  In this paper we consider the problem of recovering a signal $x \in
\mathbb{R}^N$ from its power spectrum assuming that the signal is sparse with
respect to a generic basis for $\mathbb{R}^N$. Our main result is that if the
sparsity level is at most $\sim\! N/2$ in this basis then the generic sparse
vector is uniquely determined up to sign from its power spectrum. We also prove
that if the sparsity level is $\sim\! N/4$ then every sparse vector is
determined up to sign from its power spectrum. Analogous results are also
obtained for the power spectrum of a vector in $\mathbb{C}^N$ which extend
earlier results of Wang and Xu \cite{arXiv:1310.0873}.
