Cramer-Rao bounds for coupled harmonics in noise

When a signal consisting of harmonics (and possibly stationary mixing noise) is passed through a non-linearity, the output consists of frequency and phase coupled harmonics. There exist several techniques (e.g. based on periodograms, poly-periodograms and their slices) which may be used to detect and estimate the coupled frequencies (and then to infer properties of the non-linearity). In order to assess the performance of these algorithms, we derive the corresponding Cramer-Rao bounds, constrained by the coupling equations. Special cases such as quadratic, cubic and self coupling are then considered. These results are used to evaluate the asymptotic relative efficiency of some previously proposed algorithms. Several extensions of the basic results are also given.