Fourier Series Representation Research Articles

Modeling and Analysis of Harmonic Spurs in DLL-Based Frequency Synthesizers

Periodic jitter raises the harmonic spurs at frequency synthesizer output spectrum, down-converting the out-of-band interferers into the desired band and corrupting the wanted signal. This paper proposes a comprehensive behavioral model for spur characterization of edge-combining delay-locked loop (DLL)-based synthesizers, which includes the effects of delay mismatch, static phase error (SPE), and duty cycle distortion (DCD). Based on the proposed model and utilizing Fourier series representation of DLL output phases, an analytical model which formulates the synthesizer spur-to-carrier ratio (SCR) is developed. Moreover, from statistical analysis of the analytical derivations, a closed-form expression for SCR is obtained, from which a spur-aware synthesizer design flow is proposed. Employing this flow and without Monte Carlo (MC) method, one can determine the required stage-delay standard deviation (SD) of a DLL-based synthesizer, at which a certain spurious performance demanded by a target wireless standard is satisfied. A design example is presented which utilizes the proposed design flow to fulfill the SCR requirement of -45 dBc for WiMedia-UWB standard. Transistor-level MC simulation of the synthesizer SCR for a standard 65-nm CMOS implementation exhibits good compliance with analytical models and predictions.

Analysis of non-Fourier heat conduction in a solid sphere under arbitrary surface temperature change

In this paper, the non-Fourier heat conduction in a solid sphere under arbitrary surface thermal disturbances is solved analytically. Four cases including sudden, simple harmonic periodic, triangular and pulse surface temperature changes are investigated step-by-step. The analytical solutions are obtained using the separation of variables method and Duhamel’s principle along with the Fourier series representation of an arbitrary periodic function and the Fourier integral representation of an arbitrary non-periodic function. Using these analytical solutions, the temperature profiles of the solid sphere are analyzed, and the differences in the temperature response between the “hyperbolic” and “parabolic” are discussed. These solutions can be applicable to all kinds of non-Fourier heat conduction analyses for arbitrary boundary conditions occurred in technology. And as application examples, particular attention is devoted to the cases of triangular surface temperature change and pulse surface temperature change. The examples presented in this paper can be used as benchmark problems for future numerical method validations.

Double‐Diffusive Convective Transport in a Nanofluid‐Saturated Porous Layer with Cross Diffusion and Variation of Viscosity and Conductivity

AbstractThe onset of double‐diffusive nanofluid convection in a fluid‐saturated horizontal porous layer is studied with thermal conductivity and viscosity dependent on the nanoparticle volume fraction. The Darcy model has been used for the porous medium, while the nanofluid incorporates the effects of Brownian motion along with thermophoresis. The nanofluid is assumed to be diluted and this enables the porous medium to be treated as a weakly heterogeneous medium with variation in the vertical direction of conductivity and viscosity. In addition, the thermal energy equation includes regular diffusion and cross diffusion terms. The linear stability analysis is based on the normal mode technique, while for nonlinear analysis, minimal representation of the truncated Fourier series representation involving only two terms has been used. It is found that for the stationary mode the Soret parameter, Dufour parameter, viscosity ratio, and conductivity ratio have a stabilizing effect, while the solutal Rayleigh number destabilizes the system. For the oscillatory mode, the Soret parameter, Dufour parameter, and viscosity ratio have a stabilizing effect while the solutal Rayleigh number and conductivity ratio destabilize the system. For steady finite amplitude motions, the heat and mass transport decreases with an increase in the values of the Dufour parameter and solutal Rayleigh number. The Soret parameter enhances the solute concentration Nusselt number while it retards the thermal Nusselt number and concentration Nusselt number. The viscosity ratio and conductivity ratio enhances the heat and mass transports. We also study the effect of time on transient Nusselt numbers which is found to be oscillatory when time is small. However, when time becomes very large, all three transient Nusselt values approach a steady value. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(7): 628–652, 2014; Published online 11 November 2013 in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21102

