Integer Periods Research Articles

A novel method for dominant oscillation estimation and harmony quantification through spectral leakage suppression

Several methods exist for the quantification of the underlying oscillation within a time series in terms of frequency, amplitude and phase. The evaluation of the discrete Fourier transform is limited to integer periods and therefore prone to spectral leakage. Other approaches detect the dominant oscillation on a continuous frequency range, but lead to a nonlinear optimization problem, resulting in high requirements for the solving algorithm. We propose a novel direct method for the quantification of the dominant oscillation based on a characteristic value – called “DOH” – which is defined both in frequency and time domain. This enables the assessment of the harmony and periodicity and is discussed for different signal-to-noise ratios and the number of periods. By iteratively calculating the DOH for a successive truncated time series, the method – called “LSDOH” – is extended to a high frequency resolution. The algorithm suppresses spectral leakage, though it does not require any windowing. The method is compared to the DFT based on a Monte Carlo simulation. It is shown that the proposed identification method is highly accurate in determining the oscillation parameters of a time series overlaid by different noise levels, especially for high sampling frequencies.

Improved Repetitive Control for an LCL-Type Grid-Tied Inverter with Frequency Adaptive Capability in Microgrids

Repetitive control (RC), which can track any periodic signal with a known integer period with zero steady-state error, is widely used for current control of grid-tied inverters in microgrids. However, the inherent one fundamental period time delay, leads to poor dynamic performance. Furthermore, the performance of conventional RC (CRC) will degrade when operating at a high variation grid frequency. Therefore, this paper proposes a frequency adaptive improved RC (FA-IRC) for grid-tied inverters. The improved RC (IRC) consists of a repetitive controller with a modified internal model filter, plus a proportional controller. In comparison to the CRC, the IRC has a good dynamic response, because it provides a higher gain and a wider bandwidth at the resonant frequency. Moreover, to achieve the frequency adaptability of the IRC, a fractional delay, based on a finite impulse response (FIR) filter, is built into the IRC system, to ensure that the resonant frequency of the IRC is approximately equal to the actual grid frequency and harmonic frequency. Stability analysis and characteristic analysis of the FA-IRC system are reported in this paper. Simulations are conducted, to demonstrate the validity of the proposed method.

An 8-bit integer true periodic orbit PRNG based on delayed Arnold’s cat map

Designing a pseudo-random number generator (PRNG) exhibiting true periodic orbits with large periods is promising for embedded applications. Implementing chaotic behaviors on a finite precision digital platform using conventional unsigned integer or floating-point arithmetic yields short period limit-cycles. It leads the output to diverge from the true chaotic orbits. This paper presents an unsigned integer encoded PRNG based on the delayed quantized Arnold’s Cat Map (QACM) that exhibits true periodic orbits with large periods. The proposed PRNG is obtained by delaying one of the 2-D QACM outputs with a linear feedback shift register (LFSR) thereby increasing its period. By investigating its period, we show that the periodicity increases with an increase in the delay and bit-precision. We also present the proposed PRNG unit’s hardware architecture and its synthesis results targeting on a Xilinx Zynq 7000 Field Programmable Gate Array (FPGA) with 8-bit precision and delay δ=58. Further, we synthesize the design using 90 nm CMOS technology for ASIC realization. The synthesis results demonstrate the throughput as 2.208 Gbps and 5.46 Gbps at 138 MHz and 342 MHz operating frequency on FPGA and ASIC respectively. We experimentally evaluate the performance of random numbers using the NIST 800-22-1 tests along with the correlation and key sensitivity tests.

When Ramanujan meets time-frequency analysis in complicated time series analysis

To handle time series with complicated oscillatory structure, we propose a novel time-frequency (TF) analysis tool that fuses the short time Fourier transform (STFT) and periodic transform (PT). Since many time series oscillate with time-varying frequency, amplitude and non-sinusoidal oscillatory pattern, a direct application of PT or STFT might not be suitable. However, we show that by combining them in a proper way, we obtain a powerful TF analysis tool. We first combine the Ramanujan sums and $l_1$ penalization to implement the PT. We call the algorithm Ramanujan PT (RPT). The RPT is of its own interest for other applications, like analyzing short signal composed of components with integer periods, but that is not the focus of this paper. Second, the RPT is applied to modify the STFT and generate a novel TF representation of the complicated time series that faithfully reflect the instantaneous frequency information of each oscillatory components. We coin the proposed TF analysis the Ramanujan de-shape (RDS) and vectorized RDS (vRDS). In addition to showing some preliminary analysis results on complicated biomedical signals, we provide theoretical analysis about RPT. Specifically, we show that the RPT is robust to three commonly encountered noises, including envelop fluctuation, jitter and additive noise.

