  Reconfigurable intelligent surfaces (RISs) represent a pioneering technology
to realize smart electromagnetic environments by reshaping the wireless
channel. \textcolor[rgb]{0,0,0}{Jointly designing the transceiver and RIS
relies on the channel state information (CSI), whose feedback has not been
investigated in multi-RIS-assisted frequency division duplexing systems.} In
this study, the limited feedback of the RIS-assisted wireless channel is
examined by capitalizing on the ability of the RIS in channel customization.
\textcolor[rgb]{0,0,0}{By configuring the phase shifters of the surfaces using
statistical CSI, we customize a sparse channel in rich-scattering environments,
which significantly reduces the feedback overhead in designing the transceiver
and RISs. Since the channel is customized in terms of singular value
decomposition (SVD) with full-rank, the optimal SVD transceiver can be
approached without a matrix decomposition and feeding back the complete channel
parameters. The theoretical spectral efficiency (SE) loss of the proposed
transceiver and RIS design is derived by considering the limited CSI
quantization. To minimize the SE loss, a bit partitioning algorithm that splits
the limited number of bits to quantize the CSI is developed.} Extensive
numerical results show that the channel customization-based transceiver with
reduced CSI can achieve satisfactory performance compared with the optimal
transceiver with full CSI. Given the limited number of feedback bits, the bit
partitioning algorithm can minimize the SE loss by adaptively allocating bits
to quantize the channel parameters.
