Monopulse Estimation of Unknown Polarization for Conformal Phased Array with Arbitrarily Oriented Dipoles

2021 ◽  
Author(s):  
Yuhao Deng ◽  
Julan Xie ◽  
Zishu He ◽  
Jun Li
Keyword(s):  
Least Squares ◽  
Phased Array ◽  
Least Squares Method ◽  
Likelihood Function ◽  
Hessian Matrix ◽  
Mean Square ◽  
Similar Technique ◽  
Dipole Orientation ◽  
Initial Direction ◽  
The Difference

Abstract In this paper, a novel monopulse estimator is proposed to surmount obstacles caused by unknown polarized pattern and the difference among each dipole orientation. Polarized pattern often alters the phased array response and we could hardly recover it if we known nothing about polarized parameters. The sum and difference beamforming of conformal phased array is affected due to the difference among each dipole orientation. Therefore the conventional monopulse estimator is dumped in this circumstance. The method proposed in this paper is a remarkable estimator based on maximum likelihood methodology. In this method, polarized parameters are considered as a part of desired signal and the least-squares solution of desired signal is obtained. With the desired signal solution, the likelihood function with respect to direction is derived at first. Then from the above, Jacobian and Hessian matrix of likelihood function is deduced. The boresight is considered as the initial direction value and the estimator of desired signal direction is obtained by Newton's formula. Finally, the polarized parameters are estimated by least-squares method using the direction estimator. The root-mean-square error (RMSE) of angle estimation is acceptable when prior polarized information is completely unknown. Polarized parameters are estimated by similar technique after we find out azimuth and elevation. Our research fills a gap of monopulse estimation with arbitrary polarization and diverse dipole arrangement.

scholarly journals Comparing Parameter Estimation of Random Coefficient Autoregressive Model by Frequentist Method

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 62 ◽  
Author(s):  
Autcha Araveeporn
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Mean Square Error ◽  
Maximum Likelihood Method ◽  
Least Squares Method ◽  
Likelihood Function ◽  
Likelihood Method ◽  
Mean Square ◽  
Average Mean Square Error ◽  
Frequentist Method

This paper compares the frequentist method that consisted of the least-squares method and the maximum likelihood method for estimating an unknown parameter on the Random Coefficient Autoregressive (RCA) model. The frequentist methods depend on the likelihood function that draws a conclusion from observed data by emphasizing the frequency or proportion of the data namely least squares and maximum likelihood methods. The method of least squares is often used to estimate the parameter of the frequentist method. The minimum of the sum of squared residuals is found by setting the gradient to zero. The maximum likelihood method carries out the observed data to estimate the parameter of a probability distribution by maximizing a likelihood function under the statistical model, while this estimator is obtained by a differential parameter of the likelihood function. The efficiency of two methods is considered by average mean square error for simulation data, and mean square error for actual data. For simulation data, the data are generated at only the first-order models of the RCA model. The results have shown that the least-squares method performs better than the maximum likelihood. The average mean square error of the least-squares method shows the minimum values in all cases that indicated their performance. Finally, these methods are applied to the actual data. The series of monthly averages of the Stock Exchange of Thailand (SET) index and daily volume of the exchange rate of Baht/Dollar are considered to estimate and forecast based on the RCA model. The result shows that the least-squares method outperforms the maximum likelihood method.

scholarly journals On the Geometric Mean Method for Incomplete Pairwise Comparisons

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1873
Author(s):  
Konrad Kułakowski
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Geometric Mean ◽  
Computational Method ◽  
Pairwise Comparisons ◽  
Similar Relationship ◽  
Final Solution ◽  
The Difference ◽  
The Relationship ◽  
Theoretical Considerations

One of the most popular methods of calculating priorities based on the pairwise comparisons matrices (PCM) is the geometric mean method (GMM). It is equivalent to the logarithmic least squares method (LLSM), so some use both names interchangeably, treating it as the same approach. The main difference, however, is in the way the calculations are done. It turns out, however, that a similar relationship holds for incomplete matrices. Based on Harker’s method for the incomplete PCM, and using the same substitution for the missing entries, it is possible to construct the geometric mean solution for the incomplete PCM, which is fully compatible with the existing LLSM for the incomplete PCM. Again, both approaches lead to the same results, but the difference is how the final solution is computed. The aim of this work is to present in a concise form, the computational method behind the geometric mean method (GMM) for an incomplete PCM. The computational method is presented to emphasize the relationship between the original GMM and the proposed solution. Hence, everyone who knows the GMM for a complete PCM should easily understand its proposed extension. Theoretical considerations are accompanied by a numerical example, allowing the reader to follow the calculations step by step.

scholarly journals Comparison of Relaxation Modulus Converted from Frequency- and Time-Dependent Viscoelastic Functions through Numerical Methods

2018 ◽  
Vol 8 (12) ◽  
pp. 2447 ◽  
Author(s):  
Weiguang Zhang ◽  
Bingyan Cui ◽  
Xingyu Gu ◽  
Qiao Dong
Keyword(s):  
Least Squares ◽  
Collocation Method ◽  
Creep Compliance ◽  
Least Squares Method ◽  
Relaxation Modulus ◽  
Prony Series ◽  
Significant Difference ◽  
Time Region ◽  
The Difference ◽  
Reduced Time

