Negative Frequency Components Research Articles

A Stable Forward Modeling Approach in Heterogeneous Attenuating Media Using Reapplied Hilbert Transform

In the field of geological exploration and wave propagation theory, particularly in heterogeneous attenuating media, the stability of numerical simulations is a significant challenge for implementing effective attenuation compensation strategies. Consequently, the development and optimization of algorithms and techniques that can mitigate these numerical instabilities are critical for ensuring the accuracy and practicality of attenuation compensation methods. This is essential to reveal subsurface structure information accurately and enhance the reliability of geological interpretation. We present a method for stable forward modeling in strongly attenuating media by reapplying the Hilbert transform to eliminate increasing negative frequency components. We derived and validated new constant-Q wave equation (CWE) formulations and a stable solving method. Our study reveals that the original CWE equations, when utilizing the analytic signal, regenerate and amplify negative frequencies, leading to instability. Implementing our method maintains high accuracy between analytical and numerical solutions. The application of our approach to the Chimney Model, compared with results from the acoustic wave equation, confirms the reliability and effectiveness of the proposed equations and method.

Observation of Doppler shift modulated by the internal kink mode using conventional reflectometry in the EAST tokamak

In this paper we present a new experimental observation using a conventional reflectometry technique, poloidal correlation reflectometry (PCR), in the Experimental Advanced Superconducting Tokamak (EAST). The turbulence spectrum detected by the PCR system exhibits an asymmetry and induced Doppler shift during the internal kink mode (IKM) rotation phase. This Doppler shift is the target measurement of Doppler reflectometry, but captured by conventional reflectometry. Results show that the Doppler shift is modulated by the periodic changes in the effective angle between the probing wave and cutoff layer normal, but not by plasma turbulence. The fishbone mode and saturated long-lived mode are typical IKMs, and this modulation phenomenon is observed in both cases. Moreover, the value of the Doppler shift is positively correlated with the amplitude of the IKM, even when the latter is small. However, the positive and negative frequency components of the Doppler shift can be asymmetric, which is related to the plasma configuration. A simulated analysis is performed by ray tracing to verify these observations. These results establish a clear link between and IKM rotation, and are helpful for studying the characteristics of IKM and related physical phenomena.

Parameter Estimation of Power System Oscillation Signals under Power Swing Based on Clarke–Discrete Fourier Transform

Accurate knowledge of oscillation parameters (i.e., frequency, amplitude, phase, and damping factor) is crucial for control strategies of power systems under power swing. This paper presents a method for the parameter estimation of power system oscillation signals under power swing based on Clarke–DFT. The proposed method provides accurate parameter estimation of damped sinusoidal signals for both balanced and unbalanced systems, which performs well even in the presence of harmonics. In the meantime, the negative frequency components in the spectra of the damped sinusoidal signals, which are caused by system imbalance, are calculated accurately using complex-valued interpolated DFT. To verify the performance of the proposed method, simulations are performed under balanced and unbalanced conditions. The results of the simulations confirm the effectiveness of the proposed method either in unbalanced or harmonic conditions.

Signal-Flow-Graph Analysis of Weakly Nonlinear Microwave Circuits Around a Large-Signal Operating Point

A signal flow graph (SFG) representation of small-signal responses of nonlinear microwave circuits around a large-signal operating point is developed using the X-parameters. It is shown that, unlike the SFGs for linear circuits, negative-frequency nodes need to be included explicitly. The development elucidates the circuit-operational meaning of the elusive T-type small-signal X-parameters, which represent the interaction between positive-and negative-frequency components. As an example, such an SFG is used to derive a closed-form expression of the output power of an amplifier as a function of the load reflection coefficient. It is then used to plot approximate load-pull power contours. The result is consistent with the expressions of the optimum load reflection coefficient derived by Root et al. (EuMIC 2017) and power contours derived by Peláez-Pérez et al. (TMTT 2013). SFGs provide an alternative systematic means to derive closed-form expressions in terms of X-parameters and gain illuminating insights into the workings of weakly nonlinear circuits.

