Spectral cross-correlation function

As seen from the above theoretical developments, accessing geometrical (and stereochemical) information implies at least an estimation of the dynamical part of the various relaxation parameters. The latter is represented by spectral densities which rest on the calculation of the Fourier transform of auto- or cross-correlation functions. These calculations require necessarily a model for describing molecular reorientation... 

Experimental Setup. An obvious extension of the one-color pump-probe experiments is the application of two-color experiments in which two independently tunable dye lasers share the same pump laser. One can use the same high repetition rate and obtain spectral evolutions on excitation at selected wavelengths. The measurements are performed in essentially the same way as one-color experiments.A disadvantage is the broadened instrument function (cross-correlation function) caused by time jitter between the two pulses, since they are not obtained from the same dye laser. This leads to a full-width half-maximum (fwhm) value of the instrument function of approximately 5-10 psec. 

Cross-correlation and spectral analysis have proven invaluable tools for quantifying the frequency dependent characteristics of the human subject. The cross-spectral density function, or cross-spectrum Sxyif), can be obtained from the random target x t) and random response y t) by taking the Fourier transform of the cross-correlation function Vxyir), that is, Sxyif) = Ffr yfr), or in the frequency domain via Sxy if) = X(/) y(/), or by a nonparametric system identification approach (e.g., spa.m in Matlab ). The cross-spectrum provides estimates of the relative amphtude (i.e., gain) and phase-lag at each frequency. Gain, phase, and remnant frequency response curves provide objective measures of pursuit... 

The auto- and cross-correlation functions of two simultaneously monitored signals X (r) and y (t) in the time domain, and their corresponding spectral power density and cross-power spectral density in the frequency domain, are defined by the following equations. The auto-correlation function Rxxit) of the signal x (t) in the time domain is defined as... 

Figure 4 Typical cross-correlation function used to measure the radial velocity. This figure represents the mean of the spectral lines of the star 51 Peg. The position of the Gaussian function fitted (solid line) is a precise measurement of the Doppler shift to an accuracy of about 15 m s The width of the cross-correlation function reflects the star s rotational velocity. Reproduced with permission of Macmillan Magazines Ltd. from Mayor M and Queioz D (1995) Nature 378 355.

We expect the effective-mode models described here to be versatile tools that can predict general trends, and that can be used in conjunction with microscopic information provided from other sources, i.e., spectral densities, energy gap correlation functions, and possibly cross-correlation functions. Further, model parametrizations could be provided by QM/MM type simulations, and the model-based dynamics could be employed to analyse the wealth of microscopic information provided by such simulations. Such complementary strategies would bridge the gap between system-bath theory approaches and explicit multi-dimensional simulations for ultrafast photochemical processes in various types of environments. 

Let Sf(zi, Z2, co) be the space-time cross-spectral density function, this function is connected to the cross-correlation function by the Wiener-Khintchine relations ... 

Cross-correlation can also be used for spectral reconstruction. If a x) is the absorbance or reflectance at a specific wavelength, and c(x) is the concentration of the desired component in the xth sample, then the cross-correlation function, Cadd),... 

The cross-correlated DD-CSA (or DD-CSR) spectral densities, giving rise to differential line broadening and to the order transfer phenomena summarized by Eq. (20), can in principle be complex functions. The line-broadening... 

Spectral lineshapes were first expressed in terms of autocorrelation functions by Foley39 and Anderson.40 Van Kranendonk gave an extensive review of this and attempted to compute the dipolar correlation function for vibration-rotation spectra in the semi-classical approximation.2 The general formalism in its present form is due to Kubo.11 Van Hove related the cross section for thermal neutron scattering to a density autocorrelation function.18 Singwi et al.41 have applied this kind of formalism to the shape of Mossbauer lines, and recently Gordon15 has rederived the formula for the infrared bandshapes and has constructed a physical model for rotational diffusion. There also exists an extensive literature in magnetic resonance where time-correlation functions have been used for more than two decades.8... 

Emission decays were obtained from a fs Tl sapphire laser uorescence upconversion spectrometer whose construction is reported elsewhere [8]. Here we only note that the overall temporal response used in these studies was between 112-125 fs as measured by the FWHM of the cross correlation between the pump and gate pulses. Decays (0-200 ps with a variable step size) were collected at a series of ten emission wavelengths (8 nm bandpass) which were then used to reconstruct time-evolving emission spectra in the manner described in Refs. 8 and 9. From these spectra the solvation dynamics was extracted in the form of the spectral response function,... 

Measurements. CILS spectra are described by a photon scattering cross section, a [Eqn. (2)], the spectral function, g(oj T) [Eqn. (3)], or the Fourier transformation of a correlation function, 0(f) [Eqn. (4)]. For the most part, CILS spectra are diffuse and show little structure so that previously it was often considered sufficient to specify a few spectral moments. 

Cross-correlation and spectral analysis have proven invaluable tools for quantifying the frequency-dependent characteristics of the human subject. The cross-spectral density function, or cross-spectrum can be obtained from the random target x t) and random response y(t) by taking the... 

Statistics of the Estimates. The mean and the variance of the sample estimates of the coherence function ate derived in Ref. 4, and I only reproduce the final results here. In general, the cross-spectral density is evaluated by doing the Fourier transform of a windowed (using a lag window) sequence of cross-correlation estimates. The choice of the smoothing window therefore determines the variance in the estimates of cross-spectral density (numerator in the expression of coherence). The variance of the smoothed coherence estimator is given by ... 

Nucleic acids > ca. 10 bp long are not spherically symmetric. To a good approximation they are equivalent to circular cylinders with a hydrodynamic diameter of 20-23 A for DNA (33-35) and 25 A for RNA (35). The correlation function for such symmetric top molecules consist of three exponentials, whose arguments are combinations only of the correlation time for end over end tumbling (tl) and for rotation about the principal symmetiy axis (ts). Thus for anisotropic motion, two independent correlation times are needed to describe the rotational diffusion. The spectral density function also depends on the angle (0) the interproton vector makes with the principal axis. J(0), and hence the cross-relaxation rate constant, varies as a function of this angle according to (.16) ... 

A remark applies to the experimental values of deduced from experimental bandshapes. Since we deal with total (multimolecular) correlation functions, their moments might also contain contributions arising from the intercorrelations of different molecules - or multimolecular effects - since cross terms in the statistical averages do not necessarily vanish. On this account it is necessary to estimate the extent of high-frequency collective effects if any, whenever torques are to be computed from the spectral moments. 

NMR relaxation data depend on dipolar ( N and C) and quadrupolar ( H) interactions on chemical shift anisotropy and cross-correlation effects. It is well known that the NMR relaxations can be written as functions of the spectral densities of the magnetic interactions, and this is the intersecting point between macroscopic and microscopic descriptions The spectral densities are calculated within the theoretical framework describing the dynamics of the system. 

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