The Gaussian Erasure Channel

This paper finds the capacity of linear time-invariant systems observed in additive Gaussian noise through a memoryless erasure channel. This problem 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. We show that the optimum normalized power spectral density is the water filling solution for reduced signal-to-noise ratio, where the gap to the actual signal-to-noise ratio depends on both the erasure probability and the channel transfer function. We find asymptotic expressions for the capacity in the sporadic erasure and sporadic non-erasure regimes as well as the low and high signal-to-noise regimes.