Triangular function interferogram

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.

It is found that multiplication of the Fourier transform of the data by a carefully chosen window function is very effective in removing the artifacts around peaked functions. This process is called apodization. Apodization with the triangular window function is often applied to Fourier transform spectroscopy interferograms to remove the ringing around the infrared... 

One window function widely used is the triangular window function. This function is shown in Fig. 14(d). Multiplying the interferogram of Fig. 13(a) by this window function produces the altered interferogram shown in Fig. 

Fig. 14 Four window functions used to multiply the interferogram. (a) Gaussian window, (b) Triangular window, (c) Triangular window of greater slope than (b). (d) Triangular window tapering to zero at the end point of the interferogram (used for apodiza-tion).

Fig. 17 Effectiveness in removing the artifacts from the spectrum of multiplying the interferogram by the proper window function before extending the interferogram by a finite number of points, (a) Cosine interferogram of Fig. 13(a) premultiplied by the triangular window function of Fig. 14(b) before extending by 50 data points, (b) Restored spectral line.

Thus far, the discussion has been restricted to triangular window functions. However, it has been discovered that windows of many other functional forms are capable of bringing about improvement in the spectral lines. In this research the author has found that the window of Gaussian shape has produced the best overall results. With the same interferogram and extension by the same amount as in the previous example, premultiplication by the Gaussian window function shown in Fig. 14(a) produced the restored interferogram shown in Fig. 18(a). The restored spectral line shown in Fig. 18(b) has a resolution much improved over that of Fig. 17(b), where the triangular window function was used, yet the artifacts are no worse. The researcher should explore the various functional forms of the window function to find the one best suited for his or her particular data. 

If we multiply the interferogram by the triangular window function of Fig. 14(b) before extension, we obtain the interferogram and spectral lines shown in Fig. 29. The artifacts have been considerably reduced, but a slight loss of resolution has occurred. The best overall results are obtained by premultiplying the interferogram by the Gaussian window function of Fig. 14(a) before extension. These results are shown in Fig. 30. The spectral lines of... 

An additional consequence of finite retardation is the appearance of secondary extrema or "wings" on either side of the primary features. The presence of these features is disadvantageous, especially when it is desired to observe a weak absorbance in proximity to a strong one. To diminish this problem the interferogram is usually multiplied by a triangular apodization function which forces the product to approach zero continuously for s = + Fourier transformation of the... 

In the transformation the physical units are inverted. When the interferogram is expressed in optical path difference units (cm), the spectrum is obtained in wave-numbers (cm-1) and when the interferogram is expressed in time units (s) the spectrum is in frequency units (s 1). Apart from sine and cosine functions, box-car and triangular, etc. functions are also known, for which the Fourier transformation can be calculated. When applying the Fourier transformation over the whole area + oo, the arm of the interferometer also would have to be moved from — co to +co. When making a displacement over a distance of +L only, the interferogram has to be multiplied by a block function, which has the value of 1 between + and —I and the value 0 outside. I then influences the resolution that can be obtained. 

Fig. 10. Interferogram I (s) of a continuous spectrum versus path difference s (upper part), the corresponding spectrum I (f) versus wave number v, and the instrument line-shape function (for triangular apodization)...

The Forman phase correction algorithm, presented in Chap. 2, is shown in Fig. 3.6. Initially, the raw interferogram is cropped around the zero path difference (ZPD) to get a symmetric interferogram called subset. This subset is multiplied by a triangular apodization function and Fourier transformed. With the complex phase obtained from the FFT a convolution Kernel is obtained, which is used to filter the original interferogram and correct the phase. Finally the result of the last operation is Fourier transformed to get the phase corrected spectrum. This process is repeated until the convolution Kernel approximates to a Dirac delta function. 

From Section 2.3 we know that when a cosine wave interferogram is unweighted, the shape of the spectral line is the convolution of the true spectrum and a sine function [i.e., the transform of the boxcar truncation function, 0(8)]. If instead of using the boxcar function, we used a simple triangular weighting function of the form... 

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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