Cosine function correlation

Conversely, a rf field is totally correlated because it is represented by a sine (or cosine) function and, as a consequence, its value at any time t can be predicted from its value at time zero. The efficiency of a random field at a given frequency co can be appreciated by the Fourier transform of the above correlation function... 

Field-based similarities are usually evaluated by the cosine or correlation function similarity measure employed initially by Carbo and co-workers (67) to compute molecular similarities based upon quantum mechanical wavefunctions. Such a measure, which is usually called a Carbo similarity index, is given by... 

Fig. 5.5 Cosine transform correlation function computed from the SAXS data for a PE-PEE diblock diblock (M = 23kgmor, /pE = 0.49). with an interpretation of the distances based on extrapolation of the small r slope and subsequent maxima and minima (Ryan ei al. 1995). Here am = amorphous (rubbery) and xl = crystalline.

The NMR frequencies at two times separated by the time tm, and thus the corresponding orientations [cf. Eq. (15)] are correlated via cosine functions. The correlation function CSin(fm tp), where the cosine functions are replaced by sine functions, may also be accessible modifying pulse lengths and pulse phases in an appropriate way. This is possible for both CSA and Q interactions. Rotational jumps of the molecules during the mixing time tm lead to 0)>(0) / tm), and hence to a decay of CCOSjSin(fm tp). Therefore, these correlation functions provide access to the details of the molecular reorientation dynamics. 

It has been assumed that the movement correlates to a sine function and the velocity to a cosine function. As this would lead to considerable integration errors the instrument referred to has been provided with a cosine corrector this corrects the measuring signal coming from the photomultiplier after it has been amplified. According to Ebel et al.17), however, the velocity of the oscillating movement corresponds to an entirely different function (formula and figure can be found in the... 

In panel a, p(x)p(0) is a constant plus a cosine function with a period of a. This correlation function is observed when p(x) changes sinusoidally. The Fourier transform converts the constant into d(k) and cos 2irx/a) into Sik-lv/a). In part b, (p(jc)p(0)) has a harmonic sAk = Air/a. The density correlation is slightly distorted from the cosine function. 

The amplitudes of the cosine and sine components, respectively, are given by the trigonometric identities, ) (v) = (v) cos j0(v) and > (v) = y(v) sin / (v). For the purpose of demonstration, the observation interval is set to be sufficiently long compared to the periodicity of the cosine function, that is, - fmin 1 /S. We also focus our attention only to the range of the cross correlation function with a relatively small value of correlation time, T cross correlation function vanishes to zero as t approaches the value... 

The substitution of Equation (F25) into Equation (E6) yields the expression for the cross correlation function between two cosine functions ... 

A correlation analysis is applied to the individual sine and cosine functions in the manner similar to the derivation for Equations (F28) and (F29), which in turn generates a number of synchronous and asynchronous spectra, that is, cospectra 4>s quad-spectra j/, for individual Fourier components with different frequencies s. 

The most prevalent among the similarity coefficients is the so-called cosine similarity index or correlation coefficient. For the field functions discussed in Subheading 2.4. it is usually called the Carbo similarity index, and this nomenclature will be used here as well,... 

Classical force-force correlation functions and their (cosine) Fourier transformations are shown in the left and middle column of Fig. 5 for five different trajectories. Here we have defined a ( function as... 

---