Harmonic Representation Research Articles

Musicianship Modulates Cortical Effects of Attention on Processing Musical Triads

Background: Many studies have demonstrated the benefits of long-term music training (i.e., musicianship) on the neural processing of sound, including simple tones and speech. However, the effects of musicianship on the encoding of simultaneously presented pitches, in the form of complex musical chords, is less well established. Presumably, musicians’ stronger familiarity and active experience with tonal music might enhance harmonic pitch representations, perhaps in an attention-dependent manner. Additionally, attention might influence chordal encoding differently across the auditory system. To this end, we explored the effects of long-term music training and attention on the processing of musical chords at the brainstem and cortical levels. Method: Young adult participants were separated into musician and nonmusician groups based on the extent of formal music training. While recording EEG, listeners heard isolated musical triads that differed only in the chordal third: major, minor, and detuned (4% sharper third from major). Participants were asked to correctly identify chords via key press during active stimulus blocks and watched a silent movie during passive blocks. We logged behavioral identification accuracy and reaction times and calculated information transfer based on the behavioral chord confusion patterns. EEG data were analyzed separately to distinguish between cortical (event-related potential, ERP) and subcortical (frequency-following response, FFR) evoked responses. Results: We found musicians were (expectedly) more accurate, though not faster, than nonmusicians in chordal identification. For subcortical FFRs, responses showed stimulus chord effects but no group differences. However, for cortical ERPs, whereas musicians displayed P2 (~150 ms) responses that were invariant to attention, nonmusicians displayed reduced P2 during passive listening. Listeners’ degree of behavioral information transfer (i.e., success in distinguishing chords) was also better in musicians and correlated with their neural differentiation of chords in the ERPs (but not high-frequency FFRs). Conclusions: Our preliminary results suggest long-term music training strengthens even the passive cortical processing of musical sounds, supporting more automated brain processing of musical chords with less reliance on attention. Our results also suggest that the degree to which listeners can behaviorally distinguish chordal triads is directly related to their neural specificity to musical sounds primarily at cortical rather than subcortical levels. FFR attention effects were likely not observed due to the use of high-frequency stimuli (>220 Hz), which restrict FFRs to brainstem sources.

Harmonic flow-field representations of quantum bits and gates

We describe a general procedure for mapping arbitrary n-qubit states to two-dimensional (2D) vector fields. The mappings use complex rational function representations of individual qubits, producing classical vector field configurations that can be interpreted in terms of 2D inviscid fluid flows or electric fields. Elementary qubits are identified with localized defects in 2D harmonic vector fields, and multiqubit states find natural field representations via complex superpositions of vector field products. In particular, separable states appear as highly symmetric flow configurations, making them both dynamically and visually distinct from entangled states. The resulting real-space representations of entangled qubit states enable an intuitive visualization of their transformations under quantum logic operations. We demonstrate this for the quantum Fourier transform and the period finding process underlying Shor's algorithm, along with other quantum algorithms. Due to its generic construction, the mapping procedure suggests the possibility of extending concepts such as entanglement or entanglement entropy to classical continuum systems, and thus may help guide new experimental approaches to information storage and nonstandard computation. Published by the American Physical Society 2024

Characterizing grain boundary network length features through a harmonic representation

When modeling and characterizing grain boundary networks (GBNs), there are situations where local descriptors such as triple junction fractions (TJFs) and special boundary fractions are identical between two microstructures, but the performance or properties between the two are distinct. These differences are caused by higher length-scale features that cannot be identified using local structural descriptors. Spectral graph theory (SGT) has been used previously to encode network length features, but did not enable direct quantitative comparisons between the structures that cause property differences. In this paper, we derive a harmonic representation of diffusion on GBNs based on SGT. This method enables direct quantitative comparisons between GBNs for microstructures with different morphologies, and identifies network length features responsible for structural and performance differences, which local descriptors cannot explain. We show an interpretation of the eigenmodes generated by this method that explains long-range structural causes of certain property differences. We apply this method to a large library of microstructures, and identify structural classes through clustering. We show that equal proportioned TJF and J1 dominated microstructures are the most sensitive to network length differences because of boundary configurations, while J2-J3 dominated structures are the least sensitive. This method also identifies network length features that result in anomalous percolation/non-percolation compared to predictions based on local correlations alone.

