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To address the issues of high peak-to-average power ratio and high power backing-off in orthogonal frequency-division multiplexing (OFDM) relaying systems, a cyclic prefixed single-carrier (CP-SC) relaying system employing best terminal selection (BTS) is considered in this paper. Under a system power constraint, the problem of joint optimal power allocation to the source and relay is investigated. For a two-hop amplify-and-forward (AF) relaying protocol, the optimal power allocation is first obtained by maximizing the achievable spectral efficiency. After applying the obtained optimal power allocations to the source and relay, a destination terminal that has the best effective end-to-end signal-to-noise ratio is selected. With the help of the statistical properties of circulant channel matrices in the relay links, closed-form bounds for the maximum achievable spectral efficiency, outage probability, and average symbol error rate (ASER) are derived. Further, an asymptotic analysis of the outage probability and ASER is conducted and it is shown that both the number of terminals in the system and the number of channel taps being supported by the CP length play key roles in determining the overall diversity gain. Monte Carlo simulation results verify the derived closed-form analytical expressions.