序列凸优化法实现最大稀疏阵列设计

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本文主要探讨了"Maximally Sparse Arrays Via Sequential Convex Optimizations"这一主题，发表在2012年的IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS上。在现代无线通信和信号处理领域，设计能高效辐射聚焦波束图案且满足给定功率限制的极稀疏阵列是一个日益受到关注的研究课题。传统的阵列设计问题通常涉及对解决方案空间（即源数量）的非凸约束，这使得找到最优解成为一项具有挑战性的任务，因为这类问题属于NP-hard问题。为了克服这个难题，研究者们提出了将非凸约束转化为可优化的凸形式，其中一种常见的策略是通过最小化向量的-norm来实现。尽管这种方法在某些情况下有效，但并不总是能够达到最稀疏的解决方案。因此，文章的核心贡献在于提出了一种创新的合成算法，该算法不仅考虑了阵列激励权重的设计，还同时优化了传感器的位置，目标是实现发射聚焦的针状波束模式。这种新颖的算法基于序列化的凸优化，其中关键步骤包括使用加权-norm的最小化。作者通过引用基准问题进行了一系列数值测试，结果表明这种方法能够在保持性能的同时显著提高阵列的稀疏度。相比于传统的技术，这种方法的优势在于它能够在解决复杂优化问题时找到更接近于全局最优的稀疏解决方案，这对于降低能耗、提升系统效率以及减少干扰等方面具有重要意义。总结来说，这篇论文提供了关于如何利用序列化凸优化技术设计出既能满足功率限制又能达到最大稀疏度的阵列设计方法，这对于现代无线通信系统的高效、紧凑设计具有实际应用价值。通过这种创新策略，研究人员能够在保证信号质量的同时，显著简化硬件设计，对于无线通信设备的小型化和多天线系统的发展具有推动作用。

192 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 11, 2012

Maximally Sparse Arrays Via Sequential Convex

Optimizations

Giancarlo Prisco and Michele D’Urso

Abstract—The d esign of sparse arrays able to radiate focused

beam patterns satisfying a given upper-bound power

mask with

the minimum number of sources is a research area of increasing

interest. The related synthesis problem can be formulated with

proper constraints on the cardinality of th

e solution space, i.e., its

-norm. Unfortunately, s uch a nonconvex constraint requires to

solve an NP-hard problem. Interesting ideas to relax the above

constraint in a convex way have been succe

ssfully proposed. A

possible solution is based on the minimization of the

-norm.

This strategy is not always able to achieve a maximally sparse

solution. In the following, an innovat

ive synthesis scheme that

optimizes both excitation weights and sensor positions of an

array radiating pencil beam-patterns is discussed. The solution

algorithm is based on sequentia

l convex optimizations including

a reweighted

-norm minimization. Numerical tests, referred to

benchmark problems, show that the proposed synthesis method is

able to achieve maximally spa

rse linear arrays, also compared to

the best results reported in the literature, obtained by means of

global optimization schemes.

Index Terms—Compressed sensing, global optimizations,

sequential convex optimi

zation, sparse array.

I. INTRODUCTION

GOOD deal of attention has been devoted to the problem

of synthesizing arb

itrary far-ﬁeld pattern subject to given

far-ﬁeld constraints by using the minimal number of sensors.

The problem is of particular interest in t hose application s where

the anten na weigh

t and size m ust be limited [1], [2]. The tools

and algorithms proposed in the open literature can be subdi-

vided in two classes: deterministic and stochastic approaches.

To the ﬁrst clas

s belong the methods proposed in [1 ]–[7], where

simple formulas are derived for synthesizing given far-ﬁeld pat-

terns by using uniform amplitude sparse arrays. In [1]–[3], in

particular

, a large num ber of sensors is often required to ful-

ﬁll the design constraints. The number of senso rs is an input

of the solution methods, and it is n o t derived once th e far-ﬁeld

performan

ces to ach ieve have been given [1]–[3]. A novel al-

gorithm has been proposed in [ 4]–[6]. The proposed approach

allows to determine the minimum number of sensors to ﬁt, with

a ﬁxed a

ccuracy, a reference pattern, given in am plitude and

phase. Concerning [5] and [6], note that th e ﬁeld distributio n

satisfying the given design constraints is n ot un iqu e, and dif-

feren

t choices for the reference ﬁeld pattern conduct to different

results in terms of number of required sensors. In particular, the

ﬁeld distribution to achieve the minimum num ber of sen sors is

Manuscript received November 03, 2011; revised January 06, 2012; accepted

January 27, 2012. Date of publication February 03, 2012; date of current version

March 19, 2012.

The authors are with the SELEX Sistemi Integrati S.p.A., 80014 Giugliano,

Italy (e-mail: gprisco@selex-si.com; mdurso@selex-si.com).

Dig

ital Object Identiﬁer 10.1109/LAWP.2012.2186626

aprioriunknown, and its determination requires to apply the

procedure in [5] and [6] for each ﬁel d distributio n b

elonging to

the solution set. Furthermore, regarding the reference ﬁeld to a

set achieved by arbitrarily ﬁxing the sensor positions (equally

spaced half-wavelength array; see [5] and [

6]), there is no guar-

antee that the number of antenn as of the resulting array is the

minimum one.

Therefore, the problem to aprioriestabli

sh the reference

far-ﬁeld p attern, so as to achieve the solution with the minimum

number of sensor, arises. Such a problem represents the main

drawback also for the compressive s

ensing (CS)- and Bayesian

CS (BCS)-based approaches [8]–[10], wherein the reference

ﬁeld to ﬁt in order to obtain the maximally sparse array is

aprioriassumed known.

To the second class belong the stochastic ap-

proaches [11]–[14], where different global opti mi zation

methods are exploited for syn

thesizing given radiation patterns

while minimizing the number of required sensors. A nonconvex

properly weighted cost functional is introduced and minimized,

whose global solution, wh

en reached, gives back the minimum

number of required sensors and the corresponding complex

excitations.

In this letter, the synthe

sis problem involving the minimum

number of sensors is addressed from a new perspective. The

array weights can be seen as a sup erposition of spikes, located

in the sensor positio

ns on the array supp ort [15]. The sparse

array i s then synthesized by extracting the sensor positions and

weights from a predeﬁned dense grid, resembling a continu ous

source. Herein, th

e number of required sensors is minimized by

minimizing the cardinality of the current distribution through its

-norm. On the other hand, such a constraint leads to compu-

tationally int

ractable problem s; the p rop osed method is based

on the minimization of a convex measure of the

-norm, as

introduced in [16]. Such a measure outperforms the common

-norm, l a rg

ely exploited in the framework of t he CS-based

optimization prob lems. As a furth er contribution, note that t he

existence of a non-su per-directive source whose radiation pat-

tern fulﬁl

ls the design constr a int s is veriﬁed by taking advantage

from the results in [17] before applying the proposed procedure.

Notably, if the feasibility of the problem is not aprioriveri-

ﬁed, a su

per-directive array may be found as a result, as in [18].

In the proposed synthesis a pproach, differently from [5], [6],

and [8]–[10], only the upper boun d of the p ower pattern is as-

signe

d (and not a reference ﬁeld, in amplitude and phase), thus

allowing to determine the minimum number of required sensors

independently on the particular reference ﬁeld.

This l

etter is organized as follows. In Section II, the addressed

synthesis p rob lem is formulated. In Section III, the prop osed

solution method is presented. In Section IV, numerical synthesis

res

ults are p resented and discussed. Conclusions follow.

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