  A noncoherent two-way relaying system is developed using physical-layer
network coding for improved throughput over conventional relaying in a fading
channel. Energy-efficient noncoherent operation is achieved using multitone
frequency shift keying (FSK). A novel soft-output demodulator is developed for
the relay, and corresponding achievable exchange rates are found for Rayleigh
fading and AWGN channels. Bit-error rate performance approaching the achievable
rate is realized using a capacity-approaching channel code and a receiver
architecture that iterates between demodulation and channel decoding. Iterative
decoding is performed feeding information back from the channel decoder to the
demodulator. Additionally, error-rate performance is made to approach the
achievable rate more closely by optimizing LDPC codes for this system. The
energy efficiency improvement obtained by increasing the modulation order is
more dramatic for the proposed physical-layer network coding scheme than it is
for a conventional point-to-point system. Using optimized LDPC codes, the
bit-error rate performance is improved by as much as 1.1 dB over a widely known
standardized LDPC code, and comes to within 0.7 dB of the limit corresponding
to the achievable rate. Throughout this work, performance for physical-layer
network coding is compared to conventional network coding. When noncoherent FSK
is used, physical-layer network coding enables higher achievable rates, and
conventional network coding exhibits better energy efficiency at low rates.
