Periodic Waveforms Research Articles

Anharmonic Waves in a Mindlin–Herrmann Rod Immersed in a Nonlinearly Elastic Medium

Propagation of longitudinal waves in a Mindlin–Herrmann rod immersed in a nonlinearly elastic medium is studied. Various options of the stiffness ratio of the rod and external medium in which the rod is placed are considered and three limiting cases are obtained. It is shown that, if the stiffness of the external medium significantly exceeds the rod stiffness, then the evolutionary equation becomes the Ostrovsky equation that is well-known in the nonlinear dynamics. The equation has no exact solutions, but allows a qualitative study when the senior derivative is equal to zero. In this case, the solution in the form of a nonlinear periodic stationary wave is found and analyzed. If the stiffness of external medium is significantly lower than the rod stiffness, then the evolutionary equation is the equation different from the Ostrovsky equation in the nonlinear part. In this case, it is demonstrated soliton propagation of a classical profile in the rod is possible. It is pointed out that, when the stiffnesses of external medium and rod are of the same order of magnitude, the nonlinear stationary waves cannot be generated.

Fourier analysis: the simplification of a complex waveform into simple component sine waves of different amplitudes and frequencies

Fourier analysis is the simplification of a complex waveform into simple component sine waves of different amplitudes and frequencies. A discussion on Fourier analysis necessitates reiteration of the physics of waves. A wave is a series of repeating disturbances that propagate in space and time. Frequency: the number of oscillations, or cycles per second. It is measured in Hertz and denoted as 1/time or s-1. Fundamental frequency: the lowest frequency wave in a series. It is also known as the first harmonic. Every other wave in the series is an exact multiple of the fundamental frequency. Harmonic: whole number multiples of the fundamental frequency. Amplitude: the maximum disturbance or displacement from zero caused by the wave. This is the height of the wave. Period: time to complete one oscillation. Wavelength: physical length of one complete cycle. This can be between two crests or two troughs. The higher the frequency, the shorter the wavelength. Velocity: frequency x wavelength. Phase: displacement of one wave compared to another, described as 0°–360°. A sine wave is a simple wave. It can be depicted as the path of a point travelling round a circle at a constant speed, defined by the equation ‘y = sinx’. Combining sine waves of different frequency, amplitude and phase can yield any waveform, and, conversely, any wave can be simplified into its component sine waves. Fourier analysis is a mathematical method of analysing a complex periodic waveform to find its constituent frequencies (as sine waves). Complex waveforms can be analysed, with very simple results. Usually, few sine and cosine waves combine to create reasonably accurate representations of most waves. Fourier analysis finds its anaesthetic applications in invasive blood pressure, electrocardiogram (ECG) and electroencephalogram (EEG) signals, which are all periodic waveforms. It enables monitors to display accurate representations of these biological waveforms. Fourier analysis was developed by Joseph Fourier, a mathematician who analysed and altered periodic waveforms. It is done by computer programmes that plot the results of the analysis as a spectrum of frequencies with amplitude on the y-axis and frequency on the x-axis.

A Simple Scheme for Photonic Generation of Microwave Waveforms Using a Dual-drive Mach–Zehnder Modulator

In this paper, a novel photonic scheme for the generation of square and triangular waveforms is proposed and experimentally demonstrated by using a dual-drive Mach–Zehnder modulator (DDMZM) without any optical signal processing. By properly setting the modulation index of the DDMZM, the square and triangular microwave waveforms could be easily achieved in conjunction with a 90° hybrid coupler. Furthermore, the modulation index of the DDMZM could be tuned for a low radio frequency (RF) signal power requirement by tuning the phase difference of the RF signals applied to the two arms of the DDMZM. The proposed scheme for the photonic generation of microwave waveforms is theoretically analyzed and experimentally demonstrated. The periodical triangular and square waveforms with repetition rates of 5 GHz, 8 GHz, and 10 GHz are successfully obtained. The root-mean-square errors between the generated and theoretical triangular and square waveforms with a repetition of 5 GHz are 0.042 and 0.053, respectively.

