Optimal Relay Selection and Beamforming Strategy in Cooperative Systems with Limited Feedback and Asymptotically Large Number of Relays

We consider a cooperative wireless network with a single source, a single destination, and $M$ cooperative relays. We show that, with only limited feedback, $Mbar{B}$ bits, from the destination, using all the relays to perform beamforming may not be optimal. Assuming that the destination selects the relays whose power gains exceed a threshold and feeds back $Mbar{B}$ bits to specify a precoding matrix from a randomly generated pre-coder codebook, we find the optimal threshold that maximizes the capacity of a Decode-and-Forward (DF) system for a given $bar{B}$, asymptotically in the number of relays, $M$. We show that the system using the optimal threshold has a higher capacity than a system using all relays for all $bar{B}$, especially for small values of $bar{B}$. Furthermore, using a fixed fraction ($beta^{ast}$ which is obtained from the optimal threshold) of relays in systems with finite number of relays also leads to a significant improvement in outage probability, comparing to that of the systems using all relays, in both DF and Amplify-and-Forward (AF) cooperating schemes.