On a reverse ℓ2-inequality for sparse circular convolutions

In this paper we show that convolutions of sufficiently sparse signals always admit a non-zero lower bound in energy if oversampling of its Fourier transform is employed. This bound is independent of the signals and the ambient dimension and is determined only be the sparsity of both input signals. This result has several implications for blind system and signal identification and detection, noncoherent communication of sporadic and short-message type user data and strategies for its compressive reception. Furthermore, we give some first insights into the combinatorial nature of this problem, its scaling behavior and present numerical results as well.