Sidelobe Decay 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.

Two-dimensional Hanning self-convolution window for enhancing Moiré fringe alignment in lithography

The current image processing techniques heavily rely on classic windows for suppressing spectral leakage. However, the limited capability of these classic windows to suppress spectral leakage restricts their application in the field of high-precision measurement. This paper proposes a novel family of windows capable of suppressing spectral leakage associated with the Moiré fringes pattern, thereby enhancing the accuracy of lithography alignment. Specifically, a novel two-dimensional Hanning self-convolution window is introduced, which is derived from the Hanning window to inherit its advantage of ultra-rapid sidelobe decay. The behaviors of the main lobe and desirable sidelobes of the 1st- to 4th-order are examined. Additionally, a novel fringe phase extraction algorithm is established based on the proposed window, which demonstrates high accuracy and efficiency. Simulation and experimental results validate the feasibility and practicality of the proposed window. In particular, the proposed window outperforms classic windows in suppressing spectral leakage, thereby successfully enhancing the phase extraction accuracy to achieve high-precision misalignment measurement at the sub-2-nm level (1.47 nm @ 3σ criterion). The proposed method can be easily extended to other real engineering applications utilizing fringe to achieve high-precision measurements, such as fringe projection profilometry, damage detection, and visual mechanical vibration monitoring.

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.

New Class of Cosine-Sum Windows

Window functions are required to achieve excellent frequency resolution, peak sidelobe ratio, and sidelobe decay. The cosine-sum window function satisfies these requirements. However, conventional cosine-sum windows limit the number of terms and coefficient values. Therefore, this study proposes a new class of cosine-sum windows that has more terms than those in the conventional cosine-sum window and considers positive and negative coefficients. For such windows, we present three design examples. The first is a window function with low sidelobe decay, which reduce the sidelobes by adding correction terms to the conventional windows without damaging the sidelobe decay. The second is a window function that improves the frequency resolution by emphasizing the discontinuities of the extracted signal. The third is a window function that reduces sidelobes further without deteriorating the frequency resolution by adding correction terms to the window functions, which are designed in advance. We confirmed the effectiveness of these examples of the proposed window function.

Performance analysis of N-continuous OFDM systems with low-interference signal

N-continuous orthogonal frequency division multiplexing (NC-OFDM) is a promising multi-carrier transmission waveform conceived for improving sidelobe suppression performance. To reduce the severe inband interference in traditional NC-OFDM, we have proposed low-interference signal modeling for NC-OFDM. However, spectral leakage and error may be undesirably increased owing to the limited continuous differentiability of the smooth signal. In this paper, the low-interference scheme is investigated in terms of power spectrum density (PSD) and error performance, under the parameters of the highest derivative order (HDO) and the length of the smooth signal, to prove and quantify its advantages over traditional NC-OFDM. In the context of PSD, sidelobe decay is evaluated upon considering two discontinuous points due to the finite continuity of the smooth signal and its higher-order derivatives. Furthermore, it was shown that the low-interference scheme incurs small signal-to-noise ratio (SNR) loss and bit error rate (BER) for a short length of the smooth signal or a small HDO compared to traditional NC-OFDM. Meanwhile, due to the cyclostationarity loss imposed by the smooth signal, an effective solution is suggested for the time synchronization in a practical system. Based on analyses and simulation results, the tradeoffs between sidelobe suppression and BER are studied.

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.

Research on Cosine-Sum Windows with Maximum Side-Lobe Decay for High Precision ADC Spectral Testing

Achieving coherent sampling has always been a major challenge in analog-to-digital converter (ADC) spectral testing. If the coherent sampling condition cannot be met, leakage appears in the spectrum, which results in inaccurate parameters. The windowing method is widely used to process ADC data to eliminate leakage. However, this method requires prior knowledge about the type of window, and some commonly used windows cannot provide accurate results for a high precision ADC. In this paper, some general principles for optimizing the cosine-sum window and some windows with maximum side-lobe decay have been presented. A test method for eliminating the leakage caused by non-coherent sampling is also proposed. The proposed method can accurately evaluate the dynamic parameters of an ADC with arbitrary non-coherency. Various simulation results and measurement data demonstrate the functionality and robustness of the proposed method. The proposed method significantly relaxes the condition of coherent sampling and decreases the test cost.

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.

Voltage unbalance factor detection based Kaiser- maximum sidelobe decay convolution window and amplitude method

The wide application of distributed energy and nonlinear load causes the voltage unbalance of power grid. This paper establishes an accurate method based on the Kaiser- maximum sidelobe decay (K-MSD) convolution window and three-spectral line interpolated FFT algorithm and the amplitude method to detect the negative sequence voltage unbalance factor. The voltage fundamental wave amplitude is calculated by improved FFT algorithm, which improves the calculation accuracy. Then, the negative sequence voltage unbalance factor is calculated by the amplitude method. It avoids complex phasor calculations. Based on the proposed method the voltage unbalance factor detection system is developed and implemented in a platform. The simulation experiment results show that the proposed algorithm in this paper can effectively realize the accurate detection of unbalance factor under the interference of fundamental wave, harmonics and noise. Compared with the method recommended by the IEC61000-4-27, it is easy to implement and can achieve the same level of precision.

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.

Design of Windows for Efficient Spectral Resolution and Harmonic Suppression [Tips and Tricks

This article discusses the requirements for a window function in applications aiming at efficient spectral resolution along with high harmonic suppression. A brief review of potential windows that tend to meet these requirements is presented, and it is noted that polynomial windows are best suited for such applications. A design procedure is described to synthesize a polynomial window with a 24-dB/octave sidelobe decay and minimal main-lobe width.

A Versatile Hyperbolic Cosine Window for Spectral Resolution

A simple but versatile hyperbolic cosine window function is presented in this paper, which has two terms in the time domain. A parameter p in the window can be varied to make the mainlobe width of the window function to approach that of a rectangular window ( $$\pm 1/T,$$ ) while maintaining higher sidelobe decay (12 dB/octave.) Even though such behavior has been demonstrated by the two-term polynomial window, it suffers from the limitation that only some discrete values of mainlobe widths can be achieved in the range, $$\pm 1/T$$ to $$\pm 1.5/T$$ . The proposed hyperbolic cosine window has no such limitation; one can achieve any desired value of mainlobe width in the above range. The proposed window can be employed for applications involving spectral resolution.

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.