Window functions

Having recorded the FID, it is possible to treat it mathematically in many ways to make the information more useful by a process known as apodization (Ernst, 1966 Lindon and Ferrige, 1980). By choosing the right window function and multiplying the digitized FID by it, we can improve either the signal-to-noise ratio or the resolution. Some commonly used apodization functions are presented in Fig. 1.36. 

To increase the signal-to-noise ratio, we need to multiply the FIDs by a window function that will reduce the noise and lead to a relative increase in signal strength. Since most of the signals lie in the head of the FID while its tail contains relatively more noise, we multiply the FID by a mathematical function that will emphasize the head of the FID and suppress its tail. ... 

Figure 1.36 Various selected apodization window functions (a) an unweighted FID (b) linear apodization (c) increasing exponential multiplication (d) trapezoidal multiplication (e) decreasing exponential multiplication (f) convolution differ-...

The weighting functions used to improve line shapes for such absolute-value-mode spectra are sine-bell, sine bell squared, phase-shifted sine-bell, phase-shifted sine-bell squared, and a Lorentz-Gauss transformation function. The effects of various window functions on COSY data (absolute-value mode) are presented in Fig. 3.10. One advantage of multiplying the time domain S(f ) or S(tf) by such functions is to enhance the intensities of the cross-peaks relative to the noncorrelation peaks lying on the diagonal. 

Figure 3.10 Effect of different window functions (apodization functions) on the appearance of COSY plot (magnitude mode), (a) Sine-bell squared and (b) sine-bell. The spectrum is a portion of an unsymmetrized matrix of a H-COSY I.R experiment (400 MHz in CDCl, at 303 K) of vasicinone. (c) Shifted sine-bell squared with r/4. (d) Shifted sine-bell squared with w/8. (a) and (b) are virtually identical in the case of delayed COSY, whereas sine-bell squared multiplication gives noticeably better suppression of the stronger dispersion-mode components observed when no delay is used. A difference in the effective resolution in the two axes is apparent, with Fi having better resolution than F. The spectrum in (c) has a significant amount of dispersion-mode line shape.

The sine-bell functions are attractive because, having only one adjustable parameter, they are simple to use. Moreover, they go to zero at the end of the time domain, which is important when zero-filling to avoid artifacts. Generally, the sine-bell squared and the pseudoecho window functions are the most suitable for eliminating dispersive tails in COSY spectra. 

What are the effects of various window functions on the shapes of peaks... 

If phase-sensitive spectra are not required, then magnitude-mode Pico) (or absolute-mode ) spectra may be recorded by combining the real and imaginary data points. These produce only positive signals and do not require phase correction. Since this procedure gives the best signal-to-noise ratio, it has found wide use. In heteronuclear experiments, in which the dynamic range tends to be low, the power-mode spectrum maybe preferred, since the S/N ratio is squared and a better line shape is obtained so that wider window functions can be applied. 

There are generally three types of peaks pure 2D absorption peaks, pure negative 2D dispersion peaks, and phase-twisted absorption-dispersion peaks. Since the prime purpose of apodization is to enhance resolution and optimize sensitivity, it is necessary to know the peak shape on which apodization is planned. For example, absorption-mode lines, which display protruding ridges from top to bottom, can be dealt with by applying Lorentz-Gauss window functions, while phase-twisted absorption-dispersion peaks will need some special apodization operations, such as muliplication by sine-bell or phase-shifted sine-bell functions. 

The sine-bell, sine-bell squared, phase-shifted sine-bell, and phase-shifted sine-bell squared window functions are generally used in 2D NMR spectroscopy. Each of these has a different effect on the appearance of the peak shape. For all these functions, a certain price may have to be paid in terms of the signal-to-noise ratio, since they remove the dispersive components of the magnitude spectrum. This is illustrated in the following COSY spectra ... 

Ifourth(fd, 2 Q) was multiplied with a window function and then converted to a frequency-domain spectrum via Fourier transformation. The window function determined the wavenumber resolution of the transformed spectrum. Figure 6.3c presents the spectrum transformed with a resolution of 6cm as the fwhm. Negative, symmetrically shaped bands are present at 534, 558, 594, 620, and 683 cm in the real part, together with dispersive shaped bands in the imaginary part at the corresponding wavenumbers. The band shapes indicate the phase of the fourth-order field c() to be n. Cosine-like coherence was generated in the five vibrational modes by an impulsive stimulated Raman transition resonant to an electronic excitation. 

