# Windowed Fourier Transform

I am having some difficulties with this question.

We define the windowed Fourier transform of $f \in L^2(\mathbb{R})$ as $$Sf(\mu,\xi)=\int_\mathbb{R}f(t)g(t-\mu)e^{-i\xi t}dt$$

where $g$ is some real, symmetric and finite supported function such that it vanishes outside a finite interval. Prove that for $f=e^{i\eta_0t}$, we have $$Sf(\mu,\eta)=e^{-i\mu(\eta-\eta_0)}\hat{g}(\eta-\eta_0)$$

Could someone also provide me with the integral definition of the windowed Fourier transform? For instance the Fourier transform is defined as, $$\hat{f}(\omega)=\int_\mathbb{R}f(x)e^{-ixw}dx$$

• For $f$ a single complex sine then $Sf$ is what you wrote, this is obvious from the integral definition. – reuns Nov 26 '17 at 14:21
• Substitute $t'=t-\mu$, pull out the exponential factor that is independent of $t,$ and then recognize the remaining integral as the Fourier transform of $g$ at $\eta-\eta_{0}$. – RideTheWavelet Nov 26 '17 at 14:33