Search results for: Error analysis

Authors: Jinfeng Wang, Yuanhong Bi, Hong Li, Yang Liu, Meng Zhao

Abstract:

In this paper, a new time discontinuous expanded mixed finite element method is proposed and analyzed for two-order convection-dominated diffusion problem. The proofs of the stability of the proposed scheme and the uniqueness of the discrete solution are given. Moreover, the error estimates of the scalar unknown, its gradient and its flux in the L1( ¯ J,L2( )-norm are obtained.

Keywords: Convection-dominated diffusion equation, expanded mixed method, time discontinuous scheme, stability, error estimates.

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Authors: Akram A. Mustafa

Abstract:

In this paper an approaches for increasing the effectiveness of error detection in computer network channels with Pulse-Amplitude Modulation (PAM) has been proposed. Proposed approaches are based on consideration of special feature of errors, which are appearances in line with PAM. The first approach consists of CRC modification specifically for line with PAM. The second approach is base of weighted checksums using. The way for checksum components coding has been developed. It has been shown that proposed checksum modification ensure superior digital data control transformation reliability for channels with PAM in compare to CRC.

Keywords: Pulse-Amplitude Modulation, checksum, transmission, discrete.

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Authors: Shilpi Lavania, Deepak Nagaria

Abstract:

A mixed method for model order reduction is presented in this paper. The denominator polynomial is derived by matching both Markov parameters and time moments, whereas numerator polynomial derivation and error minimization is done using Genetic Algorithm. The efficiency of the proposed method can be investigated in terms of closeness of the response of reduced order model with respect to that of higher order original model and a comparison of the integral square error as well.

Keywords: Model Order Reduction (MOR), control theory, Markov parameters, time moments, genetic algorithm, Single Input Single Output (SISO).

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Authors: P. Soda, G.Iannello

Abstract:

In this paper we address the issue of classifying the fluorescent intensity of a sample in Indirect Immuno-Fluorescence (IIF). Since IIF is a subjective, semi-quantitative test in its very nature, we discuss a strategy to reliably label the image data set by using the diagnoses performed by different physicians. Then, we discuss image pre-processing, feature extraction and selection. Finally, we propose two ANN-based classifiers that can separate intrinsically dubious samples and whose error tolerance can be flexibly set. Measured performance shows error rates less than 1%, which candidates the method to be used in daily medical practice either to perform pre-selection of cases to be examined, or to act as a second reader.

Keywords: Artificial neural networks, computer aided diagnosis, image classification, indirect immuno-fluorescence, pattern recognition.

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Authors: Hanita Daud, Noorhana Yahya, Vijanth Sagayan, Muizuddin Talib

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This paper discusses on the use of Spline Interpolation and Mean Square Error (MSE) as tools to process data acquired from the developed simulator that shall replicate sea bed logging environment. Sea bed logging (SBL) is a new technique that uses marine controlled source electromagnetic (CSEM) sounding technique and is proven to be very successful in detecting and characterizing hydrocarbon reservoirs in deep water area by using resistivity contrasts. It uses very low frequency of 0.1Hz to 10 Hz to obtain greater wavelength. In this work the in house built simulator was used and was provided with predefined parameters and the transmitted frequency was varied for sediment thickness of 1000m to 4000m for environment with and without hydrocarbon. From series of simulations, synthetics data were generated. These data were interpolated using Spline interpolation technique (degree of three) and mean square error (MSE) were calculated between original data and interpolated data. Comparisons were made by studying the trends and relationship between frequency and sediment thickness based on the MSE calculated. It was found that the MSE was on increasing trends in the set up that has the presence of hydrocarbon in the setting than the one without. The MSE was also on decreasing trends as sediment thickness was increased and with higher transmitted frequency.

Keywords: Spline Interpolation, Mean Square Error, Sea Bed Logging, Controlled Source Electromagnetic

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Authors: Omar Al Denali, Abdelaziz Badi

Abstract:

The ASTM A516 Grade 70 steel is a suitable material used for the fabrication of boiler pressure vessels working in moderate and lower temperature services, and it has good weldability and excellent notch toughness. The post-weld heat treatment (PWHT) or stress-relieving heat treatment has significant effects on avoiding the martensite transformation and resulting in high hardness, which can lead to cracking in the heat-affected zone (HAZ). An adaptive neuro-fuzzy inference system (ANFIS) was implemented to predict the material tensile strength of PWHT experiments. The ANFIS models presented excellent predictions, and the comparison was carried out based on the mean absolute percentage error between the predicted values and the experimental values. The ANFIS model gave a Mean Absolute Percentage Error of 0.556%, which confirms the high accuracy of the model.

