Maximum Likelihood Parameter Estimation of Superimposed Signals by Dynamic Programming

The problem of fitting a model composed of a number of superimposed signals to noisy data using the maximum likelihood criterion is considered. It is shown, using the Cramer-Rao bound for the estimation accuracy, that in many instances useful models for the composite signal can be restricted without loss of generality to component signals that directly interact only with one or two of their closest neighbors in parameter space. It is shown that for such models, the global extremum of the criterion can be found efficiently by dynamic programming. The computation requirements are linear in the number of signals, rather than exponential as in the case of exhaustive search. For example, for sinusoids the computation is O(mq⁶), where m is their number, and q is the resolution frequency, amplitude, and phase. The technique applies for arbitrary sampling of the signals. The dynamic programming method is easily adapted to determine the number of signals as well, as demonstrated using the minimum description length (MDL) principle. Computer simulation results of several examples are presented.