First principles investigations to study the impact of hydrostatic pressure on optoelectronic properties and stability of Rb2PdBr6

In this article, the effect of hydrostatic pressure upon the optoelectronic performances of the perovskite Rb2PdBr6 was investigated by using first principles calculations. The calculated results imply that the band gap of the perovskite Rb2PdBr6 can change with hydrostatic pressure. The energy band gap of Rb2PdBr6 can be decreased to 1.341 eV when 1.2 GPa hydrostatic pressure is applied. This band gap value greatly approaches the optimal band gap value (1.34 eV) of PSCs. After finding the optimal band gap value of perovskite Rb2PdBr6, we compare the optical properties of perovskite Rb2PdBr6 under the hydrostatic pressure of 0 GPa, 1 GPa, 1.2 GPa, 1.5 GPa, respectively. The results show that the band gap reduction of Rb2PdBr6 under hydrostatic pressure leads to some superior optical properties such as improved light-absorbing ability, superior dielectric capability, and preferable charge transportation performance. The calculated lattice parameters show that the perovskite Rb2PdBr6 maintains a stable cubic spatial structure under hydrostatic pressure, and the elastic modulus complies with the criterion of mechanical stability. Meanwhile, there is no negative frequency component in the computed phonon dispersion spectra. The superior optoelectronic performance along with beneficial stability under hydrostatic pressure make Rb2PdBr6 more suitable for PSCs.

Phase-based motion estimation in complex environments using the illumination-invariant log-Gabor filter

The phase-based motion estimation method is promising because the phase is sensitive to the motion information embedded in the pixel values. Unfortunately, complex environments such as extra textures and illumination changes exist in practice, whereas their effects have not been studied. This study theoretically discovers that the large spatial size and DC component of the filter adversely affect the performance of the motion estimation method under complex environments. The traditional method adopting the Gabor filter with a large bandwidth may suppress extra textures while cause a significant drift in the estimated motion due to the negative frequency and DC components. To tackle this, a log-Gabor-based new motion estimation method is proposed, in which the log-Gabor filter intrinsically without the DC and negative frequency components can extract the local phase and filter out the illumination change. A localization index based on the amplitude map is designed to determine the optimal parameters of the log-Gabor filter and maximize its spatial localization capability. Numerical and experimental examples demonstrate that the proposed method outperforms the traditional Gabor filter in motion estimation under the complex environment.

Sub-Dechirp Receiving Technology Based on the Double-Sideband Frequency Modulation

A practical sub-dechirp receiving technology for the double-sideband linear frequency modulation continuous wave (DSB-FMCW) lidar has been developed and experimentally demonstrated. Different from the traditional dechirp method, the local oscillator (LO) in the proposed technology has an independent modulator. This design can flexibly control the frequency of interference signal by adjusting the modulation parameters. At the same time, through the small difference between the modulation rate of LO and transmitted signal, the positive and negative frequency components of far and near targets can be distinguished, so as to effectively avoid overlap and ambiguity. Therefore, the proposed technology is more suitable for remote applications, and the imaging range can reach 3km.

Geophysical Tutorial: The significance of negative frequencies in Fourier transforms

Negative frequencies play a fundamental role in Fourier transforms, although they are often neglected in practice when displaying spectra of seismic traces. We can study their role quite easily when looking at the relationship of a seismic trace and its analytic counterpart that has no negative frequency components. This analytic trace, comprising the original trace and its Hilbert transform, is well known and used frequently to calculate attributes for seismic interpretation. Using Python code snippets and a simple Ricker wavelet as exemplary seismic signal, this tutorial explores and demonstrates the significance of negative frequencies in both a mathematical and a graphical way.

