Cooley–Tukey FFT algorithm

This algorithm, including its recursive application, was invented around 1805 by 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. Tukey reportedly came up with the idea during a meeting of President Kennedy's Science Advisory Committee discussing ways to detect nuclear-weapon tests in the Soviet Union by employing seismometers located outside the country. These sensors would generate seismological time series. However, analysis of this data would require fast algorithms for computing DFTs due to the number of sensors and length of time. This task was critical for the ratification of the proposed nuclear test ban so that any violations could be detected without need to visit Soviet facilities. Another participant at that meeting, Richard Garwin of IBM, recognized the potential of the method and put Tukey in touch with Cooley. However, Garwin made sure that Cooley did not know the original purpose. Instead, Cooley was told that this was needed to determine periodicities of the spin orientations in a 3-D crystal of helium-3. Cooley and Tukey subsequently published their joint paper, and wide adoption quickly followed due to the simultaneous development of Analog-to-digital converters capable of sampling at rates up to 300 kHz. The fact that Gauss had described the same algorithm (albeit without analyzing its asymptotic cost) was not realized until several years after Cooley and Tukey's 1965 paper. == The radix-2 DIT case ==

A radix-2 decimation-in-time (DIT) FFT is the simplest and most common form of the Cooley–Tukey algorithm, although highly optimized Cooley–Tukey implementations typically use other forms of the algorithm as described below. Radix-2 DIT divides a DFT of size N into two interleaved DFTs (hence the name "radix-2") of size N/2 with each recursive stage. The discrete Fourier transform (DFT) is defined by the formula: : X_k = \sum_{n=0}^{N-1} x_n e^{-\frac{2\pi i}{N} nk}, where n is an integer ranging from 0 to N-1. Radix-2 DIT first computes the DFTs of the even-indexed inputs (x_{2m}=x_0, x_2, \ldots, x_{N-2}) and of the odd-indexed inputs (x_{2m+1}=x_1, x_3, \ldots, x_{N-1}), and then combines those two results to produce the DFT of the whole sequence. This idea can then be performed recursively to reduce the overall runtime to O(N log N). This simplified form assumes that N is a power of two; since the number of sample points N can usually be chosen freely by the application (e.g. by changing the sample rate or window, zero-padding, etcetera), this is often not an important restriction. The radix-2 DIT algorithm rearranges the DFT of the function x_n into two parts: a sum over the even-numbered indices n={2m} and a sum over the odd-numbered indices n={2m+1}: : X_k = \sum_{m=0}^{N/2-1} x_{2m}e^{-\frac{2\pi i}{N} (2m)k} + \sum_{m=0}^{N/2-1} x_{2m+1} e^{-\frac{2\pi i}{N} (2m+1)k} One can factor a common multiplier e^{-\frac{2\pi i}{N}k} out of the second sum, as shown in the equation below. It is then clear that the two sums are the DFT of the even-indexed part x_{2m} and the DFT of odd-indexed part x_{2m+1} of the function x_n. Denote the DFT of the Even-indexed inputs x_{2m} by E_k and the DFT of the Odd-indexed inputs x_{2m + 1} by O_k and we obtain: : X_k= \underbrace{\sum \limits_{m=0}^{N/2-1} x_{2m} e^{-\frac{2\pi i}{N/2} mk}}_{\text{DFT of even-indexed part of } x_n} {} + e^{-\frac{2\pi i}{N}k} \underbrace{\sum \limits_{m=0}^{N/2-1} x_{2m+1} e^{-\frac{2\pi i}{N/2} mk}}_{\text{DFT of odd-indexed part of } x_n} = E_k + e^{-\frac{2\pi i}{N}k} O_k\qquad\text{ for }k=0,\dots,\frac N2-1. Note that the equalities hold for k=0,\dots,N-1, but the crux is that E_k and O_k are calculated in this way for k=0,\dots,\frac N2-1 only. Thanks to the periodicity of the complex exponential, X_{k+\frac{N}{2}} is also obtained from E_k and O_k: : \begin{align} X_{k + \frac{N}{2}} & = \sum \limits_{m=0}^{N/2-1} x_{2m} e^{-\frac{2\pi i}{N/2} m(k + \frac{N}{2})} + e^{-\frac{2\pi i}{N}(k + \frac{N}{2})} \sum_{m=0}^{N/2-1} x_{2m+1} e^{-\frac{2\pi i}{N/2} m(k + \frac{N}{2} )} \\ & = \sum_{m=0}^{N/2-1} x_{2m} e^{-\frac{2\pi i}{N/2} mk} e^{-2\pi m i} + e^{-\frac{2\pi i}{N}k}e^{-\pi i} \sum_{m=0}^{N/2-1} x_{2m+1} e^{-\frac{2\pi i}{N/2} mk} e^{-2\pi m i} \\ & = \sum_{m=0}^{N/2-1} x_{2m} e^{-\frac{2\pi i}{N/2} mk} - e^{-\frac{2\pi i}{N}k} \sum_{m=0}^{N/2-1} x_{2m+1} e^{-\frac{2\pi i}{N/2} mk} \\ & = E_k - e^{-\frac{2\pi i}{N}k} O_k \end{align} We can rewrite X_k and X_{k+\frac{N}{2}} as: : \begin{align} X_k & = E_k + e^{-\frac{2\pi i}{N}{k}} O_k \\ X_{k+\frac{N}{2}} & = E_k - e^{-\frac{2\pi i}{N}{k}} O_k \end{align} This result, expressing the DFT of length N recursively in terms of two DFTs of size N/2, is the core of the radix-2 DIT fast Fourier transform. The algorithm gains its speed by re-using the results of intermediate computations to compute multiple DFT outputs. Note that final outputs are obtained by a +/− combination of E_k and O_k \exp(-2\pi i k/N), which is simply a size-2 DFT (sometimes called a butterfly in this context); when this is generalized to larger radices below, the size-2 DFT is replaced by a larger DFT (which itself can be evaluated with an FFT). This process is an example of the general technique of divide and conquer algorithms; in many conventional implementations, however, the explicit recursion is avoided, and instead one traverses the computational tree in breadth-first fashion. 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 (In many textbook implementations the depth-first recursion is eliminated in favor of a nonrecursive breadth-first approach, although depth-first recursion has been argued to have better memory locality.) Several of these ideas are described in further detail below. == Idea ==