Oscillatory Rayleigh–Bénard Convection in elasto-viscoplastic gels

Abstract Rayleigh–Benard convection (RBC) is studied in an elasto-viscoplastic (EVP) gel, the widely used and studied Carbopol gel. The transition from conductive to convective regime is determined by the Schmidt–Milverton principle which leads to a critical value of the inverse of the Yield number 1 ∕ Y c . However, an oscillatory motion in the gel is observed below 1 ∕ Y c ≈ 60 when smooth and untreated walls are used. Particles within the gel describe periodical loops close to their initial positions for 16 . 5 1 ∕ Y 25 . For larger values of 1 ∕ Y ≈ 50 1 ∕ Y c , loops are still obtained but particles move away progressively from their initial position. The oscillations take the form of periodic traveling waves (TW). We demonstrate that elastic effects play the major role in the occurrence of oscillations.

Estimation of Modulation Parameters of LPI Radar Using Cyclostationary Method

Low Probability of intercept (LPI) radars work on the principle of low peak-power and wide bandwidth. To achieve this, the LPI radars use special type of modulated waveforms which are difficult to intercept. The main tasks of intercept receivers are to detect, estimate and classify LPI signals even in the presence of high noise. Accurate estimation of the modulation parameters also provides information about threat to radar so that necessary counter measures could be initiated against the enemy radars. In the present work, Cyclo-stationary (CS) method is used to estimate the parameters of Polytime coded LPI signals. CS method is efficient for the analysis of periodic waveforms like LPI signals and finds applications in many fields such as array signal processing, estimation of direction of arrival, signal detection. Cyclic auto correlation function and the spectral correlation density function (SCD) which are the basis for CS analysis are computed for all the four types of polytime codes. Generation of all the four types of Polytime codes and computation of SCD coefficients are carried out using MATLAB. From the contour plot of SCD function, the parameters carrier frequency, bandwidth and code rate are measured. It is assumed that the received signal is not corrupted with any noise. The simulation results are compared with the values available in the literature. The maximum error of estimation is well within ± 6% for all the codes in the proposed method whereas the maximum error reported in the literature is 25.7%. Hence the proposed method is better. Since CS method is sensitive to sampling frequency, the analysis is repeated for three different sampling frequencies and in all cases, the estimation error is less than 6%.

Practical interrogation scheme for the abrupt taper Mach-Zehnder interferometer refractive index sensor.

A practical interrogation scheme of a refractive index (RI) sensing system based on the abrupt fiber taper Mach-Zehnder interferometer sensor is designed, implemented, and demonstrated by experiment. The broadband light source and optical spectrum analyzer in the conventional design are replaced by two single-wavelength laser diodes modulated by a periodical square waveform and a simple photodetector (PD), which can significantly lower the cost and lead to easier integration. The photocurrent of the PD output signal is used as the indicator of the surrounding RI. Automatic data acquisition and processing is realized by LabVIEW programming. The experiment proves the feasibility of the new scheme and shows a high sensitivity (2371 mV/RIU) and high stability.

Propagation of periodic wave trains along the magnetic field in a collision-free plasma

In this work, a systematic study, examining the propagation of periodic and solitary waves along the magnetic field in a cold collision-free plasma, is presented. Employing the quasi-neutral approximation and the conservation of momentum flux and energy flux in the frame co-traveling with the wave, the exact analytical solution of the stationary solitary pulse is found analytically in terms of particle densities, parallel and transverse velocities, as well as transverse magnetic fields. Subsequently, this solution is generalized in the form of periodic waveforms represented by cnoidal-type waves. These considerations are fully analytical in the case where the total angular momentum flux L, due to the ion and electron motion together with the contribution due to the Maxwell stresses, vanishes. A graphical representation of all associated fields is also provided.

A phase linearisation–based modulation signal bispectrum for analysing cyclostationary bearing signals

Bearings are used as the most important load-carrying transmission components in various machines, thus subjecting to a number of faults including wear, fatigue pitting, cracks and so on. Fault detection and diagnosis of bearings can effectively prevent the machine from such typical failures and subsequent consequences. The faults in bearings can lead to the vibration signals that exhibit cyclostationary characteristics due to the inevitably random phase noise (or slippage between bearing components). In this article, a phase linearisation–based modulation signal bispectrum is proposed to tune up the cyclostationary bearing signal into a periodic waveform by linearizing the instantaneous phase of the narrow frequency band signals. In this way, the signal becomes more deterministic and modulation signal bispectrum can be effectively applied to suppression noise and obtain accurate and robust diagnosis results. As a result, this fault detector can achieve high performance in characterising the nonstationary bearing vibration signals and hence diagnose the bearing faults even in the case of extremely low signal-to-noise ratio (<−20 dB), which is benchmarked by the method of conventional modulation signal bispectrum in both simulation and experiment studies.