Keywords: Prediction, post-weld heat treatment, adaptive neuro-fuzzy inference system, ANFIS, mean absolute percentage error.

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Authors: M. A. M. El-Bendary, H. Kazimian, A. E. Abo-El-azm, N. A. El-Fishawy, F. El-Samie, F. Shawki

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This paper studies a vital issue in wireless communications, which is the transmission of images over Wireless Personal Area Networks (WPANs) through the Bluetooth network. It presents a simple method to improve the efficiency of error control code of old Bluetooth versions over mobile WPANs through Interleaved Error Control Code (IECC) technique. The encoded packets are interleaved by simple block interleaver. Also, the paper presents a chaotic interleaving scheme as a tool against bursts of errors which depends on the chaotic Baker map. Also, the paper proposes using the chaotic interleaver instead of traditional block interleaver with Forward Error Control (FEC) scheme. A comparison study between the proposed and standard techniques for image transmission over a correlated fading channel is presented. Simulation results reveal the superiority of the proposed chaotic interleaving scheme to other schemes. Also, the superiority of FEC with proposed chaotic interleaver to the conventional interleavers with enhancing the security level with chaotic interleaving packetby- packet basis.

Keywords: Mobile Bluetooth terminals, WPANs, Jackes' model, Interleaving technique, chaotic interleaver

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Authors: N. R. N. Idris, S. Baharom

Abstract:

A meta-analysis may be performed using aggregate data (AD) or an individual patient data (IPD). In practice, studies may be available at both IPD and AD level. In this situation, both the IPD and AD should be utilised in order to maximize the available information. Statistical advantages of combining the studies from different level have not been fully explored. This study aims to quantify the statistical benefits of including available IPD when conducting a conventional summary-level meta-analysis. Simulated meta-analysis were used to assess the influence of the levels of data on overall meta-analysis estimates based on IPD-only, AD-only and the combination of IPD and AD (mixed data, MD), under different study scenario. The percentage relative bias (PRB), root mean-square-error (RMSE) and coverage probability were used to assess the efficiency of the overall estimates. The results demonstrate that available IPD should always be included in a conventional meta-analysis using summary level data as they would significantly increased the accuracy of the estimates.On the other hand, if more than 80% of the available data are at IPD level, including the AD does not provide significant differences in terms of accuracy of the estimates. Additionally, combining the IPD and AD has moderating effects on the biasness of the estimates of the treatment effects as the IPD tends to overestimate the treatment effects, while the AD has the tendency to produce underestimated effect estimates. These results may provide some guide in deciding if significant benefit is gained by pooling the two levels of data when conducting meta-analysis.

Keywords: Aggregate data, combined-level data, Individual patient data, meta analysis.

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Authors: N. S. Pradeep, M. Balasingh Moses, V. Aarthi

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Any signal transmitted over a channel is corrupted by noise and interference. A host of channel coding techniques has been proposed to alleviate the effect of such noise and interference. Among these Turbo codes are recommended, because of increased capacity at higher transmission rates and superior performance over convolutional codes. The multimedia elements which are associated with ample amount of data are best protected by Turbo codes. Turbo decoder employs Maximum A-posteriori Probability (MAP) and Soft Output Viterbi Decoding (SOVA) algorithms. Conventional Turbo coded systems employ Equal Error Protection (EEP) in which the protection of all the data in an information message is uniform. Some applications involve Unequal Error Protection (UEP) in which the level of protection is higher for important information bits than that of other bits. In this work, enhancement to the traditional Log MAP decoding algorithm is being done by using optimized scaling factors for both the decoders. The error correcting performance in presence of UEP in Additive White Gaussian Noise channel (AWGN) and Rayleigh fading are analyzed for the transmission of image with Discrete Cosine Transform (DCT) as source coding technique. This paper compares the performance of log MAP, Modified log MAP (MlogMAP) and Enhanced log MAP (ElogMAP) algorithms used for image transmission. The MlogMAP algorithm is found to be best for lower Eb/N0 values but for higher Eb/N0 ElogMAP performs better with optimized scaling factors. The performance comparison of AWGN with fading channel indicates the robustness of the proposed algorithm. According to the performance of three different message classes, class3 would be more protected than other two classes. From the performance analysis, it is observed that ElogMAP algorithm with UEP is best for transmission of an image compared to Log MAP and MlogMAP decoding algorithms.

