Gaussian shape function

The line shapes of the resonance absorption curves were rather well described by a Gaussian shape, as is shown in Fig. 14. Two parameters are necessary in fitting a recorder derivative with a Gaussian shape function the maximum value of the derivative signal (dx"/3 f)m. and the width between points of maximum slope of the absorption, which are... 

Figure 2 A. Fit of a Gaussian shape function to the front half of an experimental chromatogram. B. Residual between the Gaussian and the experimental curve. (Reproduced with permission from Ref. 1. Copyright 1984, Elsevier.)...

Eq represents the amplitude of the electromagnetic field, lj the laser frequency, and s t) the shape function of the laser pulse. Here, a Gaussian shape function approximates the experiments rather well [62, 81, 306, 379]. The functions (z =X, B, 1 E )) denote the time-dependent wave packets... 

In order to properly take into account the instrumental broadening, the function describing the peak shape must be considered. In the case of Lorentzian shape it is Psize = Pexp - instr while for Gaussian shape p = Pl -Pl tr- In the case of pseudo-Voigt function, Gaussian and Lorentzian contributions must be treated separately [39]. 

Fig. 44.23. Some common neighbourhood functions, used in Kohonen networks, (a) a block function, (b) a triangular function, (c) a Gaussian-bell function and (d) a Mexican-hat shaped function. In each of the diagrams is the winning unit situated at the centre of the abscissa. The horizontal axis represents the distance, r, to the winning unit. The vertical axis represents the value of the neighbourhood function. (Reprinted with permission from [70]).

Data from phase-modulation fluorometry have been analyzed using an alternative approach to those described above, as expounded by Gratton and co-workers(14 12 13,22) and Lakowicz et al. W> Here, Lorentzian or Gaussian distribution functions with widths and centers determined by least-squares analysis are used to model the unknown distribution function. While this approach may introduce assumptions about the shape of the ultimate distribution function since these trial functions are symmetric, it has the advantage of minimizing the number of parameters involved in the fit. Here, a minimum x2 is sought, where... 

The presence of the central spot (the primary beam) and diffuse rings Idiff from the film support brings significant errors into estimated intensities. The shape of the primary beam feam can be approximated by one of several peak-shape functions such as pseudo-Voigt, Gaussian or Lorentzian [16], The diffuse background can be described by a polynomial function of order 12. Then equation (1) becomes... 

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 complete powder XRD profile (either for an experimental pattern or a calculated pattern) is described in terms of the following components (1) the peak positions, (2) the background intensity distribution, (3) the peak widths, (4) the peak shapes, and (5) the peak intensities. The peak shape depends on characteristics of both the instrument and the sample, and different peak shape functions are appropriate under different circumstances. The most common peak shape for powder XRD is the pseudo-Voigt function, which represents a hybrid of Gaussian and Lorentzian character, although several other types of peak shape function may be applicable in different situations. These peak shape functions and the types of function commonly used to describe the 20-dependence of the peak width are described in detail elsewhere [22]. 

When the optical length of the sample is much shorter than the other factors such as the length of the electron pulse, the response function takes on a Gaussian shape. As the optical length increases, the shape of the response function becomes trapezoidal. Furthermore, a thick sample causes the prolongation of the electron pulse by electron scattering, which leads to the degradation of time resolution. Therefore the experiment to observe ultrafast phenomena requires the use of a thin sample. 

Let us establish the required relationships more precisely. Consider a narrow idealized rectangular absorption line AT(x) = rect(x/2 AxL) having half-width AxL and centered at x = 0. Its variance is easily found to be <7l = (2 Axl/3)2. Its area is 2 AxL. Now, let us assume that this line is being used to measure an instrument response function exp( —x2/2cr2) that has Gaussian shape and variance ... 

Thus far, the discussion has been restricted to triangular window functions. However, it has been discovered that windows of many other functional forms are capable of bringing about improvement in the spectral lines. In this research the author has found that the window of Gaussian shape has produced the best overall results. With the same interferogram and extension by the same amount as in the previous example, premultiplication by the Gaussian window function shown in Fig. 14(a) produced the restored interferogram shown in Fig. 18(a). The restored spectral line shown in Fig. 18(b) has a resolution much improved over that of Fig. 17(b), where the triangular window function was used, yet the artifacts are no worse. The researcher should explore the various functional forms of the window function to find the one best suited for his or her particular data. 

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