Gauss-Hermite Quadrature Formula Research Articles

Computing integrals with an exponential weight on the real axis in floating point arithmetic

The aim of this work is to propose a fast and reliable algorithm for computing integrals of the type∫−∞∞f(x)e−x2−1x2dx, where f(x) is a sufficiently smooth function, in floating point arithmetic. The algorithm is based on a product integration rule, whose rate of convergence depends only on the regularity of f, since the coefficients of the rule are “exactly” computed by means of suitable recurrence relations here derived. We prove stability and convergence in the space of locally continuous functions on R equipped with weighted uniform norm. By extensive numerical tests, the accuracy of the proposed product rule is compared with that of the Gauss–Hermite quadrature formula w.r.t. the function f(x)e−1x2. The numerical results confirm the effectiveness of the method, supporting the proven theoretical estimates.

MIMO Underwater Visible Light Communications: Comprehensive Channel Study, Performance Analysis, and Multiple-Symbol Detection

This paper presents a comprehensive study of underwater visible light communications (UVLC), from channel characterization, performance analysis, and effective transmission and reception methods. To this end, we first simulate the fading-free impulse response (FFIR) of UVLC channels using Monte Carlo numerical procedure to take into account the absorption and scattering effects; and then to characterize turbulence effects, we multiply the aforementioned FFIR by a fading coefficient modeled as a lognormal random variable (RV) for weak oceanic turbulence. Based on this general channel model, we analytically study the bit error rate (BER) performance of UVLC systems with binary pulse position modulation (BPPM). In the next step, to mitigate turbulence effects, we employ multiple transmitters and/or receivers, i.e., we apply spatial diversity technique over UVLC links. Closed-form expressions for the system BER are provided, when an equal gain combiner is employed at the receiver side, thanks to the Gauss-Hermite quadrature formula as well as approximation to the sum of lognormal RVs. We further apply saddle-point approximation, an accurate photon-counting method, to evaluate the system BER in the presence of shot noise. Both laser-based collimated and light emitting diode based diffusive links are investigated. Additionally, in order to reduce the intersymbol interference (ISI), introduced by the multiple-scattering effect of UVLC channels on the propagating photons, we also obtain the optimal multiple-symbol detection (MSD) algorithm, as well as the suboptimal generalized MSD (GMSD) algorithm. Our numerical analysis indicate good matches between the analytical and photon-counting results implying the negligibility of signal-dependent shot noise, and also between the analytical results and numerical simulations confirming the accuracy of our derived closed-form expressions for the system BER. Besides, our results show that spatial diversity significantly mitigates fading impairments while (G)MSD considerably alleviates ISI deterioration.

Performance Analysis of PN Code Acquisition with MIMO Scheme for an UWB TH/CDMA System

This paper presents performance and its simulation results of pseudonoise (PN) code acquisition scheme with MIMO scheme for an ultra‐wideband time‐hopping/code‐division‐multiple‐access (UWB TH/CDMA) system. The transmission channel is modelled as a frequency selective lognormal fading channel. In almost practical PN code acquisition system, the existence of more than two synchronous cells in the uncertainty region of the search process is possible due to multipath effect. Therefore, based on deriving the detection probability, false alarm rate, miss detection probability, and mean acquisition time, the acquisition performance is analyzed under the hypothesis of multiple synchronous states (cells) in the uncertainty region of the PN code. And the code acquisition performance is evaluated when the correlator outputs are noncoherently combined by using equal gain combining (EGC) scheme. In this procedure, the closed form for the conditional probability of decision variable is derived using the Gauss‐Hermite quadrature formula. The performance comparison of the scheme mentioned above shows that the code acquisition performance with the diversity combining technique, especially when increasing the number of antenna, is more robust than that using no diversity. And code acquisition performance comparison also shows that if the detection threshold is set inappropriately, the performance might be degraded, even if an antenna diversity method is applied. It is also shown that Tx diversity can improve the acquisition performance but not as much as Rx diversity does. And Rx diversity can be applied to the acquisition system for additional diversity gain if the complexity of the receiver can be accepted.

Performance Analysis of Multi-Hop Underwater Wireless Optical Communication Systems

In this letter, we analytically evaluate the end-to-end bit error rate (BER) of point-to-point multi-hop underwater wireless optical communication (UWOC) systems with respect to all degrading effects of the UWOC channel, namely absorption, scattering, and turbulence-induced fading. To do so, we first derive the BER expression of a single-hop UWOC link as the building block for end-to-end BER evaluation. We also apply Gauss–Hermite quadrature formula to obtain the closed-form solution for the system BER in the case of lognormal underwater fading channel. Numerical results demonstrate that multi-hop transmission, by alleviating channel impairments, can significantly improve the system performance and extend the viable end-to-end communication distance.

