Noise analysis of an algorithm for uncertain frequency identification

This note presents a noise analysis for an algorithm to identify the uncertain frequency of periodic signals or disturbances. This algorithm is based on the time-varying states of an internal model principle controller which can be mapped nonlinearly to the frequency and the magnitude or energy of the periodic signal or disturbance. This note provides an analysis of the ´measurement´ of this frequency in the presence of white noise. In the case of an additive white noise, we develop some formulas to calculate the means and variances of the measured difference between the true frequency and nominal frequency for high and low signal-to-noise ratio (SNR). When an integral controller is used to eliminate this difference, we prove that this frequency estimation is unbiased. The formulae to calculate the mean and variance are also given for the output of the integral controller. The simulations verify the validity of approximations used in our noise analysis.