Multidimensional transform

In mathematical analysis and applications, multidimensional transforms are used to analyze the frequency content of signals in a domain of two or more dimensions.

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:

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

The multidimensional Fourier transform for continuous-domain signals is defined as follows:

Similar properties of the 1-D FT transform apply, but instead of the input parameter being just a single entry, it's a Multi-dimensional (MD) array or vector. Hence, it's x(n1,…,nM) instead of x(n).

if ${\displaystyle x_{1}(n_{1},\ldots ,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X_{1}(\omega _{1},\ldots ,\omega _{M})}$ , and ${\displaystyle x_{2}(n_{1},\ldots ,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X_{2}(\omega _{1},\ldots ,\omega _{M})}$ then,

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