Fourier transform coefficients

In the past, the large amount of spectral data generated by NIR instruments challenged the ability of computers to provide computations within a reasonable time frame. Mathematical techniques, therefore, which offered a reduction of the raw data, but with minimum loss of information, were often employed. The decomposition of 1000 spectral data points to 100 Fourier transform coefficients provides a great saving in computational time with very little loss of... 

In a second approach, the spectral data are expressed in terms of a vector - for example, using Hadamard or Fourier transform coefficients of IR spectra - each element of which is treated as a coordinate in multidimensional space. Each spectrum occupies a point in hyperspace and the similarity between an unknown and a reference entry is measured by the distance between the two points. Once again, the output is a rank-ordered list of structures corresponding to the spectra producing the smallest distance to the query. 

Fast Fourier Transformation is widely used in many fields of science, among them chemoractrics. The Fast Fourier Transformation (FFT) algorithm transforms the data from the "wavelength" domain into the "frequency" domain. The method is almost compulsorily used in spectral analysis, e, g., when near-infrared spectroscopy data arc employed as independent variables. Next, the spectral model is built between the responses and the Fourier coefficients of the transformation, which substitute the original Y-matrix. 

In practical applications, x(t) is not a continuous function, and the data to be transformed are usually discrete values obtained by sampling at intervals. Under such circumstances, I hi discrete Fourier transform (DFT) is used to obtain the frequency function. Let us. suppose that the time-dependent data values are obtained by sampling at regular intervals separated by [Pg.43]

Obviously, the theory outhned above can be applied to two- and three-dimensional systems. In the case of a two-dimensional system the Fourier transforms of the two-particle function coefficients are carried out by using an algorithm, developed by Lado [85], that preserves orthogonality. A monolayer of adsorbed colloidal particles, having a continuous distribution of diameters, has been investigated by Lado. Specific calculations have been carried out for the system with the Schulz distribution [86]... 

The resolution of infra-red densitometry (IR-D) is on the other hand more in the region of some micrometers even with the use of IR-microscopes. The interface is also viewed from the side (Fig. 4d) and the density profile is obtained mostly between deuterated and protonated polymers. The strength of specific IR-bands is monitored during a scan across the interface to yield a concentration profile of species. While in the initial experiments on polyethylene diffusion the resolution was of the order of 60 pm [69] it has been improved e.g. in polystyrene diffusion experiments [70] to 10 pm by the application of a Fourier transform-IR-microscope. This technique is nicely suited to measure profiles on a micrometer scale as well as interdiffusion coefficients of polymers but it is far from reaching molecular resolution. 

The first Fourier transformation of the FID yields a complex function of frequency with real (cosine) and imaginary (sine) coefficients. Each FID therefore has a real half and an imaginary half, and when subjected to the first Fourier transformation the resulting spectrum will also have real and imaginary data points. When these real and imaginary data points are arranged behind one another, vertical columns result. This transposed data... 

The combination of PCA and LDA is often applied, in particular for ill-posed data (data where the number of variables exceeds the number of objects), e.g. Ref. [46], One first extracts a certain number of principal components, deleting the higher-order ones and thereby reducing to some degree the noise and then carries out the LDA. One should however be careful not to eliminate too many PCs, since in this way information important for the discrimination might be lost. A method in which both are merged in one step and which sometimes yields better results than the two-step procedure is reflected discriminant analysis. The Fourier transform is also sometimes used [14], and this is also the case for the wavelet transform (see Chapter 40) [13,16]. In that case, the information is included in the first few Fourier coefficients or in a restricted number of wavelet coefficients. 

In summary, the Fourier transform of a continuous signal digitized in 2A/ + 1 data points returns N real Fourier coefficients, N imaginary Fourier coefficients and the average signal, also called the DC term, i.e. in total 2N + 1 points. The relationship between the scales in both domains is shown in Fig. 40.9. 