Figure 9. Data reduction and data analysis in EXAFS spectroscopy. (A) EXAFS spectrum x(k) versus k after background removal. (B) The solid curve is the weighted EXAFS spectrum k3x(k) versus k (after multiplying (k) by k3). The dashed curve represents an attempt to fit the data with a two-distance model by the curve-fitting (CF) technique. (C) Fourier transformation (FT) of the weighted EXAFS spectrum in momentum (k) space into the radial distribution function p3(r ) versus r in distance space. The dashed curve is the window function used to filter the major peak in Fourier filtering (FF). (D) Fourier-filtered EXAFS spectrum k3x (k) versus k (solid curve) of the major peak in (C) after back-transforming into k space. The dashed curve attempts to fit the filtered data with a single-distance model. (From Ref. 25, with permission.)...

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)...

User-defined functions such as rkint and f-ruteeq can be accessed by clicking on user-defined functions under the equations menu. Once within this window, functions can be imported and edited as required. User-defined functions are specific to a particular file. Thus, they must be imported into each problem file where they are used. 

Assuming that the maximum target velocity is equal to 3M (lOOOm/s), the maximum value of the time-bandwidth product is limited to 150,000. The maximum processing gain is then equal to 51.7 dB. The use of a windowing function will decrease this value by a few dB. The maximum detection range for the noise radar is given by the formula... 

Fig. 13. 13Ca-1HN planes from the HN(CO)CANH-TROSY (a) and HN(CO)CA-TROSY (b) spectra. Spectra were recorded on uniformly 15N, 13C, 2H enriched, 30.4 kDa protein Cel6A at 800 MHz at 277 K. The data were measured using identical parameters and conditions, using 8 transients per FID, 48, 32, 704 complex points corresponding acquisition times of 8, 12, and 64 ms in tly t2, and <3, respectively. A total acquisition time was 24 h per spectrum. The data were zero-filled to 128 x 128 x 2048 points before Fourier transform and phase-shifted squared sine-bell window functions were applied in all three dimensions. 

A pre-requisite for the successful extraction of key NMR parameters from an experimental spectrum is the way it is processed after acquisition. The success criteria are low noise levels, good resolution and flat baseline. Clearly, there are also experimental expedients that can further these aims, but these are not the subject of this review per se. In choosing window functions prior to FT, the criteria of low noise levels and good resolution run counter to one another and the optimum is just that. Zero filling the free induction decay (FID) to the sum of the number acquired in both the u and v spectra (in quadrature detection) allow the most information to be extracted. 

The final density calculation step is the stage at which the 2-D weighting density window function is first employed. The 2-D filter of the correct... 

Figure 7 demonstrates the effects of window functions on the signal to noise ratio of the 5r,r spectrum (for D-HMBC) and 5r,i spectrum. 5r,r spectra shown in fig. 7(c) revealed marked decrease of -noise in the range between 0.8 to 1.1 ppm. Processing of S r r spectral data, however, requires somewhat tedious phase adjustment of the Fi spectrum. In order to achieve this easily, it is recommended to take a ID-NMR spectrum at first under... 

It is found that multiplication of the Fourier transform of the data by a carefully chosen window function is very effective in removing the artifacts around peaked functions. This process is called apodization. Apodization with the triangular window function is often applied to Fourier transform spectroscopy interferograms to remove the ringing around the infrared... 

For FTS data, artifact removal is a consideration that is as important as resolution improvement for most researchers in this field. Interferogram continuation methods are not as yet widely known in this area. Methods currently in widespread use that are effective in artifact removal involve the multiplication of the interferogram by various window functions, an operation called apodization. A carefully chosen window function can be very effective in suppressing the artifacts. However, the peaks are almost always broadened in the process. This can be understood from the uncertainty principle. A window that reduces the function most strongly closest to the end points will yield a transform for the modified function that must be broader than it was originally. Alternatively we may employ the convolution... 

One window function widely used is the triangular window function. This function is shown in Fig. 14(d). Multiplying the interferogram of Fig. 13(a) by this window function produces the altered interferogram shown in Fig. 

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).

Fig. 17 Effectiveness in removing the artifacts from the spectrum of multiplying the interferogram by the proper window function before extending the interferogram by a finite number of points, (a) Cosine interferogram of Fig. 13(a) premultiplied by the triangular window function of Fig. 14(b) before extending by 50 data points, (b) Restored spectral line.

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. 

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... 

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