeat the basic equations producing the noise field


We normalize to one by considering


Now taking the ensemble average we have


and therefore we take so that


In the above we have assumed that and are independent.

Now consider the mean


The ensemble average is




where is given by (7).

For the calculation of the first few steps as above lead to


which gives


The standard deviation is therefore


which, combining (33) and (35) leads to


where is given by (9). Also, noting that


one has


Further, if we use (19) we then have


4. The Spread in Wave Number

We now consider the spatial spectrum. For the total wave we define


and for the elementary wave we define


The standard wave number moments are defined by


Substituting into (42), we have




Now consider the ensemble average of the wave number


and hence




and further


Thus we see that the ensemble of the wave number moments are the moments of the individual pulse. This is the case because our model deals with random spatial translations only.

4.1. Example

For the example we described in Section 2.1 we substitute (26) into (33) and (39) to obtain


Also using (40) we have


To be concrete we now take a particular distribution for


in which case


and therefore we have


5. Wigner Spectrum Approach

We now show that an effective method to study these types of problems is using phase-space methods. The advantage is that one can study nonstationary noise in a direct manner. Suppose we have a random function one can think of a particular realization and substitute into the Wigner distribution and then take the ensemble average of it [1619]


is called the Wigner spectrum and satisfies the marginal conditions


where is the Fourier transform of As standard we define the autocorrelation function by way of


and hence the Wigner spectrum can be written as


Taking the inverse


and letting and we also have that


However in this paper we are dealing with spatial noise and hence we have a spatial random function We define the spatial Wigner spectrum by


The spatial autocorrelation is defined by


and therefore


To specialize to our case we have


and we can substitute (2) into (65). However, it is more effective if we work in the Fourier domain. The Wigner spectrum can be written as


where is given by (42). Now combining (13) and (46), we have


and therefore


Substituting this into (66), we obtain


Setting we have


and taking the Fourier inverse we have that


Substituting this back into (69), we obtain


This expresses the Wigner spectrum at an arbitrary time given the spectrum at time zero.

One can further simplify by defining the Wigner distribution for each pulse as


Now consider


Now taking the ensemble average we have


and therefore we have that in general


Setting we also have


We now aim at expressing in terms of Using (72), we have that




This allows for the calculation of the Wigner spectrum in a simple and direct manner. We also point out that may be considered as the characteristic function


and hence we may write (79) as


Our previous results can be obtained using this Wigner spectrum. For example, consider the first conditional moment


which evaluates to (33).

6. Conclusion

We emphasize that the above model assumed that the only random variable was the mean of the individual pulses. In a future paper we will consider more general cases and of particular interest is to allow the standard deviation of each pulse to be a random variable. For example instead of (2) we can write


where is inserted for normalization purposes. The combination of and allows us to make both the means and standard deviation of each pulse random variable.

We now discuss our main result, namely, (37) and (40) which we repeat here for convenience


First we point out that one can consdier the case for the ensomble as one particle having an initial mean given and an initial standard deviation

For the case of no dispersion then and therefore


For large times the dominant term is and since the coefficient of is manifestly positive we have that


Hence for large times the spread, is a linear function of time.


  1. Faure P: Theoretical model of reverberation noise. Journal of the Acoustical Society of America 1964, 36: 259-266. 10.1121/1.1918943

    Article  MathSciNet  Google Scholar 

  2. Ol'shveskii VV: Characteristics of Sea Reverberation. Consultants Bureau, New York, NY, USA; 1967.

    Google Scholar 

  3. Middleton D: A statistical theory of reverberation and similar first-order scattered fields, part I. IEEE Transactions on Information Theory 1967, 13: 372-392.

    Article  MathSciNet  MATH  Google Scholar 

  4. Middleton D: A statistical theory of reverberation and similar first-order scattered fields, part II. IEEE Transactions on Information Theory 1967, 13: 393-414.

    Article  MathSciNet  Google Scholar 

  5. Middleton D: A statistical theory of reverberation and similar first-order scattered fields, part III. IEEE Transactions on Information Theory 1972, 18: 35-67. 10.1109/TIT.1972.1054754

    Article  MathSciNet  MATH  Google Scholar 

  6. Middleton D: A statistical theory of reverberation and similar first-order scattered fields, part IV. IEEE Transactions on Information Theory 1972, 18: 68-90. 10.1109/TIT.1972.1054755

    Article  MathSciNet  MATH  Google Scholar 

  7. Cohen L: The history of noise. IEEE Signal Processing Magazine 2005, 22(6):20-45.

    Article  Google Scholar 

  8. Jackson JD: Classical Electrodynamics. John Wiley & Sons, New York, NY, USA; 1992.

    MATH  Google Scholar 

  9. Lighthill J: Waves in Fluids. Cambridge University Press, Cambridge, UK; 1978.

    MATH  Google Scholar 

  10. Morse PH, Ingard KU: Theoretical Acoustics. McGraw-Hill, New York, NY, USA; 1968.

    Google Scholar 

  11. Tolstoy I, Clay CS: Ocean Acoustics: Theory and Experiment in Underwater Sound. Acoustical Society of America; 1966.

    Google Scholar 

  12. Whitham GB: Linear and Nonlinear Waves. John Wiley & Sons, New York, NY, USA; 1974.

    MATH  Google Scholar 

  13. Graff K: Wave Motion in Elastic Solids. Oxford University Press, Oxford, UK; 1975.

    MATH  Google Scholar 

  14. Cohen L: Why do wave packets sometimes contract? Journal of Modern Optics 2002, 49(14-15):2365-2382. 10.1080/0950034021000011248

    Article  MathSciNet  MATH  Google Scholar 

  15. Cohen L, Loughlin P, Okopal G: Exact and approximate moments of a propagating pulse. Journal of Modern Optics 2008, 55(19-20):3349-3358. 10.1080/09500340802428280

    Article  MATH  Google Scholar 

  16. Mark WD: Spectral analysis of the convolution and filtering of non-stationary stochastic processes. Journal of Sound and Vibration 1970, 11(1):19-63. 10.1016/S0022-460X(70)80106-7

    Article  MathSciNet  MATH  Google Scholar 

  17. Martin W: Time-frequency analysis of random signals. Proceedings of the International Conference on Acoustics, Speech and Signal Processing (ICASSP '82), 1982, Paris, France 1325-1328.

    Chapter  Google Scholar 

  18. Martin W, Flandrin P: Wigner-ville spectral analysis of nonstationary processes. IEEE Transactions on Acoustics, Speech, and Signal Processing 1985, 33(6):1461-1470. 10.1109/TASSP.1985.1164760

    Article  Google Scholar 

  19. Cohen L: Time-Frequency Analysis. Prentice-Hall, Upper Saddle River, NJ, USA; 1995.

    Google Scholar 

Download references


This paper is supported by the Office of Naval Research.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Leon Cohen.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Cohen, L. The Spread of a Noise Field in a Dispersive Medium. EURASIP J. Adv. Signal Process. 2010, 484753 (2010).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: