SYLLABUS Previous: 4.2 Linear equations.
Up: 4 FOURIER TRANSFORM
Next: 4.4 Non-linear equations.
Slide : [ Aliasing - Filters - Convolution || VIDEO login]
One of the beauties when using Fourier transforms, is the ability to work with a spectrum of modes and act on each of the components individually with a filter. By sampling the function over a period with a finite number of values , where is the size of the sampling interval, the spectrum gets truncated at the shortest wavelength called Nyquist critical wavelength: this corresponds to exactly 2 mesh points per wavelength, which does however not mean that shorter wavelengths do not contribute to the Fourier coefficients (4.1#eq.1). Figure 4.3#fig.1 illustrates how they get aliased back into the lower components of the spectrum.4.3 shows that even with the greatest precautions, such an aliasing can sometimes not be avoided, an needs then to be correctly interpreted.
Although they are present right from the beginning when the initial condition is first discretized (try to initialize and propagate an aliased cosine with ICWavelength=1.05 mesh points per wavelength using the JBONE applet above), aliases do not actually interfere with the resolution of linear equations. The story is however different for spatial non-linearities such as the quadratic term that is responsible for the wave-breaking (1.3.4#eq.1). Indeed, think of the convolution theorem, which tells that the Fourier transform of a convolution is just the product of the individual Fourier transforms . The converse is unfortunately also true: what can be viewed as a simple product in X-space becomes a convolution in K-space
or in discrete form
For the quadratic wave-breaking non-linearity, this shows that a short wavelength component such as in a sampling with 64 points, will incorrectly ``pollute'' a long wavelength channel through aliasing: . A simple cure for this, is to expand the size of the arrays by a factor two before the convolution takes place and pad them with zeros; changing the representation to calculate the multiplication of arrays twice the original size, the upper part of the spectrum is then simply discarded after the data has been transformed back. The entire procedure is illustrated in the coming section, where the non-linear Korteweg-DeVries (1.3.4#eq.3) and Burger equations (1.3.4#eq.2) are solved with a convolution in Fourier space.
|Copyright © Lifelong-learners at 11:00:17, February 17th, 2019|