. One performs smaller 1d DFTs along the N2 direction (the non-contiguous direction), then multiplies by phase factors (twiddle factors), and finally performs 1d DFTs along the N1 direction. The transposition step can be performed in the middle, as shown here, or at the beginning or end. This is done recursively for the smaller transforms. More generally, Cooley–Tukey algorithms recursively re-express a DFT of a composite size N = N1N2 as: • Perform N1 DFTs of size N2. • Multiply by complex roots of unity (often called the twiddle factors). • Perform N2 DFTs of size N1. Typically, either N1 or N2 is a small factor (not necessarily prime), called the radix (which can differ between stages of the recursion). If N1 is the radix, it is called a decimation in time (DIT) algorithm, whereas if N2 is the radix, it is decimation in frequency (DIF, also called the Sande–Tukey algorithm). The version presented above was a radix-2 DIT algorithm; in the final expression, the phase multiplying the odd transform is the twiddle factor, and the +/- combination (butterfly) of the even and odd transforms is a size-2 DFT. (The radix's small DFT is sometimes known as a butterfly, so-called because of the shape of the dataflow diagram for the radix-2 case.) == Variations ==

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(N2) algorithm for the prime base cases of the recursion (it is also possible to employ an N log N algorithm for the prime base cases, such as Rader's or Bluestein's algorithm). • 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 what was long the lowest known arithmetic 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.) e.g. for cache optimization or out-of-core operation, and was later shown to be an optimal cache-oblivious algorithm. The general Cooley–Tukey factorization rewrites the indices k and n as k = N_2 k_1 + k_2 and n = N_1 n_2 + n_1, respectively, where the indices ka and na run from 0..Na-1 (for a of 1 or 2). That is, it re-indexes the input (n) and output (k) as N1 by N2 two-dimensional arrays in column-major and row-major order, respectively; the difference between these indexings is a transposition, as mentioned above. When this re-indexing is substituted into the DFT formula for nk, the N_1 n_2 N_2 k_1 cross term vanishes (its exponential is unity), and the remaining terms give :X_{N_2 k_1 + k_2} = \sum_{n_1=0}^{N_1-1} \sum_{n_2=0}^{N_2-1} x_{N_1 n_2 + n_1} e^{-\frac{2\pi i}{N_1 N_2} \cdot (N_1 n_2 + n_1) \cdot (N_2 k_1 + k_2) } ::= \sum_{n_1=0}^{N_1-1} \left[ e^{-\frac{2\pi i}{N_1N_2} n_1 k_2 } \right] \left( \sum_{n_2=0}^{N_2-1} x_{N_1 n_2 + n_1} e^{-\frac{2\pi i}{N_2} n_2 k_2 } \right) e^{-\frac{2\pi i}{N_1} n_1 k_1 } ::= \sum_{n_1=0}^{N_1-1} \left( \sum_{n_2=0}^{N_2-1} x_{N_1 n_2 + n_1} e^{-\frac{2\pi i}{N_2} n_2 k_2 } \right) e^{-\frac{2\pi i}{N_1N_2} n_1(N_2k_1+k_2) } . where each inner sum is a DFT of size N2, each outer sum is a DFT of size N1, and the [...] bracketed term is the twiddle factor. An arbitrary radix r (as well as mixed radices) can be employed, as was shown by both Cooley and Tukey This analysis was erroneous, however: the radix-butterfly is also a DFT and can be performed via an FFT algorithm in O(r log r) operations, hence the radix r actually cancels in the complexity O(r log(r) N/r logrN), and the optimal r is determined by more complicated considerations. In practice, quite large r (32 or 64) are important in order to effectively exploit e.g. the large number of processor registers on modern processors, and even an unbounded radix r= also achieves O(N log N) complexity and has theoretical and practical advantages for large N as mentioned above. == Data reordering, bit reversal, and in-place algorithms ==

