Noise analysis of an approach for frequency identification

This paper 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 controller (IMC). In steady state, these states can be mapped nonlinearly into two time invariant variables: the frequency and the magnitude or energy of the periodic signal or disturbance. This paper provides an analysis of the 'measurement' of this frequency in the presence of white noise. In the case of an additive white noise, we prove this approach is unbiased and a formula to calculate the variance of the estimated frequency is given. Two limit cases are also provided, one for high SNR and the other for low SNR. The simulations verify the validity of approximations used in our noise analysis.