Taylor Series Approximation Research Articles

A Comprehensible Form of the Product of Two Gaussian Q Functions and its Usefulness in [formula omitted] Shadowed Fading Distribution

In 2007, contrary to other approximations, Karagiannidis et al. proposed a more accurate approximation of the Gaussian Q function and its integer powers, for all the positive arguments. However, when it is used to compute the symbol error probability (SEP) of various coherent digital modulation techniques over parametric fading distributions like the κ−μ shadowed fading, it loses its analytical tractability. In this paper, using Taylor series approximation of the exponential function, we comprehend the approximation of interest (with accuracy intact) which facilitates in the simplification of the key integrals used in the SEP computation of digital modulation techniques over κ−μ shadowed fading distribution. Apart from various applications, the significance of the κ−μ shadowed fading statistics lies in the fact that it unifies most of the popular fading models like one sided Gaussian, Rayleigh, Nakagami-m, Nakagami-q, Rician, Rician-shadowed, η−μ and κ−μ, under one umbrella. In order to show the utility of the proposed work, the SEP of hexagonal-QAM is calculated over Rayleigh fading channel which is one of the widely used cases of the κ−μ shadowed fading distribution. Monte Carlo simulations have also been carried out to justify the accuracy of the analysis.

Asymptotic Performance Analysis of Maximum Likelihood Algorithm for Direction-of-Arrival Estimation: Explicit Expression of Estimation Error and Mean Square Error

This paper proposes a new method to get explicit expressions of some quantities associated with performance analysis of the maximum likelihood DOA algorithm in the presence of an additive Gaussian noise on the antenna elements. The motivation of the paper is to make a quantitative analysis of the ML DOA algorithm in the case of multiple incident signals. We present a simple method to derive a closed-form expression of the MSE of the DOA estimate based on the Taylor series expansion. Based on the Taylor series expansion and approximation, we get explicit expressions of the MSEs of estimates of azimuth angles of all incident signals. The validity of the derived expressions is shown by comparing the analytic results with the simulation results.

Phase-Based 3-D Source Localization Using 1-b Quantized Measurements Under Uniform Circular Array

1-b quantization has become an important topic in modern signal processing. In this letter, we consider the problem of 3-D source localization by combining 1-b quantized measurements and the phase-based algorithm under a uniform circular array. With analysis, unquantized and 1-b quantized cross-correlation coefficient can be approximated by the arcsine law and Taylor series approximation under sufficiently low signal-to-noise ratio (SNR). Even at high SNR levels, the approximated error only slightly affects source localization. Therefore, under 1-b quantized measurements, the phase-based algorithm can be straightforwardly applied to source's 3-D localization without extra preprocessing. Numerical examples demonstrate the theoretical analysis.

Controlling the pressure of hydrogen-natural gas mixture in an inclined pipeline.

This paper discusses the optimal control of pressure using the zero-gradient control (ZGC) approach. It is applied for the first time in the study to control the optimal pressure of hydrogen natural gas mixture in an inclined pipeline. The solution to the flow problem is first validated with existing results using the Taylor series approximation, regression analysis and the Runge-Kutta method combined. The optimal pressure is then determined using ZGC where the optimal set points are calculated without having to solve the non-linear system of equations associated with the standard optimization problem. It is shown that the mass ratio is the more effective parameter compared to the initial pressure in controlling the maximum variation of pressure in a gas pipeline.

False Discovery Variance Reduction in Large Scale Simultaneous Hypothesis Tests

Statistical dependence between hypotheses poses a significant challenge to the stability of large scale multiple hypotheses testing. Ignoring it often results in an unacceptably large spread in the false positive proportion even though the average value is acceptable (Fan et al., J Amer Statist Assoc 107(499): 1019-1035, 2012; Owen J R Stat Soc Ser B 67(3): 411–426, 2005; Qiu et al., Stat Appl Genet Mol Biol 4: 32, 2005 and Schwartzman and Lin Biometrika 98(1): 199–214, 2011). However, the statistical dependence structure of data is often unknown. Using a generic signal-processing model, Bayesian multiple testing, and simulations, we demonstrate that the variance of the false positive proportion can be substantially reduced even under unknown short range dependence. We do this by modeling the data generating process as a stationary ergodic binary signal process embedded in noisy observations. We derive conditional probabilities needed for the Bayesian multiple testing by incorporating nearby observations into a second order Taylor series approximation. Simulations under general conditions are carried out to assess the validity and the variance reduction of the approach. Along the way, we address the problem of sampling a random Markov matrix with specified stationary distribution and lower bounds on the top absolute eigenvalues, which is of interest in its own right.

