On the exact and asymptotic ser of receive diversity with multiple amplify-and-forward relays

Receive diversity with multiple amplify-and-forward (AF) relays is analyzed over unbalanced hops with independent but not necessarily identically distributed Rayleigh fading. We consider two main AF protocols: 1) fixed-gain relaying, which requires partial channel-state information (CSI), and 2) variable-gain relaying, which requires full CSI. Exact closed-form expressions are derived for the moment-generating function of the end-to-end signal-to-noise ratio (SNR). Our analysis is valid for arbitrary numbers of relays and arbitrary numbers of antennas at the destination. Based on the derived results, we determine the symbol error rate with $M$-ary phase-shift keying. In addition, we carry out an asymptotic analysis in the high-SNR regime. Our results provide fundamental insight into the impact of the number of antennas on the performance of multiple AF relaying. We explicitly reveal that the diversity order is equal to the sum of two parameters: 1) the number of relays and 2) the number of antennas at the destination.