Spectral density function Fourier transform

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

A stochastic process is also characterized by its spectral density, the Fourier transform of its autocorrelation function. The autocorrelation function of a (stationary stochastic process) measures the correlation of the process at different time intervals while the spectral density measures the amplitudes of the component waves of different frequencies. A white noise process has a constant spectral density (i.e., the same amplitude for all frequencies) and the band-limited noise has a frequency band over which the spectral density is nearly constant. 

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... 

In practice, the Fourier transform of the correlation function, denoted as spectral density function, /(cu), is more often used to express the interrelationships of parameters 3 ... 

The spectral density function is defined to be the temporal Fourier transform of G(t) so that... 

Now we can show the explicit relation with experiment. What is usually measured in spectroscopic or scattering experiments is the spectral density function /(to), which is the Fourier transform of some correlation function. For example, the absorption intensity in infrared spectroscopy is given by the Fourier transform of the time-dependent dipole-dipole correlation function <[/x(r), ju,(0)]>. If one expands the observables, i.e., the dipole operator in the case of infrared spectroscopy, as a Taylor series in the molecular displacement coordinates, the absorption or scattering intensity corresponding to the phonon branch r at wave vector q can be written as (Kobashi, 1978)... 

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 Fourier transform Gsr E) of the propagator is an analytic function in the complex F-plane, except on the real axis where it can have simple poles and cuts. These correspond to energy eigenvalues of the (single-particle) Hamiltonian h. Consideration of the discontinuity of Gsr(E) at the real axis yields the spectral density function... 

The relaxation dynamics of junctions in polymer networks have not been well-known until the advent of solid-state NMR spin-lattice relaxation measurements in a series of poly(tetrahydrofuran) networks with tris(4-isocyanatophenyl)-thiophosphate junctions [100]. The junction relaxation properties were studied in networks with molecular weights between crosslinks. Me, ranging from 250 to 2900. The dominant mechanism for nuclear spin lattice relaxation times measured over a wide range of temperatures were fit satisfactorily by spectral density functions, /( ), derived from the Fourier transforms of the Kohlrausch stretched exponential correlation functions... 

Nuclear magnetic resonance spectroscopy is another powerful technique for probing local chain dynamics. With this tedmique, the spectral drasity of local relaxational frequendes, spin-lattice rdaxation times and oriratational correlation times are characterized. Here, the spectral density function is the Fourier transform of the OACF assodated with the investigate intemudear vector, given as... 

In addition to being characterized by a PDF, signals can be characterized in the frequency domain by their power spectral density functions, which are Fourier transforms of the autocorrelation functions. White signals, which are uncorrelated from sample to sample, have a delta function autocorrelation or a flat (constant) power spectral density. Ocean signals, in general, are much more colorful and are not limited to being stationary. 

Figures 3 and 4 show the spectral density functions respectively for L/R 4 and 1. For a helicopter with R 35 ft, we are referring to turbulence with L = 140 ft and 35 ft, typical of low altitude hovering, possibly, close to the presence of surface structures. The Inset figures In Figures 3 and 4 are the autocorrelation functions whose Fourier transforms are the spectral density...

Wiener-Khinchin theorem of statistical mechanics, the Fourier transform of the autocorrelation function of a fluctuating quantity is the power spectrum of the fluctuations, where the power at a given frequency is the mean square amplitude of the fluctuations at that frequency. To convert Mnmjkit) to a suitable function of frequency, Redfield [19] defined the spectral density function, J mjk(,co)> ... 

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. 

For a pure state density operator, the Fourier transform of this double-time Green s function yields the spectral representation of the propagator (21)... 

In Equation (5), we can first notice (i) the factor 1/r6 which makes the spectral density very sensitive to the interatomic distance, and (ii) the dynamical part which is the Fourier transform of a correlation function involving the Legendre polynomial. We shall denote this Fourier transform by (co) (we shall dub this quantity "normalized spectral density"). For calculating the relevant longitudinal relaxation rate, one has to take into account the transition probabilities in the energy diagram of a two-spin system. In the expression below, the first term corresponds to the double quantum (DQ) transition, the second term to single quantum (IQ) transitions and the third term to the zero quantum (ZQ) transition. 

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... 

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... 

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