SER Analysis and PDF Derivation for Multi-Hop Amplify-and-Forward Relay Systems

An amplify-and-forward, multi-branch, multi-hop relay system with K relays, in which the relays broadcast to other relays as well as the destination, is analyzed. An approximate symbol-error-rate (SER) expression, which is valid for any number of relays and for several modulation schemes, is found for the multi-hop system. Also, the cumulative density function (CDF) and probability density function (PDF) are found for the random variable, Z = XY/(X + Y + c), where X and Y are sums of independent, Erlang random variables, and c is a constant. The moment generating function (MGF) of Z is found for the special case in which c = 0. It is shown that these results are generalizations of previously published results for special cases of Z. The MGF of Z is used to develop the approximate SER expression. Results for the analytic SER expression are included and compared with simulation results for various values of K, for various modulation schemes, and for two choices of system parameters (channel variances). Results for the multi-hop system are also compared to results for the two-hop system (in which relays transmit only to the destination).