Autocorrelation stationary process

It is assumed that the noise voltage n(t) is the result of a real stationary process (Davenport and Root, 1958) with zero mean. Because it can be shown that the spectral density function S(f) is the Fourier transform of the autocorrelation function of the noise, it follows that the rms noise is given by... 

As remarked in II.3, strictly stationary processes do not exist in nature, let alone in the laboratory, but they may be approximately realized when a process lasts much longer than the phenomena one is interested in. One condition is that it lasts much longer than the autocorrelation time. Processes without a finite tc never forget that they have been switched on in the past and can therefore not be treated as approximately stationary. 

The diagonal elements represent autocorrelations, the off-diagonal elements are cross-correlations In case of a zero-average stationary process this equation reduces to... 

Correlated Data A random time series with low correlations between observations provides an autocorrelation function as shown in Figure 3.24. In the case of a stationary process of the first order, the function can be described by the following exponential model ... 

In axial flight (e.g. hovering), we have y = 0 and consequently a = 0 in equation (10). Then equation (10) shows that Rw(t,i) = Rw( c), that is, the RFT is a stationary process, and the expression of Rv,(i) agrees in form with that of the wind turbine studies. (For wind turbines Rw(t,T) refers to the horizontal component.) For compound helicopters with rotor off-loading by a fixed wing (y 1.6), the downwash contribution is negligible. We then have b = 0 in equation (10) and the expression reduces to the one given in reference 15. If we further stipulate, as done in references 13 and 14, that the spatial separation r is essentially determined by the X(t2) - X(tx)> then the autocorrelation Rw(t,i) simplifies to the equation... 

Suppose Y(t) is a stationary, zero-average process with a finite autocorrelation time tc then <7(r)2> is independent of time and one has... 

Note that as a consequence of the fact that the process is not stationary 0 this autocorrelation function does not depend on xl — x2 alone, and even contains the total length L. Hence the Wiener-Khinchin theorem does not apply directly yet a similar calculation of the Fourier coefficients An yields < An) = 0 and... 

The oldest and best known example of a Markov process in physics is the Brownian motion.510 A heavy particle is immersed in a fluid of light molecules, which collide with it in a random fashion. As a consequence the velocity of the heavy particle varies by a large number of small, and supposedly uncorrelated jumps. To facilitate the discussion we treat the motion as if it were one-dimensional. When the velocity has a certain value V, there will be on the average more collisions in front than from behind. Hence the probability for a certain change AV of the velocity in the next At depends on V, but not on earlier values of the velocity. Thus the velocity of the heavy particle is a Markov process. When the whole system is in equilibrium the process is stationary and its autocorrelation time is the time in which an initial velocity is damped out. This process is studied in detail in VIII.4. 

Exercise. The autocorrelation function of a stationary Markov process with zero mean is given by... 

Exercise. Apply the result to the harmonic oscillator (1.3) with frequency co2(t) = cog l H- a (t), where t) is a stationary random process with zero mean and autocorrelation time tc. The answer is... 

Assumption (a) implies that V(t) is a stationary Gaussian process. The Langevin equation when solved subject to assumption (b) yields the velocity autocorrelation function... 

A random process is weakly stationary if its mean value and autocorrelation function are independent of r. Thus, for a weakly stationary random process, the mean value is a constant [fJiy r) = fiy] and the autocorrelation function depends only on the spatial lag 6 [e.g., Ryir, r + 6) = Ry d)]. A random process is strongly stationary if the infinite collection of higher order statistical moments and joint moments are space invariant. Most geophysical phenomena are not strongly stationary. However, the random process under study must be at least weakly stationary, otherwise the results of the space- or time-series analysis can be suspect. An extensive treatment of these statistical concepts is available 45, 46). A detailed re-... 

Because we are working with stationary variables, the autocorrelation gives no information on the origin of time, so that it can only depend on the time difference s. The autocorrelation coefficient is the correlation coefficient between the process at time t and f - - s. 

On the basis of the Viner-Hinchin theorem the autocorrelation function of the stationary random process is represented by the Fourier integral ... 

