Fundamental Frequency and its Harmonics Model: A Robust Method of Estimation

In this paper we have proposed a novel robust method of estimation of the unknown parameters of a fundamental frequency and its harmonics model. Although the least squares estimators (LSEs) or the periodogram type estimators are the most efficient estimators, it is well known that they are not robust. In presence of outliers the LSEs are known to be not efficient. In presence of outliers, robust estimators like least absolute deviation estimators (LADEs) or Huber’s M-estimators (HMEs) may be used. But implementation of the LADEs or HMEs are quite challenging, particularly if the number of component is large. Finding initial guesses in the higher dimensions is always a non-trivial issue. Moreover, theoretical properties of the robust estimators can be established under stronger assumptions than what are needed for the LSEs. In this paper we have proposed novel weighted least squares estimators (WLSEs) which are more robust compared to the LSEs or periodogram estimators in presence of outliers. The proposed WLSEs can be implemented very conveniently in practice. It involves in solving only one non-linear equation. We have established the theoretical properties of the proposed WLSEs. Extensive simulations suggest that in presence of outliers, the WLSEs behave better than the LSEs, periodogram estimators, LADEs and HMEs. The performance of the WLSEs depend on the weight function, and we have discussed how to choose the weight function. We have analyzed one synthetic data set to show how the proposed method can be used in practice.