Application of Map-Based Multipactor Theory to Alternate Periodic Radio Frequency Waveforms

Multipactor is an electron discharge driven by secondary electron emission (SEE) that plagues microwave components, accelerator structures, and space-borne systems operating in vacuum conditions. Recently, a novel map-based theoretical approach for multipactor based on nonlinear dynamics and chaos theory has been introduced and applied to systems driven by a standard radio frequency (RF) sinusoidal electric field. This theory, however, is universal in that it is applicable to any periodic RF waveform. To demonstrate this, this article applies the map-based theory to periodic rectangular and triangular RF waveforms in a parallel-plate geometry. Bifurcation diagrams are constructed that concisely summarize the underlying multipactor dynamics for both waveforms. These theoretical predictions are then quantitatively verified with the particle-in-cell simulations.

Data Transmission Based on Exact Inverse Periodic Nonlinear Fourier Transform, Part II: Waveform Design and Experiment

The nonlinear Fourier transform has the potential to overcome limits on performance and achievable data rates which arise in modern optical fiber communication systems when nonlinear interference is treated as noise. The periodic nonlinear Fourier transform (PNFT) has been much less investigated compared to its counterpart based on vanishing boundary conditions. In this paper, we design a first experiment based on the PNFT in which information is encoded in the invariant nonlinear main spectrum. To this end, we propose a method to construct a set of periodic waveforms each having the same fixed period, by employing the exact inverse PNFT algorithm developed in Part I. We demonstrate feasibility of the transmission scheme in experiment in good agreement with simulations and obtain a bit-error ratio of $10^{-3}$ over a distance of 2000 km. It is shown that the transmission reach is significantly longer than expected from a naive estimate based on group velocity dispersion and cyclic prefix length, which is explained through a dominating solitonic component in the transmitted waveform. Our constellation design can be generalized to an arbitrary number of nonlinear degrees of freedom.

Measurement of periodic voltage waveforms on secondary side of high voltage measuring device using Arduino

Data acquisition on the secondary side is an important part of high voltage signal measurement. Based on Arduino microcontroller, a data acquisition (DAQ) system is developed. The frequency, root-mean-square value and DC component of periodic voltage waveforms can be measured by the system. The relative errors of the DAQ system are tested by comparing with standard voltage waveforms. The measurement range of the DAQ system is further discussed. The DAQ system in this paper is helpful to measure the periodic voltage waveforms on secondary side of high voltage measuring device.

Analysis of high-order sub-harmonically injection-locked oscillators

AbstractHigh-order sub-harmonically injection-locked oscillators have recently been proposed for low phase-noise frequency generation, with carrier-selection capabilities. Though excellent experimental behavior has been demonstrated, the analysis/simulation of these circuits is demanding, due to the high ratio between the oscillation frequency and the frequency of the input source. This work provides an analysis methodology that covers the main aspects of the circuit behavior, including the detection of the locking bands and the prediction of the phase-noise spectral density. Initially, the oscillator in the presence of a multi-harmonic input source is described with a reduced-order envelope-domain formulation, at the oscillation frequency, based on an oscillator-admittance function extracted from harmonic-balance simulations. This allows deriving an expression for the oscillation phase shift with respect to the input source, and the average of this phase shift is shown to evolve continuously in the distinct synchronization bands obtained when varying a tuning voltage. This property can be used to detect the locking bands in circuit-level envelope-domain simulations, which, as shown here, can be done through different Fourier decompositions and sampling rates. The phase noise of the high-order sub-harmonic injection-locked oscillator under an arbitrary periodic input waveform is investigated in detail. The frequency response to the noise sources is described with a semi-analytical formulation, relying on the oscillator-admittance function in injection-locked conditions. The input noise is derived from the timing jitter of the injection source and the phase-noise response is shown to exhibit a low-pass characteristic, which initially follows the up-converted input noise and then the oscillator own noise sources. A method is proposed to identify the key parameters of the derived phase-noise spectrum from envelope-domain simulations. The various analysis methodologies have been applied to a prototype at 2.7 GHz at the sub-harmonic order N = 30 which has been manufactured and measured.

Frequency combs induced by phase turbulence.