Reflection of Waves in a Two-Temperature Magneto-fiber-Reinforced Solid with Memory-Dependent Derivative Using Different Theories

Abstract Purpose The problem is concerned with the analysis of the reflection of the waves through a fiber-reinforced thermoelastic medium under the effect of the magnetic field, gravity, and the initial stress. The problem is discussed in the context of the three-phase-lag model and Green-Naghdi theory of type II and III with the memory-dependent derivative and variable thermal conductivity. Methods The harmonic representation of waves is used to find the solution to the problem. Based on the solution, it is concluded that after reflection three quasi-waves propagate through the medium. Results Numerical computations were performed using MATLAB software. The reflection coefficient ratio variations with the angle of the incident are shown graphically. Conclusion Comparisons are made with the results predicted for different values of the thermal conductivity parameter, two-temperature parameter, initial stress, gravity field, and different values of the magnetic field.

Very Accurate Time-Frequency Representation of Induction Motors Harmonics for Fault Diagnosis Under Load Variations

Very Accurate Time-Frequency Representation of Induction Motors Harmonics for Fault Diagnosis Under Load Variations

Revaluation of occupancy duration for live load using big data of enterprise credit information

The occupancy duration is an essential variable in establishing the time-varying model of the live load. The manual questioning and inquiries, which are the common survey methods for occupancy duration, suffer from being expensive and time-consuming. This has hindered the updating of the data basis and progress in live load research. This study proposes a novel big data survey method, which utilizes 210 million enterprise credit information to collect occupancy duration samples. Along with the big data survey method, an address-oriented mining method is presented to efficiently extract duration samples of the target type from big data. Based on the actual distribution of the extracted duration data, a more general time-varying model, called the compound renewal process, is suggested to describe the live load. The stochastic harmonic function representation method and probability density evolution method are then adopted to access the maximum load distribution of the live load process. The maximum load distributions of the sustained and combined loads are calculated and compared with previous studies to evaluate the impact of the new survey results. Comparison shows that the occupancy duration of office buildings is significantly shorter, resulting in an increase in the maximum load. The proposed data survey method enables continuous updating of duration samples, and the compound renewal process allows better compatibility for new survey data. The up-to-date duration data and a realistic time-varying model provide new ideas for determining the design live load.

Reduced Representation of Spatial Harmonics for Motor Model Order Reduction Using Block Arnoldi Method

Reduced Representation of Spatial Harmonics for Motor Model Order Reduction Using Block Arnoldi Method

Nonsingular Calculation of Global Magnetic Gradient Tensor Based on Oblate Spheroidal Harmonics Representation

Nonsingular Calculation of Global Magnetic Gradient Tensor Based on Oblate Spheroidal Harmonics Representation

Fast multi-compartment Microstructure Fingerprinting in brain white matter.

We proposed two deep neural network based methods to accelerate the estimation of microstructural features of crossing fascicles in the white matter. Both methods focus on the acceleration of a multi-dictionary matching problem, which is at the heart of Microstructure Fingerprinting, an extension of Magnetic Resonance Fingerprinting to diffusion MRI. The first acceleration method uses efficient sparse optimization and a dedicated feed-forward neural network to circumvent the inherent combinatorial complexity of the fingerprinting estimation. The second acceleration method relies on a feed-forward neural network that uses a spherical harmonics representation of the DW-MRI signal as input. The first method exhibits a high interpretability while the second method achieves a greater speedup factor. The accuracy of the results and the speedup factors of several orders of magnitude obtained on in vivo brain data suggest the potential of our methods for a fast quantitative estimation of microstructural features in complex white matter configurations.

Mie scattering with 3D angular spectrum method.

