= Short-time Fourier transform
{title2=$V_gf(\tau,\omega)$}
The short-time Fourier transform analyzes the local frequency content of a signal by translating a window and then taking a <Fourier transform>. For a window $g$ and suitable signal $f$, one angular-frequency convention is
$$
V_gf(\tau,\omega)=\int_{\mathbb R}f(t)\overline{g(t-\tau)}e^{-i\omega t}\,dt.
$$
The time coordinate $\tau$ selects a neighborhood through the <window function>, while $\omega$ tests a local oscillation. The complex conjugation and Fourier normalization are part of the convention.
For $f(t)=e^{i\omega_0t}$ and an integrable window, substitution $u=t-\tau$ gives $V_gf=e^{-i(\omega-\omega_0)\tau}\widehat{\overline g}(\omega-\omega_0)$. Its magnitude is centered on the tone frequency according to the window transform. A narrower time window improves temporal localization but spreads its Fourier transform, reflecting the <uncentred Fourier uncertainty principle>. The squared magnitude gives a <spectrogram>, which displays changing frequencies rather than only one whole-record spectrum.
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