Recovering a sparse signal from an undersampled set of random linear
measurements is the main problem of interest in compressed sensing. In this
paper, we consider the case where both the signal and the measurements are
complex. We study the popular reconstruction method of LASSO. The precise
asymptotic performance of this algorithm in the complex setting is not yet
known. We extend the approximate message passing (AMP) algorithm to the complex
signals and measurements and obtain the complex approximate message passing
algorithm (CAMP). We then generalize the state evolution framework recently
introduced for the analysis of AMP, to to the complex setting. Using the state
evolution, we derive accurate formulas for the phase transition and noise
sensitivity of both LASSO and CAMP.