Apodizing functions Fourier transforms

Clicking the Execute button the window function is applied to the FID whilst clicking the Exec./FT button performs both apodization and Fourier transformation. Clicking the Window button using the left mouse button applies the window function currently defined in the Window Function dialog box to the FID. Check it 3.2.33 demonstrates how the lineshape of the zero filled FIDs in Check it 3.2.3.2 can be improved using apodization. 

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... 

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... 

It is found that multiplication of the Fourier transform of the data by a carefully chosen window function is very effective in removing the artifacts around peaked functions. This process is called apodization. Apodization with the triangular window function is often applied to Fourier transform spectroscopy interferograms to remove the ringing around the infrared... 

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... 

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. 

A(8) is called the boxcar function. This limit on the retardation leads to a limit on the resolution of 1/2L, so if L - 100 cm, the highest resolution attainable is 0.005 cm-1. By the convolution theorem, the product of two functions in one space is the same as the convolution of the Fourier transforms of the two functions in the reciprocal space. The effect of multiplying by this boxcar function is to convolve each point in the reciprocal wavenumber space with a sine function [sinc(x) = sin(x)/x Figure 4], An undesirable feature of the sine function as a lineshape is the large amplitude oscillation (the first minimum is -22% of the maximum). This ringing can make it difficult to get information about nearby peaks and leads to anomalous values for intensities. This ringing can be removed by the process known as apodization. 

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.

The Fourier transform necessary to convert I (s) into /(l ) may be executed either in an analogue or in a digital way. In this section, we shall concentrate on the first possibility. One potential way is illustrated in Fig. 19 the movable mirror of the Michelson interferometer is moved according to a triangular wave function (Fig. 19 upper left). This can be done by means of an appropiate mechanical cam system. Problems arise only at the points of reversal, but usually these difficulties are bypassed by the apodization. During half a period the mirror moves with constant velocity, say uq- Hence s=2vat and, for monochromatic radiation, there is an a.c. component in the signal recorded by the detector ... 

The Polytec FIR 30 provides the parabola fit and the Coderg FS 4000 a special electronic phase error correction. All instruments with the fast Fourier transform (FFT) correct phase errors in the interferogram mathematically according to a method first proposed by M. Forman 88,70) xhis correction procedure was outlined in detail in Section 5.3 (cf. Fig. 41). In addition to Fourier transform and phase error correction, it is advisable to use apodization in Fourier spectroscopy (cf. Sections 2.3 and 3.2). In all commercial instruments, the operator has the choice among a number of different apodization functions. 

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. 

Load the configuration fiie ch3235.cfg. Seiect the Go I Run Experiment command from the pull-down menubar to start the simulation. Process the FID prior to Fourier transformation using zero filling (Sl(r+i) 64K) and apodization (wdw function EM, LB 2.0 Hz). The Fourier transformed FID should resemble the spectrum in Fig. 3.10a. 

Fourier ripples. Constantly spaced bumps in the frequency spectrum found on either side of a peak. In a 1 -D spectrum or a half-transform 2-D data matrix, these ripples are found when the apodized intensity has not faded to the level of the background noise by the time the digitization of the FID ceases. In a fully transformed 2-D spectrum, Fourier ripples parallel to the f, frequency axis are observed when an inappropriate t apodization function is used prior to conversion of the t, time domain to the f, frequency domain. 

The use of certain apodization functions improves the frequency resolution we obtain in our Fourier-transformed spectrum, but caution should be exercised when employing this technique. The use of negative line broadening and shifted Gaussian or squared sine bells (with the maximum to the right of the start of the FID) can be used to resolve a small peak that formerly appeared as the shoulder of a larger peak, but supervisors and reviewers frown upon the excessive application of these methods the starting NMR spectroscopist would do well to exercise restraint in this area. 

Resolution enhancement The application ofan apodization function that emphasizes the later portions of the digitized FID (at the expense of the signal-to-noise ratio) that, upon Fourier transformation, will generate a spectrum with peaks whose widths are narrower than their natural line widths. 

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