Application of the harmonic mean statistics to the end-to-end performance of transmission systems with relays

Closed-form expressions for the statistics of the harmonic mean of two independent and identically distributed gamma variates are presented. The probability density function of the harmonic mean of two F variates is also derived. These statistical results are then applied to study the performance of wireless communication systems with non-regenerative relays over flat Nakagami (1960) fading channels. It is shown that these results can either be exact or tight lower bounds on the performance of these systems depending on the choice of the relay gain. More specifically, outage probability formulas for noise limited systems as well as systems affected by interference are obtained. Furthermore, average bit error rate and outage capacity expressions are derived. Finally, comparisons between regenerative and non-regenerative systems are presented. Numerical results show that the former systems clearly outperform the latter ones for low average signal-to-noise-ratio (SNR). They also show that the two systems have similar performance at high average SNR.