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In this paper, we propose a novel multi-hop relaying scheme to improve the performance and coverage of impulseradio-based ultra-wideband (IR-UWB) systems. With regard to a simple practical realization, we focus on a non-coherent system setup in conjunction with amplify-and-forward (A&F) relaying. In particular, we propose to employ a multiple-differential encoding scheme at the source node and single differential decoding at each relay and at the destination node, respectively, so as to efficiently limit intersymbol-interference effects at the destination node. For a dual-hop system we derive a closed-form expression for the signal-to-noise ratio (SNR) at the destination node, and for the general multi-hop case we provide a simple recursive formula for SNR calculation. Based on these SNR results, we obtain a closed-form expression for the optimal transmit power allocation to the source node and the relay for a dual-hop system and a simple recursive suboptimal power allocation scheme for the multi-hop case, which permits a semi-distributed implementation with limited feedback between nodes. Simulation results illustrate the excellent performance of the proposed multiple-differential encoding scheme with A&F relaying for both uncoded and coded transmission compared to various alternative coherent and non-coherent schemes based on A&F relaying and decode-andforward (D&F) relaying. Furthermore, our simulations confirm the (near-)optimal performance of the proposed power allocation solutions.Index Terms-Ultra-wideband (UWB) communications, impulse radio (IR), amplify and forward (A&F) relaying, multiple hops, differential encoding, performance analysis, power allocation.