Gaussian convolution

FIGURE 27. The photoelectron spectra of compounds 40 and 42. The solid line refers to the experimental data the dashed line shows the Gaussian components the points refer to the convoluted Gaussian components. Reproduced by permission of Elsevier Science from Reference 141... 

Fig. 88. Distribution of the Eu hyper-fine field in amorphous EuqjqAuqjo at 4.2 K (top part) and distribution of Eu hyperfine field calculated under the assumptions of non-collinear and (broken curve) and of convoluted Gaussian distributions (AH = 2 T) on H and (full curve). The curves were constructed from the data published by Friedt et al. (1982).

Equation (Bl.8.6) assumes that all unit cells really are identical and that the atoms are fixed hi their equilibrium positions. In real crystals at finite temperatures, however, atoms oscillate about their mean positions and also may be displaced from their average positions because of, for example, chemical inlioniogeneity. The effect of this is, to a first approximation, to modify the atomic scattering factor by a convolution of p(r) with a trivariate Gaussian density function, resulting in the multiplication ofy ([Pg.1366]

Sometimes it is not possible to improve the resolution of a complex mixture beyond a certain level and, under these circumstances, the use of some de-convolution technique may be the only solution. The algorithms in the software must contain certain tentative assumptions in order to analyze the peak envelope. Firstly, a particular mathematical function must be assumed that describes the peaks. The function used is usually Gaussian and, in most cases, no account is taken of the possibility of asymmetric peaks. Furthermore it is also assumed that all the peaks can be described by the same function (i.e. the efficiency of all the peaks are the same) which, as has already been discussed, is also not generally true. Nevertheless, providing the composite peak is not too complex, de-convolution can be reasonably successful. 

It seen that the de-convolution is likely to be successful as the position of the peak maximum, and the peak width, of the major component is easily identifiable. This would mean that the software could accurately determine the constants in the Gaussian equation that would describe the profile of the major component. The profile of the major component would then be subtracted from the total composite peak leaving the small peak as difference value. This description oversimplifies the calculation processes which will include a number of iteration steps to arrive at the closest fit for the two peaks. 

Figure 16. Experimental and calculated IR resonance enhanced photodissociation spectra of Fe" (CH4)3 and Fe" (CH4)4. Experimental spectra were obtained by monitoring loss of CH4. Calculated spectra are based on vibrational frequencies and intensities calculated at the B3LYP/ 6-311+G(d,p) level. Calculated frequencies are scaled by 0.96. The calculated spectra have been convoluted with a 10-cm full width at half-maximum (FWHM) Gaussian. The D2d geometries of Fe (CH4)4 are calculated to have very similar energies, and it appears that both isomers are observed in the experiment.

Figure 5.11 shotvs the temporal profile of the intensity change in the SFG signal at the peak of the Vco mode (2055 cm ) at OmV induced by visible pump pulse irradiation. The solid line is the least-squares fit using a convolution of a Gaussian function for the laser profile (FWFJ M = 20 ps) and a single exponential function for the recovery profile. The SFG signal fell to a minimum within about 100 ps and recovered... 

The Voigt function is a convolution product ( ) between L and G. As the convolution is expensive from a computational point of view, the pseudo-Voigt form is more often used. The pseudo-Voigt is characterized by a mixing parameter r], representing the fraction of Lorentzian contribution, i.e. r] = 1(0) means pure Lorentzian (Gaussian) profile shape. Gaussian and Lorentzian breadths can be treated as independent parameters in some expressions. 

It is also clear from Eq. (2.5.1) that the linewidth of the observed NMR resonance, limited by 1/T2, is significantly broadened at high flow rates. The NMR line not only broadens as the flow rate increases, but its intrinsic shape also changes. Whereas for stopped-flow the line shape is ideally a pure Lorentzian, as the flow rate increases the line shape is best described by a Voigt function, defined as the convolution of Gaussian and Lorentzian functions. Quantitative NMR measurements under flow conditions must take into account these line shape modifications. 

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... 

Fig. 6 Gaussian convolution of an experimental profile (a) raw spectrum (b) after convolution by a Gaussian of width 5 points (c) after convolution by a Gaussian of 30 points. The ordinate scale is arbitrary. 

The remaining terms in Eq. (4-24) are the nth-order corrections to approximate the real system, in which the expectation value ( c is called cumulant, which can be written in terms of the standard expectation value ( by cumulant expansion in terms of Gaussian smearing convolution integrals ... 

