Analysis of Estimation Bias for Direct Frequency Estimators of Complex Sinusoid

Frequency estimation for complex sinusoid is a fundamental problem in signal processing. A simple and effective way is to directly interpolate the discrete Fourier transform coefficients around the peak of the magnitude spectrum. The shortcoming of this method is the non-uniform estimation bias across the frequency. This paper theoretically analyzes the estimation bias for some of the most popular and well-performed estimators and shows that the bias can be accurately predicted by a polynomial. Applying the polynomial, three additional results are derived. First, the optimal scaling factor to reduce the bias is found. Second, the theoretical expression for the threshold between the bias limiting region and Cram'er Rao bound limiting region is derived. Third, we proposed a new estimation method that uses the roots of the polynomial to reduce the bias. Experiments show that the new method can reduce the bias by more than three orders when the number of samples are more than 64.