Numerical approximation of millimeter-wave frequency sharing between cellular systems and fixed service systems

This paper presents numerical analysis and simulation results in order to study the impact of interference between fixed service (FS) systems and 5G cellular networks at 28 GHz, 38 GHz, and 60 GHz millimeter-wave (mmWave) bands. For this study, two different scenarios were considered, i.e., the aggregation of interference from small cells into an FS receiver from (i) base stations (BSs) to their associated user equipment (UE) (downlink); (ii) from UEs to their associated BSs (uplink). The simulation results determined how much interference rejection is required to protect the operation of the FS. This study is essential when using mmWave technologies in cellular networks, to determine whether the newly deployed cellular systems can co-exist with pre-deployed FS systems or not. This paper presents closed-form numerical approximation results with Taylor series approximation along with intensive simulation results. Finally, this paper confirmed that the numerical approximation results were precise, i.e., that here were only marginal differences to the intensive simulation results.

Performance evaluation of Monte Carlo simulation: case study of Monte Carlo approximation versus analytical solution for a chi-squared distribution

The Guide to the Expression of Uncertainty in Measurement (GUM) describes the law of propagation of uncertainty for linear models based on the first-order Taylor series approximation of Y = f (X1, X2, …, XN). However, for non-linear models this framework leads to unreliable results while estimating the combined standard uncertainty of the model output [u(y )]. In such instances, it is possible to implement the method(s) described in Supplement 1 to GUM—Propagation of distributions using a Monte Carlo Method. As such, a numerical solution is essential to overcome the complexity of the analytical approach to derive the probability density functions of the output. In this paper, Monte Carlo simulations are performed with the aim of providing an insight into the analytical transformation of the probability density function (PDF) for Y = X2 where X is normally distributed and a detailed comparison of analytical and Monte Carlo approach results are provided. This paper displays how the used approach enables to find PDF of Y = X2 without the use of special functions. In addition, the singularity of the PDF and the nonsymmetric coverage interval are also discussed.

Fast Generation of Deceptive Jamming Signal Against Space-Borne SAR

Due to advantages, such as its low power consumption and higher concealment, deceptive jamming against the space-borne synthetic aperture radar has received extensive attention during the past decades. To reduce the computational complexity in the deceptive jamming operation, this article proposes a new approach to generate the jamming signal efficiently. Based on the Taylor series expansion and approximation, we derive the recursive formula, which can be used to calculate the current system function of jammer (JSF) from that in the previous pulse repetition interval. After the calculation of the initial JSF, the deceptive jamming signal can be quickly generated through the recursive algorithm. Then the validity and effective region of the algorithm is estimated through the theoretical analysis of the slant distance error. Finally, the simulation results confirm the superior effect of the proposed algorithm. Compared with the segmented modulation algorithm, the most commonly used fast algorithm for deceptive jamming, this algorithm improves the focusing performance slightly, while reducing the computational complexity by at least 6.25%.

