  Physical-layer Network Coding (PNC) makes use of the additive nature of the
electromagnetic (EM) waves to apply network coding arithmetic at the physical
layer. With PNC,the destructive effect of interference in wireless networks is
eliminated and the capacity of networks can be boosted significantly. This
paper addresses a key outstanding issue in PNC: synchronization among
transmitting nodes. We first investigate the impact of imperfect
synchronization (i.e., finite synchronization errors) in a 3-node network. It
is shown that with QPSK modulation, PNC still yields significantly higher
capacity than straightforward network coding when there are synchronization
errors. Significantly, this remains to be so even in the extreme case when
synchronization is not performed at all. Moving beyond a 3-node network, we
propose and investigate a synchronization scheme for PNC in a general chain
network. At last, numerical simulation verifies that PNC is robust to
synchronization errors. In particular, for the mutual information performance,
there is about 0.5dB loss without time synchronization and there is at most 2dB
loss without phase synchronization.
