Compute the Discrete Fourier Transform (DFT) by means of FFTW of data stored on a 2D grid. This action can operate on any other action that outputs scalar data on a two-dimensional grid.
Compute the Discrete Fourier Transform (DFT) by means of FFTW of data stored on a 2D grid.
This action can operate on any other action that outputs scalar data on a two-dimensional grid.
Up to now, even if the input data are purely real the action uses a complex DFT.
Just as a quick reference, given a 1D array \f$\mathbf{X}\f$ of size \f$n\f$, this action computes the vector \f$\mathbf{Y}\f$ given by
\f[
Y_k = \sum_j=0^{n-1} X_j \exp{2\pi\, j k \sqrt{-1}/n}.
Y_k = \sum_{j=0}^{n-1} X_j e^{2\pi\, j k \sqrt{-1}/n}.
\f]
This can be easily extended to more than one dimension. All the other details can be found at http://www.fftw.org/doc/What-FFTW-Really-Computes.html#What-FFTW-Really-Computes.
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@@ -49,12 +51,17 @@ This can be easily extended to more than one dimension. All the other details ca
The keyword "FOURIER_PARAMETERS" deserves just a note on the usage. This keyword specifies how the Fourier transform will be normalized. The keyword takes two numerical parameters (\f$a,\,b\f$) that define the normalization according to the following expression
\f[
\frac{1}{n^{1-a}/2} \sum_j=0^{n-1} X_j \exp{2\pi b\, j k \sqrt{-1}/n}
\frac{1}{n^{(1-a)/2}} \sum_{j=0}^{n-1} X_j e^{2\pi b\, j k \sqrt{-1}/n}
\f]
The default values of these parameters are: \f$a=1\f$ and \f$b=1\f$.
\par Examples
The following example tells Plumed to compute the complex 2D 'backward' Discrete Fourier Transform by taking the data saved on a grid called 'density', and normalizing the output by \f$ \frac{1}{\sqrt{N_x\, N_y}}\f$, where \f$N_x\f$ and \f$N_y\f$ are the number of data on the grid (it can be the case that \f$N_x \neq N_y\f$):
keys.add("optional","FT_TYPE","choose what kind of data you want as output on the grid "
" possible values are: ABS - compute the complex modulus of Fourier coefficients (DEFAULT) "
" NORM - compute the norm (i.e. ABS^2) of Fourier coefficients "
" COMPLEX - store the FFTW complex output on the grid (as a vector) ");
keys.add("compulsory","FOURIER_PARAMETERS","default","what kind of normalization is applied to the output and if the Fourier transform in FORWARD or BACKWARD "
"this keyword takes the form FOURIER_PARAMETERS=A,B "
"where A and B can be 0, 1 or -1. "
"the default values are A=1 (no normalization at all) and B=1 (forward FFT) "
"other possible choices for A are: "
" A=-1: normalize by the number of data "
" A=0: normalize by the square root of the number of data (one forward and followed by backward FFT recover the original data) ");
keys.add("optional","FT_TYPE","choose what kind of data you want as output on the grid. Possible values are: ABS = compute the complex modulus of Fourier coefficients (DEFAULT); NORM = compute the norm (i.e. ABS^2) of Fourier coefficients; COMPLEX = store the FFTW complex output on the grid (as a vector).");
keys.add("compulsory","FOURIER_PARAMETERS","default","what kind of normalization is applied to the output and if the Fourier transform in FORWARD or BACKWARD. This keyword takes the form FOURIER_PARAMETERS=A,B, where A and B can be 0, 1 or -1. The default values are A=1 (no normalization at all) and B=1 (forward FFT). Other possible choices for A are: "
"A=-1: normalize by the number of data, "
"A=0: normalize by the square root of the number of data (one forward and followed by backward FFT recover the original data). ");
