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The multidimensional [[discrete Fourier transform]] (DFT) is a sampled version of the discrete-___domain FT by evaluating it at sample frequencies that are uniformly spaced.<ref>Dudgeon and Mersereau, Multidimensional Digital Signal Processing,2nd edition,1995</ref> The {{nowrap|''N''<sub>1</sub> × ''N''<sub>2</sub> × ... ''N''<sub>''M''</sub>}} DFT is given by:
:<math> Fx(K_1,K_2,\ldots,K_n)= \sum_{n_1=0}^{N_1-1} \cdots \sum_{n_m=0}^{N_m-1} fx(n_1,n_2,\ldots,n_N) e^{-i \frac{2 \pi}{N_1} n_1 K_1 -i \frac{2 \pi}{N_2} n_2 K_2 \cdots -i \frac{2 \pi}{N_m} n_m K_m} </math>
for {{nowrap|0 ≤ ''K<sub>i</sub>'' ≤ ''N<sub>i</sub>'' − 1}}, {{nowrap|''i'' {{=}} 1, 2, ..., ''m''}}.
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The inverse multidimensional DFT equation is
:<math> fx(n_1,n_2,\ldots,n_m)= \frac{1}{N_1 \cdots N_m} \sum_{K_1=0}^{N_1-1} \cdots \sum_{K_m=0}^{N_m-1} Fx(K_1,K_2, \ldots ,K_m) e^{i \frac{2 \pi}{N_1} n_1 K_1 +i \frac{2 \pi}{N_2} n_2 K_2\cdots+i \frac{2 \pi}{N_m} n_m K_m} </math>
for {{nowrap|0 ≤ ''n''<sub>1</sub>, ''n''<sub>2</sub>, ... , ''n''<sub>''m''</sub> ≤ ''N''<sub>(1, 2, ... , ''m'')</sub> – 1}}.
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