Weighting functions exponential

During the second time interval, the weighting functions oscillate around zero but with smaller amplitudes as compared to their values in the first time interval. Therefore, the effect of disturbances during these time intervals can be expected to contribute less to the estimation errors. This can also be seen from Figures 2.16-2.18. In the final time interval, the weighting functions exponentially decay to zero and their amplitudes remain relatively small throughout the interval. Therefore, disturbances in the process step... 

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. 

The Newtonian viscosity of some polymers increases essentially linearly with the weight average molecular weight, and for other polymers the Newtonian viscosity increases with an exponential power of the molecular weight. The exponential power is found to be about 3.4, but this power does deviate for some polymers. These two transitions, Newtonian to pseudo-plastic and linear to 3.4 power in the Newtonian range are often related to molecular structure as demonstrated in Fig. 3.31 [22]. The polystyrene data used to develop the Adams-Campbell viscosity function showed almost no shear thinning at [18]. That is why the power law slope, s, is a function of and M. At the slope is zero and the material would be essentially Newtonian. 

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. 

The weighted function employed is the Fermi distribution-like function. Namely, sampling point density is dense around a nucleus and decay exponentially beyond the atomic radius. Then, the weighted volume cos(rk) is derived. 

Deconvolution is manipulation of the interferogram with an exponential weighting function. It may be viewed as the inverse of smoothing. In deconvoluted... 

Here L (u)z) are the unweighted Laguerre polynomials and u) is a scaling factor. These polynomials are orthogonal over the semi-infinite interval (O,00) with respect to the exponential weighting function. In our case, however, z has a range of... 

It is interesting to note that the (L-N-B)-model leads to similar expressions for the moduli like the VTG-model apart from the first summand of Eq. (38). However, contrary to the semi-empirical weighting functions W(y6) of the VTG-model, the corresponding density distribution function/la(y) in the (L-N-B)-model is related to the morphological structure of the filler network, i.e., the distribution of singly connected bonds in a percolation network. Unfortunately, this distribution function is not known, exactly. Therefore, a simple exponential... 

The linear polymerization of DVB was accomplished with Pd(PPh3)2(BF4)2 in acetonitrile-chloroform solvent at temperatures ranging from 40 to 70° C. The polymer contained structural units derived from both head-tail and head-head coupling. The polymerization proceeded by a step-growth mechanism. The molecular weight increased exponentially as a function of time. This reaction was also used for a one-pot synthesis... 

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 ... 

A typical choice for this function - called a weighting function - is an exponential ... 

The decay due to the weighting function on its own would give a linewidth of (R] b/x) Hz. Combining the two terms describing the exponential decays gives... 

The basic idea is simple. All we need to do is to multiply the FID by a weighting function which increases with time, for example a rising exponential ... 

The usual approach is to specify Rre in terms of the linewidth that it would create on its own if it were used to specify a decaying exponential. Recall that in such a situation the linewidth, L, is given by R. /tt. Thus Rre = ttL. The weighting function can therefore be rewritten... 

For compatibility with the linebroadening role of a decaying exponential it is usual to define the exponential weighting function as exp(—jrLt) so that... 

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 ... 

Tiziani et al. reported an extensive study on the comparison of two weighting functions, sine-bell and sine-bell combined with exponential, and their effects on the 2D /-resolved spectrum resolution and reproducibility. Their tests with dog urine, fish liver extract and leukaemia cell extract samples indicated that combined use of sine-bell and exponential apodization resulted in a better resolution and reproducibility, which should be beneficial in quantitative studies. 

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