Cramér–Rao Bounds for a Coupled Mixture of Polynomial Phase and Sinusoidal FM Signals

This letter introduces a new coupled mixture of polynomial phase signal (PPS) and sinusoidal frequency modulated (FM) signal, motivated by real-world applications, for example, contactless linear encoders. Specifically, the coupling is introduced to express the sinusoidal FM frequency as a function of the PPS-related parameters. Given the coupling mixture, it generalizes two existing models: the pure PPS model and the independent mixture model. Performance bounds of parameter estimation for the coupled mixture model are established in terms of the Cramér-Rao bound (CRB). Unlike the pure PPS case, the derived CRB shows its dependence on the PPS-related and sinusoidal FM-related parameters due to the coupling mixture. On the other hand, the derived CRBs for the PPS-related parameters are lower than their counterparts of the independent mixture model, as the sinusoidal FM frequency provides additional information on the PPS parameters.