Sub-pixel displacement measurement based on Taylor expansion theory

Traditional vision measurement algorithms have many shortcomings, such as complex process, multi-parameter adjustment and the integer pixel accuracy. To solve these problems, a sub-pixel displacement measurement algorithm based on Taylor expansion is proposed. The displacement vector between adjacent frames is modeled in the first step, and then shift equation of the template image is constructed based on assumptions of constant brightness and spatial consistency. After performing a first-order Taylor series approximation, the displacement vector is obtained by least squares estimation. Finally, through iteratively updating the position of template image, displacement is rapidly extracted. In order to verify the accuracy and effectiveness of the Taylor expansion algorithm, the accuracy test of the grating ruler platform in the presence of the target and the vibration measurement experiment on vibration motor in the absence of the target are designed. Experimental results show that, compared with template matching algorithm based on normalized cross correlation, for the experiments mentioned in this paper, the proposed algorithm can control the calculation time for one frame in less than 1 ms, and achieve higher sub-pixel computation accuracy. Compared with the actual set frequency of 10.78 Hz, the measured frequency of the motor is very consistent with a relative error of 1.1%. It also verifies that the Taylor expansion algorithm proposed in this paper is a reliable and effective non-contact measurement method with important practical application value for vibration measurement.

Efficient Implementations of First-Order Steerable Differential Microphone Arrays With Arbitrary Planar Geometry

We present a spatial filtering approach to first-order steerable Differential Microphone Arrays (DMAs) with arbitrary planar geometry. In particular, the design of the spatial filter is based on a recently proposed frequency-domain design methodology that approximates, in a least-square sense, a target beampattern using the Jacobi-Anger expansion involving Bessel functions. Despite the generality of that approach, however, its computational cost turns out to be excessive when working with limited processing resources. The beamforming technique proposed in this manuscript overcomes this issue by exploiting the fact that in DMAs the spacing between sensors is typically smaller than the smallest wavelength of audio signals of interest. This allows us to substitute zero- and first-order Bessel functions with their Taylor series approximation truncated to the first order. Moreover, we show that this approximation allows us to derive an efficient discrete-time-domain implementation of first-order steerable differential beamformers based on arrays with arbitrary geometries.

Fitted difference approach for differential equations with delay and advanced parameters

A difference scheme involving acceptable fitting parameters is suggested for differential equations with delay and advanced terms, the solutions of which show boundary layer behaviour. First, the original problem is reshaped into asymptotically comparable second order singular perturbation problem using Taylor series approximation for the retarded terms. In order to obtain precise solution, fitting parameters are introduced in difference scheme using modified upwind differences for the first order derivatives. Thomas procedure is used to solve the resulting tri-diagonal difference system. The method is tested on numerical examples for various values of the perturbation, delay and advance parameters. Computed maximum absolute errors are tabulated. Numerical experiments are shown in graphs and the effects of small shifts have been studied on the boundary layer region. Also, convergence has been established of the proposed method.

A novel generalized thermoelasticity with higher-order time-derivatives and three-phase lags

PurposeIn this work, a modified thermoelastic model of heat conduction, including higher order of time derivative, is constructed by extending the Roychoudhuri model (TPL) (Choudhuri, 2007). In this new model, Fourier’s law of heat conduction is replaced by using Taylor series expansions, including three different phase lags for the heat flux, the thermal displacement and the temperature gradient. The generalized thermoelasticity models of Lord–Shulman (Lord and Shulman, 1967), Green and Naghdi (1991), dual-phase lag (Tzou, 1996) and three-phase lag (TPL) (Choudhuri, 2007) are obtained as special cases. The paper aims to discuss these issues.Design/methodology/approachThe aim of this work is to establish a new generalized mathematical model of thermoelasticity that includes TPL in the vector of heat flux, and in the thermal displacement and temperature gradients extending TPL model (Li et al., 2019e). In this model, Fourier law of heat conduction is replaced by using Taylor series expansions to a modification of the Fourier law with introducing three different phase lags for the heat flux vector, the temperature gradient, and the thermal displacement gradient and keeping terms up with suitable higher orders.FindingsThe established high-order three-phase-lag heat conduction model reduces to the previous models of thermoelasticity as special cases.Originality/valueIn this paper, a TPL thermoelastic model is developed by extending the Roychoudhuri (Sherief and Raslan, 2017) model (TPL) considering the Taylor series approximation of the equation of heat conduction. This model is an alternative construction to the TPL model. The new model includes high order of TPL in the vector of heat flux, and in the thermal displacement and temperature gradients.

