Nonlinear Waveform Research Articles

Waveform Model to Characterize Time-Domain Pulses Resulting in EMI on Static Energy Meters

This article presents a time-domain waveform model developed to characterize pulsed, nonlinear, current waveforms resulting in electromagnetic interference on static energy meters. The waveform model is calculated by fitting the sampled waveform data into a linear piece-wise function through a process that involves applying algorithms of pulse extraction, change-point detection, and redundancy elimination. The model is applied to data from laboratory experiments that have indicated critical current waveforms resulting in electromagnetic interference problems with static meters. Afterwards, the parameters of the modeled waveforms are calculated in order to correlate them to metering errors. The most relevant parameters that are correlated to significant errors are the maximum slope, crest factor, pulse duration, and charge. The waveform model provides an accurate description of the complex nonlinear waveforms through simplified analytical expressions that reproduce the significant features of the interfering waveforms. This waveform modeling approach could be used to standardize the artificial test signals that are representative of realistic devices and scenarios.

A novel study on perception–cognition scenario in music using deterministic and non-deterministic approach

In the last few decades, nonlinear science and chaos theory has provided several robust non-deterministic tools by means of which the complexity of a nonlinear audio waveform can be measured precisely. On the other hand, sound signal analysis in linear deterministic approach has reached a new dimension where a number of well equipped software have been developed which can minutely measure and control the basic parameters of sound like pitch, intensity, tempo etc. The main objective of the present work is to quantitatively study the changes in acoustic signal complexity (measured using chaos based fractal technique) with individual variation in pitch, loudness and timbre of a sound signal. EEG (Electroencephalography) was also performed on 10 participants to see how the neuro-cognitive attributes of a sound change, i.e. when these basic components — pitch, loudness and timbre of the sound vary, one at a time. Single strokes of a piano were recorded where pitch and loudness of the sound signals were varied one at a time keeping the other parameters fixed. Then the sounds of 14 different musical instruments playing the same pitch at same loudness were recorded, which effectively served the purpose of timbre variation. EEG experiment was conducted with these audio signals as stimuli for the participants. The multifractal spectral widths were calculated for all the music signals as well as the corresponding EEG signals using Multifractal Detrended Fluctuation Analysis (MFDFA) and compared with each other. The results point towards the direction of a correlation between the conventional linear parameters and the latest nonlinear features in the acoustic domain, while the changes in the multifractal values of the different EEG waves reveal new information about the cognition of the basic features of sound in human brain. This study is a novel attempt to provide new data in engulfing apparent objective (acoustics) - subjective (EEG) connection, which is highly needed for building any model for perception–cognition connectivity.

Application of Higher-Order Interpolation Perturbation Theory on Postprandial Plasma Glucose Waveform of a Single Lunch Based on GH-Method: Math-Physical Medicine (No. 426)

In this particular research note, the author arbitrarily selects the postprandial plasma glucose (PPG) data and waveforms from his lunch on 3/31/21 as the reference base. He then applies the higher-order, including first-order, second-order, and third-order interpolation perturbation theory to generate three approximated PPG waveforms to compare them against the measured nonlinear PPG waveform. The two major purposes of this study is to check the waveform shape similarity via correlation coefficients and the prediction accuracy via comparison of the averaged PPG values using the “higher order” interpolation perturbation theory. One obvious conclusion can be drawn from this article is that the higher-order perturbation equation provides a better approximate solution. However, in reality, there are very few patients who understand the perturbation theory of quantum mechanics, let alone to be able to conduct the complex mathematical calculations in order to achieve their desired glucose projection.

A lattice method for seismic wave propagation in nonlinear viscoelastic media

SUMMARYNonlinear wave propagation is an important consideration in several geophysical problems as heterogeneities within the subsurface give rise to nonlinear stress–strain relationships in rocks. To address these problems, where the inclusion of subsurface complexity is accounted for, numerical solutions are required. Here, we outline a discrete particle or lattice numerical method that can simulate dynamic and static deformation in an isotropic viscoelastic nonlinear medium where the nonlinearity includes the fourth-order elastic tensor coefficients. The relationship between the local interaction constants in the lattice method and the macroscopic nonlinear coefficients is derived and the relationships presented. The theoretical dispersion properties of the system is determined from a nonlinear perturbation method and are validated against some numerical results. There is an excellent agreement between the theory and numerical results and the method reproduces the common features of nonlinear wave propagation, for example, harmonics, waveform distortion and spectra shifts. As expected the system is shown to be dependent on the wave amplitudes and an example is presented which shows the complexity that can arise in both dynamic and static deformation in a complex model where nonlinearity is included.

