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The generation process of difference beam is analyzed from the perspective of beam superposition. By analyzing the continuous linear source, a differential beam synthesis method, which can control the sidelobes level, is proposed with a simple form. The sidelobes level of the difference pattern is determined by controlling the direction of the subbeam and applying the traditional synthesis method, such as Chebyshev method and Taylor method. The proposed synthesis method is theoretical and explicit. Examples are presented for assessing the capabilities of the synthesis of difference patterns. The general distribution function is given.

The techniques on the generation of multiple solutions in shaped-beam pattern synthesis are standardly focused on the use of patterns with complex nature as input. Otherwise, in order to derive a symmetric pure real distribution from the canonical pattern synthesis techniques, a generation of a pure-real pattern has to be imposed. In the present work, the exploitation of the multiplicity of the shaped pattern generated by this symmetric pure real distribution is proposed, without constraining the solutions to necessarily meet the pure-real pattern requirement. Therefore, an increase on the degrees of freedom is produced and a greater number of continuous distributions (presenting different natures) is achieved, by omitting the restrictions found in the state-of-the-art methodologies. Thus, a general multiplicity of solutions can be reached and the design protocol can increase its number of alternatives for facing different feeding network structures. In such a way, this article is devoted to illustrate the improvements in terms of number of feasible solutions reached by the general method, including alternative symmetric pure real distributions as input within the procedure. In this manner, two different approaches, constraining the pattern to present the same number of ripples or a similar main beam width, are discussed. Examples of both Taylor distributions linear and circular are illustrated.

An iterative reweighted approach based on convex optimization for the array pattern synthesis is presented. The algorithm consists in solving the problem of auto-specifying the mainlobe width. In the optimization procedure, a slack variable is introduced in the region near the mainlobe region of a pattern. The value of the slack variable will change as well as the width of the region will be narrower based on iterations. The result is that the width of the region is equal to the width of mainlobe region. Representative simulation is provided to demonstrate the effectiveness of the proposed method in beampattern synthesis.

Mobile communications and spatial detection require antennas with high directive radiation pattern. The main aim of this paper is to achieve directive by reducing side lobes for linear antenna array. As the derivative based optimization is prone to level maxima, invasive weed optimization has been with tuning of vital parameters (d, noEl, noff). Being derivative free IWO with parameters tuning has achieved maximum reduction of side lobes from −13.2 db to −19.8 db. The result is encouraging slowly supervision over standard IWO. Another further is possible by Chao theory.

This paper presents an approach to pattern synthesis based on the difference coarray of minimum hole arrays (MHAs), based on the relations between the weights on the physical array and the virtual weights on the difference coarray. From these relations, some virtual weights are selected as design variables. The remaining virtual weights and the weights on the physical array are then determined by these design variables. Numerical examples show that, if these design variables are samples of a Kaiser window, the proposed approach can achieve smaller beamwidth or lower sidelobe levels than some existing weight designs for the MHA.

A hybrid method that combines the iterative Fourier transform (IFT) and the algorithm of differential evolution (DE) is proposed to address the synthesis of low sidelobe sparse linear array (SLA) including many elements. Firstly, a thinned linear array (TLA) with the lattices spaced at half wavelength is obtained by the IFT. Then, for the elements of the TLA whose left or right spacing are greater than half wavelength, their placements are selected as the candidates which will be further optimized by the DE, as long as the interelement spacing is not less than half wavelength. Consequently, the convergence pressure of DE is greatly alleviated for the reason that the selected elements only accounts for a small part of the total. Therefore, the SLA with improved sidelobe performance can be obtained at relative low hardware cost. Several numerical instances confirmed the effectiveness of the proposed method.

