Some Comments on sampling of ergodic process, an Ergodic theorem and turbulent pressure fluctuations

We present a new proof of an Ergodic theorem for Wide-Sense Stationary Random Processes added with a new canonical sampling theorem of mine for finite time duration signals in the frequency domain (periodograms) which is free from the Nyquist interval restriction.We point out the usefulness of such theorem in the context of a model of random vibrations transmission (pressure-explosive fluctuations).Replacement due to inadvertent and wrong acknowledgments


Introduction
A basic problem in the applications of stochastic processes is the estimation of a signal x(t) in the presence of an additive interference f (t) (noise). The available information (data) is the sum S(t) = x(t) + f (t) and the problem is to establish the presence of x(t) or to estimate its form. The solution of this problem depends on the state of our prior knowledge concerning of the noise statistics. One of the main results on the subject is the idea of maximize the output signal-to-noise ratio ( [1]), the matched filter system. One of the most important results on the subject is that if one knows a priori the signal in the frequency domain X(w) and, mostly important, the frequency domain expression for the noise correlation statistics function S f f (w), one has, at least in the theoretical grounds, the exactly expression for the optimum filter transference function H opt (w) = k(X * (w))(S f f (w)) −1 · e −iwt , here t denotes a certain time on the observation interval process [−A, A], supposed to be finite here.
As a consequence, it is important to have estimators and analytical expressions for the correlation function for the noise, specially in the physical situation of finite-time duration noise observation. Another very important point to be remarked is that in most of the cases of observed noise (as in the turbulence research ( [2]); one should consider the noise (at least in the context of a first approximation) as an Ergodic Random Process.
We aim in this note to present in Section 2 [of a more mathematical oriented nature] a rigorous functional analytic proof of an Ergodic Theorem stating the equality of time-averages and Ensenble-averages for wide-sense mean continuous Stationary Random Processes. In Section 3 somewhat of electrical Engineering oriented nature, we present a new approach for sampling analysis of noise, which leads to canonical and invariant analytical expressions for the Ergodic noise correlation function S f f (w) already taking into account with the finite time-observation parameter which explicitly appears in the structure formulae, a new result on the subject, since it does not require the existence of Nyquist critical frequency on the sampling rate, besides of removing, in principle, the aliasing problem in the computer-numerical sampling evaluations.
In Section 4 we present an mathematical-theoretical application of the above theoretical results for a model of Turbulent Pressure fluctuations.

A Rigorous Mathematical proof of the Ergodic theorem for Wide-Sense Stationary Stochastic Process
Let us start our section by considering a wide-sense mean continuous stationary real-valued Here Ω is the event space and dµ(λ) is the underlying probability measure.
It is well-know ( [1]) that one can always represent the above mentioned wide-sense stationary process by means of a unitary group on the Hilbert Space {L 2 (Ω), dµ(λ)}. Namely [in the quadratic-mean sense in Engineering jargon] here we have used the famous spectral Stone-theorem to re-write the associated time-translation unitary group in terms of the spectral process dE(w)X(0), where H denotes the infinitesimal unitary group operator U(t). We have supposed too that the σ-algebra generated by the X(t)process is the whole measure space Ω, and X(t) is a separable process.
Let us, thus, consider the following linear continuous functional on the Hilbert (complete) space {L 2 (Ω), dµ(λ)} -the space of the square integrable random variables on Ω By a straightforward application of the R.A.G.E. theorem ( [3]), namely: Here P Ker(H) is the (ortoghonal projection) on the kernel of the unitary-group infinitesimal generator H (see eq (1)).
By a straightforward application of the Riesz-representation theorem for linear functionals on Hilbert Spaces, one can see that P Ker(H) (X(0))dµ(λ) is the searched time-independent ergodic-invariant measure associated to the ergodic theorem statement, i.e.: For any square In general grounds, for any real bounded borelian function it is expected the result (not proved here) For the auto-correlation process function, we still have the result for the translated time ζ fixed (the lag time) as a direct consequence of eq(1) or the process' stationarity property It is important remark that we still have the probability average inside the ergodic timeaverages eq(4)-eq(6). Let us call the reader attention that in order to have the usual Ergodic like theorem result -without the probability average E on the left-hand side of the formulae, we proceed [as it is usually done in probability text-books ( [1])] by analyzing the probability convergence of the single sample stochastc-variables below [for instance] is a bounded function of the time-lag, or, if the variance below written goes to zero at T → ∞ one has that the random variables as given by eq (7)-eq(8) converge at T → ∞ to the lefthand side of eq(4)-eq(6) and producing thus an ergodic theorem on the equality of ensembleprobability average of the wide sense stationary process {X(t), −∞ < t < ∞} and any of its The above written formulae will be analyzed in next Electrical Engeneering oriented section.

