Laguerre Coefficients Research Articles

Explicit formulae for the coefficients of integrated expansions of Laguerre and Hermite polynomials and their integrals

Two new formulae expressing explicitly the integrals of Laguerre (Hermite) polynomials of any degree and for any order in terms of the Laguerre (Hermite) polynomials themselves are proved. Another two new explicit formulae relating the Laguerre (Hermite) coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function are also established. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed.

Efficient computation of passage time densities and distributions in Markov chains using Laguerre method

The Laguerre method for the numerical inversion of Laplace transforms is a well known approach to the approximation of probability density functions (PDFs) and cumulative distribution functions (CDFs) of first passage times in Markov chains. Results are presented that relate the Laguerre generating functions and Laguerre coefficients of a PDF with those of the corresponding complementary CDF. This enables the ability to compute the PDF or CDF from the Laplace transform of either at the cost of computing only one set of Laguerre coefficients.

Moment Matching Model Order Reduction in Time Domain via Laguerre Series

A new time-domain model order reduction method based on the Laguerre function expansion of the impulse response is presented. The Laguerre coefficients of the impulse response of the reduced-order model, which is calculated using a projection whose matrices form basis of appropriate Krylov subspaces, match, up to a given order, those of the original system. In addition, it is shown that the obtained reduced-order model in time-domain, is equivalent to the one obtained by the classical moment matching around a single expansion point in frequency-domain. Accordingly, a new time-domain interpretation for the rational interpolation problem is deduced.

Explicit expression of the parameter bias in identification of Laguerre models from step responses

This paper delivers an analysis of the least-square estimation of the Laguerre coefficients of a linear discrete-time system from step response data. The original contribution consists in an explicit formula for the bias error on the estimated coefficients due to the under-modelling of the system. The formula, jointly with some a-priori information on the neglected dynamics, can be used to construct bounds on this error. The results presented in this paper are illustrated with a simulation example.

Detection and classification of neurotoxins using a novel short-term plasticity quantification method

A tissue-based biosensor is described for screening chemical compounds that rapidly affect the nervous system. The proposed sensor is an extension of a previous work on cultured hippocampal slices [Biosens. Bioelectron. 16 (2001) 491]. The detection of the chemical compounds is based on a novel quantification method of short-term plasticity (STP) of the CA1 system in acute hippocampal slices, using random electrical impulse sequences as inputs and population spike (PS) amplitudes as outputs. STP is quantified by the first and the second order kernels using a variant of the Volterra modeling approach. This approach is more specific and time-efficient than the conventional paired pulse and fixed frequency train methods [J. Neurosci. Methods 2 (2002) 111]. Describing the functional state of the biosensor, the kernels changed accordingly as chemical compounds were added. The second order kernel was decomposed into nine Laguerre functions. The corresponding Laguerre coefficients along with the first order kernel were used as features for classification purposes. The biosensor was tested using picrotoxin (100 μM), trimethylopropane phosphate (10 μM), tetraethylammonium (4 mM), valproate (5 mM), carbachol (5 mM), DAP5 (25 μM), CNQX (3 μM), and DNQX (0.15, 1.5, 3, 5 and 10 μM). Each chemical compound gave a different feature profile corresponding to its pharmacological class. The first order kernel and the Laguerre coefficients formed the input to an artificial neural network (ANN) comprised of a single layer of perceptrons. The ANN was able to classify each tested compound into its respective class.

On the connection coefficients and recurrence relations arising from expansions in series of Laguerre polynomials

A formula expressing the Laguerre coefficients of a general-order derivative of an infinitely differentiable function in terms of its original coefficients is proved, and a formula expressing explicitly the derivatives of Laguerre polynomials of any degree and for any order as a linear combination of suitable Laguerre polynomials is deduced. A formula for the Laguerre coefficients of the moments of one single Laguerre polynomial of certain degree is given. Formulae for the Laguerre coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Laguerre coefficients are also obtained. A simple approach in order to build and solve recursively for the connection coefficients between Jacobi–Laguerre and Hermite–Laguerre polynomials is described. An explicit formula for these coefficients between Jacobi and Laguerre polynomials is given, of which the ultra-spherical polynomials of the first and second kinds and Legendre polynomials are important special cases. An analytical formula for the connection coefficients between Hermite and Laguerre polynomials is also obtained.

A digital-signal-processing technique for ultrasonic signal modeling and classification

In this paper, we face the problem of constructing a robust feature space for automatic classification of signals from narrow-band in-air ultrasonic sensors. In consideration of the existing sensor bandwidth restrictions, the importance of selecting a suitable signal descriptor is highlighted. We assume that the characteristics of the ultrasonic sources which produce the signals are impressed in the shape of their echo envelopes. A technique based on orthonormal Laguerre polynomials is applied to the echo envelopes for constructing the feature space. Different methods for computing the Laguerre coefficients are discussed, and the properties of the resulting feature space are investigated. For the experimental verification of the method, a set of acoustic sources is synthesized by submitting a high-frequency piezoelectric transducer to varying levels of electrical damping. How some factors, i.e., the signal-to-noise ratio of the return signals, and the sampling rate to digitize them, affect the achievable recognition rate is discussed. High recognition rates are obtained in our experiments, in spite of the fact that, by visual inspection, the shapes of the signals from the synthesized sources are very similar to one another.

