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  "body": "\n## Summary\n\nThe discrete-time Fourier transform turns a sequence $(x_n)_{n \\in \\Z}$ into a $2\\pi$-periodic function $X(\\omega) = \\sum_n x_n e^{-i \\omega n}$ of a continuous frequency $\\omega$.\nIt is the Fourier transform of the group $\\Z$, whose dual group is the circle.\nRead backwards, it is a [[Fourier series|G12KXDHPPS]]: up to the sign of the index, the sequence consists of the Fourier coefficients of $X$.\nIt is the basic tool for analysing digital filters and sampled signals.\nSampling the frequency variable at $N$ points leads to the [[discrete Fourier transform|TST0GH66A5]].\nSee the [[definition|65QTEF1FTN]].\n\n## Explanations\n\n### L1 human\n\nA digital signal is a list of numbers indexed by time steps, going on in both directions.\nIts discrete-time Fourier transform says how much of each frequency between $-\\pi$ and $\\pi$ radians per step it contains.\nFrequencies outside this range look the same as frequencies inside it, because time is sampled; this is why the transform is periodic.\nFor the signal that is $1$ at time $0$ and $0$ elsewhere, the transform is the constant $1$.\n",
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