Gram-Schmidt Orthogonalization Procedure Research Articles

Characterization of minimal network structures modeling stochastic processes

Identifying the underlying structure of a network from observed data is an important problem across various disciplines. Given the general ill-posed nature of the problem, since in many cases, multiple plausible network models can explain the data, this article concentrates on characterizing classes of models providing possible explanations. Specifically, we explore linear models that can account for observed data in the form of wide-sense stationary processes accommodating the potential presence of feedback loops and direct feedthroughs. To achieve this, we leverage key insights from the theory of graphical models. In particular, we extensively employ Pearl-Verma Theorem in causal discovery which allows one to recover all minimal network structures compatible with the observed data. We adapt such a result to deal with stochastic processes and reinterpret it as a Gram-Schmidt orthogonalization procedure in a suitable Hilbert space. This reinterpretation allows us to characterize all minimal networks explaining a set of data, which have the property of not having any algebraic loops.

Modeling Infectious Disease Trend using Sobolev Polynomials

Trend analysis plays an important role in infectious disease control. An analysis of the underlying trend in the number of cases or the mortality of a particular disease allows one to characterize its growth. Trend analysis may also be used to evaluate the effectiveness of an intervention to control the spread of an infectious disease. However, trends are often not readily observable because of noise in data that is commonly caused by random factors, short-term repeated patterns, or measurement error. In this paper, a smoothing technique that generalizes the Whittaker-Henderson method to infinite dimension and whose solution is represented by a polynomial is applied to extract the underlying trend in infectious disease data. The solution is obtained by projecting the problem to a finite-dimensional space using an orthonormal Sobolev polynomial basis obtained from Gram-Schmidt orthogonalization procedure and a smoothing parameter computed using the Philippine Eagle Optimization Algorithm, which is more efficient and consistent than a hybrid model used in earlier work. Because the trend is represented by the polynomial solution, extreme points, concavity, and periods when infectious disease cases are increasing or decreasing can be easily determined. Moreover, one can easily generate forecast of cases using the polynomial solution. This approach is applied in the analysis of trends, and in forecasting cases of different infectious diseases.

An Extrapolation Approach for RF-EMF Exposure Prediction in an Urban Area using Artificial Neural Network

The prediction of the electric (E) field plays an important role in the monitoring of the radiofrequency electromagnetic field (RF-EMF) exposure induced by cellular networks. In this paper, we present an approach to extrapolate the E field in an urban area using artificial neural network. We first apply a moving average method over a sliding window to average out the EMF random fluctuations and remove the noise produced during the drive test recording of E field measurements along the route. Using public accessed datasets, i.e., cartoradio and OpenStreetMap, we then extract relevant features, including the ones that have a relation with the number of active antennas and those used by Bertoni-Walfisch propagation model. By applying the Gram-Schmidt Orthogonalization procedure, we select the best subset of the extracted features as inputs to the artificial neural network (ANN). In this work, two disjoint subsets are selected for the learning and testing phases to evaluate the performance of our proposal in the extrapolation of the E field measurements and to quantify the uncertainty generated by the proposed predictor due to the dynamicity usage of cellular networks and geolocalization inaccuracies.

Stochastic replicator dynamics and evolutionary stability.

To develop the concept of evolutionary stability in a stochastic environment, we investigate the continuous-time dynamics of a two-phenotype linear evolutionary game with generally correlated random payoffs in pairwise interactions. By using the Gram-Schmidt orthogonalization procedure and Itô's formula, we deduce a stochastic differential equationfor the phenotype frequencies that extends the replicator equation, called the stochastic replicator equation. We give conditions for stochastic stability of a fixation state or a constant interior equilibrium point with respect to the stochastic dynamics of the two phenotypes. We show that, if a fixation state is stochastically stable, then the pure strategy corresponding to this fixation state must be stochastically evolutionarily stable with respect to mixed strategies. However, this is not the case for a mixed strategy that corresponds to a stochastically stable constant interior equilibrium point with respect to the two phenotypes.

On linear and nonlinear bending of functionally graded graphene nanoplatelet reinforced composite beams using Gram-Schmidt-Ritz method

The current study examined the linear and nonlinear bending results of functionally graded graphene nanoplatelet reinforced composite beams. The equation system was constructed using Reddy's third order shear deformation theory and a von Kármán type nonlinear strain-displacement relationship. The nonlinear bending results of this system were solved using the Gram-Schmidt-Ritz method in conjunction with the direct iteration technique. By employing the Gram-Schmidt orthogonalization procedure for generating shape functions in the method, numerically stable functions for nonlinear bending analysis of beams were obtained. Our solutions were validated for accuracy by comparing them to some published results. The numerical results indicate that the geometrical nonlinearity of beams subjected to uniform distributed load is greater than that of beams subjected to sinusoidal, linear, or parabolic distributed loads. New results on several significant effects of reinforcement patterns, graphene nanoplatelet weight fraction, and different types of distributed loads are presented and discussed in this study, in which the findings can serve as a benchmark for future research.

