Phase-only Model Research Articles

BDS-3/Galileo multi-frequency phase-only model for long baseline with best integer equivariant estimation

Currently, long baselines generally adopt phase-code models that include both phase and pseudorange observations, such as dual-frequency ionosphere-free (PC-DFIF) and multi-frequency uncombined (PC-MFUC) models. The more significant unmodeled errors in pseudorange are likely to reduce ambiguity resolution (AR) performance, and the phase-only model with only phase observations involved is expected to avoid this problem. In view of the fact that existing phase-only research is mainly focuses on GPS/BDS-2 triple-frequency signals, a BDS-3/Galileo multi-frequency phase-only (MFPO) model is proposed in this paper. Four linearly independent extra-wide-lane (EWL) phase ranges (i.e. IR1, IR2, EWL3 and EWL4) are determined stepwise with a sliding window, among which the first two belong to ionosphere-reduced (IR) combination. In the determination of EWL3/EWL4, in addition to theoretical precision, the amplified phase multipath is also considered. The aforementioned four EWL phase ranges are then equivalently transformed to wide-lane (WL) combinations with smaller coefficients, and further incorporated with narrow-lane (NL) equation to construct MFPO model. Meanwhile, the best integer equivariant (BIE) is introduced to improve AR performance. The MFPO model is evaluated with real-tracked long baseline data. The performance of EWL phase ranges can be significantly improved after phase multipath is considered. The float solution, fixed solution and BIE solution of MFPO model all achieve centimeter-level accuracy. However, during convergence stage, incorrect fixing might exist in fixed solution, while BIE solution can improve this situation. The MFPO model is further compared with commonly used PC-DFIF and PC-MFUC models. The accuracy of the three models after convergence is basically the same, while the MFPO model has higher accuracy during convergence stage.

A study on multi-GNSS phase-only positioning

The global navigation satellite systems (GNSS) carrier phase measurements form the basis of high-precision satellite positioning. These measurements are often accompanied by their code counterparts to enable one to compute single-epoch ambiguity-resolved positioning solutions. To avoid unwanted code modelling errors, such as code multipath, one may opt for a phase-only solution and take recourse to carrier phase measurements of two successive epochs. In this paper we study the ambiguity resolution performance of a dual-epoch phase-only model, upon which the unknown positioning parameters are assumed to be completely unlinked in time. With the aid of closed-form analytical results, it is investigated how ambiguity resolution performs when dealing with high-rate phase data. It is thereby shown that multi-GNSS integration makes near real-time centimetre-level phase-only positioning possible. Our analytical analysis is supported by means of numerical results.

Dynamic causal modelling of phase-amplitude interactions

Models of coupled phase oscillators are used to describe a wide variety of phenomena in neuroimaging. These models typically rest on the premise that oscillator dynamics do not evolve beyond their respective limit cycles, and hence that interactions can be described purely in terms of phase differences. Whilst mathematically convenient, the restrictive nature of phase-only models can limit their explanatory power. We therefore propose a generalisation of dynamic causal modelling that incorporates both phase and amplitude. This allows for the separate quantifications of phase and amplitude contributions to the connectivity between neural regions. We show, using model-generated data and simulations of coupled pendula, that phase-amplitude models can describe strongly coupled systems more effectively than their phase-only counterparts. We relate our findings to four metrics commonly used in neuroimaging: the Kuramoto order parameter, cross-correlation, phase-lag index, and spectral entropy. We find that, with the exception of spectral entropy, the phase-amplitude model is able to capture all metrics more effectively than the phase-only model. We then demonstrate, using local field potential recordings in rodents and functional magnetic resonance imaging in macaque monkeys, that amplitudes in oscillator models play an important role in describing neural dynamics in anaesthetised brain states.

Energy flux enhancement, intermittency and turbulence via Fourier triad phase dynamics in the 1-D Burgers equation

We present a theoretical and numerical study of Fourier-space triad phase dynamics in the one-dimensional stochastically forced Burgers equation at Reynolds number $Re\approx 2.7\times 10^{4}$. We demonstrate that Fourier triad phases over the inertial range display a collective behaviour characterised by intermittent periods of synchronisation and alignment, reminiscent of the Kuramoto model (Chemical Oscillations, Waves, and Turbulence, Springer, 1984) and directly related to collisions of shocks in physical space. These periods of synchronisation favour efficient energy fluxes across the inertial range towards small scales, resulting in strong bursts of dissipation and enhanced coherence of the Fourier energy spectrum. The fast time scale of the onset of synchronisation relegates energy dynamics to a passive role: this is further examined using a reduced system with the Fourier amplitudes fixed in time – a phase-only model. We show that intermittent triad phase dynamics persists without amplitude evolution and we broadly recover many of the characteristics of the full Burgers system. In addition, for both full Burgers and phase-only systems the physical-space velocity statistics reveals that triad phase alignment is directly related to the non-Gaussian statistics typically associated with structure-function intermittency in turbulent systems.

