This work considers an asynchronous $\textsf{K}_a$-active-user unsourced multiple access channel (AUMAC) with the worst-case asynchronicity. The transmitted messages must be decoded within $n$ channel uses, while some codewords are not completely received due to asynchronicities. We consider a constraint of the largest allowed delay of the transmission. The AUMAC lacks the permutation-invariant property of the synchronous UMAC since different permutations of the same codewords with a fixed asynchronicity are distinguishable. Hence, the analyses require calculating all $2^{\textsf{K}_a}-1$ combinations of erroneously decoded messages. Moreover, transmitters cannot adapt the corresponding codebooks according to asynchronicity due to a lack of information on asynchronicities. To overcome this challenge, a uniform bound of the per-user probability of error (PUPE) is derived by investigating the worst-case of the asynchronous patterns with the delay constraint. Numerical results show the trade-off between the energy-per-bit and the number of active users for different delay constraints. In addition, although the asynchronous transmission reduces interference, the required energy-per-bit increases as the receiver decodes with incompletely received codewords, compared to the synchronous case.

翻译：本文考虑一个具有最坏情况异步性的异步$\textsf{K}_a$活跃用户无源多址接入信道（AUMAC）。传输消息必须在$n$个信道使用内解码，但由于异步性，部分码字未能完整接收。我们考虑了传输最大允许延迟的约束条件。AUMAC 缺乏同步 UMAC 的置换不变性，因为对于固定异步性的相同码字，不同置换是可区分的。因此，分析需要计算所有$2^{\textsf{K}_a}-1$种错误解码消息组合。此外，由于缺乏异步性信息，发送方无法根据异步性调整相应的码本。为克服这一挑战，通过研究具有延迟约束的异步模式的最坏情况，推导出每用户错误概率（PUPE）的一致界。数值结果表明，在不同延迟约束下，每比特能量与活跃用户数之间存在权衡。此外，尽管异步传输减少了干扰，但与同步情况相比，由于接收端使用不完整接收的码字进行解码，所需的每比特能量更高。