In this paper, the problem of pattern synthesis with antenna arrays is addressed, and new approaches based on the recently developed accurate array response control ( $ {\textrm {A}}^{2}\textrm {RC} $ ) algorithm are presented. It is shown that the array weight vector obtained by the $ {\textrm {A}}^{2}\textrm {RC} $ algorithm to control the normalized response at a single direction in each step belongs to a specific set. Thus, an appropriate weight vector chosen from the intersection of weight vector sets corresponding to the desired responses at multiple directions is capable of simultaneously controlling those responses. This results in the so-called multipoint accurate array response control ( $ {\textrm {MA}}^{2}\textrm {RC} $ ) algorithm. Moreover, in order to avoid possible beam axis shift in pattern synthesis, a modified $ {\textrm {MA}}^{2}\textrm {RC} $ ( $ {\textrm {M}}^{2}{\textrm {A}}^{2}\textrm {RC} $ ) algorithm is proposed by imposing a derivative constraint on the direction of beam axis. Representative numerical examples are provided to demonstrate the effectiveness of the proposed $ {\textrm {MA}}^{2}\textrm {RC} $ and $ {\textrm {M}}^{2}{\textrm {A}}^{2}\textrm {RC} $ algorithms for multipoint responses control and pattern synthesis.

Numerical pattern synthesis (NPS) algorithm based on adaptive array theory is an effective spatial power pattern synthesis method for antenna arrays. In this letter, instead of using the artificial interferences weighted on the total power pattern in the conventional NPS algorithm, a generalized two-dimensional (2-D) NPS algorithm is proposed in which two weighting functions are assigned to control the copolarization and cross-polarization patterns, respectively. The weighting functions are iteratively updated until both the sidelobe and cross-polarization levels reach their prescribed ones. Two numerical examples for synthesizing the patterns of a linearly polarized planar array and a circularly polarized cylindrical array are provided to validate effectiveness and advantages of the proposed generalized 2-D NPS algorithm.

In this paper we propose a new and fast hybrid technique for pattern synthesis and peak side lobe (PSL) control of thinned array antennas. This technique is based on the using difference sets (DS) to thin an array and convex programming (CP) to synthesize the desired pattern. Based on this technique two arrays with sum and difference patterns and low PSL are synthesized. The simulation results show that the PSL of synthesized sum and difference patterns are equal to −17.3 dB and −16.1 dB, respectively. The presented technique in this paper is a fast synthesis method which leads to an appropriate pattern for thinned arrays.

Method of least squares is employed for synthesis of non-uniform spherical arrays using Euler rotation of coordinate axes for reduction of side-lobe levels, controllability of 3-db beam-width of main-lobe etc. It is useful for synthesis of conformal arrays, where pattern multiplication is not applicable and array synthesis is involved. This method for synthesis of spherical arrays is general that determines radius of sphere, longitude and latitude angles of elements and their current excitations (amplitudes and phases). The authors investigate several examples to demonstrate its capability. Studies on conformal arrays deal with non-uniform excitation currents, leading to complicated input networks. Therefore the authors suggest non-uniform spacing, which shows that array synthesis by variable element spacing will realise any specifications, same as non-uniform excitations. An error function is constructed for pattern synthesis, which is a function of element spacing and excitations. The authors use combination of global and local minimisation algorithms. In examples of spherical array synthesis, 2.5 and 5.5 dB reduction of side-lobe levels have been achieved in the E- and H-plane patterns, respectively, relative to the available design data. In addition, reduction of beam-width of main-beams in E- and H-planes has been obtained by 39% and 34%, respectively.

This paper proposes a two-step array pattern synthesis approach for phase-only array antennas. The proposed algorithm estimates the phase-only weights which minimize the error between the estimated array pattern and a desired (synthesized) array pattern by using a least square error method. The estimated weight vector is useful as the initial point for other iterative optimization methods. Simulation results are presented to illustrate the proposed algorithm using a planar array antenna.

A new method for the synthesis of linear and planar arrays having prescribed beamwidth and sidelobe levels and a minimum number of elements is proposed. In the method, the number of elements in an array is minimized while constraining the amplitude-response error in the mainlobe region, the attenuation in the sidelobe region, and the array dimensions. An iterative constrained optimization method is used where the amplitude-response error is linearly approximated at each iteration while concurrently minimizing a re-weighted L1 norm of the array coefficients. To ensure robustness of the array, we constrain a sensitivity parameter, namely, the white noise gain, to be above a prescribed level. Furthermore, the method also provides the additional flexibility of controlling the array dimensions, symmetry properties, and element positions of the array. Two variants have been developed: In the first variant, both the array coefficients and the positions of the elements are optimized; in the second variant, only the array coefficients are optimized while the elements are fixed at predefined positions. Experimental comparisons with several state-of-the-art competing methods show that the proposed method provides greater flexibility of controlling the robustness, beampattern response error, array dimensions, and element positions while at the same time the number of elements is less than or equal to that of the competing methods.