Mie theory is a powerful method to model electromagnetic scattering from a multilayered sphere. Usually, the incident beam is expanded to its vector spherical harmonic representation defined by beam shape coefficients, and the multilayer sphere scattering is obtained by the T-matrix method. However, obtaining the beam shape coefficients for arbitrarily shaped incident beams has limitations on source locations and requires different methods when the incident beam is defined inside or outside the computational domain or at the scatterer surface. We propose a 3D angular spectrum method for defining beam shape coefficients from arbitrary source field distributions. This method enables the placement of the sources freely within the computational domain without singularities, allowing flexibility in beam design. We demonstrate incident field synthesis and spherical scattering by comparing morphology-dependent resonances to known values, achieving excellent matching and high accuracy. The proposed method has significant benefits for optical systems and inverse beam design. It allows for the analysis of electromagnetic forward/backward propagation between optical elements and spherical targets using a single method. It is also valuable for optical force beam design and analysis.

Adjoint-based control of three dimensional Stokes droplets

We develop a continuous adjoint formulation and implementation for controlling the deformation of clean, neutrally buoyant droplets in Stokes flow through farfield velocity boundary conditions. The focus is on dynamics where surface tension plays an important role through the Young-Laplace law. To perform the optimization, we require access to first-order gradient information, which we obtain from the linearized sensitivity equations and their corresponding adjoint by applying shape calculus to the space-time tube formed by the interface evolution. We show that the adjoint evolution equation can be efficiently expressed through a scalar adjoint transverse field. The optimal control problem is discretized by high-order boundary integral methods using Quadrature by Expansion coupled with a spherical harmonic representation of the droplet surface geometry. We show the accuracy and stability of the scheme on several tracking-type control problems.

Capturing Statistical Isotropy Violation with Generalized Isotropic Angular Correlation Functions of Cosmic Microwave Background Anisotropy

The exquisitely measured maps of fluctuations in the cosmic microwave background (CMB) present the possibility of systematically testing the principle of statistical isotropy of the Universe. A systematic approach based on strong mathematical formulation allows any nonstatistical isotropic (nSI) feature to be traced to the nature of physical effects or observational artifacts. Bipolar spherical harmonics (BipoSH) representation has emerged as an overarching general formalism for quantifying the departures from statistical isotropy for a field on a 2D sphere. We adopt a little-known reduction of the BipoSH functions, dubbed minimal harmonics in the original paper by Manakov et al. We demonstrate that this reduction technique of BipoSH leads to a new generalized set of isotropic angular correlation functions referred to here as minimal BipoSH functions that are observable quantifications of nSI features in a sky map. This paper presents a novel observable quantification of deviation from statistical isotropy in terms of generalized angular correlation functions that are compact and complementary to the BipoSH spectra that generalize the angular power spectrum of CMB fluctuations.

Efficient representation of head-related transfer functions in continuous space–frequency domains

Utilizing spherical harmonic (SH) domain has been established as the default method of obtaining continuity over space in head-related transfer functions (HRTFs). This paper concerns different variants of extending this solution by replacing SHs with four-dimensional (4D) continuous functional models in which frequency is imagined as another physical dimension. Recently developed hyperspherical harmonic (HSH) representation is compared with models defined in spherindrical coordinate system by merging SHs with one-dimensional basis functions. The efficiency of both approaches is evaluated based on the reproduction errors for individual HRTFs from HUTUBS database, including detailed analysis of its dependency on chosen orders of approximation in frequency and space. Employing continuous functional models defined in 4D coordinate systems allows HRTF magnitude spectra to be expressed as a small set of coefficients which can be decoded back into values at any direction and frequency. The best performance was noted for HSHs and SHs merged with reverse Fourier–Bessel series, with the former featuring better compression abilities, achieving slightly higher accuracy for low number of coefficients. The presented models can serve multiple purposes, such as interpolation, compression or parametrization for machine learning applications, and can be applied not only to HRTFs but also to other types of directivity functions, e.g. sound source directivity.

