In this paper, we revisit the problem of finding the average capacity of the Gaussian feedback channel. First, we consider the problem of finding the average capacity of the analog Gaussian noise channel where the noise has an arbitrary spectral density. We introduce a new approach to the problem where we solve the problem over a finite number of transmissions and then consider the limit of an infinite number of transmissions. Further, we consider the important special case of stationary Gaussian noise with finite memory. We show that the channel capacity at stationarity can be found by solving a semi-definite program, and hence computationally tractable. We also give new proofs and results of the non stationary solution which bridges the gap between results in the literature for the stationary and non stationary feedback channel capacities. It’s shown that a linear communication feedback strategy is optimal. Similar to the solution of the stationary problem, it’s shown that the optimal linear strategy is to transmit a linear combination of the information symbols to be communicated and the innovations for the estimation error of the state of the noise process.

In this paper, we investigate the topology convergence problem for the gossip-based Gradient overlay network. In an overlay network where each node has a local utility value, a Gradient overlay network is characterized by the properties that each node has a set of neighbors containing higher utility values, such that paths of increasing utilities emerge in the network topology. The Gradient overlay network is built using gossiping and a preference function that samples from nodes using a uniform random peer sampling service. We analyze it using tools from matrix analysis, and we prove both the necessary and sufficient conditions for convergence to a complete gradient structure, as well as estimating the convergence time. Finally, we show in simulations the potential of the Gradient overlay, by building a more efficient live streaming peer-to-peer (P2P) system than one built using uniform random peer sampling.