Although the abstract Cooley–Tukey factorization of the DFT, above, applies in some form 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 best-known reordering technique involves explicit bit reversal for in-place radix-2 algorithms. Bit reversal is the permutation where the data at an index n, written in binary with digits b4b3b2b1b0 (e.g. 5 digits for N=32 inputs), is transferred to the index with reversed digits b0b1b2b3b4 . 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 E_k and O_k are combined with a size-2 DFT, those two values are overwritten by the outputs. However, the two output values should go in the first and second halves of the output array, corresponding to the most significant bit b4 (for N=32); whereas the two inputs E_k and O_k are interleaved in the even and odd elements, corresponding to the least significant bit b0. Thus, in order to get the output in the correct place, b0 should take the place of b4 and the index becomes b0b4b3b2b1. And for next recursive stage, those 4 least significant bits will become b1b4b3b2, If you include all of the recursive stages of a radix-2 DIT algorithm, all the bits must be reversed and thus one must pre-process the input (or post-process the output) with a bit reversal to get in-order output. (If each size-N/2 subtransform is to operate on contiguous data, the DIT input is pre-processed by bit-reversal.) Correspondingly, if you perform all of the steps in reverse order, you obtain a radix-2 DIF algorithm with bit reversal in post-processing (or pre-processing, respectively). The logarithm (log) used in this algorithm is a base 2 logarithm. The following is pseudocode for iterative radix-2 FFT algorithm implemented using bit-reversal permutation. algorithm iterative-fft is input: Array a of n complex values where n is a power of 2. output: Array A the DFT of a. bit-reverse-copy(a, A) n ← a.length for s = 1 to log(n) do m ← 2s ωm ← exp(−2πi/m) for k = 0 to n-1 by m do ω ← 1 for j = 0 to m/2 – 1 do t ← ω A[k + j + m/2] u ← A[k + j] A[k + j] ← u + t A[k + j + m/2] ← u – t ω ← ω ωm return A The bit-reverse-copy procedure can be implemented as follows. algorithm bit-reverse-copy(a,A) is input: Array a of n complex values where n is a power of 2. output: Array A of size n. n ← a.length for k = 0 to n – 1 do A[rev(k)] := a[k] Alternatively, some applications (such as the computation of convolution products) work equally well on bit-reversed data, so one can perform forward transforms, processing, and then inverse transforms all without bit reversal to produce final results in the natural order. 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, 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 algorithm Even greater potential SIMD advantages (more consecutive accesses) have been proposed for the Pease 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 (as in the pseudocode above) and can be argued to have cache-oblivious locality benefits on systems with hierarchical memory. 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. == References ==