Exponential Fourier series

To assess the extent to which the exponentiated Fourier series has appreciable Fourier amplitudes, and set the sampling grid accordingly, further development of formula (21) is needed. We first rewrite... 

It is therefore possible to rewrite Eq. (6.39) as an exponential Fourier series ... 

I exercise 6.20 I Write the formula for finding the coefficients for an exponential Fourier series. Is there any difference in the formulas for odd functions, even functions, or functions that are neither odd nor even What conditions must the function obey to be represented by an exponential Fourier... 

As the limit L oo is taken, k becomes a continuously variable quantity. In this limit, an exponential Fourier series becomes an integral, which is called a Fourier integral or a Fourier transform. 

If we write the Euler definitions for cosO and sin0, we obtain the complex form of the Fourier series known either as the Complex Fourier Series or the Exponential Fourier Series of f(x). It is represented as... 

However, some aspects of the spin dynamics are better described using functions other than Fourier series. For example, the magnetization decay in a CPMG [28] experiment follows an exponential form,... 

Eq. (2.11) can be solved by developing it as a complex series of sines and cosines according to relation (2.9). This is a Fourier series [14]. Thus, an exponential in the time domain, F ((), and a Lorentzian in the frequency domain, f (op, are Fourier transforms of each other [15-17],... 

Superficially, except for the sign change (in the exponential term) that accompanies the transform operation, this equation appears identical to Eq. (5.9), a general three-dimensional Fourier series. But here, each Fhkl is not just one of many simple numerical amplitudes for a standard set of component waves in a Fourier series. Instead, each Fhkl is a structure factor, itself a Fourier series, describing a specific reflection in the diffraction pattern. ("Curiouser and curiouser," said Alice.)... 

In words, the difference Patterson function is a Fourier series of simple sine and cosine terms. (Remember that the exponential term is shorthand for these trigonometric functions.) Each term in the series is derived from one reflection hkl in both the native and derivative data sets, and the amplitude of each term is (IFHpI — IFpl)2, which is the amplitude contribution of the heavy atom to structure factor FHp. Each term has three frequencies h in the u-direction, k in the v-direction, and l in the w-direction. Phases of the structure factors are not included at this point, they are unknown. 

Although this equation is rather forbidding, it is actually a familiar equation (5.15) with the new parameters included. Equation (7.8) says that structure factor Fhk[ can be calculated (Fc) as a Fourier series containing one term for each atom j in the current model. G is an overall scale factor to put all Fcs on a convenient numerical scale. In the /th term, which describes the diffractive contribution of atom j to this particular structure factor, n- is the occupancy of atom j f- is its scattering factor, just as in Eq. (5.16) Xj,yjt and zf are its coordinates and Bj is its temperature factor. The first exponential term is the familiar Fourier description of a simple three-dimensional wave with frequencies h, k, and / in the directions x, y, and 7. The second exponential shows that the effect of Bj on the structure factor depends on the angle of the reflection [(sin 0)/X]. 

The convolution theorem states diat /, g and h are Fourier transforms of F, G and H. Hence linear filters as applied directly to spectroscopic data have their equivalence as Fourier filters in die time domain in other words, convolution in one domain is equivalent to multiplication in die other domain. Which approach is best depends largely on computational complexity and convenience. For example, bodi moving averages and exponential Fourier filters are easy to apply, and so are simple approaches, one applied direct to die frequency spectrum and die other to die raw time series. Convoluting a spectrum widi die Fourier transform of an exponential decay is a difficult procedure and so die choice of domain is made according to how easy the calculations are. 

Notice that if we had taken the convention for the Fourier series Equation (44) that puts the global vector position of the atom, R/y = R/ + Xy, in the exponential term the Fourier coefficients T y could not be identified with magnetic moments because of the phase factor containing the atom positions. 

Integral transforms were discussed, including Fourier and Laplace transforms. Fourier transforms are the result of allowing the period of the function to be represented by a Fourier series to become larger and larger, so that the series approaches an integral in the limit. Fourier transforms are usually written with complex exponential basis functions, but sine and cosine transforms also occur. Laplace transforms are related to Fourier transforms, with real exponential basis functions. We presented several theorems that allow the determination of some kinds of inverse Laplace transforms and that allow later applications to the solution of differential equations. 

More rapid convergence of the Fourier series can be achieved if one uses a more efficient damping factor, g , rather than a simple exponential damping, e nt,... 

Recall that sines and cosines can be expressed in terms of complex exponential functions, according to Eqs. (4.51) and (4.52). Accordingly, a Fourier series can be expressed in a more compact form ... 

Converting the cosine/sine form to the complex exponential form allows many manipulations that would be very difficult otherwise (for an example, see Section 2 of Appendix A Convolution and DFT Properties). But, if you re totally uncomfortable with complex numbers and Euler s Identity (or with the identities of IS"" century mathematicians in general), then you can write the DFT in real number terms as a form of the Fourier Series ... 

The discrete-time Fourier series (DTPS) for discrete-time waveforms x n) of period N can also be given in three forms however, the complex exponential form is by far the most common. 

At fixed 5, the terms in (5.180) represent the modes of the f-shape of the potential. The Bessel function oscillates and thus a s are analogous to frequencies in the Fourier series expansion. Equation (5.180) shows that each mode exponentially decays with the distance 5 the characteristic damping length is / = l/a , or, in the dimensional form... 

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