## UNIVERSITY

## OF TRENTO

**DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE**

38123 Povo – Trento (Italy), Via Sommarive 14 http://www.disi.unitn.it

### ADS-BASED GUIDELINES FOR THINNED PLANAR ARRAYS

### G. Oliveri, L. Manica, and A. Massa

### January 2011

**ADS-Based Guidelines for Thinned Planar Arrays**

G. Oliveri,*Student Member, IEEE*, L. Manica, and A. Massa,*Member, IEEE*

*ELEDIA Research Group*

Department of Information and Communication Technologies, University of Trento, Via Sommarive 14, 38050 Trento - Italy Tel. +39 0461 882057, Fax +39 0461 882093

*E-mail: [email protected],*

*{giacomo.oliveri, luca.manica}@dit.unitn.it*

**ADS-Based Guidelines for Thinned Planar Arrays**

G. Oliveri, L. Manica, and A. Massa

**Abstract**

This paper proposes an analytical technique based on Almost Difference Sets (ADSs) for

thinning planar arrays with well controlled sidelobes. The method allows one to synthesize

bidimensional arrangements with peak sidelobe levels (P SLs) predictable and deducible

from the knowledge of the array aperture, the filling factor, and the autocorrelation function

of theADS at hand. The numerical validation, concerned with both small and very large

apertures, points out that the expectedP SLvalues are significantly below those of random

arrays and comparable with those from different sets (DSs) although obtainable in a wider

range of configurations.

**Key words: Array Antennas, Planar Arrays, Thinned Arrays, Sidelobe Level Control, Almost**

**1**

**Introduction**

Antenna arrays for radar tracking, remote sensing, biomedical imaging, satellite and ground communications have often to support three-dimensional scanning with a suitable beampattern shape in the whole angular region [1]. Towards this end, planar arrays have to be used and large apertures are necessary to provide satisfactory angular resolutions along both azimuth and elevation [1]. On the other hand, the inter-element spacing should not exceed half-wavelength to avoid the presence of grating lobes [1]. These requirements usually result in very inefficient, heavy, and expensive solutions consisting of planar geometries with several thousands close elements.

In order to reduce the number of elements while keeping the radiation properties of the original structures, thinning techniques have been successfully introduced [2]. Designing thinned planar arrays is an important research topic since decades (see [2][3][4][5][6][7][8] and the references cited therein). As a matter of fact, a suitable thinning allows one to reduce the array costs, its weight, and the power consumption. However, it causes the loss of the control of the peak sidelobe level (P SL) [6] to be properly counteracted. To this end, several techniques has been proposed in order to fully exploit the advantages of thinned arrangements while minimizing their drawbacks. First attempts have been conceived to require low computational resources (see Tab. I in [9]), but they have provided no significant improvements when compared with random placements [2][9] extensively employed in practice [2].

More recently, the availability of large computational resources has justified the use of
opti-mization techniques such as dynamic programming [10], genetic algorithms [5][11][8],
simu-lated annealing [12][13], and particle swarm optimizers [7]. Thinned arrays synthesized with
optimization tools turn out to be very effective [10][12][7][8], even though it is not possible
*to a-priori estimate the expected performances for a given array aperture and thinning factor*
[6]. Furthermore, computational and convergence issues make the application of stochastic
op-timizers difficult and expensive when dealing with1Dlarge apertures [6] and, even more, when
planar arrangements are considered.

In order to overcome such drawbacks, an alternative approach for thinning large arrays has been introduced [4][6][14]. Such an approach relies on the exploitation of binary sequences

derived from Difference Sets (DSs), which exhibit a two-level autocorrelation function [4]. Besides their analytic nature and the arising inexpensive generation, DS-based thinned arrays have several interesting properties. They are deterministically designed and present predictable [6] and low P SLs (3 dB and 1.5 dB below random arrays for the linear case and the planar one, respectively). However, only a limited number ofDSsequences exist and the whole set of aperture sizes and thinning values [6][15] cannot be dealt with.

The problem of obtaining sub-optimal sequences (in terms of autocorrelation levels) has been recently addressed in information theory and “close” sequences to DSs have been looked for. Almost Difference Sets (ADSs) [16][17][18] are a wide class of binary sequences with

*three-valued autocorrelations [16][17][18]. They represent the closest sets to*DSs [16][17][18]
(three-levels vs. two-levels) and large repositories of explicit sequences (e.g., [19]) are
avail-able.

