Given a sequence of sinusoidal signals s[n] with additive noise in the time domain, a fundamental problem is to estimate its frequency. It has a wide range of applications, such as biomedical signal processing and radar analysis.
The maximum likelihood estimate for the frequency is a reliable and widely used method. It finds a peak (maximum) magnitude in the frequency domain. A standard method consists of two stages: coarse search and fine search. In the coarse search, the discrete Fourier transform (DFT) is first applied to the sequence of signals to obtain another sequence S[k] in the frequency domain. Then an index (integer) kp is determined so that the magnitude of S[kp] reaches the peak among S[k]. This gives a rough estimate, since the sequence is discrete. In the fine search, an adjustment d, between -0.5 and 0.5, to the index kp is found to improve the estimate obtained by the coarse search.
This paper concerns the fine search stage. In the absence of noise, using interpolation, the authors derive a closed formula for the adjusted signal S[kp+d] (the key equation (9) in the paper). Applying the key equation, the authors unify various existing estimators and present closed formulas of the adjustment d for those estimators. The results are very interesting. Theoretically, this paper provides a unified view and better understanding of various existing methods. Practically, the closed formulas can be used to compute the adjustment. It seems that the key equation may be used to develop new methods for finding the adjustment. The authors, however, by applying the key equation, point out that the possibility of new significant estimators is limited due to some constraints.
This paper is very useful for understanding and comparing existing frequency estimators. The limitation of this work is that the key equation assumes the absence of noise. Future work should integrate noise into the analysis.