Apodization functions

Having recorded the FID, it is possible to treat it mathematically in many ways to make the information more useful by a process known as apodization (Ernst, 1966 Lindon and Ferrige, 1980). By choosing the right window function and multiplying the digitized FID by it, we can improve either the signal-to-noise ratio or the resolution. Some commonly used apodization functions are presented in Fig. 1.36. 

The signal-to-noise ratio can be increased by treating the data so as to bias the spectrum in favor of the signals and against the noise. This can be done by multiplying the FIDs by the proper apodization functions. What would happen if the spectrum is recorded without apodization ... 

The apodization functions mentioned earlier have been applied extensively in ID NMR spectra, and many of them have also proved useful in 2D NMR spectra. Before discussing the apodization functions as employed in 2D NMR spectra we shall consider the kind of peak shapes we are dealing with. 

Some of the apodization functions described in Section 1.3.11 can also be adopted for use in 2D spectra. The types of functions used will vary according to the experiment and the information wanted. 

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.

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. 

It is instructive to consider a specific example of the method outline above. The triangle fimction (l/l) a (x/l) was discussed in Section 11.1.2. It was pointed out there that it arises in dispersive spectroscopy as the slit function for a monochromator, while in Fourier-transform spectroscopy it is often used as an apodizing function. Its Fourier transform is the function sine2, as shown in Fig. (11-2). The eight points employed to construct the normalized triangle fimction define the matrix... 

An apodizing function is employed to reduce oscillations in an observed spectrum due to discontinuities at the ends of an interferogram. 

There are many other apodization functions which are used for specific types of NMR data, in fact you can make up your own if you want to but for most data sets, the canned ones that are shipped with the instrument are more than adequate. 

FIGURE 32. H NMR spectrum of filipin III, 3 mM in DMSO-dg, recorded at 400 MHz and 25 °C. The expanded region contains nine hydroxylic proton resonances that fully exchange with deuterium oxide and correspond to the nine hydroxyl groups of filipin III. No apodization functions were applied prior to the Fourier transformation. Reproduced by permission of John Wiley Sons from Reference 50... 

Good method for general smoothing, but must select an appropriate apodization function. Best method for removing specific periodic features in the raw data provided the corresponding frequencyCies) can be identified in the apodization domain. 

The discussion in the present section concerns only dispersive optical spectroscopy. We shall not treat the resolution characteristic of a Fourier interferometric spectrometer, which is determined by the optical path difference scanned and the apodizing function used. Nevertheless, these... 

Although a variety of apodization functions have been examined 34,35,36), triangular apodization has the following form... 

An additional consequence of finite retardation is the appearance of secondary extrema or "wings" on either side of the primary features. The presence of these features is disadvantageous, especially when it is desired to observe a weak absorbance in proximity to a strong one. To diminish this problem the interferogram is usually multiplied by a triangular apodization function which forces the product to approach zero continuously for s = + Fourier transformation of the... 

The result for one such apodization function is shown in the right-hand column of Figure 3. The final frequency-domain excitation magnitude spectrum is considerably smoother than without apodization. 

Spectral Manipulation Techniques. Many sophisticated software packages are now available for the manipulation of digitized spectra with both dedicated spectrometer minicomputers, as well as larger main - frame machines. Application of various mathematical techniques to FT-IR spectra is usually driven by the large widths of many bands of interest. Fourier self - deconvolution of bands, sometimes referred to as "resolution enhancement", has been found to be a valuable aid in the determination of peak location, at the expense of exact peak shape, in FT-IR spectra. This technique involves the application of a suitable apodization weighting function to the cosine Fourier transform of an absorption spectrum, and then recomputing the "deconvolved" spectrum, in which the widths of the individual bands are now narrowed to an extent which depends on the nature of the apodization function applied. Such manipulation does not truly change the "resolution" of the spectrum, which is a consequence of instrumental parameters, but can provide improved visual presentations of the spectra for study. 

Figure 4. Apodization functions and their Fourier transforms. The top left function is the boxcar function and its FT is the sine function. Note the large amplitude of the secondary minimum and the narrow full width at half maximum, Ao. The bottom pair of figures show the Hamming function and its FT. The secondary oscillations are smaller but the width has grown.

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. 

Figure 14 Stopped-flow TOCSY CEC-NMR spectra of paracetamol glucuronide. Acquisition parameters number of 144 scans for each of 256 increments. Spectral width of 4716 Hz, number of points 4096. Processing parameters zero filled and multiplied by apodization function of 3 Hz in both dimensions. (From Ref. 52 reproduced with permission from The Royal Society of Chemistry.)...

The physical meaning of Eq. (3.7) is that the true spectrum I (i> ) is scanned with a line-shape function or spectral window 5 (i> — i> ) as in the case of the diffraction grating (see Fig. 10). As mentioned already, in contrast to the grating spectrometer, the spectral window can be varied according to the choice of the apodization function. The advantage of apodization is easily seen for a narrow laser line (cf. Fig. 6). 

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