Wiener-Khintchine theorem function

It is a well known fact, called the Wiener-Khintchine Theorem [gardi85], that the correlation function and power spectrum are Fourier Transforms of one another ... 

Wiener-Khintchine theorem). The right-hand side of this equation is often called the power spectrum. It is given by the autocorrelation function, Eq. 2.55. The Fourier transform of the autocorrelation function is related to the spectral moments,... 

The velocity, which equilibrates at large times, is not an aging variable. Thus, Fourier analysis and the Wiener-Khintchine theorem can be used equivalently to obtain the equilibrium correlation function Cvv(t — t2). 

As stated above, the Langevin force F(t) can be viewed as corresponding to a stationary random process. Clearly, the same is true of the solution v(f) of the generalized Langevin equation (22), an equation which is valid once the limit ti —> —oo has been taken. Thus, Fourier analysis and the Wiener-Khintchine theorem can be used to obtain the velocity correlation function, which only depends on the observation time Cvv(t, t2) = Cvv(t —12). As in the classical case, the velocity does not age. 

Applying the Wiener-Khintchine theorem, one obtains the velocity correlation function as the inverse Fourier transform of Cvv ( ), that is, in terms of the noise spectral density CFF([Pg.299]

Equation (41a) means that the function B( r) is equivalent to the volume integral of the density matrix y(ri, ri) under the condition of r = r - r, and Eq. (41b) means that B(r) is the autocorrelation function of the position wave function (r). The latter is an application of the Wiener-Khintchin theorem (Jennison, 1961 Bracewell, 1965 Champeney, 1973), which states that the Fourier transform of the power spectrum is equal to the autocorrelation function of a function. Equation (41c) implies not only that B(r) is simply the overlap integral of a wave function with itself separated by the distance r (Thulstrup, 1976 Weyrich et al., 1979), but also that the momentum density p(p) and the overlap integral S(r) are a pair of the Fourier transform. The one-dimensional distribution along the z axis, B(0, 0, z), for example, satisfies... 

Now, we consider the important limit of weak laser intensity. In this limit, the Wiener-Khintchine theorem relating the line shape to the one-time correlation function holds. As we shall show now, a three-time correlation function is the central ingredient of the theory of fluctuations of SMS in this limit. In Appendix B, we perform a straightforward perturbation expansion with respect to the Rabi frequency Q in the Bloch equation, Eq. (4.6), to find... 

In this chapter, we developed a stochastic theory of single molecule fluorescence spectroscopy. Fluctuations described by Q are evaluated in terms of a three-time correlation function C iXi, X2, T3) related to the response function in nonlinear spectroscopy. This function depends on the characteristics of the spectral diffusion process. Important time-ordering properties of the three-time correlation function were investigated here in detail. Since the fluctuations (i.e., Q) depend on the three-time correlation function, necessarily they contain more information than the line shape that depends on the one-time correlation function Ci(ti) via the Wiener-Khintchine theorem. 

TJie Fourier transform of the first-order correlation function G r) represents the normalized frequency spectrum of the incident light-wave intensity I (o>) (Wiener-Khintchine theorem) [930, 935]. 

These records have been transformed into the frequency domain by a "Fast-Fourier Transformation" (FFT) and ensemble averaged there. The Fourier Transformation of the resulting array back into the time domain gives the correlation function (4) by the Wiener Khintchine theorem. 

Both types of data analysis are equivalent because the spectral power function Pj (oj) is a Fourier transform of the corresponding autocorrelation function (Wiener-Khintchine-theorem Wiener 1930 Khintchine 1934). In the following, only PCS will be discussed. 

The cross correlation function given in Equation (F6) may be further modified to a form better suited for the two-dimensional correlation analysis with the help of the Wiener-Khintchine theorem [5]. This theorem conveniently relates the cross correlation function with the corresponding Fourier transforms. In the first step, the expression for the dynamic spectrum y(v2, t -I- t) in Equation (F6) is rewritten in terms of the inverse of Fourier transform of y(v2, s )-... 

In general, the energy spectrum is calculated by using the auto-correlation function Rt, (r) based on Wiener-Khintchine s theorem as follows ... 

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 ... 

Theorem 5.3 Wiener-Khinchin Theorem. The autocovariance and the spectral density function are related as follows (Khintchine 1934) ... 

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