A Study of Subtractive Digital Dither in Single-Stage and Multi-Stage Quantizers

The effect of subtractive digital dither on single-stage and multi-stage quantizers is examined. In the single stage case, it is proved that the conditional density of the quantization error becomes asymptotically independent of the input and uniformly distributed as the number of digital dither levels increases. For a fixed number of levels, the conditional expectation of a class of piecewise constant functions is shown to be independent of the input. A Fourier series representation of quantization error densities is used to construct a linear periodically time varying model of multistage quantization and dithering systems. For a homogeneous multi-stage quantization system, it is shown that when the least common multiple of the numbers of digital dither levels at successive stages is large, the statistical dependence of the quantization noise on the input disappears almost entirely. Computer simulations confirm this analysis.

Using Fourier series to analyse mass imperfections in vibratory gyroscopes

When a vibrating structure is subjected to a rotation, the vibrating pattern rotates at a rate (called the precession rate) proportional to the inertial angular rate. This is known as Bryanʼs effect and it is employed to calibrate the resonator gyroscopes that are used to navigate in outer space, the stratosphere and under the polar cap. We study Bryanʼs effect for a non-ideal resonator gyroscope, using the computer algebra system (CAS) Mathematica to do the analysis involved, rendering this work accessible to undergraduate students with a working knowledge of college calculus and basic physics or mechanics (such as senior Engineering Mathematics students). In this paper the density of a slowly rotating vibrating annular disc is assumed to have small variations circumferentially, enabling a Fourier series representation of the density function. Using a CAS, the Lagrangian of the system of vibrating particles in the disc is calculated and, employing the CAS on the Euler–Lagrange equations, the equations of motion of the vibrating, rotating system are calculated in terms of “fast” variables, enabling us to demonstrate that the mass anisotropy induces a frequency splitting (beats). Unfortunately the fast variables are difficult to analyse (even with the aid of a CAS) and consequently a transformation from fast to slow variables is achieved. These slow variables are the principal and quadrature vibration amplitudes, precession rate and a phase angle. The transformation yields a system of four nonlinear ordinary differential equations (ODEs). This system of ODE demonstrates that the Fourier coefficients of the density function influence the precession rate and consequently a gyroscope manufactured from such a disc cannot use Bryanʼs effect for calibration purposes. Indeed, the CAS visualises that a capture effect occurs with the precession angle that appears to vary periodically and not increase linearly (Bryanʼs effect) as it would for a perfect structure. Keeping in mind that manufacturing imperfections will always be present in the real-world, the analysis shows how such density variations may be minimised. Using a symbolic manipulator such as Mathematica to do the “book-keeping” eliminates the plethora of technical detail that arises during calculations of a highly technical nature. This allows the aforementioned students to focus on the salient parts of the analysis, producing results that might have been beyond their capabilities without the aid of a CAS.

Integrated photonic threshold comparator based on square-wave synthesis

A photonic threshold comparator is presented. A step-like electrical-to-optical (E/O) response is obtained by employing Fourier series synthesis in which a set of sine-wave responses of different amplitudes and phases are superimposed according to the Fourier series representation of a square-wave. The proposed comparator does not rely on optical material non-linearity; rather it consists of multimode interference (MMI) couplers and phase shifters.

Boundary element elastic stress analysis of 3D generally anisotropic solids using fundamental solutions based on Fourier series

The authors have very recently proposed an efficient, accurate alternative scheme to numerically evaluate etc. Green’s function, U(x), and its derivatives for three-dimensional, general anisotropic elasticity. These quantities are necessary items in the formulation of the boundary element method (BEM). The scheme is based on the double Fourier series representation of the explicit, exact, algebraic solution derived by Ting and Lee (1997) [Ting, T.C.T., Lee, V.G., 1997. The three-dimensional elastostic Green’s function for general anisotropic linear elastic solid. Q. J. Mech. Appl. Math. 50, 407–426] expressed in terms of Stroh’s eigenvalues. By taking advantage of some its characteristics, the formulations in this double Fourier series approach are revised and several of the analytical expressions are re-arranged in the present study. This results in significantly fewer terms to be summed in the series thereby improving further the efficiency for evaluating the Green’s function and its derivatives. These revised Fourier series representations of U(x) and its derivatives are employed in a BEM formulation for three-dimensional linear elastostatics. Some numerical examples are presented to demonstrate the veracity of the implementation and its applicability to the elastic stress analysis of generally anisotropic solids. The results are compared with known solutions in the literature where possible, and with those obtained using the commercial finite element code ANSYS. Excellent agreement is obtained in all cases.

