The collaboration of multiple Reconfigurable Intelligent Surfaces (RISs) and Access Points (APs) enjoys advantages of capacity enhancement, power saving, etc., making the RIS-assisted cell-free network an important architecture for future communications. Similar to most existing works on RIS-assisted communications, the multi-hop link among RISs, i.e., the reflecting link including more than one RISs, is usually ignored in RIS-assisted cell-free networks. In these scenarios, however, since multiple RISs are closely deployed, we find that the multi-hop channels should not be simply ignored due to their potential for capacity improvement. Unfortunately, to the best of our knowledge, there is no work exploring the multi-hop transmission of RIS-assisted cell-free networks. To fill in this blank, we investigate the multi-hop transmission of RIS-assisted cell-free networks, including the multi-hop channel model and the corresponding beamforming design. Specifically, we propose a general multi-hop transmission model, which takes the direct links, single-reflecting links, and multi-hop links into account. Based on this model, we formulate a beamforming design problem in an RIS-assisted cell-free network, which allows us to maximize the multi-user sum-rate with considering the impact of multi-hop channels. To address the non-convexity of the formulated problem, a joint active and passive beamforming scheme is proposed to solve the problem. Particularly, by utilizing fractional programming, we decouple the coupled beamforming parameters in the problem, and then these parameters are alternately optimized until the convergence of the sum-rate. Simulation results verify that the consideration for multi-hop links is necessary, and the capacity performance of the proposed scheme is 20% higher than those of the existing schemes.
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