On The Cramer-Rao Lower Bound For Affine Phase Retrieval And Its Algorithms

Affine phase retrieval is an important research field in Engineering Mathematics,and the discussion for its stability and the implementation of its algorithm are a research hotspot in recent years.In the aspect of stability,this paper study the affine phase retrieval with non-additive white Gaussian noise.Firstly,we derive the Cramer-Rao lower bound.Then,we simulate the difference of Cramer-Rao lower bound and mean square error produced by Phase Lift of phase retrieval and affine phase retrieval in additive white Gaussian noise and non-additive white Gaussian noise.The results show that the reconstruction mean square error of signal with non-additive white Gaussian noise is nearer the Cramer-Rao lower bound than that with additive white Gaussian noise.In the aspect of algorithm,we show a sufficient condition in which a signal from affine phase retrieval could be recovered with high probability,and get the lower bound for recovering the signal successfully from affine phase retrieval with Phase Max algorithm.Similarly,we also use Numerical simulations,and they demonstrate that after obtaining a proper approximation vector from spectral initialization,the Phase Max algorithm can effectively solve the affine phase retrieval problems.and they also show that the selection of the values of the spectral initialization method has little influence on the successful recover of affine phase retrieval.