Fourier series analysis of a cylindrical pressure vessel subjected to axial end load and external pressure

This paper presents the comparison of a reliability technique that employs a Fourier series representation of random axisymmetric and asymmetric imperfections in a cylindrical pressure vessel subjected to an axial end load and external pressure, with evaluations prescribed by the ASME Boiler and Pressure Vessel Code, Section VIII, Division 2 Rules. The ultimate goal of the reliability technique described herein is to predict the critical buckling load associated with the subject cylindrical pressure vessel. Initial geometric imperfections are shown to have a significant effect on the calculated load carrying capacity of the vessel. Fourier decomposition was employed to interpret imperfections as structural features that can be easily related to various other types of defined imperfections. The initial functional description of the imperfections consists of an axisymmetric portion and a deviant portion, which are availed in the form of a double Fourier series. Fifty simulated shells generated by the Monte Carlo technique are employed in the final prediction of the critical buckling load. The representation of initial geometrical imperfections in the cylindrical pressure vessel requires the determination of respective Fourier coefficients. Multi-mode analyses are expanded to evaluate a large number of potential buckling modes for both predefined geometries in combination with asymmetric imperfections as a function of position within the given cylindrical shell. The probability of the ultimate buckling stress exceeding a predefined threshold stress is also calculated. The method and results described herein are in stark contrast to the “knockdown factor” approach as applied to compressive stress evaluations currently utilized in industry. Further effort is needed to improve on the current design rules regarding column buckling of large diameter pressure vessels subjected to an axial end load and external pressure designed in accordance with ASME Boiler and Pressure Vessel Code, Section VIII, Division 2 and ASME STS-1.

Truncated Fourier-series approximation of the time-domain radiative transfer equation using finite elements

The radiative transfer equation (RTE) is widely accepted to accurately describe light transport in a medium with scattering particles, and it has been successfully applied as a light-transport model, for example, in diffuse optical tomography. Due to the computationally expensive nature of the RTE, most of these applications have been in the frequency domain. In this paper, an efficient solution method for the time-domain RTE is proposed. The method is based on solving the frequency-domain RTE at multiple modulation frequencies and using the Fourier-series representation of the radiance to obtain approximation of the time-domain solution. The approach is tested with simulations. The results show that the method can be used to obtain the solution of the time-domain RTE with good accuracy and with significantly fewer computational resources than are needed in the direct time-domain solution.

A Comparative Study on the Performance of Bayesian Partially Linear Models

In this paper, we consider Bayesian approaches to partially linear models, in which a regression function is represented by a semiparametric additive form of a parametric linear regression function and a nonparametric regression function. We make a comparative study on the performance of widely used Bayesian partially linear models in terms of empirical analysis. Specifically, we deal with three Bayesian methods to estimate the nonparametric regression function, one method using Fourier series representation, the other method based on Gaussian process regression approach, and the third method based on the smoothness of the function and differencing. We compare the numerical performance of three methods by the root mean squared error(RMSE). For empirical analysis, we consider synthetic data with simulation studies and real data application by fitting each of them with three Bayesian methods and comparing the RMSEs.

