On lower bounding the information capacity of amplify and forward wireless relay channels with channel estimation errors

We formulate a capacity lower bound for the dual-hop wireless relay channel which employs an amplify-and-forward (AF) protocol at the relay node. In AF relaying, even when the fading channel in both hops is complex Gaussian distributed, the overall dual-hop channel is non-Gaussian. We highlight that there is a fundamental difference between Gaussian and non-Gaussian channels in terms of deriving their capacity lower bound. Specifically for non-Gaussian channels, the channel estimation error variance depends on the received pilot signal and is, in general, different from the average error variance. Whereas for Gaussian distributed channels, which have been predominantly studied in the literature, the channel estimation error variance conditioned on the observed pilot signal coincides with the average error variance. We provide an example using the AF dual-hop channel to exhibit the numerical difference between the true capacity lower bound and that obtained by using the average instead of the conditional error variance.