On finite alphabet compressive sensing

This paper considers the problem of compressive sensing over a finite alphabet, where the finite alphabet may be inherent to the nature of the data or a result of quantization. We show that there are significant benefits to analyzing the problem while incorporating its finite alphabet nature, versus ignoring it and employing a conventional real alphabet based toolbox. Specifically, when the alphabet is finite, our techniques have a lower sample complexity compared to real-valued compressive sensing for low levels of sparsity, facilitate constructive designs of sensing matrices based on coding-theoretic techniques, and allow for lesser amount of data storage.