Electroelastic modeling and experimental validations of piezoelectric energy harvesting from broadband random vibrations of cantilevered bimorphs

We present electroelastic modeling, analytical and numerical solutions, and experimental validations of piezoelectric energy harvesting from broadband random vibrations. The modeling approach employed herein is based on a distributed-parameter electroelastic formulation to ensure that the effects of higher vibration modes are included, since broadband random vibrations, such as Gaussian white noise, might excite higher vibration modes. The goal is to predict the expected value of the power output and the mean-square shunted vibration response in terms of the given power spectral density (PSD) or time history of the random vibrational input. The analytical method is based on the PSD of random base excitation and distributed-parameter frequency response functions of the coupled voltage output and shunted vibration response. The first of the two numerical solution methods employs the Fourier series representation of the base acceleration history in an ordinary differential equation solver while the second method uses an Euler–Maruyama scheme to directly solve the resulting electroelastic stochastic differential equations. The analytical and numerical simulations are compared with several experiments for a brass-reinforced PZT-5H bimorph under different random excitation levels. The simulations exhibit very good agreement with the experimental measurements for a range of resistive electrical boundary conditions and input PSD levels. It is also shown that lightly damped higher vibration modes can alter the expected power curve under broadband random excitation. Therefore, the distributed-parameter modeling and solutions presented herein can be used as a more accurate alternative to the existing single-degree-of-freedom solutions for broadband random vibration energy harvesting.

Statistical moment equations for stochastic spin dynamics in phase space: A uniaxial paramagnet subjected to a dc bias field of arbitrary orientation

Spin dynamics in a dissipative environment are treated via the evolution (master) equation for the quasiprobability density function of spin orientations in the phase space of the polar and azimuthal angles in the weak spin-bath coupling and high-temperature limits. The explicit solution is written for an arbitrary spin Hamiltonian as a finite series of spherical harmonics analogous to the (infinite) Fourier series representation of the classical case governed by the Fokker--Planck equation. Therefore, the expansion coefficients, i.e., the statistical averages of the spherical harmonics, may be determined as before from a differential-recurrence relation, yielding the stochastic spin dynamics for arbitrary spin number $S$. For large $S$ the differential-recurrence relations reduce to those generated by the Fokker--Planck equation. Thus, the spin dynamics may be treated in a manner transparently linking to the classical representations, thereby providing quantum corrections to classical averages. The method is illustrated via the magnetization relaxation of a uniaxial paramagnet with a dc field ${\mathbf{H}}_{0}$ applied at an arbitrary angle to the easy axis, which is the quantum version of the most basic model in classical superparamagnetism.

Investigation of the Convergence Behavior with Fluctuation Features in the Fourier Modal Analysis of a Metallic Grating

We observe that the transmission and reflection efficiencies of a one-dimensional metallic grating under transverse-magnetic illumination calculated using the Fourier modal method (FMM) with the Fourier factorization rules have peculiar fluctuations, albeit small in magnitude, as the number of field harmonics increases. It is shown that when the number of Fourier terms for the electromagnetic field is increased from that in the conventional FMM, the fluctuations due to non-convergent highly evanescent eigenmodes can be eliminated. Our examination reveals that the fluctuations originate from the Gibbs phenomenon inherent in the Fourier-series representation of a permittivity function with discontinuities, and from non-convergence of highly evanescent internal Bloch eigenmodes.

Recovery of the ERG from very low SNR recordings using a robust frequency domain approach: an Internet open source implementation

AbstractPurpose To present a statistically robust method of estimating the transient electroretinogram (ERG) in extremely poor recordings (very low SNR) contaminated by continuous noise (electrode noise, EMG, exogenous electrical interference) and discontinuous noise (eye‐movements and blinks). The transient ERG (ERG) is estimated by ensemble averaging with standard errors and p value, derived by either Magnitude Square Coherence or Bootstrap Resampling. However, both techniques are susceptible to noise. A robust estimator noise is required.Methods A series of ERG’s was recorded in normal subjects, containing numbers of prescribed eye movements and blinks. AR models were derived using the Burg algorithm over a range of explicit SNR’s. The ERG record was recovered as its Fourier Series representation over 9 harmonics. p values were derived for all estimates. An Excel implementation is presented which accesses remote Internet resources using MatSOAP©.Results The robust estimator was validated in Monte Carlo realisations of the AR model with expected high levels of performance for amplitudes and p values. In estimations in the raw data sets, high levels of statistical agreement were achieved for each subject across no‐noise:noise series. The recovery by ensemble averaging performed poorly.Conclusion The robust estimation in the ERG as presented here is novel and offers an important advance in clinical ERG measurement. The complexity is hidden from the User by providing a familiar Excel© spreadsheet interface.

