Gaussian function Fourier transform

Mathematical appendice-s introduce (5-functions, Fourier transform, or d-dimensional integration over Gaussian functions, respectively. 

If a gaussian function is chosen for the charge spread function, and the Poisson equation is solved by Fourier transformation (valid for periodic... 

It can be shown that the right-hand side of Eq. (3-208) is the -dimensional characteristic function of a -dimensional distribution function, and that the -dimensional distribution function of afn, , s n approaches this distribution function. Under suitable additional hypothesis, it can also be shown that the joint probability density function of s , , sjn approaches the joint probability density function whose characteristic function is given by the right-hand side of Eq. (3-208). To preserve the analogy with the one-dimensional case, this distribution (density) function is called the -dimensional, zero mean gaussian distribution (density) function. The explicit form of this density function can be obtained by taking the i-dimensional Fourier transform of e HsA, with the result.45... 

The time resolution of the instrument determines the wavenumber-dependent sensitivity of the Fourier-transformed, frequency-domain spectrum. A typical response of our spectrometer is 23 fs, and a Gaussian function having a half width... 

Fourier transforms boxcar function 274 Cauchy function 276 convolution 272-273 Dirac delta function 277-279 Gaussian function 275-276 Lorentzian function 276-277 shah function 277-279 triangle function 275 fraction, rational algebraic 47 foil width at half maximum (FWHM) 55, 303... 

Dice the Gaussian, discussed above, the function shah is its own Fourier transform. Thus,... 

It is also possible to play other mathematical tricks with the FID. For example, we may want to make our signals appear sharper so we can see small couplings. In this case, we want our FID to continue for longer (an infinite FID has infinitely thin lines when Fourier transformed). To do this we use Gaussian multiplication . This works exactly the same way as exponential multiplication but uses a different mathematical function (Figure 4.2). 

On the other hand, lattice distortions of the second kind are considered. Assuming [127] that ID paracrystalline lattice distortions are described by a Gaussian normal distribution go (standard deviation ay, its Fourier transform Gd (.S ) = exp (—2n2ols2) describes the line broadening in reciprocal space. Utilizing the analytical mathematical relation for the scattering intensity of a ID paracrys-tal (cf. Sect. 8.7.3 and [127,128]), a relation for the integral breadth as a function of the peak position s can be derived [127,129]... 

Fig. 12 C -detected C CSA patterns of the SHPrP109 i22 fibril sample. The upper and lower traces correspond to the experimental and simulated spectra, respectively. Simulations correspond to the evolution of a one-spin system under the ROCSA sequence. The only variables are the chemical shift anisotropy and the asymmetry parameter. A Gaussian window function of 400 Hz was applied to the simulated spectmm before the Fourier transformation. (Figure and caption adapted from [164], Copyright (2007), with permission from Elsevier)...

As a approaches zero, f(x) becomes oo at x = 0, when it is called a "unit impulse", or a delta function. Its transform g(k) = 1. Of equal importance is the Fourier transform of the Gaussian function... 

Figure 5.1 illustrates the Fourier transform (FT) of a simple function, viz., a Gaussian. The relatively sharp Gaussian function with the exponent a = 1 depicted in Figure 5.1a, yields a diffuse Gaussian (in dotted line) in momentum space. A flat Gaussian function in position space with a = 0.1, transforms to a sharp one (cf. Figure 5.1b). Connected by an FT, the wave functions in position and momentum...

The inverse Fourier transform of G again yields a Gaussian function g(t) = linewidth Tg of which can be numerically calculated from... 

The most obvious drawback of Fourier space approaches is the computational cost of the Fourier transformation itself. However, this can be circumvented in some virtual screening applications. Gaussian functions are frequently used to approximate electron densities. Interestingly, the Fourier transform of a Gaussian function is again a Gaussian function and hence amenable to analytic transformation. 

It is noted that both the probability distribution of Eq. (2.16) and the temperature factor of Eq. (2.19) are Gaussian functions, but with inversely related mean-square deviations. Analogous to the relation between direct and reciprocal space, the Fourier transform of a diffuse atom is a compact function in scattering space, and vice versa. 

Similarly to non-selective experiments, the first operation needed to perform experiments involving selective pulses is the transformation of longitudinal order (Zeeman polarization 1 ) into transverse magnetization or ly). This can be achieved by a selective excitation pulse. The first successful shaped pulse described in the literature is the Gaussian 90° pulse [1]. This analytical function has been chosen because its Fourier transform is also a Gaussian. In a first order approximation, the Fourier transform of a time-domain envelope can be considered to describe the frequency response of the shaped pulse. This amounts to say that the response of the spin system to a radio-frequency (rf) pulse is linear. An exact description of the... 

The Gaussian function is its own Fourier transform that is, when we define... 

Model correlation functions. Certain model correlation functions have been found that model the intracollisional process fairly closely. These satisfy a number of physical and mathematical requirements and their Fourier transforms provide a simple analytical model of the spectral profile. The model functions depend on the choice of two or three parameters which may be related to the physics (i.e., the spectral moments) of the system. Sears [363, 362] expanded the classical correlation function as a series in powers of time squared, assuming an exponential overlap-induced dipole moment as in Eq. 4.1. The series was truncated at the second term and the parameters of the dipole model were related to the spectral moments [79]. The spectral model profile was obtained by Fourier transform. Levine and Birnbaum [232] developed a classical line shape, assuming straight trajectories and a Gaussian dipole function. The model was successful in reproducing measured He-Ar [232] and other [189, 245] spectra. Moreover, the quantum effect associated with the straight path approximation could also be estimated. We will be interested in such three-parameter model correlation functions below whose Fourier transforms fit measured spectra and the computed quantum profiles closely see Section 5.10. Intracollisional model correlation functions were discussed by Birnbaum et a/., (1982). 

Figure 7.3.1.7 A 13C-decoupled HMQC spectrum of 54 mM chloroquine diphosphate in D2O acquired with an inverse detection microcoil probe. The 740-nl F0bs contained 40 nmol (13 jig) of chloroquine. The data, 32 transients per slice, 1024 x 128 (x2, hypercomplex) points, were acquired in 3.6h. The data were zero-filled to 256 points in the 13C dimension. A 40° shifted sinebell function was applied, followed by Gaussian multiplication prior to Fourier transformation...

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