Interpolated Discrete Fourier Transform 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.

An approach for electrical harmonic analysis based on interpolation DFT

An approach for electrical harmonic analysis based on interpolation DFT

Improved Analytic Energy Operator and Novel Three-Spectral Line Interpolation DFT Method for Parameter Estimation of Voltage Flicker

An improved analytical energy operator(IAEO) and novel three-spectral line interpolation discrete Fourier method is proposed in this paper to estimate voltage flicker parameter. Firstly, the analytic energy operator is improved based on Taylor series to better extract the amplitude and frequency features of voltage flicker envelope signal, and simplify the envelope extraction formula to avoid the square root operation. Then, a novel three-spectral line interpolation DFT algorithm is constructed by using Kaiser window and Blackman-Harris mutual convolution, and the amplitude and frequency correction formulas of voltage flicker are derived based on the novel three-spectral line interpolation DFT. Finally, the voltage flicker parameter estimation is realized on the virtual instrument based on the proposed method. The applicability and accuracy of the IAEO and novel three-spectral line interpolation DFT method for parameter estimation of voltage flicker are verified by a series of simulations and experiments.

Computationally Efficient Synchrophasor Estimation: Delayed In-Quadrature Interpolated DFT

Computationally Efficient Synchrophasor Estimation: Delayed In-Quadrature Interpolated DFT

An Improved Interpolated DFT-Based Parameter Identification for Sub-/Super-Synchronous Oscillations With Synchrophasors

The sub-synchronous resonance oscillations (SSO) caused by considerable renewable energy generations and power electronic devices seriously affect power system stability. Synchrophasor-based SSO parameter identification can effectively monitor SSOs. Aiming at the fundamental and frequency-coupled sub-/super-synchronous components that appear simultaneously in SSOs, an improved parameter identification algorithm is proposed in this paper based on the original interpolation Discrete Fourier Transform (DFT) and Hann-window. As the complement to the original algorithm, the improvements proposed in this paper additionally focus on the influence between the positive-/negative-frequency parts of the sub-/super-synchronous and fundamental components. Thus, the frequency, amplitude, and phase of the fundamental, sub-synchronous, and super-synchronous components can be accurately obtained by the proposed algorithm. This algorithm is compared to the original interpolated DFT and Prony analysis with both the simulated Phasor Measurement Unit (PMU) data and PMU data recorded during an actual SSO incident to verify its correctness and effectiveness. In the improved algorithm, the additional consideration for the super-synchronous component and the additional identified parameters significantly enhance the practical value of the interpolated DFT algorithm.

Velocity Estimation by Two Sample DFT Interpolation and Weighted Least Square Fusion for Dual-Frequency Radar

Velocity measurements of moving targets based on radar systems have extensive applications in many engineering fields. This article proposes an optimized velocity estimation strategy for dual-frequency radar by introducing a new discrete Fourier transform (DFT) interpolation scheme and the fusion of the observations acquired by two receivers. First, a new two-sample DFT interpolation method based on a phase detector is contributed. The new interpolation method can achieve an accurate estimation for each single receiver. Additionally, a weighted least square (WLS) estimation scheme is further proposed through the fusion of the observations of dual receivers. The fusion boundaries of dual receivers are investigated, and an optimized fusion strategy is finally presented. Numerical and experimental results show that the proposed strategies equip good noise resistance and outperform existing methods under various conditions.

Comparative Study of Phasor and Frequency Estimation Algorithms for Distribution Networks

Abstract: Next-generation phasor measurement units (PMUs) are critical for proper distribution network monitoring in future smart grids. In the highly dynamic environments of future distribution systems, more advanced algorithms for detecting amplitude, phase, and frequency changes in power waveforms will be required. In this context, the goal of this study is to discuss the advantages and disadvantages of commonly used PMU algorithms. Many commonly used approaches for power system phasors and frequency estimation algorithms for distribution networks are reviewed and evaluated in this work from the perspective of phasor and frequency identification. According to various performance criteria, e-IpDFT (Enhanced Interpolated Discrete Fourier Transform) and DDPE (Dictionary-based Dynamic Phasor Estimator) provided the best performance among the algorithms. This study provides some insights into the shortcomings of current phasor estimation algorithms and will be useful in the development of future estimation techniques.