Transceiver Imbalances Compensation and Monitoring by Receiver DSP

In-phase and quadrature (IQ) imbalances in optical transceiver, such as amplitude imbalance, phase imbalance, and IQ skew, can cause serious system performance degradation, especially when high order modulation formats are used. In this work, we present a method to isolate and separately monitor these various impairments from received signals by coherent digital signal processing (DSP). Receiver amplitude and phase imbalances could be compensated and monitored by Gram-Schmidt orthogonalization procedure (GSOP) and receiver skew could be compensated and monitored by a 4 x 4 or 4 x 2 multiple-input and multiple-output (MIMO) equalizer. Transmitter imperfections compensation and monitoring take place behind the residual frequency offset compensation and carrier phase recovery by a 4 x 4 MIMO equalizer based on minimal mean square error (MMSE) criterion. By these methods, we analyze and investigate the interaction of transceiver's impairments impact on each monitor numerically. After that, we conduct an experiment to verify the feasibility of the proposed method. All of these results show that each monitor could operate very robust even when multiple co-exist.

Linear and nonlinear static bending of sandwich beams with functionally graded porous core under different distributed loads

In this investigation, linear and nonlinear bending analyses of sandwich beams with functionally graded cores are determined under different types of distributed loads. These sandwich beams are composed of two isotropic faces and a porous core with different gradients of internal pores. The governing formulation used to describe the beam’s linear and nonlinear behavior is constructed from Reddy's third-order shear deformation theory and nonlinear strain–displacement relations of von Kármán. The Gram-Schmidt orthogonalization procedure is adopted to generate numerically stable functions for the displacement field to solve the beam problems with various boundary conditions. Then, the Ritz method is utilized to find out linear and nonlinear bending results in conjunction with the iterative technique. The accuracy of our solutions is validated, and our numerical results agree well with some cases available in the literature. New results of the sandwich beams based on several effects of porosity coefficient, slenderness ratio, loading types, porous distributions of the core, etc., are presented in graphical and tabular forms, serving as a benchmark solution for future studies.

Joint estimation of transmitter and receiver IQ imbalances based on GSOP and GCM for optical coherent systems

In this paper, we propose an in-phase/quadrature (IQ) imbalance estimation scheme for mixed transmitter (Tx) and receiver (Rx) IQ imbalance in the optical coherent transmission system. The proposed scheme is based on traditional IQ imbalance compensation algorithm Gram-Schmidt orthogonalization procedure (GSOP) and the proposed geometric construction method (GCM). In the actual transmission system, frequency offset (FO) and phase noise (PN) are recognized as part of received signal, which has an effect on the estimation of IQ imbalance. At the beginning of the proposed scheme, the GSOP algorithm is used to estimate and compensate Rx IQ imbalance according to the statistical characteristics of the received signal, then standard frequency offset compensation algorithm and k-means clustering aided blind phase search (BPS) algorithm are used to compensate FO and PN respectively. Finally, the proposed method GCM is used to estimate Tx IQ imbalance. 34 Gbaud QPSK and 16QAM signals are experimentally performed to demonstrate the proposed IQ imbalance estimation scheme, the results show the proposed scheme has accurate estimation capability with low estimation algorithm complexity.

Compensation for In-Phase/Quadrature Phase Mismatch in Coherent Free-Space Optical QPSK Communication Systems

The Gram–Schmidt orthogonalization procedure (GSOP) and Löwdin symmetric orthogonalization procedure (SYOP) are the two mainstream algorithms for the compensation of phase mismatch in an imperfect optical 90° hybrid. In this paper, we put forward an algorithm switching orthogonalization procedure (ASOP) according to the quality of in-phase and quadrature signals based on the Q value of the eye diagram with less computation. If the quality of the in-phase and quadrature signals has a significant difference, we use the GSOP and select the signal branch with better quality as the initial reference vector for orthogonalization. If they are of about the same quality, then we use the SYOP. We present computer simulations for a coherent free-space optical (FSO) quadrature phase-shift keying (QPSK) communication system and demonstrate the system improvement that can be achieved using the ASOP. Finally, we also show that the proposed ASOP scheme can contribute to the frequency offset and phase estimation of the FSO system in the environment of atmospheric turbulence.