Adaptive Laguerre network realization

An adaptive scheme is developed to realize the set of linear discrete-time (DT) systems by the Laguerre networks using gradient search or least mean-squares and steepest descent (LMS/SD) algorithms. Both Laguerre parameter and Laguerre coefficients are optimally evaluated by the adaptive scheme. Two suboptimal procedures called approximate and modified methods are addressed and introduced, respectively, to initialize the Laguerre parameter in its adaptive evaluation. The Laguerre coefficients are also evaluated by two adaptive algorithms presented as bilinear and direct methods. This adaptive scheme simply realizes the linear DT systems either in time or frequency domain. The application of linear DT modeling and filter design is illustrated via some examples.

The Riemann zeta function used in the inversion of the Laplace transform

Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Laguerre polynomials. The main contribution of our work is the development of a new and very effective method to evaluate the Laguerre coefficients with the use of the Riemann zeta function. Some examples are illustrated.

Numerical inversion of multidimensional Laplace transforms by the Laguerre method

Numerical transform inversion can be useful to solve stochastic models arising in the performance evaluation of telecommunications and computer systems. We contribute to this technique in this paper by extending our recently developed variant of the Laguerre method for numerically inverting Laplace transforms to multidimensional Laplace transforms. An important application of multidimensional inversion is to calculate time-dependent performance measures of stochastic systems. Key features of our new algorithm are: (1) an efficient FFT-based extension of our previously developed variant of the Fourierseries method to calculate the coefficients of the multidimensional Laguerre generating function, and (2) systematic methods for scaling to accelerate convergence of infinite series, using Wynn's ϵ-algorithm and exploiting geometric decay rates of Laguerre coefficients. These features greatly speed up the algorithm while controlling errors. We illustrate the effectiveness of our algorithm through numerical examples. For many problems, hundreds of function evaluations can be computed in just a few seconds.

On the Laguerre Method for Numerically Inverting Laplace Transforms

The Laguerre method for numerically inverting Laplace transforms is an old established method based on the 1935 Tricomi–Widder theorem, which shows (under suitable regularity conditions) that the desired function can be represented as a weighted sum of Laguerre functions, where the weights are coefficients of a generating function constructed from the Laplace transform using a bilinear transformation. We present a new variant of the Laguerre method based on: (i) using our previously developed variant of the Fourier-series method to calculate the coefficients of the Laguerre generating function, (ii) developing systematic methods for scaling, and (iii) using Wynn's ϵ-algorithm to accelerate convergence of the Laguerre series when the Laguerre coefficients do not converge to zero geometrically fast. These contributions significantly expand the class of transforms that can be effectively inverted by the Laguerre method. We provide insight into the slow convergence of the Laguerre coefficients as well as propose a remedy. Before acceleration, the rate of convergence can often be determined from the Laplace transform by applying Darboux's theorem. Even when the Laguerre coefficients converge to zero geometrically fast, it can be difficult to calculate the desired functions for large arguments because of roundoff errors. We solve this problem by calculating very small Laguerre coefficients with low relative error through appropriate scaling. We also develop another acceleration technique for the case in which the Laguerre coefficients converge to zero geometrically fast. We illustrate the effectiveness of our algorithm through numerical examples.

FORTRAN program for a numerical solution of the nonsinglet Altarelli-Parisi equation

We investigate a numerical solution of the flavor-nonsinglet Altarelli-Parisi equation with next-to-leading-order α s corrections by using Laguerre polynomials. Expanding a structure function (or a quark distribution) and a splitting function by the Laguerre polynomials, we reduce an integrodifferential equation to a summation of finite number of Laguerre coefficients. We provide a FORTRAN program for Q 2 evolution of nonsinglet structure functions ( F 1, F 2, and F 3) and nonsinglet quark distributions. This is a very effective program with typical running time of a few seconds on SUN-IPX or on VAX-4000 500 . Accurate evolution results are obtained by taking approximately twenty Laguerre polynomials.