As regards to1Dgeometries, the sidelobe characteristics ofADS-based arrays have been ana-lyzed in [20] and good performances have been predicted and numerically verified dealing with both small and large apertures. Because of these results and its deterministic nature, an ADS -based technique seems to be a good candidate for thinning planar arrangements of radiating elements and it will be presented in this paper. More specifically, the objective is not to de-fine the “optimal” thinning method, but rather to provide a simple and reliable technique which guarantees to the designer predictable performances to be taken into account during the feasibil-ity study. Towards this end, theP SLbehavior ofADS-based planar arrays will be analytically investigated and different bounds will be provided. It should be pointed out that, despite the linear case [20] where the Blahut’s theorem [20] has been applied, a different mathematical analysis is here necessary. TheP SLbounds are then derived starting from the properties of the

2Ddiscrete Fourier transform.

The paper is organized as follows. After a short overview on ADSs (Sect. 2), a set of suitable bounds of the P SL are analytically determined in Sect. 3. Section 4 provides a selected set of numerical results aimed at validating the obtainedP SLestimators as well as comparing the

ADS performances with both random techniques and state-of-the-art optimization approaches. The exploitation of directive elements is also considered in order to point out the flexibility of

theADSthinning theory. Finally, some conclusions are drawn (Sect. 5).

**2**

**Two-Dimensional Almost Difference Sets**

With reference to the 2D problem, let us define a (P Q, K,Λ, t)-almost difference set as aK
-subset D _{=} _{{}d_{k} _{∈}G_{, k}_{= 0}_{, ..., K}_{−}_{1}_{}} _{of the Abelian group} G _{of order} _{P Q}_{(}G , _{Z}P _{⊗}
ZQ(1),P andQbeing chosen according the Kronecker Decomposition Theorem [21]) for which
the multiset

M_{=}_{{}m_{j} _{= (}d_{h}_{−}d_{ℓ}_{)}_{,}d_{ℓ} _{6}_{=}d_{h}_{;}_{j} _{= 0}_{, ..., K}_{(}_{K}_{−}_{1)}_{−}_{1}_{}}

containstnonzero elements ofG_{each exactly}_{Λ}_{times and the remaining}_{P Q}_{−}_{1}_{−}_{t}_{nonzero}

elements each exactly Λ + 1times [18]. Therefore, an ADS satisfies the following existence condition [17][18]:

K(K_{−}1) =tΛ + (P Q_{−}1_{−}t)(Λ + 1) (1)

whereK _{≥} Λ + 1,0 _{≤} K _{≤} P Q, and0 _{≤} t _{≤} P Q_{−}1. Moreover, it is worth noticing that

DSs areADSs for whicht =P Q_{−}1ort = 0[18].

If D_{is a} _{(}_{P Q, K,}_{Λ}_{, t}_{)}_{-}_{ADS}_{, then it is possible to derive a two dimensional binary sequence}
W _{=} _{{}_{w}_{(}_{p, q}_{) = 1(0)} _{if}_{(}_{p, q}_{)}_{∈}_{(}_{∈}_{/}_{)}D_{;} _{p}_{= 0}_{, ..., P} _{−}_{1}_{, q}_{= 0}_{, ..., Q}_{−}_{1}_{}} _{whose} _{2}_{D} _{}

peri-odic autocorrelation function [6]Aw(p, q)(p∈[0, P −1],q ∈[0, Q−1]being its periodicity)
is athree-level function[16][18]
AADS_{w} (p, q) = (K_{−}Λ)δ(p, q) + Λ +
N−1−t
X
r=1
δ(p_{−}lr,1, q−lr,2), (2)

where K _{≥} Λ + 1, δ(p, q) = 1ifp = q = 0and δ(p, q) = 0 otherwise, and (lr,1, lr,2) , lr
is an element of the setL _{=}l_{r} _{∈}_{Z}P _{⊗}_{Z}Q_{, r} _{= 1}_{, ..., N} _{−}_{1}_{−}_{t} _{. For descriptive purposes,}

let us consider the ADSs in Tab. I [18][19]. The plots ofW _{and of the three-level function}

AADS

w (p, q)in correspondence withDi (i= 1,2,3) are shown in Fig. 1.
(1) _{The} _{symbol} _{⊗} _{stands} _{for} _{the} _{direct}_{sum}_{of} _{Z}P _{and}

