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  "body": "\n## Summary\n\nThe short-time Fourier transform of a function $f$ on $\\R$ is a [[Fourier transform|DPVM34ADZV]] taken through a window.\nOne multiplies $f$ by a window $g$ shifted to a point $x$, and transforms the product.\nIt is a function of position $x$ and frequency $\\omega$, and so it shows which frequencies are present near each time.\nIt is the basic tool of [[time–frequency analysis|GQFPVXEWK1]]; its squared modulus is the spectrogram.\nFor a window in $L^2$ it is an isometry up to the factor $\\norm{g}_2$, so no information is lost.\n\n## Explanations\n\n### L1 human\n\nListen to a piece of music through a small moving window in time.\nAt each position of the window, ask which pitches are sounding: this is a Fourier transform of the windowed piece.\nCollecting the answers for all positions gives a two-dimensional picture, time against pitch.\nA narrow window shows well when notes start but blurs their pitch, and a wide window does the opposite.\n",
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