[Paper Review] Diversity Multiplexing Tradeoff of Asynchronous Cooperative Relay Networks

R. N. Krishnakumar, Naganaboina Naveen+1 more|ArXiv.org|Jul 1, 2008

Cooperative Communication and Network Coding13 references4 citations

TL;DR

This paper proposes slotted Amplify-and-Forward (SAF) protocols for asynchronous cooperative relay networks with arbitrary propagation delays and slot offsets, using distributed space-time codes based on Cyclic Division Algebras (CDA). It proves that these protocols asymptotically achieve the optimal transmit diversity gain for all multiplexing gains, even under timing asynchronism, by ensuring the diversity-multiplexing tradeoff (DMT) approaches the theoretical upper bound as block length increases.

ABSTRACT

The assumption of nodes in a cooperative communication relay network operating in synchronous fashion is often unrealistic. In the present paper, we consider two different models of asynchronous operation in cooperative-diversity networks experiencing slow fading and examine the corresponding diversity-multiplexing tradeoffs (DMT). For both models, we propose protocols and distributed space-time codes that asymptotically achieve the transmit diversity bound for all multiplexing gains and for any number of relays.

Motivation & Objective

• To address the performance degradation in cooperative relay networks caused by timing asynchronism, which undermines the diversity gains of synchronous protocols.
• To design distributed space-time codes and protocols that maintain full transmit diversity under arbitrary propagation delays and slot offsets.
• To establish the diversity-multiplexing tradeoff (DMT) for asynchronous two-hop relay networks with and without a direct link.
• To demonstrate that the proposed protocols achieve the theoretical transmit diversity bound asymptotically, even in the presence of relay interference due to asynchronism.

Proposed method

• Proposes a slotted Amplify-and-Forward (SAF) protocol with guard time to mitigate interference from overlapping relay transmissions due to propagation delays.
• Models the network under two asynchronous scenarios: (1) propagation-delay model with variable source-to-relay and relay-to-destination delays, and (2) slot-offset model with misaligned transmission slots.
• Derives the channel matrix $ H $ as a banded lower-triangular matrix with random fading coefficients, capturing the effective channel seen at the destination.
• Uses lower bounds on the DMT by analyzing the diagonal and sub-diagonal entries of $ H $, focusing on the number of independent $ ho^{- ext{exponent}} $ terms in the eigenvalue distribution.
• Applies the concept of approximately universal codes from Cyclic Division Algebras (CDA) to ensure DMT optimality regardless of fading distribution.
• Establishes DMT lower bounds via probabilistic analysis of the product of $ (1 + ho | ext{channel gain}|^2) $ terms, showing convergence to the diversity bound as block length $ T $ increases.

Experimental results

Research questions

• RQ1 Can cooperative diversity protocols maintain full transmit diversity in the presence of arbitrary propagation delays between source, relays, and destination?
• RQ2 How does timing asynchronism affect the diversity-multiplexing tradeoff (DMT) in amplify-and-forward two-hop relay networks?
• RQ3 Can distributed space-time codes be designed to achieve the optimal DMT under asynchronous operation, even when relays interfere due to timing offsets?
• RQ4 What is the asymptotic DMT performance of SAF protocols under arbitrary delay profiles, and does it approach the theoretical transmit diversity limit?

Key findings

• The proposed SAF protocol with guard time achieves a DMT lower bound of $ d(r) ightarrow Nig(1 - rac{T}{T - heta}rig) $ as $ T o ty $, approaching the transmit diversity bound $ N(1 - r) $.
• For the case with a direct link and relay isolation, the DMT is lower bounded by $ d_H(r) ightarrow ig(1 - rig) + Nig(1 - rig) $ for large $ M $ and $ T $, achieving the full diversity gain.
• The protocol maintains full diversity even when relay transmissions overlap due to timing offsets, by ensuring that the number of independent channel gains per symbol is preserved via guard intervals.
• The use of CDA-based codes ensures DMT optimality, as these codes are approximately universal and achieve the DMT for any fading distribution, including Rayleigh fading.
• The DMT degradation due to asynchronism diminishes with increasing block length $ T $, showing that longer codes can effectively mitigate timing offset effects.
• The analysis confirms that the diversity gain is limited only by the number of relays $ N $, not by timing offsets, provided the code length $ T $ is sufficiently large.