Fourier frequency

We recommend that the Rayleigh criterion and Fourier frequency cutoff Q be used as fundamental specifications of optical performance, but that other criteria such as full width at half maximum be used in describing widths before and after deconvolution and in specifying spread functions having other than sine-squared shape. 

If at the outset the data are very noisy and if the noise predominates in the Fourier frequency range needed to effect a restoration, constraints provide the only hope for improvement. The reason is that many of the noise values in the data would restore to physically unrealizable values by linear deconvolution. The constrained methods are inherently more robust because they must find a solution that is consistent with both data and physical reality. 

In the following pages, we trace the development of the nonlinear methods and two concurrent themes that underlie this work first the physical-realizability theme already mentioned, and then the concept that one should be able to restore the Fourier frequencies obliterated by the finite Fourier bandpass of the spectrometer, that is, frequencies not present in the data. The extent to which these two themes are closely coupled was not fully appreciated in early work. 

Sometimes the spectrometer completely obliterates the information at all Fourier frequencies co beyond some finite cutoff Q. This is specifically true of dispersive optical spectrometers, where the aperture determines 1. The cutoff Q may be extended to high Fourier frequencies by multipassing the dispersive element or employing the high orders from a diffraction grating. 

The Fourier frequency bandpass of the spectrometer is determined by the diffraction limit. In view of this fact and the Nyquist criterion, the data in the aforementioned application were oversampled. Although the Nyquist sampling rate is sufficient to represent all information in the data, it is not sufficient to represent the estimates o(k) because of the bandwidth extension that results from information implicit in the physical-realizability constraints. Although it was not shown in the original publication, it is clear from the quality of the restoration, and by analogy with other similarly bounded methods, that Fourier bandwidth extrapolation does indeed occur. This is sometimes called superresolution. The source of the extrapolation should be apparent from the Fourier transform of Eq. (13) with r(x) specified by Eq. (14). 

Because the function given by Eq. (4) neither obliterates nor strongly suppresses the high Fourier frequencies in the data, we would expect a linear method to perform relatively well. A simple iterative approach based on the direct method of Section I of Chapter 3 does, in fact, prove effective. 

A review of deconvolution methods applied to ESCA (Carley and Joyner, 1979) shows that Van Cittert s method has played a big role. Because the Lorentzian nature of the broadening does not completely obliterate the high Fourier frequencies as does the sine-squared spreading encountered in optical spectroscopy (its transform is the band-limiting rect function), useful restorations are indeed possible through use of such linear methods. Rendina and Larson (1975), for example, have used a multiple filter approach. Additional detail is given in Section IV.E of Chapter 3. 

The summation over the squared sines and cosines, respectively, in the denominator of Eqs. (A.l) and (A.2) essentially acts as a weighting function applied to each Fourier coefficient. Calculating these summations for the coefficients restored over several iterations, the author observed that this weighting function, generally, varied very slowly over the Fourier frequencies... 

Note that here z is not the usual Laplace frequency (used in rest of the review), but it is a Fourier frequency. Equation (226) can be written in the Fourier-Laplace space (z) as... 

Equation (228) is the normalized density correlation function in the Fourier frequency plane and has the same structure as Eq. (210), which is the density correlation function in the Laplace frequency plane. i/ (z) in Eq. (228) is the memory function in the Fourier frequency plane and can be identified as the dynamical longitudinal frequency. Equation (228) provides the expression of the density correlation function in terms of the longitudinal viscosity. On the other hand, t]l itself is dependent on the density correlation function [Eq. (229)]. Thus the density correlation function should be calculated self-consistently. To make the analysis simpler, the frequency and the time are scaled by (cu2)1 2 and (cu2)-1 2, respectively. As the initial guess for rjh the coupling constant X is considered to be weak. Thus rjt in zeroth order is... 

Figure 4. The imaginary part of the calculated viscosity is plotted as a function of the Fourier frequency at the triple point (solid line). Also shown is the prediction of the Maxwell viscoelastic model (dashed line), given by Eq. (239). The viscosity is scaled by a2/ /(mkBT) and the frequency is scaled by x l, where xsc = [ma2/kBT l/2. For more details see the text.

The situation is far more complex for reactions in high viscous liquids. The frequency-dependent friction, (z) [in the case of Fourier frequency-dependent friction C(cu)], is clearly bimodal in nature. The high-frequency response describes the short time, primarily binary dynamics, while the low-frequency part comes from the collective that is, the long-time dynamics. There are some activated reactions, where the barrier is very sharp (i.e., the barrier frequency co is > 100 cm-1). In these reactions, the dynamics is governed only through the ultrafast component of the total solvent response and the reaction rate is completely decoupled from the solvent viscosity. This gives rise to the well-known TST result. On the other hand, soft barriers... 

Em, Hm Fourier frequency components of E (t) and H(t) the o> subscript is dropped during derivation... 

Formally, the sum of random electromagnetic-field fluctuations in any set of bodies can be Fourier (frequency) decomposed into a sum of oscillatory modes extending through space. The "shaky step" in this derivation, already mentioned, is that we treat the modes extending over dissipative media as though they were pure sinusoidal oscillations. Implicitly this treatment filters all the fluctuations and dissipations to imagine pure oscillations only then does the derivation transform these oscillations into the smoothed, exponentially decaying disturbances of random fluctuation. 

Zhongcheng Wang, Trigonometrically-fitted method with the Fourier frequency spectrum for undamped Dulfing equation. Computer Physics Communications, 2006, 174, 109-118. 

We have used the fact tliat A ° = 0, in accordance with tlie zeroth-order wave function being equal to 0). It was demonstrated by Olsen and Jprgensen that the time-dependence of the A parameters could be determined by applying Are Ehrentest theorem to the operators A , and require the equation to hold for each order of the perturbation and for each Fourier frequency [46]. 

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