  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.
