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element indexing seems more likely in this way |
m clean up spacing around commas and other punctuation, replaced: ,and → , and , ,N → , N (2), ,k → , k (29), ,r → , r (5), ,u → , u |
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We suppose the 2-D DFT is defined
:<math>X(k_1,k_2) = \sum_{n_1=0}^{N_1-1} \sum_{n_2=0}^{N_2-1} x[n_1, n_2] \cdot W_{N_1}^{k_1 n_1} W_{N_2}^{k_2 n_2}, </math>
where <math>k_1 = 0,\dots,N_1-1</math>, and <math>k_2 = 0,\dots,N_2-1</math>, and <math>x[n_1, n_2]</math> is an <math>N_1 \times N_2</math> matrix, and <math>W_N = \exp(-j 2\pi /N)</math>.
For simplicity, let us assume that <math>N_1=N_2=N</math>, and the radix-<math>(r\times r)</math> is such that <math>N/r</math> is an integer.
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* <math>k_i=r u_i+v_i</math>, where <math>u_i=0,\ldots,(N/r)-1; v_i = 0,\ldots,r-1;</math>
where <math>i = 1</math> or <math>2</math>, and the DFT equation can be written as:<ref name=Chan92/>
:<math> X(r u_1+v_1,r u_2+v_2)=\sum_{p_1=0}^{N/r-1} \sum_{p_2=0}^{N/r-1} \left[ \sum_{q_1=0}^{r-1} \sum_{q_2=0}^{r-1} x[p_1+q_1 N/r,
== Other approaches ==
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