Reducing the Number of Elements in Linear and Planar Antenna Arrays With Sparseness Constrained...

8/6/2019 Reducing the Number of Elements in Linear and Planar Antenna Arrays With Sparseness Constrained Optimization

http://slidepdf.com/reader/full/reducing-the-number-of-elements-in-linear-and-planar-antenna-arrays-with-sparseness 1/6

1

Abstract - The synthesis of sparse antenna arrays has many

practical applications in situations where the weight, sizeand cost of antennas are limited. In this communication

the antenna array synthesis problem, with minimum

number of elements, is studied from the new perspective of

sparseness constrained optimization. The number of

antenna elements in the array can be efficiently reduced by

casting the array synthesis problem into the framework of

sparseness constrained optimization and solving with the

Bayesian compressive sensing (BCS) inversion algorithm.

Numerical examples of both linear and planar arrays are

presented to show the high efficiency of achieving the

desired radiation pattern with a minimum number of

antenna elements.

Index Terms- sparse antenna array synthesis, sparseness

constrained optimization, linear array, planar array

I. INTRODUCTION

HE antenna array synthesis problem is related to thecalculation of the excitations and positions for all antenna

elements that produce a desired radiation pattern. Reducing of the number of antenna elements in the array is particularlyuseful in many applications where the weight, size and cost of the antennas are limited, such as phased array radar, satellitecommunication and MIMO radar system [1-4, 8].

Up to present many analytical formulations have been

derived for the antenna array synthesis problem, such as theDolph-Chebyshev and Taylor methods for uniformly spacedantenna arrays [10]. These methods are generally based on theassumption that the array elements are equally spaced withuniform distribution which results in a large number of antenna elements to synthesize the desired radiation pattern.An alternative option to reduce the number of elements in thearray is to use unequally spaced and non-uniform excitationstrategies. The analysis of unequally spaced antenna arraysoriginated with the work of Unz, who developed a matrixformulation to obtain the current distribution necessary togenerate a prescribed radiation pattern with an unequallyspaced linear array [6]. The unequally spaced non-uniform

antenna array synthesis problem in practice is complex andgenerally could not be efficiently solved with analyticalmethods. Therefore, global optimization methods are goodoptions for solving this problem. Among them the geneticalgorithm (GA), particle swarm optimization (PSO) methodand simulated annealing (SA) have already been successfully

used for the synthesis of non-uniform linear, planar andcircular arrays [4, 7, 9-12]. Recently, a novel non-iterativesynthesis algorithm based on the matrix pencil method hasbeen proposed which efficiently reduces the number of elements in a linear antenna array in a very short computationtime [3, 8].

The objective of an efficient antenna array synthesis is tosynthesize a desired radiation pattern with the minimumnumber of elements. Conventional methods generally cast theantenna array synthesis as a minimum L2 norm optimizationproblem and solve this with global optimization methods.These kinds of methods have two main problems. First, theglobal optimization is time consuming and requires largecomputation resources, especially for electrically large arrays.Second, the L2 minimization is an energy constraint

optimization which synthesizes the desired antenna radiationpattern but does not ensure the minimum number of antennaelements for the array. In this paper the antenna synthesisproblem is studied from the new perspective of sparsenessconstrained optimization. Sparseness constrained optimizationseeks as few non-zero solution as possible in a linear matrixequation [13, 15]. This coincides with the objective of theantenna array synthesis problem which seeks as few elementsas possible in the array, under the constraint of the prescribedarray radiation pattern. Assume that there are N equally spacedantenna elements in the array and that not each elementnecessarily radiates EM waves. The antenna element whichdoes not radiate waves is equivalent to having no antenna

element in the supposed location. Thus we exploit very usefula priori information that the antenna location space is sparse,i.e., the number of elements which radiate waves is far lessthan the assumed total number of elements in the array. Theobjective of antenna array synthesis problem is equivalent tofinding the smallest number of elements whose excitation doesnot equal zero. Then the array synthesis is cast as sparsenessconstrained optimization problem and solved with Bayesiancompressive sensing (BCS) inversion algorithm which seeksas few non-zero element as possible in the solution space.

The organization of the remainder of this paper is listed asfollows. In Section II, the synthesis of linear and plannararrays with sparseness constrained optimization is presented.