A variable chirp rate stepped frequency linear frequency modulation waveform designed to approximate wideband non-linear radar waveforms

The non-linear frequency modulation (NLFM) waveform is one of the existing waveforms that can be used in high range resolution radar applications. However, a high sampling frequency and consequently an expensive ADC are required. To overcome this drawback while taking advantage of the features of the NLFM waveform, we suggest approximating the wideband NLFM waveform by a piecewise linear waveform and using it in a stepped frequency (SF) framework. Thus, a variable chirp rate SF-LFM waveform is proposed where SF is combined with a train of LFM pulses having different chirp rates, durations, and bandwidths. In this paper, these parameters are derived from a tangent-based NLFM waveform. At the receiver, a generalized version of the time domain (TD) algorithm is proposed to process the received echoes. Our purpose is to obtain the high range resolution profile (HRRP) whose properties are of the same magnitude orders as those obtained using a tangent-based NLFM waveform. These properties are the peak sidelobe ratio, the integrated sidelobe, and the range resolution. Toward this goal, a multi-objective optimization issue is addressed to deduce the parameters of the proposed waveform by using two types of approaches based on evolutionary algorithms. Their relevance is compared. Our analysis and simulations show that the proposed approaches attain the targeted performance goals with a smaller sampling frequency at the receiver.

Nonlinear full waveform inversion of wide-aperture OBS data for Moho structure using a trans-dimensional Bayesian method

SUMMARY Seismic full waveform inversion (FWI) is a powerful method for estimating quantitative subsurface physical parameters from seismic data. As the FWI is a nonlinear problem, the linearized approach updates model iteratively from an initial model, which can get trapped in local minima. In the presence of a high-velocity contrast, such as at Moho, the reflection coefficient and recorded waveforms from wide-aperture seismic acquisition are extremely nonlinear around critical angles. The problem at the Moho is further complicated by the interference of lower crustal (Pg) and upper mantle (Pn) turning ray arrivals with the critically reflected Moho arrivals (PmP). In order to determine velocity structure near Moho, a nonlinear method should be used. We propose to solve this strong nonlinear FWI problem at Moho using a trans-dimensional Markov chain Monte Carlo (MCMC) method, where the earth model between lower crust and upper mantle is ideally parametrized with a 1-D assumption using a variable number of velocity interfaces. Different from common MCMC methods that require determining the number of unknown as a fixed prior before inversion, trans-dimensional MCMC allows the flexibility for an automatic estimation of both the model complexity (e.g. the number of velocity interfaces) and the velocity–depth structure from the data. We first test the algorithm on synthetic data using four representative Moho models and then apply to an ocean bottom seismometer (OBS) data from the Mid-Atlantic Ocean. A 2-D finite-difference solution of an acoustic wave equation is used for data simulation at each iteration of MCMC search, for taking into account the lateral heterogeneities in the upper crust, which is constrained from traveltime tomography and is kept unchanged during inversion; the 1-D model parametrization near Moho enables an efficient search of the trans-dimensional model space. Inversion results indicate that, with very little prior and the wide-aperture seismograms, the trans-dimensional FWI method is able to infer the posterior distribution of both the number of velocity interfaces and the velocity–depth model for a strong nonlinear problem, making the inversion a complete data-driven process. The distribution of interface matches the velocity discontinuities. We find that the Moho in the study area is a transition zone of 0.7 km, or a sharp boundary with velocities from around 7 km s−1 in the lower crust to 8 km s−1 of the upper mantle; both provide nearly identical waveform match for the field data. The ambiguity comes from the resolution limit of the band-limited seismic data and limited offset range for PmP arrivals.

