  This letter analyzes the performances of a simple reconstruction method,
namely the Projected Back-Projection (PBP), for estimating the direction of a
sparse signal from its phase-only (or amplitude-less) complex Gaussian random
measurements, i.e., an extension of one-bit compressive sensing to the complex
field. To study the performances of this algorithm, we show that complex
Gaussian random matrices respect, with high probability, a variant of the
Restricted Isometry Property (RIP) relating to the l1 -norm of the sparse
signal measurements to their l2 -norm. This property allows us to upper-bound
the reconstruction error of PBP in the presence of phase noise. Monte Carlo
simulations are performed to highlight the performance of our approach in this
phase-only acquisition model when compared to error achieved by PBP in
classical compressive sensing.
