Approximations of the packet error rate under slow fading in direct and relayed links

Most of the existing literature on performance evaluation of fading channels is concentrated on either the symbol error rate of the links, or the outage capacity. We present in this technical report approximation techniques for the packet error rate of quasi-static fading channels. This model applies when the channel coherence time is larger than the signaling time and leads to multiple symbols experiencing the same fading state during a packet transmission. We start by giving a closed-form upper bound on the asymptotic coding gain of the packet error rate in quasi-static fading channels. Using the fact that the asymptotic formulation is inversible with respect to the mean signal-to-noise ration (SNR), we present a packet error outage metric, with applications to channels where links are subject to both fading and log-normal shadowing effects simultaneously. We then show that a unit step approximation can be derived for the packet error rate of block fading channels, that closely matches the numerical computation of the packet error rate in a tractable closed-form expression. Using the asymptotic approximation method, we derive the optimal power allocation for different relaying protocols, especially when optimal combination of the symbols received at the destination may not be possible. Key-words: relay channels, packet error rate, quasi-static fading ha l-0 08 34 90 4, v er si on 1 17 J un 2 01 3 Approximations du taux d’erreur paquet des liens directs et des canaux à relais dans les canaux à évanouissements quasi-statiques Résumé : Voir résumé anglais Mots-clés : canal à relais, taux d’erreur paquet, évanouissements quasi-statiques ha l-0 08 34 90 4, v er si on 1 17 J un 2 01 3 Approximations of the PER under slow fading in direct and relayed links 3 Most of the existing literature on performance evaluation of fading channels is concentrated on either the symbol error rate of the links, or the outage capacity. A good review of these results as well as an interesting framework for the evaluation of symbol error rates in fading channels is available in the book of Simon and Alouini [Simon and Alouini, 2004]. The results presented here root themselves in the work of Wang and Giannakis [Wang and Giannakis, 2003] who presented an asymptotic approximation of symbol error rates in fading channels using a Taylor expansion cut on the first term. The second section presents this framework. This approach is well suited to channels whose probability density function is approximately polynomial near zero, but fails for certain models of fading such as log-normal shadow fading. The asymptotic approximation of Wang and Giannakis have been extended to general amplifyand-forward relays by Ribeiro et al. [Ribeiro et al., 2005], who showed that in that case the optimal selection criterion for relays is to maximize the harmonic mean of the source-relay and relay-destination links. Liu et al. [Liu et al., 2009] further extended these results and produces a comprehensive treatment of the end-to-end symbol error rates in relay channel for both amplifyand-forward and decode-and-forward protocols, including asymptotic approximations, and for a variety of modulation schemes [See in particular Liu et al., 2009, Ch.5]. Using the results from Zheng and Tse [Zheng and Tse, 2003], the study of symbol error rates at high SNR can be related to the study of the capacity outage probability at high SNR. Related works thus include asymptotic approximations of the outage probability in fading channels. In a manner similar to the approach we present here, Annavajjala et al. treated the asymptotical outage probability of direct links as well as relayed amplify-and-forward and decode-and-forward protocols [Annavajjala et al., 2007]. These results are focused on the symbol error rate of fading channels ; when the fading is relatively fast compared to the symbol transmission duration, with proper interleaving one can extend them to packet error rates. On the other hand, when the fading is much slower than the symbol transmission time one has to consider that most symbols in the packet will experience the same fading state – a model known as block fading or quasi-static fading. Wang and Giannakis’ approach has been extended recently by Xi et al. for the packet error rate of block fading channels [Xi et al., 2010], including the asymptotic performance of decode-and-forward in an ideal relay channel. The work presented in this paper is based on the results of Xi et al. We strengthen their results by giving a closed-form approximation of the asymptotic coding gain rather than a numerical evaluation, for two usual forms of bit error rate expressions used in realistic cases. Using the fact that the asymptotic formulation is inversible with respect to the mean signal-to-noise ration (SNR), we present a packet error outage metric, with applications to channels where links are subject to both fading and log-normal shadowing effects simultaneously. We then show that a unit step approximation can be derived for the packet error rate of block fading channels, that closely matches the numerical computation of the packet error rate in a tractable closed-form expression. Using the asymptotic approximation method, we derive the optimal power allocation for different relaying protocols, especially when optimal combination of the symbols received at the destination may not be possible. We then present immediate work perspectives of interest based on our results. 1Lemma 5 in Zheng and Tse’s paper states that at high SNR, a detection error is very likely, conditioned on a channel outage RR n° 8316 ha l-0 08 34 90 4, v er si on 1 17 J un 2 01 3 4 Ferrand & Gorce & Goursaud