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In this paper, we derive and analyze a Bayesian detector for a multiple access ultra-wideband (UWB) system, which only requires the knowledge of the signature code of the user of interest. The receiver structure is equivalent to a single user matched filter (MF) receiver followed by an adaptive threshold. Since the multiple access interference (MAI) is not Gaussian, this detector outperforms the conventional MF detector at the expense of the additional complexity necessary for calculating the adaptive threshold. In the two-user case, as the AWGN tends to zero, the probability of error of the Bayesian detector goes to zero for any finite-length time-hopping codes. This is in contrast to the MF receiver, which requires infinitely long codes to achieve the same result. Finally, we propose a linear MMSE adaptive receiver to mitigate MAI with low complexity.