Boxcar truncation

This forced termination is known as boxcar truncation, and it effects on the spectrum are known as ringing (figure A.5.)... 

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)... 

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.

The choice of a particular apodization depends on what one is aiming at. If the optimum resolution of 0.605// is mandatory, the boxcar truncation (no apodization at all) should be chosen. If a loss of resolution of 50% compared to the boxcar can be tolerated, the Happ-Genzel, or even better, the Blackman-Harris three term apodization is recommended. Since the Blackman-Harris window shows the highest side lobe suppression and is furthermore nearly zero at the interval ends, it can be considered the top performer. 

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 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.)...

Figure 10.1. Unapodized spectra of carbon monoxide from 2110 to 2075 cm showing the effect of zero filling spectra were measured at an effective resolution of 2cm with boxcar truncation (a) one independent and one interpolated point per resolution element (i.e., N zeros were added to the A-point interferogram) (b) one independent and seven interpolated points per resolution element, or 3A zeros added to the A-point interferogram. (Reproduced from [10], by permission of the Society for Applied Spectroscopy 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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