Norton-Beer function

Simulated spectra, as well as measured ones, are apodised with the Norton-Beer strong function [10], in order to reduce the interference of nearby lines. 

Figure 5.3. Various apodization functions (left) and the instrumental lineshape produced by them (right) (a) boxcar truncation (b) triangular (c) trapezoidal (d) Norton-Beer weak, medium, and strong (e) Happ-Gen-zel (f) Blackman-Harris 3-term and 4-term. The maximum retardation is set to / = 1. In the Fourier transform the FWHH of the main lobe is indicated.

Effect of the Finite Record Length Leakage and Apodization 47 Table 5.1. Coefficients of the Norton-Beer apodization functions. 

To reduce the artifacts due to the discrete Fourier transformation (see also Chapter 5) choose an appropriate function on the Apodization page depicted in Fig. 10.42. For standard measurements of liquid or solid samples, the Black-man-Harris-3-term is recommended. To obtain the highest resolution, choose no apodization (Boxcar) or if necessary a weak apodization (Norton-Beer-weak). 

Figure 10.46. The Raman spectrum of sulphur in the spectral range 100-280 cm calculated by Fourier transformation of the interferogram top, no apodization (boxcar) bottom, apodization function Norton-Beer weak. In both cases, a zerofilling factor of 2 and the power spectrum for phase correction were chosen. Further parameter used Store page selected frequencies for file first 9394 and last 5894 Limit data page limit resolution to 4 cm, limit phase resolution to 32 cm, direction both, data points both Peak search page mode absolute largest value, symmetry of the interferogram automatic.

The FWHH of the ILS function given by the Happ-Genzel function is close to that of the Norton-Beer strong apodization function. Another common function that strongly suppresses the sidelobe amplimde is the Blackman-Harris function,... 

The FWHH of this function is even wider than that of the Norton-Beer strong function and so in general is not recommended. (A similar function is the Blackman function that uses the first three terms instead of four, but the coefficients are different.)... 

Table 2.1. Coefficients Q for the Norton-Beer Apodization Functions...

There are other apodization functions in use besides the Norton-Beer and the Happ-Genzel. These functions have not been tested as thoroughly however, it has been noted by the authors that the Blackman-Harris and Blackman apodization functions (Section 2.4) are considerably broader than the Norton-Beer medium or Happ-Genzel apodization functions. It is expected that these functions will not adhere to Beer s law as well as the Norton-Beer medium function. 

It is possible to purchase variable-pathlength liquid transmission cells, which commonly have a pathlength that can be varied between 5 pm and 5 mm. A suitable sample must be chosen, preferably a neat solvent that has relatively wide peaks, the resolution of the spectrometer must be set so that the resolution parameter, p, is no greater than 0.5, and a Norton-Beer medium apodization function should be applied. A series of measurements at increasing pathlengths should yield a linear Lambert plot if the instrument is functioning correctly. Some instruments will exhibit linear Lambert plots up to 4 AU however, this is unusual, and an acceptable... 

The most common apodization functions are those proposed by Norton and Beer (1976) and the triangle function (Griffiths and de Haseth 2(X)7). Norton and Beer tested over 1000 functions of the general form... 

Figure 2.9. Normalized height of the largest secondary lobe for a series of apodization functions studied by Norton and Beer (S, ftom Figure 2.8) as a function of their half-width relative to the corresponding sine function. (Reproduced from [9], by permission of the Optical Society of America copyright 1977.)...

Figure 2.10. (a) Weak, medium, and strong apodization functions proposed by Norton and Beer (b) corresponding lineshape functions with the sine function shown for reference. 

Beer, R. Norton, R. H. (1988). Analysis of spectra using correlation functions. Applied Optics, 27, 1255-61. 

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