Sine bell function, shifted

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. 

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

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.

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 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 ... 

MHz, solvent CDC13 and homodecoupling spectra processing with sine bell function (sb = 1) tables with chemical shift in ppm and Hz for the determination of coupling constants... 

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 ... 

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). 

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

Most modem spectrometers have interactive software that permits the operator to vary a particular weighting parameter and immediately observe the effect of this change on the final spectmm. The first increment generally is selected to see what the effects of weighting parameters are on the 2 data. With the first FID displayed, the operator can then shift the midpoint of the sine bell function toward the S/N (left) part of the FID, until the best compromise is reached between sensitivity and resolution. It is important for the operator to remember that the weighting function must have a zero value at the right end of the FID so that it drives the FID to zero intensity at the end of (i) in 2 and (ii) the effective (ni/swi) in i. 

Two-Dimensional Semiquantitative NMR. Spectra were collected on a Varian Unity 400 Fourier transform NMR spectrometer at 161.90 MHz. The phosphates were prepared at 3-5 weight per cent in water and a DjO insert used for locking purposes. Homonuclear 2DJ-resolved spectra were accumulated using an 8K X 0.2X data set with an acquisition time in the F dimension of 0.946 sec, four steady state pulses, 128 transients, and 200 increments in the F domain. Spectra were analyzed with zero-filling to 16K X 0.5K and application of a sine bell or shifted sine bell weighting function on a Sun Microsystems Sparc 1+ computer. 

Fig. 4. High-frequency shifted portions of the 500 MHz H NMR spectra of sperm whale metMbiCN ) at pH 9.2, 30 "C. (A) In 90 10 = H2O H2O. (B) In 49 51 = H2O H2O. The expanded plots show the three resolved heme methyl signals after resolution enhancement by sine-bell function the vertical scale is arbitrary, frequency scale magnified by a factor of 5. Asymmetry in the expanded peaks is due to overlapping lines of different width. (From ref. 79, 1987, with permission from the publishers.)... 

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. 

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). 

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