Refinement of Optimal Interpolation Factor for DFT Interpolated Frequency Estimator

Frequency estimation is a fundamental problem in many areas. The previously proposed
q
-shift estimator (QSE), which interpolates the discrete Fourier transform (DFT) coefficients by a factor of
q
, enables the estimation accuracy to approach the Cramér-Rao lower bound (CRLB). However, it becomes less effective when the number of samples is small. In this letter, we provide an in-depth analysis to unveil the impact of
q
on the convergence of QSE, and derive the bounds of a refined region of
q
that ensures the convergence of QSE to the CRLB even with a small number of samples. Simulations validate our analysis, showing that the refined interpolation factor is able to reduce the estimation mean squared error of QSE by up to 13.14 dB when the sample number is as small as 8.