The Role of Financial Development and Institutional Quality in Remittance-Growth Nexus in Asia

Workers' remittances represent up to 47 percent of gross domestic product in some Asian countries. Also, workers' remittances to the Asian region were estimated to reach $256 billion out of $483 billion received by developing countries. However, the growth effect of workers' remittance in Asia has, over the years, remained an open empirical question. To this end, this study examines the impact of workers' remittance and its interactive effects with financial development and institutional quality on economic growth. The study used a partial Gram-Schmidt orthogonalization procedure, which enable us to examine whether the marginal effect of workers' remittances depends on the degree of financial development and institutional quality. It also used a Pooled Mean Group (PMG) estimator on panel dataset for a period of 1984-2010 from fifteen (15) remittance-receiving Asian countries. The results of this study shows that, workers' remittance impacts positively on economic growth in remittance-receiving Asian countries. But, the growth effect is higher in Asian countries with improved financial development but relatively weak institutions. According to the findings a percentage point increase in per capita workers' remittance leads to a long-run increase in per capita income by 0.103 percentage points. Moreover, the marginal effect of workers' remittance on economic growth rises with increase in the degree of financial development. But it declines with improvement in institutional environment. The partial derivative of economic growth with respect to workers' remittance at the mean level of financial development is 9.733. Similarly, the derivative evaluated at the minimum and maximum levels of financial development are found to be 1.355, and 32.087 respectively. The proportions of these derivatives at different levels of financial development showed that the marginal growth effect of workers' remittances rises with increase in the degree of financial development in remittance-receiving Asian countries. However, the partial derivative of economic growth with respect to workers' remittance at the mean level of institutional quality is –12.068. But, the partial derivative computed at the maximum and minimum levels of institutional quality take the value of –15.945 and –6.785 respectively. This suggests that the better institutional environment is in a country the lesser will be the growth impact of workers' remittance. Overall, the result suggests that workers' remittance and financial development work as complements in enhancing economic growth. However, the former and institutional quality act as substitutes in promoting economic growth.

Multi-Dimensional, Wide-Range, and Modulation-Format-Transparent Transceiver Imbalance Monitoring

We propose and experimentally demonstrate a dual-polarization (DP) transceiver imbalance monitoring scheme, which is applicable for the coherent system using square QAM signals. The scheme can separate/monitor various kinds of transceiver imbalances and has technical advantages of multi-dimension, wide monitoring range, and modulation-format-transparency. In the proposed scheme, a polarization scrambler is configured between transmitter (Tx) and receiver (Rx). After coherent detection, the proposed digital signal processing (DSP) modules are implemented to monitor transceiver imbalance based on the received signals. To simplify the DSP modules, the implementation sequence of DSP modules is in reverse order compared to the order in which the sequence of impairments is introduced. Firstly, Godard timing error detection (TED) and Gram-Schmidt orthogonalization procedure (GSOP) is used to monitor and compensate Rx imbalances. After compensating the Rx imbalances, Tx imbalance monitoring is implemented. Complex/real maximum likelihood independent component analysis (ML-ICA) and Godard-TED are used to monitor Tx imbalance. Finally, the amplitude ratios of the final output signal and Tx in-phase/quadrature (IQ) amplitude imbalance are fed back to mitigate the interaction of Tx imbalances on the signal power and improve monitoring accuracy. The transceiver IQ and x-polarization/y-polarization (XY) imbalances are separated for the first time by using the frequency offset (FO) naturally existing in the system and the continuous polarization rotation induced by a polarization scrambler. The separation method combined with the reverse algorithm design and amplitude ratio feedback mechanism makes the proposed scheme achieve the most multi-dimensional imbalance monitoring by far. The simulation and experimental results verify that the proposed scheme is modulation-format-transparent and can separate/monitor multi-dimensional transceiver imbalances within a wide range.

Chaotic modes of a non-linear fractional oscillator

A non-linear fractional oscillator is a generalization of a classical non-linear oscillator in consideration of the hereditary or the memory effect. The memory effect is a property of a dynamic system in which its current state depends on a finite number of its previous states. Therefore, a non-linear fractional oscillator can be mathematically described using integro-differential equations with difference kernels or fractional order derivatives. In this paper, a fractional non-linear oscillator has been investigated to identify chaotic oscillatory modes. The quantitative measure of chaotic regimes is the largest (maximal) Lyapunov exponents. For calculating the maximal Lyapunov exponents, the Wolff algorithm based on the Gram-Schmidt orthogonalization procedure has been selected, using both numerical solution for an initial fractional dynamical system by using variational equations. The Wolff algorithm also makes it possible to plot the spectrum of Lyapunov exponents as a function of control parameters for the initial dynamical system. It has been shown that some spectra of Lyapunov exponents contain positive values indicating the existence of chaotic modes, which are also confirmed by the corresponding phase trajectories.

