Wiener-Kinchine theorem

From the Wiener-Kinchine theorem, the integrated PSD yields the r.m.s. roughness vs the lenglh/scale / ... 

In order to calculate the spectrum of the scattered light we must use the Wiener-Kinchine theorem. One form of this theorem is... 

If Ej[t) is the amplitude of the scattered field at some time, t, then ,(t + T) is the amplitude of the mattered field at some time t+r later. If one is interested in the frequency spectrum in the range 10 to 10 Hertz then one needs to obtain correlations in the amplitude of the scattered wave fi om approximately 10" to 10" seconds. As we will see later, this can be done experimentally, but now we will show that the autocorrelation function has a simple mathematical function for simple, monodisperse polymer solutions. The use of the Wiener-Kinchine theorem is of prime importance in the theory and readers who are not familiar with this statistical theorem will find a simple explanation and derivation in almost any statistical mechanics text. 

Thus if one wishes to calculate the spectrum of light scattered from moving particles one must do the calculation in three steps (1) Construct Gs(l o, r) the Van Hove space-time correlation function. For many types of motion such a construction is quite straightforward from the physical significance of G (Ro,z). (2) Evaluate the integral, Eq.(39) using Gs(Ro> previously constructed to obtain Co(t) the auto-correlation function. Since this quantity can in fact be measured one may wish to stop here. (3) Using the Wiener-Kinchine theorem, Eq. (36), obtain the spectrum I(co). 

In the theoretical section above we showed how the spectrum of light, I(co) could be obtained from the correlation function Co(t), by means of the Wiener-Kinchine theorem, Eq. (32). For particles undergoing Brownian motion we found that the autocorrelation function was given by... 

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