# Performance of Dynamic Time Division Broadcast Protocol with Rateless Coding
## 1 Introduction  
Two-way relaying is an efficient way to improve the throughput performance of conventional one-way cooperative communications with half-duplex relay [1], where two sources exchange information with the aid of a relay (Considering the scenario where two sources \(S_{1}\) and \(S_{2}\) exchange information with the aid of relay \(R\), four time slots, i.e., the time slots on the \(S_{1}\to R\), \(R\to S_{2}\), \(S_{2}\to R\), and \(R\to S_{1}\) links' transmission, are typically entailed for one round information exchange between \(S_{1}\) and \(S_{2}\), leading to throughput loss. To improve throughput performance, two-way relaying is proposed with network coding at \(R\), where both \(S_{1}\) and \(S_{2}\) transmit simultaneously to \(R\), \(R\) then network-encodes the information received for \(S_{1}\) and \(S_{2}\) (e.g., through bit level XOR operation) and broadcasts to them, with less time slots.). There are two well-known protocols for two-way decode and forward relaying, i.e., physical-layer network coding (PNC) and time division broadcast protocol (TDBC) [2, 3]. For one round information exchange between the two sources, TDBC consumes three time slots, with lower throughput than PNC which only needs two time slots. In this paper, we focus on two-way relaying channel with direct link between the two sources (often encountered in many practical applications) and the TDBC protocol, and aim to improve its throughput. To enhance throughput performance in relaying channel, incremental redundancy (IR)-based retransmission is commonly used by utilizing one-bit feedback indicating successful or failed decoding at the destination [4; 5; 6], and it has been incorporated in TDBC (i.e., TDBC-IR) to improve the throughput performance of TDBC [7; 8] (Note that one retransmission is used in PNC in [9] to form a new protocol named PNC-IR, aiming to improve the outage performance. However, not only the outage performance but also the throughout performance of PNC-IR are improved by introducing one retransmission when compared with PNC.). Although IR can improve throughput, we argue that TDBC-IR is not the best way since fixed two or three time slots are still entailed, leading to underutilizing good channel conditions (or unable to adapt to the varying channel dynamics). Motivated by rateless coding, which adapts to varying channel dynamics and achieves (or approaches) channel capacity automatically [10], we propose a dynamic TDBC with rateless coding as a kind of incremental redundancy transmission (termed DTDBC-RC for short). In DTDBC-RC, the two sources first transmit with a given maximum allowable transmission time, and then the relay retransmits or not based on the decoding results. To reveal the advantages of DTDBC-RC, we analyze its performance comprehensively in terms of outage probability, expected rate, and diversity-multiplexing trade-off (DMT). We also present a subslot realization scheme for DTDBC-RC (i.e., sub-DTDBC-RC) for the DMT derivation of DTDBC-RC since it cannot be obtained directly based on conventional outage probability and expected rate analysis. It is revealed that although TDBC-IR and DTDBC-RC have the same outage performance, the latter outperforms the former in both expected rate and DMT performance. The rest of the paper is organized as follows. Section 2 first presents the system models of DTDBC-RC and sub-DTDBC-RC. Their performance is then analyzed in Section 3 while simulation and numerical results are shown in Section 4. Finally, conclusions are drawn in Section 5. _Notation_: \(\log(\bullet)\) is the logarithm function with base 2. \(P_{r}\left(\bullet\right)\) is the probability of a random event. \(\left(x\right)^{+}=\max(x,0)\). \(f\left(x\right)\doteq\rho^{-x}\) is exponential equality and means that \(\lim_{\rho\rightarrow+\infty}\frac{\log f\left(x\right)}{\log\rho}=-x\).