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  "body": "\n## Definition\n\nLet $N \\ge 1$ be an integer and let $\\omega_N = e^{-2\\pi i / N}$, a primitive $N$-th [[root of unity|0YV2CKP2AV]], where $e^{z}$ is the [[complex exponential|07GJV36896]].\nThe *discrete Fourier transform* of a vector $x = (x_0, x_1, \\dots, x_{N-1}) \\in \\C^N$ is the vector $X = (X_0, X_1, \\dots, X_{N-1}) \\in \\C^N$ given by\n$$X_k = \\sum_{n=0}^{N-1} x_n\\, \\omega_N^{kn} = \\sum_{n=0}^{N-1} x_n\\, e^{-2\\pi i k n / N}, \\qquad k = 0, 1, \\dots, N - 1.$$\n\n## Remarks\n\nThe map $x \\mapsto X$ is linear and invertible, with inverse\n$$x_n = \\frac{1}{N} \\sum_{k=0}^{N-1} X_k\\, e^{2\\pi i k n / N}.$$\nThis follows from $\\sum_{k=0}^{N-1} e^{2\\pi i k m / N} = N$ if $N$ divides $m$ and $0$ otherwise.\nIt also gives the Parseval identity $\\sum_k \\abs{X_k}^2 = N \\sum_n \\abs{x_n}^2$, so $x \\mapsto N^{-1/2} X$ is unitary.\n\nExample: for $N = 2$, $X_0 = x_0 + x_1$ and $X_1 = x_0 - x_1$; so $(3, 1)$ has transform $(4, 2)$, and the inverse formula gives back $\\frac12 (4 + 2, 4 - 2) = (3, 1)$.\n\nNon-example: an infinite sequence $(x_n)_{n \\in \\Z}$ is not a vector of $\\C^N$, so the definition does not apply; its analogue is the discrete-time Fourier transform, a function of a continuous frequency.\n\nIndices are read modulo $N$, so the transform is the Fourier transform on the group $\\Z / N\\Z$.\nIt turns the cyclic convolution $(x * y)_n = \\sum_{m} x_m y_{n - m}$ into the product $X_k Y_k$, and therefore diagonalizes every circulant matrix.\nSome authors put the factor $\\frac1N$ or $\\frac{1}{\\sqrt N}$ in the forward transform, or use $e^{+2\\pi i k n/N}$.\n",
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