Synthesis of sum and difference patterns of a planar array is introduced in this article by means of a clustering algorithm. The whole array aperture is divided into several sub-arrays, which will reduce the cost and complexity of manufacturing the array antenna. The sub-array configurations and the corresponding sub-array weights are defined to minimize the maximum level of the side-lobes of the corresponding sum and difference patterns. Two kinds of arrays are considered here: in the first, the whole aperture is divided into several sub-arrays to get sum and difference patterns; in the other, the compromise difference pattern is obtained. Some numerical experiments are given to assess the feasibility and effectiveness of the proposed method.

The design of monopulse radar systems requires the synthesis of both a sum pattern and a difference pattern, which is quite flexible and hard to be optimized for subarrayed antennas. In this study, innovative approaches to synthesize the optimal sum and difference patterns independently for subarrayed monopulse arrays are presented. By exploiting the relationship of the subarray partition and optimal excitations, the synthesis problem is formulated as the clustering of a set induced by the optimal excitations. Thus, the optimal subarray partition is transformed to a clustering problem. A clustering method is presented to solve the array synthesis with uniform excitation at element level. Furthermore, an improved approach is proposed to synthesize low sidelobe patterns. When compared with several classical approaches, various numerical experiments validate the effectiveness and potentiality of the proposed methods.

Adaptive beamformers will degrade in the presence of model mismatch. Because a wider beamwidth has higher ability against steering vector errors, and lower sidelobe levels can improve the robustness against fast moving interferences, in this work an iterative fast Fourier transform (FFT) based adaptive beamformer is proposed with constraints on beamwidth and peak sidelobe level. The adaptive beamforming is transformed to a weighted pattern synthesis problem. This weighted pattern is a product of the array pattern and a weighting function. Because the weighting function has shape peaks at the direction of interferences, it will have nulls in the array pattern at the directions of interferences by reducing the peak sidelobe level of this weighted pattern. A modified iterative FFT algorithm is proposed to synthesize this weighted pattern. Thanks to the efficiency of FFT, the nonconvex problem of power pattern synthesis can be solved efficiently. This method is demonstrated through several simulation examples. The results show the advantages of the proposed method in obtaining high output SINRs against moving target signals and steering vector errors.

This paper presents a new array pattern synthesis algorithm by reweighted l1-norm minimization and convex optimization. A virtual uniform array with much smaller element spacing and a weight matrix are created first; then, we obtain a weight vector by solving a weighted l1-norm minimization convex problem, and update the weight matrix by the weight vector. Some entries of the weight vector are so small and thus negligible without significantly changing the array pattern performance. Based on this notion, the pruned weight vector becomes a sparse weight vector, and a non-uniform array is formed according to the nonzero valued positions of the sparse weight vector. Determining the weight vector, updating the weight matrix and creating a non-uniform array are repeated until the final array synthesis performance is satisfactory. After optimizing the array geometric construction, we obtain the optimal weight vector of the non-uniform array by using convex optimization. One numerical example is presented to show the high efficiency of achieving the desired radiation pattern with the minimum number of antenna elements.

PSO algorithm is intensively applied in array pattern synthesis. While the choice of inertia factor in classical PSO algorithm is a cumbersome task, improper inertia factor will result in slow convergence or falling into the local minima. This paper presents an improved algorithm based on LDW-PSO to avoid converging to the local minima. Simulation results show that the algorithm has faster speed and higher robustness compared with LDW-PSO algorithm.

Reducing the computational cost of large array pattern synthesis is attractive in many applications. In this paper, a fast pencil beam pattern synthesis method for large nonuniform antenna arrays is proposed. This method is based on an interpolation in a least square sense and iterative fast Fourier transform (FFT), i.e., interpolate the nonuniform distributed elements into uniform virtual elements, and then apply FFT to synthesis the uniform array. Thanks to the efficiency of FFT, the proposed method is much faster and can handle much larger arrays than the existing methods. To guarantee the interpolation accuracy, the choice of the interpolation parameters is discussed. Both linear and planar nonuniform array examples are shown to validate the advantages of the proposed method.

