Orthogonal Series Expansion Research Articles

Analysis of switching loads in networks using operational matrices

This letter presents an approach for rapid calculation of the steady state of time-varying electric networks based on the use of orthogonal series expansions. The method takes advantage of the operational properties available to most orthogonal series expansions, i.e., integration, differentiation, product, and coefficients matrices. An important feature of this approach is that an analytical expression for the steady state is provided. The approach may be seen as an extension of phasor analysis commonly used for the analysis of linear time invariant electric networks to periodic networks.

New behavioral-level simulation technique for RF/microwave applications. Part II: Approximation of nonlinear transfer functions

The instantaneous quadrature technique is an efficient tool for nonlinear behavioral-level simulation of RF/microwave circuits or systems over wide frequency and dynamic ranges. In order to obtain accurate simulation results, accurate approximation/representation of the nonlinear transfer functions (or factors) as well as accurate measurement (or circuit-level simulation) of the amplitude (AM–AM) and phase (AM–PM) nonlinearities are required. In this paper, we consider how to approximate these transfer functions (factors) using splines, orthogonal and nonorthogonal series expansions, and evolutionary programming techniques (genetic algorithm and neural networks) with viewpoint of the simulation accuracy. The influence of AM–AM and AM–PM measurement (or simulation) inaccuracy and noise on the entire simulation accuracy is also discussed. Series expansion methods are proposed as a tool to filter out the measurement noise. © 2000 John Wiley & Sons, Inc. Int J RF and Microwave CAE 10: 238–252, 2000.

Natural frequencies of fluid-filled transversely isotropic cylindrical shells

The free vibration of a simply supported, transversely isotropic cylindrical shell filled with compressible fluid is exactly analyzed based on three-dimensional elasticity. The displacement separation technique as well as the orthogonal series expansion method are employed. Bessel functions with complex arguments are directly used in the solution for the case of complex eigenvalues. Frequency equation of the coupled vibration problem is obtained by taking account of the effect of the inner fluid. Numerical results are finally presented and compared with those based on three typical shell theories.

Estimation of Optical Phase Aberrations of Micro-Optics Components by the Irradiance Transport Equation

A mathematical method for the recovery of the optical phase by using the transport intensity equation is proposed. The method, based on phase decomposition of in orthogonal series expansion, allows us to calculate the least-squares estimation of each mode coefficient of the phase as a weighted integral over intensity measurements. Experimental results of focal length and the third and fifth order spherical aberration of planar microlenses are presented. These results are compared with those of other experimental methods.

The use of switching functions and Walsh series to calculate waveform distortion in thyristor controlled compensated power circuits

A new model based on the Walsh series for assessing the steady state operation of three-phase thyristor controlled reactors (TCRs) is presented in this paper. The model represents the TCR as an admittance matrix which interfaces easily with other power plant components in the network. Clearly, the admittance matrix representation is not confined to Walsh domain but rather it is valid for any orthogonal approximation of functions having the operational properties used in this paper. However, the switching functions which characterise the operation of TCRs are better represented by the Walsh series and they have been used in this paper. This is a new development in power systems analysis. Other orthogonal series expansions such as Fourier series and Hartley series have been used elsewhere. The Walsh transform possesses the important operational properties of the product, coefficient and integration matrices. The relevant aspects of the interaction between the TCR and the electric network are analysed. A three-phase TCR and a transmission network represented by a Thevenin equivalent are used for this purpose.

A numerical-analytic method of studying the dispersion of elastic waves in a fixed orthotropic cylinder with a simply connected curvilinear section

We give a method of constructing the dispersion equations for waveguides made of recitilinearly orthotropic materials and having a section that is either elliptic or approximately that of a regular 2n-gon (n>-2) with rounded corners. The equations of motion for the waveguide can be integrated in series of basic particular solutions of exponential type. The dispersion functions are obtained in the form of reduced infinite determinants from homogeneous functional boundary conditions on the lateral surface of the waveguide after these conditions have been algebraized using orthogonal series expansions.

