Coefficients Fourier

Flarker D and Kasper J S 1948 Phases of Fourier coefficients directly from crystal diffraction data Aota Crystallogr. 70-5... 

Each logarithm in the last temi can now be expanded and the (—n)th Fourier coefficient arising fi om each logarithm is — jn) zk-Y- To this must be added the n = 0 Fourier coefficient coming from the first, f-independent term and that arising from the expansion of second term as a periodic function, namely. 

For the Fourier coefficients of the modulus and the phase we note that, because of the time-inversion invariance of the amplitude, the former is even in f and the latter is odd. Therefore the former is representable as a cosine series and the latter as a sine series. Formally,... 

We use the sine series since the end points are set to satisfy exactly the three-point expansion [7]. The Fourier series with the pre-specified boundary conditions is complete. Therefore, the above expansion provides a trajectory that can be made exact. In addition to the parameters a, b and c (which are determined by Xq, Xi and X2) we also need to calculate an infinite number of Fourier coefficients - d, . In principle, the way to proceed is to plug the expression for X t) (equation (17)) into the expression for the action S as defined in equation (13), to compute the integral, and optimize the Onsager-Machlup action with respect to all of the path parameters. 

In the derivation we used the exact expansion for X t), but an approximate expression for the last two integrals, in which we approximate the potential derivative by a constant at Xq- The optimization of the action S with respect to all the Fourier coefficients, shows that the action is optimal when all the d are zero. These coefficients correspond to frequencies larger than if/At. Therefore, the optimal solution does not contain contributions from these modes. Elimination of the fast modes from a trajectory, which are thought to be less relevant to the long time scale behavior of a dynamical system, has been the goal of numerous previous studies. 

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. 

The most important feature of the Fourier analysis is the reduction of the multicoUi-nearity and ike dimension of ike original specira. However, ihe Fourier coefficients hear no. simple relationship to individual features of the spectrum so that it will not he clear what information is being used in calibration."... 

I compute each Fourier coefficient X kST) (of which there are M) it is therefore necessary to aluate the summation A (n5f) exp[—27riwfc/M] for that value of k. There will be M... 

We can recover the free-particle result (i.e. zero potential) from Equation (3.98) by setting all i)f Ihe Fourier coefficients Uq to zero, in which case the equation reduces to ... 

Because of this form, the applications of the elimination method for N2 — 1 times to Cf. ih ) as a function of the argument = ih for fixed k permit us to find a solution of problem (13) by means of formula (15). As can readily be observed, the calculations of the Fourier coefficients p. and solutions 2/jj can be carried out by the same formulas related to common sums of the special type... 

Information concerning size distribution and strain profile can be obtained from the cosine Fourier coefficients, which describe the symmetric peak broadening. 

Another typical problem met in this kind of analysis is known as the hook effect . It is due to an overestimation of the background line to the detriment of the peak tails. As a consequence, the low order Fourier coefficients of the profile are underestimated. In the fitting procedure by pseudo-Voigt functions, this problem occurs if the Gauss content is so high that the second derivative of the Fourier coefficients is negative this is obviously physically impossible because it represents a probability density. 

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. 

Summarizing, two complementary representations of a signal have been derived f(0 in the time domain and [A - jB ] in the frequency domain. The imaginary Fourier coefficients, represent the frequencies of the sine functions and the real... 

The Fourier coefficients can be combined in a so called power spectrum which is defined as 7 = i/[X/4 4- )]. One should realize that because the contributions... 

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. 

In Section 40.3.4 we have shown that the FT of a discrete signal consisting of 2N + 1 data points, comprises N real, N imaginary Fourier coefficients (positive frequencies) and the average value (zero frequency). We also indicated that N real and N imaginary Fourier coefficients can be defined in the negative frequency domain. In Section 40.3.1 we explained that the FT of signals, which are symmetrical about the / = 0 in the time domain contain only real Fourier coefficients. 

A shift or translation of f(r) by results in a modulation of the Fourier coefficients by exp(-ycofo). Without shift f(r) is transformed into F(fo). After a shift by tQ, fit - tg) is transformed into exp(-ja)tQ) F(sine wave) of the Fourier coefficients. The frequency cotg of the... 

The phase spectrum 0(n) is defined as 0(n) = arctan(A(n)/B(n)). One can prove that for a symmetrical peak the ratio of the real and imaginary coefficients is constant, which means that all cosine and sine functions are in phase. It is important to note that the Fourier coefficients A(n) and B(n) can be regenerated from the power spectrum P(n) using the phase information. Phase information can be applied to distinguish frequencies corresponding to the signal and noise, because the phases of the noise frequencies randomly oscillate. 

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. 

Ideally, any procedure for signal enhancement should be preceded by a characterization of the noise and the deterministic part of the signal. Spectrum (a) in Fig. 40.18 is the power spectrum of white noise which contains all frequencies with approximately the same power. Examples of white noise are shot noise in photomultiplier tubes and thermal noise occurring in resistors. In spectrum (b), the power (and thus the magnitude of the Fourier coefficients) is inversely proportional to the frequency (amplitude 1/v). This type of noise is often called 1//... 

Many other filter functions can be designed, e.g. an exponential or a trapezoidal function, or a band pass filter. As a rule exponential and trapezoidal filters perform better than cut-off filters, because an abrupt truncation of the Fourier coefficients may introduce artifacts, such as the annoying appearance of periodicities on the signal. The problem of choosing filter shapes is discussed in more detail by Lam and Isenhour [11] with references to a more thorough mathematical treatment of the subject. The expression for a band-pass filter is H v) = 1 for v j < v < else... 

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.

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