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  "body": "\n## Summary\n\nThe discrete Fourier transform turns a finite list of $N$ complex numbers into another list of $N$ numbers, its frequency components.\nIt is the Fourier transform of the finite cyclic group $\\Z / N\\Z$, built from the $N$-th [[roots of unity|0YV2CKP2AV]].\nIt is invertible, it preserves energy up to the factor $N$, and it turns cyclic convolution into pointwise multiplication.\nThe fast Fourier transform computes it in $O(N \\log N)$ operations, which makes it one of the most used algorithms in numerical computing and signal processing.\nSee the [[definition|TJ9XPJ65YX]].\n\n## Explanations\n\n### L1 human\n\nTake $N$ samples of a signal at equally spaced times.\nThe discrete Fourier transform tells how much of each of the $N$ basic waves, which complete $0, 1, \\dots, N - 1$ full turns over the sampling window, is present in the samples.\nFor $N = 4$, the list $(1, 1, 1, 1)$ is a pure constant, and its transform is $(4, 0, 0, 0)$: only the frequency $0$ is present.\nThe list $(1, 0, 0, 0)$, a single spike, has transform $(1, 1, 1, 1)$: every frequency is present equally.\n",
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