Maximum Sidelobe Decay Windows Research Articles

Frequency and damping factor estimation of real-valued damped sinusoids by means of an improved two-point Interpolated DFT algorithm

The contribution of this paper is two-fold. Firstly, the well-known two-point Interpolated Discrete Fourier Transform (IpDFT) algorithm based on a Maximum Sidelobe Decay (MSD) window is generalized to the estimation of the frequency and the damping factor of a real-valued damped sinusoids. Analytical expressions for the detrimental contributions on the obtained estimators due to the image frequency component and wideband noise are then derived and their accuracies are verified through computer simulations. Secondly, leveraging on the derived expression, an improved version of the IpDFT algorithm is proposed which ensure more accurate estimates by compensating the contribution of the image frequency component. The performance of the improved IpDFT algorithm is compared with those provided by various state-of-the-art algorithms through meaningful computer simulations.

Generalized multi-point interpolated DTFT frequency and damping factor estimators of real-valued damped sinusoids

In this paper a generalized multi-point interpolated Discrete-Time Fourier Transform (DTFT) is proposed for frequency and damping factor estimation of real-valued damped sinusoids. The acquired data are weighted by a Maximum Sidelobe Decay (MSD) window and the interpolation function is defined as the ratio between two linear combinations of DTFT samples located one bin apart, which is generalized exploiting a tuning parameter. Analytical expressions for the estimated parameter errors due to the interference of the fundamental image component are derived. Leveraging on these expressions an improved version of the algorithm is also proposed and optimized by selecting a suitable value for the tuning parameter. The developed theory is applied to both a three-point and a four-point interpolated Discrete Fourier Transform (IpDFT) estimators published in the literature and the achieved accuracy improvements are verified through simulations.

Supraharmonic Detection Algorithm Based on Interpolation of Self-Convolutional Window All-Phase Compressive Sampling Matching Pursuit

With the increase in the use of high-frequency power electronic devices, the harmonics injected into the power grid show a trend of high-frequency development. The continuous rise of the supraharmonic emission level in the distribution network has become one of the power quality problems that needs to be solved urgently in the power grid. In this paper, an algorithm based on the Interpolation of the Self-convolutional Window All-phase Compressive Sampling Matching Pursuit (ISWApCoSaMP) is proposed. Firstly, the self-convolution operation is used for the maximum sidelobe decay (MSD) window, and then the compressed sampling matching pursuit model based on the All-phase is constructed, leading to the All-phase Compressive Sampling Matching Pursuit (ApCoSaMP). Finally, the four-spectrum-line interpolation is combined to utilize spectrum line information to improve the accuracy of signal parameter detection in the frequency domain. The introduced All-phase greatly improves the phase measurement accuracy because the initial phase of the supraharmonic signal is selected for phase estimation. In addition, the self-convolutional window and four-spectrum-line interpolation make full use of the information in the time and frequency domains, thus optimizing the measurement results of amplitude and frequency. The algorithm achieves high accuracy in the measurement results of simulated signals and accurately measures supraharmonics.

Fast interpolated DTFT estimators of frequency and damping factor of real-valued damped sinusoids

This paper deals with the estimation of real-valued damped sinusoid frequency and damping factor. To that aim an Interpolated Discrete Time Fourier Transform (IpDTFT) algorithm is proposed which extends to real-valued signal an algorithm proposed in the literature for complex-valued damped sinusoids. The proposed algorithm provides fast estimates of the unknown parameters of either undistorted or distorted real-valued noisy damped sinusoids with accuracy generally better than the one ensured by state-of-the-art estimators, especially when few sinusoid cycles are analysed. Accuracy improvement is obtained by both weighting the acquired signal using a Maximum Sidelobe Decay (MSD) window and compensating the contribution of the spectral image component on the estimated parameters. Analytical expressions for the estimator variances due to wideband noise are also derived.

