| | 127 | [[Image(Screen Shot 2015-01-13 at 12.05.47 PM.png, width=400)]] |
| | 128 | |
| | 129 | === MultiDimensional FFTs === |
| | 130 | A multidimensional DFT can be thought of as a sequence of 1D transforms in the way that a multidimensional Fourier transform is a sequence of 1D integrals. That is |
| | 131 | |
| | 132 | [[latex($F(\mathbf{k})= \displaystyle \int_{y=0}^{L_y} \displaystyle \int_{x=0}^{L_x} e^{i\mathbf{k}\cdot \mathbf{x}} f(\mathbf{x}) dx dy = \displaystyle \int_{y=0}^{L_y} e^{ik_y y} \left [ \displaystyle \int_{x=0}^{L_x} e^{ik_xx} \left [ f(\mathbf{x}) \right ] dx \right ] dy $)]] |
| | 133 | |
| | 134 | And again, the grid will be wrapped around in both x and y. This means it will look like |
| | 135 | |
| | 136 | Now since the normalization of the wavenumber goes like [[latex($\frac{2\pi}{L}$)]], if the region is not a square/cube, the spacing of the points in fourier space will not be uniform. For example, if [[latex($L_y = L_x/2$)]] then [[latex($\Delta y = 2 \Delta x$)]] |