Some numerical results are presented in Section III to showthe efficiency of sparse linear and planar arrays synthesis withthe proposed method. Finally, some conclusions are drawn inSection IV.

II. ANTENNA ARRAY SYNTHESIS with SPARSENESSCONSTRAINED OPTIMIZATION

The antenna array synthesis problem can be described asfollows [3]:

Reducing the Number of Elements in Linear and Planar

Antenna Arrays with Sparseness Constrained Optimization

Wenji Zhang, Lianlin Li, and Fang Li

T

This work was supported by the National Natural Science Foundation of China under Grants 60701010 and 40774093.

The authors are with the Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China (e-mail: [email protected],[email protected], [email protected]).



2

( )

{ }( )

1

cos

,2

min

. min i

i i i Q

Q jkd

i R d

i

Q

const F R eθ θ ε

=

− ≤

∑

(1)

where ( )F θ is a the radiation pattern at different observation

angles, Q is the number of identical antenna elements in the

array, Ri is the excitation of the ith element located at d i, k isthe wavenumber in the freespace. The objective of the

problem is to synthesize the desired radiation pattern ( )F θ

with the minimum number of elements under a small toleranceerror ε [3].

A. Linear array

Suppose the array is composed of N identical antennaelements. For linear antenna arrays, the array factor is givenby

( ) cos

1

i

N jkd

i

i

F R eθ θ

=

= ∑ (2)

where Ri is the excitation coefficient of the ith element locatedat x = d i, k is the wave number in the freespace. Suppose thatall the antenna elements are symmetrically distributed within arange of – xs to xs along the x direction as shown in Figure 1.

Fig1. Geometry of nonuniformly spaced linear symmetric array

Considering the symmetric geometry of the array, thepattern of the array can be written as

( )[ ] / 2

2 cos( cos ) N

i i

i

F R kd θ θ = ∑ (3)

where[ ] / 2 N is the minimum integer no less than / 2 N . In

order to solve the above equation we can assume that all theantenna elements are equally spaced from – xs to xs with asmall interelement spacing ∆d . Although it is supposed thatthere is one element at each position, not each antenna elementis necessarily radiating waves or excited with current. All the

antenna elements can be in two states: “on” state (when theelement is in the supposed position or has an excitation) or“off” state (when there is no element in the supposed positionor without an excitation). Through discretization, (3) can bewritten in a matrix form

[ ] [ ] [ ]1 1m m n nF M r

× × ×= (4)

where m is the number of sampled antenna radiation pattern, n

is the minimum integer no less than 2 / s

x d , F is a vector

containing the sampled radiation pattern at different angles,

( ) ( ) ( )1 2, ,T

mF F F θ θ θ = F , M is an m n×

matrix with the (i, j)-th element 2cos( cos )ij i j M kd θ = .

Choosing m n< , M forms an overcomplete dictionary. r isthe excitation vector, Ri = 0 is equivalent to the “off” statewhich means the antenna in the ith position is not excited or isabsent from the supposed position.

All of the existing L2 minimization algorithms for antennaarray synthesis do not exploit a prior sparsity informationabout the antenna location space. Although many equallyspaced antennas are supposed to be placed in the array, noteach antenna is required to be excited. The antenna elementwithout excitation is equivalent to being absent in thesupposed position. Thus we can exploit very useful a priori information that the antenna location space is sparse. Thespasity of the antenna location space is not well exploited incurrent antenna array synthesis methods. In this paper theproblem of antenna array synthesis with minimum number of antenna elements is cast as sparseness constrainedoptimization problem

1min . .s t ε − <r F Mr (5)

where ε is a small tolerance error. In the above equation weseek to find the smallest number of non-zero elements in theexcitation vector r. Comparing with traditional antennasynthesis problem solved with L2 minimization, this papercasts the array synthesis problem as a sparseness constrainedoptimization problem and changes the L2 minimization to L1 minimization. The L2 minimization is a quadratic optimizationproblem which requires a large computation resource and doesnot ensure the minimum number of elements in the array. Incontrast, the sparseness constrained optimization problemsolves a convex, nonquadratic optimization problem and seeksthe sparsest solution to the linear equation, which is also much

more computationally efficient than the global minimizationmethods for solving L2 minimization problems. In this paperthe Bayesian compressive sensing inversion algorithm forsolving the L1 minimization problem is employed to solve theantenna array synthesis problem [14]. A brief summarizationof Bayesian compressive sensing algorithm is given inAppendix A. Empirical studies demonstrate that the Bayesianlearning algorithm has a significantly accelerated rate of convergence compared to other existing L1 minimizationalgorithms.

