Phase-corrected interferogram

Fig. 2.16. Phase corrected 22.63 MHz PFT 1 C NMR spectrum of mutarotated D-galactose 100 mg/mL deuterium oxide 30 C proton decoupled 512 accumulated pulse interferograms ...

The measuring and calculation time (Fourier transformation and phase correction for the 8 K interferogram/ 4 K spectrum) was less than 100 s in (b) and (c). 

Due to its greatly enhanced sensitivity in comparison to CW NMR, the PFT method has made 13C NMR into a routine method of structure analysis for all molecules having the natural 13C abundance of 1.1%. Additionally, phase-corrected PFT NMR spectra contain all spectral details without the lineskewing and ringing observed in CW spectra. Finally, short-lived molecules can be measured by PFT NMR, and sensitivity enhancement by accumulation of interferograms before Fourier transformation requires much less time than the accumulation of CW NMR spectra, due to the short time required for acquisition of FID signals. 

The position of ZPD (Zero Path Difference) is critical to the Fourier Transform calculation, since the algorithm assumes that the central burst in the interferogram is in fact the ZPD. However, due to the refractive index properties of the beamsplitter material, the ZPD is not at the same position for every wavelength measured. There are several ways to overcome these phase differences. The most common method is to use a correction factor, which is known as phase correction. This correction factor is calculated for every wavelength, based on a double sided interferogram, since this tends to minimize the effects of phase difference. In practice, most infrared spectrometers collect single sided interferograms, since this halves the mirror movement, and consequently the number of datapoints to be Fourier transformed. 

Connes accuracy is the result of the ability to carefully measure an accurate interferogram. If the data acquisition and phase correction is done properly it is possible to measure transitions in a species to very high accuracy. It is common to see line positions accurate to 0.0005 cm-1 or better. Accurate intensity measurements are possible as well [13],... 

Fig. 6.3-5 shows the spectra taken with a CdSe calibration plate on a Bruker IF.S-66 FT-IR spectrometer. Theoretically four curves should result. For calibration purposes, however, it is only necessary to record two curves, as the two other curves are mirror images of the latter. The curves shown in the figure were taken with only a single orientation of the CdSe calibration plate, but one with parallel and another with crossed polarizers. They are the results of 32 coadded interferograms transformed without apodization. As result we chose a power spectrum, since this is less noisy. A further advantage is that we do not need to use the (in the case of calibration spectra) complicated phase correction. The curves cross at certain points. Interpolating between those crossings, we get a curve with which we can multiply our spectra for the correction of Bessel function dependance. We also clearly can identify the node of the Bessel function at about 2450 cm . The modulator was tuned to quarter wave retardation at 1111 cm. ... 

Phase correction in contrast to the theoretical expectation, the measured interferogram is typically not symmetric about the centerburst (.v = 0). This is a consequence of experimental errors, e.g., frequency-dependent optical and electronic phase delays. One remedy is to measure a small part of the interferogram doublesided. Since the phase is a weak function of the wavenumber, one can easily interpolate the low resolution phase function and use the result later for phase correction. If there is considerable background absorption, phase errors may falsify the intensities of bands in the difference spectra. To avoid such phase errors for difference spectroscopy, the background absorbance should therefore be less than one. 

Another purpose of the measurement of the pseudosample is that it provides a phase correction needed for Fourier transform of the AC interferogram. The selfcorrection conventionally used in Fourier transformation for one-sign absorption... 

In practice, the interferogram measured is not mirror symmetrical about the point d = 0. Call up the interferogram of the file MIR GLYCIN or ACQUIS in order to verify that. This asymmetry originates from experimental errors, e.g., wavenumber-dependent phase delays of the optics, the detector/ amplifier unit, or the electronic filters. The Fourier transformation of such an asymmetrical interferogram generally yields a complex spectrum C(v) rather than a real spectrum S(r) as known from spectrometers based on the dispersive technique. That is why phase correction is necessary. 

The purpose of the phase correction procedure is to determine the amplitude spectrum 5(r) from the complex output C(v) of the FT of the interferogram. This can be performed either by calculating the square root of the power spectrum... 

This procedure is helpful for transforming interferograms after measurement by allowing different parameters to be used for apodization, phase correction and zero-filling. Utilizing this procedure you can play with the Fourier transformation. 

Interferograms recorded in forward/backward mode can be processed using either the scans during the forward or the backward movement of the mirror. If the two directions of the mirror travel should be evaluated, the forward and the backward scan will be transformed separately, followed by a phase correction and calculation of the average spectrum. 

The Phase Correction (see Fig. 10.44) can be thought of as symmetrizing the interferogram, which is always necessary due to the asymmetry of any measured interferogram. Several phase correction methods are available ... 

Figure 10.44. The Interferogram to Spectrum dialog box Phase Correction page.

Figure 10.46 represents the Raman spectra of sulphur calculated by Fourier transformation of the interferogram of the file SULPHUR using no apodization (boxcar) and the apodization fiuiction Norton-Beer weak, respectively, a zerofilling factor of 2, and the power spectrum for the phase correction. See if you obtain the same spectra. 

The symmetric Fourier transformation is used if a phase correction is not necessary, because the data contains one half of a symmetric or antisymmetric interferogram. This may occur for example, if the interferogram is generated by an inverse FT. 

Fig. 2.6 Forman phase correction method. The real and imaginary part of the spectrum corresponding to the transmission of the atmosphere from 0 to 42 cm has been distorted (top-left) with a linear phase error (top-right). The measured interferogram is not symmetric anymore (centre-left). After extracting the convolution kernel (centre-right) and applying the correction method 5 times, the interferogram symmetry is improved (bottom-left). Fourier transforming the corrected interferogram, the spectrum is recovered (bottom-right) and is real...

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. 

The interferogram acquired when a single slit is used to cover the source is known, and the spectrum of the source can be recovered by Fourier transforming and phase correcting the data. However, when more complex sources (i.e. extended sources) are used, it is difficult to discern if the final spectrum is actually the one desired. Moreover, the final goal of the system is to detect unknown sources, so previous spectral data will not be available. 

Current FTS phase correction algorithms are also dependent on a unique ZPD. However, the instrumental phase errors, if linear, will affect in the same way different regions of an interferogram. This fact can be used to redesign some of the existing phase correction algorithms, for example the Forman phase correction one. 

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. 

A more detailed discussion of phase correction is given in Chapter 4. At this point it is sufficient to say that if the phase angle 0 in an interferogram measured with a continuous broadband source varies only slowly with wavenumber (as is... 

Figure 3.11. (a) Chirped interferogram (b) phase-corrected spectrum of the source calculated from this interferogram (c) almost completely unchirped interferogram reconstructed from this spectrum. (Reproduced from [7], by permission of the Society for Applied Spectroscopy copyright 1974.)... 

Appropriate transformation of interferograms to produce accurate spectra is not restricted to Fourier transform algorithms, and one of the additional procedures, phase correction, is presented in the following section. 

When is transposed from one side to the other of Eq. 4.33, only the real terms are retained from the trigonometric expansion because the tme and amplitude spectra are real functions. Equation 4.34 represents a phase correction algorithm for double-sided interferograms. 

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