Multidimensional transform: Difference between revisions

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=== MD DFT ===

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''₁ × ''N''₂ × ... ''N''_''Mm''}} DFT is given by:

:<math> Fx(K_1,K_2,\ldots,K_nK_m)= \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_i'' ≤ ''N_i'' &minus; 1}}, {{nowrap|''i'' {{=}} 1, 2, ..., ''m''}}.