This algorithm, including its recursive application, was already known around 1805 to Carl Friedrich Gauss, who used it to interpolate the trajectories of the asteroids Pallas and Juno, but his work was not widely recognized (being published only posthumously and in neo-Latin); Gauss did not analyze the asymptotic computational time, however. Various limited forms were also rediscovered several times throughout the 19th and early 20th centuries. FFTs became popular after J. W. Cooley of IBM and J. W. Tukey of Princeton published a paper in 1965 reinventing the algorithm and describing how to perform it conveniently on a computer (including how to arrange for the output to be produced in the natural ordering).

Because the Cooley-Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT. For example, Rader's or Bluestein's algorithm can be used to handle large prime factors that cannot be decomposed by Cooley-Tukey, or the Prime-factor algorithm can be exploited for greater efficiency in separating out relatively prime factors. See also the fast Fourier transform for information on other FFT algorithms, specializations for real and/or symmetric data, and accuracy in the face of finite floating-point precision.

Table of contents |

2 General factorizations 3 Data reordering, bit reversal, and in-place algorithms 4 References |

Recall that the DFT is defined by the formula:

Radix-2 DIT first computes the Fourier transforms of the even-indexed numbers
More explicitly, let us write *n*' = *n*/2 and denote the DFT of the even-indexed numbers *x*'_{0} = *x*_{0}, *x*'_{1} = *x*_{2}, ..., *x*'_{n'-1} = *x*_{n-2} by *f*_{0}', ..., *f* '_{n'-1}; and the DFT of the odd-indexed numbers *x*''_{0} = *x*_{1}, *x*''_{1} = *x*_{3}, ..., *x*''_{n'-1} = *x*_{n-1} by *f*_{0}'', ..., *f* ''_{n'-1}. Then it follows:

The above re-expression of a size-*n* DFT as two size-*n*/2 DFTs is sometimes called the **Danielson-Lanczos** lemma, since the identity was noted by those two authors in 1942 (influenced by Runge's 1903 work). They applied their lemma in a "backwards" recursive fashion, repeatedly *doubling* the DFT size until the transform spectrum converged (although they apparently didn't realize the linearithmic asymptotic complexity they had achieved). The Danielson-Lanczos work predated widespread availability of computing machines and required hand calculation; they reported a computation time of 140 minutes for a size-64 DFT operating on real inputs (see below) to 3-5 significant digits. Cooley and Tukey's 1965 paper reported a running time of 0.02 minutes for a size-2048 complex DFT on an IBM 7094 (probably in 36-bit single precision, ~8 digits). Rescaling the time by the number of operations, this corresponds roughly to a speedup factor of around 800,000. The more modern FFT library FFTW, on a 2GHz Pentium 4 in 64-bit double precision (~16 digits), can compute a size-64 real-input DFT in 0.5μs and a size-2048 complex DFT in 50μs, speedups of about 16,000,000,000 and 20,000 over Danielson & Lanczos and Cooley & Tukey, respectively, not even including the considerable improvements in accuracy.

(140 minutes for size 64 may sound like a long time, but it corresponds to an average of at most 16 seconds per floating-point operation, around 20% of which are multiplications...this is a fairly impressive rate for a human being to sustain for almost two and a half hours, especially when you consider the bookkeeping overhead.)

- Perform
*n*_{1}DFTs of size*n*_{2}. - Multiply by complex roots of unity called
**twiddle factors**. - Perform
*n*_{2}DFTs of size*n*_{1}.

