Fourier analysis series

Time domains and frequeney domains are related through Fourier series and Fourier transforms. By Fourier analysis, a variable expressed as a funetion of time may be deeomposed into a series of oseillatory funetions (eaeh with a eharaeteristie frequeney), whieh when superpositioned or summed at eaeh time, will equal the original expression of the variable. This... 

P R Griffiths and J A de Haseth, Fourier Transform Infrared Spectroscopy — Chemical Analysis Series, Vol. 83, Wiley, Chichester, 1986... 

Instead of this methodology, we have chosen to use Fourier analysis of the entire peak shape. By this procedure all of the above problems are eliminated. In particular, we focus on the cosine coefficients of the Fourier series representing a peak. The instrumental effects are readily removed, and the remaining coefficient of harmonic number, (n), A, can be written as a product ... 

Fig. 3.47 is comparable to Fig. 3.41 for sinusoidal ac polarography if the tilted shape provides a net compensation of the charging current one obtains a symmetric bell-shaped curve of I in the square-wave polarogram, similar to that depicted in Fig. 3.42. In fact, virtually all of the statements made before on the sinusoidal technique are valid for the square-wave mode except for the rigid shape of its wave this conclusion is according to expectation, especially as Fourier analysis reveals the square wave to be a summation of a series of only...

Bloomfield, P. (1976). Fourier Analysis of Time Series An Introduction. Wiley, New York. 

Fourier analysis is a procedure in which a curve is decomposed into a sum of sine and cosine terms, called a Fourier series. To analyze the curve in Figure 20-24, which spans the interval xx = 0 to x2 = 10, the Fourier series has the form. 

There are problems in determining crystallite size from line broadening alone, since factors other than crystallite size contribute to the broadening, including local strain in the crystallites and shape anisotropy. Some of these problems can be overcome by the use of Fourier analysis of the peak shape. The cosine coefficients of the Fourier series can be used to determine a surface weighted average size for the crystallites. 

The protein structure is not known at high-resolution for any P-type pump, as crystals of a size and quality suitable for X-ray analysis have not yet been produced. However, the discovery that ordered two-dimensional semicrystalline arrays of ATPase molecules can be formed in situ in the membrane by incubation with tightly binding inhibitors such as vanadate (Skriver et al., 1981 Dux and Martonosi, 1983) has been of great help in obtaining a relatively detailed picture of the overall shape of the protein by electron microscopy. Fourier analysis of tilt series of electron micrographs permitted 3D-reconstructions of the Na+-K+- and Ca2+- ATPase protein structures to a resolution of 20-25 A (Figure 12a). Recently, this type of reconstruction analysis has been taken a step further to a resolution of about 14 A, and detailed structural features have become visible, even for the transmembrane domain (Toyoshima et al., 1993). The model of the Ca2+-ATPase peptide shown in... 

We want to learn how to quantize the radiation field. As a first step, consider a continuous elastic system. Any classical continuous elastic system in one dimension can be treated by a normal-mode analysis. Consider an elastic string of length a [m], tied at both ends to some fixed objects, with density per unit length p [kg m ], and tension, or Hooke s law force constant kH [N m-1]. The transverse displacements of the string along the x axis can be described by a transverse stretch y(x, t) at any point x along the string and at a time t. One can describe the y(x, t) as a Fourier sine series in x ... 

For quasi-periodic trajectories, like those for the normal-mode Hamiltonian in Eq. (69), I to) consists of a series of lines at the frequencies for the normal modes of vibration. In contrast, a Fourier analysis of a chaotic trajectory results in a multitude of peaks, without identifiable frequencies for particular modes. An inconvenience in this approach is that for a large molecule with many modes, a trajectory may have to be integrated for a long time T to resolve the individual lines in a power spectrum for a quasi-periodic trajectory. Moreover, in the presence of a resonance between different modes, the interpretation of the power spectrum may become misleading. 

Fourier analysis makes it possible to analyse a sectionally continuous periodic function into an infinite series of harmonics. For a non-periodic absolutely integrable function, the summation over discrete frequencies becomes an integral... 

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... 

We will now consider how to simulate this method of image formation in the X-ray diffraction experiment where we have to use a mathematical replacement for the objective lens. The studies by Porter are of great importance because they show how the Bragg reflections give the amplitude components of a Fourier series representing the electron density in the crystal (the electron-density map). In effect, Fourier analysis takes place in the diffraction experiment, so that the scattering of X rays by the electron density in the crystal produces Bragg reflections, each with a different amplitude F hkl) and relative phase Qhkl-... 

