Showing papers on "Closed-form expression published in 2006"

Closed-Form Expression for the Multi-Modal Unsaturated Conductivity Function

Abstract: In this note we derive a closed-form expression representing the hydraulic conductivity function for soils with a multi-modal pore size distribution. By combining the multi-modal representation of the retention function of Durner with the conductivity representation model of Mualem and following van Genuchten, we derive a simple analytical expression for the conductivity of soils with heterogeneous pore systems. Examples for the representation of bi-modal and tri-modal retention and corresponding conductivity curves demonstrate the applicability of the simple analytical expressions. It is concluded that the usefulness of the multi-modal representation of retention functions is increased by providing an analytical expression to directly calculate unsaturated hydraulic conductivities.

Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations. Working paper #2

Abstract: Maximum-likelihood estimates of the parameters of stochastic differential equations are consistent and asymptotically efficient, but unfortunately difficult to obtain if a closed form expression for the transitional probability density function of the process is not available. As a result, a large number of competing estimation procedures have been proposed. This paper provides a critical evaluation of the various estimation techniques. Special attention is given to the ease of implementation and comparative performance of the procedures when estimating the parameters of the Cox-Ingersoll-Ross and Ornstein-Uhlenbeck equations respectively.

Two-dimensional Green's function theory for the electrodynamics of a rotating medium

Abstract: We derive an exact spectral representation for the Green's function of Maxwell equations in a two-dimensional homogeneous and rotating environment. The formulation is developed in the medium (noninertial) rest frame, and it represents the response to a point source, where both the source and observation points rotate together with the medium. The closed form expression for the Green's function is derived for (nonrelativistic) slowly rotating media at finite distances. An approximate expression for the efficient evaluation of the Green's function, that avoids laborious summation of rotating-medium spherical harmonics, is provided and tested against the exact expression. Furthermore, it is shown that our spectral theory can provide a broad view of the optical response of rotating systems, from which the classical Sagnac effect is obtained as a special case.

Integral Evaluation Using the Mellin Transform and Generalized Hypergeometric Functions: Tutorial and Applications to Antenna Problems

Abstract: This is a tutorial presentation of the Mellin-transform (MT) method for the exact calculation of one-dimensional definite integrals, and an illustration of the application of this method to antenna/electromagnetics problems. Once the basics have been mastered, one quickly realizes that the MT-method is extremely powerful, often yielding closed-form expressions very difficult to come up with other methods or to deduce from the usual tables of integrals. Yet, as opposed to other methods, the MT-method is very straightforward to apply; it usually requires laborious calculations, but little ingenuity. In fact, the MT-method is used by Mathematica to symbolically calculate definite integrals. The first part of this paper is a step-by-step tutorial, proceeding from first principles. It includes basic information on Mellin-Barnes integrals and generalized hypergeometric functions, and summarizes the key ideas of the MT-method. In the remaining parts, the MT-method is applied to three examples from the antenna area. The results here are believed to be new, at least in the antenna/electromagnetics literature. In our first example, we obtain a closed-form expression, as a generalized hypergeometric function, for the power radiated by a constant-current circular-loop antenna; this quantity has been extensively discussed recently. Our second example concerns the admittance of a 2-D slot antenna. In both these examples, the exact closed-form expressions are applied to improve upon existing formulas in standard antenna textbooks. In our third example, finally, we obtain a very simple expression for an integral arising in recent, unpublished studies of unbounded, biaxially anisotropic media

Exact polynomial reproduction for oscillatory radial basis functions on infinite lattices

Abstract: Until now, only nonoscillatory radial basis functions (RBFs) have been considered in the literature It has recently been shown that a certain family of oscillatory RBFs based on J-Bessel functions gives rise to nonsingular interpolation problems and seems to be the only class of functions not to diverge in the limit of flat basis functions for any node layout This paper proves another interesting feature of these functions: exact polynomial reproduction of arbitrary order on an infinite lattice in @?^n First, a closed form expression is derived for calculating the expansion coefficients for any order polynomial in any dimension Then, a proof is given showing that the resulting interpolant, using this class of oscillatory RBFs, will give exact polynomial reproduction Examples in one and two dimensions are presented It is specifically noted that such closed form expressions cannot be derived for other classes of RBFs due to the fact that J-Bessel RBFS reproduce polynomials via a different mechanism

A flexible method for envelope estimation in empirical mode decomposition

Abstract: A flexible and efficient method for finding the envelope within the empirical mode decomposition (EMD) is introduced. Unlike the existing (deterministic) spline based strategy, the proposed envelope is a result of an optimisation precess and sought as a minimum of a quadratic cost function. A closed form solution of this optimisation problem is obtained and it is shown that by choosing free parameters, we can fine-tune the frequency resolution or the number of intrinsic mode functions (IMFs) as well as the shape of the envelopes. Computer simulations on both the synthetic and real-world electro-encephalogram (EEG) data support the analysis.

