Interpolated Discrete Fourier Transform Algorithm 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.

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.

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.

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.

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.

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.

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>

Fast heart rate extraction using CW Doppler radar with interpolated discrete Fourier transform algorithm

Doppler radar-based fast heart rate (HR) extraction has great application potential in stress and emotion recognition, anxiety treatments, etc. However, fast extraction of HR is still a great challenge for the vital sign detection applying the traditional discrete Fourier transform (DFT) method based on a continuous-wave (CW) Doppler radar sensor. When applying traditional DFT to the baseband signal analysis, the spectrum resolution will become insufficient if the time window is less than 10 s. In this paper, an interpolated DFT algorithm based on a poly-item cosine window is introduced to achieve fast extraction of HR with only 3 s data length. In addition, the leakage and grid effect phenomenon is presented to show the defects caused by the traditional DFT method. To verify the effectiveness of the proposed method, simulations are performed and experiments are executed using a 10-GHz CW Doppler radar sensor platform. Compared with the traditional DFT method, the interpolated DFT method reduces the average HR error from 8.48% to 1.87% based on the Hanning window and from 8.48% to 1.45% based on the rectangular window.

Accurate Frequency Estimation by Using Three-Point Interpolated Discrete Fourier Transform Based on Rectangular Window

Windowed interpolated discrete Fourier transform (DFT) has been well-studied to deal with the spectral leakage and picket fence effects in frequency estimation. When the number of acquired sine-wave cycles is small, the long-range spectral leakage of the negative frequency is the main cause of low accuracy. In this article, a novel frequency estimator using three-point interpolated DFT based on rectangular window is proposed. The spectral superposition of positive and negative frequencies is adopted in the proposed interpolation algorithm to remove both short- and long-range spectral leakage. The influence of white noise on frequency estimation can be reduced because the adopted rectangular window has the narrowest mainlobe width or the smallest equivalent noise bandwidth among all windows. Simulation and experimental results confirm that the proposed method outperforms existing windowed interpolated DFT algorithms.

Generalization of interpolation DFT algorithms and frequency estimators with high image component interference rejection

This paper focuses on the problem of frequency estimation of noise-contaminated sinusoidal. A basic tool to solve this problem is the interpolated discrete Fourier transform (DFT) algorithms, in which the influences of the spectral leakage from negative frequency are often neglected, resulting in significant errors in estimation when the signals contained small cycles. In this paper, analytic expressions of the interference due to the image component are derived and its influences on the traditional two-point interpolated DFT algorithms are analyzed. Based on the achieved expressions, the interpolated DFT algorithms are generalized and a novel frequency estimator with high image component interference rejection is proposed. Simulation results show that the frequency errors returned by the new algorithm are very small even though only one or two cycles are obtained. Comparative studies indicate that the new algorithm also has a good performance in the noise condition. With the advantages of high precision and strong robustness against additive noise, the proposed algorithm is a good choice for frequency estimation when the negative frequency interference is the dominant error source.

A Scale Factor-Based Interpolated DFT for Chatter Frequency Estimation

In this paper, a scale factor-based interpolated discrete Fourier transform (IpDFT) algorithm is proposed to estimate the dominant chatter frequency. In practical measurement systems of machining processes, the number of acquired sine-wave cycles (NASCs) is small. Therefore, the significant spectral interference from the negative frequency leads to the degradation of conventional IpDFTs that neglect such interference. The proposed IpDFT algorithm overcomes this problem by completely removing the long-range leakage of the negative frequency of the investigated component. We establish statistical properties of the approach contaminated with white noise, including upper and lower bounds of the theoretical variance. The simulation results demonstrate that our IpDFT algorithm outperforms existing IpDFT algorithms, especially when the NASC approaches zero. The proposed IpDFT algorithm has better antinoise capability and computational efficiency than the optimization-based algorithm by Radil. Furthermore, it is illustrated that the simulation results agree well with upper and lower bounds of the theoretical variance. Cutting force signals are collected to evaluate the algorithm experimentally.