Double-Resolved Beam Steering by Metagrating-Based Tamm Plasmon Polariton.

We consider Tamm plasmon polariton in a subwavelength grating patterned on top of a Bragg reflector. We demonstrate dynamic control of the phase and amplitude of a plane wave reflected from such metagrating due to resonant coupling with the Tamm plasmon polariton. The tunability of the phase and amplitude of the reflected wave arises from modulation of the refractive index of a transparent conductive oxide layer by applying the bias voltage. The electrical switching of diffracted beams of the ±1st order is shown. The possibility of doubling the angular resolution of beam steering by using asymmetric reflected phase distribution with integer and half-integer periods of the metagrating is demonstrated.

A Novel Phase Difference Measurement Method for Coriolis Mass Flowmeter Based on Correlation Theory

Aiming at the poor precision problem in phase difference measurements with unknown frequencies in engineering practice, a new phase difference measurement method is proposed for Coriolis mass flowmeter based on correlation theories. Firstly, the signal frequency was estimated by using an adaptive notch filter, which was applied to filter the waves and determined the integer period of the sampling signals, and the non-integer period sampling signals needed to be extended. Then, the Hilbert transformation was conducted relative to the extended signals, and the correlation functions of these extended signals with the transformed signals can be computed. Finally, the formula of phase difference can be obtained by utilizing the sinusoidal function. Compared to traditional methods, such as the correlation method, the Hilbert transformation method, and sliding Goertzel algorithm, the proposed method is suitable for both integer period and non-integer period sampling signals, and its accuracy, real-time, and dynamic performance is superior. Simulation and experiment results validate the superiority and effectiveness of the proposed method.

Time-domain Landau-Zener-Stückelberg-Majorana interference in an optical lattice clock

The interference between a sequence of Landau-Zener (LZ) transitions can produce Rabi oscillations(LZROs). This phenomenon is a kind of time-domain Landau-Zener-Stuckelberg-Majorana (LZSM) interference. However, experimental demonstrations of this LZSM interference induced Rabi oscillation are extremely hard due to the short coherence time of the driven quantum system. Here, we study theoretically the time-domain LZSM interference between the clock transition in one-dimensional (1D) Sr-87 optical lattice clock (OLC) system. With the help of both the adiabatic-impulse model and Floquet numerical simulation method, the LZROs with special step-like structure are clearly found both in fast- and slow-passage limit in the real experiment parameter regions. In addition, the dephasing effect caused by the system temperature can be suppressed with destructive interference in the slow-passage limit. Finally, we discuss the possible Bloch-Siegert shift while the pulse time is away from the half-integer and integer periods. Our work provides a clear roadmap to observe the LZROs on the OLC platform.

Universal Evolutionary Model for Periodical Species

Real-world examples of periodical species range from cicadas, whose life cycles are large prime numbers, like 13 or 17, to bamboos, whose periods are large multiples of small primes, like 40 or even 120. The periodicity is caused by interaction of species, be it a predator-prey relationship, symbiosis, commensalism, or competition exclusion principle. We propose a simple mathematical model, which explains and models all those principles, including listed extremal cases. This rather universal, qualitative model is based on the concept of a local fitness function, where a randomly chosen new period is selected if the value of the global fitness function of the species increases. Arithmetically speaking, the different interactions are related to only four principles: given a couple of integer periods either (1) their greatest common divisor is one, (2) one of the periods is prime, (3) both periods are equal, or (4) one period is an integer multiple of the other.

Yielding in an Integer Automaton Model for Amorphous Solids under Cyclic Shear.

We present results on an automaton model of an amorphous solid under cyclic shear. After a transient, the steady state falls into one of three cases in order of increasing strain amplitude: (i)pure elastic behavior with no plastic activity, (ii)limit cycles where the state recurs after an integer period of strain cycles, and (iii)irreversible plasticity with longtime diffusion. The number of cycles N required for the system to reach a periodic orbit diverges as the amplitude approaches the yielding transition between regimes (ii) and (iii)from below, while the effective diffusivity D of the plastic strain field vanishes on approach from above. Both of these divergences can be described by a power law. We further show that the average period T of the limit cycles increases on approach to yielding.

