# An Efficient Single-Anchor Localization Method Using Ultra-Wide Bandwidth Systems
## 3 Single-Anchor Localization Method
### (Quasi-)simultaneous Response  
In a common multi-agent measurement process [47], the anchor can only accept one message each time the receiver is turned on. Thus, for an \(N_{\text{a}}\)-agent localization system, the anchor needs to open the receiver \(N_{\text{a}}\) times to receive massages of all agents, which is time-consuming and inefficient. We next introduce the concept of (quasi-)simultaneous response, which enables the anchor to receive all the agents' responses in one estimated CIR and is the key enabler of the proposed SAL method. With the proposed method, the anchor only needs to open the receiver one time to serve \(N_{\text{a}}\) agents. The main advantage of the proposed method is that the anchor serves all the agents simultaneously. For brevity, we introduce our method by taking the two-agent scenario as an example, which can be extended to multi-agent cases. Note that our method is not limited to a specific type of UWB chip or device, we only need to follow the corresponding manual to adjust the operating process. As shown in Figure 1b, the anchor broadcasts the first two messages to initiate the measurement process. In the payload of the Init message, the anchor writes the fixed time delay \(t_{0}^{I}\) between the Init and Sync messages. The anchor also assigns a fixed delay \(\delta_{i},i\in\mathcal{N}_{\text{a}}\) to agent \(i\) and writes them into the payload. The purpose of introducing \(\delta_{i}\) is to distinguish messages from different agents. Note that the capacity of the proposed SAL system is affected by \(\delta_{i}\), since \(\delta_{i}\) represents an extra delay. However, MPCs may overlap with subsequent responses, which requires us to assgin larger delays and therefore limit the capacity of the system. In order to serve more agents in one waveform window with no waveform overlapping, we take the value of \(\delta_{i}\) in tens of nanoseconds considering the general CIR register can receive a waveform with the period of 1016 nanoseconds [46]. An illustration of the received CIR is shown in Figure 3.  
The proposed (quasi-)simultaneous response has similar concepts to the work in [35], but has different advantages. In [35], after multiple anchors send messages (quasi-)simultaneously, one agent receives the responses in a single estimated CIR, and uses the TDOA to locate itself. Since the anchor cannot know the agent's location, this is an agent-based approach. In our method, by allowing agents to respond (quasi-)simultaneously, a single anchor can complete measurement process of all agents, thus the anchor can implement multi-agent localization, which is an anchor-based approach. After receiving two messages, the agent can determine the clock drift ratio between its clock and the anchor's clock, given by  
\[\frac{\xi_{0}}{\xi_{i}}=\frac{t_{0}^{I}}{t_{i}^{I}}. \tag{6}\]  
Then, the agent \(i\) sends a Resp message after a predetermined delay \(\Delta r+\delta_{i}\), where the clock drift ratio (6) and the delay \(t_{i}^{II}\) are written in the payload. When the anchor receives all the Resp messages, the measuring process completes. We denote that the first response corresponds to \(i=1\). Thus, the distance between agent 1 and the anchor is \(d_{1}=c\frac{t_{0}^{II}-(\xi_{0}/\xi_{i})t_{i}^{II}}{2}\). The distance between the agent \(i\) (\(i\
eq 1\)) and the anchor is given by  
\[d_{i}=c\frac{t_{0}^{II}+\Delta t_{1i}-(\xi_{0}/\xi_{i})t_{i}^{II}}{2}. \tag{7}\]  
In this way, the clock drift ratio can be estimated and the asynchronism effect can be mitigated, which leads to a high-accuracy distance estimation. Note that the estimation of the clock drift ratio is performed during each ranging process, which can reduce the effect of the clock drift in each measurement. The main advantages of this (quasi-)simultaneous response method are as follows:* The anchor is able to extract the arrival time-stamp and phase of all agents in one CIR waveform, which significantly improves the efficiency of the TOA estimation.