Convolution square

In the field of scattering the autocorrelation is also known by the name convolution square . 

The real space function p(r) is directly linked to the spatial autocorrelation function (convolution square) of electron-density fluctuations inside the particle. 

Q(3 is the frequent distribution of the distances connecting all possible pairs of electrons in the sample, i.e. it has a large value, if ttere is a large number of electrons in the sample, that can be connected by the vector x. The symbol Z, means convolution square, a special kind of the... 

The symbol abbreviates the convolution integral. The Q-funrtion, the convolution square of p(x) is identical with the convolution product of p (x) with its mirror image ... 

Remarks The above procedure can be modified in the case of an antisymmetric function po (-x) = -Po W and also to multidimensional convolution squares of functions possessing inversion symmetry or antisymmetry. 

The two external integrals / d / d R R have the same effect, as taking a mean of convolution squares over a sample of plane membrane stacks. Finally we have to... 

Consider pm as a typical example. Evidently, the (R, z) dependence of the convolution square of a wavy building block pm will be ... 

Glatter, O. (1981). Convolution square root of band-limited symmetrical functions and its application to small-angle scattering data. J. Appl. Crystallogr., 14,101-108. 

The results previously discussed clearly demonstrate that electrochemical techniques are powerful tools to study compoimds of interest to human and animal health. Particularly, linear, cyclic, convolution, square wave voltammetries, and controlled potential bulk electrolysis allow inferring the reaction mechanism and perform a full thermodynamic and kinetics of redox couples controlled by diffusion, adsorption as well as those which show a mixed control diffusion/adsorption. On the other hand, the square wave voltammetry coupled to adsorptive accumulation of redox couples which are both electroactive, and show specific interactions with the electrode smface allows detecting and quantifying substrates at trace levels. 

We have seen that the intensities of diffraction of x-rays or neutrons are proportional to the squared moduli of the Fourier transfomi of the scattering density of the diffracting object. This corresponds to the Fourier transfomi of a convolution, P(s), of the fomi... 

We have seen that the intensities of diffraction are proportional to the Fourier transfomi of the Patterson fimction, a self-convolution of the scattering matter and that, for a crystal, the Patterson fimction is periodic in tln-ee dimensions. Because the intensity is a positive, real number, the Patterson fimction is not dependent on phase and it can be computed directly from the data. The squared stmcture amplitude is... 

The important information about the properties of smectic layers can be obtained from the relative intensities of the (OOn) Bragg peaks. The electron density profile along the layer normal is described by a spatial distribution function p(z). The function p(z) may be represented as a convolution of the molecular form factor F(z) and the molecular centre of mass distribution f(z) across the layers [43]. The function F(z) may be calculated on the basis of a certain model for layer organization [37, 48]. The distribution function f(z) is usually expanded into a Fourier series f(z) = cos(nqoz), where the coefficients = (cos(nqoz)) are the de Gennes-McMillan translational order parameters of the smectic A phase. According to the convolution theorem, the intensities of the (OOn) reflections from the smectic layers are simply proportional to the square of the translational order parameters t ... 

Madden, H. H., Comments on the Savitzky-Golay Convolution Method for Least-Squares Fit Smoothing and Differentiation of Digital Data, Anal. Chem. 50, 1978, 1383-1386. 

Gorry, P. A., General Least-Squares Smoothing and Differentiation by the Convolution (Savitzky-Golay) Method, Anal. Chem. 62, 1990, 570-573. 

Ratzlaff, K. L., Computation of Two-Dimensional Polynomial Least-Squares Convolution Smoothing Integers, Ana/. Chem. 61, 1989, 1303-1305. 

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... 

Another class of methods such as Maximum Entropy, Maximum Likelihood and Least Squares Estimation, do not attempt to undo damage which is already in the data. The data themselves remain untouched. Instead, information in the data is reconstructed by repeatedly taking revised trial data fx) (e.g. a spectrum or chromatogram), which are damaged as they would have been measured by the original instrument. This requires that the damaging process which causes the broadening of the measured peaks is known. Thus an estimate g(x) is calculated from a trial spectrum fx) which is convoluted with a supposedly known point-spread function h(x). The residuals e(x) = g(x) - g(x) are inspected and compared with the noise n(x). Criteria to evaluate these residuals are Maximum Entropy (see Section 40.7.2) and Maximum Likelihood (Section 40.7.1). 

In order to deduce Scherrer s equation first an infinite crystal is considered that is, second, restricted (i.e multiplied) by a shape function (cf. p. 17). Thus from the Fourier convolution theorem (Sect. 2.7.8) it follows that in reciprocal space each reflection is convolved by the Fourier transform of the square of the shape function - and Scherrer s equation is readily established. 

We turn now to the effect of using the Savitzky-Golay convolution functions. Table 57-1 presents a small subset of the convolutions from the tables. Since the tables were fairly extensive, the entries were scaled so that all of the coefficients could be presented as integers we have previously seen this. The nature of the values involved caused the entries to be difficult to compare directly, therefore we recomputed them to eliminate the normalization factors and using the actual direct coefficients, making the coefficients more easily comparable we present these in Table 57-2. For Table 57-2 we also computed the sums of the squares of the coefficients and present them in the last row. 

However for several of the molecules shown in Figures 1 and 2, DNA has only a small effect on the observed fluorescence lifetime. These molecules include trans-7,8-dihydroxy-7,8-dihydro-BP (15,18,19), trans-4,5-dihydroxy-4,5-dihydro-BP (15,18), BPT (7,18), 1,2,3,4-tetrahydro-BA (12), 8,9,10,11-tetrahydro-BA (14), 5,6-dihydro-BA (12), anthracene (12) and DMA (14). Typical decay profiles obtained in fluorescence lifetime measurements of trans-7,8-dihydroxy-7,8-dihydro-BP and of 8,9,10,11-tetrahydro-BA are shown in Figure 6. The lifetimes extracted from the decay profiles shown here have been obtained by using a least-squares de-convolution procedure which corrects for the finite duration of the excitation lamp pulse (77). 

The relation between the least-squares minimization and the residual density follows from the Fourier convolution theorem (Arfken 1970). It states that the Fourier transform of a convolution is the product of the Fourier transforms of the individual functions F(f g) = F(f)F(g). If G(y) is the Fourier transform of 9(x)-... 

Very popular is the Savitzky-Golay filter As the method is used in almost any chromatographic data processing software package, the basic principles will be outlined hereafter. A least squares fit with a polynomial of the required order is performed over a window length. This is achieved by using a fixed convolution function. The shape of this function depends on the order of the chosen polynomial and the window length. The coefficients b of the convolution function are calculated from ... 

With the aid of Eq. (48), we can show that 6ik (o) = (k + l)N(co) for t(co) = 0. The object estimate consists of noise at frequencies that t does not pass. The noise grows with each iteration. This problem can be alleviated if we bandpass-filter the data to the known extent of z to reject frequencies that t is incapable of transmitting. Practical applications of relaxation methods typically employ such filtering. Least-squares polynomial filters, applied by finite discrete convolution, approximate the desired characteristics (Section III.C.5). For k finite and t 0, but nevertheless small,... 

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