Sine-shaped function

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. 

Fig. 5a-e FID of compound 1. a Original data b multiplied by a negative line broadening function (-0.3 Hz) c multiplied by a shaped sine bell function (SSB = 1) d multiplied by a positive line broadening function (0.8 Hz) e multiplied by a positive line broadening function (1.9 Hz)... 

The other plots are made with the software TABLECURVE. The special function F2 used there is a log-normal relation and F3 is a sine-wave function. Usually a ratio of low degree polynomials also provides a good fit to bell-shaped curves here five constants are needed. The Gamma distribution needs only one constant, but the fit is not as good as some of the other curves. The peak, especially, is missed. 

One should note that the phase shift becomes time-independent and maximal for a = 1, i.e., at the resonance condition v = vG. The frequency spectrum 4>(a) bears a sine shape with a bandwidth inversely proportional to the number of oscillations of the gradient field (Fig. 4). Such a behaviour was also predicted in Ref. 15. Recording in a systematic way the phase shift as a function of vG without space encoding would be a very fast and efficient method to scan in a whole object the possible frequencies of spin motions. 

Fig. 19, an unapodized spectrum [response function (sin nx)/nx = sinc(x)] is shown in trace (b). For such a spectrum there will be sidelobes and negative absorption if the natural linewidths are narrower than the full width of the sine-shaped response function. These are seen in Fig. 19, where the linewidth is three points and the response function width eight points. Here the phrase instrument response function may have a slightly different definition, but the meaning is clear. For such a response function, the direct deconvolution methods fall short. 

The raw bandpass of an AOTF has a sine squared function line shape with sidebands, which if ignored may amount to 10% of the pass optical energy in off-centre wavelengths. This apodisation issue is normally addressed by careful control of the transducer coupling to the crystal. 

FT of the sudden step within the FID, the result of which is described by the function (sin x)/x, also known as sine. r. Fig. 3.16 shows that this produces undesirable ringing patterns that are symmetrical about the base of the resonance, often referred to as sine wiggles . To avoid this problem it is essential to either ensure the acquisition time is sufficiently long, to force the FID to decay smoothly to zero with a suitable shaping function (Section 3.2.7) or to artificially extend the FID by linear prediction. 

A sine-shape has side lobes which impair the excitation of a distinct slice. Other pulse envelopes are therefore more commonly used. Ideally, one would like a rectangular excitation profile which results from a sine-shaped pulse with an infinite number of side lobes. In practice, a finite pulse duration is required and therefore the pulse has to be tmneated, which causes oscillations in the excitation profile. Another frequently used pulse envelope is a Gaussian function ... 

We have introduced apodisation as a weighting of the signal, but we can just as well view it as a weighting of the Fourier basis functions. The sines and cosines become squeezed down at the ends, as illustrated by Fig. 24. To the left it shows a sine base function, a gaussian apodisation that is chosen narrow in order to amplify its effect, and the resulting apodised base function that has the shape of a ripple. 

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 spectral results of the simulation are shown in Fig. 5.7 (left) for the central pixel of the gaussian source (blue), the point source (green) and the central pixel of the elliptical source (red). It can be observed that the emission and absorption line positions are detected but present a sine-shape this is due to the boxcar function... 

Figure 2. The sine x function instrumental line shape function of a perfectly aligned Michelson interferometer with no apodization...

Figure 2.5. (a) Fourier transform of a boxcar function of unit amplitude extending from +A to —A this function has the shape of a sin x/x or sine x, function, (b) Fourier transform of an unweighted sinusoidal interferogram generated by a monochromatic line at wavenumber Vi the maximum retardation for this interferogram was A centimeters. 

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 ... 

Sine-beU An apodization function employed for enhancing resolution in 2D spectra displayed in the absolute-value mode. It has the shape of the first halfcycle of a sine function. 

Fig. 1.10 Soft rf pulses (left) in the shape of a sine (sin x/x) function, and their Fourier transforms (right), being equivalent to the excited slice in the presence of a constant magnetic field gradient. The well defined sine function (top) produces an excitation that is a slice...

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