A Modified Complex Variational Mode Decomposition Method for Analyzing Nonstationary Signals with the Low-Frequency Trend

Complex variational mode decomposition (CVMD) has been proposed to extend the original variational mode decomposition (VMD) algorithm to analyze complex-valued data. Conventionally, CVMD divides complex-valued data into positive and negative frequency components using bandpass filters, which leads to difficulties in decomposing signals with the low-frequency trend. Moreover, both decomposition number parameters of positive and negative frequency components are required as prior knowledge in CVMD, which is difficult to satisfy in practice. This paper proposes a modified complex variational mode decomposition (MCVMD) method. First, the complex-valued data are upsampled through zero padding in the frequency domain. Second, the negative frequency component of upsampled data are shifted to be positive. Properties of analytical signals are used to get the real-valued data for standard variational mode decomposition and the complex-valued decomposition results after frequency shifting back. Compared with the conventional method, the MCVMD method gives a better decomposition of the low-frequency signal and requires less prior knowledge about the decomposition number. The equivalent filter bank structure is illustrated to analyze the behavior of MCVMD, and the MCVMD bi-directional Hilbert spectrum is provided to give the time–frequency representation. The effectiveness of the proposed algorithm is verified by both synthetic and real-world complex-valued signals.

On the construction of the orthonormal wavelet in the Hardy space H2(ℝ)

We are concerned with the orthonormal wavelet [Formula: see text] in the Hardy space [Formula: see text] which is a closed subspace of [Formula: see text] without negative frequency components. It is well known that there does not exist an [Formula: see text]-wavelet such that [Formula: see text] is continuous on [Formula: see text] and satisfies [Formula: see text] for some [Formula: see text]. The aim of this paper is to find a critical decay rate in the existing [Formula: see text]-wavelet under the condition that [Formula: see text] is continuous on [Formula: see text]. Moreover, we also construct a concrete [Formula: see text]-wavelet having infinite vanishing moments.

Nonlinear characteristic investigation of tribo-crack-dynamic rotor system based on full spectrum analysis

The tribo-crack-dynamic model and experimental method to research dynamic mechanism are proposed based on the full spectrum in this paper. In view of the nonlinear characteristics from rub-impact forces, oil-film forces and time-varying stiffness of crack shaft, the tribo-crack-dynamic rotor model is established to serve the analysis of dynamic mechanism and characteristics. The numerical simulation is developed to obtain the precession signal and bifurcation diagram for the tribo-crack-dynamic rotor system. The results show that the negative frequency components are more abundant and their amplitudes increase in the full spectrum, showing obvious reverse precession characteristics when cracks and rub-impact faults occur at the same time. Experiment investigation on nonlinear dynamics is designed, which verifies the tribo-crack-dynamic model and the positive and negative precession characteristics in multi-fault rotor system.

An analytic signal-based accurate time-domain viscoacoustic wave equation from the constant-Q theory

We have introduced a concise time-domain wave equation to accurately simulate wave propagation in viscoacoustic media. The central idea behind this work is to dismiss the negative frequency components from a time-domain signal by converting the signal to its analytic format. The negative frequency components of any analytic signal are always zero because we can construct the viscoacoustic wave equation to honor the relaxation property of the medium for positive frequencies only. The newly proposed complex-valued wave equation (CWE) represents the wavefield with its analytic signal, whose real part is the desired physical wavefield, whereas the imaginary part is the Hilbert transform of the real component. Specifically, this CWE is accurate for weakly and strongly attenuating media in terms of the dissipation and dispersion, and the attenuation is precisely linear with respect to the frequencies. The CWE easily and flexibly models dispersion-only, dissipation-only, or dispersion-plus-dissipation seismic waves. We have verified these CWEs by comparing the results with analytical solutions, and we achieved nearly perfect matches. When extending to heterogeneous media, the results are consistent with those computed from the nonstationary operator-based Fourier integral method and from the standard linear solid equations.

