Correlation function delay time

All these features were observed experimentally for solutions of 3-amino-/V-methylphthalimide, 4-amino-/V-methylphthalimide, and for nonsubstituted rhoda-mine. The results were observed for cooled, polar solutions of phthalimides, in which the orientational relaxation is delayed. Exactly the same spectral behavior was observed [50] by picosecond spectroscopy for low viscosity liquid solutions at room temperature, in which the orientational relaxation rate is much higher. All experimental data indicate that correlation functions of spectral shifts Av-l(t), which are used frequently for describing the Time Dependent Stokes Shift, are essentially the functions of excitation frequency. 

The time delay between two signals can be estimated using different methods among them the most popular is finding the maximum of the cross-correlation function... 

The above equation is very similar to the cross-ambiguity function, but here the time-delay is introduced only in the transmitted signal. This form of the transform is more convenient for digital implementation and will be referred to below as a range-Doppler correlation function. 

The intensity of the sum frequency light /sum at a given delay time r between the probe pulse and the fluorescence beam co( is proportional to the correlation function of the fluorescence intensity with the intensity of the probe pulse co ... 

The phase-dependent directionality of photocurrents produced by such a detector entails advantageous properties of the photocurrents cross correlations in nonoverlapping time intervals or spatial regions (considered in Section 4.2.2). These directional time-dependent correlations are measured with one detector only. They involve solely terms dependent on LO phases, in contrast to similar correlations measured by conventional photocounters, which inevitably contain terms depending on photon fluxes such as the LO excess noise. Owing to these properties, the mean autocorrelation function of the SL quadrature is shown in the schemes considered here to be measurable without terms related to the LO noise. LO shot noise, which affects the degree of accuracy to which this autocorrelation is measured (i.e., its variance) is easily obtainable from zero time delay correlations because the LO excess noise is suppressed. The combined measurements of cross correlations and zero time delay correlations yield complete information on the SL in these schemes. 

In the treatment of a rigid dumbbell, where the whole time-correlation functions (TCF) can be solved exactly, Stockmayer and Burchard21 disclosed the origin for the discrepancy between theory and experiments. They recognized that all measurements of the TCF can be carried out down only to a limiting minimum delay time. With common instruments, this lower limit lies at about 100 ns but the lowest time is often much higher under conditions such that the TCF should have decayed to e"2 at channel 8Q220). These experimental condition imply that only an apparent first cumulant is determined defined by... 

This tells us how much energy in the original signal remains after some arbitrary delay time t. Next we compute the cross-correlation function Rxy(t), which contains information about energy shared between channel X and channel Y. 

However, delayed dissipation typically occurs after the s-region has settled into its thermal equilibrium, with RDOp peq, and the more familiar equations for the dissipative rate can be used. Then yVl( -u)(/, //) is obtained to second order from a p-s coupling Hps = A(p B(SJ>, with trs(r(s>Bs3>) = 0 chosen for convenience, and can be written in terms of the time-correlation functions Cp-7 t ))) = trs[B t)B t )feq] of the atomic... 

At the present time, in most PCS instruments, dust is handled in two ways an experimentally measured, delayed baseline and/or a dust term in the calculation. The latter method usually assumes dust to be infinitely large with a zero diffusion coefficient. This leads to a constant, which is another way of saying a baseline. The problem with adjusting the baseline is that even a very small baseline uncertainty can lead to rather large errors in the distribution parameters as shown in the Appendix. A better procedure would be to reject dust before it contributed to the correlation function. 

In a dynamic light scattering experiment, the measured intensity-intensity time-correlation function g<2)(tc), where tc is the delay time, is related to the normalized electric field correlation function g(1)frc), representative of the motion of the particles, by the Siegert relation [18] ... 

The autocorrelation (or correlation) function is obtained by multiplying each y (f) by y (t — t°), where t° is a time delay, and summing the products over all points [43]. Examination of the sum plotted as a function of t° reveals the level of dependency of data points on their neighbors. The correlation time is the value of t° for which the value of the correlation function falls to exp (—1). When the correlation function falls abruptly to zero, that indicates that the data are without a deterministic component a slow fall to zero is a sign of stochastic or deterministic behavior when the data slowly drop to zero and show periodic behavior, then the data are highly correlated and are either periodic or chaotic in nature [37,43]. 

The preceding discussion has considered processes that are fast, or at least closely correlated in time. Other functional processes in biopolymers may require delay times, which in some cases may imply lower bounds on the roughness of the energy landscape. 

Fig. 14. Effects of small-amplitude reorientation on 2H NMR experiments, as calculated by means of RW simulations. In the model, C-2H bonds (<5 = 2n 125 kHz, rj = 0) perform rotational random jumps on the surface of a cone with a full opening angle % = 6°. (a) 2H NMR spectra for various solid-echo delays tp (tj = t = 30 pis), and (b) 2H NMR correlation functions Fcos(tm) for various evolution times tp (tj = t = 10ms). (Adapted from Ref. 76.)...

Software for cumulant analysis from Brookhaven Instruments was used to fit the measured correlation function C(f) to equation 9 by using a weighted second-order polynomial nonlinear regression. The measured base line B of the correlation function C t) was determined from the average of four delay channels (1029-1032) multiplied by the sample time Af. The calculated and measured base lines were within 0.1% for all runs used in the analysis. 

Fig. 4.2.3 [Bliil] Time conventions for three-pulse excitation. In 3D correlation spectroscopy, the pulse seperations t/ are used as parameters. In nonlinear system theory, the parameters are the time delays at of the cross-correlation function corresponding to the arguments r, of the response kernels.

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