Fourier analysis theory

Matthee, K. and Visser, K. (1995) Background correction in atomic emission spectrometry using repetitive harmonic wavelength scanning and applying Fourier analysis theory, Spectrochimica Acta, 50B, pp823-835. 

Spectral modelling techniques are the legacy of the Fourier analysis theory. Originally developed in the nineteenth century, Fourier analysis considers that a pitched sound is made up of various sinusoidal components, where the frequencies of higher components are integral multiples of the frequency of the lowest component. The pitch of a musical note is then assumed to be determined by the lowest component, normally referred to as the fundamental frequency. In this case, timbre is the result of the presence of specific components and their relative amplitudes, as if it were the result of a chord over a prominently loud fundamental with notes played at different volumes. Despite the fact that not all interesting musical sounds have a clear pitch and the pitch of a sound may not necessarily correspond to the lower component of its spectrum, Fourier analysis still constitutes one of the pillars of acoustics and music. 

Spiegel, Murray R., Schaum s Outline of Theory and Problems of Fourier Analysis, McGraw-Hill Book Company, New York (1974). 

This book is organized into five sections (1) Theory, (2) Columns, Instrumentation, and Methods, (3) Life Science Applications, (4) Multidimensional Separations Using Capillary Electrophoresis, and (5) Industrial Applications. The first section covers theoretical topics including a theory overview chapter (Chapter 2), which deals with peak capacity, resolution, sampling, peak overlap, and other issues that have evolved the present level of understanding of multidimensional separation science. Two issues, however, are presented in more detail, and these are the effects of correlation on peak capacity (Chapter 3) and the use of sophisticated Fourier analysis methods for component estimation (Chapter 4). Chapter 11 also discusses a new approach to evaluating correlation and peak capacity. 

Felinger, A., Pasti, L., Dondi, F. (1990). Fourier analysis of multicomponent chromatograms. Theory and models. Anal. Chem. 62, 1846. 

A shortcut solution for the analysis of anisotropic data is found by mapping scattering images to scattering curves as has been devised by Bonart in 1966 [16]. Founded on Fourier transformation theory he has clarified that information on the structure in a chosen direction is not related to an intensity curve sliced from the pattern, but to a projection (cf. p. 23) of the pattern on the direction of interest. 

As an example of the insufficiency of present usefulness and self-consistency as grounds for belief in a scientific construct, it may be in order to recall some scientific history. In our own field we have the familiar example of phlogiston and in astronomy the example of epicycles. By the use of epicycle superimposed on epicycle, the geocentric theory was able to give a self-consistent, popular, and accurate description of apparent planetary motions. The epicycle treatment is analogous to a Fourier analysis of the motions and its accuracy did not guarantee the physical reality of epicycles. 

Using functions of R, 6, Z (radius, angle, height), Leurkens proposed surface integrals that could be used to describe three dimensional shapes. In a simplified space described by R and 6, a trace of the particle perimeter produces a plot that can be analyzed using Fourier analysis. Leurkins extends the theory to describe particles with three-dimensional complex shapes. 

The off-diagonal elements depend on the x and y components of the local field, which contain many fluctuating components oscillating at different frequencies. The parts which oscillate at the ESR frequency ( ) induce transition between the a- and P-states. By making a Fourier analysis of V(f) and using time-dependent perturbation theory [22], the transition probability between the a- and p-states (P< ) is given by... 

Any periodic function (such as the electron density in a crystal which repeats from unit cell to unit cell) can be represented as the sum of cosine (and sine) functions of appropriate amplitudes, phases, and periodicities (frequencies). This theorem was introduced in 1807 by Baron Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist who pioneered, as a result of his interest in a mathematical theory of heat conduction, the representation of periodic functions by trigonometric series. Fourier showed that a continuous periodic function can be described in terms of the simpler component cosine (or sine) functions (a Fourier series). A Fourier analysis is the mathematical process of dissecting a periodic function into its simpler component cosine waves, thus showing how the periodic function might have been been put together. A simple... 

Fourier series are used in crystal structure analysis in several ways. An electron-density map is a Fourier synthesis with measured values of F hkl) and derived values of phase angles 0 1. A Fourier analysis is the breakdown to component waves, as in the diffraction experiment. Fourier transform theory allows us to travel computationally between real space, p xyz), and reciprocal space, F hkl). 

Fourier transform is a well-known technique for signal analysis, which breaks down a signal into constituent sinusoids of different frequencies. Another way to think of Fourier analysis is as a mathematical technique for transforming a descriptor from the spatial domain into a frequency domain. The theory of Fourier transforms is described in several textbooks, such as Boas [53], and is not discussed here in detail. 

In theory it is not required that the periodic wave be sinusoidal. Any repetitive wave of this type can be resolved by Fourier analysis into a series of pure sine waves which, in turn, can be used to compute the response. However, this detracts from the simplicity of the analysis. A sine wave is preferred to avoid a complex mathematical treatment with results difficult to interpret. 

In the drop shape technique sinusoidal area changes can be easily generated via changes of the drop volume in a very accurate way. The Fourier analysis of the surface tension response however shows that besides the main mode with the period T of the generated oscillation there are also modes with periods of 3T/2, T/2, T/4 and T/8. The origin of these modes is not yet fully understood but certainly caused by deviation of the area changes from harmonicity and surface layer compression/expansion beyond the limits of a linear theory. 

If the current level is low enough for linear conditions, the resultant current density in the tissue is the linear summation of the two current densities (see Chapter 8.2.2). According to the superposition theorem in network theory and Fourier analysis, the new waveform f(t) does not contain any new frequencies. The linear summation of two currents at two different frequencies remains a current with a frequency spectrum with just the two frequencies no current at any new frequency is created. 

Medicine and Pharmaceuticals. Applied mathematics, particularly the use of Fourier analysis and wavelet theory, has sparked explosive growth in the fields of medicine and pharmaceutical development. The enhanced analytical methods available to chemists and bioresearchers through NMR and Fourier transform infrared (FTIR) spectroscopy feciUtate the identification and investigation of new compounds that have potential pharmaceutical applications. In addition, advanced statistical methods, computer modeling, and epidemiological studies provide the foundation for unprecedented levels of research. 

To obtain excitation energies and properties within the time-dependent Kohn-Sham framework, it is possible to propagate in time the time-dependent electron density, through the solution of Eq. (4.60), and then extract energies and oscillator strengths from a Fourier analysis of the results [98-102]. Alternatively, the excited-state properties can be determined through the linear response theory. This is an efficient approach which avoids the direct solution of the time-dependent Kohn-Sham equations and is often used in practical applications. 

Additive synthesis is deeply rooted in the theory of Fourier analysis. The technique assumes that any periodic waveform can be modelled as a sum of sinusoids at various amplitude envelopes and time-varying frequencies. An additive synthesiser hence functions by... 

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