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  "body": "\n## Definition\n\nLet $x = (x_n)_{n \\in \\Z}$ be an absolutely summable sequence of complex numbers, that is, $x \\in \\ell^1(\\Z)$, a case of the [[$L^p$ spaces|SASP14MA1Z]].\nThe *discrete-time Fourier transform* of $x$ is the function $X \\colon \\R \\to \\C$ given by\n$$X(\\omega) = \\sum_{n \\in \\Z} x_n\\, e^{-i \\omega n}, \\qquad \\omega \\in \\R,$$\nwhere $e^{-i\\omega n}$ is the [[complex exponential|07GJV36896]].\n\n## Remarks\n\nThe series converges absolutely and uniformly in $\\omega$, so $X$ is continuous and $2\\pi$-periodic, with $\\abs{X(\\omega)} \\le \\sum_n \\abs{x_n}$.\nThe sequence is recovered by\n$$x_n = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} X(\\omega)\\, e^{i \\omega n}\\, d\\omega,$$\nso $x_{-n}$ is the $n$-th coefficient of the [[Fourier series|G12KXDHPPS]] of $X$.\n\nExample: for $\\abs{a} \u003c 1$, let $x_n = a^n$ for $n \\ge 0$ and $x_n = 0$ for $n \u003c 0$.\nThen $\\sum_n \\abs{x_n} = 1/(1 - \\abs{a}) \u003c \\infty$, and the geometric series gives $X(\\omega) = \\sum_{n \\ge 0} (a e^{-i\\omega})^n = \\frac{1}{1 - a e^{-i\\omega}}$.\n\nNon-example: the constant sequence $x_n = 1$ is not absolutely summable, and $\\sum_n e^{-i\\omega n}$ diverges for every $\\omega$ because its terms do not tend to $0$.\nAs a distribution its transform is a Dirac comb of period $2\\pi$, scaled by $2\\pi$.\n\nBy the Riesz–Fischer and Parseval theorems the transform extends to an isometry, up to the factor $2\\pi$, from $\\ell^2(\\Z)$ onto $L^2$ of a period, with the series converging in $L^2$.\nSome authors use the frequency $f = \\omega / (2\\pi)$, which gives period $1$.\n",
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