Networks of Gaussian channels with applications to feedback systems

This paper discusses networks (directed graphs) having one input node, one output node, and an arbitrary number of intermediate nodes, whose branches are noisy communications channels, in which the input to each channel appears at its output corrupted by additive Gaussian noise. Each branch is labeled by a non-negative real parameter which specified how noisy it is. A branch originating at a node has as input a linear combination of the outputs of the branches terminating at that node. The channel capacity of such a network is defined. Its value is bounded in terms of branch parameter values and procedures for computing values for general networks are described. Explicit solutions are given for the class Dc which includes series-parallel and simple bridge networks and all other networks having r paths, b branches, and v nodes with r = b u + 2, and for the class D1 of networks which is inductively defined to include Dc and all networks obtained by replacing a branch of a network in D1 by a network in DI. The general results are applied to the particular networks which arise from the decomposition of a simple feedback system into successive forward and reverse (feedback) channels. When the feedback channels are noiseless, the capacities of the forward channels are shown to add. Some explicit expressions and some bounds are given for the case of noisy feedback channels.