We address the problem of sampling and reconstruction of sparse signals with finite rate of innovation. We derive general conditions under which perfect reconstruction is possible for sampling kernels satisfying Strang-Fix conditions. Previous results on the subject consider two particular cases; when the kernel is able to reproduce (complex) exponentials, or when it has the polynomial reproduction property. In this work we extend such analysis to the case where both properties could be found in the sampling kernel and show that the former two sitations can be regarded as special cases. As a result of our analysis, we provide general conditions under which perfect recovery in the noiseless case is possible. In practice, a given sampling kernel might not satisfy Strang-Fix conditions. When dealing with arbitrary sampling kernels we propose a unified view for sampling and reconstruction in the frequency domain. Our formulation generalizes previous approaches and provides new insights for devising optimal reconstruction schemes. We also propose a novel algorithm for denoising treating the problem as a particular instance of structured low-rank approximation. Finally, we provide some numerical experiments and a comparison between different state-of-the-art methods showing the improved estimation performance of the proposed approach.​