Concise Derivation of Complex Bayesian Approximate Message Passing via Expectation Propagation

Abstract—In this paper, we address the problem of recovering complex-valued signals from a set of complex-valued linear measurements. Approximate message passing (AMP) is one state-ofthe-art algorithm to recover real-valued sparse signals. However, the extension of AMP to complex-valued case is nontrivial and no detailed and rigorous derivation has been explicitly presented. To fill this gap, we extend AMP to complex Bayesian approximate message passing (CB-AMP) using expectation propagation (EP). This novel perspective leads to a concise derivation of CBAMP without sophisticated transformations between the complex domain and the real domain. In addition, we have derived state evolution equations to predict the reconstruction performance of CB-AMP. Simulation results are presented to demonstrate the efficiency of CB-AMP and state evolution.