There are many other variations on the Cooley-Tukey algorithm. **Mixed-radix** implementations handle composite sizes with a variety of (typically small) factors in addition to two, usually (but not always) employing the O(*n*^{2}) algorithm for the prime base cases of the recursion. **Split-radix** merges radices 2 and 4, exploiting the fact that the first transform of radix 2 requires no twiddle factor, in order to achieve a minimal operation count for power-of-two sizes. (On present-day computers, performance is determined more by cache and CPU pipeline considerations than by strict operation counts; well optimized FFT implementations often employ larger radices and/or hard-coded base-case transforms of significant size.) Another way of looking at the Cooley-Tukey algorithm is that it re-expresses a size *n* one-dimensional DFT as an *n*_{1} by *n*_{2} two-dimensional DFT (plus twiddles), where the output matrix is transposed. The net result of all of these transpositions, for a radix-2 algorithm, corresponds to a bit reversal of the input (DIF) or output (DIT) indices. If, instead of using a small radix, one employs a radix of roughly √*n* and explicit input/output matrix transpositions, it is called a **four-step** algorithm (or *six-step*, depending on the number of transpositions), initially proposed for memory locality (cache) optimization (Gentleman and Sande, 1966; also Bailey, 1990).

The general Cooley-Tukey factorization rewrites the indices *j* and *k* as *j* = *n*_{2} *j*_{1} + *j*_{2} and *k* = *n*_{1} *k*_{2} + *k*_{1}, respectively, where the indices *j*_{a} and *k*_{a} run from 0..*n*_{a}-1 (for *a* of 1 or 2). That is, it re-indexes the input (*k*) and output (*j*) as *n*_{1} by *n*_{2} two-dimensional arrays in column-major and row-major order, respectively. When this reindexing is substituted into
the DFT formula for *jk*, the *n*_{2}*j*_{1}*n*_{1}*k*_{2} cross term vanishes (its exponential is unity), and the remaining terms give

The 1965 Cooley-Tukey paper noted that one can employ an arbitrary radix *r* (as well as mixed radices), but failed to realize that the radix butterfly is itself a DFT that can use FFT algorithms. Hence, they reckoned the complexity to be O(*r*^{2} *n*/*r* log_{r}*n*), and erroneously concluded that the optimal radix was 3 (the closest integer to *e*). Gauss also derived the algorithm for arbitrary radices, and gave explicit examples of both radix-3 and radix-6 steps.

Although the abstract Cooley-Tukey factorization of the DFT, above, applies to all implementations of the algorithm, much greater diversity exists in the techniques for ordering and accessing the data at each stage of the FFT. Of special interest is the problem of devising an in-place algorithm that overwrites its input with its output data using only O(1) auxiliary storage.

The most well-known reordering technique involves explicit **bit reversal** for in-place radix-2 algorithms. Bit reversal is the permutation where the data at an index *k*, written in binary with digits *b*_{4}*b*_{3}*b*_{2}*b*_{1}*b*_{0} (e.g. 5 digits for *n*=32 inputs), is transferred to the index with reversed digits *b*_{0}*b*_{1}*b*_{2}*b*_{3}*b*_{4} . Consider the last stage of a radix-2 DIT algorithm like the one presented above, where the output is written in-place over the input: when *f _{j}*' and

Many FFT users, however, prefer natural-order outputs, and a separate, explicit bit-reversal stage can have a non-negligible impact on the computation time, even though bit reversal can be done in O(*n*) time and has been the subject of much research (e.g. Karp, 1996; Carter, 1998; and Rubio, 2002). Also, while the permutation is a bit reversal in the radix-2 case, it is more generally an arbitrary (mixed-base) digit reversal for the mixed-radix case, and the permutation algorithms become more complicated to implement. Moreover, it is desirable on many hardware architectures to re-order intermediate stages of the FFT algorithm so that they operate on consecutive (or at least more localized) data elements. To these ends, a number of alternative implementation schemes have been devised for the Cooley-Tukey algorithm that do not require separate bit reversal and/or involve additional permutations at intermediate stages.

The problem is greatly simplified if it is **out-of-place**: the output array is distinct from the input array or, equivalently, an equal-size auxiliary array is available. The **Stockham auto-sort** algorithm (Stockham, 1966) performs every stage of the FFT out-of-place, typically writing back and forth between two arrays, transposing one "digit" of the indices with each stage, and has been especially popular on SIMD architectures (Swarztrauber, 1982). Even greater potential SIMD advantages (more consecutive accesses) have been proposed for the **Pease** (1968) algorithm, which also reorders out-of-place with each stage, but this method requires separate bit/digit reversal and O(*n* log *n*) storage. One can also directly apply the Cooley-Tukey factorization definition with explicit (depth-first) recursion and small radices, which produces natural-order out-of-place output with no separate permutation step and can be argued to have cache-oblivious locality benefits on systems with hierarchical memory (Singleton, 1967; see also FFTW).

