# Shift property of the Short-term Fourier transform (STFT)

Properties of STFT (Short-term Fourier transform) say that it preserves time shift up to modulation. Does it mean it is sensitive to time-shift in signal?

• I saw that you've removed your other question. Just ask it again including your own efforts to answer it, even if you got stuck somewhere. If we understand where and why you're stuck we can probably provide a useful answer. – Matt L. Aug 21 '16 at 17:30

Of course the STFT changes when the signal is shifted. According to the definition we have

$$\text{STFT}\{x(t)\}=X(\tau,\omega)=\int_{-\infty}^{\infty}x(t)w(t-\tau)e^{-j\omega t}dt\tag{1}$$

If we define a shifted signal $y(t)=x(t-T)$, the corresponding STFT becomes

\begin{align}\text{STFT}\{y(t)\}=Y(\tau,\omega)&=\int_{-\infty}^{\infty}x(t-T)w(t-\tau)e^{-j\omega t}dt\\&=\int_{-\infty}^{\infty}x(u)w(t-(\tau-T))e^{-j\omega u}e^{-j\omega T}du\\&=e^{-j\omega T}X(\tau - T,\omega)\tag{2}\end{align}

From $(2)$ you see that time shifting results in a phase factor and a shift in the STFT. The magnitudes of the STFTs are related by just a time shift, which is of course identical to the time shift of the signal:

$$|\text{STFT}\{x(t-T)\}|=|Y(\tau,\omega)|=|X(\tau-T,\omega)|\tag{3}$$

• Would you agree with the notion of time-shift-covariance? – Laurent Duval Aug 21 '16 at 13:17
• @LaurentDuval: Yes, for the magnitude of the STFT. – Matt L. Aug 21 '16 at 14:10
• @user23380: STFT indeed preserves time-shifts up to a complex phase factor $e^{-j\omega T}$. – Matt L. Aug 21 '16 at 14:22