A Sampling Theorem for Ergodic Process
Let us start this section by considering F (w) as an entire complex-variable function such that and such its restriction to real domain w = x + i0 satisfies the square integrable condition By the famous Wienner Theorem (ref. [4]) there exists a function The As much as in the usual proof of the Shanon results eq(15), we introduce eq(16) into eq (14) and by using the Lebesgue convergence theorem, since f (t) ∈ L 1 (−A, A) either, we get the somewhat (canonical) interpolating formula without any Nyquist -like restrictive sampling frequency condition on the periodogran F (w) It is worth call attention that the sampling coefficients as given by eq(17) are exactly the values of the Fourier transform of the signal g(θ) = f (A sen θ) cos θ for − π 2 < θ < θ 2 , i.e.: In the general case of a quadratic mean continuous wide-sense stationary process {X t , 0 < t < ∞} ([1]), we still have the quadratic mean result analogous to the above exposed result for those process with time-finite sampling ([1]) with the spectral process possesing the canonical form (in the quadratic mean sense) Since we have observed the sampling-continuous function {f (t)} only for the time-limited , it apears quite convenient at this point to re-write eq(21) in the frequency domain (as a somewhat generalized process) (see [1]) where the T → ∞ limit is already evaluated It is worth point out that eq(22) has already "built" in its structure, the infinite time-ergodic evalution lim with The above written equations are the main results of this Section 3.
It is worth call the reader attention that after passing the random signal f (t) by a causal linear system with transference function H f f (w) ( [6]), one obtain the output self-correlation function statistics in the standard formulae Finally, let us comment on the evaluation of eq(23) on the time-domain. This step can be implemented throught the use of the well-known formula for Bessel functions (see [8] -eq 6.626), analytically continued in the relevant parameters formulae We have, thus, the result: where with here, the explicitly expressions T n (t) , for 0 < t < 1 and n = 2k + 1 0, for 0 < 1 < t 2(2k + 1) The effective equation governing the "outside" pressure in our model is thus given by (d ≤ with the boundary condition (2 nd Newton's law) Here the beam's deflection W (x, t) is assumed to be given by the beam small linear deflection equation with B denoting the bending rigidity, m the mass per unit lenght of the beam and p(x, z, t) the (supersonic) boundary-layer pressure fluctuation of the exterior medium.
The searched induced pressure p 2 (x, z, t) in the fluid interior medium (0 ≤ z ≤ d) is governed by The solution of eq.(31) and eq.(32) is straightforward obtained in the Fourier domaiñ where the deflection beamW (k, w) is explicitly given bỹ W (k, w) = (P (k, w) −p 2 (k, w, d) At this point, we solve our problem of determining the pressurep 2 (k, w, z) in the interior domain eq.(34)-eq.(35) if one knows the pressurep 2 (k, w, d) on the boundary z = d. Let us, thus, consider the Taylor's serie in the z-variable (0 ≤ z ≤ d) around z = d, namelỹ From the boundary condition eq.(35), we have the explicitly expression for the secondderivative on the depth z ∂p 2 (k, w, z) ∂z z=d = +ρ 2 w 2W (k, w) = + ρ 2 w 2 [P (k, w) −p 2 (k, w, d)] × Bk 4 − mw 2 + The second z-derivative of the interior pressure (and the higher ones!) are easily obtained recursively from the wave equation (34) (k ≥ 0, k ∈ Z + ) and eq(39) Let us finally make the connection of this random transmission vibration model with the section 3 by calling the reader attention that the general turbulent pressure is always assumed to be expressible in a integral form where F (w) is the Fourier Transform of a time-limited finite duration sample function of an Ergodic Process ( [1]) simulating the stochastic-turbulent nature of the pressure field acting on the fluid with the exactly interpolating formulae eq.(17)-eq.(18). Numerical studies of this random vibration transmission will be presented elsewhere.