You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
This paper investigates coherent detection for physical-layer network coding (PNC) with short packet transmissions in a two-way relay channel (TWRC). PNC turns superimposed EM waves into network-coded messages to improve throughput in a relay system. To achieve this, accurate channel information at the relay is a necessity. Much prior work applies preambles to estimate the channel. For long packets, the preamble overhead is low because of the large data payload. For short packets, that is not the case. To avoid excessive overhead, we consider a set-up in which short packets do not have preambles. A key challenge is how the relay can estimate the channel and detect the network-coded messages jointly based on the received signals from the two end users. We design a coherent detector that makes use of a belief propagation (BP) algorithm to do so. For concreteness, we focus on frequency-shift-keying (FSK) modulation. We show how the BP algorithm can be simplified and made practical with Gaussian-mixture passing. In addition, we demonstrate that prior knowledge on the channel distribution is not needed with our framework. Benchmarked against the detector with prior knowledge of the channel distribution, numerical results show that our detector can have nearly the same performance without such prior knowledge.

Cited Abstract(s):
[1]: In this work we extend the existing concept of De- Noise and Forward (DNF) for bidirectional relaying to utilise non-coherent modulation schemes. This is done in order to avoid the requirement of phase tracking in coherent detection. As an example BFSK is considered, and through analysis the decision regions for the denoise operation in DNF are identified. The throughput performance of BFSK in DNF is compared to BPSK.

[2]: This paper investigates the noncoherent detection in a two-way relay channel operated with physical-layer network coding (PNC), assuming FSK modulation and short-packet transmissions. For noncoherent detection, the detector has access to the magnitude but not the phase of the received signal. For conventional communication in which a receiver receives the signal from a transmitter only, the phase does not affect the magnitude, and hence the performance of the noncoherent detector is independent of the phase. PNC, on the other hand, is a multiuser system in which a receiver receives signals from multiple transmitters simultaneously. The relative phase of the signals from different transmitters affects the received signal magnitude through constructive-destructive interference. In particular, for good performance, the noncoherent detector of a multiuser system such as PNC must take into account the influence of the relative phase on the signal magnitude. Building on this observation, this paper delves into the fundamentals of PNC noncoherent detector design. To avoid excessive overhead, we assume a set-up in which the short packets in the PNC system do not have preambles. We show how the relative phase can be deduced directly from the magnitudes of the received data symbols, and that the knowledge of the relative phase thus deduced can in turn be used to enhance performance of noncoherent detection. Our overall detector design consists of two components: 1) a channel gains estimator that estimates channel gains without preambles; and 2) a detector that builds on top of the estimated channel gains to jointly estimate relative phase and detect data using a belief propagation algorithm. Numerical results show that our detector performs nearly as well as a “fictitious” optimal detector that has perfect knowledge of the channel gains and relative phase. Although this paper focuses on PNC with FSK modulation, we believe that the insight of this paper applies generally to noncoherent detection in other multiuser systems with other modulations. Specifically, our insight is that the relative phase of overlapped signals affects the signal magnitude in multiuser systems, but fortunately the relative phase can be deduced from the magnitudes and this knowledge can be used to improve the detection performance.

[3]: A channel-coded physical-layer network coding strategy is refined for practical operation. The system uses frequency-shift keying (FSK) modulation and operates noncoherently, providing advantages over coherent operation: there are no requirements for perfect power control, phase synchronism, or estimates of carrier-phase offset. In contrast with analog network coding, which relays received analog signals plus noise, the system relays digital network codewords, obtained by digital demodulation and channel decoding at the relay. The emphasis of this paper is on the relay receiver formulation. Closed-form expressions are derived that provide bitwise log-likelihood ratios, which may be passed through a standard error-correction decoder. The role of fading-amplitude estimates is investigated, and an effective fading-amplitude estimator is developed. Simulation results are presented for a Rayleigh block-fading channel, and the influence of block length is explored. An example realization of the proposed system demonstrates a 32.4 throughput improvement compared to a similar system that performs network coding at the link layer. By properly selecting the rates of the channel codes, this benefit may be achieved without requiring an increase in transmit power.

[4]: This paper investigates noncoherent detection in a two-way relay channel operated with physical- layer network coding (PNC), assuming FSK modulation and short-packet transmissions. For noncoherent detection, the detector has access to the magnitude but not the phase of the received signal. For conventional communication in which a receiver receives the signal from a transmitter only, the phase does not affect the magnitude, hence the performance of the noncoherent detector is independent of the phase. PNC, on the other hand, is a multiuser system in which a receiver receives signals from multiple transmitters simultaneously. The relative phase of the signals from different transmitters affects the received signal magnitude through constructive-destructive interference. In particular, for good performance, the noncoherent detector of a multiuser system such as PNC must take into account the influence of the relative phase on the signal magnitude. Building on this observation, this paper delves into the fundamentals of PNC noncoherent detector design. To avoid excessive overhead, we assume a set-up in which the short packets in the PNC system do not have preambles. We show how the relative phase can be deduced directly from the magnitudes of the received data symbols, and that the knowledge of the relative phase thus deduced can in turn be used to enhance performance of noncoherent detection. We design a noncoherent detector that jointly estimates relative phase and detects data using a belief propagation algorithm. Numerical results show that our detector performs as well as a “fictitious” optimal detector that has perfect knowledge of the relative phase. Although this paper focuses on PNC with FSK modulation, we believe the insight of this paper applies generally to noncoherent detection in other multiuser systems with other modulations. Specifically, our insight is that the relative phase of overlapped signals affects the signal magnitude in multiuser systems, but fortunately the relative phase can be deduced from the magnitudes and this knowledge can be used to improve detection performance.

[5]: Physical-layer network coding (PNC) is an attractive technique because it can significantly enhance the throughput of a wireless network with a relatively simple implementation, as compared with conventional network coding (NC) schemes. In this paper, the authors first investigate PNC with frequency shift keying (FSK) through the derivation of its bit-error rate (BER) and capacity, as well as the impact of imperfect synchronization to the overall performance. Based on the discussions on PNC with 2FSK, the authors then further expand the analysis to PNC with M -ary FSK (MFSK). Theoretical results indicate that the BER performance of PNC with 2FSK is a bit worse than that of PNC with 2PSK. As the constellation dimension M increases, the BER performance of PNC with M -ary FSK improves at the expense of bandwidth efficiency degradation.
There have been prior investigations on FSK-PNC detection. Specifically, [2]0 studied coherent detection and [2] [2]5 [1] [3] studied noncoherent detection. A coherent detector has access to the magnitude and the phase of the received signals, whereas a noncoherent detector only has access to the magnitude of the received signals. Since a coherent detector has all the signal information that a noncoherent detector has, but not vice versa, the coherent detector potentially has better performance. In this paper, we focus on coherent FSK-PNC detection.