Information Velocity of Cascaded AWGN Channels with Feedback

We consider a line network of nodes connected by additive white Gaussian noise channels and equipped with local feedback. We study the velocity at which information spreads over this network. For the transmission of a data packet, we derive an explicit positive lower bound on the velocity for any packet size. Furthermore, we consider streaming, that is, transmission of data packets that is generated at a given average arrival rate. We show that a positive velocity exists as long as the arrival rate is below the individual Gaussian channel capacity and provide an explicit lower bound. Our analysis involves applying pulse-amplitude modulation to the data (successively in the streaming case) and using linear mean-squared error estimation at the network nodes. Due to the analog-linear nature of the scheme, the results extend to any additive noise. For general noise, we derive exponential error-probability bounds. Moreover, for (sub-)Gaussian noise, we show doubly-exponential behavior, which reduces to the celebrated Schalkwijk-Kailath scheme when considering a single node. By viewing the constellation as an 'analog source', we also provide bounds on the exponential decay of the mean-squared error of source transmission over the network.