Dynamical current correlations in Cooper pair splitters based on proximized quantum dots

Entanglement of electrons is studied by means of current–current correlations in two Cooper pair splitter devices: with one and two proximized quantum dots (1QD and 2QD), in presence of intra- and inter-dot Coulomb interactions, and weakly coupled with metallic electrodes. The 1QD system, where Cooper pairs can be transmitted to the same or split to different normal electrodes, is contrasted with the 2QD device, where double occupancy of a single quantum dot is forbidden and transport is only through an inter-dot singlet due to non-local crossed Andreev reflection processes deep in the superconducting energy gap. Separating the current correlation function into components for partial currents of electrons and holes through various Andreev bound states, one can see bunching and antibunching of split particles: inter-level components between electron and hole currents flowing to different electrodes are positive, while intra-level electron–electron or hole–hole components are negative, respectively. Spectral decomposition of the frequency-dependent current cross-correlation is performed to get better insight into mechanisms of entanglement and dynamics of split Cooper pairs, and to extract various charge fluctuation processes with different relaxation times, related to electron and hole currents flowing through the Andreev bound states. Only low frequency polarization fluctuations are seen in the current cross-correlations, while various negative and positive high frequency (charge fluctuations) components compensate each other in the symmetric system.

Multi-Scale Modeling and Simulation of DFIG-Based Wind Energy Conversion System

A multi-scale transients model of a doubly fed induction generator (DFIG)-based wind energy conversion system (WECS) is developed, implemented, and validated. All ac circuit and control quantities of the electrical part are modeled by analytic signals rather than just real signals. In addition to the real parts, the analytic signals also comprise orthogonal imaginary parts. While measured Fourier spectra of real ac quantities involve positive and negative frequency components, they only involve positive frequency components when extended and represented as analytic signals. With the introduced shift frequency operator, the analytic signal can now be shifted in the frequency domain to reduce its maximum frequency contents and thus allow for a larger time-step size in accordance with the Nyquist criterion. If the shift frequency is set equal to the ac fundamental frequency, then the affiliated ac voltages and currents become dynamic phasors. At a zero shift, however, instantaneous signals can be tracked as in an electromagnetic transients program (EMTP). This is illustrated here for the voltage sourced converters (VSC) of the WECS. By appropriate selection of the shift frequency, both electromagnetic and electromechanical transients are simulated efficiently. Studies involving wind power fluctuations, three-phase-to-ground fault and single-phase-to-ground fault confirm these claims.

An Improved Harmonic Suppression Control Strategy for the Hybrid Microgrid Bidirectional AC/DC Converter

The nonlinear load in AC/DC hybrid microgrid seriously affects the power transmission quality of bidirectional AC/DC interlinking converter (BIC). In this article, an improved Fryze-Buchholz-Depenbrock (FBD) harmonic current detection method combined with droop control is designed to suppress harmonic current control strategy. The decoupled double synchronous reference frame phase locked loop (DDSRF-PLL) with self decoupled network is used to eliminate the negative sequence fundamental frequency component, and the accurate phase detection can also be realized when affected by nonlinear load. The least mean square (LMS) algorithm adaptive filter can enhance the dynamic response ability when the load current suddenly changes. To solve the problem that the cut-off frequency of low pass filter (LPF) can not take into account the detection accuracy and response speed of FBD harmonic current detection method. Finally, Matlab/Simulink is used for simulation verification. The simulation results show that the improved control strategy not only realizes the bidirectional flow of power and maintains the DC bus voltage stability, but also reduces the total harmonic distortion (THD) at point of common coupling (PCC) by 0.45% compared with the traditional FBD control strategy. And the effectiveness and feasibility of the design method are proved.

A new parameter estimation method for damped real-value sinusoidal signals in noise

To reduce the spectrum leakage influence of the negative frequency component of damped real-value sinusoidal signals, a high precision parameter estimation method is proposed in this paper. The fast Fourier transform algorithm is adopted to improve the real-time performance, and the negative frequency influence is suppressed by a spectrum subtract strategy based on an interpolation method. Then, the accurate frequency, damping factor, initial amplitude and phase of the sampled signals are obtained by an iterative procedure. The theoretical bias and variance are analyzed, and the analytic results indicate that the proposed method is an asymptotic unbiased estimation method. The simulation testing is performed, and the numerical results demonstrate that the proposed method suppresses the spectrum leakage influence of the negative frequency component and improves the estimation performance. Moreover, the efficacy of the proposed method is validated by experimental testing on a Coriolis mass flowmeter.

