In mathematical analysis and applications, multidimensional transforms are used to analyze the frequency content of signals in a domain of two or more dimensions.
Multidimensional Fourier transform One of the more popular multidimensional transforms is the Fourier transform, which converts a signal from a time/space domain representation to a frequency domain representation. The discrete-domain multidimensional Fourier transform (FT) can be computed as follows:
F ( w 1 , w 2 , … , w m ) = ∑ n 1 = − ∞ ∞ ∑ n 2 = − ∞ ∞ ⋯ ∑ n m = − ∞ ∞ f ( n 1 , n 2 , … , n m ) e − i w 1 n 1 − i w 2 n 2 ⋯ − i w m n m {\displaystyle F(w_{1},w_{2},\dots ,w_{m})=\sum _{n_{1}=-\infty }^{\infty }\sum _{n_{2}=-\infty }^{\infty }\cdots \sum _{n_{m}=-\infty }^{\infty }f(n_{1},n_{2},\dots ,n_{m})e^{-iw_{1}n_{1}-iw_{2}n_{2}\cdots -iw_{m}n_{m}}}
where F stands for the multidimensional Fourier transform, m stands for multidimensional dimension. Define f as a multidimensional discrete-domain signal. The inverse multidimensional Fourier transform is given by
f ( n 1 , n 2 , … , n m ) = ( 1 2 π ) m ∫ − π π ⋯ ∫ − π π F ( w 1 , w 2 , … , w m ) e i w 1 n 1 + i w 2 n 2 + ⋯ + i w m n m d w 1 ⋯ d w m {\displaystyle f(n_{1},n_{2},\dots ,n_{m})=\left({\frac {1}{2\pi }}\right)^{m}\int _{-\pi }^{\pi }\cdots \int _{-\pi }^{\pi }F(w_{1},w_{2},\ldots ,w_{m})e^{iw_{1}n_{1}+iw_{2}n_{2}+\cdots +iw_{m}n_{m}}\,dw_{1}\cdots \,dw_{m}}
The multidimensional Fourier transform for continuous-domain signals is defined as follows:
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