Accuracy of spike-train Fourier reconstruction for colliding nodes

We consider a signal reconstruction problem for signals F of the form F(x) = Σ^d_j=1 a_jδ(x-x_j) from their Fourier transform F(F)(s) = ∫^∞_-∞ F(x)e^-isxdx. We assume F(F)(s) to be known for each s ε [-N,N] with an absolute error not exceeding ε > 0. We give an absolute lower bound (which is valid with any reconstruction method) for the “worst case” reconstruction error of F from F(F) for situations where the x_j nodes are known to form an I elements cluster contained in an interval of length h <;<; 1. Using “decimation” algorithm of [6], [7] we provide an upper bound for the reconstruction error, essentially of the same form as the lower one. Roughly, our main result states that for h of order 1/N ϵ^1/2l-1 the worst case reconstruction error of the cluster nodes is of the same order 1/N ϵ^1/2l-1, and hence the inside configuration of the cluster nodes (in the worst case scenario) cannot be reconstructed at all. On the other hand, decimation algorithm reconstructs F with the accuracy of order 1/N ϵ^1/2l.