Apodization Weight function

Apodization Weighting function applied to a truncated time-... 

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. 

Tiziani et al. reported an extensive study on the comparison of two weighting functions, sine-bell and sine-bell combined with exponential, and their effects on the 2D /-resolved spectrum resolution and reproducibility. Their tests with dog urine, fish liver extract and leukaemia cell extract samples indicated that combined use of sine-bell and exponential apodization resulted in a better resolution and reproducibility, which should be beneficial in quantitative studies. 

This function is shown in Figure 2.7. As we saw from the discussion of the Rayleigh criterion of resolution, this is also the shape of the ILS function of a diffraction-limited grating monochromator. It can be seen that the amplitude of the sidelobes has been considerably reduced from that of the sidelobes for the sine function. Suppression of the magnitude of these oscillations is known as apodization, and functions such as Ai(S), which weight the interferogram for this purpose, are known as apodization functions. 

This equation holds strictly only for the apodization function A(5) = 1. However, the interferogram can only be recorded over a finite interval, from — max to -l- max, and not beyond these limits. The function A((5) is introduced and set equal to one for 1 1 < Vax and equal to zero for 1 > Vax-The reconstructed spectrum is, therefore, not the true spectrum but the convolution of the true spectrum with the Fourier transform of the rectangular weighting function A(<5). The Fourier transform of the... 

Multiplication with a Processing Function s(t) fCt) Weighting , Filtering , Apodization ... 

Apodization is a mathematical procedure used to overcome the fact that a recorded inter-ferogram is truncated (i.e., does not extend to an infinite distance) and to ensure that the interferogram to be Fourier-transformed terminates smoothly without a step. An explanation of apodization is given in Section 4.4.1. In Figure 5.12 , a trapezoidal apodization function is shown overlaid with a measured interferogram to be weighted by this apodization function. 

On the left is the mean value of 7(v), within the interval Av, weighted by the instrument function, which may have a nonrectangular shape in this case. Narrow-band interference filters are often bell-shaped grating spectrometers and Michelson interferometers with triangular apodization have a (sin a /a ) response function. If one defines I(v) as the spectrum weighted by the narrow instrument function one obtains... 

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