OverviewSplit a signal S(SPACE,TIME) into its downward/upward space propagating and stationary components via a 2D Fourier decomposition. S is a (SPACE,TIME) matrix. We eventually proceed to a space and/or time filtering. DT,DX are temporal and space step. PERIOD=[Tmin Tmax] and WAVELENGTH=[Xmin Xmax] are time and space band-pass filters specifications.DownloadWith SVN:The fortran code is here: svn checkout http://code.google.com/p/guillaumemaze/source/browse/trunk/fortran/WSEdecomp gmaze_WSEdecomp The Matlab version here: Direct to the file: Matlab or at Matlab Central: DocumentationTheorie of the 2D Fourier analysis
We can express a variable
First, for each time of the record we compute cosine and sine coefficients from Fourier
decomposition of the variable, ie we express
Next, for each point we compute cosine and sine coefficients from the Fourier decomposition of
A and B:
Last, inserting 2 and 3 into 1 we obtain the
and following the notation from Park et al. (2004): with the westward/eastward sine and cosine coefficients:
and phases of East and West waves: ![]() ![]()
We can rewrite equation 5 in the more significative form:
where we can identify West/East waves having amplitude ![]() ![]() ![]() ![]() We state that a stationnary wave is the sum of an eastward and a westward wave having the same amplitude. This allow us to rewrite expression 6 separating standing from propagating (eastward or westward, it depends on the relative amplitude of each part) wave components.
If
and if
First terms on the right-hand-side of equations 7 and 8 are standing wave parts of
the harmonic while second term is an Eastward or Westward wave.
We reconstruct the signal by summing harmonics over the two directions
Finally, this method allow to plot a very powerfull diagram: a 2D power spectral density. Contours of
amplitudes
Bibliography
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