Analytical investigation of modulated spin-torque oscillators in the framework of coupled differential equations with variable coefficients

Modulation of spin torque oscillators (STOs) is investigated by analytically solving the time-dependent coupled equations of an auto-oscillator. A Fourier-series solution is proposed, leading to the coefficients being determined with a linear set of equations, from which a nonlinear amplitude and frequency modulation (NFAM) scheme is obtained. In this framework, the NFAM features are related to the intrinsic STO parameters, revealing a frequency dependence of the harmonic-dependent modulation index that allows a modulation bandwidth to be defined for these devices. The presented results expose a rich parameter space, where the modulation and the STO's operation conditions define the observed modulation features. The Fourier-series representation of the time signal is suitable for studying periodic perturbations on the auto-oscillator equation.

X-ray cross-correlation analysis applied to disordered two-dimensional systems

Angular x-ray cross-correlation analysis (XCCA) is an approach to study the structure of disordered systems using the results of coherent x-ray scattering experiments. Here, we present the results of simulations that validate our theoretical findings for XCCA obtained in a previous paper [M. Altarelli et al., Phys. Rev. B 82, 104207 (2010)]. We consider as a model two-dimensional (2D) disordered systems composed of non-interacting colloidal clusters with fivefold symmetry and with orientational and positional disorder. We simulate a coherent x-ray scattering in the far field from such disordered systems and perform the angular cross-correlation analysis of calculated diffraction data. The results of our simulations show the relation between the Fourier series representation of the cross-correlation functions (CCFs) and different types of correlations in disordered systems. The dependence of structural information extracted by XCCA on the density of disordered systems and the degree of orientational disorder of clusters is investigated. The statistical nature of the fluctuations of the CCFs in the model `single-shot' experiments is demonstrated and the potential of extracting structural information from the analysis of CCFs averaged over a set of diffraction patterns is discussed. We also demonstrate the effect of partial coherence of x-rays on the results of XCCA.

Onset of Darcy–Brinkman convection in a binary viscoelastic fluid saturated porous layer

The onset of Darcy–Brinkman double-diffusive convection in a binary viscoelastic fluid-saturated porous layer is studied using both linear and weakly nonlinear stability analyses. The Oldroyd-B model is employed to describe the rheological behavior of the fluid. An extended form of Darcy–Oldroyd law incorporating the Brinkman’s correction and time derivative is used to describe the fluid flow and the Oberbeck–Boussinesq approximation is invoked. The onset criterion for stationary and oscillatory convection is derived analytically. The effects of rheological parameters, Darcy number, normalized porosity, Lewis number, solute Rayleigh number, and Darcy–Prandtl number on the stability of the system is investigated. The results indicated that there is a competition among the processes of thermal, solute diffusions and viscoelasticity that causes the convection to set in through the oscillatory modes rather than the stationary. The Darcy–Prandtl number has a dual effect on the threshold of oscillatory convection. The nonlinear theory based on the method of truncated representation of Fourier series is used to find the transient heat and mass transfer. Some existing results are reproduced as the particular cases of present study.

