Harmonic Pairs Research Articles

PRODUCTION OF FINE STRUCTURES IN TYPE III SOLAR RADIO BURSTS DUE TO TURBULENT DENSITY PROFILES

Magnetic reconnection events in the corona release energetic electron beams along open field lines, and the beams generate radio emission at multiples of the electron plasma frequency f{sub p} to produce type III solar radio bursts. Type III bursts often exhibit irregularities in the form of flux modulations with frequency and/or local temporal advances and delays, and a type IIIb burst represents the extreme case where a type III burst is fragmented into a chain of narrowband features called striae. Remote and in situ spacecraft measurements have shown that density turbulence is ubiquitous in the corona and solar wind, and often exhibits a Kolmogorov power spectrum. In this work, we numerically investigate the effects of one-dimensional macroscopic density turbulence (along the beam direction) on the behavior of type III bursts, and find that this turbulence produces stria-like fine structures in the dynamic spectra of both f{sub p} and 2 f{sub p} radiation. Spectral and temporal fine structures in the predicted type III emission are produced by variations in the scattering path lengths and group speeds of radio emission, and in the locations and sizes of emitting volumes. Moderate turbulence levels yield flux enhancements with much broader half-power bandwidths in f{submore » p} than 2 f{sub p} emission, possibly explaining the often observed type IIIb-III harmonic pairs as being where intensifications in 2 f{sub p} radiation are not resolved observationally. Larger turbulence levels producing trough-peak regions in the plasma density profile may lead to broader, resolvable intensifications in 2 f{sub p} radiation, which may account for the type IIIb-IIIb pairs that are sometimes observed.« less

On Primitives and Conjugate Harmonic Pairs in Hermitian Clifford Analysis

The notion of a conjugate harmonic pair in the context of Hermitian Clifford analysis is introduced as a pair of specific harmonic functions summing up to a Hermitian monogenic function in an open region $$\Omega $$ of $$\mathbb C ^n$$ . Hermitian monogenic functions are special monogenic functions, which are at the core of so-called Clifford analyis, a straightforward generalization to higher dimension of the holomorphic functions in the complex plane. Under certain geometric conditions on $$\Omega $$ the conjugate harmonic to a given specific harmonic is explicitly constructed and the potential or primitive of a Hermitian monogenic function is determined.

Analysis of Detection Methods of RF Vibrometer for Complex Motion Measurement

In this paper, the detection technique of the previously reported wavelength-division-sensing RF vibrometer is thoroughly analyzed by investigating the Bessel function coefficient of harmonics in the baseband spectrum. The analysis provides guidelines on the selection of harmonic pairs to help improve detection accuracy and reliability. A new detection method that utilizes multiple harmonic pairs under a fixed carrier frequency proves to be a more reliable approach for RF vibrometers. The sensitivity of harmonic amplitude ratio to the excess phase angle is explored as well. The results show that by using insensitive harmonic pairs, information about the characteristics of the vibration system and the external driving force are still attainable even when the phase angles are unknown. Real-time monitoring of various harmonic vibrations has been realized through a LabVIEW program with a user interface that provides visualization of the vibration patterns. The function of through-wall detection of the RF vibrometer has also been verified by conducting all measurements behind a wall with a detection distance of 1.5 m.

Harmonic wavelet-based data filtering for enhanced machine defect identification

A filter construction technique is presented for enhanced defect identification in rotary machine systems. Based on the generalized harmonic wavelet transform, a series of sub-frequency band wavelet coefficients are constructed by choosing different harmonic wavelet parameter pairs. The energy and entropy associated with each sub-frequency band are then calculated. The filtered signal is obtained by choosing the wavelet coefficients whose corresponding sub-frequency band has the maximum energy-to-entropy ratio. Experimental studies using rolling bearings that contain different types of structural defects have confirmed that the developed new technique enables high signal-to-noise ratio for effective machine defect identification.

