We investigate different decoding strategies at the relay over a frequency selective two-way relay channel with physical-layer network coding (PNC). The incorporation of the PNC scheme enables two users exchange information via a relay in two transmission phases. We study two approaches at the relay to decode the XOR of the transmitted codewords namely; a) an approximately optimal decoding scheme which is implemented using a list decoding algorithm, and b) a minimum mean square error (MMSE) based detector followed by a PNC decoder. The list decoding scheme selects L most likely pairs of sequences corresponding to the transmitted codewords and sorts them in the order of decreasing a-posteriori probabilities. From this list, estimates of the highly likely network coded sequences are obtained. Numerical examples show that a joint detector/physical-layer network coded sequence decoder (JD/PNCD) has a performance similar to the list decoding scheme and can be implemented with a lower complexity. The detection scheme based on the MMSE criterion is a suboptimal approach that generates soft information corresponding to the linear sum of the received symbols and is particularly useful in scenarios where the number of channel taps is large.

A two-way relay channel is considered where two users exchange information via a common relay in two transmission phases using physical-layer network coding (PNC). We consider an optimal decoding strategy at the relay to decode the network coded sequence during the first transmission phase, which is approximately implemented using a list decoding (LD) algorithm. The algorithm jointly decodes the codewords transmitted by the two users and sorts the L most likely pair of sequences in the order of decreasing a-posteriori probabilities, based on which, estimates of the most likely network coded sequences and the decoding results are obtained. Using several examples, it is observed that a lower complexity alternative, that jointly decodes the two transmitted codewords, has a performance similar to the LD based decoding and offers a near-optimal performance in terms of the error rates corresponding to the XOR of the two decoded sequences. To analyze the error rate at the relay, an analytical approximation of the word-error rate using the joint decoding (JD) scheme is evaluated over an AWGN channel using an approach that remains valid for the general case of two users adopting different codebooks and using different power levels. We further extend our study to frequency selective channels where two decoding approaches at the relay are investigated, namely; a trellis based joint channel detector/physical-layer network coded sequence decoder (JCD/PNCD) which is shown to offer a near-optimal performance, and a reduced complexity channel detection based on a linear receiver with minimum mean squared error (MMSE) criterion which is particularly useful where the number of channel taps is large.

We study two-receiver degraded binary broadcast channels (DBBCs), focusing on the capacity region and the encoding schemes that achieve its boundary. First, we derive the conditions for a general binary broadcast channel to be degraded, and show that only a very limited subset of the possible degraded configurations have been investigated in the previous literature. Then, we show how to design a capacity-achieving encoding scheme for a general DBBC, and give a detailed specific example. The designed scheme turns out to be very general and to include, as special cases, the encoding schemes that achieve the capacity boundary of the DBBCs previously studied in the literature.

Unequal error-protection schemes obtained by means of two-level superposition coding are considered. Their performance over Gaussian broadcast channels (GBCs) is investigated with optimal maximum-likelihood decoding as well as with a suboptimal decoding strategy based on interference cancellation. We focus on GBCs without fading and, assuming that linear codes are used, we evaluate, for both decoding strategies, analytical approximations of the word-error rate based on a suitable application of the union bound. As in the case of turbo codes and turbo-coded modulations in schemes without superposition, the derivation of the approximations exploits the concept of uniform interleaving. The analytical expressions obtained are in excellent agreement with the simulation results, and thus provide a useful tool for analysis and design of practical superposition-coding schemes. Unlike the existing design tools, which rely on the assumption of infinite-length superposition codes, the proposed approach allows us to study the effectiveness of finite-length coding schemes with known distance spectrum.

We consider unequal error-protection schemes obtained by means of two-level superposition coding. The performance over additive white Gaussian noise channels is investigated for optimal maximum-likelihood decoding as well as for a suboptimal decoding strategy based on interference cancellation. Assuming that linear codes are used, we evaluate, for both strategies, analytical approximations of the word-error rate, based on the union bound. As in the case of turbo codes and turbo-coded modulations, the derivation exploits the concept of uniform interleaving, and the bounds are in excellent agreement with the simulation results obtained using iterative decoding. The analytical expressions are useful for code design and for the selection of decoding strategies providing a suitable performance/complexity tradeoff.