Truncated sine wave

The cosine Fourier transform of a truncated sine wave has the form shown in Figure 2.11. In general, the shape of the ILS is intermediate between this function and the sine function that results from the cosine transform of a truncated cosine wave. The process of removing these sine components from an interferogram, or removing their effects from a spectrum, is known as phase correction. 

Fig. 4. Iron EXAFS Fourier isolates (solid line) and the fits to the data (dotted line), Fourier transforms, and the Fe-S structures found in (a) IFe Fe-S protein rubredoxin, (b) 2Fe-2S plant ferredoxin, and (c) 4Fe-4S bacterial ferredoxin. In (a) the EXAFS spectrum from the IFe protein rubredoxin shows a single damped sine wave indicating the presence of one major Fe-S distance. This is reflected in the Fourier transform which shows only one major peak. The peak at 1.5 A is due to residual background and/or Fourier truncation. In (b) and (c) and EXAFS spectra show a beat pattern indicating the presence of Fe-S and Fe-Fe distances. The Fourier transforms shows two peaks that are due to backscattering from S and Fe atoms, respectively. [Adapted from B.-K. Teo and R. G. Shulman, in Iron-Sulfur Proteins (T. G. Spiro, ed.), p. 343. Wiley, New York, 1982.]...

The truncated time interval also defines the lowest frequency that can be analyzed nothing can be said about sine wave components with half-periods longer than the time analyzed. 

This will result in a cancellation of the decay, and the Fourier domain signal becomes a pure tmncated cosine wave, as shown in Figure 10.10. The Fourier transform of a pure truncated cosine wave is a sine function, as shown in Section 2.3. The sine function has a narrower FWHH than almost any other spectral waveform however, it does have large sidelobes. Of course, these could be removed with apo-dization (Section 2.4), but it is usually easier to change the rate of decay in Eq. 10.7. If the Fourier domain array is multiplied by an exponential function with a different FWHH, y such that y < y, that is. 

Figure 10.10. Full Fourier self-deconvolution of a Lorentizan band. The Lorentzian band (a) undergoes Fourier transformation to yield the Fourier domain signal b which has a decay, exp(—ya ). The signal in (b) is multiplied by the inverse exponential decay (c), expC-t-yx), to produce a truncated cosine wave (d). Upon inverse Fourier transformation of (d) a sine waveform is produced (e). The sine waveform has a narrower FWHH than that if the original Lorentzian band.

In Table 5.9, we have listed the electronic energies of truncated Cl wave functions for the water molecule at Rtts and 2R,ef. Sinee the Cl model is variational, the FCI energy is approached monotonically from above. From the weights, we see how the FCI wave function is approached as higher and higher excitations are included in the expansion. 

Sometimes the FID doesn t behave as we would like. If we have a truncated FID, Fourier transformation (see Section 4.4) will give rise to some artefacts in the spectrum. This is because the truncation will appear to have some square wave character to it and the Fourier transform of this gives rise to a Sine function (as described previously). This exhibits itself as nasty oscillations around the peaks. We can tweak the data to make these go away by multiplying the FID with an exponential function (Figure 4.1). 

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... 

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