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In ultra wide band (UWB) positioning systems, the key problem is to detect the timing of transmitted UWB pulses. Narrow pulses are required for high precision of positioning, thus demanding high sampling rate. To alleviate the difficulty for analog-to-digital converters (ADC) and utilize the feature of time sparsity of UWB pulses, a compressed sensing based scheme is proposed, in which the received signal is mixed using distributed amplifiers, sampled using a bank of low rate ADCs and then reconstructed. To combat the detrimental effect of Gaussian noise, Bayesian compressed sensing is applied, using a structure similar to the iterative decoder for turbo codes to exploit the redundancy in time, history and space. Numerical simulation results demonstrate that the Bayesian compressed sensing using redundancy achieves substantial performance gain (UWB pulse recovery rate and positioning precision), compared with traditional compressed sensing approaches.