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Reference.org
Prime-factor FFT algorithm
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<p>The prime-factor algorithm (PFA), also called the Good–Thomas algorithm (1958/1963), is a <a href="/facts/Fast_Fourier_transform/caT7UUCh">fast Fourier transform</a> (FFT) algorithm that re-expresses the <a href="/facts/Discrete_Fourier_transform/uK9Tjsbl">discrete Fourier transform</a> (DFT) of a size <i>N</i> = <i>N</i>1<i>N</i>2 as a two-dimensional <i>N</i>1×<i>N</i>2 DFT, but <i>only</i> for the case where <i>N</i>1 and <i>N</i>2 are <a href="/facts/Relatively_prime/bnCxCXxs">relatively prime</a>. These smaller transforms of size <i>N</i>1 and <i>N</i>2 can then be evaluated by applying PFA <a href="/facts/Recursion/CxSKxJoW">recursively</a> or by using some other FFT algorithm. </p><p>PFA should not be confused with the <i>mixed-radix</i> generalization of the popular <a href="/facts/Cooley%E2%80%93Tukey_FFT_algorithm/03VB7JrC">Cooley–Tukey algorithm</a>, which also subdivides a DFT of size <i>N</i> = <i>N</i>1<i>N</i>2 into smaller transforms of size <i>N</i>1 and <i>N</i>2. The latter algorithm can use <i>any</i> factors (not necessarily relatively prime), but it has the disadvantage that it also requires extra multiplications by roots of unity called <a href="/facts/Twiddle_factor/NSUVICjF">twiddle factors</a>, in addition to the smaller transforms. On the other hand, PFA has the disadvantages that it only works for relatively prime factors (e.g. it is useless for <a href="/facts/Power_of_two/MA6WEpTt">power-of-two</a> sizes) and that it requires more complicated re-indexing of the data based on the <a href="/facts/Additive_group/djWVpp9D">additive group</a> <a href="/facts/Group_isomorphism/LC4N1aDw">isomorphisms</a>. Note, however, that PFA can be combined with mixed-radix Cooley–Tukey, with the former factorizing <i>N</i> into relatively prime components and the latter handling repeated factors. </p><p>PFA is also closely related to the nested Winograd FFT algorithm, where the latter performs the decomposed <i>N</i>1 by <i>N</i>2 transform via more sophisticated two-dimensional convolution techniques. Some older papers therefore also call Winograd's algorithm a PFA FFT. </p><p>(Although the PFA is distinct from the Cooley–Tukey algorithm, <a href="/facts/I._J._Good/nRWnP9KP">Good</a>'s 1958 work on the PFA was cited as inspiration by Cooley and Tukey in their 1965 paper, and there was initially some confusion about whether the two algorithms were different. In fact, it was the only prior FFT work cited by them, as they were not then aware of the earlier research by Gauss and others.) </p>