- Research Article
- Open Access
The Spread of a Noise Field in a Dispersive Medium
© Leon Cohen. 2010
- Received: 31 January 2010
- Accepted: 17 May 2010
- Published: 17 June 2010
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.
- Ensemble Average
- Random Parameter
- Elementary Signal
- Dispersive Medium
- Spatial Spectrum
We then evolve into in a medium with dispersion giving (2) and calculate the appropriate ensemble averaged moments of
2.1. An Exactly Solvable Case
In the above we have assumed that and are independent.
where is given by (7).
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.
This expresses the Wigner spectrum at an arbitrary time given the spectrum at time zero.
which evaluates to (33).
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.
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
Hence for large times the spread, is a linear function of time.
This paper is supported by the Office of Naval Research.
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