Cebirsel düşünme üzerine son yıllarda yapılan araştırmaların çoğunda cebirsel düşünme ve işlem becerilerinin yetersiz olduğu görülmüştür. Bu durum ise cebirsel düşünme becerisi üzerine araştırma yapmaya yöneltmiştir. Araştırmanın amacı; cebirsel düşünme becerisi ile ilgili yapılan çalışmaları, meta-sentez yöntemi ile cebirsel düşünmede ön koşul beceriler ve kritik süreç açısından incelemektir. Bu bağlamda, Yüksek Öğretim Kurulu Ulusal Tez Merkezi resmi sitesinin veri tabanından `cebir` ve `cebirsel düşünme` anahtar kelimeleri kullanılarak yapılan tarama sonucu yayınlandığı 2005-2016 yılları arasında cebir ile ilgili erişime açık 472 yüksek lisans tezi ile erişime açık 160 doktora tezine, Eğitim Araştırma ve Bilgi Merkezi olan ERIC veri tabanından `algebraic thinking` anahtar kelimesi aranarak erişime açık olan 59 teze, `algebra` kelimesi aranarak ise erişime açık 958 teze, Ulusal Akademik Ağ ve Bilgi Merkezi olan ULAKBİM resmi sitesinin veri tabanından `cebirsel düşünme` anahtar kelimesi kullanılarak 55 teze ulaşılmıştır. Ulaşılan tezlerden dahil edilme kriterlerine uygun olan 23 çalışma incelenmiş ve sonuçlar araştırmacı önerileriyle desteklenerek sunulmuştur. İncelenen çalışmaların büyük kısmında cebirsel düşünmede, ön koşul beceriler için örüntü genellemelerinin ve kritik süreç olarak da 4-12 yaş aralığının belirtildiği görülmüştür. Bu nedenle, cebirsel düşünme becerisinin gelişimi için okul öncesinden itibaren cebirsel düşünmenin temeli olan örüntü genelleme etkinlikleriyle cebire giriş yapılması yararlı olacaktır. Ayrıca okul öncesi ve ilkokul müfredatlarında örüntülere kavramsal ve işlemsel anlamayı geliştirecek şekilde kapsamlı olarak yer verilmelidir. Algebraic thinking skills and algebraic achievements seem to be inadequate in the most of the researches in recent years. This leds to the research into algebraic thinking skills. Purpose of research, examine algebraic thinking in terms of prerequisite skills and critical process. In this study, meta-synthesis method was used. In this context, it has been reached published 472 masters dissertation and 160 doctoral dissertation related to algebra which published between 2005 and 2016 by using keywords `algebra` and `algebraic thinking` from the database of Council of Higher Education National Science Center. By searching key word `algebraic thinking` from database ERIC reached 59 dissertation and by searching key word `algebra` reached 958 dissertation. Using key word `algebraic thinking` from database ULAKBİM reached 55 dissertation. 23 research which suitable for inclusion criteria in study are examined and presented with support of researcher. In the large part of studies examined, it has been found that both pattern generalizations for prerequisite skills and critical period is between 4-12 years for algebraic thinking. To improve algebraic thinking skills, it will be useful to begin algebra with pattern activities from beginning of the pre-school because pattern activities have important role in development of algebraic thinking. Therefore, patterns should be comprehensively included in pre-school and primary school curriculum to develop both conceptual and procedural meaning. 91

For the flat-top array beampattern synthesis, a conceptually simple closed-form method without iterative work and heavy numerical calculations is desired. We are therefore motivated to look for a good template function for flat-top beampattern synthesis, such as Dolph-Chebyshev Function (DCF) for pencil-beam pattern synthesis. In this paper, two new template functions, Exponential Dolph-Chebyshev Function (EDCF) and Generalized Dolph-Chebyshev Function (GDCF) are derived from the original DCF, and the corresponding synthesis procedures in the Quadratic Programming (QP) method are also presented. From the simulation results using the proposed templates, it is shown that although the resultant beampatterns are not equi-ripple, both the mainbeam width and the Side-Lobes Level (SLL) of the synthesized beampatterns are controlled well, simultaneously, and the proposed methods can also be applied to synthesize arrays of different geometries including Uniform Linear Arrays (ULA) and Uniform Circular Arrays (UCA). The comparison between the two proposed template functions is also presented so that one can choose the proper synthesis template based on specified requirements.

