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In this paper, we are interested in resource allocation strategies for wireless time-hopping ultra-wide band (TH-UWB) communications with multiple rate capabilities between users. Multiple rates are achieved by assigning different processing gains, i.e. Nf, to users. For this purpose, the multiple-access interference (MAI) variance accounting for multi-rate is needed. It is a challenging task due to the lack of a suitable closed-form expression for the MAI variance in a multi-rate context. We further study the multi-rate resource allocation problem in uplink TH-UWB systems for which an optimal search cannot be envisaged due to the exponential complexity induced. Our contribution lies in three-fold: i) A new intercode correlation expression accounting for multi-rate communications is derived, and the variance of the MAI averaging over the codes is obtained. ii) The multi-rate resource allocation problem is tackled by relaxing the integer constraint on the processing gains and modeled via a signomial programming problem. iii) Based on this, a branch and bound (BB) algorithm is derived for the allocation of the processing gains in TH-UWB systems. We also propose a really simple heuristic with linear complexity for the Nf allocation. We show that the algorithm proposed outperforms the BB algorithm in average throughput and average starvation rate.