Apodization functions boxcar

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.

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.

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.

Apodization is the modification of the interferogram by multiplication with an apodization function (Griffiths and de Haseth 2007). If the interferogram is unweighted, the shape of a spectral line is the convolution of the spectrum with a sine function, which is the Fourier transform of the boxcar truncation function. 

Fig. 2.5 Apodization functions (left) and corresponding Instrumental Line Shape (right) boxcar (blue), triangular (green) and squared triangular (red)...

FD giving the Fourier spectrum bl. This spectrum is then multiplied by an apodization function (a boxcar function is shown in b2). Finally, the Fourier transform of the truncated Fourier spectrum is computed, yielding the smoothed spectrum. 

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. 

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 2.7. The sinc instrument lineshape function computed for triangularly apodized interferograms note that its full width at half-height is greater than that of the sine function resulting ftom boxcar truncation of the same interferogram.

Figure 8.1. Variation of the measured, or apparent, absorbance, Ap, at the peak of a Lorentzian band as a function of the true peak absorbance, Ap j, plotted on a logarithmic scale for Lorentzian bands measured with no apodization (boxcar truncation). A, p = 0 B, p = 1.0 C, p = 3 D, p = 10 E, p = 25 F, p = 50. (Reproduced from [2], by permission of the American Chemical Society copyright 1975.)...

Boxcar truncation of the interferogram results in a sine function which has side lobes. The reduction in the side lobes on the spectral lines observed can be accomplished by apodization. Triangular apodization gives a sine function with the side lobes considerably reduced. The reduction in side lobes is accomplished at the expense of a some loss in spectral resolution. 

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