  We compare the maximum achievable rates in single-carrier and OFDM modulation
schemes, under the practical assumptions of i.i.d. finite alphabet inputs and
linear ISI with additive Gaussian noise. We show that the Shamai-Laroia
approximation serves as a bridge between the two rates: while it is well known
that this approximation is often a lower bound on the single-carrier achievable
rate, it is revealed to also essentially upper bound the OFDM achievable rate.
We apply Information-Estimation relations in order to rigorously establish this
result for both general input distributions and to sharpen it for commonly used
PAM and QAM constellations. To this end, novel bounds on MMSE estimation of PAM
inputs to a scalar Gaussian channel are derived, which may be of general
interest. Our results show that, under reasonable assumptions, optimal
single-carrier schemes may offer spectral efficiency significantly superior to
that of OFDM, motivating further research of such systems.