A Fast Algorithm for Computing the Fourier Spectrum of a Fractional Period.

Directly computing Fourier power spectra at fractional periods of real sequences can be beneficial in many digital signal processing applications. In this article, we present a fast algorithm to compute the fractional Fourier power spectra of real sequences. For a real sequence of length of we may deduce its congruence derivative sequence with a length of l. The discrete Fourier transform of the original sequence can be calculated by the discrete Fourier transform of the congruence derivative sequence. The relation of discrete Fourier transforms between the two sequences may derive the special features of Fourier power spectra of the integer and fractional periods for a real sequence. It has been proved mathematically that after calculating the Fourier power spectrum (FPS) at an integer period, the Fourier power spectra of the fractional periods related this integer period can be easily represented by the computational result of the FPS at the integer period for the sequence. Computational experiments using a simulated sinusoidal data and protein sequence show that the computed results are a kind of Fourier power spectra corresponding to new frequencies that cannot be obtained from the traditional discrete Fourier transform. Therefore, the algorithm would be a new realization method for discrete Fourier transform of the real sequence.

Improved accuracy in high-frequency AC transport measurements in pulsed high magnetic fields.

We show theoretically and experimentally that accurate transport measurements are possible even within the short time provided by pulsed magnetic fields. For this purpose, a new method has been devised, which removes the noise component of a specific frequency from the signal by taking a linear combination of the results of numerical phase detection using multiple integer periods. We also established a method to unambiguously determine the phase rotation angle in AC transport measurements using a frequency range of tens of kilohertz. We revealed that the dominant noise in low-frequency transport measurements in pulsed magnetic fields is the electromagnetic induction caused by mechanical vibrations of wire loops in inhomogeneous magnetic fields. These results strongly suggest that accurate transport measurements in short-pulsed magnets are possible when mechanical vibrations are well suppressed.

Chains of Planets in Mean Motion Resonances Arising from Oligarchic Growth

Abstract Exoplanet systems with multiple planets in mean motion resonances have often been hailed as a signpost of disk-driven migration. Resonant chains like Kepler-223 and Kepler-80 consist of a trio of planets with the three-body resonant angle librating and/or with a two-body resonant angle librating for each pair. Here we investigate whether close-in super-Earths and mini-Neptunes forming in situ can lock into resonant chains due to dissipation from a depleted gas disk. We simulate the giant impact phase of planet formation, including eccentricity damping from a gaseous disk, followed by subsequent dynamical evolution over tens of millions of years. In a fraction of simulated systems, we find that planets naturally lock into resonant chains. These planets achieve a chain of near-integer period ratios during the gas-disk stage, experience eccentricity damping that captures them into resonance, stay in resonance as the gas disk dissipates, and avoid subsequent giant impacts, eccentricity excitation, and chaotic diffusion that would dislodge the planets from resonance. Disk conditions that enable planets to complete their formation during the gas-disk stage enable those planets to achieve tight period ratios ≤2 and, if they happen to be near-integer period ratios, lock into resonance. Using the weighting of different disk conditions deduced by MacDonald et al. and forward modeling Kepler selection effects, we find that our simulations of in situ formation via oligarchic growth lead to a rate of observable trios with integer period ratios and librating resonant angles comparable to observed Kepler systems.

Frequency Adaptive Proportional-Repetitive Control for Grid-Connected Inverters

The repetitive control (RC) scheme, which can achieve zero steady-state error tracking of any periodic signal with a known integer period, is widely employed in grid-connected inverters to mitigate feed-in current harmonics. However, the conventional RC (CRC) scheme cannot exactly compensate harmonics with fractional period caused by grid frequency variations. Furthermore, the small RC gain of the CRC scheme causes slow transient response. In this article, a proportional-repetitive control (PRC) scheme, which can accommodate a larger RC gain than the CRC scheme, is proposed to provide fast transient response for RC systems. In order to enhance the frequency adaptability of the PRC scheme with a fixed sampling rate, the Thiran all-pass filter, which is a type of infinite impulse response filter for approximating the fractional delay (FD), is used as the FD filter to achieve fast online tuning of the FD and the fast update of the coefficients. The proposed frequency adaptive PRC (FA-PRC) scheme provides grid-connected inverters with a control solution with excellent dynamic performance and accurate frequency adaptability to eliminate harmonic distortion even in the presence of grid frequency variations. A comprehensive series of experiments have demonstrated the effectiveness of the proposed FA-PRC scheme.

