  We consider the reconstruction problem in compressed sensing in which the
observations are recorded in a finite number of bits. They may thus contain
quantization errors (from being rounded to the nearest representable value) and
saturation errors (from being outside the range of representable values). Our
formulation has an objective of weighted $\ell_2$-$\ell_1$ type, along with
constraints that account explicitly for quantization and saturation errors, and
is solved with an augmented Lagrangian method. We prove a consistency result
for the recovered solution, stronger than those that have appeared to date in
the literature, showing in particular that asymptotic consistency can be
obtained without oversampling. We present extensive computational comparisons
with formulations proposed previously, and variants thereof.