Accurate Frequency Estimator for Real Sinusoid Based on DFT

An accurate frequency estimator for real sinusoid based on Discrete Fourier Transform (DFT) is proposed. The proposed estimator is based on the interpolation of the maximum DFT spectral line and two Discrete-Time Fourier Transform (DTFT) spectral lines and can operate with both the rectangular window and the maximum sidelobe decay (MSD) window. In the coarse estimation step, the proposed estimator with the MSD window is used. According to the value of the coarse estimate, the negative frequency spectral component is removed. In the fine estimation step, the proposed estimator with the rectangular window is utilized to achieve the Cramer–Rao lower bound (CRLB). Simulation results show that the performance of this algorithm is better than that of the AM algorithm, Candan algorithm, Djukanovic algorithm, and FDIAM algorithm.

Noniterative method for frequency estimation based on interpolated DFT with low-order harmonics elimination

The paper describes a new interpolated discrete Fourier transform (IpDFT) method for the frequency estimation of the real sine wave distorted by harmonics. The method complements recent work on the direct-formula IpDFT methods by expanding the model of the signal included in the interpolation. Spectral superposition for a total of four frequency components is accounted for in a one-step formula. Alongside the positive and negative components of the fundamental frequency, the influence from the closest low-order real harmonic is eliminated. By using the generalized maximum sidelobe decay (GMSD) window family, the interference from more distant harmonics and interharmonics is suppressed. This approach allows us to perform the measurement with a shortened data record compared to previous methods and improves accuracy in the case that a second-order harmonic is present in the signal. The study provides a five-point analytical, one-step formula that is verified by simulations and experiments in terms of accuracy and complexity. The experimental results are presented featuring straightforward C implementation on the system with an affordable microcontroller.

Interpolated DFT Algorithm for Frequency Estimation by Using Maximum Sidelobe Decay Windows

A sinusoidal frequency estimator based on interpolated Discrete Fourier Transform (DFT) algorithm by using Maximum Sidelobe Decay (MSD) windows is proposed in this paper. Firstly, the received sinusoid is weighted by an appropriate MSD window. Then DFT is carried out on the weighted sinusoid and the coarse estimation is acquired by finding the position of the maximum DFT sample. Different from all the existing algorithms, the presented estimator adopts the maximum DFT sample and two Discrete Time Fourier Transform (DTFT) spectral lines which are on the same side of the maximum DFT sample in the fine estimation step. MSE formulas of the presented estimator in additive white noise are derived. Simulation results indicate that the presented estimator outperforms the competing estimators.

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

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.

Fast procedures for accurate parameter estimation of sine-waves affected by noise and harmonic distortion

In this paper two procedures are proposed for the estimation of the frequency, amplitude, and phase of sine-waves affected by either wideband noise or by both noise and harmonic distortion, respectively. The procedures are very simple and suitable for real-time applications. According to them, firstly the signal parameters are estimated by means of the new Corrected Interpolated Discrete Fourier transform (IpDFTc) algorithm, which compensates the contribution of the spectral interference from the fundamental image component and harmonics on the parameter estimates returned by the classical IpDFT algorithm. Then, a linear sine-fit algorithm is applied to optimize the noise robustness of the estimators, which is worsened by signal windowing employed to reduce the effect of spectral leakage on the IpDFTc estimates. In the paper, the Hann window is applied since it exhibit both the highest noise robustness among the Maximum Sidelobe Decay (MSD) windows and a good reduction of long-range spectral leakage. It has been shown that both proposed procedures almost attain the Cramér-Rao Lower Bounds (CRLBs) for unbiased estimators when at least about 1.5 sine-wave cycles are observed so that the effect of interfering tones on the IpDFTc estimates can be effectively compensated by windowing. The performances of both procedures are analysed through computer simulations and experimental results.

Accurate and Fast Amplitude Estimation of Signal Distorted by Noise and Harmonics for Control of VSI

Arriving at an accurate and fast amplitude estimation of the sine-wave distorted by harmonics and noise is important for the control of voltage source inverters in grid-connected photovoltaic system. In this article, the three-point interpolation discrete Fourier transform method based on the maximum sidelobe decay windows is proposed for accurate amplitude estimation. The proposed amplitude estimator is speeded-up by using the guaranteed stable recursive method. The influence of noise on the proposed amplitude estimator is investigated, and the amplitude estimation error caused by the spectral interference is also analyzed. The accuracy and effectiveness of the proposed method are verified through simulation and experiments, where the state-of-art amplitude estimation methods and the boundaries specified in the Standard IEEE C37.118.1-2011 are considered.