In this paper, we propose a multi-beam model for antenna array pattern synthesis (AAPS) problem. The model uses a conic trust region algorithm (CTRA) similarly proposed in this paper to optimize its cost function. Undoubtedly, whole algorithm e-ciency ultimately lies on the CTRA, thereof, we propose a method to improve the iterative algorithm's e-ciency. Unlike traditional trust region methods that resolve sub-problems, the CTRA e-ciently searches a region via solving a inequation, by which it identifles new iteration points when a trial step is rejected. Thus, the proposed algorithm improves computational e-ciency. Moreover, the CTRA has strong convergence properties with the local superlinear and quadratic convergence rate under mild conditions, and exhibits high e-ciency and robustness. Finally, we apply the combinative algorithm to AAPS. Numerical results show that the method is highly robust, and computer simulations indicate that the algorithm excellently performs AAPS problem.

An improved nonlinear least-square method is presented in this paper. This method changes the traditional least-square method's shortness of being sensitive to its initial conditions. Pattern synthesis for concentric circular arrays using nonlinear least-square method is introduced. The excitation amplitudes and phases of the array elements are optimized. This method can make the design of the feeding network much easier because the excitation amplitudes of the elements placed on the same ring are equal. The number of parameters to be optimized is reduced which leads to a faster simulation speed and makes the simulation results much more accurate. Also, the cost of designing the feeding network is reduced. The simulation results show the good agreement between the synthesized and desired radiation pattern. Also, the peak side lobe level (PSLL) of the synthesized radiation pattern is quite low.

In this letter, a modified differential evolution (DE) based on harmony search algorithm is developed for the linear aperiodic array synthesis. By adding the parameter selection strategy into the classic differential evolution, the modified DE combines the good local search capability of the classic DE and the great search diversity of harmony search algorithm. The pattern optimization simulations of the linear aperiodic array verify that it converges faster and requires less computation.

A new strategy based on the multiple measurement vectors FOCUSS (M-FOCUSS) is presented for the synthesis of reconfigurable sparse arrays radiating multiple patterns. In this method, the array synthesis problem is cast into a framework of sparse reconstruction with multiple measurement vectors (MMV). A proper sparseness constraint is placed on the solution vectors to ensure not only the sparseness of the array but also the uniqueness of element position distribution for different patterns. Numerical results show the high precision and efficiency of this method to achieve multiple desired radiation patterns with sparse nonuniform linear array.

An efficient approach for array synthesis including the effect of mutual coupling is presented. The method is based on an iterative convex optimization scheme and on a fast full-wave simulation technique. The embedded element patterns (EEP), obtained by full-wave simulation at each iteration of the convex optimization, are exploited to take into account the mutual coupling. The final results, therefore, not only satisfy the pre-defined constraints but also guarantee the performance of the array. Numerical results of linear focused beam arrays are presented to show the effectiveness of the approach.

A novel algorithm on beam pattern synthesis for linear aperiodic arrays with arbitrary geometrical configuration is presented in this paper. Linear aperiodic arrays are attractive for their advantages on higher spatial resolution and lower sidelobe. However, the advantages are attained at the cost of solving a complex non-linear optimization problem. In this paper, we explain the Improved Genetic Algorithm (IGA) that simultaneously adjusts the weight coefficients and inter-sensor spacings of a linear aperiodic array in more details and extend the investigations to include the effects of mutual coupling and the sensitivity of the Peak Sidelobe Level (PSL) to steering angles. Numerical results show that the PSL of the synthesized beam pattern has been successfully lowered with the IGA when compared with other techniques published in the literature. In addition, the computational cost of our algorithm can be as low as 10% of that of a recently reported genetic algorithm based synthesis method. The excellent performance of IGA makes it a promising optimization algorithm where expensive cost functions are involved.

This letter proposes an optimal difference pattern synthesis with polarization control ability for arbitrary arrays. The joint synthesis of the polarization and spatial power patterns are formulated as a convex optimization form that can be efficiently solved by existing numerical methods. The proposed synthesis method, which has no limitation on the array geometry and element pattern, is flexible and applicable to arbitrary arrays. Two typical examples, the linear arrays for circular polarization synthesis and the cylindrical arrays for elliptical polarization synthesis, are presented to testify the potentiality of the proposed method.

