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We consider a low data rate non-coherent impulse radio ultra-wideband (IR-UWB) system based on binary pulse position modulation (2-PPM), which applies sparse codes to enable code division multiple access (CDMA). The suitability of sparse codes (i.e., codes with a low code weight) is investigated considering multi-user interference (MUI) and multipath propagation. The decoding of the particular CDMA code takes place after non-coherent combining. Different sparse codes such as time hopping (TH) random codes, TH codes constructed from M-sequences, and optical orthogonal codes are employed. We propose a semi-analytical performance analysis method using Gaussian approximation and the statistics of the code collisions to obtain the bit error rate expressions, which is much more accurate than the code correlation function. The multiple access performance is analyzed in terms of the signal-to-noise ratio (SNR) as well as the number of supported users.