  A general class of nonlinear Least Square Error (LSE) precoders in multi-user
multiple-input multiple-output systems is analyzed using the replica method
from statistical mechanics. A single cell downlink channel with $N$ transmit
antennas at the base station and $K$ single-antenna users is considered. The
data symbols are assumed to be iid Gaussian and the precoded symbols on each
transmit antenna are restricted to be chosen from a predefined set
$\mathbb{X}$. The set $\mathbb{X}$ encloses several well-known constraints in
wireless communications including signals with peak power, constant envelope
signals and finite constellations such as Phase Shift Keying (PSK). We
determine the asymptotic distortion of the LSE precoder under both the Replica
Symmetry (RS) and the one step Replica Symmetry Breaking (1-RSB) assumptions.
For the case of peak power constraint on each transmit antenna, our analyses
under the RS assumption show that the LSE precoder can reduce the peak to
average power ratio to 3dB without any significant performance loss. For PSK
constellations, as $N/K$ grows, the RS assumption fails to predict the
performance accurately and therefore, investigations under the 1-RSB assumption
are further considered. The results show that the 1-RSB assumption is more
accurate.
