Sine-bell window function

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. 

Below is the symmetrized, magnitude, COSY-60 spectrum of camphor with the one-dimensional spectrum plotted to the same scale. This spectrum was collected using the full phase cycling normally applied to suppress artefacts. Each FID consisted of IK data points, and 256 increments were used. The data were multiplied by a sine bell window function in both dimensions to improve the appearance of the contour plot. 

The size of the window must be carefully fit to the FID being processed. Varian uses the parameter sb to describe the width (in seconds) of the sine-bell window from the 0° point to the 90° point. Thus for an unshifted sine-bell function, we want the 0° to 180° portion of the sine function (2 sb) to just fit over the time duration of the FID at). This is accomplished by setting the value of sb to one-half the acquisition time sb = at/2. Since the sine-bell is not shifted, the sine-bell shift (sbs) is set to zero. For a cosine-bell or 90° shifted sine-bell window, we want the portion of the sine function from 90° to 180° (or sb, since the 0° to 90° portion is of the same duration as the 90° to 180° portion) to just fit over the FID (duration at) sb = at. In addition, the whole sine function is shifted to the left side by the duration of the FID, so we set the parameter sbs (sine-bell shift) equal to —at (left shift corresponds to a negative number). In F we do not have a parameter for acquisition time at) in t, but we know that the maximum t value is just the number of data points times the sampling delay ... 

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. 

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. 

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 ... 

A very simple window function for resolution enhancement is the sine bell (Fig. 3.34), which is just the function sin(x) for x = 0 to 180°. This function grows for the first half of the FID and then brings the signal smoothly to zero during the second half. We saw examples of this window in Chapter 2 (Figs. 2.9 and 2.10). We will see that the sine-bell family of... 

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. 

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. 

Span of sine-bell function in seconds (0° to 90°). Amount of right shift of sine-bell function in seconds. Window function (SINE, QSINE, etc). 

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... 

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