Gabor frame

Summary

A Gabor frame is a frame of $L^2(\R)$ made of time shifts and frequency shifts of one window function $g$. The shifts run over a lattice $a\Z \times b\Z$ of the time–frequency plane. It gives a stable, discrete version of the short-time Fourier transform. A function is determined by the samples of $V_g f$ on the lattice, and can be rebuilt from them. A Gabor frame can exist only when $ab \le 1$. At $ab = 1$, the Balian–Low theorem shows that the window cannot be well localized in both time and frequency.