The design of nonuniformly spaced linear array antennas using Particle Swarm Optimization method is considered. The purpose is to match a desired radiation pattern and improve the performance of these arrays in terms of sidelobe level. This performance criterion determines how well the system is suitable for wireless communication applications and interference reduction. Two approaches are considered: in the first, the design of element placement with the constraint of array length being imposed is performed. The second is based on element position perturbation starting from a uniform element distribution. Many examples are treated to show the effectiveness of the designs and the effect some other parameters might have on the overall performance of the array.

A numerically-efficient technique based on the Bayesian compressive sampling (BCS) for the design of maximally-sparse linear arrays is introduced. The method is based on a probabilistic formulation of the array synthesis and it exploits a fast relevance vector machine (RVM) for the problem solution. The proposed approach allows the design of linear arrangements fitting desired power patterns with a reduced number of non-uniformly spaced active elements. The numerical validation assesses the effectiveness and computational efficiency of the proposed approach as a suitable complement to existing state-of-the-art techniques for the design of sparse arrays.

A method of designing a cylindrical conformal array with shaped-beam and reconflgurable dual-beam using a modifled particle swarm optimization algorithm is proposed in this paper. The proposed algorithm is easy to implement and e-cient to be used in synthesizing conformal arrays with digital attenuators and digital phase shifters. Moreover, the proposed synthesis has taken the actual active element patterns into account, which can reduce the error between computation and realization. Good agreement can be obtained between the desired patterns and the synthesized patterns.

title/>Patterns consisting of point symbols are commonly applied to symbolize land cover. Tree symbols, for example, indicate forested areas; small irregular dots show quarries and pits; or regular line symbols represent a vineyard. This paper presents an automatic method for synthesizing patterns consisting of one or more point symbols. Symbols of varying sizes or graphical appearance can be combined according to user-defined ratios. The automatic method generates patterns of regularly or irregularly distributed symbols. It ensures that symbols do not overlap and do not graphically conflict with other map features. The method has been implemented in a free plug-in for the Adobe Illustrator vector graphics editor.

This paper proposes a design method of array pattern optimization for sidelobe level (SLL) reduction. The element positions and their weights are determined by electromagnetism-like (EM)-like algorithm. Simulation results show that the EM-like algorithm is efficient in SLL reduction for beamwidth restriction. Synthesis of array pattern using EM-like algorithm is very straightforward and efficient. It does not require any gradient operation and is very easy in both formulation and programming. Therefore, the EM-like algorithm can be applied to many other applications in array signal processing.

In the sum and difference pattern synthesis problem of the equi-spaced monopulse tracking linear array antennas, an efficient numerical approach to deriving the relative excitation current weights of antenna elements is presented for the desired patterns. This method is based on the optimum perturbation of null points which are inherent to the Schelkunoffs polynomial representing the pattern array factor. Accordingly, opposite to the conventional method in which the excitation weights are directly optimized, this method is advantageous in that the patterns with the desired individual sidelobe levels(SLLs) and the corresponding excitation weights are easily synthesized by the control of null points. Furthermore, it is showed that two types of difference patterns can be synthesized as imposing the different initial values of null points in the optimization process. The proposed method is numerically validated by synthesizing the patterns with the arbitrary SLLs and substituting the extracted results into the array factor equation.

Desired far-field radiation patterns of 5 × 11 conformal antenna array are synthesized using a hybrid genetic algorithm (HGA), which combines the simplified quadratic interpolation (SQI) method and the real-coded genetic algorithm (RCGA). This hybrid genetic algorithm is shown to outperform standard genetic algorithm (GA) when used to synthesize amplitude weights of the elements to satisfy specified deep notches, nulls and average sidelobe level constraints. The HGA procedure appears to be a high effective means to compensate the mutual coupling effects on the individual element patterns for the conformal antenna array.