ZQ, that is G =

gi= (αj, βh), αj ∈ZP, βh∈ZQ, i= 0, ..., P Q−1, j = 0, ..., P −1, h= 0, ..., Q−1 andGis equipped
with the component-wise operations derived fromZP_{and}_{Z}Q_{, that is}_{g}

As regards to the closeness of the ADS to the DS sequences, likewise 1D arrangements, the bidimensional autocorrelation function of a (P Q, K,Λ, t)-ADS differs from ADS

w (p, q) =

K δ(p, q) + Λ[6] by a unity in onlyP Q_{−}1_{−}tpoints [16][18] [Eq. (2)]. Moreover, theADS

autocorrelation function still remains unaltered after cyclic shifts of the reference sequence
[16][18] since ifD_{is an}_{ADS}_{, then}

D(σx,σy)_{=}

{((p+σx)mod P,(q+σy)mod Q); (p, q)∈D, σx, σy ∈Z} (3)
is still anADS. As a consequence, starting from a(P Q, K,Λ, t)-ADS, it is always possible to
buildP _{×}Qdifferent(P Q, K,Λ, t)-ADSs by applying cyclic shifts to its elements.

**3**

**ADS-Based Planar Arrays - Mathematical Formulation**

Let us consider a planar array of P _{×}Q elements located, according to the binary sequence

W_{, on a bidimensional lattice of points spaced by} _{d}_{x} _{and} _{d}_{y} _{wavelenghts along the} _{x} _{and}_{y}

directions, respectively. The array factor of such an elements arrangement turns out to be [6][1]

AF_{{}W_{}},_{W}_{AF}_{(}_{u, v}_{) =}
P−1
X
p=0
Q−1
X
q=0
w(p, q)exp[2πj(pdxu+qdyv)] (4)

whereu = sin(θ)cos(φ)and v = sin(θ)sin(φ). Moreover, WAF(u, v)can be also expressed
in terms of the2DDiscrete Time Fourier transform (DT F T) of the sequenceW_{,}

DT F T_{{}W_{}},_{T}_{(}_{α, β}_{) =}
P−1
X
p=0
Q−1
X
q=0
w(p, q)exp[_{−}j(pα+qβ)], (5)
as follows
WAF(u, v) =T(−2πdxu,−2πdyv). (6)

Furthermore, by applying the Sampling Theorem [22] to the function_{T}(α, β),

T(α, β) =
P−1
X
k=0
Q−1
X
l=0
F(k, l) sin
αP
2 −kπ
P sin α
2 −
kπ
P
sin βQ_{2} _{−}lπ
Q sinβ_{2} _{−} lπ_{Q}
, (7)

Fbeing 2D Discrete Fourier Transform (DF T) of the sequenceW _{(}_{DF T}_{{}W_{}} , _{F}_{(}_{k, l}_{) =}
PP−1
p=0
PQ−1
q=0 w(p, q)exp
h
−2πjpk_{P} + ql_{Q}i), it results that
WAF(u, v) =
P−1
X
k=0
Q−1
X
l=0
F(k, l) sin(πdxuP +kπ)
P sin πdxu+ kπ_{P}
sin(πdyvQ+lπ)
Q sinπdyv+ lπ_{Q}
. (8)

Such a relationship states that the samples of the array factor atu = k dxP,v =

l

dyQ are equal to

the values of theDF T of the weighting sequenceW_{in}_{(}_{k, l}_{)}

WAF
−_{d}k
xP
,_{−} l
dyQ
=F(k, l). (9)

For illustrative purposes, Figure 2 shows the plot of the array factor (dx = dy = 1_{2}) and the
samples of the DF T ofW_{in correspondence with the set}D

1 *[Fig. 2(a)] and the set* D2 [Fig.
*2(b)].*

As regards to the peak sidelobe level (P SL), it is defined as

P SLnD(σx,σy)o, max(u,v)∈/R|W (σx,σy) AF (u, v)|2 |W(σx,σy) AF (0,0)|2 (10) whereW(σx,σy)

AF (u, v)is the array factor coming from the shifted setD

(σx,σy)_{and}_{R}_{is the }

main-lobe region of extension (see Appendix)

R ,
(u, v)_{∈} [_{−}1,1]_{×}[_{−}1,1] : u2+v2 _{≤}1, uv _{≤} K
4P Qdxdymax(k,l)∈H0|F(k, l)|
(11)
with_{H}0 ,G\(0,0)(2)