Double‐adaptive chirplet transform for radar signature extraction

This study presents a novel method for time-frequency (TF) signal analysis and chirp rate estimation dedicated for electromagnetic spectrum sensing, electronic warfare, electronic intelligence and/or passive bistatic radar purposes in which frequency modulated signals occur. The approach is based on the double-adaptive chirplet transform, providing optimal analysis window parameters, which is a crucial problem during TF processing. The presented methodology is based on a Gaussian window with two degrees of freedom, which results in a strong concentration of energy on the TF plane around the main component, even if the initial window parameters were mismatched. As an example of the method's usefulness, different types of real-life radar pulses were processed: firstly, two types of non-linear frequency-modulated waveforms were examined as a suitable illustration of the changeability of the analysis window parameters. Secondly, the linear frequency modulated signals were analysed in the presence of strong interference and multipath propagation. The waveforms were processed and compared, creating individual radar signatures, which may allow transmitter classification and signal reconstruction to be carried out in further processing. Moreover, the estimation limitations were compared to the Cramer-Rao lower bound, and an appendix organising mathematical fundamentals for the analysed methods is provided.

To Predict a Shock Pulse using Non Linear Dynamic Model of Rubber Waveform Generator

Drop test produces a variety of shock pulses, these shock pulses can accurately be predicted using polynomial based nonlinear empirical relations. The aim of this study is to monitor and improve the reliability of such systems/structures/equipment that might face critical installations or unlikely shocks during transportation, otherwise catastrophic failure is difficult to avoid. Rubber Wave Generators (RWGs) in Drop Test Machines (DTMs) are generally used for producing sinusoidal shock pulse of specific time and height. Three types of RWGs are used for analyzing the behavior of the generated shock pulse. The nonlinear equation expressing the restoring and damping forces through cubic and quadratic nonlinear terms is formulated. Harmonic Balance Method (HBM) is used to solve the nonlinear equation. Each RWG is excited harmonically through its base. The nonlinear stiffness and damping terms are obtained experimentally by the use of suitable curve fitting techniques. The nonlinear equation is reformulated to predict the shock pulse during drop test. Drop tests ranges from low g to 156g are performed at different heights and weights. Results of both experimental and nonlinear equation waveforms are within prescribed range defined in JEDEC and MIL standards for all three RWGs. The results also show that the presented method has the ability to accurately predict the shock pulse shape, time and width at a range of heights and weights.

Nonlinear Equalization Based on Artificial Neural Network in DML-Based OFDM Transmission Systems

This article reports the application of an equalizer based on an artificial neural network (ANN), in the form of nonlinear waveform regression, to mitigate nonlinear impairments in directly modulated laser (DML)-based orthogonal frequency-division multiplexing (OFDM) optical transmission. Experiments involving transmission over 0-200 km demonstrate that using an ANN with one hidden layer can greatly reduce nonlinear distortion. The proposed scheme outperformed a Volterra nonlinear equalizer at transmission distances exceeding 25 km. Using a 10G-class DML, the proposed scheme achieved the following data rates: 39.2 Gbps at 100 km (an improvement of 59%) and 33.5 Gbps at 150 km (an improvement of 57%). We also modified the cost function of the ANN during the training procedure to overcome the poor signal-to-noise ratio of the original ANN at low frequencies. This resulted in >30-Gbps transmission over 0-200 km.

Stretched-Pulse Soliton Kerr Resonators.

Kerr resonators support novel nonlinear wave phenomena including technologically important optical solitons. Fiber Kerr resonator solitons enable wavelength and repetition-rate versatile femtosecond-pulse and frequency-comb generation. However, key performance parameters, such as pulse duration, lag behind those from traditional mode-locked laser-based sources. Here we present new pulse generation in dispersion-managed Kerr resonators based on stretched-pulse solitons, which support the shortest pulses to date from a fiber Kerr resonator. In contrast to established Kerr resonator solitons, stretched-pulse solitons feature Gaussian temporal profiles that stretch and compress each round trip. Experimental results are in excellent agreement with numerical simulations. The dependence on dispersion and drive power are detailed theoretically and experimentally and design guidelines are presented for optimizing performance. Kerr resonator stretched-pulse solitons represent a new stable nonlinear waveform and a promising technique for femtosecond pulse generation.