Wave instability-the process that gives rise to turbulence in hydrodynamics1-represents the mechanism by which a small disturbance in a wave grows in amplitude owing to nonlinear interactions. In photonics, wave instabilities result in modulated light waveforms that can become periodic in the presence of coherent locking mechanisms. These periodic optical waveforms are known as opticalfrequency combs2-4. In ring microresonator combs5,6, an injected monochromatic wave becomes destabilized by the interplay between the resonator dispersion and the Kerr nonlinearity of the constituent crystal. By contrast, in ring lasers instabilities are considered to occur only under extreme pumping conditions7,8. Here we show that, despite this notion, semiconductor ring lasers with ultrafast gain recovery9,10 can enter frequency comb regimes at low pumping levels owing to phase turbulence11-an instability known to occur in hydrodynamics, superconductors and Bose-Einstein condensates. This instability arises from the phase-amplitude coupling of the laser field provided by linewidth enhancement12, which produces the needed interplay of dispersive and nonlinear effects. We formulate the instability condition in the framework of the Ginzburg-Landau formalism11. The localized structures that we observe share several properties with dissipative Kerr solitons, providing a first step towards connecting semiconductor ring lasers and microresonator frequency combs13.

An analytical method of switching waveform design for selective harmonic elimination

The paper proposes an analytical approach to the design of switching waveforms to eliminate harmonics and thereby improve THD (Total Harmonic Distortion) in power converters. Generally, numerical solutions are used to obtain the switching angles to eliminate targeted harmonics and the solution requires guessing of an initial value and a valid modulation index, m (a ratio of the fundamental to the dc input). Both of these values affect solution convergence whereas, such difficulties do not arise in analytical solutions. The closed-form solutions obtained through the analytical approach also determine the valid range of m. Due to the closed form, this method is suitable for real-time applications where a microcontroller can be programmed, thus making it attractive for low-cost applications. A PWM (Pulse Width Modulation) signal with even–odd symmetry has been selected to minimize the complexity of the solution. The approach requires the solution of transcendental equations associated with Fourier coefficients of a periodic waveform. Chebyshev expansion is used to convert the transcendental equations to power-sum non-linear polynomials. A successive polynomial simplification method has been used that lead to a set of elementary functions in closed-form, which are used to generate a polynomial and its roots are the desired optimal switching angles. In this paper, the proposed approach has been examined through a fourth-order, three-phase system for bipolar and multistep waveforms. The approach is equally applicable to other types of PWM systems. Simulation results verify the harmonic cancellations.

Extracting new information from old waveforms: Symmetric projection attractor reconstruction: Where maths meets medicine.

What is the topic of this review? Symmetric Projection Attractor Reconstruction (SPAR) is a relatively new mathematical method that can extract additional information pertaining to the morphology and variability of physiological waveforms, such as arterial pulse pressure. Herein, we describe the potential utility of the method for more sensitive quantification of cardiovascular changes. What advances does it highlight? We use a simple example of a human tilt table to illustrate these concepts. SPAR can be used on any approximately periodic waveform and may add value to experimental and clinical settings, where such signals are collected routinely. Periodic physiological waveform data, such as blood pressure, pulse oximetry and ECG, are routinely sampled between 100 and 1000Hz in preclinical research and in the clinical setting from a wide variety of implantable, bedside and wearable monitoring devices. Despite the underlying numerical waveform data being captured at such high fidelity, conventional analysis tends to reside in reporting only averages of minimum, maximum, amplitude and rate, as single point averages. Although these averages are undoubtedly of value, simplification of the data in this way means that most of the available numerical data are discarded. In turn, this may lead to subtle physiological changes being missed when investigating the cardiovascular system over time. We have developed a mathematical method (symmetric projection attractor reconstruction) that uses all the numerical data, replotting and revisualizing them in a manner that allows unique quantification of multiple changes in waveform morphology and variability. We propose that the additional quantification of these features will allow the complex behaviour of the cardiovascular system to be mapped more sensitively in different physiological and pathophysiological settings.

Two types of ictal phase-amplitude couplings in epilepsy patients revealed by spectral harmonicity of intracerebral EEG recordings

