Beampattern optimization by minimization of quartic polynomial

Aittomaki, Tuomas; Koivunen, Visa

doi:10.1109/ssp.2009.5278546

Cited by 33 publications

(26 citation statements)

References 9 publications

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“…Theorem 5.4 If C(x, x) is square-free, then (C m ) admits a polynomial-time randomized approximation algorithm with relative approximation ratio 1 2 τ (C m ).…”

Section: Conjugate Form In the M-th Roots Of Unitymentioning

confidence: 99%

“…Maricic et al [20] proposed a quartic polynomial model for blind channel equalization in digital communication. Aittomäki and Koivunen [1] discussed the problem of beam-pattern synthesis in array signal processing problem and formulated it to be a complex quartic minimization problem. Chen and Vaidyanathan [7] studied MIMO radar waveform optimization with prior information of the extended target and clutter, by relaxing a quartic complex model.…”

Section: Introductionmentioning

confidence: 99%

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Approximation methods for complex polynomial optimization

Jiang

Zhang

2014

Comput Optim Appl

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Complex polynomial optimization problems arise from real-life applications including radar code design, MIMO beamforming, and quantum mechanics. In this paper, we study complex polynomial optimization models whereby the objective function takes one of the following three forms: (1) multilinear; (2) homogeneous polynomial; (3) a conjugate symmetric form. On the constraint side, the decision variables belong to one of the following three sets: (1) the m-th roots of complex unity; (2) the complex unity; (3) the Euclidean sphere. We first discuss the multilinear objective function. Polynomial-time approximation algorithms are proposed for such problems with assured worst-case performance ratios, which depend only on the dimensions of the model. Then we introduce complex homogenous polynomial functions and establish key linkages between complex multilinear form and the complex polynomial functions. Approximation algorithms for the above-mentioned complex polynomial optimization models with worst-case performance ratios are presented.

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“…Theorem 5.4 If C(x, x) is square-free, then (C m ) admits a polynomial-time randomized approximation algorithm with relative approximation ratio 1 2 τ (C m ).…”

Section: Conjugate Form In the M-th Roots Of Unitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Approximation methods for complex polynomial optimization

Jiang

Zhang

2014

Comput Optim Appl

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“…3 for example, the interested regions are denoted by three main lobes, then beampattern matching is to synthesize a beampattern that well matches the desired shape (Generally speaking, the desired (or ideal) beampatterns are often at rectangular shapes because we always pursue that the interested area is illuminated uniformly by a radar). At present, most of the literature for BMD [14][15][16][17][18][19] is mainly to synthesize beampatterns from the perspective of shape approximation by minimizing MSE directly. The main defect of these SAMs is using excessive mathematical derivations and is not very efficient, especially when using iterative algorithms.…”

Section: Main Ideamentioning

confidence: 99%

“…To this end, there has been extensive work on transmit beampattern synthesis. And the work is mainly divided into two categories: one is to minimize the sidelobe level as its main objective [13,14], and the other is to match the beampattern to a known shape [14][15][16][17][18][19]. For the former design case, [13,14] proposed a minimum-sidelobe-based method to design the transmit beampattern in steradian space.…”

Section: Introductionmentioning

confidence: 99%

“…For the latter design case, most of the literature is mainly to synthesize beampatterns from the perspective of shape approximation. The existing work either uses convex optimization [14] or uses iterative algorithms [15][16][17][18][19] to do shape approximation by minimizing mean square error (MSE) between the designed and desired beampatterns. The main defect of these shape-approximation-based methods (SAMs) is using excessive mathematical derivations and is not very efficient, especially when using iterative algorithms.…”

Section: Introductionmentioning

confidence: 99%

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Mimo Radar Transmit Beampattern Synthesis via Minimizing Sidelobe Level

Xu¹,

Wang²,

Yuan³

et al. 2013

PIER B

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Abstract-In multi-input multi-output (MIMO) radar transmit beampattern synthesis, most current literature formulates the problems in steradian space. However, since the beampattern and its parameters are both measured and defined in radian space, from the view point of physical meaning, it will be better to reformulate the problems in radian space rather than in steradian space. In this paper, we propose methods in the radian space to synthesize beampatterns based on minimizing sidelobe level for the two main designs in MIMO radar, i.e., minimum sidelobe beampattern design (MSBD) and beampattern matching design (BMD). For MSBD, the design criteria considering both peak sidelobe level and integrated sidelobe level is proposed. By this we can have a good tradeoff between the intensity and power distribution in beampattern synthesis. After a two-step converting, the formulation of the criteria is transformed into a convex programming, where a global optimal solution can be obtained. For BMD, instead of minimizing mean square error directly as in conventional methods, we propose a power-approximation-based method by minimizing integrated sidelobe level. Finally, numerical comparisons with classical methods demonstrate that the proposed MSBD maintains for all range of main lobe width and the proposed BMD has smoother main lobes with maximal power focused in.

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On decompositions and approximations of conjugate partial-symmetric tensors

Fu¹,

Jiang

2021

Calcolo

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Hermitian matrices have played an important role in matrix theory and complex quadratic optimization. The high-order generalization of Hermitian matrices, conjugate partial-symmetric (CPS) tensors, have shown growing interest recently in tensor theory and computation, particularly in application-driven complex polynomial optimization problems. In this paper, we study CPS tensors with a focus on ranks, computing rank-one decompositions and approximations, as well as their applications. We prove constructively that any CPS tensor can be decomposed into a sum of rank-one CPS tensors, which provides an explicit method to compute such rank-one decompositions. Three types of ranks for CPS tensors are defined and shown to be different in general. This leads to the invalidity of the conjugate version of Comon’s conjecture. We then study rank-one approximations and matricizations of CPS tensors. By carefully unfolding CPS tensors to Hermitian matrices, rank-one equivalence can be preserved. This enables us to develop new convex optimization models and algorithms to compute best rank-one approximations of CPS tensors. Numerical experiments from data sets in radar wave form design, elasticity tensor, and quantum entanglement are performed to justify the capability of our methods.

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Beampattern optimization by minimization of quartic polynomial

Cited by 33 publications

References 9 publications

Approximation methods for complex polynomial optimization

Approximation methods for complex polynomial optimization

Mimo Radar Transmit Beampattern Synthesis via Minimizing Sidelobe Level

On decompositions and approximations of conjugate partial-symmetric tensors

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