Relay subset selection in cooperative systems with beamforming and limited feedback

We consider a cooperative wireless network with a single source, a single destination and M cooperative relays and show that with limited feedback bits from the destination, using all the relays to perform beamforming may not be optimal. We further assume that the destination chooses N relays with power gains exceeding a threshold to do transmit beamforming and feeds back B bits to specify a precoding vector from a randomly generated precoder codebook. For the given average feedback bits per relay, \bar{B}=B/M, we find the optimal threshold that maximizes the capacity of a decode-and-forward (DF) system when the number of relays is asymptotically large. We show that the system using this asymptotically optimal threshold has a larger capacity than a system using all relays for finite \bar{B}, especially for small values of \bar{B}. We also obtain numerically the optimal thresholds for systems with a finite number of relays. A simplified approach that uses a fixed number of relays is also proposed and leads to significant improvements. Given a fixed power budget, the power allocation between two phases is also examined in networks with asymptotically large numbers of relays, and the optimal power allocation fraction can be found numerically.