B. Planar Array

Fig. 2 Geometry of 2 N x × 2N y elements symmetrical array [4]



3

Planar antenna arrays are currently under study forspacecraft communication and phased array radar systems dueto the low side lobe level (SLL) and ease of deployment [4, 5].Here we consider a uniformly spaced 2 N x × 2N y elementsrectangular planar array with uniform phase excitation shownin Figure 2. The excitation of the (n, m)-th antenna element,located at r nm = ( xnm, ynm), is denoted as I nm. d x and d y are theinter-element spacing between the antennas in x and y

directions. Assume that the excitation and inter-elementspacing are symmetric with respect to both x and y axes, thearray factor can be written as [8, 12]

( ) ( ) ( )1 1

, 4 cos cos y x

N N

nm nm nm

n m

F u v I kx u ky v= =

= ∑∑ (6)

wheresin cosu θ ϕ = , sin sinv θ ϕ = (7)

Through discretization, (6) can be written in following matrixform

=F MI (8)Assuming that N u and N v samples are taken in the u and v

planes and by vectoring ( ),i jF u v , which is sampled at

different angles, into a column vector the k th element of F canbe written as

[ ] ( ), , ( 1)i j vk

F F u v k j i N = = + − (9)

I is the excitation vector whose lth element is

[ ] , ( 1)nm yl I I l m n N = = + − (10)

The (k, l)-th element of the matrix M can be derived as

[ ] ( ) ( )4cos cos

( 1) , ( 1)

nm i nm jkl

y v

M kx u ky v

l m n N k j i N = + − = + −

=(11)

F is an N u × N v dimension column vector and M is an ( N u × N v)× ( N x × N y) matrix. Similar to the linear antenna array

synthesis problem, the sparse planar array synthesis can besolved as the following sparseness constrained optimizationproblem

1min . .s t ε − <I F MI (12)

Although all elements in the planar array are assumed to beequally spaced with very small inter-element spacing, not eachelement is necessarily radiating EM waves or excited withcurrent. If the excitation of the lth element is 0 (“off” state), itis equivalent to there being no element in the correspondinglocation. The sparseness constrained optimization problem in(12) seeks as few non-zero element as possible in theexcitation vector I, which coincides with the objective of the

antenna array synthesis with minimum number of elements.III NUMERICAL RESULTS

In this section some numerical results are presented to showthe effectiveness of sparseness constrained optimization forboth linear and planar sparse antenna array synthesis.

A. Synthesis of Chebyshev Pattern with linear array

The first example synthesizes a twenty elements uniformlyspaced Chebyshev array. The side lobe level (SLL) of the

Chebyshev array is SLL = -30dB. If not specified, in all thefollowing simulations xs = 10 λ and the inter-element spacingis chosen to be ∆d = λ /10. Table1 gives the antenna locationand excitation amplitude in [3] and the result obtained by thesparseness constrained optimization in this paper. Figure 3 isthe reconstructed radiation pattern with sparseness constrainedoptimization and compared with the matrix pencil methodresult in [3]. It is observed that for cos(θ ) between 0.85 and

1.0 the side lobe level is a bit higher than that of the desiredradiation pattern. This is due to the reason that in thecomputation we discarded one antenna element with excitationamplitude smaller than 1/20 of the maximum excitationamplitude. In order to characterize the difference between thedesired pattern and the synthesized pattern the mean squarederror (MSE) is used in this paper. The MSE is defined as normof the error between the desired pattern and the synthesizedpattern, normalized to the norm of the desired pattern. TheMSE in this example is 1.57e-4. From the result we can findthat through sparseness constrained optimization only 14antenna elements are required to achieve the desired radiationpattern. The number of antenna elements is reduced by 30% ascompared to the uniformly spaced array and the synthesizedpattern is quite satisfactory for practical use.