Spherical Harmonic Representation of Energetic Neutral Atom Flux Components Observed by IBEX

The Interstellar Boundary Explorer (IBEX) images the heliosphere by observing energetic neutral atoms (ENAs). The IBEX-Hi instrument on board IBEX provides full-sky maps of ENA fluxes produced in the heliosphere and very local interstellar medium through charge exchange of suprathermal ions with interstellar neutral atoms. The first IBEX-Hi results showed that, in addition to the anticipated globally distributed flux (GDF), a narrow and bright emission from a circular region in the sky, dubbed the IBEX ribbon, is visible in all energy steps. While the GDF is mainly produced in the inner heliosheath, ample evidence indicates that the ribbon forms outside the heliopause in the regions where the interstellar magnetic field is perpendicular to the lines of sight. The IBEX maps produced by the mission team distribute the observations into 6° × 6° rectangle pixels in ecliptic coordinates. The overlap of the GDF and ribbon components complicates qualitative analyses of each source. Here, we find the spherical harmonic representation of the IBEX maps, separating the GDF and ribbon components. This representation describes the ENA flux components in the sky without relying on any pixelization scheme. Using this separation, we discuss the temporal evolution of each component over the solar cycle. We find that the GDF is characterized by larger spatial scale structures than those of the ribbon. However, we identify two isolated, small-scale signals in the GDF region that require further study.

Spherical harmonics representation of the gravitational phase shift

We investigate the general relativistic phase of an electromagnetic wave as it propagates in the gravitational field of the Earth, which is modeled as an isolated, weakly aspherical gravitating body. We introduce coordinate systems to describe light propagation in the Earth's vicinity along with the relevant coordinate transformations, and discuss the transformations between proper and coordinate times. We represent the Earth's gravitational field using Cartesian symmetric trace-free (STF) mass multipole moments. The light propagation equation is solvable along the trajectory of a light ray to all STF orders $\ensuremath{\ell}$. Although we focus primarily on the quadrupole ($\ensuremath{\ell}=2$), octupole ($\ensuremath{\ell}=3$), and hexadecapole ($\ensuremath{\ell}=4$) cases, our approach is valid to all orders. We express the STF moments via spherical harmonic coefficients of various degree and order, ${C}_{\ensuremath{\ell}k},{S}_{\ensuremath{\ell}k}$. The result is the gravitational phase shift expressed in terms of the spherical harmonics. These results are new. We also consider contributions due to tides and the Earth's rotation. We estimate the characteristic magnitudes of each term of the resulting overall gravitational phase shift. The terms assessed are either large enough to impact current-generation clocks or will become significant as future-generation clocks offer greater sensitivity. Our formulation is useful for many practical and scientific applications, including space-based time and frequency transfers, relativistic geodesy and navigation, as well as quantum communication links and space-based tests of fundamental physics.

Mathematical representation of harmonic resonance phenomenon and harmonic compensation in PMSG based wind farms under feedforward compensation of the grid voltages

This paper mainly focuses on harmonic resonance analysis and harmonic compensation in electrical power grid including PMSG-based wind turbines. In this way, first, harmonic resonance representation and impedance model of the study system is presented by taking the grid-side converter control into account. Then, it is mainly dealt with the harmonic resonance analysis in the study electrical power grid under feedforward compensation of the grid voltages. In this paper, under feedforward compensation of the grid voltage, impacts of the bandwidths of the voltage/current measurement filters on harmonic resonance response of the system are investigated by using frequency scan and harmonic resonance mode analysis (HRMA) approaches. It is shown that by increasing the voltage measurement filter bandwidth, resonance damping is weakened and the magnitudes of driving point impedances increase. Hence, feedforward compensation of the grid voltage in a limited frequency range results in better resonance damping in comparison with feedforward compensation in a wide frequency range. On the other hand, by increasing the current measurement filter bandwidth, the new resonant modes are not added to the network. It is also shown that employing passive filters significantly reduces the driving point impedances of all buses at the different harmonic orders.

