Abstract: Systems and methods relating to estimating data symbols encoded in a received signal that has been transmitted at a faster than Nyquist rate. The present invention uses a heuristic method for non-convex problems and involves an input matrix and received samples from the received signal. These are preconditioned and the preconditioned input matrix is factorized. The method then iterates a three-step process that estimates the sequence of data symbols based on the current estimate, the preconditioned input matrix, the preconditioned samples vector, a multiplier vector, and an auxiliary vector. The process then calculates the next multiplier vector and the next auxiliary vector. If the result indicates a minimum as compared to the best estimate, then the result is used as the best estimate. Multiple iterations of the process are performed, and the multiple iterations are repeated for multiple random initializations of the estimate.