Toward Multi-Input Control: A Dual-Feedback Loop Model of the Mammalian Circadian Clock

Many mathematical models have been used to describe the mammalian circadian clock, ranging from phase-only models to complex systems of differential equations. Here, we develop a 14 state model of the molecular clock, which provides sufficient detail to consider multi-input control. Assuming mass action and Michaelis-Menten kinetics, we derive a model of the transcription-translation feedback loops in the mammalian clock. Using a genetic algorithm, we fit the parameters of the model to capture the appropriate peak to trough ratios, relative abundances, and phase differences of the model species. We show that this in silico model captures much of the behavior that is observed in vitro under knockout conditions of different molecular species, validating our model. Thus, this model gives a simple but valid model of both feedback loops in the mammalian clock, providing the opportunity for enhanced control of both the phase and amplitude of the clock through multi-input control in a combination of small molecules which act on the species in both the positive and negative feedback loops.

A Focal Curve Design Method for Rotman Lenses With Wider Angular Scanning Range

This letter describes a novel focal curve design method for two-dimensional Rotman lenses. The proposed design method offers some additional degrees of freedom in the design of such beamformers and enables to extend significantly their angular scanning range, providing a solution to overcome the resulting focal arc and inner focal lens contour obstruction. A specific design, based on a phase-only model, is described, and numerical results are discussed. Good scanning performance over a very wide angle (± 50°) is demonstrated with phase-aberrations one order of magnitude lower than those of an equivalent standard Rotman lens. This improved beamformer design is of interest for low Earth orbit communication satellite antennas with a field-of-view ranging typically from ±40° to ±60°.

Complex‐valued analysis of arterial spin labeling–based functional magnetic resonance imaging signals

Cerebral blood flow-dependent phase differences between tagged and control arterial spin labeling images are reported. A biophysical model is presented to explain the vascular origin of this difference. Arterial spin labeling data indicated that the phase difference is largest when the arterial component of the signals is preserved but is greatly reduced as the arterial contribution is suppressed by postinversion delays or flow-crushing gradients. Arterial vasculature imaging by saturation data of activation and hypercapnia conditions showed increases in phase accompanying blood flow increases.An arterial spin labeling functional magnetic resonance imaging study yielded significant activation by magnitude-only, phase-only, and complex analyses when preserving the whole arterial spin labeling signal. After suppression of the arterial signal by postinversion delays, magnitude-only and complex models yielded similar activation levels, but the phase-only model detected nearly no activation. When flow crushers were used for arterial suppression, magnitude-only activation was slightly lower and fluctuations in phase were dramatically higher than when postinversion delays were used.Although the complex analysis performed did not improve detection, a simulation study indicated that the complex-valued activation model exhibits combined magnitude and phase detection power and thus maximizes sensitivity under ideal conditions. This suggests that, as arterial spin labeling imaging and image correction methods develop, the complex-valued detection model may become helpful in signal detection.

Circadian Response Reduction in Light and Response Restoration in Darkness: A “Skeleton” Light Pulse PRC Study in Mice (Mus musculus)

Entrainment may involve responses to dawn, to dusk, and to the light in between these transitions. Previous studies showed that the circadian system responds to only 2 light pulses, one at the beginning and one at the end of the day, in a similar way as to a full photoperiod, as long as the photoperiod is less than approximately 1/2 tau. The authors used a double 1-h light pulse protocol with different intervals of darkness in between (1, 2, 4, 7, 10, and 16 h) to study the phase responses of mice. The phase response curves obtained were compared to full light pulse PRCs of corresponding durations. Up to 6 hours, phase responses induced by double light pulses are virtually the same as by a corresponding full light pulse. The authors made a simple phase-only model to estimate the response reduction due to light exposure and response restoration due to dark exposure of the system. In this model, they assumed a 100% contribution of the first 1-h light pulse and fitted the reduction factor for the second light pulse to yield the best fit to the observations. The results suggest that after 1 h of light followed by less than 4 h of darkness, there is a considerable reduction in response to the second light pulse. Full response restoration requires more than 10 h of darkness. To investigate the influence of the duration of light on the response saturation, the authors performed a second series of experiments where the duration of the 2 light pulses was varied from 4 to 60 min each with a fixed duration of the stimulus (4 h). The response to 2 light pulses saturates when they are between 30 and 60 min long. In conclusion, double pulses replace single full light pulses of a corresponding duration of up to 6 h due to a response reduction during light, combined with response restoration during darkness. By the combined response reduction and response restoration, mice can maintain stable entrainment to the external LD cycle without being continuously exposed to it.

