  This paper investigates various subtleties of applying linear physical-layer
network coding (PNC) with q-level pulse amplitude modulation (q-PAM) in two-way
relay channels (TWRC). A critical issue is how the PNC system performs when the
received powers from the two users at the relay are imbalanced. In particular,
how would the PNC system perform under slight power imbalance that is
inevitable in practice, even when power control is applied? To answer these
questions, this paper presents a comprehensive analysis of q-PAM PNC. Our
contributions are as follows: 1) We give a systematic way to obtain the
analytical relationship between the minimum distance of the signal
constellation induced by the superimposed signals of the two users (a key
performance determining factor) and the channel-gain ratio of the two users,
for all q. In particular, we show how the minimum distance changes in a
piecewise linear fashion as the channel-gain ratio varies. 2) We show that the
performance of q-PAM PNC is highly sensitive to imbalanced received powers from
the two users at the relay, even when the power imbalance is slight (e.g., the
residual power imbalance in a power-controlled system). This sensitivity
problem is exacerbated as q increases, calling into question the robustness of
high-order modulated PNC. 3) We propose an asynchronized PNC system in which
the symbol arrival times of the two users at the relay are deliberately made to
be asynchronous. We show that such asynchronized PNC, when operated with a
belief propagation (BP) decoder, can remove the sensitivity problem, allowing a
robust high-order modulated PNC system to be built.
