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The Spread of a Noise Field in a Dispersive Medium
EURASIP Journal on Advances in Signal Processing volume 2010, Article number: 484753 (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.
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 [1–7]. 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 [8–13],
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
The standard deviation is therefore
which, combining (33) and (35) leads to
where is given by (9). Also, noting that
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
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.
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 [16–19]
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
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
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 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).
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.
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This paper is supported by the Office of Naval Research.
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Cohen, L. The Spread of a Noise Field in a Dispersive Medium. EURASIP J. Adv. Signal Process. 2010, 484753 (2010). https://doi.org/10.1155/2010/484753
- Ensemble Average
- Random Parameter
- Elementary Signal
- Dispersive Medium
- Spatial Spectrum