  In this paper we consider the following real-valued and finite dimensional
specific instance of the 1-D classical phase retrieval problem. Let ${\bf
F}\in\mathbb{R}^N$ be an $N$-dimensional vector, whose discrete Fourier
transform has a compact support. The sign problem is to recover ${\bf F}$ from
its magnitude $|{\bf F}|$. First, in contrast to the classical 1-D phase
problem which in general has multiple solutions, we prove that with sufficient
over-sampling, the sign problem admits a unique solution. Next, we show that
the sign problem can be viewed as a special case of a more general piecewise
constant phase problem. Relying on this result, we derive a computationally
efficient and robust to noise sign recovery algorithm. In the noise-free case
and with a sufficiently high sampling rate, our algorithm is guaranteed to
recover the true sign pattern. Finally, we present two phase retrieval
applications of the sign problem: (i) vectorial phase retrieval with three
measurement vectors; and (ii) recovery of two well separated 1-D objects.