In order to overcome drawbacks of standard particle swarm optimization (PSO) algorithm, such as prematurity and easily trapping in local optimum, a modified PSO algorithm which adopts a global best perturbation, is used to optimize the pattern of cylindrical conformal antenna array for sidelobe level (SLL) suppression and null control in certain directions.The convergence speed and accuracy of the algorithm are improved.Compared with genetic algorithm and simulated annealing, The PSO algorithm is much easier to understand and implement.Firstlypattern formula of conformal array is provided, then, the standard and modified PSO algorithm are introduced, at last, application examples and simulation results are presented.The results show that the Modified PSO algorithm is an effective and efficient method to solve multi-dimension and nonlinear problem.

Simple and efficient procedures are presented to synthesize antenna array patterns (focused or shaped beams) while controlling the excitations. This synthesis problem is of great practical interest as it makes active arrays more attractive by reducing their complexity, cost, and weight. With a proper regularization of the array pattern synthesis solution, the excitations can be optimized in order to design uniform amplitude sparse arrays, arrays whose excitations have a low dynamic range ratio or a smooth amplitude variation. More specifically, the synthesis procedures are composed of the minimization of a norm (mixed $\ell _{1}/\ell _{\infty }$ norm, total variation norm, or a combination of the $\ell _{1}$ - and $\ell _{\infty }$ -norms) associated with radiation pattern constraints. Various representative array synthesis problems are considered. Examples including the coupling between antennas and a comparison with a deterministic approach confirm both the usefulness and effectiveness of the proposed procedures.

Given an antenna array of N elements, we deal with the problem of finding an array pattern that approximates a desired pattern, while: a) null constraints are imposed in M(les N - 2) assigned directions; b) an upper bound D is imposed on the dynamic range ratio (DRR) of the excitations. At first we show that the system of constraint equations has solutions if and only if D ges D0, where D0 is a parameter depending on the null directions, and describe a method to evaluate D0. Then we propose two simple and fast algorithms to solve the synthesis problem and show their effectiveness by numerical examples. Statistical results show that usually D0 = 1 when M is small compared to N, thus exact nulls can be formed for any D ges 1. This result provides an insight into the numerical results.

This correspondence proposes a method for array pattern synthesis of a concentric ring array that yields invariant array pattern over a certain frequency band for beamforming in three dimensions. The proposed method uses the Fourier-Bessel series to determine the weights from different rings to form the overall array pattern. Simulation examples are given to illustrate the proposed method.

This letter presents a new array pattern synthesis algorithm using the two-step least-squares method. The algorithm utilizes the randomness of the desired pattern phase to design an array pattern. Several numerical examples are presented to illustrate this algorithm.

We present a new technique to solve array pattern synthesis problems by using semidefinite programming. We first formulate (or reformulate) the array design problems into semidefinite programming problems, and then use the recently developed efficient numerical algorithms and software to compute the numerical solution of antenna array weights. Using this approach, we can directly solve not only the standard synthesis problems for nonuniform arrays, but also the synthesis problems for arrays having power restrictions and uncertainties. Numerical examples are presented to illustrate our approach.

In this paper, the classic PSO algorithm is applied to the pattern synthesis of antenna array using 4-bit digital phase shifters. In order to directly apply the existed PSO algorithms to the pattern synthesis using digital phase shifters, this paper proposes a simple truncation rounded number strategy to transfer the discrete problem to the continuous one, which not only produces no quantization error, but also avoids the limitation of the discrete PSO algorithm. The optimized results are better than that of the discrete PSO algorithm.

The solutions of pattern synthesis are derived for circular sensor arrays based on the criterion of minimizing the mean square error between the desired and synthesized beampatterns. Specifically, the optimal weighting vector, the output beam, and the minimum mean square error are all expressed in closed-form exactly when the desired beampattern is properly formulated. These results provide a more effective and convenient scheme for designing practical frequency-invariant beamformers. Simulations and experimental results demonstrate the performance of the proposed approach.

In this paper, we tackle the problem of tiling a domain with a set of deformable templates. A valid solution to this problem completely covers the domain with templates such that the templates do not overlap. We generalize existing specialized solutions and formulate a general layout problem by modeling important constraints and admissible template deformations. Our main idea is to break the layout algorithm into two steps: a discrete step to lay out the approximate template positions and a continuous step to refine the template shapes. Our approach is suitable for a large class of applications, including floorplans, urban layouts, and arts and design.