Low-complexity nonlinear equalizer based on absolute operation for C-band IM/DD systems

The intensity-modulation/direct-detection transmission system operating in the C-band suffers from nonlinear waveform distortions induced by fiber chromatic dispersion due to the square-law detection. The Volterra nonlinear equalizers (VNLEs) can be used at the receiver to compensate for such distortions. However, the major concern about the equalizers is their huge implementation complexity. In this paper, we propose and demonstrate a low-complexity nonlinear equalizer based on the absolute operation for a cost-sensitive IM/DD system. In this equalizer, the cross-beating product terms (required in VNLE) are replaced with the absolute operation of the sum of two input samples. We evaluate the performance of the proposed equalizer over a 56-Gb/s 4-ary pulse amplitude modulation transmission system implemented by using 1.5-µm directly modulated laser or electro-absorption modulated laser. The results show that the proposed equalizer performs similar to the 2nd-order diagonally-pruned VNLE, but lowers the implementation complexity by >20%. We also show that the proposed equalizer outperforms the VNLE when the implementation complexities of the two nonlinear equalizers are similar.

Poroacoustic Traveling Waves under the Rubin–Rosenau–Gottlieb Theory of Generalized Continua

We investigate linear and nonlinear poroacoustic waveforms under the Rubin–Rosenau– Gottlieb (RRG) theory of generalized continua. Working in the context of the Cauchy problem, on both the real line and the case with periodic boundary conditions, exact and asymptotic expressions are obtained. Numerical simulations are also presented, von Neumann–Richtmyer “artificial” viscosity is used to derive an exact kink-type solution to the poroacoustic piston problem, and possible experimental tests of our findings are noted. The presentation concludes with a discussion of possible follow-on investigations.

Kink-based mirrorless quasi-bistability in resonantly absorbing media

Optical bistability is the basic nonlinear phenomenon mediating the control of light by light, being of ultimate demand for a plethora of applications in laser information technologies. The two most general designs are driven by the presence or absence of an external feedback, among which the mirrorless architecture seems promising, being free of bulky mirrors, resonant cavities, photonic crystals, etc. In this Letter, we propose a method to achieve optical quasi-bistability governed by the formation of specific nonlinear waveforms called “kinks.” We show that a thin layer of the relatively dilute resonant medium specially designed to support kinks could serve as a platform for compact, ultrafast, low-power optical switching. This mechanism does not require high densities of resonant particles specific for the feedback-free devices driven by dipole–dipole interactions, and enhances the overall practical relevance of such devices for optical computing.

Revisiting terrain-following Boussinesq equations on a highly variable periodic bed

On re-deriving the Boussinesq-type equations in a terrain-following coordinate system based on the conformal map for an arbitrarily (submerged) periodic bed, it is pointed out that some aspects of the existing approach need to be clarified, specifically the use of velocity potential gradient in the mapped plane in connection with the transformation of velocity vector, and the appropriate frequency–wavenumber relation that is from the solution to the linearized Boussinesq equations. It is shown that as the bed undulation height increases, this relationship increasingly departs from the flat-bottom dispersion relation that has previously been assumed for the terrain-following Boussinesq systems. Over a highly variable periodic bed, the waveforms of linear time harmonic waves already have features reminiscent of nonlinear waveforms, which should be distinguished from the subsequent nonlinear evolution.

Patient asynchrony modelling during controlled mechanical ventilation therapy

Background and objectiveMechanical ventilation therapy of respiratory failure patients can be guided by monitoring patient-specific respiratory mechanics. However, the patient's spontaneous breathing effort during controlled ventilation changes airway pressure waveform and thus affects the model-based identification of patient-specific respiratory mechanics parameters. This study develops a model to estimate respiratory mechanics in the presence of patient effort. MethodsGaussian effort model (GEM) is a derivative of the single-compartment model with basis function. GEM model uses a linear combination of basis functions to model the nonlinear pressure waveform of spontaneous breathing patients. The GEM model estimates respiratory mechanics such as Elastance and Resistance along with the magnitudes of basis functions, which accounts for patient inspiratory effort. Results and discussionThe GEM model was tested using both simulated data and a retrospective observational clinical trial patient data. GEM model fitting to the original airway pressure waveform is better than any existing models when reverse triggering asynchrony is present. The fitting error of GEM model was less than 10% for both simulated data and clinical trial patient data. ConclusionGEM can capture the respiratory mechanics in the presence of patient effect in volume control ventilation mode and also can be used to assess patient-ventilator interaction. This model determines basis functions magnitudes, which can be used to simulate any waveform of patient effort pressure for future studies. The estimation of parameter identification GEM model can further be improved by constraining the parameters within a physiologically plausible range during least-square nonlinear regression.

