Many applications involve sparse signals with unknown, continuous parameters; a common example is a signal consisting of several sinusoids of unknown frequency. Applying compressed sensing techniques to these signals requires a highly oversampled dictionary for good approximation, but these dictionaries violate the RIP conditions and produce inconsistent results. We consider recovering both a sparse vector and parameter perturbations from an initial set of parameters. Joint recovery allows for accurate reconstructions without highly oversampled dictionaries. Our algorithm for sparse recovery solves a series of linearized subproblems. Recovery error for noiseless simulated measurements is near zero for coarse dictionaries, but increases with the oversampling. This technique is also used to reconstruct Radio Frequency data. The algorithm estimates sharp peaks and transmitter frequencies, demonstrating the potential practical use of the algorithm on real data.