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A characteristic function method is proposed for precisely calculating the bit error probability of time-hopping ultra-wideband systems with multiple access interference. The analytical expressions are validated by simulation and used to assess the accuracy of the Gaussian approximation. The sensitivity of the performance of the modulation schemes to the system parameters is also addressed through numerical examples. The performances of time-hopping pulse position modulation and binary phase-shift keying modulation schemes are accurately compared based on the new analytical results.