Radar Target Recognition Based on Late Time Representation: Closed-Form Expression for Criterion

Abstract: In the previous paper, we have defined the normalized estimation error and presented the radar target recognition scheme in frequency domain. A time-domain version of the previous paper is considered and a closed-form expression for the defined normalized estimation error in terms of Z-plane natural frequencies and transient response is derived. Evaluation of the closed-form expression of the normalized estimation error is shown in the numerical results, where it is shown that the performance of the radar target recognition scheme improves with an increase of the number of the natural frequencies and with an increase of signal-to-noise ratio. Furthermore, the mean and the variance of the square of the numerator of the normalized estimation error are obtained, and the validity of the statistics is shown using the numerical results

BER performance of BPSK transmissions over multipath channels

Abstract: A new closed form expression for the bit error rate (BER) performance of binary phase shift keying (BPSK) transmissions over frequency selective channels is presented. The expression is obtained through a novel approximation of the Gaussian Q(·) function by a fixed series of sinusoids with exponentially decreasing amplitudes. Numerical results demonstrate the accuracy of the derived expression.

Application of Closed-Form Solution for Normal Displacements on Impacted Half Space: Quantification Impact-Echo Signals

Abstract: In order to investigate the feasibility of deducing a simulated transfer function based on the Rayleigh wave form in an Impact-Echo signal, the analytical solution for the normal sur- face displacement due to a heaviside force at the half-space was reviewed and used to compute the surface displacement responses resulted from various types of impulse forces. Based on a series of numerical studies on the characteristics of Rayleigh wave form in the surface displace- ment responses, this paper presents the idea of using an equivalent impact force to derive an in- tentionally scaled transfer function. The pseudo force can be obtained from using Rayleigh wave form as a pseudo force or by generating an equivalent half-sine impact force accordingly. The effect of using such pseudo and equivalent force functions was discussed in details. In the pro- posed method, the force amplitude was first estimated from an amplitude curve established from numerical simulations using half-sine force functions. The recovery of a simulated transfer func- tion was next achieved via the use of an estimated force amplitude and a selected force function. The proposed procedure also results in steady thickness amplitudes when measurements on two concrete plates were taken for various impacts associated with different steel balls and different impact locations. The success in recovering constant thickness amplitudes for plate-like struc- tural members proved that the derivation of simulated transfer function is a useful tool in ex- tending the Impact-Echo test. The quantitative evaluation of the interfacial property of the sub- strate layer will also benefit from this simulated transfer function.

On the bit-error rate for generalized gamma fading channels

Abstract: The recent paper by Aalo et al., (2005), derived closed-form expressions for the bit-error rate (BER) of binary digital modulation schemes. The given expressions involve the Fox's H and Meijer's G functions and so their applicability is limited (e.g. we are not aware of any computer program for evaluating the Fox's H function). In this paper, we derive simpler expressions for the BER in terms of well known hypergeometric functions

Seeing the wood for the trees: A critical evaluation of methods to estimate the parameters of stochastic differential equations

Abstract: Maximum likelihood (ML) estimates of the parameters of stochastic differential equations (SDEs) are consistent and asymptotically efficient, but unfortunately difficult to obtain if a closed form expression for the transitional density of the process is not available. As a result, a large number of competing estimation procedures have been proposed. This paper provides a critical evaluation of the various estimation techniques. Special attention is given to the ease of implementation and comparative performance of the procedures when estimating the parameters of the Cox-IngersollRoss and Ornstein-Uhlenbeck equations respectively.

How to represent continuous piecewise linear functions in closed form

Abstract: This paper presents a simple approach to representing continuous piecewise linear functions in closed form for arbitrary dimensions. The absolute value function is used to embed non-linearity, its usage increasingly nested as the dimension is increased. Copyright © 2006 John Wiley & Sons, Ltd.