Technical note: Uncertainty in multi-source partitioning using large tracer data sets

Abstract. The availability of large tracer data sets opened up the opportunity to investigate multiple source contributions to a mixture. However, the source contributions may be uncertain and, apart from Bayesian approaches, to date there are only solid methods to estimate such uncertainties for two and three sources. We introduce an alternative uncertainty estimation method for four sources based on multiple tracers as input data. Taylor series approximation is used to solve the set of linear mass balance equations. We illustrate the method to compute individual uncertainties in the calculation of source contributions to a mixture, with an example from hydrology, using a 14-tracer set from water sources and streamflow from a tropical, high-elevation catchment. Moreover, this method has the potential to be generalized to any number of tracers across a range of disciplines.

Temperature and Emissivity Smoothing Separation with Nonlinear Relation of Brightness Temperature and Emissivity for Thermal Infrared Sensors

Aiming at low spectral contrast materials, the Optimized Smoothing for Temperature Emissivity Separation (OSTES) method was developed to improve the Temperature and Emissivity Separation (TES) algorithm based on the linear relationship between brightness temperature and emissivity features, but there was little smoothing improvement for higher spectral contrast materials. In this paper, a new nonlinear-relationship based algorithm is presented, focusing on improving the performance of the OSTES method for materials with middle or high spectral contrast. This novel approach is a two-step procedure. Firstly, by introducing atmospheric impact factor, the nonlinear relationship is mathematically proved using first-order Taylor series approximation. Moreover, it is proven that nonlinear model has stronger universality than linear model. Secondly, a new method named Temperature and Emissivity Separation with Nonlinear Constraint (TESNC) is proposed based on the nonlinear model for smoothing temperature and emissivity retrieval. The key procedure of TESNC is the lowest emissivity smoothing estimation based on nonlinear model and retrieved by minimizing the reconstruction error of the Planck radiance. TESNC was tested on a series of synthetic data with different kinds of natural materials representing several multispectral and hyperspectral infrared sensors. It is shown that, especially for materials with higher spectral contrast, the proposed method is less sensitive to changes in atmospheric conditions and sample temperatures. Furthermore, the standard Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) products in different kind of atmospheric conditions were used for verifying the improvement. TESNC is more accurate and stable with the decrease of emissivity and changes of atmospheric conditions compared with TES, Adjusted Normalized Emissivity Method (ANEM), and OSTES methods.

DETERMINATION OF THE STANDARD RESISTOR TEMPERATURE COEFFICIENTS AND THEIR UNCERTAINTIES

<p>SNSU TK-BSN’s capability in determining the temperature coefficients of a standard resistor has been improved. The temperature coefficient is one of the important parameter in determining the definition of the standard resistor. Currently, the measurement result has been reported together with the measurement uncertainty. The determination itself is based on a numerical approach of Taylor Series Approximation (TSA) instead of based on a fitting to a certain equation. And by this determination, the uncertainty was calculated. The determination was validated by comparing the measurement result committed by SNSU TK-BSN to that of by the manufacturer. The equation for the temperature coefficient follows the parabolic equation with an alpha coefficient of -5.30 x 10-8 Ω/Ω/°C and beta coefficient of -4.70 x 10-8 Ω/Ω/°C<sup>2</sup>, with the respective uncertainties of 2.4 x 10-8 Ω/Ω/°C and 1.6x 10-8 Ω/Ω/°C<sup>2</sup>, respectively. SNSU TK-BSN measurement results in determining the temperature coefficient in agreement with the manufacturer's measurement results show an appropriate value. This correspondence has an equivalent degree of 0.20 for the alpha temperature coefficient and 0.27 for the beta coefficient.</p>

Designing of internal model control proportional, integral, and derivative controller with second‐order filtering for lag‐ and delay‐dominating processes based on suitable dead time approximation

AbstractIn practice, majority of the industrial chemical processes contain considerable dead time and significant process lag. Depending on the value of dead time to time constant ratio, industrial processes can be classified as lag dominating and delay dominating in nature. Here, an attempt has been made to find out an improved dead time estimation technique from the available methodologies like Taylor series and Pade's approximation relations so that a more accurate process model can be obtained and, consequently, an enhanced model‐based internal model control based proportional, integral, and derivative (IMC‐PID) controller may be designed. In addition, to ensure an improved closed loop performance, a second‐order filter is incorporated in IMC‐PID structure where damping term of the filter is calculated from dead time to time constant ratio of the concerned process. Efficacy of the proposed scheme is verified through simulation study on first‐order plus dead time, second‐order plus dead time, higher order, and nonminimum phase processes in comparison with recently reported model‐based PID settings. In addition to simulation study, a real‐time experimental verification is also made on a level control loop under transient and steady state operating phases.