Nonlinear wave propagation in media with attenuation and dispersion modeled using a superposition of relaxation mechanisms

Michael Buckingham has long held interest in modeling wave propagation in media with nontraditional attenuation and dispersion, especially unconsolidated marine sediments exhibiting power-law attenuation [Buckingham, J. Acoust. Soc. Am. 138, 2871 (2015)]. The present authors have examined the implications of power-law attenuation and single relaxation mechanisms for modeling nonlinear propagation of compressional and shear waves [Cormack and Hamilton, Wave Motion 85, 13 (2019)]. Reported here are results for media that are modeled using multiple relaxation mechanisms. Below the lowest relaxation frequency, attenuation increases as frequency squared, and above the highest relaxation frequency, attenuation is constant. In between, the superposition may be tailored to approximate power-law attenuation that increases in proportion to frequency raised to an exponent between 0 and 2. Regardless of the source frequency and the number of relaxation mechanisms incorporated, there exists a critical source amplitude above which the mathematical model is incapable of offsetting nonlinear waveform distortion sufficiently to stabilize shock formation beyond the distance where an infinite gradient first appears in the waveform. Results are presented for birelaxing media corresponding to seawater and atmosphere, followed by models based on multiple relaxation mechanisms that approximate power-law attenuation over limited frequency ranges in sediment and tissue.

Aberration correction using nonlinear backscattered signals from the focus of an ultrasound beam

High intensity focused ultrasound (HIFU) therapies are often affected by aberrations, which can severely limit treatments. For multi-element arrays, applying phase corrections for each element could mitigate these effects. Here, an aberration correction method based on cross-correlation of nonlinear HIFU waves, backscattered from the focus is proposed and tested. A spherically focused 1.5 MHz 256-element HIFU array powered by Verasonics system was used for both emitting and receiving backscattered ultrasound waves. A short (3-cycle) high-amplitude pulse was transmitted through an aberrating layer into an ex vivo clotted bovine blood sample. The backscattered signal was recorded by individual array elements and high-pass filtered at 4 MHz. The layer introduced different acoustic path lengths for individual elements leading to a decrease in the peak-positive pressure of ~70%. The time delay required for each element to restore the nonlinear waveform at the focus was calculated based on the cross-correlation between the backscattered filtered signals. The results were confirmed by replacing the sample with a hydrophone. Using the corrected time delays, peak-positive focal pressure increased over 75% from uncorrected aberrated levels. These results demonstrate that the proposed focal “nonlinear beacon” can be used for phase correction feedback in HIFU treatments of soft tissue. [Work supported by NIH R01GM122859, R01EB007643, and RSF19-12-00148.]

Simulation of nonlinear trans-skull focusing and formation of shocks in brain using a fully populated ultrasound array with aberration correction.

Multi-element high-intensity focused ultrasound phased arrays in the shape of hemispheres are currently used in clinics for thermal lesioning in deep brain structures. Certain side effects of overheating non-targeted tissues and skull bones have been revealed. Here, an approach is developed to mitigate these effects. A specific design of a fully populated 256-element 1-MHz array shaped as a spherical segment (F-number, F# = 1) and filled by randomly distributed equal-area polygonal elements is proposed. Capability of the array to generate high-amplitude shock fronts at the focus is tested in simulations by combining three numerical algorithms for linear and nonlinear field modeling and aberration correction. The algorithms are based on the combination of the Rayleigh integral, a linear pseudo-spectral time domain Kelvin-Voigt model, and nonlinear Westervelt model to account for the effects of inhomogeneities, aberrations, reflections, absorption, nonlinearity, and shear waves in the skull. It is shown that the proposed array can generate nonlinear waveforms with shock amplitudes >60 MPa at the focus deep inside the brain without exceeding the existing technical limitation on the intensity of 40 W/cm2 at the array elements. Such shock amplitudes are sufficient for mechanical ablation of brain tissues using the boiling histotripsy approach and implementation of other shock-based therapies.

