An Exact Analysis of Fine Resolution Frequency Estimation Method from Three DFT Samples: Windowed Data Case

Frequency estimation for a single complex sinusoid in noise is a fundamental problem in signal processing. A suboptimal but simple frequency estimator, known as Jacobsen estimator, which is based on three discrete Fourier transform (DFT) samples, gives good bias performance without the need to increase the DFT size. Candan has modified the Jacobsen estimator by adding a so-called bias correction factor to further reduce the bias of the estimator. In addition to bias considerations, a number of asymptotic variance expressions of the estimators were performed in the literature. However, these expressions are valid only for signal frequencies located very near a DFT bin index. In this paper, with the use of a simple variance analysis technique, an accurate general variance expression for arbitrary frequency locations is derived for the case of windowed data. A general method for calculating the bias correction factor is also proposed. The variance expression is examined for the cosine-sum window family. An approximate variance formula for sufficiently large data record lengths is also given for windows from this family. Computer simulations are included to validate the theoretical results.