Leon Cohen

Abstract:A simple and mathematically tractable model of a nonstationary process is developed.The process is the sum of waves where the parameters of the waves are random.Explicit expressions for the mean and autocorrelation function at each position as a function of time are obtained.In the case of infinite time,the model evolves into a stationary process.The time-frequency distribution at each position is also obtained.An explicit example is given where the initial waves are Gaussian.The case where there is dispersion in the propagation is also discussed.

Keywords:nonstationary process;mean and autocorrelation function;time-frequency distribution;Gaussian waves

1 Introduction

Most signals in nature,deterministic and random,are nonstationary.By nonstationary,we mean that the frequency content is changing in time.Many methods have been developed to study nonstationary deterministic signals,and many examples are easy to construct.The situation is quite different for the case of random signals.We aim to develop a simple model of a pulse train that produces nonstationary random processes.

Consider a space-time signal,V(x,t),composed of the sum of elementary signals,un,

whereunis a deterministic function,andεnare random parameters.These random parameters will be discussed subsequently.All theun’s are taken to have the same functional form,u(x,t),except for the parameters that will be random.Expressions like Eq.(1) are sometimes called FOM(Faure,Ol’shveskii,Middleton) models of noise production [1−3]and have appeared as early as 1900 [4].The pulse train model has been modified in a variety of ways [5].

As standard we define the spatial spectrum,S(k,t),of the elementary signalu(x,t) by

in which case

All integrals range from–∞ to ∞ unless otherwise noted.

For the sake of notational simplicity we shall write the pulse and its spectrum at time zero as

Also we suppress the random parameter notation in Eq.(1) and write

For the time evolution we take the spatial spectrum to evolve by way of

whereS(k) is the spectrum at time zero and whereω(k) is called the dispersion relation.The initial spatial spectrum is obtained from the initial,u(x),by way of

These types of waves arises in the study of many wave equations [6,7].Substituting Eq.(7)into Eq.(3) we have

We can write

whereK(x −x′,t) is the propagator and satisfies

The propagator for Eq.(9) is given by [8,9]

It is a function of only two variables and we generally write

In what follows we take the dispersion relation to be

and hence

Now suppose we consider initial pulses that are of the form

whereais the mean.In general the mean is calculated by

where each pulse is normalized to one

In that case we have

One can also write that

1.1 Random Factors

We take the random factors to be the mean of the initial pulsea,the constant pulse speedc,and thekdependent part of the dispersion relation,η(k).Therefore we write

Alternatively we can writeV(x,t) in terms of the spectrum at zero

1.2 Random Quantities

We now discuss the important random quantities to be calculated.For simplicity we assume thatηn(k) has a parameterγwhich is random.For the mean ofV(x,t) we have

where the ensemble average is done with the probability distribution,P(a,c,γ),of the mean,candγ

The two time autocorrelation function,〈V(x,t1)V ∗(x,t2)〉,is given by

Equivalently we may write the two time autocorrelation function for timetand the difference in time,τ=t2−t1as

forn≠k.

For the square we have

The dispersion of the process is given by

2 Model

In this paper we consider the simplest case where the only random parameter is the initial mean and where there is no dispersion.That is,we take

From Eq.(20) it follows then that

We take the mean to be distributed according to the following probability distribution

That is,att=0,Nwaves are generated at random positionsanchosen from a uniform distribution ranging from−Ato 0.The waves move to the right with a constant group velocity,c.At positionxand timetwe observe the random process given byV(x,t) of Eq.(34).In Appendix A,we show that the mean ofV(x,t) is given by

For the autocorrelation we obtain

Takingτ=0 gives the mean of the square,

The standard deviation of the process at positionxis calculated to be

2.1 Constant Density

Suppose we takeNandAlarge in such a manner that the density,ρ,is constant

in which case we also have that

Then,for the mean we have that

and for the autocorrelation function we obtain

The autocorrelation function depends on bothtandτand therefore the process is non-stationary.This is the case for each position,x.For the square and standard deviation we obtain respectively

Note the simplicity of Eqs.(42) and (45).These can be thought of as a time dependent generalization of Campbell’s theorem [10,11].