Phase and Period Responses of the Circadian System of Mice (Mus musculus) to Light Stimuli of Different Duration

To understand entrainment of circadian systems to different photoperiods in nature, it is important to know the effects of single light pulses of different durations on the free-running system. The authors studied the phase and period responses of laboratory mice (C57BL6J//OlaHsd) to single light pulses of 7 different durations (1, 3, 4, 6, 9, 12, and 18 h) given once per 11 days in otherwise constant darkness. Light-pulse duration affected both amplitude and shape of the phase response curve. Nine-hour light pulses yielded the maximal amplitude PRC. As in other systems, the circadian period slightly lengthened following delays and shortened following advances. The authors aimed to understand how different parts of the light signal contribute to the eventual phase shift. When PRCs were plotted using the onset, midpoint, and end of the pulse as a phase reference, they corresponded best with each other when using the mid-pulse. Using a simple phase-only model, the authors explored the possibility that light affects oscillator velocity strongly in the 1st hour and at reduced strength in later hours of the pulse due to photoreceptor adaptation. They fitted models based on the 1-h PRC to the data for all light pulses. The best overall correspondence between PRCs was obtained when the effect of light during all hours after the first was reduced by a factor of 0.22 relative to the 1st hour. For the predicted PRCs, the light action centered on average at 38% of the light pulse. This is close to the reference phase yielding best correspondence at 36% of the pulses. The result is thus compatible with an initial major contribution of the onset of the light pulse followed by a reduced effect of light responsible for the differences between PRCs for different duration pulses. The authors suggest that the mid-pulse is a better phase reference than lights-on to plot and compare PRCs of different light-pulse durations.

Low-energy phase-only action in a superconductor: A comparison with theXYmodel

The derivation of the effective theory for the phase degrees of freedom in a superconductor is still, to some extent, an open issue. It is commonly assumed that the classical XY model and its quantum generalizations can be exploited as effective phase-only models. In the quantum regime, however, this assumption leads to spurious results, such as the violation of the Galilean invariance in the continuum model. Starting from a general microscopic model, in this paper we explicitly derive the effective low-energy theory for the phase, up to fourth-order terms. This expansion allows us to properly take into account dynamic effects beyond the Gaussian level, both in the continuum and in the lattice model. After evaluating the one-loop correction to the superfluid density we critically discuss the qualitative and quantitative differences between the results obtained within the quantum XY model and within the correct low-energy theory, both in the case of s-wave and d-wave symmetry of the superconducting order parameter. Specifically, we find dynamic anharmonic vertices, which are absent in the quantum XY model, and are crucial to restore Galilean invariance in the continuum model. As far as the more realistic lattice model is concerned, in the weak-to-intermediate-coupling regime we find that the phase-fluctuation effects are quantitatively reduced with respect to the XY model. On the other hand, in the strong-coupling regime we show that the correspondence between the microscopically derived action and the quantum XY model is recovered, except for the low-density regime.

Electron-like quasiparticles drive the superconductor-insulator transition in homogeneously disordered thin films

Abstract Transport data on Bi, MoGe and PbBi/Ge homogeneously disordered thin films demonstrate that the critical resistivity Rc at the nominal insulator-superconductor transition is linearly proportional to the normal sheet resistance RN. In addition, the critical magnetic field scales linearly with the superconducting energy gap and is well approximated by Hc2 Because RN is determined at high temperatures and Hc2 is the pair-breaking field, the two immediate consequences are that electron-like quasiparticles populate the insulating side of the transition and that standard phase-only models are incapable of describing the destruction of the superconducting state. As gapless electronic excitations populate the insulating state, the universality class is no longer the three-dimensional XY model, thereby relaxing the constraints that this model imposes on the critical exponents as well as on the critical resistance, namely Rc = RQ = h/4e2. The lack of a unique critical resistance in homogeneously disordered films can be understood in this context. In the light of recent experiments which observe an intervening metallic state separating the insulator from the superconductor in homogeneously disordered MoGe thin films, we argue that the two transitions that accompany the destruction of superconductivity are superconductor to Bose metal in which phase coherence is lost and Bose metal to localized electron insulator via pair breaking.

