Contents I. Introduction
Real-valued and complex-valued sine-waves are used in many application areas such as communications, radar, power systems, and instrumentation. Often their parameters need to be accurately estimated in real-time using simple and low-cost processing systems. To that aim the so-called Interpolated Discrete Time Fourier Transform (IpDTFT) algorithms are frequently employed since they provide fast and accurate parameter estimates [1]–[13]. This class of algorithms compensates the effect of the finite number of processed samples on the returned frequency estimates by interpolation two or more nearby Discrete Time Fourier Transform (DTFT) samples [14]. IpDFT algorithms for the estimation of the parameters of either real-valued [1]–[8] or complex-valued pure sine-waves [9]–[13] have been proposed. In the latter case the returned estimates are not affected by the spectral interference from the image of the fundamental component that raise because of spectral leakage. For that reason signal windowing is not required and a simple rectangular window is used, thus leading to better robustness with respect to wideband noise superimposed to the acquired signal [9]–[11]. Recently, an IpDTFT estimator of complex-valued sine-wave frequency based on parabolic interpolation has been proposed in [15]. It is called Parabolic IpDTFT (PIpDTFT) algorithm in the following. That algorithm interpolates three adjacent DTFT samples and it almost attains the Cramer-Rae Lower Bound (CRLB) for unbiased frequency estimators. It uses the rectangular window and the initial frequency estimate is provided by the improved three-point IpDFT algorithm proposed in [11]. However, the choice of the distance between adjacent DTFT interpolation points is simply based on the analysis of simulation results. No criterion has been proposed for the selection of such a distance. In this paper the PIpDTFT algorithm is extended to windowed signals and an expression for the bias of the PIpDTFT frequency estimator based on the Maximum Sidelobe Decay (MSD) windows is derived. The MSD windows have been considered due to their good effectiveness in reducing spectral leakage [16], [17]. Leveraging on the obtained expression, an upper bound on the distance between the DTFT interpolation points that ensures a negligible frequency estimator bias is derived.