ADMM-based multiparameter wavefield reconstruction inversion in VTI acoustic media with TV regularization

SUMMARYFull waveform inversion (FWI) is a nonlinear waveform matching procedure, which suffers from cycle skipping when the initial model is not kinematically accurate enough. To mitigate cycle skipping, wavefield reconstruction inversion (WRI) extends the inversion search space by computing wavefields with a relaxation of the wave equation in order to fit the data from the first iteration. Then, the subsurface parameters are updated by minimizing the source residuals the relaxation generated. Capitalizing on the wave-equation bilinearity, performing wavefield reconstruction and parameter estimation in alternating mode decomposes WRI into two linear subproblems, which can be solved efficiently with the alternating-direction method of multiplier (ADMM), leading to the so-called iteratively refined WRI (IR–WRI). Moreover, ADMM provides a suitable framework to implement bound constraints and different types of regularizations and their mixture in IR–WRI. Here, IR–WRI is extended to multiparameter reconstruction for vertical transverse isotropic (VTI) acoustic media. To achieve this goal, we first propose different forms of bilinear VTI acoustic wave equation. We develop more specifically IR–WRI for the one that relies on a parametrization involving vertical wave speed and Thomsen’s parameters δ and ϵ. With a toy numerical example, we first show that the radiation patterns of the virtual sources generate similar wavenumber filtering and parameter cross-talks in classical FWI and IR–WRI. Bound constraints and TV regularization in IR–WRI fully remove these undesired effects for an idealized piecewise constant target. We show with a more realistic long-offset case study representative of the North Sea that anisotropic IR–WRI successfully reconstruct the vertical wave speed starting from a laterally homogeneous model and update the long wavelengths of the starting ϵ model, while a smooth δ model is used as a passive background model. VTI acoustic IR–WRI can be alternatively performed with subsurface parametrizations involving stiffness or compliance coefficients or normal moveout velocities and η parameter (or horizontal velocity).

Investigation of the effect of the bed slope on extreme waves using First Order Reliability Method

The effect of bed slope on the force statistics and shape of the force time history around the force peak is investigated numerically with a fully nonlinear wave model and FORM analysis. The numerical model setup is validated by comparison of averaged experimental force and free surface elevation time series and the FORM results for the given force levels. The fully nonlinear FORM solution shows a good improvement over first-order and second-order results by increasing the asymmetry of the force history through the peak. The average deviation between FORM and the experimental curve is found to be at the level of 10% of the maximum force value. Next, the order statistics for force peaks from experimental and numerical 3 h realization are compared. Bootstrapping is applied to estimate the expected mean value of the inline forces for a given exceedance probability and a good match between the numerical and experimental order statistics is found.FORM predictions of the force peak histories for a cylinder on flat or sloping bed are next compared. The diameter, depth at the structure and significant wave height are kept identical between the two cases. It is found that the force histories are not changed significantly by the presence of the slope in the sea states with lower Ursell number. For the larger Ursell number sea states, the corresponding time histories of the free surface elevation, however, show a larger skewness for the flat bed case. Further, from the FORM analysis, the exceedance probabilities for a given force level at sloped bed is found to be larger than for flat bed. The ratio of exceedance probability is found to increase with force level except for the sea states with largest Ursell number where the numerical results are affected by strong nonlinearity.The higher-harmonic content of the force histories is further analyzed by the four-phase separation method. The analysis confirms that the relative contribution from the higher-harmonic force components increases with the force peak level and further shows that the contribution from the first, second and third harmonics are very similar between flat bed and sloped bed. The analysis further highlights the presence of spurious second-harmonic waves from the linear wave generation method and shows that these are stronger for the flat bed case. The phase shift between the different harmonics of free surface elevation and inline force time series is observed to be constant for different sea states and target forces irrespective of the bed slope.Besides the direct findings of the study, the paper demonstrates the applicability of the FORM method for determination of design waves based on certain load effects. Although difficulties may occur for breaking waves or very strong nonlinearity, the combination of FORM and a fully nonlinear wave model enables average probability levels and time histories for extreme events to be determined and applied as design waves specific to a given load level or response level.