  The estimation of parameters in a linear model is considered under the
hypothesis that the noise, with finite second order statistics, can be
represented in a given deterministic basis by random coefficients. An extended
underdetermined design matrix is then considered and an estimator of the
extended parameters is proposed with minimum $l_1$ norm. It is proved that if
the noise variance is larger than a threshold, which depends on the unknown
parameters and on the extended design matrix, then the proposed estimator of
the original parameters dominates the least-squares estimator in the sense of
the mean square error. A small simulation illustrates the behavior of the
proposed estimator. Moreover it is shown experimentally that the proposed
estimator can be convenient even if the design matrix is not known but only an
estimate can be used. Furthermore the noise basis can eventually be used to
introduce some prior information in the estimation process. These points are
illustrated by simulation by using the proposed estimator for solving a
difficult inverse ill-posed problem related to the complex moments of an atomic
complex measure.