Phase retrieval with the transport-of-intensity equation II Orthogonal series solution for nonuniform illumination

In a previous paper [ J. Opt. Soc. Am. A12, 1932 ( 1995)] we presented a method for phase recovery with the transport-of-intensity equation by use of a series expansion. Here we develop a different method for the solution of this equation, which allows recovery of the phase in the case of nonuniform illumination. Though also based on the orthogonal series expansion, the new method does not require any separate boundary conditions and can be more easily adjusted for apertures of various shapes. The discussion is primarily for the case of a circular aperture and Zernike polynomials, but we also outline the solution for a rectangular aperture and Fourier harmonics. The latter example may have some substantial advantages, given the availability of the fast Fourier transform.

SFEM for Reliability of Structures with Material Nonlinearities

A stochastic finite element method (SFEM) is developed to study the reliability of structures with material nonlinearities. Bilinear stress-strain relationships that involve two correlated random fields are used to model the material nonlinearities. Random fields are represented as continuous functions by using general orthogonal series expansions, and are incorporated directly into the nonlinear finite element analysis and first-order reliability formulation. The proposed method is illustrated by two examples involving a fixed-end beam and a frame. Random spatial fluctuations in the material properties or distributed loads may have a significant effect on reliability if correlation lengths of random fields are less than the lengths of members, meaning that local fluctuations in the fields are relatively high.

Convergence Rates of SNP Density Estimators

The SNP density estimator is a truncation (or sieve) estimator based upon a Hermite series expansion. Other nonparametric estimators based on orthogonal series expansions have been proposed.

Error Measure for Reliability Studies Using Reduced Variable Set

A reduction in the number of random variables carried in a structural-reliability analysis can be achieved by an orthogonal transformation of the basic random vector to a vector of independent standard normal variables, followed by a reduction procedure similar to that employed in modal analysis in structural dynamics. This paper investigates the error introduced by this reduction in the representation of random fields through orthogonal-series expansions. It is found that the error in the second-moment representation of a random field is bounded by the reduction error, which is easily computed, when the correlation length of the random field is short; this often is the case in stochastic finite-element analysis.

Green function for wave motion in acoustic media with large randomness

Fundamental solutions, or Green functions, for waves propagating in acoustic media that exhibit random material properties, besides being useful in their own right, also serve as the basic building block for integral equation formulations that can be used for the numerical solution of acoustic wave scattering problems of practical importance. The present work serves as an extension of earlier derivations of boundary integral equation statements based on the perturbation approach by removing the assumption of small fluctuations of key medium properties about their mean values. The solution methodology is to assume time harmonic conditions and employ an orthogonal polynomial series expansion of the Green function. The position-dependent coefficients of this expansion are found through an eigensolution of first order differential equations. Finally, results for a series of representatives cases are presented and the sensitivity of the solution to the number of terms employed in the expansion is investigated.

Orthogonal Series Expansions of Random Fields in Reliability Analysis

A new approach for first‐order reliability analysis of structures with material parameters modeled as random fields is presented. The random field is represented by a series of orthogonal functions, and is incorporated directly in the finite‐element formulation and first‐order reliability analysis. This method avoids the difficulty of selecting a suitable mesh for discretizing the random field. A general continuous orthogonal series expansion of the random field is derived, and its relationship with the Karhunen‐Loeve expansion used in recent stochastic finite‐element studies is examined. The method is illustrated for a fixed‐end beam with bending rigidity modeled as a random field. A set of Legendre polynomials is used as the orthogonal base to represent the random field. Two types of correlation models are considered. The Karhunen‐Loeve expansion leads to a lower truncation error than does the Legendre expansion for a given number of terms, but one or two additional terms in the Legendre expansion yield...

Non-parametric identification with errors in independent variables

The problem of non-parametric identification of a static system is considered in the presence of errors both in dependent and independent variables. Identification is based on the orthogonal series expansion. Errors in independent variables are modelled in two ways, namely as pointwise errors and as errors caused by an averaging. Sufficient conditions for MISE consistency are proved in both cases.