Keywords: AWGN, BER, DCT, Fading, MAP, UEP.

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Authors: Jingbo Yang, Hong Li, Yang Liu, Siriguleng He

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In this paper, a nonconforming mixed finite element method is studied for semilinear pseudo-hyperbolic partial integrodifferential equations. By use of the interpolation technique instead of the generalized elliptic projection, the optimal error estimates of the corresponding unknown function are given.

Keywords: Pseudo-hyperbolic partial integro-differential equations, Nonconforming mixed element method, Semilinear, Error estimates.

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Authors: Maria Hussin, Malik Zawwar Hussain, Mubashrah Saddiqa

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We present a trigonometric scheme to approximate a circular arc with its two end points and two end tangents/unit tangents. A rational cubic trigonometric Bézier curve is constructed whose end control points are defined by the end points of the circular arc. Weight functions and the remaining control points of the cubic trigonometric Bézier curve are estimated by variational approach to reproduce a circular arc. The radius error is calculated and found less than the existing techniques.

Keywords: Control points, rational trigonometric Bézier curves, radius error, shape measure, weight functions.

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Authors: N. Z. Azlan, H. Yamaura

Abstract:

The experimental study of position control of a light weight and small size robotic finger during non-contact motion is presented in this paper. The finger possesses fingertip pinching and self adaptive grasping capabilities, and is made of a seven bar linkage mechanism with a slider in the middle phalanx. The control system is tested under the Proportional Integral Derivative (PID) control algorithm and Recursive Least Square (RLS) based Feedback Error Learning (FEL) control scheme to overcome the uncertainties present in the plant. The experiments conducted in Matlab Simulink and xPC Target environments show that the overall control strategy is efficient in controlling the finger movement.

Keywords: Anthropomorphic finger, position control, feedback error learning, experimental study

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Authors: V. T. S. Dao, T. G. Etoh, C. Vo Le, H. D. Nguyen, K. Takehara, T. Akino, K. Nishi

Abstract:

We present an explicit expression to estimate driving voltage attenuation through RC networks representation of an ultrahigh- speed image sensor. Elmore delay metric for a fundamental RC chain is employed as the first-order approximation. By application of dimensional analysis to SPICE simulation data, we found a simple expression that significantly improves the accuracy of the approximation. Estimation error of the resultant expression for uniform RC networks is less than 2%. Similarly, another simple closed-form model to estimate 50 % delay through fundamental RC networks is also derived with sufficient accuracy. The framework of this analysis can be extended to address delay or attenuation issues of other VLSI structures.

Keywords: Dimensional Analysis, Elmore model, RC network, Signal Attenuation, Ultra-High-Speed Image Sensor.

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Authors: Young-Su Ryu, Su-Hyun Jung, Myoung-Jin Kim, Hyoung-Kyu Song

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In this paper, improved performance scheme for joint transmission (JT) is proposed in downlink (DL) coordinated multi-point (CoMP) in case of the constraint transmission power. This scheme is that a serving transmission point (TP) requests the JT to an inter-TP and it selects a precoding technique according to the channel state information (CSI) from user equipment (UE). The simulation results show that the bit error rate (BER) and the throughput performances of the proposed scheme provide the high spectral efficiency and the reliable data at the cell edge.

Keywords: CoMP, joint transmission, minimum mean square error, zero-forcing, zero-forcing dirty paper coding.

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Authors: Abdelghani Dendouga, Nour-Eddine Bouguechal, Souhil Kouda, Samir Barra

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A behavioral model of a second order switchedcapacitor Sigma-Delta modulator is presented. The purpose of this work is the presentation of a behavioral model of a second order switched capacitor ΣΔ modulator considering (Error due to Clock Jitter, Thermal noise Amplifier Noise, Amplifier Slew-Rate, Non linearity of amplifiers, Gain error, Charge Injection, Clock Feedthrough, and Nonlinear on-resistance). A comparison between the use of MOS switches and the use transmission gate switches use is analyzed.

