Interferogram finite

Let the discrete spectrum, which consists of the coefficients of u(k) and v(k), be denoted by U(n) and V(n), respectively. The low-frequency spectral components U(n) are most often given by the most noise-free Fourier spectral components that have undergone inverse filtering. For these cases V(n) would then be the restored spectrum. However, for Fourier transform spectroscopy data, U(n) would be the finite number of samples that make up the interferogram. For these cases V(n) would then represent the interferogram extension. 

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.

Fig. 24 Interferogram of the two monochromatic sources that would be obtained for a finite maximum path difference of the interferometer. (a) Finite interferogram. (b) Recorded spectrum. The two lines are completely merged into one.

Fig. 27 Finite interferogram of the four monochromatic sources of Fig. 26 with Gaussian noise of rms amplitude 0.1 superimposed and the resulting degraded spectral lines, (a) Interferogram of 30 data points, (b) Merged and distorted spectral lines.

For a finite sampling interval A<5, more than one superposition of cosine/sine waves can give rise to the recorded interferogram. For the transformed spectrum to be unique, the sampling interval A<5 must be sufficiently small to detect modulations in the interferogram due to the shortest wavelength present in the spectrum, the so-called Nyquist criterion [66] ... 

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... 

Figure 4. Appearance of spectrum obtained by Fourier transformation of (a) an infinite interferogram (b) a finite interferogram and (c) a finite interferogram with...

We can conclude from this that the resolution of a Michelson interferometer is proportional to the maximum path difference up to which the interferogram has been measured. When we now consider the case of three narrow lines, we must remember that we have to calculate I (v) from I (s) by means of a Fourier transform [see Eqs. (2.10) and (2.12)]. However, the Fourier integral cannot be executed over s from — oo to - -oo, since the interferogram I s) can be determined experimentally only over a finite range ( —Smax s -j-Smax)- Therefore, the integration too can be performed only over a finite range. 

Fig. 6. Finite interferogram and resolution. (Example Three narrow lines of different intensities) upper infinite interferogram I (v) and corresponding spectrum middle finite interfero-...

Next, let us consider that the finite dimensions of the source impose a limit to the resolving power of the Michdson interferometer in the same way as the finite slit width does for the grating spectrometer. The finite size of the source means that also rays enter the interferometer which are inclined by an angle d to the optical axis (see Fig. 34) hence, a displacement s/2 of the movable mirror produces the path difference s cos d instead of s. The total interferogram is the sum of the contributions from all points of the (circular ) source... 

When the Michelson interferometer with finite aperture is not properly adjusted nonlinear phase errors arise These phase errors are no longer linearly dependent on the wave number v, and they cause an asymmetric distortion of the interferogram (Figs, 40b and 41). It should be noted that all illustrations in connection with errors (Figs. 39, 40 and 41) have been produced by computer simulation (cf. Appendix 1). In order to make the essential features as clear as possible the effects of finite resolution etc. are left out where they have not necessarily to be included. In these cases, the resolution width /d is given in the figure (Figs. 39a—c). In Fig. 41, the error correction is demonstrated with finite... 

The Fourier transform of a finite interferogram without apodization is represented by the following expression ... 

In fact, the interferogram is never totally symmetric about x = 0 and to recover the full spectral information, it is necessary to take the complex rather than the cosine Fourier transform. The interferogram is recorded to a finite path difference L rather than infinity. It is actually recorded by sampling it at discrete intervals At. 

Small, G. W., Harms, A. C., Kroutil, R. T., Ditillo, J. T. Loerop, W. R. (1990) Design of optimized finite impulse-response digital-filters for use with passive Fourier-transform infrared interferograms. Ana/. Chem. 62, 1768-1777. 

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... 

The relationship between an interferogram and the corresponding spectrum is the Fourier transform (or cosine transform, as the interferogram is real). However, the interferogram is discretely sampled and finite. For this reason, a discrete Fourier transform (DFT) needs to be performed. 

The second factor is related to the fact that in real life the interferogram is truncated at finite optical path difference. In addition, in the fast Fourier transform (FFT) algorithm, according to Cooley and TTikey [30], which is used to perform the Fourier transform faster than the classical method, certain assumptions and simplifications are made. The result is that the FFT of a monochromatic source is not an infinitely narrow line. 

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