Some large linear systems in communications: Estimation, information and statistical physics

Linear systems are frequently used to model physical channels in communication problems. In such systems, the goal is to recover the information bearing input signal to the linear transformation based on noisy observations of the output. This paper investigates the performance of linear systems randomly selected from some ensembles in the large-dimension limit. Previous analysis of such systems using statistical physics techniques is strengthened, where it is shown that the posterior of each individual input symbol is asymptotically identical to the posterior of a Gaussian channel with the same input. Interestingly, the replica symmetry solution takes an identical form as the rigorous result obtained for an ensemble of sparse linear systems. Some challenges to the information theory and statistical physics communities are highlighted.