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A harmonic decomposition is obtained from a discrete Fourier transform
(DFT) assuming a regular mesh and a periodic domain of length
Using the notations
forward and backward transformations are defined as
If is a power of 2, the total number of operations can be dramatically reduced from to in a fast Fourier transform (FFT), by applying recursively the decomposition
with until a sum of DFT of length is obtained. Figure (1.4.5#fig.1) illustrates how the approximation of a periodic square wave converges with an increasing number of Fourier components. Note how the function overshoots in the neighborhood of sharp edges: this is the Gibbs phenomenon and it will always be there for a finite number of terms in the sum.
It should not be surprizing to hear that a harmonic decomposition is particularly well suited for smooth global functions with long wavelengths , since they result in a narrow spectrum . Note that the convergence with an increasing number of Fourier components is faster than any polynomial once the smallest structure has been resolved. The implementation of non-periodic boundary conditions, however, can sometimes be problematic.
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