  The classical shift retrieval problem considers two signals in vector form
that are related by a shift. The problem is of great importance in many
applications and is typically solved by maximizing the cross-correlation
between the two signals. Inspired by compressive sensing, in this paper, we
seek to estimate the shift directly from compressed signals. We show that under
certain conditions, the shift can be recovered using fewer samples and less
computation compared to the classical setup. Of particular interest is shift
estimation from Fourier coefficients. We show that under rather mild conditions
only one Fourier coefficient suffices to recover the true shift.