A\xa0duration approach for the measurement of biometric risks in life insurance

Duration concepts are standard methods for measuring interest rate risks of portfolios, liabilities or other cash flows. Macaulay duration, effective duration and key-rate duration are widely used in practice according to different types of yield curves. In this paper, we will present a formulation for a forward rate duration measure by using multi-dimensional Taylor series approximation. It allows to measure the interest rate risk based on arbitrary forward rates. This approach can easily be adopted for biometric risk management, allowing the definition of a biometric duration. The biometric duration will be applied to actuarial present values of premiums and benefits as well as the actuarial reserve to quantify the corresponding biometric risk. It proves to be an easy-to-use tool in the actuarial practise.

REGRESSION-IN-RATIO ESTIMATORS IN THE PRESENCE OF OUTLIERS BASED ON REDESCENDINGM-ESTIMATOR

In this paper, a robust redescending M-estimator is used to construct the regression-inratio estimators to estimate population when data contain outliers. The expression of mean square error of proposed estimators is derived using Taylor series approximation up to order one. Extensive simulation study is conducted for the comparison between the proposed and existing class of ratio estimators. It is revealed form the results that proposed regression-in-ratio estimators have high relative efficiency (R.E) as compared to previously developed estimators. Practical examples are also cited to validate the performance of proposed estimators.

Traveltime approximation for converted waves in elastic orthorhombic media

The moveout approximations are commonly used in seismic data processing such as velocity analysis, modeling, and time migration. The anisotropic effect is very obvious for a converted wave when estimating the physical and processing parameters from the real data. To approximate the traveltime in an elastic orthorhombic (ORT) medium, we defined an explicit rational-form approximation for the traveltime of the converted [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-waves. To obtain the expression of the coefficients, the Taylor-series approximation is applied in the corresponding vertical slowness for three pure-wave modes. By using the effective model parameters for [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-waves, the coefficients in the converted-wave traveltime approximation can be represented by the anisotropy parameters defined in the elastic ORT model. The accuracy in the converted-wave traveltime for three ORT models is illustrated in numerical examples. One can see from the results that, for converted [Formula: see text]- and [Formula: see text]-waves, our rational-form approximation is very accurate regardless of the tested ORT model. For a converted [Formula: see text]-wave, due to the existence of cusps, triplications, and shear singularities, the error is relatively larger compared with PS-waves.

High-Accuracy Multiscale Simulation of Three-Dimensional Squeezing Carbon Nanotube-Based Flow inside a Rotating Stretching Channel

Enhancing the heat transfer rate using nanofluids is of great interest to engineers and scientists. This research aims to study the heat and mass transfer analysis of three-dimensional squeezing carbon nanotube- (CNT-) based nanofluid flow inside a rotating stretching channel. The upper wall of the channel is assumed to have a reciprocating movement, and the lower wall is assumed to be stationary and permeable. Also, radiative effects are taken into account using the Taylor series approximation. The momentum and energy equations are transformed into a coupled system of nonlinear ordinary differential equations utilizing similarity solutions. A new multiscale and accurate method was developed to solve the achieved nonlinear systems of equations. Water is chosen as the base fluid; single-wall carbon nanotubes (SWCNTs) and multiwall carbon nanotubes (MWCNTs) are added to it, and then two types of nanofluids were created. The effect of different variables such as the concentration of nanotubes, nanotube’s type, suction parameter, rotation parameter, squeezing number, Eckert number, and radiation parameter on the velocity and temperature profiles is investigated. Our results reveal that the temperature profile is an increasing function of the squeezing number, suction, rotation, and radiation parameters when the upper wall moves towards the lower one.