A typical strategy for in-place algorithms without auxiliary storage and without separate digit-reversal passes involves small matrix transpositions (which swap individual pairs of digits) at intermediate stages, which can be combined with the radix butterflies to reduce the number of passes over the data (Johnson & Burrus, 1984; Temperton, 1991; Qian *et al.*, 1994; Hegland, 1994).

- James W. Cooley and John W. Tukey, "An algorithm for the machine calculation of complex Fourier series,"
*Math. Comput.***19**, 297–301 (1965). - Carl Friedrich Gauss, "Nachlass: Theoria interpolationis methodo nova tractata,"
*Werke*band**3**, 265–327 (Königliche Gesellschaft der Wissenschaften, Göttingen, 1866). See also M. T. Heideman, D. H. Johnson, and C. S. Burrus, "Gauss and the history of the fast Fourier transform,"*IEEE ASSP Magazine***1**(4), 14–21 (1984). - G. C. Danielson and C. Lanczos, "Some improvements in practical Fourier analysis and their application to X-ray scattering from liquids,"
*J. Franklin Inst.***233**, 365–380 and 435–452 (1942). - W. M. Gentleman and G. Sande, "Fast Fourier transforms—for fun and profit,"
*Proc. AFIPS***29**, 563–578 (1966). - P. Duhamel and M. Vetterli, "Fast Fourier transforms: a tutorial review and a state of the art,"
*Signal Processing***19**, 259–299 (1990). - David H. Bailey, "FFTs in external or hierarchical memory,"
*J. Supercomputing***4**(1), 23–35 (1990). - Alan H. Karp, "Bit reversal on uniprocessors,"
*SIAM Review***38**(1), 1–26 (1996). - Larry Carter and Kang Su Gatlin, "Towards an optimal bit-reversal permutation program,"
*Proc. 39th Ann. Symp. on Found. of Comp. Sci. (FOCS)*, 544–553 (1998). - M. Rubio, P. Gómez, and K. Drouiche, "A new superfast bit reversal algorithm,"
*Intl. J. Adaptive Control and Signal Processing***16**, 703–707 (2002). - T. G. Stockham, "High speed convolution and correlation",
*Spring Joint Computer Conference, Proc. AFIPS***28**, 229–233 (1966). - P. N. Swarztrauber, "Vectorizing the FFTs", in G. Rodrigue (Ed.),
*Parallel Computations*(Academic Press, New York, 1982), pp. 51–83. - M. C. Pease, "An adaptation of the fast Fourier transform for parallel processing",
*J. ACM***15**(2), 252–264 (1968). - Richard C. Singleton, "A method for computing the fast Fourier transform",
*Commun. of the ACM***10**(1967), 647–654. - H. W. Johnson and C. S. Burrus, "An in-place in-order radix-2 FFT,"
*Proc. ICASSP*, 28A.2.1–28A.2.4 (1984). - C. Temperton, "Self-sorting in-place fast Fourier transform,"
*SIAM J. Sci. Stat. Comput.***12**(4), 808–823 (1991). - Z. Qian, C. Lu, M. An, and R. Tolimieri, "Self-sorting in-place FFT algorithm with minimum working space,"
*IEEE Trans. ASSP***52**(10), 2835–2836 (1994). - M. Hegland, "A self-sorting in-place fast Fourier transform algorithm suitable for vector and parallel processing,"
*Numerische Mathematik***68**(4), 507–547 (1994). - Matteo Frigo and Steven G. Johnson:
*FFTW*, http://www.fftw.org/. A free (GPL) C library for computing discrete Fourier transforms in one or more dimensions, of arbitrary size, using the Cooley-Tukey algorithm. Also M. Frigo and S. G. Johnson, "FFTW: An adaptive software architecture for the FFT,"*Proc. ICASSP***3**, 1381–1384 (1998).