Resynchronization of islanded unbalanced ADNs: Control, synchrocheck and experimental validation

The problem of securely reconnecting active distribution networks (ADNs) – e.g. microgrids – to their upstream grids at the point of common coupling (PCC) has been extensively discussed by the existing literature. The latter is commonly referred to as resynchronization and has to be done with care in order to avoid large transient current flows resulting from differences of nodal voltage phasors at both sides of the PCC. The active resynchronization process can be split into two tasks: the PCC-control and the synchrocheck. The PCC-control refers to the process used to steer the PCC nodal voltage at the ADN’s side (i.e. downstream) towards the PCC nodal voltage at the upstream-grid’s side (i.e. upstream). The synchrocheck refers to the algorithm used to check the synchronization (i.e. phasor alignment within tolerances) of the upstream and downstream PCC nodal voltages. Methods for PCC-control and synchrocheck presented in the literature commonly ignore the ADN’s operational constraints and rely on the assumption of a balanced system. In this respect, the contribution of this paper is twofold. First, an approximated optimal-power-flow is proposed to control ADNs’ resources in order to rapidly steer their PCC downstream nodal voltages close to their non-controllable upstream counterparts. Second, an Interpolated-Discrete-Fourier-Transform (IpDFT)-based synchrocheck that verifies the alignment of all three-phases of both upstream and downstream nodal voltages at the PCC, is proposed. The algorithms associated to both contributions are experimentally validated on the CIGRE-low-voltage-benchmark-microgrid at the Distributed Electrical Systems Laboratory (DESL) at the École Polytechnique Fédérale de Lausanne (EPFL) where the results of the developed synchrocheck are further benchmarked against the Schneider Electric’s Micom P143 grid relay.

Iterative Two-Point Interpolated DFT Algorithm for Accurate Frequency Estimation

The literature on the subject of frequency estimation algorithms has discussed the use of interpolated discrete Fourier transform (IpDFT) for the unbalanced/balanced three-phase voltage signal in a power system. It is difficult to estimate frequency of the unbalanced signal with high accuracy since the unknown parameters of the unbalanced signal make the spectrum different from the real-valued sinusoid signal. Moreover, the spectral leakage of the spectrum of the unbalanced signal also brings large estimation errors when the acquired signal cycles become very small. In this paper, we propose a two11 point iterative IpDFT algorithm for frequency estimation of the unbalanced/balanced three-phase voltage signal. We decompose the unbalanced signal into a real-valued sinusoid signal, and then search the system frequency iteratively with the minimum error by bisection method. Both long- and short- range leakages are compensated to estimate the frequency accurately. Simulation results prove the effectiveness of our method, compared with other iterative/noniterative IpDFT methods.

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.

Fundamental frequency estimation by an interpolated DFT algorithm eliminating negative-image component interference in arbitrary windows

Interpolated discrete Fourier transform (IpDFT) is an effective algorithm for spectrum analysis. However, the interference effect from the negative-image component significantly influences the accuracy of spectrum analysis for a small number of acquired sine wave cycles. In this study, a high-accuracy IpDFT algorithm is proposed for fundamental frequency estimation, wherein the interference effect of the fundamental negative-image component is theoretically eliminated. The proposed IpDFT algorithm uses the real or imaginary parts of three DFT bins with the largest amplitudes to establish the interpolation formula, which is employed to correct the frequency deviation caused by the picket fence effect. Additionally, this algorithm is suitable for arbitrary windows. The calculation process is simplified using the lookup-table method. Moreover, the effectiveness of the proposed IpDFT algorithm is verified by simulation and experimental results. The frequency estimation accuracy of the proposed IpDFT algorithm is comparatively analyzed with those of several state-of-the-art IpDFT algorithms for both noisy and noisy-and-harmonically distorted sine waves.