Keywords: Charge injection, clock feed through, Sigma Deltamodulators, Sigma Delta non-idealities, switched capacitor.

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Authors: Hermenegilde Nkurunziza, Albrecht Gebhardt, Juergen Pilz

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The focus in this work is to assess which method allows a better forecasting of malaria cases in Bujumbura ( Burundi) when taking into account association between climatic factors and the disease. For the period 1996-2007, real monthly data on both malaria epidemiology and climate in Bujumbura are described and analyzed. We propose a hierarchical approach to achieve our objective. We first fit a Generalized Additive Model to malaria cases to obtain an accurate predictor, which is then used to predict future observations. Various well-known forecasting methods are compared leading to different results. Based on in-sample mean average percentage error (MAPE), the multiplicative exponential smoothing state space model with multiplicative error and seasonality performed better.

Keywords: Burundi, Forecasting, Malaria, Regressionmodel, State space model.

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Authors: Lívia B. Meirelles, Erika C. A. N. Chrisman, Flávia B. de Andrade, Lilian C. M. de Oliveira

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True Boiling Point distillation (TBP) is one of the most common experimental techniques for the determination of petroleum properties. This curve provides information about the performance of petroleum in terms of its cuts. The experiment is performed in a few days. Techniques are used to determine the properties faster with a software that calculates the distillation curve when a little information about crude oil is known. In order to evaluate the accuracy of distillation curve prediction, eight points of the TBP curve and specific gravity curve (348 K and 523 K) were inserted into the HYSYS Oil Manager, and the extended curve was evaluated up to 748 K. The methods were able to predict the curve with the accuracy of 0.6%-9.2% error (Software X ASTM), 0.2%-5.1% error (Software X Spaltrohr).

Keywords: Distillation curve, petroleum distillation, simulation, true boiling point curve.

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Authors: S. Wongkoon, M. Pollar, M. Jaroensutasinee, K. Jaroensutasinee

Abstract:

This study aimed at developing a forecasting model on the number of Dengue Haemorrhagic Fever (DHF) incidence in Northern Thailand using time series analysis. We developed Seasonal Autoregressive Integrated Moving Average (SARIMA) models on the data collected between 2003-2006 and then validated the models using the data collected between January-September 2007. The results showed that the regressive forecast curves were consistent with the pattern of actual values. The most suitable model was the SARIMA(2,0,1)(0,2,0)12 model with a Akaike Information Criterion (AIC) of 12.2931 and a Mean Absolute Percent Error (MAPE) of 8.91713. The SARIMA(2,0,1)(0,2,0)12 model fitting was adequate for the data with the Portmanteau statistic Q20 = 8.98644 ( x20,95= 27.5871, P>0.05). This indicated that there was no significant autocorrelation between residuals at different lag times in the SARIMA(2,0,1)(0,2,0)12 model.

Keywords: Dengue, SARIMA, Time Series Analysis, Northern Thailand.

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Authors: Elragig Aiman, Dreiwi Hanan, Townley Stuart, Elmabrook Idriss

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In this paper, we reconsider the analysis of the Oregonator model. We highlight an error in this analysis which leads to an incorrect depiction of the parameter region in which diffusion driven instability is possible. We believe that the cause of the oversight is the complexity of stability analyses based on eigenvalues and the dependence on parameters of matrix minors appearing in stability calculations. We regenerate the parameter space where Turing patterns can be seen, and we use the common Lyapunov function (CLF) approach, which is numerically reliable, to further confirm the dependence of the results on diffusion coefficients intensities.

Keywords: Diffusion driven instability, common Lyapunov function (CLF), turing pattern, positive-definite matrix.

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Authors: Seungwon Choi, Seungri Jin, Ayoung Heo, Jung-Hyun Park, Dong-Jo Park

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This paper deals with wireless relay communication systems in which multiple sources transmit information to the destination node by the help of multiple relays. We consider a signal forwarding technique based on the minimum mean-square error (MMSE) approach with multiple antennas for each relay. A source-relay-destination joint design strategy is proposed with power constraints at the destination and the source nodes. Simulation results confirm that the proposed joint design method improves the average MSE performance compared with that of conventional MMSE relaying schemes.

Keywords: minimum mean squre error (MMSE), multiple relay, MIMO.

