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  "body": "\n## Definition\n\nLet $f \\in L^2(\\R)$ (see [[Lp space|SASP14MA1Z]]).\nThe *Gabor transform* of $f$ is the function $G f \\colon \\R^2 \\to \\C$,\n$$G f(\\tau, \\omega) = \\int_{\\R} f(t) \\, e^{-\\pi (t - \\tau)^2} \\, e^{-2\\pi i \\omega t} \\, dt,$$\nfor $\\tau \\in \\R$ (time) and $\\omega \\in \\R$ (frequency).\n\n## Remarks\n\nIt is the [[short-time Fourier transform|3DHNP5NEWH]] $V_g f$ with the Gaussian window $g(t) = e^{-\\pi t^2}$.\nSome authors allow a Gaussian $e^{-\\alpha \\pi (t - \\tau)^2}$ of any width $\\alpha \u003e 0$, or normalize the window to have $L^2$ norm $1$.\n\nExample: let $f(t) = e^{-\\pi t^2}$.\nThen $f(t) e^{-\\pi (t - \\tau)^2} = e^{-\\pi \\tau^2/2} e^{-2\\pi (t - \\tau/2)^2}$, and the Fourier transform of $e^{-2\\pi s^2}$ at $\\omega$ is $2^{-1/2} e^{-\\pi \\omega^2/2}$.\nSo $G f(\\tau, \\omega) = 2^{-1/2} e^{-\\pi \\tau^2/2} e^{-\\pi \\omega^2/2} e^{-\\pi i \\tau \\omega}$, and $\\abs{G f}$ is a Gaussian bump centred at the origin of the time–frequency plane.\n\nNon-example: the short-time Fourier transform with the window $1_{[-1/2, 1/2]}$ is not a Gabor transform, since its window is not Gaussian.\n",
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