تبدیل فوریه زمان کوتاه

${\displaystyle \mathbf {STFT} \{x(t)\}(\tau ,\omega )\equiv X(\tau ,\omega )=\int _{-\infty }^{\infty }x(t)w(t-\tau )e^{-j\omega t}\,dt}$

${\displaystyle \operatorname {spectrogram} \{x(t)\}(\tau ,\omega )\equiv |X(\tau ,\omega )|^{2}}$

${\displaystyle \mathbf {STFT} \{x[n]\}(m,\omega )\equiv X(m,\omega )=\sum _{n=-\infty }^{\infty }x[n]w[n-m]e^{-j\omega n}}$

${\displaystyle \int _{-\infty }^{\infty }w(\tau )\,d\tau =1}$

${\displaystyle \int _{-\infty }^{\infty }w(t-\tau )\,d\tau =1\quad \forall \ t}$

و

${\displaystyle x(t)=x(t)\int _{-\infty }^{\infty }w(t-\tau )\,d\tau =\int _{-\infty }^{\infty }x(t)w(t-\tau )\,d\tau }$

${\displaystyle X(\omega )=\int _{-\infty }^{\infty }x(t)e^{-j\omega t}\,dt}$

${\displaystyle X(\omega )=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }x(t)w(t-\tau )\,d\tau \right]\,e^{-j\omega t}\,dt}$

${\displaystyle =\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }x(t)w(t-\tau )\,e^{-j\omega t}\,d\tau \,dt}$

${\displaystyle X(\omega )=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }x(t)w(t-\tau )\,e^{-j\omega t}\,dt\,d\tau }$

${\displaystyle =\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }x(t)w(t-\tau )\,e^{-j\omega t}\,dt\right]\,d\tau }$

${\displaystyle =\int _{-\infty }^{\infty }X(\tau ,\omega )\,d\tau .}$

${\displaystyle x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )e^{+j\omega t}\,d\omega }$

${\displaystyle x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }X(\tau ,\omega )e^{+j\omega t}\,d\tau \,d\omega }$

یا

${\displaystyle x(t)=\int _{-\infty }^{\infty }\left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\tau ,\omega )e^{+j\omega t}\,d\omega \right]\,d\tau }$

${\displaystyle x(t)w(t-\tau )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\tau ,\omega )e^{+j\omega t}\,d\omega }$