Interpolating Value Functions in Discrete Choice Dynamic Programming Models

Abstract: Structural discrete choice dynamic programming models have been shown to be a valuable tool for analyzing a wide range of economic behavior. A major limitation on the complexity and applicability of these models is the computational burden associated with computing the high dimensional integrals that typically characterize an agent's decision rules. This paper develops a regression based approach to interpolating value functions during the solution of dynamic programming models that alleviates this burden. This approach is suitable for use in models that incorporate unobserved state variables that are serially correlated across time and correlated across choices within a time period. The key assumption is that one unobserved state variable, or error term, in the model is distributed extreme value. Additional error terms that allow for correlation between unobservables across time or across choices within a given time period may be freely incorporated in the model. Value functions are simulated at a fraction of the state space and interpolated at the remaining points using a new regression function based on the extreme value closed form solution for the expected maxima of the value function. This regression function is well suited for use in models with large choice sets and complicated error structures. The performance of the interpolation method appears to be excellent, and it greatly reduces the computational burden of estimating the parameters of a dynamic programming model.

The Analysis of distributed systems with nonlocal damping

Abstract: This paper considers the analysis of structures with nonlocal damping, where the reaction force at any point is obtained as a weighted average of state variables over a spatial domain. The model yields an integro-differential equation, and obtaining closed form solutions is only possible for a limited range of boundary conditions by the transfer function method. Approximate solutions using the Galerkin method for beams are presented for typical spatial kernel functions, for a nonlocal viscoelastic foundation model. This requires the approximation of the displacement to be defined over the whole domain. To treat more complicated problems with variable damping parameters, non-uniform section properties, intermediate supports or arbitrary boundary conditions, a finite element method for beams is developed. However, in nonlocal damping models, nodes remote from the element do have an effect on the energy expressions, and hence the damping matrix is no longer block diagonal. The expressions for these direct and cross damping matrices are obtained for separable spatial kernel functions. The approach is demonstrated on a range of examples.

Closed form expression for the error performance of noncoherent M-ary orthogonal signals over multi-branch Rayleigh fading channels with arbitrary average fading powers

Abstract: This letter presents a closed form expression for the average error performance of noncoherent M-ary orthogonal signals over nonidentically distributed Rayleigh fading channels with postdetection equal-gain diversity combining (EGC). The derived expression can be easily evaluated numerically, and is valid for the general case in which the received average fading powers over diversity branches are arbitrary and not required to be distinct. In addition, it reduces to some existing results in the literature as special cases

Closed-form expression for arbitrary modulation formats performance evaluation in optically amplified systems

Abstract: A closed-form expression to analytically estimate the performance of optically preamplified receivers with arbitrary optical filtering and arbitrary modulation formats, which requires only the eye-diagram inspection and the knowledge of the optical and electrical filters transfer functions, is proposed. Simplicity, physical insight and fast computation time are key advantages of the proposed expression. The accuracy of the expression is evaluated through the comparison with a rigorous method, for return-to-zero (RZ) coding, considering different extinction ratios, optical pulse shapes, duty-cycles, and signal propagation in an optical fiber. Maximum sensitivity differences of 0.8 dB have been found for infinite extinction ratio and signal propagation in an optical fiber. The accuracy improves with the extinction ratio reduction. The dependence of the accuracy on duty-cycle and optical pulse shape is almost insignificant. Similar accuracy has been for obtained for other modulation formats, such as the chirped RZ or the carrier-suppressed RZ.

Nonlinear Model Selection—Application to an Indirect Inductive Measurement of Conductivity

Abstract: An efficient method and algorithm for experimental data processing based on parametric inversion is proposed. This method is applied to a metallic rod conductivity measurement based on induced secondary voltage technique. First, the case of an ideal excitation circuit is studied. For this case, when the excitation signal is a current step, a model of the data can be obtained in a closed-form as an infinite sum of exponential functions whose relaxation times are related to the physical properties of the inspected material. Therefore, for time points greater than the largest relaxation constant, only one term is sufficient; but, for these times, the signal-to-noise ratio (SNR) is smaller, so the variance of the estimated conductivity is larger. When taking into account some data at earlier time points, N terms are necessary to minimize the residual modelling error. The conductivity variance is then smaller but the identification process is more complex. A tradeoff has been achieved between these two aspects. Finally, the choice of an L1 norm criterion upon the identification error is used to reject outliers from experimental data. Second, a more realistic circuit is considered (the primary time constant is taken into account). In that case, two steps are necessary to calculate the direct model. A nonlinear equation is solved and the results are put into a closed-form expression (an infinite sum of exponential functions). For both cases, a multiparameters inversion is eventually used to minimize the effect of the modifying quantities