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  "body": "\n## Definition\n\nLet $g \\in L^2(\\R)$ be a nonzero function, the *window*, and let $f \\in L^2(\\R)$ (see [[Lp space|SASP14MA1Z]]).\nThe *short-time Fourier transform* of $f$ with respect to $g$ is the function $V_g f \\colon \\R^2 \\to \\C$,\n$$V_g f(x, \\omega) = \\int_{\\R} f(t) \\, \\overline{g(t - x)} \\, e^{-2\\pi i t \\omega} \\, dt,$$\nfor $x \\in \\R$ (time) and $\\omega \\in \\R$ (frequency).\n\n## Remarks\n\nThe integral converges absolutely, since $f \\cdot \\overline{g(\\cdot - x)}$ is integrable by the Cauchy–Schwarz inequality.\nSo $V_g f(x, \\cdot)$ is the [[Fourier transform|DPVM34ADZV]] of $f \\cdot \\overline{g(\\cdot - x)}$.\nIn inner product form, $V_g f(x, \\omega) = \\ip{f}{M_\\omega T_x g}$, where $T_x g(t) = g(t - x)$ and $M_\\omega g(t) = e^{2\\pi i \\omega t} g(t)$.\nMoyal's formula states $\\norm{V_g f}_{L^2(\\R^2)} = \\norm{f}_2 \\norm{g}_2$.\nThe squared modulus $\\abs{V_g f}^2$ is the *spectrogram*.\nIn engineering texts the variable $\\omega$ often denotes angular frequency, and the kernel is $e^{-i \\omega t}$.\n\nExample: let $f = g = 1_{[0, 1]}$.\nFor $0 \\le x \\le 1$, the product $f \\cdot g(\\cdot - x)$ is the indicator of $[x, 1]$, so $V_g f(x, 0) = 1 - x$.\nFor $\\abs{x} \u003e 1$ the supports are disjoint and $V_g f(x, \\omega) = 0$.\n\nNon-example: with the constant window $g = 1$, which is not in $L^2(\\R)$, the formula gives $\\hat f(\\omega)$ for every $x$.\nIt carries no time information, and this choice is excluded by the definition.\n",
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