Spectrum Analysis, Fourier Transforms

The time domain is well known from daily life experience, it is, for example, the way a signal is recorded as a function of time as an electrocardiogram (ECG) waveform. 

From the waveform in the time domain, the components can be found in the frequency 

A periodic waveform occupies a line spectrum an aperiodic waveform occupies a continuous frequency spectrum. 

The Fourier series with discrete harmonics was introduced in Section 8.2.2. Mathematically, the Fourier transform of a function (periodic or nonperiodic) in the time domain f(t), to the corresponding function in the frequency domain f(w), is described by  

Note that by multiplying by the complex expression = cos wt + j sin wt, both the in- 

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IR analysis was carried out to evaluate structural changes of CMC and CFF and its grafted copolymers, by means main functional groups signals. IR spectra were recorded with a Perkin-Elmer Spectrum One Fourier Transform IR spectrophotometer, using an Attenuated Total Reflactance (ATR) accessory, with ZnSe plate, using 12 scans and resolution of 4 cm", ranging from 4000 to 600 cm-i. 

Fourier transform spectrum odd-harmonic components analysis... 

FIGURE 30.7 Typical (averaged) torque traces as recorded when a gum polybutadiene sample is submitted to high strain the Fourier transform (FT) spectrum exhibits accordingly significant harmonic contributions the inset table gives the results of the automatic analysis of torque and strain signals. 

The essence of analyzing an EXAFS spectrum is to recognize all sine contributions in x(k)- The obvious mathematical tool with which to achieve this is Fourier analysis. The argument of each sine contribution in Eq. (8) depends on k (which is known), on r (to be determined), and on the phase shift

characteristic property of the scattering atom in a certain environment, and is best derived from the EXAFS spectrum of a reference compound for which all distances are known. The EXAFS information becomes accessible, if we convert it into a radial distribution function, 0 (r), by means of Fourier transformation ... 

An analysis of the Fourier Transform i.r. spectrum of PF around 946 cm-1 gave rise to nine spectroscopic constants for the Vg band and five for the Vg + v -v band which allowed calculation of the wavenumbers of the Vg band with a precision of 1 x 10-3 cm-1.11... 

An adaptation of Fourier analysis to 2D separations can be established by calculating the autocovariance function (Marchetti et al., 2004). The theoretical background of that approach is that the power spectrum and the autocovariance function of a signal constitute a Fourier pair, that is, the power spectmm is obtained as the Fourier transform of the autocovariance function. 

Figure 9. Data reduction and data analysis in EXAFS spectroscopy. (A) EXAFS spectrum x(k) versus k after background removal. (B) The solid curve is the weighted EXAFS spectrum k3x(k) versus k (after multiplying (k) by k3). The dashed curve represents an attempt to fit the data with a two-distance model by the curve-fitting (CF) technique. (C) Fourier transformation (FT) of the weighted EXAFS spectrum in momentum (k) space into the radial distribution function p3(r ) versus r in distance space. The dashed curve is the window function used to filter the major peak in Fourier filtering (FF). (D) Fourier-filtered EXAFS spectrum k3x (k) versus k (solid curve) of the major peak in (C) after back-transforming into k space. The dashed curve attempts to fit the filtered data with a single-distance model. (From Ref. 25, with permission.)...

The ability of the new precursors to decompose thermally to yield singlephase CIS was investigated by powder XRD analysis and EDS on the nonvolatile solids from the TGA experiments of selected compounds. Furthermore, using TGA-evolved gas analysis (EGA), the volatile components from the degradation of the SSPs could be analyzed via real-time fourier transform infrared (FTIR) and mass spectrometry (MS), thus providing information for the decomposition mechanism.3 The real-time FTIR spectrum for 7 and 8 shows absorptions at approximately 3000,1460,1390,1300, and 1250 cm-1 (see Fig. 6.7). 

Instrumentation. Fourier transform infrared (FUR) spectra were recorded on a Nicolet 5DX using standard techniques. Spectra were measured from various sample supports, including KBR pellets, free polymer films and films cast on NaCl windows. Spectra for quantitative analysis were recorded in the absorbance mode. The height of the 639 cm 1 absorbance was measured after the spectrum was expanded or contracted such that the 829 cm 1 absorbance was a constant height. In some spectra an artifact due to instrumental response appeared near 2300 cm 1. 

ESE envelope modulation. In the context of the present paper the nuclear modulation effect in ESE is of particular interest110, mi. Rowan et al.1 1) have shown that the amplitude of the two- and three-pulse echoes1081 does not always decay smoothly as a function of the pulse time interval r. Instead, an oscillation in the envelope of the echo associated with the hf frequencies of nuclei near the unpaired electron is observed. In systems with a large number of interacting nuclei the analysis of this modulated envelope by computer simulation has proved to be difficult in the time domain. However, it has been shown by Mims1121 that the Fourier transform of the modulation data of a three-pulse echo into the frequency domain yields a spectrum similar to that of an ENDOR spectrum. Merks and de Beer1131 have demonstrated that the display in the frequency domain has many advantages over the parameter estimation procedure in the time domain. 

We discussed the fundamentals of mass spectrometry in Chapter 10 and infrared spectrometry in Chapter 8. The quadrupole mass spectrometer and the Fourier transform infrared spectrometer have been adapted to and used with GC equipment as detectors with great success. Gas chromatography-mass spectrometry (GC-MS) and gas chromatography-infrared spectrometry (GC-IR) are very powerful tools for qualitative analysis in GC because not only do they give retention time information, but, due to their inherent speed, they are also able to measure and record the mass spectrum or infrared (IR) spectrum of the individual sample components as they elute from the GC column. It is like taking a photograph of each component as it elutes. See Figure 12.14. Coupled with the computer banks of mass and IR spectra, a component s identity is an easy chore for such a detector. It seems the only real... 

FTIR is a natural for HPLC in that it (FTIR) is a technique that has been used mostly for liquids. The speed introduced by the Fourier transform technique allows, as was mentioned for GC, the recording of the complete IR spectrum of mixture components as they elute, thus allowing the IR photograph to be taken and interpreted for qualitative analysis. Of course, the mobile phase and its accompanying absorptions are ever present in such a technique and water must be absent if the NaCl windows are used, but IR holds great potential, at least for nonaqueous systems, as a detector for HPLC in the future. 

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