Subcarrier pairing for amplify-and-forward and decode-and-forward OFDM relay links

We consider a two-hop relay link in which orthogonal frequency division multiplexing (OFDM) is used on both hops. Under a joint sum-power constraint, our aim is to allocate power over subcarriers on the two hops such that the instantaneous rate of the relay link is maximized. Ordered subcarrier pairing (OSP) has been proposed in the literature to further improve the relay link rate; however, the optimality of OSP has been proven only for equal power allocation and the proof of its optimality under optimal power allocation has not been available yet. In this letter, we will provide our proof which verifies that OSP is optimal for both amplify-and-forward (AF) and decode-and-forward (DF) relay links when optimal power allocation is applied.