Automotive Radar Interference Mitigation Using Adaptive Noise Canceller

Interference among frequency modulated continues wave automotive radars can either increase the noise floor, which occurs in the most cases, or generate a ghost target in rare situations. To address the increment of noise floor due to interference, we proposed a low calculation cost method using adaptive noise canceller to increase the signal-to-interference ratio. In a quadrature receiver, the interference in the positive half of frequency spectrum is correlated to that in the negative half of frequency spectrum, whereas the beat frequencies from real targets are always present in the positive frequency. Thus, we estimated the power of the negative frequency as an indication of interference, and fed the positive frequency and negative frequency components into the primary and reference channel of an adaptive noise canceller, respectively. The least mean square algorithm was used to solve for the optimum filter solution. As a result, both the simulation and experiment showed a good interference mitigation performance.

Influence of the noise on DFT-based sine-wave frequency and amplitude estimators

This paper analyses the effect of wide-band noise on the accuracy of generic sine-wave frequency and amplitude estimators based on Discrete Fourier Transform (DFT) samples located closely to the spectrum peak. A cosine window is adopted to reduce the influence on the estimated parameters of spectral leakage from the negative frequency component. Closed-form expressions for the estimator variances are derived. Moreover, the variance of a frequency estimator expressed as a ratio of generic linear functions of DFT sample modules is specifically investigated. It is also shown that the same frequency estimation accuracy can be achieved when complex-valued DFT samples are considered. The derived expressions are verified on a specific sine-wave frequency estimator and a related amplitude estimator, using both simulation and experimental results.

Accurate Frequency Estimation of a Real Sinusoid by Three New Interpolators

Interpolators are widely employed in the frequency estimation of sinusoidal signals. However, for a real-valued sinusoidal signal, the existing interpolators do not consider the negative frequency component, which leads to a frequency estimation error floor. To solve this problem and further improve the estimation accuracy, three new interpolators are proposed in this paper. These three new interpolators consist of two initial interpolators and one fine interpolator. In the proposed algorithm, the initial interpolators consider both the positive and negative frequency components and are utilized first to eliminate this error floor and obtain an initial frequency estimate. Then, the fine interpolator for the frequency estimation of a complex sinusoid is exploited to improve the accuracy of the initial frequency estimate. The theoretical analysis demonstrates that the frequency estimation mean square error of the proposed algorithm is almost equal to the Cramer-Rao lower bound. Compared with the existing time-domain analysis algorithms, the proposed algorithm has better estimation performance, especially in the case of a low signal-to-noise ratio. Compared with the existing frequency-domain analysis algorithms, the proposed algorithm has lower computational complexity and a wider valid estimation range.

Complex Singular Spectrum Decomposition and its Application to Rotating Machinery Fault Diagnosis

In recent years, singular spectrum decomposition (SSD) has been recognized as a powerful signal decomposition technique and has become a useful tool for fault signature extraction of rotating machinery. A complex SSD (CSSD) approach is proposed in this paper, extending SSD to the complex domain. Moreover, the equivalent filter banks property of CSSD is studied by using the Gaussian noise. Based on the CSSD and Hilbert transform (HT), two fault feature extraction algorithms, named CSSD Hilbert time-frequency spectrum (CSSD-HTFS) and CSSD Hilbert envelope spectrum (CSSD-HES), are exploited. In order to fuse the information of multi-sensors and extract sufficient fault features, a novel fault diagnosis framework for rotating machinery is developed. Specifically, the vibration signals collected along the X-direction and Y-direction are firstly employed to construct a complex-valued signal. Next, the CSSD of the complex-valued signal is carried out to obtain a series of decomposed components associated with the positive and negative frequency components. Following, the CSSD-HTFS or the CSSD-HES algorithm is adopted to extract the fault signatures of the decomposed components. Finally, the fault pattern of the rotating machinery is recognized based on the extracted fault signatures. Several examples of fault diagnosis applications validate the developed methodology.