Weighting functions Gaussian

In the case of 3b, Gaussian quadrature can be used, choosing the weighting function to remove the singularities from the desired integral. A variable step size differential equation integration routine [38, Chapter 15] produces the only practicable solution to 3c. 

Heteronuclear-shift-correlation spectra, which are usually presented in the absolute-value mode, normally contain long dispersive tails that are suppressed by applying a Gaussian or sine-bell function in the F domain. In the El dimension, the choice of a weighting function is less critical. If a better signal-to-noise ratio is wanted, then an exponential broadening multiplication may be employed. If better resolution is needed, then a resolution-enhancing function can be used. 

In this section, three relaxation methods are tested and compared Jansson weighting, clipping, and Gaussian weighting characterized by weighting functions given in Chapter 6 by Eqs. (55), (58), and (57), respectively. 

Bruker uses the command EM (exponential multiplication) to implement the exponential window function, so a typical processing sequence on the Bruker is EM followed by FT or simply EE (EF = EM + FT). Varian uses the general command wft (weighted Fourier transform) and allows you to set any of a number of weighting functions (lb for exponential multiplication, sb for sine bell, gf for Gaussian function, etc.). Executing wft applies the window function to the FID and then transforms it. 

It is important to note that these weight functions are exactly the weights of the Gaussian quadrature, this fact will be useful in the discussion of error bounds. 

It is noted that the phonon wavefunction is a superposition of plane waves with q vectors centered at In the literature, several weighting functions such as Gaussian functions, sine, and exponential functions have been extensively used to describe the confinement functions. The choice of type of weighting function depends upon the material property of nanoparticles. Here, we present a brief review about calculated Raman spectra of spherical nanoparticle of diameter D based on these three confinement functions. In an effort to describe the realistic Raman spectrum more properly, particle size distribution is taken into account. Then the Raman intensity 7(co) can be calculated by ... 

Most modem spectrometers have macros that automatically set sine bell and Gaussian weighting functions on the basis of the acquisition parameters. Shift values for the various shifted functions are entered interactively, and the function is then readjusted so that it goes to zero at the end of the tfs in t2 and t. ... 

The Gaussian function is frequently used as the weight function because it is well behaved, easily calculated, and satisfies the conditions required by Parzen s estimator. Thus the probability density function for the multivariate... 

In Gaussian quadrature theory the NDF is called the weight function or measure. The weight function must be nonnegative and non-null in the integration interval and all its moments. 

From Fig. 4.11 (j) we can see that the effect of the rising exponential and the Gaussian is to give an overall weighting function which has a maximum in it. It is usual to define the Gaussian parameter a from the position of this maximum. A little mathematics shows that this maximum occurs at time tmax given by ... 

The basic sine bell is just the first part of a sin 9 for 9 = 0 to 6 = tv, this is illustrated in the top left-hand plot of Fig. 4.13. In this form the function will give resolution enhancement rather like the combination of a rising exponential and a Gaussian function (compare Fig. 4.11 (j)). The weighting function is chosen so that the sine bell fits exactly across the acquisition time mathematically the required function is ... 

V Vu w(a,p,y) w(N, r) W t) wq volume of a polymer segment. 6.1.1.3 scattering volume. 1.2.2 unit cell volume. 3.3.1 crystallite orientation distribution function. 3.6.3 end-to-end distribution of a Gaussian chain. 5.2.1 [5.12] slit-length weighting function. 5.6.1 constant value of W(t) with infinite slit approximation. 5.6.3... 

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