  In this paper, a cyclic-prefixed single-carrier (CPSC) transmission scheme
with phase shift keying (PSK) signaling is presented for broadband wireless
communications systems empowered by a reconfigurable intelligent surface (RIS).
In the proposed CPSC-RIS, the RIS is configured according to the transmitted
PSK symbols such that different cyclically delayed versions of the incident
signal are created by the RIS to achieve multipath diversity. A practical and
efficient channel estimator is developed for CPSC-RIS and the mean square error
of the channel estimation is expressed in closed-form. We analyze the bit error
rate (BER) performance of CPSC-RIS over frequency-selective Nakagami-$m$ fading
channels. An upper bound on the BER is derived by assuming the
maximum-likelihood detection. Furthermore, by resorting to the concept of index
modulation (IM), we propose an extension of CPSC-RIS, termed CPSC-RIS-IM, which
enhances the spectral efficiency. In addition to conventional constellation
information of PSK symbols, CPSC-RIS-IM uses the full permutations of cyclic
delays caused by the RIS to carry information. A sub-optimal receiver is
designed for CPSC-RIS-IM to aim at low computational complexity. Our simulation
results in terms of BER corroborate the performance analysis and the
superiority of CPSC-RIS(-IM) over the conventional CPSC without an RIS and
orthogonal frequency division multiplexing with an RIS.
