## 2 Impact of MAC Delay on Localization  
### Transmission Scheme  
The communication scheme used for the ranging can be classified according to which node transmits the packets for the ranging. Figure 1 gives a localization example to estimate the position of the AUV denoted by \(S\) with three ANs denoted by \(A1\), \(A2\), and \(A3\) respectively. As shown in Figure 0(a), if the ranging is initiated by the AUV, all the ANs receiving the packet deliver it to a certain AN to run a specific localization algorithm. In the example, \(A2\) is selected as a representative node; consequently \(A1\) and \(A3\) relay the received packet to \(A2\). Based on the collected information, \(A2\) estimates the position of \(S\) and then \(A2\) inform \(S\) the result. In this approach, MAC delays do not affect the localization because the contention for the channel occupation will not happen. However, this approach causes additional message exchange to gather the received information and to return the estimated result to the AUV. In addition, this communication overhead significantly increases when a number of nodes should be localized, leading to the spectral inefficiency. In contrast, if the ranging is initiated by the AN as shown in Figure 0(b), all the underwater nodes, which receive the packets transmitted by the ANs, can estimate their position at the same time without additional message exchange.  
Letting \(N\) and \(N_{s}\) denote the number of ANs and AUVs respectively, then the number of packets exchanged in underwater is \(2N_{s}\) in the case of ranging initiated by the AUVs. In the case of ranging initiated by the ANs, the number of packets is \(N\) regardless of the number of AUVs. Therefore, if \(N_{s}\) is considerably bigger than \(N\) (\(N_{s}\gg N\)), significant channel resources are wasted for the localization, which is not suitable in underwater communications where the available bandwidth is very limited.  
Therefore, for the underwater localization, ranging initiated by the ANs is desirable, especially a large number of nodes are deployed in underwater [11]. However, in this message exchange scheme, each AN transmits the packet at the different times for collision avoidance, which significantly degrades the localization performance.  
Figure 2 illustrates the impact of MAC delay on the time of arrival (ToA)-based ranging. The AUV is moving from left to right in the figure. \(A\)1, \(A\)2, and \(A\)3 sequentially transmits a packet including their position and transmission time at the assigned time slot upon the time division multiple access (TDMA) protocol. Here, for TDMA, the time synchronization of ANs is the crucial requirement. In this paper, we assume that all the ANs are synchronized via GPS time. Letting \(t_{i}\) denote the reception time of the packet transmitted by the \(i\)th AN, and then the distances from the AUV to the \(i\)th AN, which is denoted by \(r_{i}\). are estimated based on the time difference between transmission and reception. Meanwhile, the actual distance between the AUV and the \(i\)th AN at time \(t\) is presented by \(d_{i}(t)\). If we assume that the AUV performs localization after receiving the packets from all the ANs, i.e., \(t_{3}\), the distance vector used for the localization is \([r_{1},r_{2},r_{3}]\). It is noteworthy that due to the change of the AUV's position, \(r_{1}\) and \(r_{2}\) are not equal to the actual distances \(d_{1}(t_{3})\) and \(d_{2}(t_{3})\) respectively. In other words, because the AUV continuously moves during the MAC delay, the range cannot be measured accurately. Naturally, the longer the MAC delay and the faster the AUV movement, the greater the ranging error, resulting in degradation of localization performance.  
Figure 1: Communication scheme for the ranging.  
Figure 2: The impact of MAC delay and the AUV movement on the localization initiated by ANs’ transmission.