To render the KP theory feasible for many-body systems with N particles, we make the approximation of independent instantaneous normal mode (INM) coordinates [qx° 3N for a given configuration xo 3W [12, 13], Hence the multidimensional V effectively reduces to 3N one-dimensional potentials along each normal mode coordinate. Note that INM are naturally decoupled through the 2nd order Taylor expansion. The INM approximation has also been used elsewhere. This approximation is particularly suited for the KP theory because of the exponential decaying property of the Gaussian convolution integrals in Eq. (4-26). The total effective centroid potential for N nuclei can be simplified as ... 

As shown by Strobl [230], the integral breadths B in a series of reflections is increasing quadratically if (1) the structure evolution mechanism leads to a convolution polynomial, (2) the polydispersity remains moderate, (3) the rod-length distributions can be modeled by Gaussians (cf. Fig. 8.44). For the integral breadth it follows... 

James and Guth showed rigorously that the mean chain vectors in a Gaussian phantom network are affine in the strain. They showed also that the fluctuations about the mean vectors in such a network would be independent of the strain. Hence, the instantaneous distribution of chain vectors, being the convolution of the distribution of mean vectors and their fluctuations, is not affine in the strain. Nearly twenty years elapsed before his fact and its significance came to be recognized (Flory, 1976,... 

In the practice of solid-state bioEPR, a Lorentzian line shape will be observed at relatively high temperatures and its width as a function of temperature can be used to deduce relaxation rates, while a Gaussian line will be observed at relatively low temperatures and its linewidth contains information on the distributed nature of the system. What exactly is high and low temperature, of course, depends on the system for the example of low-spin cytochrome a in Figure 4.2, a Lorentzian line will be observed at T = 80°C, and a Gaussian line will be found at T 20°C, while at T 50°C a mixture (a convolution) of the two distributions will be detected. 

Figure 3. Calculated longitudinal projectile momentum transfer distributions convoluted to a Gaussian distribution for a Gaussian width of 0.22 a.u. Experimental data (histogram) are from Moshammeret al. [2], Theoretical results CDW results [20], CDW-EIS1 results [20], CDW-EIS2 [48],...

Figure 2. Space-scale representation of the GC content of a 10-Mbp-long fragment of human chromosome 22 when using a Gaussian smoothing filter (x) [Eq. (6)]. (a) GC content flucmations computed in adjacent 1 kbp intervals, (b) Color coding of the convolution product Wg(o)[GC](n,a) = (GC /a))(n) using 256 colors from black (0) to red (max) superimposed...

Curve fitting is an important tool for obtaining band shape parameters and integrated areas. Spectroscopic bands are typically modeled as Lorenzian distributions in one extreme and Gaussian distributions in the other extreme [69]. Since many observable spectroscopic features lie in between, often due to instrument induced signal convolution, distributions such as the Voight and Pearson VII have been developed [70]. Many reviews of curve fitting procedures can be found in the literature [71]. 

Figure 10.2 The procedure of convolution, represented graphically, (a) A one-dimensional centrosymmetric structure, (b) A Gaussian distribution, which could potentially be an atomic shape function.

The most recent calculations, however, of the photoemission final state multiplet intensity for the 5 f initial state show also an intensity distribution different from the measured one. This may be partially corrected by accounting for the spectrometer transmission and the varying energy resolution of 0.12, 0.17, 0.17 and 1,3 eV for 21.2, 40.8, 48.4, and 1253.6 eV excitation. However, the UPS spectra are additionally distorted by a much stronger contribution of secondary electrons and the 5 f emission is superimposed upon the (6d7s) conduction electron density of states, background intensity of which was not considered in the calculated spectrum In the calculations, furthermore, in order to account for the excitation of electron-hole pairs, and in order to simulate instrumental resolution, the multiplet lines were broadened by a convolution with Doniach-Sunjic line shapes (for the first effect) and Gaussian profiles (for the second effect). The same parameters as in the case of the calculations for lanthanide metals were used for the asymmetry and the halfwidths ... 

Another approach is the characterization of peaks with a well-defined model with limited parameters. Many models are proposed, some representative examples will be deaaib i. Wefl known is the Exponentially Modified Gaussian (EMG) peak, i.e. a Gaussian convoluted with an exponential decay function. Already a few decades ago it was recognized that an instrumental contribution such as an amplifier acting as a first-order low pass system with a time constant, will exponentially modify the... 

An example is the relatively simple moving average filter. In case of a digitized signal, the values of a fixed (odd) number of data points (a window) are added and divided by the number of points. The result is a new value of the center point. Then the window shifts one point and the procedure, which can be considered as a convolution of the sipal with a rectangular pulse function, repeats. Of course, other functions like a triangle, an exponential and a Gaussian, can be used. 

We have seen that convolving rectangles gives a gaussianlike function. A gaussian is, in fact, the exact result of an infinite number of convolutions provided that the functions convolved obey certain conditions. The rigorous... 

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