Phase correction of discrete Fourier transform coefficients to reduce frequency estimation bias of single tone complex sinusoid

• A phase term in DFT coefficient is a cause of bias in frequency estimation.
• Propose three different ways to correct the phase and reduce bias.
• Derive the amount of bias reduction that can be achieved by phase correction.
• Derive frequency threshold between phase correction and post-estimation correction.
• Propose a hybrid estimator which switches corrections at threshold.

Frequency estimation for single tone complex sinusoid is a fundamental problem in signal processing. A simple and effective way is to directly interpolate the discrete Fourier transform (DFT) coefficients around the peak of the magnitude spectrum. In this paper, we use theoretical analysis to show that correcting a phase term on the DFT coefficients before interpolation can reduce the estimation bias and improve the accuracy of the estimation significantly. We derive the amount of bias reduction that can be achieved by the phase correction. We then show that the phase correction can be considered as a pre-estimation correction while the effect of the previously proposed post-estimation correction is also to reduce the bias caused by the phase. In the experiments, we find that the phase correction proposed here and the post-estimation correction previously proposed is effective in different regions of frequency offset. We theoretically derive the threshold between the regions and propose a hybrid estimator which uses different corrections in different regions of frequency offset. Experiments show that the hybrid estimator is superior among direct estimators across the entire range of frequency offset at moderate and high SNR.