In this paper, an intelligent reflecting surface (IRS)-aided downlink cloud radio access network (C-RAN) is studied, wherein the baseband unit (BBU) pool communicates with the users through multiple remote radio heads (RRH). Specifically, the IRSs are deployed to assist both the wireless fronthaul link and the user access link. Due to the limitation of the fronthaul capacity, the BBU pool deals with the baseband signals by point-to-point compression or multivariate compression and sends the quantization bits to the RRHs via the fronthaul link. Considering the imperfect channel state information (CSI) for both the conventional direct channel and the IRS cascaded channel, we investigate the robust design by jointly optimizing the transmit beamformers for BBU pool and RRHs, the passive beamformer for IRSs and the fronthaul compression under point-to-point compression and multivariate compression, respectively. Specifically, we derive the closed-form user downlink rate and the achievable rate for the wireless fronthaul under imperfect CSI. To tackle the non-convex problem, we adopt the successive convex approximation (SCA) approach and transform it into a tractable form. Then utilizing the alternating optimization (AO) approach, we divide the transformed problem into three sub-problems which can be efficiently solved. Furthermore, since the fronthaul link and access link work on time division duplex (TDD) mode, we also discuss the joint optimization taking the time allocation into consideration. Finally, via numerical results, the effectiveness of the proposed robust design is verified.
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