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Authors: M. Nahid, A. Bajit, A. Tamtaoui, E. H. Bouyakhf

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To model the human visual system (HVS) in the region of interest, we propose a new objective metric evaluation adapted to wavelet foveation-based image compression quality measurement, which exploits a foveation setup filter implementation technique in the DWT domain, based especially on the point and region of fixation of the human eye. This model is then used to predict the visible divergences between an original and compressed image with respect to this region field and yields an adapted and local measure error by removing all peripheral errors. The technique, which we call foveation wavelet visible difference prediction (FWVDP), is demonstrated on a number of noisy images all of which have the same local peak signal to noise ratio (PSNR), but visibly different errors. We show that the FWVDP reliably predicts the fixation areas of interest where error is masked, due to high image contrast, and the areas where the error is visible, due to low image contrast. The paper also suggests ways in which the FWVDP can be used to determine a visually optimal quantization strategy for foveation-based wavelet coefficients and to produce a quantitative local measure of image quality.

Keywords: Human Visual System, Image Quality, ImageCompression, foveation wavelet, region of interest ROI.

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Authors: Andhe Dharani, P. S. Satyanarayana, Andhe Pallavi

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Considering payload, reliability, security and operational lifetime as major constraints in transmission of images we put forward in this paper a steganographic technique implemented at the physical layer. We suggest transmission of Halftoned images (payload constraint) in wireless sensor networks to reduce the amount of transmitted data. For low power and interference limited applications Turbo codes provide suitable reliability. Ensuring security is one of the highest priorities in many sensor networks. The Turbo Code structure apart from providing forward error correction can be utilized to provide for encryption. We first consider the Halftoned image and then the method of embedding a block of data (called secret) in this Halftoned image during the turbo encoding process is presented. The small modifications required at the turbo decoder end to extract the embedded data are presented next. The implementation complexity and the degradation of the BER (bit error rate) in the Turbo based stego system are analyzed. Using some of the entropy based crypt analytic techniques we show that the strength of our Turbo based stego system approaches that found in the OTPs (one time pad).

Keywords: Halftoning, Turbo codes, security, operationallifetime, Turbo based stego system.

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Authors: Melodi Mbuyi Mata, Blessing Olamide Taiwo, Afolabi Ayodele David

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The purpose of this research is to create empirical models for assessing the profitability of granite aggregate production in Akure, Ondo state aggregate quarries. In addition, an Artificial Neural Network (ANN) model and multivariate predicting models for granite profitability were developed in the study. A formal survey questionnaire was used to collect data for the study. The data extracted from the case study mine for this study include granite marketing operations, royalty, production costs, and mine production information. The following methods were used to achieve the goal of this study: descriptive statistics, MATLAB 2017, and SPSS16.0 software in analyzing and modeling the data collected from granite traders in the study areas. The ANN and Multi Variant Regression models' prediction accuracy was compared using a coefficient of determination (R2), Root Mean Square Error (RMSE), and mean square error (MSE). Due to the high prediction error, the model evaluation indices revealed that the ANN model was suitable for predicting generated profit in a typical quarry. More quarries in Nigeria's southwest region and other geopolitical zones should be considered to improve ANN prediction accuracy.

Keywords: National development, granite, profitability assessment, ANN models.

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Authors: Susmita Sahoo, Madhumanti Datta, Rajib Kar

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This work proposes an accurate crosstalk noise estimation method in the presence of multiple RLC lines for the use in design automation tools. This method correctly models the loading effects of non switching aggressors and aggressor tree branches using resistive shielding effect and realistic exponential input waveforms. Noise peak and width expressions have been derived. The results obtained are at good agreement with SPICE results. Results show that average error for noise peak is 4.7% and for the width is 6.15% while allowing a very fast analysis.

Keywords: Crosstalk, distributed RLC segments, On-Chip interconnect, output response, VLSI, noise peak, noise width.

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Authors: Ahmad M. Sarhan, Omar I. Al Helalat

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In this paper, an Arabic letter recognition system based on Artificial Neural Networks (ANNs) and statistical analysis for feature extraction is presented. The ANN is trained using the Least Mean Squares (LMS) algorithm. In the proposed system, each typed Arabic letter is represented by a matrix of binary numbers that are used as input to a simple feature extraction system whose output, in addition to the input matrix, are fed to an ANN. Simulation results are provided and show that the proposed system always produces a lower Mean Squared Error (MSE) and higher success rates than the current ANN solutions.

