We consider a linear time-invariant channel with given transfer function, whose output is corrupted by additive white Gaussian noise and random erasures. Determining the capacity of this channel requires obtaining the asymptotic spectral distribution of a submatrix of a nonnegative definite Toeplitz matrix obtained by retaining each column/row independently and with identical probability. This is a new problem in random matrix theory that does not follow from previously known results. We find explicitly the required asymptotic eigenvalue distribution in terms of a fixed-point equation, that yields the eta-transform of the distribution. This results represents the first known connection between the well-known theory of large Toeplitz matrices (Grenander-Szego) and random matrices. Also, we shall stress an appealing formal analogy with free-probability and the S-transform, even though freeness does not generally apply to this problem. This may suggest that freeness is a too restrictive condition for the S-transform to apply. Furthermore, we find the optimal input spectrum that achieves capacity and show that this is given by the waterfilling solution as in the case of no erasures, but computed for a scaled SNR that takes into account the presence of erasures. We find simple and easily computable upper and lower bounds to capacity, and we characterize the effect of erasures on the key quantities that determine the high-SNR and low-SNR regimes of spectral efficiency versus Eb/N0.Applications of these results are, for example, the capacity of Gaussian channels with impulsive noise, in the limit of very large impulse power, the capacity of a Wyner-model cellular system with centralized processing, where the base stations are connected to the central processor through unreliable links that can be either ``on'' or ``off'' with a certain probability, and the characterization of the mean-square error of a linear predictor (e.g., a Kalman filter) with randomly missing observations.
(joint work with Sergio Verdu, Antonia Tulino and Shlomo Shamai)