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  "body": "\n## Statement\n\nLet $N \\ge 1$, let $x \\in \\C^N$ and let $X$ be its [[discrete Fourier transform|TST0GH66A5]], $X_k = \\sum_{n=0}^{N-1} x_n e^{-2\\pi i nk/N}$.\nFor integers $m$ let $b_m = e^{\\pi i m^2/N}$, and for $0 \\le n \u003c N$ let $a_n = x_n e^{-\\pi i n^2/N}$.\nThen for $0 \\le k \u003c N$,\n$$X_k = \\overline{b_k} \\sum_{n=0}^{N-1} a_n\\, b_{k-n}.$$\nLet $M \\ge 2N - 1$, let $\\alpha \\in \\C^M$ be $a$ followed by zeros, and let $\\beta \\in \\C^M$ have $\\beta_j = b_j$ for $0 \\le j \u003c N$, $\\beta_{M-j} = b_{-j}$ for $1 \\le j \u003c N$, and zeros elsewhere.\nThen $X_k = \\overline{b_k}\\, (\\alpha \\circledast \\beta)_k$ for $0 \\le k \u003c N$, where $(\\alpha \\circledast \\beta)_k = \\sum_{j=0}^{M-1} \\alpha_j \\beta_{(k-j) \\bmod M}$ is the cyclic [[convolution|2Z29JN3VES]] of length $M$.\nTake $M$ a power of $2$ and compute the cyclic convolution with three [[radix-2 FFTs|9N8D9E5TB2]] of length $M$.\nThen the DFT of any length $N$ costs $O(N \\log N)$ operations.\n\n## Remarks\n\nThe key identity is $nk = \\frac12 \\bigl(n^2 + k^2 - (k-n)^2\\bigr)$, which gives $e^{-2\\pi i nk/N} = e^{-\\pi i n^2/N} e^{-\\pi i k^2/N} e^{\\pi i (k-n)^2/N}$.\nSo the DFT is a pointwise product with a chirp, a convolution with the conjugate chirp, and another pointwise product.\nThe index $k - n$ runs from $-(N-1)$ to $N-1$, and the zero padding to $M \\ge 2N-1$ makes the cyclic convolution agree with this linear convolution on $0 \\le k \u003c N$.\nThe cyclic convolution of length $M$ equals the inverse DFT of the product of the DFTs of $\\alpha$ and $\\beta$.\nSince one can choose $M \u003c 4N$, the constant in $O(N \\log N)$ does not depend on the arithmetic of $N$; in particular prime $N$ is allowed.\nThe same identity with $e^{-2\\pi i/N}$ replaced by any complex $z$ evaluates the chirp z-transform $\\sum_n x_n z^{-nk}$, which is the subject of the source article.\n\n## Explanations\n\n### L1 human\n\nThe fast Fourier transform is easiest when the length is a power of two.\nBluestein's trick rewrites a transform of any length, even a prime length, as a convolution.\nA convolution can be padded with zeros to a power-of-two length and computed with power-of-two FFTs.\nSo every length gets the speed $N \\log N$, at the price of a larger constant.\n\n### L2 human\n\nThe algorithm is the standard fallback in FFT libraries for lengths with large prime factors.\nCompared with Rader's algorithm, it does not need a primitive root and works for every $N$, not only primes.\nIt needs about three FFTs of length $M \\approx 2N$ to $4N$, so it is slower than a direct Cooley–Tukey FFT of a highly composite length.\nThe chirp $e^{\\pi i m^2/N}$ is the same object as in the fractional Fourier transform and in radar signal processing.\n",
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