Novel Interpolation Method of Multi-DFT-Bins for Frequency Estimation of Signal With Parameter Step Change

The interpolation discrete Fourier transform (IpDFT) method is one of the most commonly used nonparametric methods. However, when a parameter (frequency, amplitude, or phase) step changes in the DFT period, the DFT coefficients will be distorted seriously, resulting in the large estimation error of the IpDFT method. Hence, it is a key challenge to find an IpDFT method, which not only can eliminate the effect of the step-changed symbol but also can sufficiently eliminate the fence effect and the spectrum leakage. In this article, an IpDFT-based method is proposed to estimate the frequency of the single-tone signal with the step-changed parameters in the sampling signal sequence. The relationship between the DFT bins and the step-changed parameters is given by several linear equations. At most six different DFT bins are used to eliminate the effect of the symbol.

Accurate Damping Factor and Frequency Estimation for Damped Real-Valued Sinusoidal Signals

The interpolated discrete Fourier transform (IpDFT) is one of the most popular techniques to estimate the parameters of a damped real-valued sinusoidal signal (DRSS). However, its accuracy is affected by strong noise presence and short observation windows. To this end, this letter proposes a novel two-point IpDFT method, called I2pZDFT, for the parameter estimation of a DRSS. The proposed I2pZDFT uses the zero-padding technique to increase the sampling rate in the frequency domain. The conjugate symmetry and the parity of the zero-padded signal are utilized to eliminate the influence of the spectral leakage. Simulation results highlight that the proposed I2pZDFT outperforms the existing IpDFT-based methods in terms of noise immunity, especially in the case of observation windows as short as 0.5 ~ 1 cycles.

Accurate Frequency Estimation of Signals with Symbol Transition by Using Interpolated Dft

Accurate Frequency Estimation of Signals with Symbol Transition by Using Interpolated Dft

Single-Tone Frequency Estimation by Weighted Least-Squares Interpolation of Fourier Coefficients

Frequency estimation of a single complex exponential signal embedded in additive white Gaussian noise is a major topic of research in many engineering areas. This work presents further investigations on this problem with regards to the fine estimation task, which is accomplished through a suitable interpolation of the discrete Fourier transform (DFT) coefficients of the observation data. The focus is on fast real-time applications, where iterative estimation methods can hardly be applied due to their latency and complexity. After deriving the analytical expression of the Cramér-Rao bound (CRB) for general values of the system parameters, we present a new DFT interpolation scheme based on the weighted least-squares (WLS) rule, where the optimum weights are precomputed through a numerical search and stored in the receiver. In contrast to many existing alternatives, the proposed method can employ an arbitrary number of DFT samples so as to achieve a good trade-off between system performance and complexity. Simulation results and theoretical analysis indicate that, at sufficiently high signal-to-noise ratios, the estimation accuracy is close to the relevant CRB at any value of the frequency error. This provides some advantage with respect to non-iterative competing schemes, without incurring any penalty in processing requirement.