Sound source segregation by goldfish: Two simultaneous tones

The perception of two simultaneous tones was investigated in goldfish using classical respiratory conditioning and a stimulus generalization paradigm. Pairs of tones were used to make up a mixture of 150 Hz and a higher harmonic or a mistuned harmonic. Fish were conditioned to the two-tone mixture and then tested for generalization to several pure tones. The simultaneous tones tended to be segregated in perception, with the generalization gradient for single tones having two peaks corresponding to the frequencies of the tone pairs. There were no consistent differences in the generalization gradients following conditioning to harmonic or inharmonic tone pairs. In addition, experiments were carried out in which the two tones of the pair were heard on alternate trials, always as single tones, followed by generalization tests to single tones. There was more generalization in this experiment, reflecting the fact that conditioning and generalization test stimuli were both single tones. However, the shapes of the generalization gradients were similar to those in which fish were conditioned to two simultaneous tones, indicating that the simultaneity of the tones did not make them harder to segregate. As the frequency separation between the two components narrowed, segregation tended to fail.

Optimizing Gabor Filter Design for Texture Edge Detection and Classification

An effective and efficient texture analysis method, based on a new criterion for designing Gabor filter sets, is proposed. The commonly used filter sets are usually designed for optimal signal representation. We propose here an alternative criterion for designing the filter set. We consider a set of filters and its response to pairs of harmonic signals. Two signals are considered separable if the corresponding two sets of vector responses are disjoint in at least one of the components. We propose an algorithm for deriving the set of Gabor filters that maximizes the fraction of separable harmonic signal pairs in a given frequency range. The resulting filters differ significantly from the traditional ones. We test these maximal harmonic discrimination (MHD) filters in several texture analysis tasks: clustering, recognition, and edge detection. It turns out that the proposed filters perform much better than the traditional ones in these tasks. They can achieve performance similar to that of state-of-the-art, distribution based (texton) methods, while being simpler and more computationally efficient.

Generalized Moisil-Théodoresco Systems and Cauchy Integral Decompositions

Letℝ0,m+1(s)be the space ofs-vectors(0≤s≤m+1)in the Clifford algebraℝ0,m+1constructed over the quadratic vector spaceℝ0,m+1, letr,p,q∈ℕwith0≤r≤m+1, 0≤p≤q, andr+2q≤m+1, and letℝ0,m+1(r,p,q)=∑j=pq⨁ ℝ0,m+1(r+2j). Then, anℝ0,m+1(r,p,q)-valued smooth functionWdefined in an open subsetΩ⊂ℝm+1is said to satisfy the generalized Moisil-Théodoresco system of type(r,p,q)if∂xW=0inΩ, where∂xis the Dirac operator inℝm+1. A structure theorem is proved for such functions, based on the construction of conjugate harmonic pairs. Furthermore, ifΩis bounded with boundaryΓ, whereΓis an Ahlfors-David regular surface, and ifWis aℝ0,m+1(r,p,q)-valued Hölder continuous function onΓ, then necessary and sufficient conditions are given under whichWadmits onΓa Cauchy integral decompositionW=W++W−.

A new strategy on pitch estimation in polyphonic musical audio

In this paper, a new perceptually motivated approach for pitch estimation in polyphonic musical audio is presented. The proposed algorithm is based on the pair-wise analysis of spectral peaks. Starting from a spectral representation of the audio signal, audible spectral peaks are selected with the help of a simplified psychoacoustic model. The idea of the technique lies in the identification of partials with a successive harmonic number (1 and 2, 2 and 3, etc.). Since two successive harmonics of a harmonic sound have a well-defined frequency ratio, a possible fundamental can be derived from the instantaneous frequencies of the two peaks. Furthermore, under certain conditions the harmonic integer ratios 5/3, 7/5, and 9/7 are also considered. Consecutively, the identified harmonic pairs are evaluated according to a perceptually motivated rating scheme, and the resulting pitch strengths are added to a pitch spectrogram. In addition to the efficient estimation of the fundamental frequency, the presented approach results in a quantitative prediction of the perceived pitch strength, allowing the identification of the predominant voices (e.g., melody and bass line) in music recordings. The method is robust in regard to the missing fundamental frequency phenomenon and works well with quasi-harmonic sounds.