2.2 Stationary Process

If we further take the limit for time going to infinity,t→∞,we have

Notice that〈V(x,t)V ∗(x,t+τ)〉is now independent of time and depends only onτand therefore we have a stationary process.Moreover,it is stationary process for every position,x.

For the square we have

and standard deviation is given by

Nowσ2(x,t) is independent of position and time.Eq.(46) and Eq.(49) are analogous to what is usually called Campbell’s theorem but we have derived them for a particular random model.[10,11]

3 Time Varying Spectrum

We now calculate the time varying spectrum at positionx.For the time varying spectrum we use the Wigner distribution approach [12,13].The time-frequency Wigner distribution forV(x,t) at positionx,is given by

For the random case we ensemble average Eq.(50) to obtain [14−16]

whereW(t,ω;x) is called the Wigner spectrum.One may write Eq.(51) as

whereR(t+τ/2,t −τ/2;x) is the autocorrelation function at positionxand is given by

The autocorrelation function may be obtained from the Wigner spectrum by way of,

Substituting Eq.(34) into Eq.(50) we have

In Appendix B we calculate that

The inner integral is the Wigner spectrum of the functionu(z)

Therefore,we have that

ForWnmwe obtain

which may be written as

Therefore

If we apply the limit as in the previous sections,we obtain

which is independent of time as expected.

4 Gaussian Initial Pulse

We now take a specific initial pulse,

For the average,〈V(x,t)〉,we have,from Eq.(42) that

which evaluates to

where e rf(z) is the standard error function

and e rfc(z) is the complimentary error function,

For the autocorrelation function we obtain

Takingτ=0 we have that

and for the standard deviation one has that

5 Dispersion

In the considerations of Sec.2,the shape of the initial pulses were the same and kept their shape as they evolved.We now consider a situation where the shape change with time.We first outline a basic approach.Consider first the propagation of a single pulse where the initial mean position isa.The evolved pulse is then given by

Explicitly,using Eq.(20),we have that

and therefore

The advantage of writing it this way is that now we can immediately use the results of Sec.2,and write that

5.1 Quadratic Dispersion

We now take

in which case

5.2 Gaussian Initial Pulse

As in Sec.4 we take the initial pulse to be

Giving that

which evaluates to

Note that forγ=0 we obtain Eq.(68)

We therefore have

5.3 Calculation Using the Evolved Pulse

It is of interest to obtain〈V(x,t)〉directly from the evolved pulse.That is,we calculate

directly.For the initial pulse given by Eq.(83)we calculate that the initial spatial spectrum is

and therefore

which evaluates to

It is convenient to let

in which case

We note that

Using the facts that

we obtain that

This shows that the pulse is expanding and the standard deviation is given by

Now for an initial pulse centered atawe have

Taking the limit as Eq.(40)

which is the same as Eq.(68).

6 Conclusion

We have developed a simple model of a pulse train that leads to a nonstationary stochastic process.To some extent,the model is realistic since these models have been used to study reverberation noise.Further development would include takingcrandom andγrandom.Takingcrandom would simulate a common phenomenon since it models propagation in a medium where there are fluctuations of the index of refraction,such as in the atmosphere or ocean.

Appendix A

We derive Eq.(36) of the text.Consider the mean ofV(x,t) as given by Eq.(34)

which is Eq.(36) of the text.

For the autocorrelation function we have

Consider now the first term above,Eq.(103)

For the second term we have

which is Eq.(37) of the text.

For the square we takeτ=0 to obtain that

For the standard deviation of the process,we have

which simplifies to

Appendix B

We derive Eq.(59) of the text.Consider firstWnn

TakingP(an) as per Eq.(35) we have

Making the substitutionz=x −ct −a′with dz=−da′results in

The inner integral is the Wigner distribution of the functionu(z)

and therefore we can write

Now considerWnm

Using the same procedure as above we have that

Consider the inner integral.We want to express in terms of a Wigner function.In general

and hence it follows that

Therefore

which is Eq.(59) of the text.