Apodization window function

Figure 1.36 Various selected apodization window functions (a) an unweighted FID (b) linear apodization (c) increasing exponential multiplication (d) trapezoidal multiplication (e) decreasing exponential multiplication (f) convolution differ-...

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. 

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.

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. 

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... 

For FTS data, artifact removal is a consideration that is as important as resolution improvement for most researchers in this field. Interferogram continuation methods are not as yet widely known in this area. Methods currently in widespread use that are effective in artifact removal involve the multiplication of the interferogram by various window functions, an operation called apodization. A carefully chosen window function can be very effective in suppressing the artifacts. However, the peaks are almost always broadened in the process. This can be understood from the uncertainty principle. A window that reduces the function most strongly closest to the end points will yield a transform for the modified function that must be broader than it was originally. Alternatively we may employ the convolution... 

Fig. 15 Apodization, or the reduction of artifacts in the spectral line by the multiplication of the interferogram by a window function that tapers to zero at the end point of the interferogram. (a) Cosine interferogram of Fig. 13(a) multiplied by the triangular window function of Fig. 14(d). (b) Resulting spectral line, the sine-squared function.

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. 

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 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. 

Apodization function. Syn. window function. The mathematical function multiplied by the time domain signal. 

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). 

Fig. 8 The first apodized sine basis functions for. several positions of the. sliding window.

Figure 6.4 The influence of spectral resolution (RES) and zero-filling factor (ZFF) on the detectability of IR spectral features of colon tissue. In this example, identical positions of a tissue sample mounted on a CaF2 window of a thickness of 1 mm were characterised by utilising a Bruker IR Scope II IR microscope. Transmission type IR spectra were recorded using a circular aperture of 900 pm diameter and a Cassegrain objective (36 x, NA 0.5, SR ca. 25 pm). A Happ-Genzel apodization function and a first derivative Savitzky-Golay filter with nine smoothing points were applied to the spectra.

---