  In this letter, we consider a problem of reconstructing an unknown discrete
signal taking values in a finite alphabet from incomplete linear measurements.
The difficulty of this problem is that the computational complexity of the
reconstruction is exponential as it is. To overcome this difficulty, we extend
the idea of compressed sensing, and propose to solve the problem by minimizing
the sum of weighted absolute values. We assume that the probability
distribution defined on an alphabet is known, and formulate the reconstruction
problem as linear programming. Examples are shown to illustrate that the
proposed method is effective.