Electron-like quasiparticles drive the superconductor-insulator transition in homogeneously disordered thin films

Transport data on Bi, MoGe, and PbBi/Ge homogeneously-disordered thin films demonstrate that the critical resistivity, $R_c$, at the nominal insulator-superconductor transition is linearly proportional to the normal sheet resistance, $R_N$. In addition, the critical magnetic field scales linearly with the superconducting energy gap and is well-approximated by $H_{c2}$. Because $R_N$ is determined at high temperatures and $H_{c2}$ is the pair-breaking field, the two immediate consequences are: 1) electron-quasiparticles populate the insulating side of the transition and 2) standard phase-only models are incapable of describing the destruction of the superconducting state. As gapless electronic excitations populate the insulating state, the universality class is no longer the 3D XY model. The lack of a unique critical resistance in homogeneously disordered films can be understood in this context. In light of the recent experiments which observe an intervening metallic state separating the insulator from the superconductor in homogeneously disordered MoGe thin films, we argue that the two transitions that accompany the destruction of superconductivity are 1) superconductor to Bose metal in which phase coherence is lost and 2) Bose metal to localized electron insulator via pair-breaking.

Spatial distribution of charge-density-wave phase slip inNbSe3

Steady-state charge transport via motion of charge-density waves (CDW's) requires conversion between collective CDW conduction and quasiparticle conduction in the vicinity of current injection contacts. The conversion occurs via phase slip, in which CDW amplitude defects form and move in the presence of CDW strain so as to remove or add CDW phase fronts in a process analogous to phase slip in superconductors and superfluids. We have determined the spatial distribution of phase slip for the ${T}_{{P}_{1}}$ CDW in ${\mathrm{NbSe}}_{3}$ by measuring the spatial variation of the CDW current density ${j}_{c}(x)$. As the current contacts are approached, ${j}_{c}(x)$ decreases and the normal current ${j}_{n}(x)$ due to quasiparticle flow increases. The size of the region near each contact where appreciable phase slip occurs is less than $40 \ensuremath{\mu}$m for $T$ near 120 K but grows to hundreds of micrometers at lower temperatures. The current and phase-slip profiles are asymmetric with respect to driving current direction, implying an asymmetry between phase front addition and removal. Analysis of these profiles yields the local relation between the phase-slip rate ${r}_{\mathrm{ps}}(x)$ and CDW strain $\ensuremath{\epsilon}(x)$. This relation is not unique, and for a given strain the phase-slip rate increases with increasing distance from the current electrode. These results are inconsistent with the predictions of models for phase slip via homogeneous defect nucleation, and provide evidence for amplitude defect motion. The presence of substantial amounts of phase slip at large distances from the current contacts explains the loss of coherence of the sliding CDW observed at lower temperatures, and suggests that predictions of phase-only models of CDW dynamics may be of very limited use in describing the sliding CDW in ${\mathrm{NbSe}}_{3}$.

Phase-amplitude resetting of the human circadian pacemaker via bright light: a further analysis.

We present here an analysis of strong, weak, and critical bright-light resetting trials in humans, and report not only phase but also amplitude data for the first time. For this analysis, an appropriate iterative smoothing procedure for phase transition curves is introduced, in which the data are sequenced so as to minimize the perpendicular distance from the data to the smoothed fit. From these smoothed data, we create polar phase-amplitude resetting maps (PARMs) in order to fully illustrate the effects of the resetting stimuli on both circadian amplitude and phase, and thereby to determine whether these resetting results can be described by a phase-only model or whether a phase-amplitude model is required. Our results indicate that a single 5-hr episode of bright light induces weak type 1 resetting of the human circadian pacemaker. Two cycles of exposure to the same stimulus on consecutive days induce critical resetting, in which significant amplitude reduction may be observed. A three-cycle stimulus induces strong type 0 resetting with different effects on circadian amplitude, depending on the initial phase of the stimulus application. When a three-cycle stimulus is centered near the nadir of the temperature cycle, large phase shifts are achieved via amplitude suppression. However, when this stimulus is centered away from the temperature nadir, smaller phase shifts are achieved in which both small increases and small decreases in circadian amplitude are observed. These data indicate that the human circadian pacemaker is not a simple, phase-only oscillator. Instead, a full description of human circadian resetting responses to light requires analysis of both phase and amplitude data--a finding that is consistent with a phase-amplitude model of the circadian resetting mechanism.