Dynamic Phasor and Frequency Measurements by an Improved Taylor Weighted Least Squares Algorithm

One of the most accurate phasor estimation procedures recently proposed in the literature is the so-called Taylor weighted least squares (TWLS) algorithm, which relies on a dynamic phasor model of an electrical waveform at nominal frequency. In this paper, an extension of the TWLS algorithm [called generalized TWLS (GTWLS) algorithm] to a generic (not only nominal) reference frequency is described and the accuracies of the returned estimates are analyzed through meaningful simulations, performed in different steady-state and dynamic testing conditions according to the IEEE Standard C37.118.1-2011 about synchrophasor measurement for power systems and its Amendment IEEE Standard C37.118.1a-2014. It is shown that the accuracy of the total vector error (TVE), frequency error (FE), and rate of change of frequency error (RFE) normally decreases as the deviation between the reference frequency and the true waveform frequency decreases. Furthermore, a two-step procedure for accurate estimation of the phasor parameters is proposed. In the first step, the waveform frequency is estimated by a classical interpolated discrete Fourier transform (IpDFT) algorithm. The second step then returns an estimate of the phasor parameters by applying the TWLS algorithm based on the frequency estimate returned by the first step. It is shown that the proposed procedure, called the GTWLS–IpDFT algorithm, can comply with the $P$ -class or the $M$ -class of performances in all the considered testing conditions when an appropriate number of waveform cycles is considered and the most significant disturbances are removed from the analyzed waveform. Finally, uncertainties of the proposed estimators and the $\mathrm{IpD}^{2}$ FT algorithm recently presented in the literature are also compared.

Interpolated DFT for $\sin^{\alpha}(x)$ Windows

This paper describes interpolated discrete Fourier transform (IpDFT) for parameter estimation of sinusoidal and damped sinusoidal signals analyzed with a sinα(x) window. For α = 0,2,4,...sinα(x) windows are Rife-Vincent class I (RVI) windows, for which IpDFT algorithms are known. We present a new IpDFTs for α = 1,3,5,.... The bias-variance trade-off of the proposed IpDFT fits between results offered by RVI windows, e.g., for α = 1, we get higher noise immunity than Hann (RVI order 1, α = 2) window and lower bias than rectangular (RVI order 0, α = 0) window.

Fast Synchrophasor Estimation by Means of Frequency-Domain and Time-Domain Algorithms

This paper presents a performance comparison of two classes of synchrophasor estimators recently proposed in the literature, i.e., the frequency-domain algorithms known as interpolated discrete Fourier transform (IpDFT), and the time-domain algorithms based on the weighted least squares (WLSs) approach. The analysis is focused on fast phasor estimation only and it is performed under steady-state, dynamic, and transient conditions, when the length of the observation interval is equal to one or two nominal cycles of the acquired electric waveform. The considered testing conditions include not only the worst-case scenarios specified in the IEEE Standard C37.118.1-2011, but they also combine the effect of static and dynamic disturbances. As accuracy performance parameters, the total vector error as well as the amplitude and phase estimation errors are considered and evaluated. Estimator responsiveness is analyzed instead in terms of response times, when amplitude or phase steps occur. The preferable one- and two-cycles IpDFT and WLS algorithms are first determined by choosing the windows and the number of Taylor's series terms (in the WLS case only) assuring the best accuracy. It is shown that, when the time-domain approach is considered and one-cycle or two-cycle intervals are observed, the best accuracy is achieved by the WLS algorithm based on the rectangular window or the two-term one-cycle maximum image tone interference rejection (MIR) window, respectively. Conversely, when the frequency-domain approach is adopted, the best accuracy is achieved by the IpDFT-based on the MIR window related to the observed interval length. The estimates produced by the selected WLS and IpDFT algorithms are then compared against the boundaries specified in the Standard IEEE C37.118.1-2011 for P-and M-class compliance. During the discussion, advantages and disadvantages of both time-domain and frequency-domain approaches are highlighted.

Synchronization of Sampling-Based Measuring Systesm

The amplitude and the phase of a measured single-tone signal can accurately be calculated from the discrete Fourier transform (DFT) components in case of coherent sampling. When the synchronization is not perfect, spectral leakage appears and errors are introduced in the determination of the signal parameters. In this paper, a new method to synchronize the sampling frequency with the frequency of the measured signal is presented. The method is based on the information contained in the fundamental and the sideband components of the DFT of the measured signal. A single measurement gives the synchronization error and permits the correction of the sampling frequency required to become coherent. Moreover, the error on the amplitude measurement relative to a quasi-coherent sampling is also discussed and quantitatively modeled. The results of the model as well as the synchronization process are experimentally verified and compared with the interpolated DFT algorithm.

Analysis of the Exponential Signal by the Interpolated DFT Algorithm

In this paper, exponential signals in the frequency domain are accurately analyzed by an algorithm, and the peaks of the discrete Fourier transform (DFT) result are adopted to obtain parameters that include amplitudes, frequencies, dampings, and phases. There are two steps for this algorithm: interpolated DFT and leakage elimination. Interpolated DFT refers to the three neighboring spectral lines at the peak of each mode used to calculate an approximate result, with the purpose of leakage elimination being to eliminate the influence of leakage on these data. After some iteration, this algorithm will obtain accurate parameters of the exponential components that are derived from their DFT results. The comparison of different algorithms is also discussed in this paper.