Quadratic Mode Shape Components From Ground Vibration Testing

Points on a vibrating structure generally move along curved paths rather than straight lines. For example, the tip of a cantilever beam vibrating in a bending mode experiences axial displacement as well as transverse displacement. The axial displacement is governed by the inextensibility of the neutral axis of the beam and is proportional to the square of the transverse displacement; hence the name “quadratic mode shape component.” Quadratic mode shape components are largely ignored in modal analysis, but there are some applications in the field of modal-basis structural analysis where the curved path of motion cannot be ignored. Examples include vibrations of rotating structures and buckling. Methods employing finite element analysis have been developed to calculate quadratic mode shape components. Ground vibration testing typically only yields the linear mode shape components. This paper explores the possibility of measuring the quadratic mode shape components in a sine-dwell ground vibration test. This is purely an additional measurement and does not affect the measured linear mode shape components or the modal parameters, i.e., modal mass, frequency, and damping ratio. The accelerometer output was modeled in detail taking into account its linear acceleration, its rotation, and gravitational acceleration. The response was correlated with the Fourier series representation of the output signal. The result was a simple expression for the quadratic mode shape component. The method was tested on a simple test piece and satisfactory results were obtained. The method requires that the accelerometers measure down to steady state and that up to the second Fourier coefficients of the output signals are calculated. The proposed method for measuring quadratic mode shape components in a sine-dwell ground vibration test seems feasible. One drawback of the method is that it is based on the measurement and processing of second harmonics in the acceleration signals and is therefore sensitive to any form of structural nonlinearity that may also cause higher harmonics in the acceleration signals. Another drawback is that only the quadratic components of individual modes can be measured, whereas coupled quadratic terms are generally also required to fully describe the motion of a point on a vibrating structure.

Fourier-series representation and projection of spherical harmonic functions

Computations of Fourier coefficients and related integrals of the associated Legendre functions with a new method along with their application to spherical harmonics analysis and synthesis are presented. The method incorporates a stable three-step recursion equation that can be processed separately for each colatitudinal Fourier wavenumber. Recursion equations for the zonal and sectorial modes are derived in explicit single-term formulas to provide accurate initial condition. Stable computations of the Fourier coefficients as well as the integrals needed for the projection of Legendre functions are demonstrated for the ultra-high degree of 10,800 corresponding to the resolution of one arcmin. Fourier coefficients, computed in double precision, are found to be accurate to 15 significant digits, indicating that the normalized error is close to the machine round-off error. The orthonormality, evaluated with Fourier coefficients and related integrals, is shown to be accurate to O(10−15) for degrees and orders up to 10,800. The Legendre function of degree 10,800 and order 5,000, synthesized from Fourier coefficients, is accurate to the machine round-off error. Further extension of the method to even higher degrees seems to be realizable without significant deterioration of accuracy. The Fourier series is applied to the projection of Legendre functions to the high-resolution global relief data of the National Geophysical Data Center of the National Oceanic and Atmospheric Administration, and the spherical harmonic degree variance (power spectrum) of global relief data is discussed.

Development of a two-dimensional model of a compliant non-pneumatic tire

An analytical model for a compliant non-pneumatic tire on frictionless, rigid ground is presented. The tire model consists of a thin flexible annular band and spokes that connect the band to a rigid hub. The annular band is modeled using curved beam theory that takes into account deformations due to bending, shearing and circumferential extension. The effect of the spokes, which are distributed continuously in the model and act as linear springs, is accounted for only in tension, which introduces a nonlinear response. The quasi-static, two-dimensional analysis focuses on how the contact patch, vertical tire stiffness and rolling resistance are affected by the stiffness properties of the band and the spokes. A Fourier series representation of the shear strain in the annular band and the complex modulus of the material were used to predict rolling resistance due to steady state rolling. From the analysis point of view, when the wheel is loaded at its hub, the following three distinct regions develop: (1) a support region where the hub hangs by the spokes from the upper part of the flexible band, (2) a free surface region where the spokes buckle and have no effect, and (3) a contact region where the flexible band is supported by the ground without the effect of the spokes. The angular bounds of these three regions are determined by the spoke angle and the contact angle, which are respectively the angle at which the spokes start to engage in tension and the angle that defines the edge of contact. Closed-form expressions of contact stress, stress-resultants and displacements at the centroids of the cross-sections of the flexible band are expressed in terms of these angles, which must be determined numerically. A thorough parametric analysis of quantities of interest for the tire is presented, which can be used to help support the optimal and rational design of compliant non-pneumatic tires. The model was validated by comparison with two computational models using the commercial finite element software ABAQUS and by experimental rolling resistance data.