ObjectiveSpectral harmonicity of the ictal activity was analyzed regarding two clinically relevant aspects, (1) as a confounding factor producing ‘spurious’ phase-amplitude couplings (PAC) which may lead to wrong conclusions about the underlying ictal mechanisms, and (2) its role in how good PAC is in correspondence to the seizure onset zone (SOZ) classification performed by the epileptologists. MethodsPAC patterns observed in intracerebral electroencephalography (iEEG) recordings were retrospectively studied during seizures of seven patients with pharmacoresistant focal epilepsy. The time locked index (TLI) measure was introduced to quantify the degree of harmonicity between frequency bands associated to the emergence of PAC during epileptic seizures. Results(1) Harmonic and non harmonic PAC patterns coexist during the seizure dynamics in iEEG recordings with macroelectrodes. (2) Harmonic PAC patterns are an emergent property of the periodic non sinusoidal waveform constituting the epileptiform activity. (3) The TLI metric allows to distinguish the non harmonic PAC pattern, which has been previously associated with the ictal core through the paroxysmal depolarizing shifts mechanism of seizure propagation. ConclusionsOur results suggest that the spectral harmonicity of the ictal activity plays a relevant role in the visual analysis of the iEEG recordings performed by the epileptologists to define the SOZ, and that it should be considered for the proper interpretation of ictal mechanisms. SignificanceThe proposed harmonicity analysis can be used to improve the delineation of the SOZ by reliably identifying non harmonic PAC patterns emerging from fully recruited cortical and subcortical areas.

Energy-Based Periodicity Mining With Deep Features for Action Repetition Counting in Unconstrained Videos

Action repetition counting is to estimate the occurrence times of the repetitive motion in one action, which is a relatively new, significant, but challenging problem. To solve this problem, we propose a new method superior to the traditional ways in two aspects, without preprocessing and applicable for arbitrary periodicity actions. Without preprocessing, the proposed model makes our scheme convenient for real applications; processing the arbitrary periodicity action makes our model more suitable for the actual circumstance. In terms of methodology, firstly, we extract action features using ConvNets and then use Principal Component Analysis algorithm to generate the intuitive periodic information from the chaotic high-dimensional features; secondly, we propose an energy-based adaptive feature mode selection scheme to adaptively select proper deep feature mode according to the background of the video; thirdly,we construct the periodic waveform of the action based on the high-energy rules by filtering the irrelevant information. Finally, we detect the peaks to obtain the times of the action repetition. Our work features two-fold: 1) We give a significant insight that features extracted by ConvNets for action recognition can well model the self-similarity periodicity of the repetitive action. 2) A high-energy based periodicity mining rule using features from ConvNets is presented, which can process arbitrary actions without preprocessing. Experimental results show that our method achieves superior or comparable performance on the three benchmark datasets, i.e. YT_Segments, QUVA, and RARV.

Mathematical model of propagation pulse in optical fiber with power nonlinearities

An nonlinear fourth-order equation for describing the pulse propagation in an optical fiber is considered. Equation generalizes a number of well-known mathematical models in nonlinear media. A characteristic feature of the equation is that when describing the envelope of a wave packet, an arbitrary power can be taken for changing the amplitude and width of the pulse. To find exact solutions that traveling wave reduction is used to nonlinear ordinary differential equation. We also present a method for constructing exact solutions using a generalized ordinary first-order differential equation of the second degree as an auxiliary equation. It is shown that the equation under consideration has exact solutions in the form of periodic and solitary waves, which are expressed in terms of the Weierstrass and the Jacobi elliptic functions. Special cases of equation are presented. Periodic and solitary waves of the equation are demonstrated.

Two-stage reactive compensation in a three-phase four-wire systems at nonsinusoidal periodic waveforms

The improvement of the power factor in three-phase, four-wire, stationary, unbalanced and linear loads, supplied from an ideal and symmetrical source at nonsinusoidal periodic waveforms, is the subject of this article. The case of an ideal source or a source that has a negligibly low internal impedance was considered. The analysis was based on the load current decomposition, according to the Current Physical Components (CPC) theory. The paper presents fundamentals of design of reactive compensators for compensation of the reactive and unbalanced currents of linear time invariant (LTI) loads supplied with nonsinusoidal voltage in four-wire systems. A reactive balancing compensator consists of two compensators working in a star and triangle configuration. The article presents the method of calculating the parameters of such a compensator using the CPC power theory. It was proved that the reactive and unbalanced components of current can be fully compensated in the reactive compensator also in the case of nonsinusoidal waveforms in power networks.

Potential systems of a Buckley–Leverett equation: Lie point symmetries and conservation laws

In this work, we study a Buckley–Leverett equation of two-phase flow in porous media from the point of view of the Lie theory. We find that for some functions the equation has abundant exact solutions expressible in terms of Jacobi elliptic functions and consequently has many solutions in the form of periodic waves or solitary waves. We determine low-order conservation laws by using the multiplier method. From conservations laws, we construct the potential systems. From these systems we deduce potential equations. We study these equations from the point of view of Lie theory. Moreover, for these equations some nontrivial conservation laws are constructed by using the multipliers method.