Intersections of Lines and Circles

A method is presented for determining barycentric coordinates of points of intersection of a line and a circle. The method is applied specifically to the Euler line, the line of the circumcenter and incenter, the Brocard axis, and several circles, including the circumcircle, incircle, nine-point circle, and Brocard circle. The method also applies to intersections of certain pairs of lines, harmonic conjugate pairs, and to centers of similitude of pairs of circles.

Duality for Harmonic Differential Forms Via Clifford Analysis

The space HFk(Ω) of harmonic multi-vector fields in a domain \(\Omega \subset \mathbb{R}^{n}\) as introduced in [1] is closely connected to the space of harmonic forms. The main aim of this paper is to characterize the dual space of HFk(E) being \(\mathbf{E} \subset \mathbb{R}^{n}\) a compact set. It is proved that HFk(E)* is isomorphic to a certain quotient space of so-called harmonic pairs outside E vanishing at infinity.

Recovery of overlapped spectral harmonics in multiplex Fabry-Pérot interferometry

We propose an algorithm for recovering the overlapped harmonic components of a multiplex Fabry-Perot interferometer (MFPI) spectrum. A potential solution that utilizes the multiresolution properties of the MFPI spectrum is presented. The corrupted harmonic components are detected by comparing consecutive harmonic pairs at the same sampling rate and window length. The overlapped spectral components are adjusted with correction terms that have been determined from the detection stage.

The Orbital Perturbation Environment for the COBRA and COBRA Teardrop Elliptical Constellations

Elliptical, non-geostationary satellite orbit (NGSO) satellite constellations have been proposed as a solution to the problem of the shortage of slots in the Geostationary Ring for new communications satellites. These elliptical constellations have the potential to greatly multiply the available space real estate without interfering with the existing GEO satellites. These constellations employ eight-hour “leaning” elliptical orbits, operating at approximately the critical inclination of 63.4 degrees. The three active arcs of a single satellite have a ground trace that resembles a coiled cobra, hence the acronym name COBRA. Due to the exact repeat ground-track, with three orbital revolutions for each rotation of the Earth, these COBRA orbits experience deep tesseral resonance perturbations with the 3rd, 6th, and other (multiples of three) order geopotential harmonic coefficient pairs. These resonant perturbations are dependent on the RAAN of the particular ground-track. These orbits also experience strong perturbations due to the lunar-solar point masses. The semi-analytical satellite theory DSST is used to explore both the tesseral resonance and lunar-solar point mass perturbations.

Towards Attosecond Pulses with High Harmonics

The high harmonics produced by focusing an intense femtosecond laser in a gas are theoretically shown to be locked in phase. The physics of this locking is discussed and a new method based on quantum interference in two-photon, two-color ionization allowing to retrieve the relative phase of harmonic pairs is described. The main result is that the 5 harmonics of orders 11-19 produced in argon generate a train of subfemtosecond pulses with a period of 1.35 fs and a duration of 250 attoseconds.

Magnon waveforms in the presence of a vortex-pair in easy-plane antiferromagnets

In this paper we study the interactions between magnons and a pair of vortex–antivortex, in a classical two-dimensional easy-plane antiferromagnet, using elliptic coordinates. Our calculations can be applied to any system in which harmonic oscillations and vortex pairs are present.

Pattern Recognition in Images by Symmetries and Coordinate Transformations

A theory for detecting general curve families by means of symmetry measurements is presented. Symmetries are modeled by iso-gray curves of conjugate harmonic function pairs which define coordinate transformations. Harmonic function pair coordinates render the target curve patterns as parallel lines, which is defined here as linear symmetry. Detecting these lines, or generalized linear symmetry fitting, corresponds to finding invariants of Lie groups of transformations. A technique based on total least square error minimization for estimating the invariance parameters is presented. It uses the Lie infinitesimal operators to construct feature extraction methods that are efficient and simple to implement. The technique, which is shown to be an extension of the generalized Hough Transform, enables detection by voting and accumulating evidence for the searched pattern. In this approach complex valued votes are permitted, where the phase of the vote identifies the member of the family of patterns that are detectable. Experimental results illustrating the theory and its application to real as well as synthetic images are presented.

