Sparse Recovery Guarantees of Periodic Signals with Nested Periodic Dictionaries

Periodic signals admit sparse representations in nested periodic dictionaries (NPDs). While sparse recovery algorithms rooted in the theory of compressive sensing can successfully recover their underlying periods, existing recovery conditions derived for random dictionaries are of limited use in this context as they fail to explain the achievability results of said algorithms. In addition, provable achievability guarantees specific to NPDs have been heretofore lacking. In this paper, we derive exact recovery conditions for sparse periodic signals by leveraging prior information about the structure of NPDs. As instances of such dictionaries, we investigate the achievability conditions for the Farey and the Ramanujan Periodicity Transform dictionaries. Our numerical results demonstrate that the newly derived conditions can provide guarantees for exact recovery with the Farey dictionary, and in turn for exact period estimation, for large enough data lengths.