Fourier series technique

The breakdown of a given signal into a sum of oscillatory functions is accomplished by application of Fourier series techniques or by Fourier transforms. For a periodic function F t) with a period t, a Fourier series may be expressed as... 

Chu, H.S. Chen, C.K. Weng, C. Applications of Fourier series technique to transient heat transfer problems. Chem. Eng. Commun. 16 (1982) 215-225... 

Duda and Vrentas [42] used this approach and found an infinite-series analytical solution for the closed-streamline axisymmetric flow in this cylinder. In a second paper [43], the corresponding developing heat transfer problem was solved using a formal Fourier series technique. The method allowed the calculation of time-dependent Nusselt numbers up to Is/Pedet numbers of up to 400. Extension to higher I/dh was prohibited as the eigenvalues of the solution became too close together as the aspect ratio was increased. 

Since /j (r) represents a linear filter, g2 t) will be a linear filtered version ofgi (f) however, 6>2(r) and R2(t), the PM and AM components of 2(t)> wiUbe a nonlinear filtered version ofgi(t) since 02(t) and i 2(t) are nonlinear functions ofg2(t). The analysis of the nonlinear distortion is very complicated. Although many analysis techniques have been pubhshed in the literature, none has been entirely satisfactory. Pan ter (1965) gives a three-chapter summary of some of these techniques, and a classical paper is also recommended (Bedrosian and Rice, 1968). Furthermore, nonUnearities that occur in a practical system will also cause nonlinear distortion and AM-to-PM conversion effects. Nonlinear effects can be analyzed by several techniques, including power-series analysis (Couch, 1995). If a nonlinear effect in a bandpass system is to be analyzed, a Fourier series technique that uses the Chebyshev transform has been foimd to be useful (Spilker, 1977). 

These symmetry relations will be used to advantage in the application of Fourier-series techniques to the description of periodic functions. 

Bogoliubov, Shotkin (28) numerically studied centers and limit cycles in the two-temperature model. Fourier-series techniques were recently applied to the same model in two interesting papers (29, 30). 

About 1915 W.H. Bragg suggested to use Fourier series to describe the arrangement of the atoms in a crystal [1]. The proposed technique was somewhat later extended by W. Duane [2] and W.H. Zachariasen was the first who used a two-dimensional Fourier map in 1929 for structure determination [3], Since then Fourier synthesis became a standard method in almost in every structure determination from diffraction data. 

Mathematically, geometric parameters can be described by using the Fourier Series in polar coordinates (p,9). Thus, given a set of boundary points (x, y) from an object of interest, they can be transformed into the polar coordinates with respect to its geometric center (x, y). A curve fitting technique in polar coordinates can be used to fit this set of points into a Fourier Series such that any point p(0) on this boundary can be expressed by... 

An extensive study of analytical techniques used in conduction heat transfer requires a background in the theory of orthogonal functions. Fourier series are one example of orthogonal functions, as are Bessel functions and other special functions applicable to different geometries and boundary conditions. The interested reader may consult one or more of the conduction heat-transfer texts listed in the references for further information on the subject. 

In some cases cyclic events occur, dependent, for example, on time of day, season of tire year or temperature fluctuations. These can be modelled using sine functions, and are tire basis of time series analysis (Section 3.4). In addition, cyclicity is also observed in Fourier spectroscopy, and Fourier transform techniques (Section 3.5) may on occasions be combined with methods for time series analysis. 

The technique used to generate Fourier series is easily extended to develop expressions for non-periodic functions. The Fourier series then becomes a Fourier integral. The Fourier integral expression for a function /(x) is... 

The stability attributes of the aforementioned method are explored via the combined von Neumann and Routh-Hurwitz technique [2]. According to this approach, the error that appears during the computation of any field component is described by a single term of a Fourier series expansion ... 

Another technique that supplements the results of X-ray diffraction has come into wide use in recent years. It is a form of nuclear magnetic resonance (NMR) spectroscopy. In this particular application of NMR, called 2-D (two dimensional) NMR, lai e collections of data points are subjected to computer analysis (Figure 4.14b). Like X-ray diffraction, this method uses a Fourier series to analyze results. It is similar to X-ray diffraction in other ways It is a long process, and it requires considerable amounts of computing power and milligram quantities of protein. One way in which 2-D NMR differs from X-ray diffraction... 

Wilson, R.G. Fourier Series and Optical Transform Techniques in Contemporary Optics An... 

Theret et al. [1988] analyzed the micropipette experiment with endothelial cell. The cell was interpreted as a linear elastic isotropic half-space, and the pipette was considered as an axisymmetric rigid ptmch. This approach was later extended to a viscoelastic material of the cell and to the model of the cell as a deformable layer. The solutions were obtained both analytically by using the Laplace transform and numerically by using the finite element method. Spector et al. [ 1998] analyzed the application of the micropipette to a cylindrical cochlear outer hair cell. The cell composite membrane (wall) was treated as an orthotropic elastic shell, and the corresponding problem was solved in terms of Fourier series. Recently, Hochmuth [2000] reviewed the micropipette technique applied to the analysis of the cellular properties. 

Use an indirect Fourier transform technique which involves a series of Fourier transforms between the real and the reciprocal space to fit the scattering data. ° It is quite possible, from a mathematical perspective, to get negative size distributions which are difficult to interpret. 

Two additional methods use a potentiostat with a device permitting automatic determination of the impedance. Phase-sensitive detection (PSD) is used in lock-in amplifiers [83]. The most popular commercial equipment of this type was used by EG G Princeton Applied Research. In this technique, the measured ac signal, which is the potential or current, E = Eq sin( ), is multiplied by a square wave signal of the same frequency w. The square wave signal of unit amplitude can be represented by an infinite Fourier series ... 

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