Boxcar truncation function

Equation (5.9) shows that in order to measure the complete spectrum, we would have to scan the moving mirror of the interferometer an infinitely long distance, with (5 varying between -oo and +cx) centimeters. In practice, the optical path length difference is finite. By restricting the maximum retardation to /, we are effectively multiplying the complete interferogram by the boxcar truncation function (see Fig. 5.3a left)... 

By sampling a finite path difference A another instrumental effect is introduced to the interferogram. Effectively, the complete interferogram (from —oo to oo) is multiplied by a boxcar truncation function, D x), which is... 

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. 

In view of the shape of this function, D(5) is often called a boxcar truncation function. By analogy to Eq. 2.13, the spectmm in this case is given by the equation... 

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 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. 

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. 

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