Rader's FFT algorithm: Difference between revisions

Revision as of 05:10, 12 January 2004

Revision as of 22:53, 19 January 2004

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j = 0,\dots,n-1. </math>

If ''n'' is a prime number, then the set of non-zero indices ''k'' = 1,...,''n''-1 forms a [[group (mathematics)|group]] under multiplication [[modulo]] ''n''. One consequence of this is that there exists a [[generating set of a group|generator]] of the group, an integer ''g'' such that ''k'' = ''g''^''q'' (mod ''n'') for any non-zero index ''k'' and for somea unique ''q'' in 0,...,''n''-2 (forming a [[bijection]] from ''q'' to non-zero ''k''). Similarly ''j'' = ''g''^-''p'' (mod ''n'') for any non-zero index ''j'' and for somea unique ''p'' in 0,...,''n''-2, where the negative exponent denotes the multiplicative inverse of ''g''^''p'' modulo ''n''. That means that we can rewrite the DFT using these new indices ''p'' and ''q'' as:

:<math> f_0 = \sum_{k=0}^{n-1} x_k,</math>