# Adaptive Packet Coding for Reliable Underwater Acoustic Communications
## 2 System Model  
We consider a point-to-point system operating in a half-duplex manner without interference from other nodes. The sender collects \(W_{s}\) information packets. Each packet has \(N_{b}\) symbols and \(N_{b}\) is a constant number during the transmission. These information packets are first divided into multiple blocks, each containing \(W\) packets. In this paper, we only consider the FEC technique in the packet level. The \(W\) original packets in each block are then encoded based on the RLPC [14; 15; 16] to generate sufficient coded packets. We assume that the data is delay constrained, which means the receiver should receive the data before the deadline. Otherwise, the data becomes useless. We first consider an S&W ARQ scheme with RLPC, which is illustrated in Figure 1. The coded packets from different blocks are transmitted in sequence. Let \(N_{i}^{(k)}\) denotes the number of transmitted packets for the \(k\)-th block in the \(i\)-th transmission. Let \(\rho_{k,i}^{(1)}\) and \(\rho_{k,i}^{(2)}\) denote the beginning and ending time instants of the \(k\)-th block in the \(i\)-th transmission. Hence, the relationship between them is  
\[\rho_{k,i}^{(2)}=\rho_{k,i}^{(1)}+T_{p}N_{i}^{(k)} \tag{1}\]  
where \(T_{p}=N_{b}T_{s}\) is the time duration of a coded packet, \(T_{s}\) is the symbol duration. Figure 1: The S&W ARQ scheme with RLPC. Let \(T_{d}\) denote the propagation delay. Then, the transmitted signal is received after \(T_{d}\) duration, \(T_{d}=d/c\), where \(d\) is the distance between the transceiver pair, and \(c\) is the sound speed in water. Let \(\sigma_{k,i}^{(1)}\) and \(\sigma_{k,i}^{(2)}\) denote the beginning and ending time instants of the received signal at the receiver. Then, we have  
\[\left\{ \begin{array}{l}\sigma_{k,i}^{(1)}=\rho_{k,i}^{(1)}+T_{d}\\ \sigma_{k,i}^{(2)}=\sigma_{k,i}^{(1)}+T_{p}N_{i}^{(k)}\end{array} \right. \tag{2}\]  
When the receiver completes the receiving process, there is a guard interval, \(T_{g}\), for the receiver node to process the data and change the mode. Next, the receiver responds with an ACK. According to the characteristic of RLPC, the receiver can recover the original information if there are enough coded packets successfully received. Thus, we do not need to retransmit the unsuccessful packets in the following transmission. In the following transmission, the sender will send enough other coded packets. It is noted that the other coded packets are also generated from the same information packets. There is the main different from the traditional ARQ scheme. Hence, the ACK contains the number of successfully received packets \(s_{i}^{(k)}\) in the current transmission and some pilot symbols used for channel estimation at the sender. The BER \(e\) depends on the channel coding rate, the modulation order, the channel state, and the codeword length. In this paper, these parameters are fixed except the channel state. Thus, the packet error probability (PEP) \(P_{c}\) is only connected with the channel state. As mentioned above, we focus on the link level performance, so we model the channel state by a finite-state Markov chain (FSMC) [22; 24; 25]. Assume the channel state is constant during the \(i\)-th transmission. Let \(h_{i}^{(k)}\) denote the CSI when the \(k\)-th block is transmitted at the \(i\)-th transmission, which is quantized into \(L\) levels, i.e., \(h_{i}^{(k)}\in\{\delta_{1},\ldots,\delta_{L}\}\). The state transition probabilities \(P_{h}(h_{i}^{(k)}|h_{i-1}^{(k)})\) are assumed known. Let \(P_{S}(s_{i}^{(k)},h_{i}^{(k)},N_{i}^{(k)})\) denote the probability that \(s_{i}^{(k)}\) coded packets have been successfully received when \(N_{i}^{(k)}\) coded packets are transmitted under the channel state \(h_{i}^{(k)}\). Then,  
\[P_{S}(s_{i}^{(k)},h_{i}^{(k)},N_{i}^{(k)})= \begin{pmatrix}\dot{s}_{i}^{(k)}\\ N_{i}^{(k)}\end{pmatrix} (1-P_{c})^{s_{i}^{(k)}}P_{c}^{(N_{i}^{(k)}-s_{i}^{(k)})} \tag{3}\]  
Let \(\sigma_{k,i}^{(3)}\) and \(\sigma_{k,i}^{(4)}\) denote the beginning and ending time points for the ACK at the receiver.