  The need of reconstructing discrete-valued sparse signals from few
measurements, that is solving an undetermined system of linear equations,
appears frequently in science and engineering. Whereas classical compressed
sensing algorithms do not incorporate the additional knowledge of the discrete
nature of the signal, classical lattice decoding approaches such as the sphere
decoder do not utilize sparsity constraints.
  In this work, we present an approach that incorporates a discrete values
prior into basis pursuit. In particular, we address unipolar binary and bipolar
ternary sparse signals, i.e., sparse signals with entries in $\{0,1\}$,
respectively in $\{-1,0,1\}$. We will show that phase transition takes place
earlier than when using the classical basis pursuit approach and that,
independently of the sparsity of the signal, at most $N/2$, respectively
$3N/4$, measurements are necessary to recover a unipolar binary, and a bipolar
ternary signal uniquely, where $N$ is the dimension of the ambient space. We
will further discuss robustness of the algorithm and generalizations to signals
with entries in larger alphabets.
