Gaussian window function

Fig. 12 C -detected C CSA patterns of the SHPrP109 i22 fibril sample. The upper and lower traces correspond to the experimental and simulated spectra, respectively. Simulations correspond to the evolution of a one-spin system under the ROCSA sequence. The only variables are the chemical shift anisotropy and the asymmetry parameter. A Gaussian window function of 400 Hz was applied to the simulated spectmm before the Fourier transformation. (Figure and caption adapted from [164], Copyright (2007), with permission from Elsevier)...

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. 

Fig. 22 Interferogram of Fig. 19(a) multiplied by a Gaussian window function before extension in an attempt to remove the artifacts. (a) Interferogram premultiplied by the Gaussian window function of Fig. 14(a) before extending by 50 data points, (b) Restored spectral line. 

If we multiply the interferogram by the triangular window function of Fig. 14(b) before extension, we obtain the interferogram and spectral lines shown in Fig. 29. The artifacts have been considerably reduced, but a slight loss of resolution has occurred. The best overall results are obtained by premultiplying the interferogram by the Gaussian window function of Fig. 14(a) before extension. These results are shown in Fig. 30. The spectral lines of... 

Fig. 5. Spectrogram of periodically oscillating components of pump-probe signals of polyacetylene probed at 750 nm shown in Fig. 4 and calculated using a Gaussian window function with a HWHM At = 96 fs. S and D denote the stretching modes of single and double bonds respectively. Short-lived satellite-bands (S , S and D , D associated with S and D modes, respectively) indicate the modulation induced by the breather state.

Fig. 14 Four window functions used to multiply the interferogram. (a) Gaussian window, (b) Triangular window, (c) Triangular window of greater slope than (b). (d) Triangular window tapering to zero at the end point of the interferogram (used for apodiza-tion).

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. 

Figure 3.34. Some commonly employed window functions. These are used to modify the acquired FID to enhance sensitivity and/or resolution (lb = line broadening parameter, gb = Gaussian broadening parameter i.e. the fraction of the acquisition time when the function has its maximum value see text)...

Explore the effect of other window functions, such as the triangular function (see exercise 7.5-1 under instruction 24), a trapezoidal function, a Gaussian, a Lorentzian, or whatever. 

Fourier transformation of a FID, which has not decayed to zero intensity causes a distortion ("wiggles") at the base of peaks in the spectrum. By applying a suitable window function WDW the FID will decay smoothly to zero. A variety of window functions options are available, none, exponential EM, gaussian GM, sine SINE, squared sine QSINE and trapezoidal TRAP function. The best type of window function depends on the appearance of the FID and the resulting spectrum. Consequently where possible it is best to fit the window function interactively. 

The kernel defined by Eqs. (2.5) will be inefficient when pB is multimodal. In this situation we must dissect the conformation space into separate macrostate regions a, b, c,. .. and find Gaussian kernels that match pB in each region. This can be accomplished using window functions and characteristic packets, concepts that have previously been introduced in the context of global optimization [14,15], potential-energy landscape analysis... 

Wavelets are a set of basis functions that are alternatives to the complex exponential functions of Fourier transforms which appear naturally in the momentum-space representation of quantum mechanics. Pure Fourier transforms suffer from the infinite scale applicable to sine and cosine functions. A desirable transform would allow for localization (within the bounds of the Heisenberg Uncertainty Principle). A common way to localize is to left-multiply the complex exponential function with a translatable Gaussian window , in order to obtain a better transform. However, it is not suitable when <1) varies rapidly. Therefore, an even better way is to multiply with a normalized translatable and dilatable window, v /yj,(x) = a vl/([x - b]/a), called the analysing function, where b is related to position and 1/a is related to the complex momentum. vl/(x) is the continuous wavelet mother function. The transform itself is now... 

The window function, w s) = w(ti,T2, ts), will typically be a low-pass filter. An implication of the windowing function is a smoothing of the dip and azimuth estimate. We have had good experience using a Gaussian low-pass filter,... 

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