By substituting (8) in (10), it appears that

P SLnD(σx,σy)o_{=}
max(u,v)∈R/
˛
˛
˛
˛
˛
PP−1
k=0
PQ−1
l=0 F(σx,σy)(k,l)
sin(πdxuP+kπ)
P sin(πdxu+kπ_{P} )
sin(πdyvQ+lπ)
Q sin(πdyv+lπ_{Q})
˛
˛
˛
˛
˛
2
˛
˛
˛
˛
˛
PP−1
k=0
PQ−1
l=0 F(σx,σy)(k,l)
sin(kπ)
P sin(kπ_{P})
sin(lπ)
Q sin(lπ_{Q})
˛
˛
˛
˛
˛
2 =
= 1
K2max(u,v)/∈R
PP−1
k=0
PQ−1
l=0 F(σx,σy)(k, l)
sin(πdxuP+kπ)
P sin(πdxu+kπ_{P})
sin(πdyvQ+lπ)
Q sin(πdyv+lπ_{Q})
2
.
(12)

(2) _{The notation}_{G}_{\}_{(0}_{,}_{0)}_{indicates the set of elements of the Abelian group}_{G}_{without the null element,}

since _{F}(σx,σy)_{(0}_{,}_{0) =} PP−1
p=0
PQ−1
q=0 w(σx,σy)(p, q) = K, W(σx,σy) =
w(σx,σy)_{(}_{p, q}_{);} _{p} _{=}
0, ..., P_{−}1; q= 0, ..., Q_{−}1_{}}being the two-dimensional sequence derived fromD(σx,σy)_{. As it}

can be noticed, theP SLof anADS-based array is a function of the coefficients_{F}(σx,σy)_{(}_{k, l}_{)}_{.}

Unfortunately, since these coefficients cannot be expressed in closed-form (but their values are available when the generatingADSis known) and, unlikeDSs, depends on the indexeskandl, it is not possible to provide aP SLthreshold as forDSs-based planar arrays [6]. Nevertheless, the following set of inequalities holds true for sufficiently large values of P andQ(Appendix)

P SLIN F ≤P SLmin ≤P SLopt{D} ≤P SLmax ≤P SLSU P (13)

whereP SLopt_{{}_{D}_{}}_{=}_{min}
(σx,σy)
h
P SLnD(σx,σy)oi_{,}_{P SL}
min = Ω_{K}{D2},P SLmax =
Ω{D}E{Γopt_{P Q}}
K2 ,
P SLIN F =
K−Λ−
q
(t+1)(P Q−1−t)
P Q−1
K2 , P SLSU P =
“
K−Λ+√(t+1)(P Q−1−t)”E{Γopt_{P Q}}
K2 , Ω{D} =
max(k,l)∈H0
F(σx,σy)(k, l)
2
, andE
Γopt_{P Q} _{≈ −}0.1 + 1.5log10(P Q).

It is now worth pointing out that P SLmin andP SLmax can be evaluated only once theADS sequence is exactly known, since the knowledge of the term F(σx,σy)(k, l)

2

is required, while
the boundsP SLIN F andP SLSU P *can be a-priori determined starting for the knowledge of the*
characteristic parameters describing the ADS (i.e.,P,Q,K,Λ, andt).

For a preliminary validation of such an estimate criterion, let us refer to the planar array gen-erated by D

3 in Tab. I. As expected, theP SLof the setD (σx,σy)

3 depends on the values ofσx
andσy *[Fig. 3(a)] and different shift values give the same optimal*P SL,P SLopt, whose value
*lies into the range of confidence defined in (13) [Fig. 3(b)]. The multiplicity of the optimal*
solutions indicates that less thanP _{×}Qevaluations are actually needed to identify the optimal

ADS-based planar array. This is a negligible computational cost compared to the burden re-quired by stochastic optimization techniques to determine a thinned arrangement on the same aperture.

**4**

**Numerical Analysis**

In this section, the results of a numerical assessment are described and discussed to point out po-tentialities and limitations of theADS-based approach proposed as a suitable tool for predicting

the performance of an effective set of planar thinned arrays. For comparison purposes, random
arrangements [3][6] are considered as reference since, likewiseADSarrays, their performances
*can be a-priori estimated. More in detail, the estimator of the normalized peak sidelobe level*
*of planar random arrays (*RND) turns out to be [3]

P SLRN D =
−ln
1_{−}β λ
2
π2x_{0}y_{0(}P−1)(Q−1)
+ 1_{−}2
ln
1_{−}β λ
2
π2x_{0}y_{0(}P−1)(Q−1)
−1
K (14)

where β is the probability or confidence level that no sidelobe exceeds the P SLRN D value.
*Moreover, random lattice planar arrays (*RNL), whose elements are located on a
uniformly-spaced lattice of points over the aperture, exhibit the followingP SL[6]

P SLRN L=P SLRN D×(1−ν) (15)

(whereν= _{P Q}K is the thinning factor).

