  Conventional sparse phase retrieval schemes can recover sparse signals from
the magnitude of linear measurements only up to a global phase ambiguity. This
work proposes a novel approach that instead utilizes the magnitude of affine
measurements to achieve ambiguity-free signal reconstruction. The proposed
method relies on two-stage approach that consists of support identification
followed by the exact recovery of nonzero signal entries. In the noise-free
case, perfect support identification using a simple counting rule is guaranteed
subject to a mild condition on the signal sparsity, and subsequent exact
recovery of the nonzero signal entries can be obtained in closed-form. The
proposed approach is then extended to two noisy scenarios, namely, sparse noise
(or outliers) and non-sparse bounded noise. For both cases, perfect support
identification is still ensured under mild conditions on the noise model,
namely, the support size for sparse outliers and the power of the bounded
noise. Under perfect support identification, exact signal recovery can be
achieved using a simple majority rule for the sparse noise scenario, and
reconstruction up to a bounded error can be achieved using linear least-squares
(LS) estimation for the non-sparse bounded noise scenario. The obtained
analytic performance guarantee for the latter case also sheds light on the
construction of the sensing matrix and bias vector. In fact, we show that a
near optimal performance can be achieved with high probability by the random
generation of the nonzero entries of the sparse sensing matrix and bias vector
according to the uniform distribution over a circle. Computer simulations using
both synthetic and real-world data sets are provided to demonstrate the
effectiveness of the proposed scheme.