Orthogonal AMP for Massive Access in Channels With Spatial and Temporal Correlations

We address the joint device activity detection and channel estimation (JACE) problem in a massive MIMO connectivity scenario in which a large number of mobile devices are connected to a base station (BS), while only a small portion are active at any given time. The main objective is to provide an efficient transmission and detection scheme with both spatial and temporal correlations. We formulate JACE as a multiple measurement vector (MMV) problem with correlated entries in the vectors to be estimated. We propose an MMV form of the orthogonal approximate message passing algorithm (OAMP-MMV). We derive a group Gram-Schmidt orthogonalization (GGSO) procedure for the realization of OAMP-MMV. We outline a state evolution (SE) procedure for OAMP-MMV and examine its accuracy using numerical results. We also compare OAMP-MMV with existing alternatives, including AMP-MMV and GTurbo-MMV. We show that OAMP-MMV outperforms AMP-MMV when pilot sequences are generated using Hadamard pilot matrices. Such a pilot design is attractive due to the low-cost signal processing technique using the fast Hadamard transform (FHT). We also show that OAMP-MMV outperforms GTurbo-MMV in correlated channels.

Gram-Schmidt-Ritz method for dynamic response of FG-GPLRC beams under multiple moving loads

The objective of this study is to apply Gram-Schmidt orthogonalization procedure for generating displacement functions. This procedure allows us to obtain numerically stable functions to be used in vibration analysis of novel composite beams via the Ritz method. Within this study, the theoretical modeling based on third-order shear deformation beam theory is derived to deal with free and forced vibration of functionally graded graphene nanoplatelet-reinforced composite beams subjected to multiple moving loads. To solve the problems of the beams with various end supports, the Ritz method-based Gram-Schmidt polynomial series is adopted to obtain the accurate results. In forced vibration due to the applied multiple moving loads, the time integration technique of Newmark is also employed to find out time history of dynamic deflection of such beams associated with several parametric studies such as patterns of reinforcement, weight fraction of graphene nanoplatelets, and the number of moving loads. According to the numerical results, it is found that the composite beams with low content of graphene nanoplatelets can result in enhancing performance of dynamic bending resistance. Increasing number of moving loads on the beams yields the increases of dynamic deflection and critical velocity.

IQ imbalance compensation based on simplified GSOP and FPGA implementation in optical coherent QPSK receiver

Abstract In this paper, a simplified Gram-Schmidt orthogonalization procedure (SG) method was proposed for in-phase/quadrature (IQ) imbalance compensation based on the Gram-Schmidt orthogonalization procedure (GSOP) algorithm. The proposed method can compensate IQ imbalance in the optical coherent system by two steps of phase orthogonalization and amplitude balance. In addition, this method avoids square root operation compared to the GSOP in the field programmable gate array (FPGA) hardware implementation, therefore it is easier to implement in terms of resources and complexity. Two steady states are achieved by accumulating scale factor to reduce resource consumption in FPGA hardware implementation of the proposed scheme. Finally, a FPGA design with 32 parallel units is performed to demonstrate the proposed method in a polarization multiplexed quadrature phase shift keying (PM-QPSK) system, the results show the proposed algorithm simplifies FPGA implementation complexity and maintains compensation performance comparable to GSOP algorithms.

New directional vector limiters for discontinuous Galerkin methods

Second and higher order numerical approximations of conservation laws for vector fields call for the use of limiting techniques based on generalized monotonicity criteria. In this paper, we introduce a family of directional vertex-based slope limiters for tensor-valued gradients of formally second-order accurate piecewise-linear discontinuous Galerkin (DG) discretizations. The proposed methodology enforces local maximum principles for scalar products corresponding to projections of a vector field onto the unit vectors of a frame-invariant orthogonal basis. In particular, we consider anisotropic limiters based on singular value decompositions and the Gram-Schmidt orthogonalization procedure. The proposed extension to hyperbolic systems features a sequential limiting strategy and a global invariant domain fix. The pros and cons of different approaches to vector limiting are illustrated by the results of numerical studies for the two-dimensional shallow water equations and for the Euler equations of gas dynamics.