The first numerical example deals with the analysis of theP SLbounds (13) versusη , t
P Q−1
for different apertures and when ν = 0.5 (Fig. 4). As expected (Sect. 2), the upper bound
of P SL tends to P SLDS when η = 1 and η = 0 (P Q → ∞) and its value, P SLSU P, is
always below P SLRN D andP SLRN L except for a small set ofη values close toη = 0.5and
large apertures (P Q _{≥} 106_{). As a matter of fact, whatever the array dimension, the worst}
performances verify in correspondence with η = 0.5. Therefore, such an index value will be
analyzed in the following to provide “worst-case” indications onADS-based thinning.

*Figure 5(c) shows the behaviors of the* ADS bounds versus the aperture dimension (ν = η =
0.5). SinceADS are here available [19],P SLopt, P SLmin, andP SLmaxare reported, as well.
As it can be noticed, these plots confirm that P SLSU P usually overestimates the actual peak
sidelobe of theADSarray (whileP SLmin →P SLIN F) and thatP SLoptis always well below
the values exhibited by random families.

For completeness, the remaining of Fig. 5 gives an indication on the estimated behavior ofADS

arrays in correspondence with different thinning percentages [ν = 0.3*- Fig. 5(a),*ν= 0.4- Fig.
*5(b),* ν = 0.6*- Fig. 5(d)] for which* ADSs are not still available. As regards to the confidence

range∆ADS, defined as

∆ADS ,

P SLSU P

P SLIN F

, (16)

it slightly increases with P Qand shows a limited dependence on the aperture dimension (_{∼}

4dB in102 _{≤} _{P Q} _{≤} _{10}6_{) (Fig. 6). Moreover,}_{∆}

ADS(ν) = ∆ADS(1−ν)and the minimum value of∆ADS verifies forν= 0.5as it can be analytically derived.

Concerning available ADSs with ν _{6}= 0.5, Figure 7 shows the behavior of the P SLopt (and
related bounds) of the array generated from the sequence D

3 (η= 0.473andν = 0.485) whose
*power pattern and elements arrangement are given in Fig. 7(c) and Fig. 7(d), respectively.*
Despite the small aperture (3λ_{×}5λ), P SLoptstill lies in the range of values estimated by (13)
*[Figs. 7(a)-7(b)] and it appears to be significantly below the random estimates and comparable*
with the DS value at η = 1. It is also interesting to notice that the reference array derived
fromD

3allows one to determine several shifted array configurations withP SL n

D_{3}(σx,σy)o_{≤}

P SLSU P *[Fig. 3(b)] as well as multiple arrays with*P SL
n

D_{3}(σx,σy)o_{≤}_{P SL}

max. Such a feature is not only limited to D

3, but it is a common property ofADS-based arrays as confirmed by the examples in Figs. 8-10 and concerned with larger apertures. Furthermore, it should be pointed out that more than one cyclic shift of the reference ADS sequence D

i

(i = 4, ...,7) gives an array pattern with P SLnD_{i}(σx,σy)o _{=} _{P SL}

opt *[Figs. 3(b), 8(a), 9(a),*
*10(a)]. Such considerations highlight that: (a) also through an exhaustive search, less than*

P _{×}Q evaluations are actually needed to identify the optimalADS*-based planar array; (b) a*
very limited number of evaluations is enough to synthesize an ADS array with a P SLvalue
below that from random/random lattice distributions.