Keywords: ANN, Backpropagation, Gaussian, LMS, MSE, Neuron, standard deviation, Widrow-Hoff rule.

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Authors: Youguo Wang, Lenan Wu

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Stochastic resonance (SR) is a phenomenon whereby the signal transmission or signal processing through certain nonlinear systems can be improved by adding noise. This paper discusses SR in nonlinear signal detection by a simple test statistic, which can be computed from multiple noisy data in a binary decision problem based on a maximum a posteriori probability criterion. The performance of detection is assessed by the probability of detection error Per . When the input signal is subthreshold signal, we establish that benefit from noise can be gained for different noises and confirm further that the subthreshold SR exists in nonlinear signal detection. The efficacy of SR is significantly improved and the minimum of Per can dramatically approach to zero as the sample number increases. These results show the robustness of SR in signal detection and extend the applicability of SR in signal processing.

Keywords: Probability of detection error, signal detection, stochastic resonance.

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Authors: Abdelhamid Louliej, Younes Jabrane

Abstract:

Satellites are becoming indispensable assets to fifth-generation (5G) new radio architecture, complementing wireless and terrestrial communication links. The combination of satellites and 5G architecture allows consumers to access all next-generation services anytime, anywhere, including scenarios, like traveling to remote areas (without coverage). Nevertheless, this solution faces several challenges, such as a significant propagation delay, Doppler frequency shift, and high Peak-to-Average Power Ratio (PAPR), causing signal distortion due to the non-linear saturation of the High-Power Amplifier (HPA). To compensate for HPA non-linearity in 5G satellite transmission, an efficient pre-distorter scheme using Neural Networks (NN) is proposed. To assess the proposed NN pre-distorter, two types of HPA were investigated: Travelling Wave Tube Amplifier (TWTA) and Solid-State Power Amplifier (SSPA). The results show that the NN pre-distorter design presents an Error Vector Magnitude (EVM) improvement by 95.26%. Normalized Mean Square Error (NMSE) and Adjacent Channel Power Ratio (ACPR) were reduced by -43,66 dB and 24.56 dBm, respectively. Moreover, the system suffers no degradation of the Bit Error Rate (BER) for TWTA and SSPA amplifiers.

Keywords: Satellites, 5G, Neural Networks, High-Power Amplifier, Travelling Wave Tube Amplifier, Solid-State Power Amplifier, EVM, NMSE, ACPR.

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Authors: Rajesh Joseph Abraham, D. Das, Amit Patra

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This paper is concerned with the application of small rating Capacitive Energy Storage units for the improvement of Automatic Generation Control of a multiunit multiarea power system. Generation Rate Constraints are also considered in the investigations. Integral Squared Error technique is used to obtain the optimal integral gain settings by minimizing a quadratic performance index. Simulation studies reveal that with CES units, the deviations in area frequencies and inter-area tie-power are considerably improved in terms of peak deviations and settling time as compared to that obtained without CES units.

Keywords: Automatic Generation Control, Capacitive EnergyStorage, Integral Squared Error.

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Authors: Mamoun F. Al-Mistarihi, Rami Mohaisen

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Cooperative diversity (CD) has been adopted in many communication systems because it helps in improving performance of the wireless communication systems with the help of the relays that emulate the multiple antenna terminals. This work aims to provide the derivation of the performance analysis expressions of the multiuser diversity (MUD) in the two-hop cooperative multi-relay wireless networks (TCMRNs). Considering the work analysis, we provide analytically the derivation of a closed form expression of the two most commonly used performance metrics namely, the outage probability and the symbol error probability (SEP) for the fixed decode-and-forward (FDF) protocol with MUD.

Keywords: Cooperative diversity (CD), fixed decode-andforward(FDF), multiuser diversity (MUD) , two - hop cooperative multi-relay wireless networks (TCMRN).

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Authors: Zilong Feng, Hong Li, Yang Liu, Siriguleng He

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In this article, an adaptive least-squares mixed finite element method is studied for pseudo-parabolic integro-differential equations. The solutions of least-squares mixed weak formulation and mixed finite element are proved. A posteriori error estimator is constructed based on the least-squares functional and the posteriori errors are obtained.

Keywords: Pseudo-parabolic integro-differential equation, least squares mixed finite element method, adaptive method, a posteriori error estimates.

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