Autocorrelation and power spectrum

If two dynamic properties are correlated over a period of time, the function that describes such a correlation is called the correlation function (Chapter 5, Appendix B). Let the dynamic property be P t). The value of P t) at t fluctuates about the value at to. The average value of (P) at average time T is defined as 

The measured bulk property P is simply a time average property (P(to, T)). At two different times, t and t -I- x, the property can have different values  

If X is very small, P(f + x) = P(t). If x is very large, there can be no correlation. Only within a certain range of x can there be a correlation between P t) and P(f + x). A measure of this correlation is called the autocorrelation function of P and is defined 

Experimentally, the autocorrelation function is a measure of the similarity between two noise signals, P t) and P(f- -x). If the two signals are completely in phase with each other, then (P(O)P(x)) is large. If they are out of phase with each other, then the autocorrelation function (P(O)P(x)) is small (see Berne and Pecora, 1976). 

The intermediate scattering function S g, t) is the autocorrelation of density p(f), defined as 

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Figures 9.24-9.26 show, respectively, the cross-correlations, autocorrelations, and power spectrum of the detector signals for the 500-pm glass particles at = 2. In Fig. 9.24, the curves for the opposite detector pairs, 9/11 and 10/12, are of particular interest. They exhibit nearly zero values of correlation at zero time lag, indicating significant antisymmetric sloshing motion. The autocorrelations, shown in Fig. 9.25, reveal the existence of both near-periodic and random fluctuations. The power spectrum of the detector signals in Fig. 9.26 shows the dominant fluctuations of the solids motion in the bed. The dominant frequencies for the 500- and 705-mm glass particles at u,/umf = 1.5, 2, 3, and 4 are listed in Table 9.2. The...

In many cases the peaks in the RD curve are composite as in Figure 4. The bond distances may in such cases be better determined by the autocorrelation and power spectrum analysis. ... 

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