  In this paper, we consider two-way orthogonal frequency division multiplexing
(OFDM) relay channels, where the direct link between the two terminal nodes is
too weak to be used for data transmission. The widely known per-subcarrier
decode-and-forward (DF) relay strategy, treats each subcarrier as a separate
channel, and performs independent channel coding over each subcarrier. We show
that this per-subcarrier DF relay strategy is only a suboptimal DF relay
strategy, and present a multi-subcarrier DF relay strategy which utilizes
cross-subcarrier channel coding to achieve a larger rate region. We then
propose an optimal resource allocation algorithm to characterize the achievable
rate region of the multi-subcarrier DF relay strategy. The computational
complexity of this algorithm is much smaller than that of standard Lagrangian
duality optimization algorithms. We further analyze the asymptotic performance
of two-way relay strategies including the above two DF relay strategies and an
amplify-and-forward (AF) relay strategy. The analysis shows that the
multi-subcarrier DF relay strategy tends to achieve the capacity region of the
two-way OFDM relay channels in the low signal-to-noise ratio (SNR) regime,
while the AF relay strategy tends to achieve the multiplexing gain region of
the two-way OFDM relay channels in the high SNR regime. Numerical results are
provided to justify all the analytical results and the efficacy of the proposed
optimal resource allocation algorithm.
