We consider separation of finite alphabet signals in the instantaneous case with underdetermined known mixing matrix. This problem occurs in communication systems, image processing and operations research amongst others. We consider the noiseless case. However, in contrast to these papers, we do not restrict our attention to binary alphabets, but address the case of alphabets with arbitrary finite sizes. In addition, we do not assume that the signal to reconstruct is necessarily sparse. Our approach then involves a suitable sparse transform so as to introduce sparsity constraints. The resulting problem can then be relaxed into 1-norm optimization. The recovery performance of the approach is established for generic random mixing matrices satisfying appropriate asymptotic properties.