IpDFT-Tuned Estimation Algorithms for PMUs: Overview and Performance Comparison

The Interpolated Discrete Fourier Transform (IpDFT) is one of the most popular algorithms for Phasor Measurement Units (PMUs), due to its quite low computational complexity and its good accuracy in various operating conditions. However, the basic IpDFT algorithm can be used also as a preliminary estimator of the amplitude, phase, frequency and rate of change of frequency of voltage or current AC waveforms at times synchronized to the Universal Coordinated Time (UTC). Indeed, another cascaded algorithm can be used to refine the waveform parameters estimation. In this context, the main novelty of this work is a fair and extensive performance comparison of three different state-of-the-art IpDFT-tuned estimation algorithms for PMUs. The three algorithms are: (i) the so-called corrected IpDFT (IpDFTc), which is conceived to compensate for the effect of both the image of the fundamental tone and second-order harmonic; (ii) a frequency-tuned version of the Taylor Weighted Least-Squares (TWLS) algorithm, and (iii) the frequency Down-Conversion and low-pass Filtering (DCF) technique described also in the IEEE/IEC Standard 60255-118-1:2018. The simulation results obtained in the P Class and M Class testing conditions specified in the same Standard show that the IpDFTc algorithm is generally preferable under the effect of steady-state disturbances. On the contrary, the tuned TWLS estimator is usually the best solution when dynamic changes of amplitude, phase or frequency occur. In transient conditions (i.e., under the effect of amplitude or phase steps), the IpDFTc and the tuned TWLS algorithms do not clearly outperform one another. The DCF approach generally returns the worst results. However, its actual performances heavily depend on the adopted low-pass filter.

Three-Point Synchrophasor Estimator With Better Off-Nominal Frequency Leakage Reduction

Synchronous phasor measurements have long been an important tool in power flow modelling and network state estimation as well as in many other tasks that support power grid observability. Classic phasor estimation algorithms are often preferred over complex alternatives. They are based on a variant of the Discrete Fourier Transform (DFT) while relying on a sampling frequency locked to the system nominal frequency. The spectral leakage caused by the small off-nominal frequency excursions in such settings introduces errors that are handled by the estimation algorithms to a different extent. Well-known three-point averaging filter (3P) is a low-cost spectral leakage reduction technique in one of the simplest estimators based on a single-bin DFT. However, it only partially removes the oscillating components. When an independent standard-based frequency tracking is available, the errors can be further compensated with minimal cost. We propose a frequency-corrected three-point estimator (F3P) to eliminate the induced spurious spectrum progressively with the increasing sampling rate while building on the analytical expression of the remaining error. We show performance benefits of the F3P under finite sampling rates when compared to the baseline 3P and to the classical Interpolated DFT (IPDFT) in static conditions. The F3P removes most of the leakage without the need of data resampling. In less ideal conditions with additive 50-dB noise, up to a 10-fold reduction of the 3P error is achieved. The error ratio of the reference IPDFT over the F3P is approximately 1.4 under the same conditions and largely independent of the frequency excursion. The presented simulation results prove the algorithm's superior performance in most of the steady-state tests defined by the relevant Standard, whereas the dynamic performance can be brought to an acceptable level by the use of a single-cycle observation window, with the response times comparable to those of the IPDFT.

Sensitivity to Frequency Uncertainty of the Estimators Provided by the Two-Parameter Sine-Fit Algorithm

In this article, the contribution of frequency error on the sine-wave amplitude and phase estimators returned by the linear two-parameter sine-fit (2PSF) algorithm is analyzed. Expressions for the mean square errors (MSEs) of both estimators are derived in the case of pure and noisy sine waves assuming that the frequency error is small. From the derived expressions a constraint on frequency uncertainty ensuring that the MSEs of the amplitude and phase estimators reach the corresponding Cramer-Rao Lower Bounds is determined. The derived constraint is applied to two different state-of-the-art Interpolated Discrete Fourier Transform (IpDFT) frequency estimators to determine the number of observed sine-wave cycles above which the linear 2PSF algorithm provides statistical efficient amplitude and phase estimates.

Comparative of Influence of Noise on Frequency Estimation Provided by Three Points Interpolated Discrete Fourier Transform

The paper compares the accuracy of two different three points Interpolated Discrete Fourier Transform (M3IpDFT and C3IpDFT) algorithms in the presence of Gaussian white noise. The above two algorithms use the magnitude and the complex value of spectrum line for interpolation, respectively. The theoretical expressions of variances of frequency estimation due to noise are derived and then verified by simulations. From the theoretical and simulation results, the variances of frequency estimation are inversely proportional to the signal-to-noise ratio (SNR) and the length of DFT. Simulation shows that the C3IpDFT outperforms the M3IpDFT.