Due to the difficulty of obtaining relaxation modulus directly from experiments, many interconversion methods from other viscoelastic functions to relaxation modulus were developed in previous years. The objectives of this paper were to analyze the difference of relaxation modulus converted from dynamic modulus and creep compliance and explore its potential causes. The selected methods were the numerical interconversions based on Prony series representation. For the dynamic to relaxation conversion, the time spectrum was determined by the collocation method. Meanwhile, for the creep to relaxation conversion, both the collocation method and least squares method were adopted to perform the Laplace transform. The results show that these two methods do not present a significant difference in estimating relaxation modulus. Their difference mostly exists in the transient reduced time region. Calculating the average of two methods is suggested to avoid great deviation of single experiment. To predict viscoelastic responses from creep compliance, the collocation method yields comparable results to the least squares method. Thus, simply-calculated collocation method may be preferable in practice. Further, the master curve pattern is sensitive to the Prony series coefficients. The difference in transient reduced time region may be attributed to the indeterminate Prony series coefficients.

scholarly journals IN INTERPOLATING THE VALUES OF FUNCTIONS SPECIFIED BY THE TABLE EFFECTIVENESS OF THE LEAST SQUARES METHOD

2020 ◽  
pp. 68-71
Author(s):  
М.Е. Eskaliyev ◽  
◽  
А.А. Masimgazieva ◽  
N.A. Nurgali ◽  
◽  
...  
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Main Idea ◽  
Gram Matrix ◽  
Method Of Least Squares ◽  
Newton Interpolation ◽  
Wide Range ◽  
Mathematical Formulas ◽  
The Difference ◽  
Interpolation Polynomials

The article provides a general algorithm of the method of least squares (OLS) for the compilation of a program account, taking into account the features of the Gram matrix. In fact, the difference and advantage of the OLS from the long-known Langrange and Newton interpolation polynomials is emphasized. Using the general MNC algorithm, the square version of the MNC is considered. The characteristic of the full algorithm of the square version is given using special selected mathematical formulas. The scope of OLS in household and practical computational problems is indicated. The application of the least squares method has a wide range, especially for geographical forecasts, hydrometeorological control, and dosage of geological resources. Therefore, for applied calculations, DVI is used for more accurate calculation of approximate values of functions that are suitable in some values and are presented in the form of tables. Its main idea is to create a function and correct deviations caused by errors made during measurement.

Densities determined from surface and subsurface gravity measurements

Geophysics ◽  
1981 ◽  
Vol 46 (11) ◽  
pp. 1568-1571 ◽  
Author(s):  
B. A. Sissons
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Average Density ◽  
Laboratory Measurements ◽  
Density Estimates ◽  
Direct Inversion ◽  
Gravity Measurements ◽  
The Difference ◽  
Made In

A least‐squares method for the direct inversion of surface and subsurface gravity measurements to obtain in situ density estimates is presented. The method is applied to a set of measurements made in a tunnel through the flank of an andesitic volcano. Densities obtained are [Formula: see text] for material in the top 100 m increasing to [Formula: see text] at about 200 m depth. The average density for rocks penetrated by the tunnel is, from laboratory measurements, [Formula: see text] i.e., about 4 percent higher. The difference is ascribed to joints and voids present in situ and not sampled in the laboratory specimens.

‘Least squares’ method—theorem or law?

1980 ◽  
Vol 59 (9) ◽  
pp. 8
Author(s):  
D.E. Turnbull
Keyword(s):  
Least Squares ◽  
Least Squares Method

An Augmented Iteratively Re-weighted and Refined Least Squares Method for lp Minimization Problem in Inverse Modeling of Potential Field Magnetic Data

2020 ◽  
Vol 1 (3) ◽  
Author(s):  
Maysam Abedi
Keyword(s):  
Magnetic Field ◽  
Magnetic Susceptibility ◽  
Least Squares ◽  
Minimization Problem ◽  
Potential Field ◽  
Field Data ◽  
Least Squares Method ◽  
Porphyry Copper ◽  
Magnetic Data ◽  
Magnetic Field Data

The presented work examines application of an Augmented Iteratively Re-weighted and Refined Least Squares method (AIRRLS) to construct a 3D magnetic susceptibility property from potential field magnetic anomalies. This algorithm replaces an lp minimization problem by a sequence of weighted linear systems in which the retrieved magnetic susceptibility model is successively converged to an optimum solution, while the regularization parameter is the stopping iteration numbers. To avoid the natural tendency of causative magnetic sources to concentrate at shallow depth, a prior depth weighting function is incorporated in the original formulation of the objective function. The speed of lp minimization problem is increased by inserting a pre-conditioner conjugate gradient method (PCCG) to solve the central system of equation in cases of large scale magnetic field data. It is assumed that there is no remanent magnetization since this study focuses on inversion of a geological structure with low magnetic susceptibility property. The method is applied on a multi-source noise-corrupted synthetic magnetic field data to demonstrate its suitability for 3D inversion, and then is applied to a real data pertaining to a geologically plausible porphyry copper unit.  The real case study located in  Semnan province of  Iran  consists  of  an arc-shaped  porphyry  andesite  covered  by  sedimentary  units  which  may  have  potential  of  mineral  occurrences, especially  porphyry copper. It is demonstrated that such structure extends down at depth, and consequently exploratory drilling is highly recommended for acquiring more pieces of information about its potential for ore-bearing mineralization.

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