Performance Studies of Underwater Wireless Optical Communication Systems With Spatial Diversity: MIMO Scheme

In this paper, we analytically study the performance of multiple-input multiple-output underwater wireless optical communication (UWOC) systems with ON–OFF keying modulation. To mitigate turbulence-induced fading, which is amongst the major degrading effects of underwater channels on the propagating optical signal, we use spatial diversity over UWOC links. Furthermore, the effects of absorption and scattering are considered in our analysis. We analytically obtain the exact and an upper bound bit error rate (BER) expressions for both optimal and equal gain combining. In order to more effectively calculate the system BER, we apply Gauss-Hermite quadrature formula as well as approximation to the sum of lognormal random variables. We also apply the photon-counting method to evaluate the system BER in the presence of shot noise. Our numerical results indicate an excellent match between the exact and upper bound BER curves. Also, a good match between the analytical results and numerical simulations confirms the accuracy of our derived expressions. Moreover, our results show that spatial diversity can considerably improve the system performance, especially for channels with higher turbulence, e.g., a $3\times 1$ multiple-input single-output transmission in a 25 m coastal water link with a log-amplitude variance of 0.16 can introduce 8 dB performance improvement at the BER of 10−9.

Crank-Nicolson scheme and its error estimates for backward stochastic differential equations

A new numerical scheme of the Crank-Nicolson type for backward stochastic differential equations (BSDEs) is proposed. The Gauss-Hermite quadrature formula is used to approximate conditional mathematical expectations in the numerical applications, and space interpolations are used to compute values at non-space-grid points. Some numerical tests are given to validate the theoretical results.

Uncertainty Quantification for Delineation of Wellhead Protection Areas Using the Gauss‐Hermite Quadrature Approach

AbstractMissing from most wellhead protection area (WHPA) delineation studies is a measure of the uncertainty associated with model predictions. A quantitative representation of that uncertainty can be used by regulators to implement different degrees of protection for areas with different degrees of certainty.Uncertainty analysis was performed for one‐ and five‐year WHPA delineation for two municipal wells within a buried‐valley glacial‐outwash aquifer in southwestern Ohio. An approximation of the three‐point Gauss‐Hermite quadrature formula was used. This method is an alternative to simple Monte Carlo random sampling, typically requiring fewer model runs. It results in model‐prediction expected values and variances and quantifies parameter main and two‐way interactive effects. This study explored the collective impact of the uncertainties associated with six model parameters. Parameter probability density functions (PDFs) were based on field data and modified using a ground water flow model to ensure that all combinations of parameter values, within their PDFs, yielded an acceptable model calibration.The one‐year WHPA for the upgradient well had a high degree of associated uncertainty represented by the difference in size between the WHPA low and high 95% confidence interval limits. The large uncertainty was due to the uncertainties associated with model parameter values, especially effective porosity and horizontal hydraulic conductivity. River conductance also had a substantial impact on the WHPA predictions. The WHPAs for the downgradient well and the five‐year WHPA for the upgradient well were limited by upgradient hydrogeologic boundaries; their prediction was, therefore, less sensitive to the uncertainties inherent in the model parameters.

Using an approximation of the three-point Gauss–Hermite quadrature formula for model prediction and quantification of uncertainty

Computer models are commonly used by regulators and managers to make predictions regarding groundwater flow and contaminant concentrations at various locations and times. However, the uncertainty associated with those predictions is often overlooked, despite the fact that an assessment of such uncertainty is critical in the formulation of policy decisions. One method of quantifying the uncertainty of model predictions, based on the collective uncertainties of the model parameter input values, is to use an approximation of the three-point Gauss–Hermite quadrature formula. The Gauss–Hermite approximation is a convenient substitute for simple Monte Carlo sampling, because it requires fewer model runs and provides an immediate sensitivity analysis of parameter main effects and two-way interactions. For example, a model with four parameters, each with its own associated uncertainty, needs to be run only 33 times to complete the Gauss–Hermite analysis. For an application to a contaminant-transport model, the Gauss–Hermite approximation compares well to the full method, with considerable savings in computing effort. By comparison, Latin hypercube sampling can be more flexible, but it is more complex to use in some circumstances and cannot as easily generate the detailed sensitivity analysis that the Gauss–Hermite approach offers.

Abscissae and weights for the Gauss-Hermite quadrature formula

Title of program: GAUSSH Catalogue number: ABBL Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland (see application form in this issue) Computer: FACOM M-780/20; Installation: Kyushu University, Hakozaki, Higashi-ku, Fukuoka 812, Japan Operating system: IV/F4 Programming language used: FORTRAN 77 High speed storage required: 264 Kbytes No. of bits in a byte: 8 Other peripherals used: card reader, line printer No. of cards in combined program and test deck: 261 Card punching code: EBCDIC

State estimation by orthogonal expansion of probability distributions

A recursive estimation scheme suitable for real-time implementation is derived for a class of nolinear systems and observations expressed as nonlinear functions in discrete time, corrupted by a non-Gaussian mutually correlated random white noise sequence. The probability densities are expanded as a Gram-Charlier series and a Gauss-Hermite quadrature formula is used for computing the expectations. In the multidimensional case an expansion about a density of mutually independent Gaussian variables is used instead of a general multidimensional Gaussian density, which may result in a poorer performance in linear systems with Gaussian noise. However, in the case of nonlinear systems and non-Gaussian noise, the computational simplifications which result, outweigh the impairment in performance if any. A computational example is included.