These four steps are illustrated in Fig. 40.17 where two triangles (array of 32 data points) are convoluted via the Fourier domain. Because one should multiply Fourier coefficients at corresponding frequencies, the signal and the point-spread function should be digitized with the same time interval. Special precautions are needed to avoid numerical errors, of which the discussion is beyond the scope of this text. However, one should know that when J(t) and h(t) are digitized into sampled arrays of the size A and B respectively, both J(t) and h(t) should be extended with zeros to a size of at least A + 5. If (A -i- B) is not a power of two, more zeros should be appended in order to use the fast Fourier transform. 

Fig. 40.31. Data compression by a Fourier transform, (a) A spectrum measured at 512 wavelengths (b) spectrum after reconstruction with 2, 4,..., 256 Fourier coefficients.

In single-scale filtering, basis functions are of a fixed resolution and all basis functions have the same localization in the time-frequency domain. For example, frequency domain filtering relies on basis functions localized in frequency but global in time, as shown in Fig. 7b. Other popular filters, such as those based on a windowed Fourier transform, mean filtering, and exponential smoothing, are localized in both time and frequency, but their resolution is fixed, as shown in Fig. 7c. Single-scale filters are linear because the measured data or basis function coefficients are transformed as their linear sum over a time horizon. A finite time horizon results infinite impulse response (FIR) and an infinite time horizon creates infinite impulse response (HR) filters. A linear filter can be represented as... 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

An important technical development of the PFG and STD experiments was introduced at the beginning of the 1990s the Diffusion Ordered Spectroscopy, that is DOSY.69 70 It provides a convenient way of displaying the molecular self-diffusion information in a bi-dimensional array, with the NMR spectrum in one dimension and the self-diffusion coefficient in the other. While the chemical-shift information is obtained by Fast Fourier Transformation (FFT) of the time domain data, the diffusion information is obtained by an Inverse Laplace Transformation (ILT) of the signal decay data. The goal of DOSY experiment is to separate species spectroscopically (not physically) present in a mixture of compounds for this reason, DOSY is also known as "NMR chromatography."... 

The strength of the Bronsted (BAS) and Lewis (LAS) acid sites of the pure and synthesized materials was measured by Fourier transformed infrared spectroscopy (ATI Mattson FTIR) by using pyridine as a probe molecule. Spectral bands at 1545 cm 1 and 1450 cm 1 were used to indentify BAS and LAS, respectively. Quantitative determination of BAS and LAS was calculated with the coefficients reported by Emeis [5], The measurements were performed by pressing the catalyst into self supported wafers. Thereafter, the cell with the catalyst wafer was outgassed and heated to 450°C for lh. Background spectra were recorded at 100°C. Pyridine was then adsorbed onto the catalyst for 30 min followed by desorption at 250, 350 and 450°C. Spectra were recorded at 100°C in between every temperature ramp. 

The range of coefficients constitute a function say, which is the Fourier transform of f(x), e.g. 

Better procedures for determining particle sizes from X-ray diffraction are based on line profile analysis with Fourier transform methods. The average size is obtained from the first derivative of the cosine coefficients and the distribution of particle sizes from the second derivative. When used in this way, XRD offers a fundamental advantage over electron microscopy, because it samples a much larger portion of the catalyst. The reader is referred to publications by Cohen and coworkers for more details and examples [4,10,11],... 

The evaluation of the contribution of cycle diagrams to the activity coefficient is formally complete once the eigenvalues of SI have been found. Let us write the result explicitly in terms of the Fourier transforms for the important case of two defects a and / each of which is allowed to occupy a particular one of the two positions in the unit cell. In this simple system the labelling of positions in the unit cell by superscripts on Fourier transforms becomes redundant and can be omitted. The result is... 

The remaining terms in the expression for the activity coefficient all involve the function mu defined in Eq. (140). Using the asymptotic expression for the Fourier transform in Eq. (155) it is found that... 

The Fourier and inverse Fourier transforms couple all of the coefficients (UK, AK, BK), and typically represent the most expensive part of the computation. Care must also be taken to ensure that the scalar field remains bounded in composition space when applying the inverse Fourier transform. 

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