Spherical harmonics representation of the steady-state membrane potential shift induced by tDCS in realistic neuron models

Objective. We provide a systematic framework for quantifying the effect of externally applied weak electric fields on realistic neuron compartment models as captured by physiologically relevant quantities such as the membrane potential or transmembrane current as a function of the orientation of the field. Approach. We define a response function as the steady-state change of the membrane potential induced by a canonical external field of 1 V m−1 as a function of its orientation. We estimate the function values through simulations employing reconstructions of the rat somatosensory cortex from the Blue Brain Project. The response of different cell types is simulated using the NEURON simulation environment. We represent and analyze the angular response as an expansion in spherical harmonics. Main results. We report membrane perturbation values comparable to those in the literature, extend them to different cell types, and provide their profiles as spherical harmonic coefficients. We show that at rest, responses are dominated by their dipole terms (), in agreement with experimental findings and compartment theory. Indeed, we show analytically that for a passive cell, only the dipole term is nonzero. However, while minor, other terms are relevant for states different from resting. In particular, we show how and terms can modify the function to induce asymmetries in the response. Significance. This work provides a practical framework for the representation of the effects of weak electric fields on different neuron types and their main regions—an important milestone for developing micro- and mesoscale models and optimizing brain stimulation solutions.

In search of the missing resolved harmonics: Neural and behavioral studies of pitch discrimination

The pitch of harmonic complex tones (HCT) plays important roles in speech and music perception and in the perceptual organization of sound. Human listeners rely primarily on the spectral pattern of harmonics individually resolved by the frequency analysis in the cochlea to derive a pitch percept. Yet, evidence for a robust neural representation of resolved harmonics in animal models has been elusive, and some species appear to rely on temporal envelope cues rather than on resolved harmonics to discriminate pitch. We performed both behavioral experiments on pitch discrimination by rabbits and single-unit recordings from the auditory nerve (AN) and inferior colliculus (IC) using HCTs with equal-amplitude harmonics in which the fundamental frequency (F0) was varied over a wide range. We found that rabbits can discriminate the pitch of HCT using the spectral pattern of resolved harmonics for F0s within the range of conspecific vocalizations. Correspondingly, many IC neurons contribute to a rate-place code for the frequencies of resolved harmonics for F0s >500 Hz, and this code is more robust to variations in stimulus level than that found in the AN. A remaining challenge is to identify the neural mechanisms underlying enhanced rate-place coding in IC. [Work supported by NIH-NIDCD DC002258.]

Real-time sound synthesis of pass-by noise: comparison of spherical harmonics and time-varying filters

This paper proposes and compares two sound synthesis techniques to render a moving source for a fixed receiver position based on indoor pass-by noise measurements. The approaches are based on the time-varying infinite impulse response (IIR) filtering and spherical harmonics (SH) representation. The central contribution of the work is a framework for realistic moving source sound synthesis based on transfer functions measured using static far-field microphone arrays. While the SHs require a circular microphone array and a free-field propagation (delay, geometric spread), the IIR filtering relies on far-field microphones that correspond to the propagation path of the moving source. Both frameworks aim to provide accurate sound pressure levels in the far-field that comply with standards. Moreover, the frameworks can be extended to additional sources and filters (e.g. sound barriers) to create different moving source scenarios by removing the room size constraint. The results of the two sound synthesis approaches are preliminary evaluated and compared on a vehicle pass-by noise dataset and it is shown that both approaches are capable of accurately and efficiently synthesize a moving source.

Periodic LQG Wind Turbine Control with Adaptive Load Reduction

A periodic linear quadratic Gaussian (LQG) control law augmented with a reference point adaption to enable adequate rotor speed tracking and sufficient load reductions for a wind turbine is presented. The solution of the periodic LQG control problem is based on solving two periodic Ricatti differential equations in continuous time with a multiple shooting integration technique. For this, the available gridded linear time-variant description of the turbine is converted to a harmonic representation using harmonic Fourier approximation. While the periodic LQG controller provides rotor speed tracking and effective damping of the aeroelastic blade modes, the reference point adaption explicitly reduces the loads resulting from the periodic operation of the turbine rotor at the rotor rotational frequency. The performance of the proposed control system is compared against a baseline controller in realistic wind scenarios using a high fidelity nonlinear simulator. The results show a significant damage equivalent load reduction while maintaining adequate rotor speed tracking.