Upper and Lower Bounds of Two-Way Opportunistic Amplify-and-Forward Relaying Channels

With the high signal-to-noise ratio (SNR) approximation, the statistical descriptions of the two-way opportunistic relaying amplify-and-forward channels (TWOR-AF) are investigated through the appropriate mathematic proof and simulations. Firstly, we obtain the tractable closed-form expressions of the cumulative distribution function (CDF) and outage probability by defining the equivalent instantaneous end-to-end channels variance {λ _{eq - k}}(θ ), where 2 ≤ θ ≤ 3. Then, we study the performance bounds of the TWOR-AF systems. The simulations show that, in high SNR, the derivation is the tight lower bound of the outage probability when θ=2, and is the upper bound when θ=3. Finally, we present the approximate analysis of the outage probability by taking θ=2.3 and θ=2.5, respectively. The results show that we can obtain the more precise approximate evaluation of the systems performance when θ=2.3, especially in low SNR region. However, when θ=2.5, we can approximately evaluate the upper bound of the systems performance over the entire SNR region. This yields that the TWOR-AF systems can achieve about 1dB of SNR gain at 10^{-4} of outage probability over θ=3 case.