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[[File:Bailey 4-step FFT.svg|thumb|Bailey algorithm (4-step version) for a 16-point FFT]]
The '''Bailey's FFT''' (also known as a '''4-step FFT''') is a high-performance algorithm for computing the [[fast Fourier transform]] (FFT). This variation of the [[Cooley–Tukey FFT algorithm]] was originally designed for systems with hierarchical memory common in modern computers (and was the first FFT algorithm in this so called "out of core" class). The algorithm treats the samples as a two dimensional matrix (thus yet another name, a '''matrix FFT algorithm'''{{sfn|Arndt|2010|p=438}}) and executes short FFT operations on the columns and rows of the matrix, with a correction multiplication by "[[twiddle factor]]s" in between.{{sfn|Hart|Tornaría|Watkins|2010|p=191}}
The algorithm got its name after an article by [[David H. Bailey (mathematician)|David H. Bailey]], ''FFTs in external or hierarchical memory'', published in 1989. In this article Bailey credits the algorithm to W. M. Gentleman and G. Sande who published their paper, ''Fast Fourier Transforms: for fun and profit'', some twenty years earlier in 1966.{{sfn|Bailey|1989|p=2}} The algorithm can be considered a radix-<math>\sqrt n</math> FFT decomposition.{{sfn|Frigo|Johnson|2005|p=2}}
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