Frequency estimator of sinusoid by interpolated DFT method based on maximum sidelobe decay windows

Highlights

• A frequency estimator of sinusoid by interpolated Discrete Fourier Transform (DFT) method based on maximum sidelobe decay (MSD) windows is proposed. Unlike other frequency estimators which use spectral lines located in fixed positions, the proposed estimator utilizes the maximum spectral line and two auxiliary spectral lines arbitrarily located within the weighted sinusoid spectrum main lobe.

• An accurate formula for the estimator MSE due to additive white noise is derived.

• As the frequency intervals between the spectral lines are arbitrary, when non-rectangular windows are adopted, the magnitude spectrum correlation function of complex sinusoid plus complex white noise is derived for the derivation of MSE formula.

• Simulation results show that the proposed estimator is more general and performs better than the competing estimators.

Abstract

A frequency estimator of sinusoid by interpolated Discrete Fourier Transform (DFT) method based on maximum sidelobe decay (MSD) windows is proposed. Unlike other frequency estimators which use spectral lines located in fixed positions, the proposed estimator utilizes the maximum spectral line and two auxiliary spectral lines arbitrarily located within the weighted sinusoid spectrum main lobe. An accurate formula for the estimator MSE due to additive white noise is derived. To further improve the estimation performance, iterative procedure is used. It is analytically and numerically shown that the MSE increases with the increment of the interval length between the maximum spectral line and the auxiliary spectral line in additive white noise. The proposed estimator is more general and performs better than the competing estimators.

Introduction

Frequency estimation of sinusoid is a fundamentally important problem in digital signal processing and can be applied in many areas such as communications, measurement, instrumentation, radar, sonar and power systems. Many sinusoidal frequency estimators have been proposed and they can be categorized into time-domain methods [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] and frequency-domain methods [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. Time-domain methods estimate sinusoidal frequency by the maximum likelihood methods [1], [2], [3], autocorrelation methods [4], [5], [6], [7], [8], [9], linear prediction methods [10], [11], [12] and so on. These methods generally have low computational efficiency and are not suitable for real-time applications. Frequency-domain methods based on the Discrete Fourier Transform (DFT) of the sampled sinusoid, on the other hand, exhibit low computational complexity and good anti-interference performance. Therefore, they are often adopted in real-time applications. Typically, DFT-based frequency estimators are implemented in two stages. In the first stage, FFT is performed and the maximum spectral line in the DFT spectrum is found to obtain a coarse estimate. In the next stage, a fine estimate is obtained by interpolating the DFT spectral lines adjacent to the maximum spectral line. The positions of the spectral lines employed by DFT-based estimators can be fixed [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] or arbitrary [23]. Candan estimator utilizes the maximum spectral line and its two neighbors to estimate the frequency [19], and its estimation bias is the best among the non-iterative methods experimentally [20]. AM method uses the amplitudes or the complex values of two DFT samples located exactly halfway between the maximum spectral line and its two neighbors [21], and it can approach Cramer–Rao lower bound (CRLB) in two iterations. In [23], a more general form of frequency estimator utilizes two auxiliary spectral lines located at arbitrary position within the main lobe in the fine estimation.

All the above DFT-based methods [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23] estimate the frequency in white noise background and are based on rectangular window. In many engineering applications, there are interfering signals besides wide band noise. For example, due to the existence of non-linear loads, harmonic interfering signals are ubiquitous in power systems. Under this situation, when the spectral leakage effect of interfering signals prevails over the wideband noise, nonrectangular windowing methods are often used to reduce the effect of the interfering signals on estimated parameters of desired signal [24], [25], [26], [27], [28], [29], [30], [31], [32], [33]. When the signal is weighted by a non-rectangular window, the curvature around the peak of the DTFT spectrum changes significantly [24]. Several methods have been proposed to generalize the estimators based on rectangular window to the case when non-rectangular windows are adopted [24], [25], [26]. Candan extends the estimator in [19] to the case of non-rectangular windowing [24]. In [25], Jacobsen [18] and AM estimator [21] are generalized to the case when the acquired signal is multiplied by a generic cosine window.

In this paper, the frequency estimator in [23] based on rectangular window is generalized to the case when the signal is weighted by MSD windows in order to reduce the effect of spectral leakage from interfering signals on estimated frequency of the desired signal. In the fine estimation stage, the proposed estimator utilizes two auxiliary spectral lines located at arbitrary position (symmetrically located about the maximum spectral line) within the main lobe of the frequency spectrum of the signal multiplied by MSD windows. An accurate formula for the estimator MSE due to additive white noise is derived. As the frequency intervals between the spectral lines are arbitrary, when non-rectangular windows are adopted, the magnitude spectrum correlation function of complex sinusoid plus complex white noise is needed for the derivation of MSE formula. Thus, this magnitude spectrum correlation function is given. Iterative procedure is used to improve the estimation performance. In order to verify the performance and the MSE formula, simulation experiments are conducted in the case of noisy or noisy and harmonically distorted complex sinusoid. The influence of the interval length between the maximum spectral line and the auxiliary spectral lines on the MSE is analyzed through theoretical analysis and simulation experiments.

The rest of the paper is organized as follows. In Section 2, the expression of the proposed estimator is derived. In Section 3, an accurate formula for the estimator MSE due to additive white noise is derived, and the iterative procedure is described. In Section 4, the accuracy of the MSE formula and the performance of the algorithm are numerically verified. Furthermore, the proposed estimator is compared with state-of-art estimators and the CRLB. Finally, conclusions are drawn in Section 5.