Fourier analysis correlation

This book is organized into five sections (1) Theory, (2) Columns, Instrumentation, and Methods, (3) Life Science Applications, (4) Multidimensional Separations Using Capillary Electrophoresis, and (5) Industrial Applications. The first section covers theoretical topics including a theory overview chapter (Chapter 2), which deals with peak capacity, resolution, sampling, peak overlap, and other issues that have evolved the present level of understanding of multidimensional separation science. Two issues, however, are presented in more detail, and these are the effects of correlation on peak capacity (Chapter 3) and the use of sophisticated Fourier analysis methods for component estimation (Chapter 4). Chapter 11 also discusses a new approach to evaluating correlation and peak capacity. 

Whenever the frequency / becomes close to a strong periodic component of the measured signal, the terms in parentheses tend to add up and the periodogram shows a power peak around /. Between these peaks, the terms are not correlated, their sign and amplitude tend to be random, and the sum will be small. Still with reference to Fourier analysis, it is common practice to plot the power P(f) as AS(/)/2. 

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. 

Several lines of evidence indicate that active motility of tumor cells is required in the metastatic process. The metastatic potential of Dunning R-3327 rat prostatic adenocarcinoma sublines best correlates with their ability to move on a substrate, as assessed either by a visual grading system of time-lapse videomicroscopy, or by a computerized mathematical system using spatial temporal Fourier analysis (Mohler, 1993). Similarly, the high metastatic capacity of a v-fos-transfected rat fibrosarcoma cell line corre-... 

Partin, A. W., Schoeniger, J. S., Mohler, J. L. and Coffey, D. S. (1989). Fourier analysis of cell motility correlation of motility with metastatic potential. Proc. Natl. Acad. Sci. USA 86, 1254-1258. 

The phenomenon of EXAFS has been known for a considerable time (see ref. 126) but it has been applied to obtain structural information within the last decade only. From equation (2) it is seen that neighbour separation depends on the phase of the EXAFS oscillations, while the co-ordination number Nj and thermal correlation factor (Tj depend on the signal amplitude. In 1971 it was shown by Sayers, Stern, and Lytle that an appropriate Fourier analysis of the data gives a radial structure factor (j) R) from which one can locate the positions of the atoms surrounding the atom which absorbs the X-ray photon (for detailed discussion see refs. 123, 128—130). A second method of data analysis, involving curve fitting techniques, has been used also. ... 

These correlations constitute a fundamental difference of any time-frequency (or scale) resolved analysis to time independent Fourier analysis, where, neighboring frequencies are asymptotically uncorrelated. 

Fourier analysis of neutron diffraction data (time-of-flight method) was also employed by Etherington et al. (1984b) to probe the structure of the barium dizirconate glass and the combination of the X-ray and neutron methods allowed an assignment of the different peaks in the total correlation function with a good degree of... 

The correlation will be maximal if one signal can be displaced with respect to the other until they fluctuate together. The correlation function c(t) will be a more or less noisy sine wave symmetrical around t = 0. The decay of the amplitude envelope from t = 0 indicates the degree of correlation the slower the decay, the higher the correlation. If fi(t) = f2(t), autocorrelation is done by delaying a copy of the function itself and perform the integration of Eq. 8.34. The process will be much the same as a Fourier analysis, a search for periodicity. 

Bispectral analysis. Instead of phase spectra obtained in Fourier analysis where the phase relates to the start of the epoch, the bispectrum correlates the phase between different frequency components. Used in EEG. 

Van Veen and De Loos-Vollebregt reviewed various chemometric procedures which had been developed during the last decade for ICP-OES analysis. In these procedures, data reduction by techniques such as digital filtering, numerical derivatives, Fourier transforms, correlation methods, expert systems, fuzzy logic, neural networks, PCA, PLS, projection methods, Kalman filtering, MLR and generalised standard additions were discussed. 

The general properties of Fourier analysis tell us that the asymptotic trend, at high q, of the scattering intensity I q) is connected to the behavior of the y r) function at smaU r. The correlation fimction y(r), for two electron density systems, can be approximated at small r by (Porod, 1982) ... 

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