  The reflection characteristics of a reconfigurable intelligent surface (RIS)
depend on the phase response of the constituent unit cells, which is
necessarily frequency dependent. This paper investigates the role of an RIS
constituting unit cells with different phase-frequency profiles in a wide-band
orthogonal frequency division multiplexing (OFDM) system to improve the
achievable rate. We first propose a mathematical model for the phase-frequency
relationship parametrized by the phase-frequency profile's slope and
phase-shift corresponding to a realizable resonant RIS unit cell. Then,
modelling each RIS element with $b$ control bits, we propose a method for
selecting the parameter pairs to obtain a set of $2^b$ phase-frequency
profiles. The proposed method yields an RIS design that outperforms existing
designs over a wide range of user locations in a single-input, single-output
(SISO) OFDM system. We then propose a low-complexity optimization algorithm to
maximize the data rate through the joint optimization of (a) power allocations
across the sub-carriers and (b) phase-frequency profile for each RIS unit cell
from the available set. The analysis is then extended to a multi-user
multiple-input single-output (MISO) OFDM scenario. Numerical results show an
improvement in the coverage and achievable rates under the proposed framework
as compared to single-slope phase-frequency profiles.
