Autocorrelation function envelope

The angular velocity and angular momentum acfs themselves are important to any dynamical theory of molecular liquids but are very difficult to extract directly from spectral data. The only reliable method available seems to be spin-rotation nuclear magnetic relaxation. (An approximate method is via Fourier transformation of far-infrared spectra.) The simulated torque-on acfs in this case become considerably more oscillatory, and, which is important, the envelope of its decay becomes longer-lived as the field strength increases. This is dealt with analytically in Section III. In this case, computer simulation is particularly useful because it may be used to complement the analytical theory in its search for the forest among the trees. Results such as these for autocorrelation functions therefore supplement our... 

The nature of the excitation has a profound influence on the subsequent relaxation of molecular Uquid systems, as the molecular dynamics simulations show. This influence can be exerted at field-on equiUbrium and in decay transients (the deexdtation effect). GrigoUni has shown that the effect of high-intensity excitation is to slow the time decay of the envelope of such oscillatory functions as the angular velocity autocorrelation function. The effect of high-intensity pulses is the same as that of ultrafast (subpicosecond laser) pulses. The computer simulation by Abbot and Oxtoby shows that... 

When y - A2 the equivalent of the microscopic time y is = y/ -Decoupling effects are present when Uj = F. To obtain an approximate value of F we can use the experimental data as follows. First, we evaluate the value of decay of the oscillation envelopes of the angular velocity autocorrelation function as a function of Equation (14) shows that this is, approximately, a Lorentzian, the linewidth of which provides the approximate expression for F. The agreement with the numerical decoupling effect is quantitatively good when the ratio ai/uf is assumed to be equal to 8.5. Simple Markovian models cannot account for decoupling effects. 

Although the eigenstate-to-eigenstate and autocorrelation function formulations of the spectrum, Ik(w), are mathematically equivalent, they focus attention on complementary features. The most readily interpretable features in the autocorrelation function picture are early-time features (initial decay rate, the times at which partial recurrences occur, the magnitudes of the earliest and largest partial recurrences) which primarily sample the potential surface at the highly localized and a priori known initial position of the wavepacket, I (O), before it has had time to explore the entire dynamically accessible region of the potential surface. This early time information is encoded in the broad envelope (low resolution) of the Ik(u>) spectrum (see Fig. 9.2). 

Modification is performed by separating the harmonics from the spectral envelope, but this is achieved in a way that doesn t perform explicit source/filter separation as with LP analysis. The spectral envelope can be found by a number of numerical techniques. For example, Kain [244] transforms the spectra into a power spectrum and then uses an inverse Fourier transform to find the time domain autocorrelation function. LP analysis is performed on this to give an allpole representation of the spectral envelope. This has a number of advantages over standard LP analysis in that the power spectrum can be weighted so as to emphasise the perceptually important parts of the spectrum. Other techniques use peak picking in the spectrum to determine the spectral envelope. Once the envelope has been found, the harmonics can be moved in the frequency domain and new amplitudes found from the envelope. From this, the standard synthesis algorithm can be used to generate waveforms. 

Fig. 15. A comparison of the envelopes of intensity autocorrelation function for turbulent ( ) and striped ( ) states. The correlation lengths A for these states are 0.85 and 2.1, respectively.

The correlation length of a state was calculated from the intensity autocorrelation function of a given digital image. The correlation function envelope of the measured intensity /(R) was fit to an exponential,... 

The breadth of the envelope of the absorption spectrum in Figure 6(d) is, as in the case of direct dissociation, related to the width of the first peak of the autocorrelation function. It basically reflects the number of vibrational states in the upper manifold that have appreciable FC overlap with the initial-state wave function I gr., and are therefore excited,... 

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. 

In Fig. 6.75b, the upper and lower envelopes of the autocorrelation signal are plotted as a function of the normalized delay time r I AT for different values of the chirp parameter a in (6.47) [775]. 

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