Shifted squared sine bell function

Shifted squared sine bell function. An apodization function with the amplitude of a squared sinusoidal pattern starting at a maximum and dropping to zero. The first quarter of a squared cosine waveform. 

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. 

Figure 7-6 Weighting functions. (a) Sine bell, (b) Squared sine bell, (c) Shifted sine bell.

A simple way to do this is to multiply by a symmetrical shaping function, such as the sine-bell function (Marco and Wuethrich, 1976), which is zero in the beginning, rises to a mciximum, and then falls to zero again, resembling a broad inverted cone (Fig. 1.36g). One problem with this function is that we cannot control the point at which it is centered, and its use can lead to severe distortions in line shape. A modification of the function, the phase-shifted sine bell (Wagner et al, 1978) (Fig. 1.36h), allows us to adjust the position of the maximum. This leads to a lower reduction in the signal-to-noise ratio and improved line shapes in comparison to the sine-bell function. The sine-bell squared and the corresponding phase-shifted sine-bell squared functions have also been employed (see Section 3.2.2. also). 

The use of certain apodization functions improves the frequency resolution we obtain in our Fourier-transformed spectrum, but caution should be exercised when employing this technique. The use of negative line broadening and shifted Gaussian or squared sine bells (with the maximum to the right of the start of the FID) can be used to resolve a small peak that formerly appeared as the shoulder of a larger peak, but supervisors and reviewers frown upon the excessive application of these methods the starting NMR spectroscopist would do well to exercise restraint in this area. 

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, 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 ... 

So you can just set sbl = nilswl and sbsl = —sbl for a 90°-shifted sine-bell, and sbl = nil(2 x swl) and sbsl = 0 for an unshifted sine-bell. Bruker uses the parameter wdw (in both F and To) to set the window function (SINE = sine-bell, QSINE = sine-squared, etc.) and ssb for the sine-bell shift. For example, if ssb = 2, the sine function is shifted 90° (180°/ssb) and we get a simple cosine-bell window. For an unshifted sine-bell, use ssb = 0. 

Apodization is the process of multiplying the FID prior to Fourier transformation by a mathematical function. The type of mathematical or window function applied depends upon the enhancement required the signal-to-noise ratio in a spectrum can be improved by applying an exponential window function to a noisy FID whilst the resolution can be improved by reducing the signal linewidth using a Lorentz-Gauss function. ID WIN-NMR has a variety of window functions, abbreviated to wdw function, such as exponential (EM), shifted sine-bell (SINE) and sine-bell squared (QSINE). Each window function has its own particular parameters associated with it LB for EM function, SSB for sine functions etc. 

FID of row 71 and unshifted, Sine-Bell FID of row 71 and 7r/2-shifted, Sine-Bell squared window function squared window function... 

It is apparent from Check it 3.3.2.1 that the 7i/2-shifted Sine-Bell squared window function is the most appropriate apodization procedure for the 2D IR phase sensitive COSY spectrum, see Fig. 3.16. The reason that the Sine-Bell squared function is the best choice is because the last data points are zero and this type of window function ensures that there is no discontinuity in the FID. However the position of the function also has an important effect on the intensity of the data points in the first third of FID and this is why several values of SSB should be tried prior to making a final selection. 

D data processing consisted of zero-filling once and apodization by a 90°-shifted sine-square-bell function in both dimensions before Fourier transformation. Proton and carbon-13 chemical shifts are referenced to TSP as an internal standard. Phosphorus-31 chemical shifts are referenced to 85% phosphoric acid as an external standard. 

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