Forward and Reciprocal Noisy Coded Networks: Precoding, Topology, and Error Analysis
Samah A. M. Ghanem · 2019
In this paper, a new class of problems are highlighted capitalizing on recent proofs of fundamental relations between information theory and estimation theory in noisy network-coded flows [1]. In particular, when the network is represented by a directed graph G= (V,E) and under the assumption of uncorrelated noise over information flows between the directed links connecting transmitters, switches (relays), and receivers. We show that there exist closed-form relations for the gradient of the mutual information with respect to different components of - forward and reciprocal- networks' system matrix M and the MMSE, what is called the network I-MMSE. On the one hand, this result allows for studying effects of the error propagation and backward traceability with respect to effects of the network topology, topological changes when nodes are mobile or `ad-hoc' along the transmission paths into the network capacity. On the other hand, we shed light into compensation methods where a change in the precoding process can be used as an alternative change to the network topology or vice versa when optimizing coded flows or when optimizing the topology or the decoding complexity. We provide optimal precoding designs that are adapted to the network level, where a network can be harmonized and information flows optimized and adapted to the awareness of the network topology. This paper is the first to present the failure of RLNC to be optimal with topology awareness. In particular, we highlight that DLNC is optimal for a network topology with deterministic matrix.