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I need some help with applying the properties of Fourier Transform.

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

Prove that for $f=e^{i\eta_0t}$, we have $Sf(\mu,\eta)=e^{-i\mu(\eta-\eta_0)}\hat{g}(\eta-\eta_0)$

All I can obtain till now is this. $$Sf(\mu,\eta)=\int_\mathbb{R}e^{i\eta_0t}g(t-\mu)e^{-i\eta t}dt=\int_\mathbb{R}e^{-it(\eta-\eta_0)}g(t-\mu)dt$$

I know that I need to apply some properties of fourier transform but I am not sure how. Thanks in advance.

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HINT:

Just substitute $\tau=t-\mu$ in the integral and see what you get. It's just very basic math, no complicated properties of the Fourier transform. I assume that you know the definition of $\hat{g}(\eta)$.

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For one, $f=e^{i\eta_0 t}$ is not in $L_2(\mathbb{R})$. But with mild assumptions on $g$, things can become square (pun on power norms, overruled). Using standard integral tools, the hint is in Matt L.'s answer.

Second, this is an basic instance of the modulation/shift properties in the Fourier transform: if you choose $f=e^{i\eta_0 t}$, it sounds like you are computing the Fourier transform of the shifted and modulated window $e^{i\eta_0 t}g(t−\mu)$. So if you know $\hat{g}$ the Fourier transform of $g(t)$, you can use:

  • Translation / time shifting: for any real number $x_0$, if $u(x) = v(x - x_0)$, then $\hat{u}(\xi) = e^{-2i \pi x_0 \xi} \hat{v}(\xi)$.
  • Modulation / frequency shifting: for any real number $\xi_0$, if $u(x) = e^{2 i\pi \xi_0} v(x)$, then $u(\xi) = \hat{v}(\xi - \xi_0)$.

Now, combine both Fourier properties.

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