[Paper Review] Performance Analysis of Random Linear Network Coding in Two-Source Single-Relay Networks

Amjad Saeed Khan, Ioannis Chatzigeorgiou|Lancaster EPrints (Lancaster University)|Mar 19, 2015

Cooperative Communication and Network Coding10 references4 citations

TL;DR

This paper analyzes the performance of random linear network coding in a two-source single-relay network over packet erasure channels, using intra-session coding at sources and inter-session coding at the relay. It derives tight upper bounds on the destination's decoding probability and demonstrates that systematic network coding significantly improves performance, especially when uplink channel quality is good.

ABSTRACT

This paper considers the multiple-access relay channel in a setting where two source nodes transmit packets to a destination node, both directly and via a relay node, over packet erasure channels. Intra-session network coding is used at the source nodes and inter-session network coding is employed at the relay node to combine the recovered source packets of both source nodes. In this work, we investigate the performance of the network-coded system in terms of the probability that the destination node will successfully recover the source packets of the two source nodes. We build our analysis on fundamental probability expressions for random matrices over finite fields and we derive upper bounds on the system performance for the case of systematic and non-systematic network coding. Simulation results show that the upper bounds are very tight and accurately predict the decoding probability at the destination node. Our analysis also exposes the clear benefits of systematic network coding at the source nodes compared to non-systematic transmission.

Motivation & Objective

• To analyze the decoding performance of a two-source single-relay network using random linear network coding across both source and relay nodes.
• To model the system as a network-coded multiple-access relay channel with packet erasure links and derive theoretical bounds on successful decoding at the destination.
• To compare systematic and non-systematic network coding at the source nodes in terms of decoding probability and system efficiency.
• To validate theoretical bounds through simulations and assess their tightness under varying channel conditions and network parameters.

Proposed method

• The system model uses two source nodes transmitting to a destination via a relay, with links modeled as independent packet erasure channels.
• Intra-session random linear network coding is applied at each source node, using coefficients from GF(2), with systematic coding preserving original source packets in the first $ K_\ell $ transmissions.
• The relay performs inter-session network coding by combining received packets from both sources using random linear combinations over GF(2).
• The destination decodes source packets by solving a system of linear equations derived from received coded packets, with decoding success dependent on the rank of the resulting coefficient matrix.
• Theoretical upper bounds on the decoding probability are derived using properties of random block angular matrices over finite fields, particularly focusing on the rank distribution of random matrices.
• The analysis accounts for the interdependency between relay decoding and destination decoding by modeling the joint probability of successful recovery at both nodes.

Experimental results

Research questions

• RQ1 How does the use of systematic network coding at the source nodes affect the decoding probability at the destination compared to non-systematic coding?
• RQ2 What are the tightest theoretical upper bounds on the probability that the destination successfully recovers all source packets in a two-source single-relay network?
• RQ3 How does the quality of the source-to-relay and relay-to-destination channels influence the system's decoding performance?
• RQ4 How does the number of transmitted coded packets affect the accuracy of the derived upper bounds compared to simulation results?
• RQ5 What is the impact of relay cooperation on system performance when direct source-to-destination links are unreliable?

Key findings

• The derived upper bounds on the decoding probability are extremely tight, especially as the number of source packets $ K $ and transmitted coded packets $ N $ increase, with near-perfect agreement between theory and simulation for $ K=20 $, $ N=30 $.
• Systematic network coding at the source nodes significantly improves decoding performance compared to non-systematic coding, particularly when the source-to-relay channel quality is high (e.g., $ p_{\mathrm{S,R}} = 0.3 $).
• For good uplink conditions, systematic coding reduces the number of excess coded packets $ N - K $ required at the source to achieve a target decoding probability.
• The benefit of systematic coding diminishes as the source-to-relay channel quality degrades, but it still offers advantages in progressive recovery and reduced decoding complexity.
• The theoretical bounds accurately quantify the trade-off between relay transmission load $ N_{\mathrm{R}} $ and source-to-destination erasure probability $ p_{\mathrm{S,D}} $, enabling system design for target reliability.
• The analysis reveals a strong interdependency between relay decoding success and destination decoding success, which diminishes with increasing $ K $, leading to tighter bounds.