Wiener spectrum equation

Two estimates of how well the particle size distribution shows through the computed Wiener spectrum were made. First, the extrapolated value of (0) was determined. This should be proportional to the granularity of the deposit (2, 6), and according to the relation of equation (3), should in turn correlate with dm. Figure 9 shows this comparison, yielding a correlation coefficient of 0.985. Secondly, the spatial frequency at which (v) falls to the extrapolated maximum /w(0) should be correlated with the average particle size of those contributing to the spectrum. 

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,... 

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... 

Qualitative justification for the highly random nature of the power spectrum in a chaotic system is readily given. That is, IT[co, x(0)] is a function of a single trajectory. Since a single trajectory is known to depend sensitively on initial conditions, any such function is expected to show such extreme sensitivity as well. Nonetheless, it is necessary to reconcile this result with standard textbook statements28 of the Wiener-Khinchin theorem,29 which equate the power spectrum and spectral density S(co, M) ... 

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 )-... 

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