Vector-radix FFT algorithm: Difference between revisions

Revision as of 04:14, 29 May 2023

Revision as of 13:13, 29 May 2023

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* <math>S_{11}</math> is the DFT over the samples for which both <math>n_1</math> and <math>n_2</math> are odd.

Thanks to the [[List of trigonometric identities#Shifts and periodicity|periodicity of the complex exponential]], we can obtain the the following additional identities, valid for <math>0\leq k_1, k_2 < \frac{N}{2}</math>:

* <math>X\biggl(k_1+\frac{N}{2},k_2\biggr) = S_{00}(k_1,k_2) + S_{01}(k_1,k_2) W_N^{k_2} -S_{10}(k_1,k_2) W_N^{k_1} - S_{11}(k_1,k_2) W_N^{k_1+k_2}</math>;

* <math>X\biggl(k_1,k_2+\frac{N}{2}\biggr) = S_{00}(k_1,k_2) - S_{01}(k_1,k_2) W_N^{k_2} +S_{10}(k_1,k_2) W_N^{k_1} - S_{11}(k_1,k_2) W_N^{k_1+k_2}</math>;