Development of the procedure for forming non­stationary signal structures based on multicomponent LFM signals

Noise protection of existing radio lines with noise-shaped signals and digital types of modulation was studied. Analysis has shown that the use of such signals in conditions of the radio-electronic conflict does not permit to provide necessary level of noise immunity and transmission security of radio communication lines. It was explained by presence of cyclo-conditionality of the carrier oscillation in signals with digital modulation types. Such properties simplify detection and search of signals by means of spectral correlation methods of modern hostile means of electronic surveillance. To solve this problem, the use of nonstationary signal structures with variable central frequency and spectral density of power was proposed. A procedure of forming such signal structures by application of the Gram-Schmidt orthogonalization procedure to the ensemble of multicomponent LFM signals with controlled spectral characteristics was developed. It was proposed to estimate various signal structures of multicomponent signal by means of phase portraits of summed signals depending on the scaling factor value. This factor’s boundary values at which complexity of the multicomponent signal structure is ensured and degeneration of the process into classical LFM is prevented were established. Change of probability of a symbol error in a channel with the use of multicomponent orthogonal signal structures was studied depending on the signal/noise ratio. This makes it possible to estimate potential noise immunity of the radio line provided that the signal/noise ratio is determined by energy indicators of the radio channel and the spectral density of the noise of natural origin. Structural security of the developed signal structures was estimated by means of an energy detector and a cyclo-stationarity detector. It was established that in the case of energy detection, nonstationary signals, and signals of any other type of modulation are equivalent. However, probability of detecting nonstationary signal structures decreased 2–2.5 times compared to other types of signal modulation when using the cyclo-stationarity detector

Receiver IQ mismatch and phase noise suppression in PDM CO-OFDM system

Abstract Receiver in-phase/quadrature (IQ) mismatch as well as phase noise are hard to mitigate at the receiver in polarization division multiplexed (PDM) coherent optical orthogonal frequency division multiplexing (CO-OFDM) system and so far this problem hasn’t been properly solved yet. In this paper, a novel scheme is proposed to suppress receiver IQ mismatch and phase noise. In this scheme, through combining the pilot structure with Gram Schmidt orthogonalization procedure (GSOP) algorithm we can estimate the phase noise as well as channel distortion and eliminate the effect of part of the residual IQ mismatch. Numerical simulation shows that the proposed method has better performance than compared method at 100 Gb/s after 480 km transmission. What’s more, the calculation complexity is also analyzed.

The study of chaotic and regular regimes of the fractal oscillators FitzHugh-Nagumo

In this paper we study the conditions for the existence of chaotic and regular oscillatory regimes of the hereditary oscillator FitzHugh-Nagumo (FHN), a mathematical model for the propagation of a nerve impulse in a membrane. To achieve this goal, using the non-local explicit finite-difference scheme and Wolf’s algorithm with the Gram-Schmidt orthogonalization procedure and the spectra of the maximum Lyapunov exponents were also constructed depending on the values of the control parameters of the model of the FHN. The results of the calculations showed that there are spectra of maximum Lyapunov exponents both with positive values and with negative values. The results of the calculations were also confirmed with the help of oscillograms and phase trajectories, which indicates the possibility of the existence of both chaotic attractors and limit cycles.

Improving the Efficiency of Beyond-RPA Methods within the Dielectric Matrix Formulation: Algorithms and Applications to the A24 and S22 Test Sets.

Within a formalism based on dielectric matrices, the electron-hole time-dependent Hartree-Fock (eh-TDHF) and the adiabatic connection second-order screened exchange (AC-SOSEX) are promising approximations to improve ground-state correlation energies by including exchange effects beyond the random phase approximation (RPA). We introduce here an algorithm based on a Gram-Schmidt orthogonalization (GSO) procedure that significantly reduce the number of matrix elements to be computed to evaluate the response functions that enter in the formulation of these two methods. By considering the A24 test set, we show that this approach does not lead to a significant loss of accuracy and can be effectively applied to compute the small interaction energies involved in weakly bound dimers. Importantly, the GSO method significantly extends the applicability of the eh-TDHF and AC-SOSEX to large systems. This is shown by considering the S22 test set, which includes dimers with up to one hundred valence electrons requiring hundreds of thousands of plane-waves in the basis set. By comparing our results to coupled-cluster benchmark values, we show that the inclusion of exchange effects beyond the RPA significantly improves the accuracy, with mean absolute errors that decrease by almost 40% for the A24 test set and by almost 50% for the S22 test set. This approach based on dielectric matrices is particularly suited for plane-wave implementations and might be used in the future to improve the description of the correlation energy in solid state applications.