Chebyshev(N=20)

Position/excitationin [3]

Position/excitationin this paper

i d i / λ Ri d i / λ Ri d i / λ Ri

1 0.25 1.0 0 1 0.4 12 0.75 0.97010 0.8206 0.95818 1.2 0.92953 1.25 0.91243 1.6381 0.84113 2 0.79514 1.75 0.83012 2.4481 0.67176 2.8 0.61435 2.25 0.73147 3.2432 0.48115 3.6 0.41726 2.75 0.62034 4.0071 0.30046 4.5 0.24097 3.25 0.50461 4.7145 0.23345 5.0 0.08048 3.75 0.39104

9 4.25 0.2855810 4.75 0.32561

Table 1 Positions and amplitudes of the reconstructed nonuniform arrayelements for the Chebyshev radiation pattern

Fig. 3 20 elements Chebyshev pattern synthesis with 14 non-uniform antennaelements with sparseness constrained optimization

B. Synthesis of a Taylor-Kaiser Pattern with linear array

The side lobe level (SLL) of the Taylor-Kaiser array to besynthesized in this example is SLL= -25dB. Using the matrixpencil method in [3], only 17 elements are required to producethe desired pattern. Table 2 gives the positions and amplitudesof the reconstructed nonuniform array element with sparseness



4

constrained optimization. For the convenience of comparisonthe result in [3] is also given in the table. Figure 4 is thereconstructed radiation pattern with sparseness constrainedoptimization and the results obtained in [3]. The MSE betweenthe desired radiation pattern and the synthesized pattern is2.09e-4 in this example. It is observed that for cos(θ ) between0.7 and 1.0 the side lobe level is a bit higher than that of thedesired radiation pattern. This is due to the reason that in the

computation process we discard one antenna element whoseexcitation amplitude is 1/20 smaller than the maximumexcitation amplitude. From the reconstruction results we canfind that only 18 antenna elements are required to achieve thedesired radiation pattern using sparseness constrainedoptimization. It requires about 38% fewer antenna elementsthan the uniformly space antenna array. Although one moreelement is used than the matrix pencil method in [3], the resultis still very satisfactory.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-60

-50

-40

-30

-20

-10

0

cos(θθθθ)

P a t t e r n

( d B )

Fig.2 in [3]

This paper

Fig. 4 Reconstruction of the 29 element Taylor-Kaise arrays with 18 antenna

elements with sparseness constrained optimization

Position/ excitation in [1]

Position/excitationin [3]

Position/excitationin this paper

i d i / λ Ri d i / λ Ri d i / λ Ri

1 0 1 0 1 0.4 12 0.5 0.99328 0.8831 0.97859 1.2 0.964313 1 0.97329 1.7652 0.91634 2.0 0.896894 1.5 0.94063 2.6451 0.81903 2.8 0.80455 2 0.89622 3.5211 0.69547 3.6 0.694296 2.5 0.84132 4.3905 0.55651 4.4 0.571467 3 0.77748 5.2485 0.4137 5.2 0.439988 3.5 0.70645 6.0842 0.27782 6.0 0.306949 4 0.63017 6.8 0.18562

10 4.5 0.5506511 5 0.4699412 5.5 0.3900413 6 0.3128214 6.5 0.2400115 7 0.17309

Table 2 Positions and amplitudes of the reconstructed nonuniform array forthe Taylor-Kaise radiation pattern

C. Synthesis of Electrically Large Linear Array

In order to show the efficiency of sparseness constrainedoptimization for solving electrically large scale array synthesisproblem which involves a large number of unknowns in theoptimization, a 300 element Chebyshev array is tested in thisexample. In the simulation xs = 150 λ and ∆d = λ /10, the SLLof the Chebyshev pattern is -35dB. The synthesized pattern ispresented in Figure 5 and the total number of element used is

224. The MSE between the desired radiation pattern and thesynthesized pattern is 1.85e-4. The computation time of thisexample is 7.5s which is much shorter than the globaloptimization methods and is similar to the matrix pencilmethod in [3]. The reducing of the number of elements inlarge antenna arrays and short computation time show thatsparseness constrained optimization is very efficient for sparseantenna array synthesis.

