Show that: FOV = fs / (g Gf ) .

Assume quadrature sampling of the time domain signal, and adopt the symbol devinitions used in The Basics of MRI. Start with the resonance equation.

n = g B

In the presence of a magnetic field gradient the equation becomes

n = g (Bo + x Gx).

Relative to the frequency at the isocenter of the magnet, we have

n = no + g x Gx

n - no = Dn = g x Gx

To record a range of frequencies Dn a quadrature sampling rate of fs is required. In the presence of the magnetic field gradient, this range of frequencies corresponds to the distance across the image in the frequency encoding direction, or the FOV. Therefore.

fs = g FOV Gf ,

or
FOV = fs / (g Gf).