Advances in the measurement of gravity-induced quantum phase shifts

One of the significant successes in the field of neutron interferometry has been the experimental observation of the phase shift of a neutron deBroglie wave due to the action of the Earth's gravitational field. In this experiment the gravity-induced phase shift of the neutron is measured with a statistical uncertainity of order 1 part in 1000 in two different interferometers using nearly harmonic pairs of neutron wavelengths in two different interferometers. Extensions of the theoretical description of the tilt-angle gravity interferogram are discussed. A discrepancy between the theory and measured values of the phase shift due to gravity is observed at the 1% level.

Two-wavelength-difference measurement of gravitationally induced quantum interference phases

One of the significant successes in the field of neutron interferometry has been the experimental observation of the phase shift of a neutron de Broglie wave due to the action of the Earth's gravitational field. Past experiments have clearly demonstrated the effect and verified the quantum-mechanical equivalence of gravitational and inertial masses to a precision of about 1%. In this experiment the gravitationally induced phase shift of the neutron is measured with a statistical uncertainty of order 1 part in 1000 in two different interferometers. Nearly harmonic pairs of neutron wavelengths are used to measure and compensate for effects due to the distortion of the interferometer as it is tilted about the incident beam direction. A discrepancy between the theoretically predicted and experimentally measured values of the phase shift due to gravity is observed at the 1% level. Extensions to the theoretical description of the shape of a neutron interferogram as a function of tilt in a gravitational field are discussed and compared with experiment.

Novel four-bridge PWM static VAr compensator

A novel four-bridge PWM static VAr compensator is presented. Theoretical analysis shows that the harmonic pairs of [(2k + 1)fs ± f] and [(4k + 2)fs ± f)] generated in each bridge are effectively eliminated through appropriate phase shift control. To satisfy the same 5% total harmonic distortion requirement, this new compensator can work at 200 Hz, which is much lower than the previously reported 1.9 kHz of a single-bridge PWM static VAr compensator. The efficiency is increased and reliability is enhanced. The fairly low switching frequency and multiple-bridge structure facilitate its use in a high-power VAr compensator since GTO can be employed as the power switch. Theoretical analysis was verified through simulations and experimental results.

Numerical simulation of type III solar radio bursts caused by high-density electron beam

Numerical simulation for the type III solar radio bursts in meter wavelengths was made with the electron beam of a high number density enough to emit fundamental radio waves comparable in intensity with the second harmonic. This requirement is fulfilled if the optical thickness τ 1 for the negative absorption (amplification) becomes -23 to -25. Since τ 1 is roughly proportional to the time-integral of the electron flux of the beam, the intensity of the fundamental waves depends strongly on the parameters which determine the electron flux. Therefore, it is most unlikely that the harmonic pairs of type III bursts of the first and the second harmonics occur frequently with comparable intensities in a wide frequency range, say 200 MHz to 20 MHz, if we take the working hypothesis that the fundamental waves are caused by the scattering of electron plasma waves by thermal ions and amplified during the propagation along the beam. However, we cannot rule out the possibility that single type III bursts with short durations or group of such bursts are the fundamental waves emitted by the above mechanism, but only if the observed large size of the radio source can be attributed to the radio scattering alone.

Towards a theory for type III solar radio bursts

The procedure developed in Smith (1974) to model the radiation source for type 11I bursts is modified to include scattering of radiation in the source itself. Since the inhomogeneities in the source must have the same statistical properties as the inhomogeneities used in tracing radiation from the source to the observer, these two parts of the type II1 problem are no longer uncoupled. Thus we use inhomogeneities consistent with the scattering inhomogeneities of Steinberg et aL (1971) and Riddle (1974) and apply the procedure to an archetype 'fundamental-harmonic' pair observed at Culgoora on 28 September, 1973 at 0319 UT. We find that it is impossible to model this burst with a source which is homogeneous in the sense that every part of the source has the same energy density in plasma waves. The density inhomogeneities in the source severely hamper amplification of the supposed fundamental. Possible ways out of this dilemma are discussed, including second harmonic pairs and a source with an inhomogeneous distribution of plasma waves. It is concluded that none of the possibilities are completely satisfactory to explain present observations and suggested that critical observations are missing.