The autocorrelation method

A slightly different method known as the autocorrelation method can also be used. Since this has many practical advantages over the covariance method, it is the method most often nsed in practice. 

Recall from Section 10.3.9 that the autocorrelation function of a signal is defined as 

This is similar in form to Equation (12.19), and, if we can calculate the autocorrelation, we can make use of important properties of this signal. To do so, in effect we have to calculate the squared error from -oo to oo. This differs from the covariance method, in which we just consider the speech samples from a specific range. To perform the calculation from —oo to oo, we window the waveform using a banning, hamming or other window, which has the effect of setting all values outside the range 0 n Ai to 0. 

While we are calculating the error from —oo to oo, this will always be 0 before the windowed area, because all the samples are 0 and hence multiplying them by the filter coefficients will just produce 0. This will also be the case significantly after the window. Importantly, however, for P samples after the window, the error will not be 0 since it will still be influenced by the last few samples at the end of the region. Hence the calculation of the error from —oo to oo can be rewritten as 

Because this is a function of one independent variable j — k, rather than the two of (12.23), we can rewrite it as 

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In many cases, especially in female or other high pitched speech, the length of the closed phase can be very small, perhaps only 20 or 30 samples. In the autocorrelation method, the initial samples of the residual are dominated by the errors caused by calculating the residual from the zero signal before the window. The high (and erratic) error in the residual can be seen in the first few samples of the residual in Figure 12.17b. For short analysis windows this can lead to a residual dominated by these terms, and for this reason, covariance analysis is most commonly adopted for closed phase analysis. 

The FROG method provides information on the time-dependent frequency spectrum of a short pulse but cannot measure the phases of these spectral components. A newly developed technique is helpful in this case this method is called SPIDER (Spectral Phase /nterferometry for Direct Field Reconstruction). It uses the interference structure generated when two spatially separated pulses are superimposed [788]. Similar to the autocorrelation method, the two pulses are generated from the input pulse that is to be measured, using a beam splitter and a delay line which changes the time delay between the two pulses. The second pulse is therefore a copy of the first pulse with a time delay t. The electric field amplitude... 

An interferometric method was first used by Porter and Topp [1, 92] to perfonn a time-resolved absorption experiment with a -switched ruby laser in the 1960s. The nonlinear crystal in the autocorrelation apparatus shown in figure B2.T2 is replaced by an absorbing sample, and then tlie transmission of the variably delayed pulse of light is measured as a fiinction of the delay This approach is known today as a pump-probe experiment the first pulse to arrive at the sample transfers (pumps) molecules to an excited energy level and the delayed pulse probes the population (and, possibly, the coherence) so prepared as a fiinction of time. 

In an ambitious study, the AIMS method was used to calculate the absorption and resonance Raman spectra of ethylene [221]. In this, sets starting with 10 functions were calculated. To cope with the huge resources required for these calculations the code was parallelized. The spectra, obtained from the autocorrelation function, compare well with the experimental ones. It was also found that the non-adiabatic processes described above do not influence the spectra, as their profiles are formed in the time before the packet reaches the intersection, that is, the observed dynamic is dominated by the torsional motion. Calculations using the Condon approximation were also compared to calculations implicitly including the transition dipole, and little difference was seen. 

Let us start with a classic example. We had a dataset of 31 steroids. The spatial autocorrelation vector (more about autocorrelation vectors can be found in Chapter 8) stood as the set of molecular descriptors. The task was to model the Corticosteroid Ringing Globulin (CBG) affinity of the steroids. A feed-forward multilayer neural network trained with the back-propagation learning rule was employed as the learning method. The dataset itself was available in electronic form. More details can be found in Ref. [2]. 

Fluorescence intensity detected with a confocal microscope for the small area of diluted solution temporally fluctuates in sync with (i) motions of solute molecules going in/out of the confocal volume, (ii) intersystem crossing in the solute, and (hi) quenching by molecular interactions. The degree of fluctuation is also dependent on the number of dye molecules in the confocal area (concentration) with an increase in the concentration of the dye, the degree of fluctuation decreases. The autocorrelation function (ACF) of the time profile of the fluorescence fluctuation provides quantitative information on the dynamics of molecules. This method of measurement is well known as fluorescence correlation spectroscopy (FCS) [8, 9]. 

In addition to the fact that MPC dynamics is both simple and efficient to simulate, one of its main advantages is that the transport properties that characterize the behavior of the macroscopic laws may be computed. Furthermore, the macroscopic evolution equations can be derived from the full phase space Markov chain formulation. Such derivations have been carried out to obtain the full set of hydrodynamic equations for a one-component fluid [15, 18] and the reaction-diffusion equation for a reacting mixture [17]. In order to simplify the presentation and yet illustrate the methods that are used to carry out such derivations, we restrict our considerations to the simpler case of the derivation of the diffusion equation for a test particle in the fluid. The methods used to derive this equation and obtain the autocorrelation function expression for the diffusion coefficient are easily generalized to the full set of hydrodynamic equations. 

The Langevin dynamics method simulates the effect of individual solvent molecules through the noise W, which is assumed to be Gaussian. The friction coefficient r is related to the autocorrelation function of W through the fluctuation-dissipation theorem,... 

The present analysis follows the approach taken by aU three of these authors, in which SDEs are constructed by choosing the drift and diffusivity coefficients so as to yield a desired diffusion equation. Peters [13] has pioneered an alternative approach, in which expressions for the drift and diffusivity are derived from a direct, but rather subtle, analysis of the underlying inertial equations of motion, in which (for rigid systems) he integrates the instantaneous equations of motion over time intervals much greater than the autocorrelation time of the particle velocities. Peters has expressed his results both as standard Ito SDEs and in a nonstandard interpretation that he describes heuristically as a mixture of Stratonovich and Ito interpretations. Peters mixed Ito—Stratonovich interpretation is equivalent to the kinetic interpretation discussed here. Here, we recover several of Peters results, but do not imitate his method. 

To illustrate the potential of the hybrid method in describing the role of an intramolecular bath in the decay dynamics induced by a conical intersection, we consider the model of Ref. [7,8] for the S2-S1 Cl in pyrazine. Fig. 1 shows the wavepacket autocorrelation function C(t) = ( k(O)l (t) for an increasing number of bath modes. G-MCTDH hybrid calculations for 4 core (primary) modes plus nb bath (secondary) modes are compared with reference calculations by the standard MCTDH method. 

Fig. 1. The autocorrelation function C(t) = (U (O)l I (i) is shown for a wavepacket initially prepared on the upper diabatic surface [7]. Panels (a) and (b) C(t) for the four core modes calculated by the standard MCTDH method for the model Hamiltonian Hy of Eq. (9), shown on different scales in the two panels. Panel (c) G-MCTDH calculation (bold line) as compared with standard MCTDH calculation (dotted line) for the composite system with four core modes (combined into two 2-dimensional particles

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