The Cramer-Rao bound for the estimation of noisy phase signals

Non-stationary signals, i.e. signals whose spectral content varies with time, are very common in many signal processing applications. Examples include, but are not limited to, communications, sonar, radar, vibration, ultrasound imaging, speech and biological signals. Many methods of analysis of non-stationary signals focus on nonparametric time-frequency distributions. However these suffer from the well known time-frequency resolution problem and an alternative to alleviate this problem lies in the use of parametric methods. Monocomponent polynomial phase signals (pps), i.e. signals whose phase is modeled by a polynomial and of constant amplitude , have been first studied by Djurić and Kay [1]. Later, Peleg, Porat and Friedlander [2, 3] introduced the so-called polynomial phase transform as a tool for estimating the parameters of pps. Most of the work cited above assumes that the signal phase fits perfectly to a polynomial of a certain degree. However, in a practical situation this is not true and a proper model should include a noise component on the phase, i.e.