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Wireless amplify-and-forward relay networks in which the source communicates with the relays and destination in the first phase and the relays forward signals to the destination in the second phase over orthogonal and uncorrelated Rayleigh fading channels are considered. Convex programming is used to obtain optimal and approximately optimal power allocation schemes to maximize the average signal-to-noise ratios at the output of the receiver filters under two different assumptions of partial channel state information (CSI). Analysis and simulation results demonstrate the superiority of the proposed power allocation schemes over the equal-power allocation scheme. Performance comparison to the extreme cases of i) direct transmission between the source and destination and ii) having full CSI is made to illustrate the gain and loss, respectively, of the proposed schemes. The impact of power allocation between the source and relays is also investigated by computer simulation.