The Fourier Transform can be extended to multidimensions of space. This extension is quite simple to do, as all it involves is integrating with respect to each dimension for CT signals, and performing a summation for each dimension in a DT signal.
For example, the single dimension Fourier Transform only integrates with respect to x:
- $ F(\omega) = \int_{-\infty}^{\infty} f(x)e^{-2\pi j\omega x} dx $
If our two-dimensional function depends on x and y ( f(x,y)), then the two dimensional Fourier Transform can be represented as:
- $ F(u,v) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(x,y)e^{-2\pi j(ux + vy)}\,dx,dy $
The variables u and v represent the omega for each direction respectively.
For DT Signals, the jump to multiple dimensions is basically the same. With one dimension, our function only depended on n, (x[n]). The Fourier Transform was then a summation with respect to n and was thus represented as:
- $ F[\omega] = \frac{1}{N}\sum_{x=0}^{N-1} f[x]e^{-2\pi j\omega x/N} $
If the two-dimensional function depends on variable n and m. (x[n,m]) , the Fourier Transform then becomes a double summation:
- $ F[u,v] = \frac{1}{M}\sum_{y=0}^{M-1}\left( \frac{1}{N}\sum_{x=0}^{N-1} f[x,y]e^{-2\pi ju x/N} \right)e^{-2\pi jvy/M} $
Again, The variables u and v represent the omega for each direction respectively, meaning there is a seperate omega for each variable in the original function.
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