Fig.5 Reconstruction of the 300 element Chebyshev array with 224 antenna

elements with sparseness constrained optimizationD. Synthesis of square planar array

In the last example we synthesized a square planar antennaarray with sparseness constrained optimization. Planar arraysynthesis has more antenna elements, resulting in moreunknowns in the optimization than the linear array. This is avery challenging problem for the traditional globaloptimization methods to synthesize the desired radiationpattern in a short computation time while maintaining aminimum number of elements in the array. In this paper, the1024 element planar array in [4] is synthesized withsparseness constrained optimization. A total of 2500 samples

of the radiation pattern are sampled at different azimuth andelevation angles. The antenna locations in the x and y directions have total array lengths of 10 λ with an inter-elementspacing d x = d y = λ /10, resulting in 101 × 101 unknowns. Forthe large amount of unknowns the planar array synthesisproblem can still be efficiently solved within 10 seconds,which is much more computationally efficient than otherexisting planar array synthesis methods. Figure 7 gives thesynthesized 3D radiation pattern using the sparsenessconstrained optimization. The MSE between the 3D desiredradiation pattern and the synthesized pattern is 2.41e-4. Forthe convenience of comparison the radiation pattern in theE-plane is also presented in Figure 8. In the synthesized planar

array 496 elements are required to achieve the desiredradiation pattern. Approximately 50% fewer antenna elementsare required by solving the planar array synthesis assparseness constrained optimization problem.

IV. CONCLUSION

In this paper the antenna array synthesis problem is studiedfrom the new perspective of sparseness constrainedoptimization. Most of the existing antenna array synthesisalgorithms are based on L2 minimization which does notexploit any sparse information about the antenna location



5

space. By exploiting the a priori information that the antennalocation space is sparse, the antenna array synthesis problem iscast as sparseness constrained optimization problem andsolved with Bayesian compressive sensing inversion algorithm.Numerical examples are presented to show the high efficiencyof both planar and linear arrays synthesis with sparsenessconstrained optimization.

Acknowledgement:

The authors would like to thank S. Ji, Y. Xue and L. Carin forthe permission to use the Bayesian Compressive SensingCodes (BCS_demo) for solving the problems in this paper.

Fig. 6 3D radiation pattern of the 1024 elements planar array in [4]

Fig. 7 3D synthesized pattern of the 496 elements array with sparsenessconstraint optimization

Fig. 8 Reconstructed pattern on the E plan (θ=0°) with sparseness constrainedoptimization

APPENDIX A: BAYESIAN COMPRESSIVE SENSING

In the appendix, the optimization algorithm exploited byBayesian compressive sensing algorithm is brieflysummarized [14, 16, 17].

Assume that the equation relating the measurement N

R∈b and the unknowns M R∈x is nbAx += ,where

( )Inn 2,0|~ σ N is i.i.d Gaussian measurement noise, σ 2 is

the variance. From Baye’s rule, one can get the posterior

distribution of x ( ) ( ) ( )xxbbx Pr|Pr|Pr ∝ (A.1)

It is noted that Pr (x) describes the prior distribution of unknown x. A sparseness-promoting prior used in theBayesian compressive sensing is the so-called Laplace densityfunction, in particular,

( ) ( )1

exp2

Pr xx λ λ

−

=

N

(A.2)

where λ is the reduced hyperparameters. Now, one can get theestimation of unknown x via maximizing its posteriordistribution, in particular,

( 1

22minarg xbAxxx

λ σ +−=−

(A. 3)

which is the standard sparsity-promoted linear inverseproblem via L1-norm.

It is noted that, since the distribution (A.2) is not conjugatedto the conditional distribution in (A.1), it does not allow atractable Bayesian analysis. Therefore, the hierarchical priorsor Gaussian mixtures are firstly employed to describe thedistribution of

( ) ( )iiii γxγx ,0||Pr N = (A.4)

The specification of the hierarchical prior model can be

completed by defining hyperpriors over ix s. To make the

priors on the hyperparameters non-informative (i.e. flat), we

choose iγ to be very small. So in the second stage of thehierarchy the Gamma distribution are assigned to iγ , in

particular,

( )

=

2,1||Pr

λ λ ii Gamma γγ (A.5)

Now, the original unknowns x and reduced hyperparametersγ and λ can be iteratively estimated via the following

formulations, in particular, at the (n+1) iteration, one has

( ) ( )

+−= ∑

=

−−+

N

i

i

n

i

nn

1

21221 minarg xγbAxx x σ

(A.6)and

( ) ) ( ) ( )nnnnxbγγ

γ,|,,Prmaxarg,, 2

,,

11212 λ σ λ σ

λ σ

−++

−+

−=

(A.7)Additionally, this fast algorithm has been developed in [16]and [17] by analyzing the properties of the marginal likelihoodfunction in [14]. This enables a principled and efficientsequential addition and deletion of candidate basis function tomonotonically maximize the marginal likelihood. We omit the



6

detailed discussion of this fast algorithm and refer the readerto [16] and [17] for more details.