Magnetohydrodynamic hybrid simulation of Alfvén eigenmodes in Heliotron J, a low shear helical axis stellarator/heliotron

Alfvén eigenmodes destabilized by energetic particles in Heliotron J, a low magnetic shear helical axis stellarator/heliotron with four toroidal equilibrium field periods, are investigated with MEGA, a hybrid simulation code for energetic particles interacting with a magnetohydrodynamic fluid. The objectives of this study are to validate the MEGA simulation results on Alfvén eigenmodes in Heliotron J, and to clarify the properties of Alfvén eigenmodes. The experimentally observed global Alfvén eigenmode (GAE) was reproduced with MEGA simulations based on the experimental bulk plasma temperature and density profiles. Beside GAE, GAE was also observed with lower amplitude and linear growth rate. The frequency and spatial location of GAE are close to those of the experimentally observed energetic particle mode (EPM). For GAE, it was found that the three-dimensional spatial profile of the GAE was primarily composed of n = 2 harmonics while the contribution from the other toroidal mode numbers such as n = 2 ± iNfp was weak, where ‘i’ and ‘Nfp’ are arbitrary integer and toroidal field period, respectively. This indicates that the coupling of harmonics with the same toroidal mode number through toroidicity is dominant for the GAE spatial profile, and the coupling with the different toroidal mode numbers through helicity and bumpiness has minor effects. For GAE, the coupling of harmonic through toroidicity is weaker than that of GAE. The non-linear evolutions of the GAE and GAE destabilized by energetic particles with a bump-on-tail distribution and a slowing-down distribution were compared. Weak dependence of the linear growth rate for both and modes on charge-exchange time were implied.

Elastic Full Waveform Inversion With Source-Independent Crosstalk-Free Source-Encoding Algorithm

Elastic full waveform inversion (FWI) is more suitable to process multicomponent seismic data and can provide more subsurface medium information than acoustic FWI often with lower efficiency. Except for the parallel algorithms, source-encoding methods are usually adopted to improve the efficiency of FWI, but it often includes crosstalk noise. Besides, the additional source estimation process, critical for a successful FWI, would counteract the high-efficiency advantage of the source-encoding algorithm. We propose an elastic FWI with source-independent crosstalk-free encoding algorithm to solve the above problems. Arbitrary-phase harmonic sine functions are used as new source wavelets to perform the time-domain wavefield simulation regardless of the true wavelet. Treating the harmonic wavelet as the encoding operator and based on the orthogonality of trigonometric functions within integer periods, the amplitude and phase of each source are recovered from the blended source and adjoint wavefields so that the influence of crosstalk noise is avoided. With the deblended data, the proposed algorithm can be naturally applied to unfixed-spread acquisition systems. Moreover, we can conveniently perform the multiscale inversion by controlling the frequencies of simultaneous-source signals as conventional frequency-domain FWI does. Synthetic examples show that the proposed algorithm has high efficiency and accuracy with a strong robustness to the incorrect wavelets.

Effective two-level models for highly efficient inner-state enantioseparation based on cyclic three-level systems of chiral molecules

Based on cyclic three-level systems of chiral molecules, we propose two methods to realize highly efficient inner-state enantioseparations of a chiral mixture with the two enantiomers initially prepared in their ground states. Our methods work in the region where the evolutions of the two enantiomers can be described by their corresponding effective two-level models, simultaneously. The approximately $100%$-efficiency inner-state enantioseparations can be realized when the probability occupying the ground state of one enantiomer becomes zero by experiencing half-integer periods of its corresponding on-resonance Rabi oscillation and the other one stays always in the ground state approximately, under the conditions that the two enantiomers are governed by the effective on-resonance and large-detuning two-level models, respectively. Alternatively, the exactly $100%$-efficiency inner-state enantioseparation can be obtained when the probability occupying the ground state of one enantiomer becomes zero by experiencing half-integer periods of its corresponding on-resonance Rabi oscillation and in the meanwhile that of the other one becomes 1 by experiencing integer periods of its corresponding detuned Rabi oscillation, respectively.

