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A characteristic function method is proposed for precisely calculating the bit-error probability of time-hopping (TH) ultra-wideband (UWB) systems with multiple-access interference in an additive white Gaussian noise environment. The analytical expressions are validated by simulation and used to assess the accuracy of the Gaussian approximation. The Gaussian approximation is shown to be inaccurate for predicting bit-error rates (BERs) for medium and large signal-to-noise ratio (SNR) values. The performances of TH pulse position modulation (PPM) and binary phase-shift keying (BPSK) modulation schemes are accurately compared in terms of the BER. It is shown that the BPSK system outperforms the binary PPM system for all values of SNR. The sensitivity of the performance of the modulation schemes to the system parameters is also addressed through numerical examples.