Short time single polynomial phase signal using legendre function

We model non stationary signals by assuming that the phase and the amplitude are both a polynomial function of time on a short finite interval. The used functions are normalized Legendre polynomial. Applying the model to the instantaneous frequency instead of the phase and to a short time window allows the estimation with a second order polynomial only. This paper presents first results, where we study a single component model on a single short time window only. We set the model origins at time window center in order to minimize the estimation error. A maximum likelihood estimate of the parameter model leads to a non linear equation system in ℝ
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we solve by a simulated annealing technique. The appropriate Cramer-Rao bounds (CRB) are derived. Monte Carlo simulations illustrate the good performance of the proposed algorithm, which yields estimates close to the CRB, even for short time windows of 33 samples and for a non zero initial phase.