Spectral autocorrelation

Substituting this form into Eq. (10.9), and using the fact that t > f, giver j spectral autocorrelation ... 

Figure 3.9 The spectral autocorrelation function F (r) (full line), given by Eq. (71) and the EIT line shape (broken line) corresponding to it, as of Eq. (72).

Goldfisher, Autocorrelation function and power spectral density of laser-produced speckle pattern . J. Opt. Soc. Am., vol.55, p.247(1965). 

BPTI spectral densities Cosine Fourier transforms of the velocity autocorrelation function... 

Fig. 8. Spectral densities for BPTI as computed by cosine Fourier transforms of the velocity autocorrelation function by Verlet (7 = 0) and LN (7 = 5 and 20 ps ). Data are from [88].

Another view of this theme was our analysis of spectral densities. A comparison of LN spectral densities, as computed for BPTI and lysozyme from cosine Fourier transforms of the velocity autocorrelation functions, revealed excellent agreement between LN and the explicit Langevin trajectories (see Fig, 5 in [88]). Here we only compare the spectral densities for different 7 Fig. 8 shows that the Langevin patterns become closer to the Verlet densities (7 = 0) as 7 in the Langevin integrator (be it BBK or LN) is decreased. 

An important property of the time autocorrelation function CaU) is that by taking its Fourier transform, F CA(t) a, one gets a spectral decomposition of all the frequencies that contribute to the motion. For example, consider the motion of a single particle in a hannonic potential (harmonic oscillator). The time series describing the position of the... 

Thus the nth vibrational spectral moment is equal to an equilibrium correlation function, the nth derivative of the dipole moment autocorrelation function evaluated at t=0. By using the repeated application of the Heisenberg equation of motion ... 

Autocorrelation function Power spectrum (Spectral power density)... 

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 ... 

We may recall and emphasize that the autocorrelation function obtained in the three representations I, II, and III must be equivalent, from the general properties of canonical transformation which must leave invariant the physical results. Thus, because of this equivalence, the spectral density obtained by Fourier transform of (43) and (45) will lead to the same Franck-Condon progression (51). 

On the other hand, the undamped autocorrelation function (17) we have obtained within the standard approach avoiding the adiabatic approximation must lead after Fourier transform to spectral densities involving very puzzling Dirac delta peaks given by... 

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... 

Strategies for Spectral Analysis in Dissipative Systems Filter Diagonalization in the Lanczos Representation and Harmonic Inversion of the Chebychev Order Domain Autocorrelation Function. 

Wold, H., Christie, O. H. J. Anal. Chim. Acta 165, 1984, 51-59. Extraction of mass spectral information by a combination of autocorrelation and principal components models. 

Using an experimentally-optimized focusing lens (/ = 15 mm spot radius, w = 4.3 pm) and a 3 mm KNbOj crystal (cut for NCPM at 22°C and 858 nm AR-coated), up to 11.8 mW of blue average power with a spectral width up to AIsh 1-4 nm at 429 nm was generated with only 44.6 mW of incident fundamental. The maximum observed SHG conversion efficiency was as high as 30 %. The overall efficiency of the electrical-to-blue process was over 1 %, and the blue pulses were measured by autocorrelation to be -500 fs in duration. ... 

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... 

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