REFERENCES

[1] B. Kumar, G. Branner, “Design of Unequally Spaced Arrays forPerformance Improvement,” IEEE Trans. on Antennas Propag., vol. 47, no.3,pp. 511-523, 1999.[2] Ó. Teruel and E. Iglesias, “Ant Colony Optimization in Thinned ArraySynthesis with Minimum Sidelobe Level,” IEEE Antennas and Wireless

Propag. Lett., vol.5, pp. 349- 352 , 2006.[3] Y. Liu, Z. Nie, Q. Liu, “Reducing the number of elements in a linearantenna array by the matrix pencil method,” IEEE Trans. on Antennas

Propag. vol. 56, no. 9, pp. 2955-2962, 2008.[4] J. Ye and W. Pang, “Immune algorithm in array-pattern synthesis withside lobe reduction,” International Conference on Microwave and MillimeterWave Technology, vol. 3: 1127-1130, 2008.[5] B. Kumar, G. Branner, “Generalized Analytical Technique for theSynthesis of Unequally Spaced Arrays with Linear, Planar, Cylindrical orSpherical Geometry,” IEEE Trans. on Antennas Propag., vol. 53, no.2, pp.621-634, 2005.[6] H. Unz, “Linear arrays with arbitrarily distributed elements,” IEEE Trans.Antennas Propagat., vol.8, pp. 222–223, 1960.[7] K. Chen, X. Yun, Z. He and C. Han, “Synthesis of Sparse Planar ArraysUsing Modified Real Genetic Algorithm,” IEEE Trans. on Antennas Propag.,vol. 55, no. 4, pp. 1067-1073, 2007[8] Y. Liu, Q. Liu, and Z. Nie, “Reducing the Number of Elements in theSynthesis of Shaped-Beam Patterns by the Forward-Backward Matrix PencilMethod,” IEEE Trans. on Antennas Propag., vol. 58, no.2, pp. 604-608, 2010.[9] Y. Chen, S. Yang, and Z. Nie, “The Application of a Modified DifferentialEvolution Strategy to Some Array Pattern Synthesis Problems,” IEEE Trans.

on Antennas Propag., vol. 56, no.7, pp. 1919-1927, 2008.[10]C. Balanis, Antenna Theory: Analysis and Design, 3rd ed. New York:Wiley, 2005.[11] D. Kurup, M. Himdi, and A. Rydberg, “Synthesis of Uniform AmplitudeUnequally Spaced Antenna Arrays Using the Differential EvolutionAlgorithm,” IEEE Trans. Antennas and Propag., vol. 47, no. 3, pp. 511-513,1999.[12] K. Yan and Y. Lu, “Sidelobe reduction in array pattern synthesis usinggenetic algorithm,” IEEE Trans. Antennas and Propag., vol. 45: 1117–1122,1997.[13] L. Li and B. Jafarpour, A sparse Bayesian framework for conditioninguncertain geologic models to nonlinear flow measurements, Advances in

Water Resources, vol. 33, pp. 1024-1042, 2010.[14] S. Ji, Y. Xue and L. Carin, “Bayesian Compressive Sensing,” IEEETrans. Signal Processing, vol. 56, no. 6, pp. 2346-2356, 2008.[15] J. A. Tropp, “Greed is good: algorithmic results for sparseapproximation,” IEEE Trans. Inform. Theory, vol. 50, pp. 2231-2242, 2004.[16]A. C. Faul and M. E. Tipping, “Analysis of sparse Bayesian learning,” inAdvances in Neural Information Processing Systems, pp. 383–389, 2002.[17] M. E. Tipping and A. C. Faul, “Fast marginal likelihood maximizationfor sparse Bayesian models,” in Proc. 9th Int.Workshop Artificial Intelligenceand Statistics, 2003.

Reducing the Number of Elements in Linear and Planar Antenna Arrays With Sparseness Constrained...

Documents

Transcript of Reducing the Number of Elements in Linear and Planar Antenna Arrays With Sparseness Constrained...