A Theory of Network Equivalence— Part I: Point-to-Point Channels

Abstract

A family of equivalence tools for bounding network capacities is introduced. Part I treats networks built from point-to-point channels. Part II generalizes the technique to networks containing wireless channels such as broadcast, multiple access, and interference channels. The main result of part I is roughly as follows. Given a network of noisy, independent, memoryless point-to-point channels, a collection of demands can be met on the given network if and only if it can be met on another network where each noisy channel is replaced by a noiseless bit pipe with throughput equal to the noisy channel capacity.This result was known previously for the case of a single-source multicast demand. The result given here treats general demands -including, for example, multiple unicast demands -and applies even when the achievable rate region for the corresponding demands is unknown in both the noisy network and its noiseless counterpart.

Separation of Source-Network Coding and Channel Coding in Wireline Networks

“…The results derived here are consistent with those of [9], [10], [8], which prove the separation between network coding and channel coding for multicast [9], [10] and general demands [6], [8], respectively, under the assumption that messages transmitted to different subsets of users are independent. The shift here is from independent sources to dependent sources and from reliable information delivery to both lossy and lossless data descriptions.…”

“…As a result, a joint network-source code is required, and only the channel code can be separated. While the achievability of a separated strategy is straightforward, the converse is more difficult since preserving statistical dependence between codewords transmitted across distinct edges of a network of noisy links improves the end-to-end network performance in some networks [6], [7], [8].…”

“…The strategy underlying our proof follows that of [6], [8], but the details differ significantly, both due to the inclusion of dependent sources and lossy reconstruction and in the focus on discrete-alphabet channels.…”

Low SNR — When only decoding will do

“…Given abundant bandwidth and limited power, we are interested in transmission schemes that can approach the capacity, although the capacity itself may not be fully characterized. A serial of network equivalence tools have been developed in [13]- [15] and we use these tools to develop capacity upper bounds, which help us to evaluate the performance of transmission schemes.…”

“…In [13] it has been shown that the capacity of a network remains unchanged if we replace an independent point-topoint noisy channel by a noiseless bit-pipe with rate equals the capacity of the original noisy channel. Therefore, we can replace the orthogonal channels from relay node R i to D and to R j by bit-pipes with rate, respectively,…”