The Genuine Bernstein-Durrmeyer Operator on a Simplex

In 1967 Durrmeyer introduced a modification of the Bernstein polynomials as a selfadjoint polynomial operator on L2[0,1] which proved to be an interesting and rich object of investigation. Incorporating Jacobi weights Berens and Xu obtained a more general class of operators, sharing all the advantages of Durrmeyer’s modification, and identified these operators as de la Vallée-Poussin means with respect to the associated Jacobi polynomial expansion. Nevertheless, all these modifications lack one important property of the Bernstein polynomials, namely the preservation of linear functions. To overcome this drawback a Bernstein-Durrmeyer operator with respect to a singular Jacobi weight will be introduced and investigated. For this purpose an orthogonal series expansion in terms generalized Jacobi polynomials and its de la Vallée-Poussin means will be considered.These Bernstein-Durrmeyer polynomials with respect to the singular weight combine all the nice properties of Bernstein-Durrmeyer polynomials with the preservation of linear functions, and are closely tied to classical Bernstein polynomials. Focusing not on the approximation behavior of the operators but on shape preserving properties, these operators we will prove them to converge monotonically decreasing, if and only if the underlying function is subharmonic with respect to the elliptic differential operator associated to the Bernstein as well as to these Bernstein-Durrmeyer polynomials. In addition to various generalizations of convexity, subharmonicity is one further shape property being preserved by these Bernstein-Durrmeyer polynomials. Finally, pointwise and global saturation results will be derived in a very elementary way.

On the Build-Up of Oscillations in a Cylindrical Magnetron

A fully nonlinear theory of the magnetron is presented starting from the equations of classical electrodynamics. A crucial step in the development is an orthogonal-series expansion of the space charge and current densities using appropriate sets of cavity modes, the expansion coefficients being nonlinear functionals of the electron trajectories. Splitting the electric field vector into its solenoidal and irrotational (gradient of a scalar potential) parts, expanding the magnetic field and the solenoidal part of the electric field in a complete set of orthonormal solenoidal cavity modes, and substituting these expansions into Maxwell's equations, differential equations governing the time evolution of the mode amplitudes are derived. These differential equations for the mode amplitudes and the Poisson's equation for the scalar potential are solved in terms of the charge and current densities in the interaction region. Substitution of the explicit expression for the field components in terms of the charge an...

Direct method for optimal control of discrete time-varying systems

An efficient direct method of using the finite set orthogonal series expansion is presented for the finite-time optimal; control of linear discrete-time systems with a quadratic performance index. Discrete Legendre orthogonal polynomials are used as the expansion basis, owing to their unity weight and useful properties. The main advantage of this method is that the optimal control problem is simplified to finding the sequence of control that minimizes a performance index of quadratic form with linear algebraic constraints. A general formulation of the method is given and numerical examples are shown to demonstrate the effectivness of the method.

From static approximations to full‐wave analysis: The analysis of planar line discontinuities

AbstractA comparison of different methods for the characterization of planar line discontinuities with emphasis on microstrip discontinuities is given, thereby describing the present state of the art in this area of microwave circuit CAD. Quasi‐static equivalent circuit models, the magnetic wall waveguide model of microstrip discontinuities, full‐wave analysis techniques like the orthogonal series expansion and mode matching techniques, the spectral domain analysis technique using roof‐top functions, and the finite‐difference time‐domain technique are described briefly and compared with respect to their numerical efficiency and their capabilities.

Moment characterization of phase noise in coherent optical systems

The statistical characterization of the phase noise introduced by a semiconductor laser in a coherent optical transmission system is a key problem in the system performance evaluation. The authors consider the moment characterization, of the complex random process. Starting from the implicit representation of the probability density function through the Fokker-Planck equation, the authors obtain closed form analytical expressions for the moments of the filtered phase noise both in stationary and nonstationary conditions. Then the use of the moments for the computation of probability densities through orthogonal polynomial series expansion and maximum entropy approach is considered in application examples.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Fitting conditional distributions using orthogonal expansions

The paper describes a Bayes least squares linear method for fitting conditional distributions to data. The method might be used by an investigator who wishes to proceed as if the data arose by random sampling from distributions having densities adequately represented by orthogonal series expansions. In an illustrative example a conditional distribution of lung function given age and height are fitted to data.

Incoherent optical Abel inversion

In this paper we propose an incoherent optical processor which can perform Abel inversion. Construction of the processor is based on Tchebycheff and Zernike orthogonal polynomials series expansion. It is shown that the series expansion coefficients together with the series summation, can be easily computed optically (using cylindrical optics). Experimental results are presented and compared to the theoretical predictions. Use of the series expansion is proposed for other integral transformations.