Gabor transform

Remarks

It is the short-time Fourier transform $V_g f$ with the Gaussian window $g(t) = e^{-\pi t^2}$. Some authors allow a Gaussian $e^{-\alpha \pi (t - \tau)^2}$ of any width $\alpha > 0$, or normalize the window to have $L^2$ norm $1$.

Example: let $f(t) = e^{-\pi t^2}$. Then $f(t) e^{-\pi (t - \tau)^2} = e^{-\pi \tau^2/2} e^{-2\pi (t - \tau/2)^2}$, and the Fourier transform of $e^{-2\pi s^2}$ at $\omega$ is $2^{-1/2} e^{-\pi \omega^2/2}$. So $G f(\tau, \omega) = 2^{-1/2} e^{-\pi \tau^2/2} e^{-\pi \omega^2/2} e^{-\pi i \tau \omega}$, and $\abs{G f}$ is a Gaussian bump centred at the origin of the time–frequency plane.

Non-example: the short-time Fourier transform with the window $1_{[-1/2, 1/2]}$ is not a Gabor transform, since its window is not Gaussian.