The time distribution of the fluorescence photons emitted by a single dye molecule reflects its intra- and intermolecular dynamics. One example are the quantum jumps just discussed which lead to stochastic fluctuations of the fluorescence emission caused by singlet-triplet quantum transitions. This effect, however, can only be observed directly in a simple fluorescence counting experiment when a system with suitable photophysical transition rates is available. By recording the fluorescence intensity autocorrelation function, i.e. by measuring the correlation between fluorescence photons at different instants of time, a more versatile and powerful technique is available which allows the determination of dynamical processes of a single molecule from nanoseconds up to hundreds of seconds. It is important to mention that any reliable measurement with this technique requires the dynamics of the system to be stationary for the recording time of the correlation function. 

A stationary stochastic process can be characterized by the autocorrelation function of velocity i (r), where r is a time delay. The autocorrelation function measures the persistence of a given value of the random variable concerned. For particle diffusing velocities, when the particle possesses a given y-directed velocity component v t), a short time later it is likely to have a veloeity of similar magnitude and sign, v(t +1). The velocity covariance can be formed for the particle as the mean of the product of the two velocities. 

The spectral density function of the fluctuation can be calculated from the autocorrelation function by the Wiener-Khintchine relation (Wiener, 1930 Khintchine, 1934). The original formulation of the theorem refers to stationary stochastic processes for a possible generalisation see, for example, Lampard, 1954. The relationship connects the autocorrelation function to the spectrum ... 

For a wide-sense stationary noise process, the noise autocorrelation function can be estimated from an initial segment of the noisy signal that contains noise only. If the noise process is not wide-sense stationary, frames of the noisy signal must be classified as in Eq. (19.98), and the autocorrelation function of the noise process must be updated whenever a new noise frame is detected. 

A stochastic process is also characterized by its spectral density, the Fourier transform of its autocorrelation function. The autocorrelation function of a (stationary stochastic process) measures the correlation of the process at different time intervals while the spectral density measures the amplitudes of the component waves of different frequencies. A white noise process has a constant spectral density (i.e., the same amplitude for all frequencies) and the band-limited noise has a frequency band over which the spectral density is nearly constant. 

In the diffusion limit it is foimd that the combined effects of particle inertia and shear flow modify the amphtude and the time-dependence of the particle-velocity autocorrelation functions, a result which is expressed in terms of the Stokes number, St = 7/fi. The shear flow breaks macroscopic time reversibility and stationarity the autocorrelation functions of the particle velocities are stationary and the velocity correlation along the shear is symmetric in the time difference t, but the cross correlation is non-symmetric in t function in the streamwise direction is non-stationary The time decay of the velocity correlation along the flow is not a pure exponential and the imderlying stochastic process is not an Omstein-Uhlenbeck process. 

As said, the Fourier transform of a stationary random process X t) usually does not exist. However, the Fourier transform of the autocorrelation function R x) always exists. To understand what is an autocorrelation function consider a generic random spectrum like that shown in Fig. 8.39 and sample the value of the variable X(f) at two different times ti and 2 = fi -I- t. For digital data the autocorrelation function R x) is a function of the time interval x — t2 — ti defined as... 

However, in real life situations, wind flow is not deterministic but has random fluctuations about a mean value. Hence, this randomness must be included in the model in order to gain better insights on the behavior of system. Therefore, the nondimensional free stream velocity U is assumed to be a stationary Gaussian random process with the autocorrelation function as given in Eq. 55 with oy= 1 and Cq = 1 x 10 This implies a correlation length of 2.628 x 10 s. It must be... 

This relationship shows that the stationary counterpart of the multi-correlated stochastic process is a vector process, N(( ) (of order AT), with orthogonal increments. Furthermore, Gnn(Hermitian matrix function which describes the one-sided PSD function matrix of the so-called embedded stationary counterpart vector process, N(m). After some algebra it can be proved that the autocorrelation function matrix of the zero-mean Gaussian nonstationary random vector process F(t) can be obtained as... 

This set of assumptions on the statistical properties of f(t) determines the statistical properties of the solution v(0 of the stochastic differaitial equation in Equation 1.1, which are summarized saying that v(0 is a Gaussian stationary Markov stochastic process, that is, it is generally not delta-correlated. The specific results that follow from this simple mathanatical model regarding propo ties such as the velocity autocorrelation function, msd, and so on, are reviewed in standard statistical physics textbooks [48]. 

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