Introduction to linear and nonlinear programming

Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

With the demanding system requirements for the fifth-generation (5G) wireless communications and the severe spectrum shortage at conventional cellular frequencies, multibeam antenna systems operating in the millimeter-wave frequency bands have attracted a lot of research interest and have been actively investigated. They represent the key antenna technology for supporting a high data transmission rate, an improved signal-to-interference-plus-noise ratio, an increased spectral and energy efficiency, and versatile beam shaping, thereby holding a great promise in serving as the critical infrastructure for enabling beamforming and massive multiple-input multiple-output (MIMO) that boost the 5G. This paper provides an overview of the existing multibeam antenna technologies which include the passive multibeam antennas (MBAs) based on quasi-optical components and beamforming circuits, multibeam phased-array antennas enabled by various phase-shifting methods, and digital MBAs with different system architectures. Specifically, their principles of operation, design, and implementation, as well as a number of illustrative application examples are reviewed. Finally, the suitability of these MBAs for the future 5G massive MIMO wireless systems as well as the associated challenges is discussed.

Multibeam Antenna Technologies for 5G Wireless Communications

The proliferation of wireless services is driving innovative phased array solutions that are able to provide better cost/performance tradeoffs. In this context, the use of irregular array architectures provides a viable solution. This paper reviews and highlights some of the most recent advances in this field, including clustered, thinned, sparse, and time-modulated arrays, and their proposed design methodologies.

Unconventional Phased Array Architectures and Design Methodologies—A Review

From the material presented here, it is clear that the theory underlying the SNF approach is complex and involved to implement. However, it is also very elegant and provides one with many measurement options and powerful capabilities. The numerical implementation of the theory can be efficiently deployed through the use of the fast Fourier transform (FFT) enabling transforms of even electrically large antennas to be accomplished in a matter of a few seconds on a modern powerful digital computer. With the advent of commercially available SNF test systems, the user can exploit these techniques, largely unimpeded by the burden of the theory or the implementation thereof. The material presented here highlighted some of the fundamental concepts and limitations the user needs to be aware of in order to use these test systems with confidence.

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Spherical near-field antenna measurements

The joint synthesis of the spatial power pattern and polarization of arbitrary arrays is addressed. Specifically, the proposed approach gives the solution to a frequently encountered problem, namely the array design (i.e., the determination of the radiating element weightings) to achieve a pattern that is arbitrarily upper bounded, while its polarization is optimized in a given angular region. Any state of polarization (elliptical, circular and linear) can be synthesized and there is no restriction regarding the array geometry and element patterns. The synthesis problem is rewritten as a convex optimization problem, that is efficiently solved using readily available software. This ensures the optimality of the proposed solution. Various numerical results are presented to validate the proposed method and illustrate its potentialities. The synthesis of a sequentially rotated array is first addressed. Then a linear array of equispaced randomly oriented dipoles is considered. Finally, a conformal and a planar array of patches, where the mutual coupling effects are considered, are synthesized to radiate a linear and a circular polarization.

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Optimal Polarization Synthesis of Arbitrary Arrays With Focused Power Pattern

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.

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Array Pattern Synthesis With Excitation Control via Norm Minimization

The classical narrow main beam and low sidelobe synthesis problem is addressed and extended to arbitrary sidelobe envelopes in both one and two dimensional scenarios The proposed approach allows also to handle arbitrary arrays This extended basic synthesis problem is first rewritten as a convex optimization problem This problem is then transformed into either a linear program or a second order cone program that can be solved efficiently by readily available software The convex formulation is not strictly equivalent to the initial synthesis problem but arguments are given that establish that the obtained optimum is either identical or can be made as close as wanted to the desired one Finally, numerical applications and a comparison with a classical solution are presented to both confirm the optimality of the solution and illustrate the potentialities of the proposed approach

Optimal Narrow Beam Low Sidelobe Synthesis for Arbitrary Arrays

An iterative procedure for the synthesis of uniform amplitude focused beam arrays is presented. Specifically, the goal is to optimize the locations of a fixed number of array elements with known excitations in order to synthesize narrow-beam low-sidelobe patterns. Any fixed elements excitations can be handled, although uniform amplitude and equiphase excitations are aimed because of their practical interest. Moreover, the method can be applied to both linear and planar arrays, and there is no restriction regarding the element patterns. On top of being easy to implement, the proposed procedure leads to solutions that either equal or outperform the ones found by previous approaches in various cases of interest.

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Synthesis of Uniform Amplitude Focused Beam Arrays

A procedure is proposed to significantly reduce the amount of time to characterize 3-D antenna far-field patterns. The measured far field is expanded into spherical harmonics, and a sparse recovery algorithm is used to recover the spherical wave coefficients giving access to the field radiated by the antenna everywhere. A small number of measurement points are required, since the relevant information of the most antenna patterns is concentrated in only a few spherical wave coefficients. Sampling strategies enabling fast spherical scans are discussed, which makes the approach both efficient and easy to implement in existing far-field measurement facilities. Simulations are first provided to show the potentialities of this compressive sensing-based approach. The proposed strategy is then applied to characterize 3-D far-field patterns radiated by several antennas operating in different frequency bands measured in far field in direct line of sight configuration and in a compact antenna test range. Experimental results show that a saving in the number of measurement points up to 70% can be achieved compared with standard approaches. These results pave the way to a more efficient use of far-field measurement facilities.

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Benjamin Fuchs

Papers

Optimal Polarization Synthesis of Arbitrary Arrays With Focused Power Pattern

Array Pattern Synthesis With Excitation Control via Norm Minimization

Optimal Narrow Beam Low Sidelobe Synthesis for Arbitrary Arrays

Synthesis of Uniform Amplitude Focused Beam Arrays

Fast Antenna Far-Field Characterization via Sparse Spherical Harmonic Expansion