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). 

When a diffraction grating, such as a crystal, interacts with X rays, the electron density that causes this diffraction can be described by a Fourier series, as discussed in Chapter 6. The diffraction experiment effects a Fourier analysis, breaking down the Fourier series (of the electron density) into its components, that is, the diffracted beams with amplitudes, F[hkl). The relative phases a(hkl) are, however, lost in the process in all usual diffraction experiments. This loss of the phase information needed for the computation of an electron-density map is referred to as the phase problem. The aim of X-ray diffraction studies is to reverse this process, that is, to find the true relative phase and hence the true three-dimensional electron density. This is done by a Fourier synthesis of the components, but it is now necessary to know both the actual amplitude F[hkl) and the relative phase, a[hkl), in order to calculate a correct electron-density map (see Figure 8.1). We must be able to reconstruct the electron-density distribution in a systematic way by approximating, as far as possible, a correct [but so far unknown) set of phases In this way the crystallographer, aided by a computer, acts as a lens for X rays. 

Now consider the case of two nuclei with different resonance frequencies, each different from the reference frequency. Their decay patterns are superimposed, reinforcing and interfering to create a complex FID, as in Figure l-16b for the protons of methyl acetate. By the time there are four frequencies, as in the carbons of 3-hydroxybutyric acid shown in Figure l-16c, it is nearly impossible to unravel the frequencies visually. The mathematical process called Fourier analysis matches the FID with a series of sinusoidal curves and obtains from them the frequencies, line widths, and intensities of each component. The FID is... 

Fourier demonstrated that any periodic function, or wave, in any dimension, could always be reconstructed from an infinite series of simple sine waves consisting of integral multiples of the wave s own frequency, its spectrum. The trick is to know, or be able to find, the amplitude and phase of each of the sine wave components. Conversely, he showed that any periodic function could be decomposed into a spectrum of sine waves, each having a specific amplitude and phase. The former process has come to be known as a Fourier synthesis, and the latter as a Fourier analysis. The methods he proposed for doing this proved so powerful that he was rewarded by his mathematical colleagues with accusations of witchcraft. This reflects attitudes which once prevailed in academia, and often still do. 

In principle, Fourier analysis does not suffer from these shortcomings. By calculating the amplitude ratios and the phase lags for a series of frequencies, an unlimited number of equations can be obtained for the calculation of the transport parameters. However, due to the interactions that take place between the transport parameters at low frequencies and to the amplification of the experimental errors that take place at high frequencies, the advantages of the Fourier analysis on the method of moments are quite limited. Boersma-Klein et al. [85] carried... 

Fourier analysis tells us that h(6) can be written as a Fourier series... 

Deconvolution of instrumental line broadening was done before applying Fourier analysis to the 111 Bragg peak. In the case of Pd-Au-2 series, the crystallite sizes reported in Table 1 are those of the 111 reflection of the Pd-Au alloy, while for the other series, the Pd crystallite sizes are indicated. 

But we also know that any repetitive waveform, of almost arbitrary shape, can be decomposed into a sum of several sine (and cosine) waveforms of frequencies that are multiples of the basic repetition frequency f ( the fundamental frequency ). That is what Fourier analysis is all about. Note that though we do get an infinite series of terms, it is... 

We now present the main mathematical analysis and tools needed to finally cap a successful filter design. First we should recall our hazy Fourier series class. Fourier analysis is often avoided by power supply engineers, but it can go a long way toward understanding and tackling several key issues like EMI/noise, transformer proximity losses, PFC (power factor correction), and so on. 

In the case of Fourier analysis, the coherence critical value is independent of the processes to be compared, if they sufficiently well follow a linear description [1, 15]. This independency, however, holds exactly only in the limit of long time series. As wavelet analysis is a localized measure, this condition is not fullfilled. Hence, for different AR[1] processes (from white noise to almost nonstationary processes), we found a marginal dependency on the process parameters. 

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. 

A Fourier transform enables one to convert the variation of some quantity as a function of time into a function of frequency, and vice versa. Thus, if we represent the quantity that varies in time as x(f), then Fourier analysis enables us to also represent that quantity as a function X i>), where i/ is the frequency (—oo < i/ < oo). Fourier analysis is usually introduced by considering functions that vary in a periodic manner with time which can be written as a superposition of sine and cosine functions (a Fourier series see Section 110.8). If the period of the fvmction x f) is r then the cosine and sine terms in the Fourier series are functions of frequencies 27m/r, where n can take integer values 1, 2, 3, ... 

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