Accurate frequency estimation of a noisy sine-wave by means of an interpolated discrete-time Fourier transform algorithm

This paper investigates the accuracy of a recently proposed sine-wave frequency estimator based on the interpolation of Discrete-Time Fourier Transform (DTFT) spectrum samples located one frequency bin apart. Specifically, an analytical expression for the variance of the Interpolated DTFT (IpDTFT) frequency estimator due to additive wide-band noise is derived in the case when the acquired sine-wave samples are weighted by a Maximum Sidelobe Decay (MSD) window. The derived expression enables the identification of an iterative algorithm, called IpDTFT-NR procedure, that ensures the maximum Noise Rejection on the IpDTFT frequency estimator. Moreover, the accuracy of the IpDTFT-NR procedure is compared with that of another algorithm, called IpDTFT-IR procedure, which ensures the maximum Image component interference Rejection on the IpDTFT frequency estimator. To this aim, accurate expressions for the Mean Square Errors (MSEs) of the two frequency estimators are derived. Also, the value of the Signal-to-Noise Ratio (SNR) up to which the IpDTFT-NR procedure outperforms the IpDTFT-IR one is determined as a function of the number of analysed sine-wave cycles. Once the SNR and the number of observed cycles of the analysed signal are known or estimated, the knowledge of this threshold enables the definition of an algorithm, called IpDTFT-IR/NR procedure, that estimates the sine-wave frequency by using the most accurate of the two procedures above. The accuracies of the IpDTFT-IR/NR procedure and of other state-of-the-art interpolated Fourier algorithms are compared with each other using both simulation and experimental data.