Spectral density of the autocorrelation

Hence, is the spectral density of the autocorrelation function of density... 

Magnetic resonance methods have been used extensively to probe the structure and dynamics of thermotropic nematic liquid crystals both in the bulk and in confined geometry. Soon after de Gennes [27] stressed the importance of long range collective director fluctuations in the nematic phase, a variable frequency proton spin-lattice relaxation Tx) study [32] showed that the usual BPP theory [33] developed for classical liquids does not work in the case of nematic liquid crystals. In contrast to liquids, the spectral density of the autocorrelation function is non-Lorentzian in nematics. As first predicted independently by Pincus [34] and Blinc et al. [35], collective, nematic type director fluctuations should lead to a characteristic square root type dependence of the spin-lattice relaxation rate rf(DF) on the Larmor frequency % ... 

The spectral density of the autocorrelation function of the solid surface oscillator deviation (yy) ) can be derived by the relationship between the energy spectrum ((d) of the surface phonons and (yy)Q)... 

As an example, we consider these error bounds for the cumulative distribution of the spectral density of the velocity autocorrelation function,... 

If an autocorrelator is used to analyze the PM output the correlation function in Eq. (4.3.5) is directly measured. Sometimes a spectrum analyzer4 is. used instead of an autocorrelator. This device determines the spectral density of the photocurrent from the PM and thus in a homodyne experiment it determines the time Fourier transform of h t), that is,... 

In light-scattering experiments one measures the spectral density of the electric field autocorrelation function of the scattered light wave, given as... 

The dynamics of different modes of molecular librations (hindered rotations) and intramolecular vibrations in supercritical water can now be analyzed in terms of velocity autocorrelation functions for the corresponding projections (Eqns. 22-27) (Kalinichev and Heinzinger 1992, 1995 Kalinichev 1993). The velocity autocorrelation functions calculated for the quantities Qi (Eqns. 25-27) are shown in Figure 19 for two extreme cases of high-density and low-density supercritical water. The Fourier transforms of these functions result in the spectral densities of the corresponding vibrational modes. They are shown in Figure 20 for the supercritical thermodynamic states listed in Table 5. 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

Note that this last expression is nothing but the closed form [90] of the autocorrelation function obtained (as an infinite sum) in quantum representation III by Boulil et al.[87] in their initial quantum approach of indirect damping. Although the small approximation involved in the quantum representation III and avoided in the quantum representation II, both autocorrelation functions are of the same form and lead to the same spectral densities (as discussed later). 

The spectral density is the Fourier transform of this autocorrelation function, that... 

The symbol Re(K ((o)) denotes the real part of the complex spectral density, corresponding to the autocorrelation of the dipolar interactions, while Re(i (co)) is its counterpart for the scalar interaction. The symbol Re(K (a>)) denotes the spectral density describing the cross-correlation of the two parts of the hyperfine interaction. The cross-correlation vanishes at the MSB level of the theory, but in the more complicated case of the lattice containing the electron spin, the cross term may be non-zero. A general expression for the dipolar spectral density is ... 

It is assumed that the noise voltage n(t) is the result of a real stationary process (Davenport and Root, 1958) with zero mean. Because it can be shown that the spectral density function S(f) is the Fourier transform of the autocorrelation function of the noise, it follows that the rms noise is given by... 

Fig. 2. Error bounds for the cumulative frequency distribution of the spectral density for the velocity autocorrelation function using jU.0, /n2, and ja evaluated for a classical model of liquid argon.29...

Fig. 4. Spectral density for the velocity autocorrelation function for vibrations of abody-centered cubic lattice, extrapolated from 7 even moments, by the method of Section IV.

More precisely, die quantity displayed is the signal power estimated from 10ms frames. As die power spectral densities of die two types of noise exhibit a strong peak at the null frequency, the two noises were pre-whitened by use of an all-pole filter [Cappe, 1991]. This pre-processing guarantees that the noise autocorrelation functions decay sufficiently fast to obtain a robust power estimate even with short frame durations [Kay, 1993]. 

In particular, if the number of photons detected during each period is recorded as a sequential record (instead of the simpler data recording mode utilized for this work) then autocorrelation functions and spectral densities of concentration... 

The next question is to identify residues near the active site that may modulate the donor-acceptor distance. In Fig. 10 we show the active site and some nearby residues. In the spirit of the previous results, in order to predict the degree that the motion of these residues is correlated with the donor-acceptor motion, we can calculate the Fourier transform of the autocorrelation of the residue motion, and then order the residues according to the height of the peak of the spectral density.34 In Fig. 11 we show one result, the spectral densities for the motion, projected first along the residue-donor axis and then along the donor-acceptor direction, of three residues, two of them strongly correlated and one not correlated. 

The difference in the spectral density between the displacement and velocity autocorrelation functions can be understood from a normal-mode description (see Chapt. IV.F). Using Eq. 23 for the displacement autocorrelation function and differentiating it to obtain the velocity autocorrelation function, one finds that the terms in the latter are weighted by the square of the mode frequency relative to the former. Thus higher-frequency contributions are more important in the spectral density associated with the velocity autocorrelation function than the displacement autocorrelation function.153,332... 

Figure 45. Time-scale matching for protein-solvent motions. The normalized spectral density for the (a) displacement and (6) the velocity autocorrelation functions of Trp-62 N 1, and (c) for the velocity autocorrelation function of ST2 water. (Note the differences in the timescales.)...

In an "analog optical mixing experiment one measures either the time autocorrelation function of the photomultiplier output current or its corresponding spectral density. In the former case, the photomultiplier output is analyzed by an autocorrelator and in the latter, by a spectrum analyzer. 

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