Fourier transform components

We will assume that the impurity potential is a very smooth one, which implies that its Fourier transform components fall very fast with the magnitude of the reciprocal-space vectors, and therefore the only relevant matrix elements are those for G = 0 and k k. We will also assume that the expansion of the impurity wavefunction onto crystal wavefunctions needs only include wave-vectors near a band extremum at ko (the VBM for holes or the CBM for extra electrons). But for G = 0 and k = k = ko, Xkk (G) becomes... 

Conceptually, the problem of going from the time domain spectra in Figures 3.7(a)-3.9(a) to the frequency domain spectra in Figures 3.7(b)-3.9(b) is straightforward, at least in these cases because we knew the result before we started. Nevertheless, we can still visualize the breaking down of any time domain spectrum, however complex and irregular in appearance, into its component waves, each with its characteristic frequency and amplitude. Although we can visualize it, the process of Fourier transformation which actually carries it out is a mathematically complex operation. The mathematical principles will be discussed only briefly here. 

Principal component analysis has been used in combination with spectroscopy in other types of multicomponent analyses. For example, compatible and incompatible blends of polyphenzlene oxides and polystyrene were distinguished using Fourier-transform-infrared spectra (59). Raman spectra of sulfuric acid/water mixtures were used in conjunction with principal component analysis to identify different ions, compositions, and hydrates (60). The identity and number of species present in binary and tertiary mixtures of polycycHc aromatic hydrocarbons were deterrnined using fluorescence spectra (61). 

This process is continued until there is only one component. For this reason, the number N is taken as a power of 2. The vector [yj] is filled with zeroes, if need be, to make N = 2 for some p. For the computer program, see Ref. 26. The standard Fourier transform takes N operations to calculation, whereas the fast Fourier transform takes only N log2 N. For large N, the difference is significant at N = 100 it is a factor of 15, but for N = 1000 it is a factor of 100. 

Fourier transform infrared (FTIR) analyzers can be used for industrial applications and m situ measurements in addition to conventional laboratory use. Industrial instruments are transportable, rugged and relatively simple to calibrate and operate. They are capable of analyzing many gas components and determining their concentrations, practically continuously. FTIR analyzers are based on the spectra characterization of infrared light absorbed by transitions in vibrational and rotational energy levels of heteroatomic molecules. 

Frequency-domain data are obtained by converting time-domain data using a mathematical technique referred to as Fast Fourier Transform (FFT). FFT allows each vibration component of a complex machine-train spectrum to be shown as a discrete frequency peak. The frequency-domain amplitude can be the displacement per unit time related to a particular frequency, which is plotted as the Y-axis against frequency as the X-axis. This is opposed to time-domain spectrums that sum the velocities of all frequencies and plot the sum as the Y-axis against time... 

Most of the early vibration analysis was carried out using analog equipment, which necessitated the use of time-domain data. The reason for this is that it was difficult to convert time-domain data to frequency-domain data. Therefore, frequency-domain capability was not available until microprocessor-based analyzers incorporated a straightforward method (i.e.. Fast Fourier Transform, FFT) of transforming the time-domain spectmm into its frequency components. 

The frequency-domain format eliminates the manual effort required to isolate the components that make up a time trace. Frequency-domain techniques convert time-domain data into discrete frequency components using a mathematical process called Fast Fourier Transform (FFT). Simply stated, FFT mathematically converts a time-based trace into a series of discrete frequency components (see Figure 43.19). In a frequency-domain plot, the X-axis is frequency and the Y-axis is the amplitude of displacement, velocity, or acceleration. 

The phrase full Fast Fourier Transform (FFT) signature is usually applied to the vibration spectrum that uniquely identifies a machine, component, system, or subsystem at a specific operating condition and time. It provides specific data on every frequency component within the overall frequency range of a machine-train. The typical frequency range can be from 0.1 to 20,000 Hz. 

Most predictive-maintenance programs rely almost exclusively on frequency-domain vibration data. The microprocessor-based analyzers gather time-domain data and automatically convert it using Fast Fourier Transform (FFT) to frequency-domain data. A frequency-domain signature shows the machine s individual frequency components, or peaks. 

According to Eq. (2.13), the spectra we are interested in are given by a Fourier transform of the orientational correlation functions of the corresponding order Similarly to what was done in Chapter 3, the correlation functions for linear and spherical molecules may be represented as a superposition of the partial (marginal) components... 

Thus the mutual intensity at the observer is the Fourier transform of the source. This is a special case of the van Cittert-Zernike theorem. The mutual intensity is translation invariant or homogeneous, i.e., it depends only on the separation of Pi and P2. The intensity at the observer is simply / = J. Measuring the mutual intensity will give Fourier components of the object. 

The first Raman and infrared studies on orthorhombic sulfur date back to the 1930s. The older literature has been reviewed before [78, 92-94]. Only after the normal coordinate treatment of the Sg molecule by Scott et al. [78] was it possible to improve the earlier assignments, especially of the lattice vibrations and crystal components of the intramolecular vibrations. In addition, two technical achievements stimulated the efforts in vibrational spectroscopy since late 1960s the invention of the laser as an intense monochromatic light source for Raman spectroscopy and the development of Fourier transform interferometry in infrared spectroscopy. Both techniques allowed to record vibrational spectra of higher resolution and to detect bands of lower intensity. 

FIGURE 30.S Fourier transform (FT) resolution of torque signals into harmonic components notes that relative harmonic components are displayed. 

Fourier transform spectrum odd-harmonic components analysis... 

Fourier transformation of Rf pulses (which are in the time domain) produces frequency-domain components. If the pulse is long, then the Fourier components will appear over a narrow frequency range (Fig. 1.24) but if the pulse is narrow, the Fourier components will be spread over a wide range (Fig. 1.25). The time-domain signals and the corresponding frequency-domain partners constitute Fourier pairs. 

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