Contribution of an interharmonic component to the sine-wave parameters estimators returned by the interpolated Discrete Fourier transform algorithm

<p class="Abstract">This article investigates the contribution of a small-amplitude interharmonic component to the sine-wave parameter estimators returned by the classical interpolated discrete Fourier transform (IpDFT) algorithm. The analytical expressions for the frequency, amplitude, and phase estimation errors are derived herein by considering the IpDFT algorithm based on the maximum sidelobe decay (MSD) windows and by assuming the interharmonic frequency located at least one bin apart the unknown sine-wave frequency. The derived expressions allow us to analyse the impact of an interharmonic on the accuracies of the IpDFT frequency, amplitude, and phase estimators. The accuracies of the derived expressions are verified by means of both computer simulations and experimental results.</p>

Frequency Estimation in Interpolated Discrete Fourier Transform With Generalized Maximum Sidelobe Decay Windows for the Control of Power

The study presented in this article demonstrates that the sin p time window family is an extension of class I Rife–Vincent windows to a more general class, i.e., generalized maximum sidelobe decay (GMSD) windows. The presented frequency characteristics and other useful formulas derived for GMSD windows allow the derivation of a method to estimate the frequency of a sinusoidal component of a multifrequency signal based on the interpolated discrete Fourier transform. The presented estimation method is a significant generalization of methods already known. The method uses GMSD windows and takes into account the conjugate component in the spectrum. The method can be widely used, and, in particular, it allows for fast and accurate estimation of the frequency for a short data record (even below 1–2 signal periods). One of the many possible applications is frequency estimation in a power control system, which was studied in the accuracy analysis.

Effect of windowing and noise on the amplitude and phase estimators returned by the Taylor-based Weighted Least Squares

This paper analyzes the effect of windowing and additive wideband noise on the estimates returned by the real-valued Taylor-based Weighted Least Squares (TWLS) algorithm when applied to noisy sine-wave with constant amplitude and phase. The Taylor's series of the signal time-varying phasor are truncated to the second order and the two-term or three-term Minimum Sidelobe Level (MSL) and Maximum Sidelobe Decay (MSD) windows are considered. Expressions for the estimation variances are derived that hold even when the sine-wave amplitude and phase exhibit slow and limited, but significant, time variations. Then they are compared with the unbiased Cramér–Rao Lower Bounds (CRLBs) for constant parameter sine-waves in order to analyze the estimator statistical efficiency. The accuracies of the derived expressions are verified through computer simulations and validated by experimental results.

Accurate amplitude and phase estimation of noisy sine-waves via two-point interpolated DTFT algorithms

The estimation of the amplitude and the phase parameters of a noisy real-valued sine-wave by means of the two-point Interpolated Discrete-Time Fourier Transform (IpDTFT) algorithms based on the Maximum Sidelobe Decay (MSD) windows is analysed in this paper. In particular, the DTFT samples and the IpDTFT frequency estimator that minimize the contribution of either the spectral image component or wideband noise are determined and the related estimation procedures are presented. Once the number of analysed sine-wave cycles and the signal-to-noise ratio are given, a procedure for the selection of the most accurate estimator is then proposed. Finally, the accuracies of the proposed procedures and the classical two-point IpDFT algorithm are compared with each other through both simulation and experimental results.

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.

Novel Three-Point Interpolation DFT Method for Frequency Measurement of Sine-Wave

This paper proposes a novel three-point interpolation discrete Fourier transform for accurate power system frequency measurement. The accurate formula of the proposed frequency measurement method is derived by using the maximum sidelobe decay windows. Moreover, the influence of white noise on the proposed frequency measurement is analyzed by deducing the expression of frequency measurement variance. The systematic errors and variances of frequency measurement are analyzed by simulation.

