Sine wiggles

Answer The FID was truncated by the short acquisition time ( 1.3.3 and 1.3.4). The sharp cut-off at the end of the FID has led to sine wiggles around each peak. This problem can be avoided either by using a longer acquisition time or by applying an apodization function that forces the FID to zero. 

Fig. 5.3.5 [Cal2] The sine pulse, (a) The amplitude of the rf carrier is modulated by a truncated sine function, (b) The magnitude of the Fourier transform of the pulse is a rectangular function distorted by wiggles in the centre and near the edges.

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. 

Fig. 4.15 Illustration of how truncation leads to artefacts (called sine wiggles) in the spectrum. The FID on the left has been recorded for sufficient time that it has decayed almost the zero the corresponding spectrum shows the expected lineshape. However, if data recording is stopped before the signal has fully decayed the corresponding spectra show oscillations around the base of the peak.

As is shown clearly in the figure, a truncated FID leads to oscillations around the base of the peak these are usually called sine wiggles or truncation artefacts - the name arises as the peak shape is related to a sine function. The more severe the truncation, the larger the sine wiggles. It is easy to show that the separation of successive maxima in these wiggles is l/tacq Hz. 

Fourier transformation of a FID, which has not decayed to zero intensity causes a distortion ("wiggles") at the base of peaks in the spectrum. By applying a suitable window function WDW the FID will decay smoothly to zero. A variety of window functions options are available, none, exponential EM, gaussian GM, sine SINE, squared sine QSINE and trapezoidal TRAP function. The best type of window function depends on the appearance of the FID and the resulting spectrum. Consequently where possible it is best to fit the window function interactively. 

Thus, finite acquisition time causes a convolution of NMR spectrum with sine function. This manifests itself in peak broadening and presence of sine wiggles . The broadness of the NMR peak is thus dependent not only on relaxation rate but also on the maximum evolution time. Both effects correspond to the Fourier Uncertainty Principle [53] stating that, in general, the broadness of time representation and frequency representation are inversely proportional to each other. 

A similar approach was presented by Keeler in application to heteronuclear J spectra with highly truncated echo modulation [87]. Truncation of signal, used for sensitivity reasons, results in sine wiggles . These artifacts can be suppressed by apodization, although at the expense of resolution. Keeler showed that CLEAN is an inexpensive alternative to the maximum entropy method, which can also remove truncation artifacts without degrading resolution. 

Fig. 4.56. The effects of Fourier transforms on different input signals, (a) A step function is converted to sine (x), (b) a fully dampened transient signal containing only one frequency yields a single Lorentzian curve, and (c) the truncated transient can be regarded as a combination of (a) and (b) and yields a Lorentzian curve accompanied by wiggles .

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