A New Method for Calibrating Nonlinear Coefficients of PIGA on Linear Vibrator

This paper proposes a method for calibrating nonlinear coefficients of pendulous integrating gyroaccelerometer (PIGA) on linear vibrator in order to increase the calibration accuracy. First, a series of coordinate systems is established by analyzing the working mechanism of linear vibrator, and the error sources during calibration process are described. The precise input-specific forces and angular rates of PIGA are derived by homogeneous transformation algorithm. Then, the calibration error model of PIGA is proposed, which jointly considers the second-order, the third-order, and the cross-quadratic terms. The parasitic rotation errors of linear vibrator are separated from error terms of the model by vibrating within integer periods, and the static errors are suppressed by subtracting the static test outputs. Finally, the frequency errors and the parasitic translation are estimated accurately, and the error model is simplified and compensated by the analysis of measurement. It is verified by simulations that the nonlinear coefficients of PIGA are identified accurately, and the magnitude of absolute errors are less than $10^{-6}$ . The simulation results also make contributions to the design of the test plan appropriately.

IMUSIC: A Family of MUSIC-Like Algorithms for Integer Period Estimation

The MUSIC algorithm is one of the most popular techniques today for line spectral estimation. If the line spectrum is that of a periodic signal, can we adapt MUSIC to exploit the additional harmonicity in the spectrum? Important prior work in this direction includes the Harmonic MUSIC algorithm and its variations. For applications where the period of the discrete signal is an integer (or can be well approximated by an integer), this paper introduces a new and simpler class of alternatives to MUSIC. This new family, called iMUSIC, also includes techniques where simple integer valued vectors are used in place of complex exponentials for both representing the signal subspace, and for computing the pseudo-spectrum. It will be shown that the proposed methods not only make the computations much simpler than prior periodicity-adaptations of MUSIC, but also offer significantly better estimation accuracies for applications with integer periods. These advantages are demonstrated on examples that include repeats in protein and DNA sequences. The iMUSIC algorithms are based on the recently proposed Ramanujan subspaces and nested periodic subspaces. The resulting signal space bases are non-Vandermonde in structure. Consequently, many aspects of classical MUSIC that were based on the Vandermonde structure of complex-exponentials, such as guarantees for identifiability of the frequencies (periods in our case), are addressed in new ways in this paper.

Three Pathways for Observed Resonant Chains

Abstract A question driving many studies is whether the thousands of exoplanets known today typically formed where we observe them or formed further out in the disk and migrated in. Early discoveries of giant exoplanets orbiting near their host stars and exoplanets in or near mean motion resonances were interpreted as evidence for migration and its crucial role in the beginnings of planetary systems. Long-scale migration has been invoked to explain systems of planets in mean motion resonant chains consisting of three or more planets linked by integer period ratios. However, recent studies have reproduced specific resonant chains in systems via short-scale migration, and eccentricity damping has been shown to capture planets into resonant chains. We investigate whether the observed resonant chains in Kepler-80, Kepler-223, Kepler-60, and TRAPPIST-1 can be established through long-scale migration, short-scale migration, and/or only eccentricity damping by running suites of N-body simulations. We find that, for each system, all three mechanisms are able to reproduce the observed resonant chains. Long-scale migration is not the only plausible explanation for resonant chains in these systems, and resonant chains are potentially compatible with in situ formation.

Reference group influence on binary choices dynamics

The recent literature has analyzed binary choices dynamics providing interesting results. Most of these contributions consider interactions within a single group. Nevertheless, in some situations the interaction takes place not only within a single group but also between different groups. In this paper, we investigate the choice dynamics when considering two populations where one serves as a reference group. Considering this influence effect enriches the dynamics. Although the structurally stable resulting dynamics are attracting cycles only, with any positive integer period, the reference group makes the dynamics of the influenced population much more complex. We considered both the possibility that the reference group has the same or the opposite attitude toward the distribution over the choices. We show how the dynamics and the bifurcation structure are modified under the influence of the reference group. Our results illustrate how the propensity to switch choices in the reference groups may, indirectly, affect choices in the first group.