On the determination of the optimal pole position of Laguerre filters

As is known in the literature, at each stationary point of the squared error (SE) curve of a Laguerre filter at least one of a pair of certain Laguerre coefficients (weights) vanishes. The author describes a very efficient way to compute the derivatives of each one of these coefficients with respect to the pole position of the Laguerre filter. The knowledge of these derivatives makes possible the computation of high-order approximations to the coefficients in question, such as truncated Taylor series and Pade approximants. The zeros of these approximations are usually good estimates of the location of the stationary points of the SE curve nearer to the center of the approximation. In this way, the position of these stationary points, in particular local minima, can be estimated without resorting to a numerical search algorithm. Both continuous time and discrete time Laguerre filters are discussed, excited either by an impulse or by an arbitrary signal. The authors illustrate the main results of the paper with a numerical example.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Calculation of the local cross-correlation function on the basis of the Laguerre transform

A simple and computationally efficient mechanism for calculating a running or local cross-correlation function of two time-domain signals is presented. In order to obtain a running cross-correlation function, the signals must be windowed. It is argued that an appropriate window for a local cross correlation is an exponential function. To obtain a computationally efficient mechanism, the windowed functions are decomposed in a series of orthogonal functions. The set or orthogonal functions is matched to the chosen window and is a Laguerre-Fourier series. The cross correlation of the windowed functions is equal to a weighted summation of cross-correlated pattern functions. The weights are determined by cross correlating the Laguerre coefficients.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Performability analysis of distributed real-time systems

An algorithm and a methodology for the performability analysis of repairable distributed real-time systems are presented. The planning cycle of a real-time distributed system, which normally consists of several task invocations, is first identified. The performability distribution at the end of the planning cycle is determined by repeated convolutions of performability densities between task invocations. These convolution operations are efficiently carried out using the operational properties of Laguerre coefficients. The algorithm numerically determines both moments and distribution of performability in O(N/sub max//sup 3/), where N/sub max/ is the largest size of the state space between any task invocations. To illustrate the overall methodology, a simplified example of a radar system is analyzed, and the various performability measures are obtained using the algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

THE LAGUERRE TRANSFORM OF PRODUCT OF FUNCTIONS

The Lal;uerre transform developed by Keih;on, Nunn and Sumita (1979,1981, 1981) provides an algorithmic basis for mechanizing various continuum operations such as multiple convolutions, integration, differentiation, and multiplication by polynomials and exponential functions. In this paper, we develop a numerical procedure for finding the Laguerre coefficients of a product of functions in terms of the Laguerre coefficients of individual functions. The procedure enables one to evaluate multiple convolutions and other continuum operations described above even when product form functions are involved, thereby further enhancing the power of the Laguerre transform.

On PID controller tuning using orthonormal series identification

The step response of a closed loop system is identified by means of a Laguerre expansion. This offers certain advantages over ARMA models, namely lack of bias in the estimates, structural flexibility and the ability to precompute the regressors. The Laguerre coefficients of the retuned system are calculated in terms of the tuning constants, and a quadratic performance index is optimized. Results from simulations and from industrial trials are given.

Evaluation of minimum and maximum of a correlated pair of random variables via the bivariate laguerre transform

In a recent paper by Sumita and Kijima (1985), the bivariate Laguerre transform has been developed for mechanizing various bivariate continuum operations such as multiple bivariate convolutions, marginal convolutions, tail integration, partial differentiation and multiplication by bivariate polynomials. The transform has been shown to be quite useful and powerful in studying bivariate distributions and bivariate stochastic processes. In this paper, we study the minimum and maximum of a correlated pair of random variables via the bivariate Laguerre transform, thereby further enhancing the applicability of the bivariate Laguerre transform. A numerical procedure is developed for calculating the Laguerre coefficients of the probability density functions for the minimum and maximum in terms of the original bivariate Laguerre coefficients corresponding to the correlated pair of random variables. This enables one to evaluate the distributions and moments of the minimum and maximum. The joint distribution of the ...

On PID Controller Tuning Using Orthonormal Series Identification

The step response of a closed loop system is identified by means of a Laguerre expansion. This offers certain advantages over ARMA models, namely lack of bias in the estimates, structural flexibility and the ability to pre-compute the regressors. The Laguerre coefficients of the retuned system are calculated in terms of the tuning constants, and a quadratic performance index is optimized. Results from simulations and from industrial trials are given.

The Laguerre Transform and a Family of Functions with Nonnegative Laguerre Coefficients

The Laguerre transform induced in Keilson and Nunn (Keilson, J., W. R. Nunn. 1979. Laguerre transformation as a tool for the numerical solution of integral equations of convolution type. Appl. Math. Comp. 5 313–359.) and Keilson, Nunn and Sumita (Keilson, J., W. R. Nunn, U. Sumita. 1981. The bilateral Laguerre transform. Appl. Math. Comp. 8 137–174.) maps functions in L2 into discrete sequences and permits rapid numerical calculation of basic continuum operations such as differentiation, integration and convolution. In this paper we study a family of functions 𝒫 with nonnegative Fourier–Laguerre coefficients. A uniform bound for truncation error can be easily found for this family. It will be shown that the family 𝒫 is closed under positive scalar multiplication, convex mixing, and multiplication by completely monotone functions.