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A discrete-time equivalent system model is derived for differential and transmitted reference (TR) ultra-wideband (UWB) impulse radio (IR) systems, operating under heavy intersymbol-interference (ISI) caused by multipath propagation. In the systems discussed, data is transmitted using differential modulation on a frame-level, i.e., among UWB pulses. Multiple pulses (frames) are used to convey a single bit. Time hopping and amplitude codes are applied for multi user communications, employing a receiver front-end that consists of a bank of pulse-pair correlators. It is shown that these UWB systems are accurately modeled by second-order discrete-time Volterra systems. This proposed nonlinear equivalent system model is the basis for developing optimal and suboptimal receivers for differential UWB communications systems under ISI. As an example, we describe a maximum likelihood sequence detector with decision feedback, to be applied at the output of the receiver front-end sampled at symbol rate, and an adaptive inverse modeling equalizer. Both methods significantly increase the robustness in presence of multipath interference at tractable complexity.