 formula_86

On computing Fourier Transform of both sides of this expression, we get

formula_87formula_88

There are many possible definitions of the M-D discrete Fourier transform, and that all of these correspond to sets of samples of the M-D Fourier transform; these DFT's can be used to perform convolutions as long their assumed region of support contains the support for formula_83.Let us assume that formula_90 is sampled on a formula_46xformula_49x...xformula_52 rectangular lattice of samples, and let

formula_94 
Therefore, formula_95formula_96.To compute (formula_46xformula_49x...xformula_52)-point DFT's of formula_100 and formula_101 requires that both sequences have their regions of support extended with samples of value zero.