Fourier series method

Patterson A L 1934 A Fourier series method for for the determination of the components of interatomic distances in crystals Phys. Rev. 46 372-6... 

To find the specific solution for the given initial condition, A and B must be determined from the initial condition (some might have already been determined from the boundary conditions). Since the solution is expressed as a Fourier series, this method is also called the Fourier series method. 

The solution is of the type of Equation 3-51. To satisfy u x=o = Q, all the in Equation 3-51 must be zero (this is because not only collectively but also individually A sm(X x)+B cos(k x) must be zero). To satisfy u x=l = 0, must equal tm, where n = l, 2,. That is, = nn/L. This example shows why the boundary conditions must be zero for the Fourier series method, because otherwise cannot be constrained. Replacing B and into Equation 3-51 leads to... 

For a much more detailed discussion of Fourier series methods, see Robertson (1937) and Lipson and Cochran (1953). 

In the Fourier series method the weighted radial distribution func -tion, which represents the number of atoms at a distance r from any atom, weighted by the products of the diffracting powers, is given by (he expression... 

A more analytical method of stability analysis is the method of von Neumann [424, 565] (note that [424] is mostly incorrectly cited as being of the year 1951 [139]). The method focusses on an interior point along X in the grid and looks at the propagation of an error at that point, making certain reasonable assumptions, using Fourier series (which is why the method on occasion is also called the Fourier series method). 

A.L. Patterson, A Fourier series representation of the average distribution of the scattering power in crystals, Phys. Rev. 45, 763 (1934), A.L. Patterson, A Fourier series method for the determination of the components of the interatomic distances in crystals, Phys. Rev. 46, 372(1934). 

Not all bioelectric potentials are periodic. In many cases, single pulse signals are observed, such as neural response to a single stimulus. The Fourier-series method of spectral analysis does not apply in such cases as it is defined for periodic functions only. It can be extended, however, through a series of limiting processes to develop the Fourier transform integrals which are defined below. 

Before beginning the Fourier series method, it is interesting to explore lurflier the use of sinusoidal signals in the least squares method as discussed in connection with Eq. (7.12). Suppose a series of sin() and cos() functions are taken as the approximating fimctions in the form ... 

Figure 7.10 shows the resulting Fourier amplitude vs. time periods for the data with time periods up to 30 years. The figure shows a strong peak in the Fourier components at a time interval of around 11 years as expected. The identification of frequency components composing a signal is one of the most important applications of the Fourier series method. However in this example the identification of... 

---