Effect of noise and harmonics on sine-wave frequency estimation by interpolated DFT algorithms based on few observed cycles

In this paper the performances of two state-of-the-art Interpolated Discrete Fourier Transform (IpDFT) algorithms are analyzed when sine-wave frequency must be estimated over short observation intervals. The first estimator, called enhanced IpDFT (e-IpDFT) algorithm, exploits a two-point interpolation and compensates the detrimental contribution of the fundamental image component on the estimated frequency by using an iterative procedure. The second estimator, called IpDFT-EIF algorithm, eliminates that contribution by using a three-point interpolation. Both algorithms reduce the spectral leakage due to time-domain truncation by weighting the acquired signal by a Maximum Sidelobe Decay (MSD) window. The analysis is performed in the case of ideal, noisy, and noisy and harmonically distorted sine-waves. Theoretical expressions for the estimation Mean Square Errors (MSEs) due to noise and harmonics are derived and verified through simulations and experiments. The performed analysis allows the selection of the best frequency estimator for given signal-to-noise-ratio, harmonic content, and number of acquired cycles.

Impact of harmonics on the interpolated DFT frequency estimator

The paper investigates the effect of the interference due to spectral leakage on the frequency estimates returned by the Interpolated Discrete Fourier Transform (IpDFT) method based on the Maximum Sidelobe Decay (MSD) windows when harmonically distorted sine-waves are analyzed. The expressions for the frequency estimation error due to both the image of the fundamental tone and harmonics, and the frequency estimator variance due to the combined effect of both the above disturbances and wideband noise are derived. The achieved expressions allow us to identify which harmonics significantly contribute to frequency estimation uncertainty. A new IpDFT-based procedure capable to compensate all the significant effects of harmonics on the frequency estimation accuracy is then proposed. The derived theoretical results are verified through computer simulations. Moreover, the accuracy of the proposed procedure is compared with those of other state-of-the-art frequency estimation methods by means of both computer simulations and experimental results.

Frequency estimation of a sinusoidal signal via a three-point interpolated DFT method with high image component interference rejection capability

This paper proposes a three-point Interpolated Discrete Fourier Transform (IpDFT) method for frequency estimation of a discrete-time sinusoidal signal. It is based on the maximum sidelobe decay (MSD) windows and is highly effective in rejecting the detrimental effect on the estimation accuracy due to the image component of the signal spectrum. This remarkable feature is achieved by using an analytical expression based on a suitable weighting of the three largest DFT spectrum samples. The proposed method provides good results when the effect of the spectral interference due to the image component dominates other estimation error sources. The accuracy of the proposed method and of other state-of-the-art methods such as the multi-point IpDFT methods and the four-parameter sine-fitting (4PSF) algorithm are compared through both computer simulations and experimental results in the case of ideal, noisy, and harmonically distorted sinusoids. A small number of acquired cycles is assumed in order to analyze situations in which the contribution from the image component interference is significant. The performed comparison shows that the proposed method outperforms the considered multi-point IpDFT methods when the Signal-to-Noise Ratio (SNR) is higher than 30 dB and the number of acquired cycles is enough small. The proposed method outperforms also the 4PSF algorithm when the frequency estimation error is dominated by harmonics rather than wideband noise.

Accuracy analysis of the normalized frequency estimation of a discrete-time sine-wave by the average-based interpolated DFT method

The normalized frequency of a sine-wave can be estimated by the average of the two estimates achieved using the Interpolated Discrete Fourier Transform (IpDFT) method. This approach is called average-based IpDFT method. The paper investigates the effect of the spectral interference from the image component and additive wideband noise on the accuracy of the normalized frequency estimator provided by the average-based IpDFT method with maximum sidelobe decay windows in the case of a discrete-time sine-wave. The expressions of the maximum of the normalized frequency estimator error due to the spectral interference and variance of the normalized frequency estimator due to the wideband noise are derived. Also, the expression of the combined standard uncertainty of the normalized frequency estimator is derived. The accuracies of all derived expressions are confirmed by means of computer simulations. Moreover, the performance of the average-based IpDFT method is compared with those of a state-of-the-art three-point IpDFT method and IpDFT method through both computer simulations and experimental results.