*As far as the radiation patterns are concerned, Figures 7(c)-10(c) allow one to point out a *
fur-ther interesting property of ADS planar arrays. Unlike DSs, where_{|}_{F}(k, l)_{|} is a two-valued
function [6], the unequal magnitudes of the samples of_{|}_{F}(k, l)_{|}(Fig. 2) lead to a non-uniform
behavior of the array pattern outside the mainlobe region with some non-negligible variations
*of the sidelobes [see Figs. 7(c)-10(c)]. This can be exploited as an additional degree of freedom*
to be used in antenna synthesis. One efficient way to do that is to consider directive elements.
As an example, let us consider the planar arrays synthesized fromD

4with isotropic or directive
elements (e.g., λ_{2} dipoles along they*axis). Figure 11(a) gives the*P SLvalues for different shifts

of the reference set. As it can be observed, the value of P SLopt reduces (P SLdiropt = −23.66 dB vs. P SLopt = −21.79dB) thanks to the use of directive elements and, more interestingly, the optimal shift for the directive array is not equal to that with isotropic elements (σx,optdir = 5,

σdir

y,opt = 20 vs. σx,opt = 7, σy,opt = 5). This is due to the following. One has to determine the
*shift generating the lowest lobes in the whole sidelobe region when dealing with the “isotropic”*
*array [Fig. 11(b)]. Otherwise, the use of directive elements suggests to choose the* σx andσy
*values with lowest sidelobes only near the mainlobe region [Fig. 11(c)] since the element factor*
“erases” the highest sidelobes far from the mainlobe region in the resulting antenna pattern [Fig.
*11(d)].*

The last section of the numerical validation is aimed at giving some indications on the perfor-mance of theADS arrays versus those coming from state-of-the-art thinning techniques based on stochastic optimizers [7][23][5]. SinceADSs are not still available in correspondence with the thinning percentage of the test cases under analysis, the comparison cannot be considered fully fair, but it can be useful to suggest some guidelines for a fast and reliable choice of the most suitable synthesis procedure as well as on the achievableP SLresults.

Figure 12 shows theP SL*of the thinned arrays optimized with the methods in [5] [Fig. 12(a)],*
*[7] [Fig. 12(b)], and [23] [Fig. 12(c)], respectively, along with the* P SLbounds derived for
the correspondingADS-based arrays (i.e., only the values of P SLSU P andP SLIN F since the

ADS sequences, although theoretically existing, have not been yet determined). As it can be noticed,ADS-based arrays compare favourably in terms ofP SLwith global optimized designs since, even in the worst case (i.e.,η= 0.5),P SLIN F < P SLglo ≤P SLSU P .

**5**

**Conclusions**

In this paper,ADSs have been considered for the design of thinned planar arrays. The research work is aimed at identifying the descriptive parameters of theADS-based thinning technique as well as their effect on the array performances. Likewise the linear case [20], the objective of this study is to analytically define a “term of comparison” to help the array designer in identify-ing the synthesis approach allowidentify-ing the optimal trade-off between computational resources and the achievable result in terms of P SLlevel. Towards this purpose, the performances of planar

ADS-based arrays have been investigated and suitable bounds for theP SL have been deter-mined thanks to a new formulation based on the properties of the two-dimensionalDF T. Such an analysis has been validated by means of a large set of numerical experiments also aimed at comparing the predicted ADS performances with those of random distributions or stochas-tically optimized arrays. The obtained results have pointed out the following features of the

ADS thinning technique:

• theP SL*of the synthesized pattern is a-priori known when the*ADS sequence is

avail-able in an explicit form, while suitavail-able bounds are predictavail-able, otherwise;

• because of the three-level autocorrelation function,ADS sequences guarantee additional

degrees-of-freedom (compared to the DS case) to be profitably exploited (e.g., using

directive elements) for fitting the design constraints;

• unlike iterative optimization or trial-and-test random synthesis techniques, the approach

determines the array configuration just through simple shifts of a reference ADS

se-quence;

• thanks to the availability of rich repositories ofADSs also concerned with largeP andQ

indexes, wide apertures (impracticable for stochastic optimizers) can be dealt with;

• the use of ADS does not prevent their integration with optimization techniques, vice

versa it could represent a way (to be explored in successive researches) to improve the convergence rate of iterative methods or for enabling stochastic searches in thinning large arrays by means of a suitableADS-based initialization.

**Acknowledgements**

A. Massa wishes to thank E. Vico for being and C. Pedrazzani for her continuous help and patience. Moreover, the authors are very grateful to M. Donelli for kindly providing some numerical results of computer simulations.

**Appendix**

* - Definition of the Mainlobe Region*R

Starting from (12) as for planar DS arrays [6], it can be proved that the P SLof ADS
-based arrays is close to the values of the samples of the array factor at u = u_{m+}1

2 ,
m+1/2
P x0 ,
v =v_{n+}1
2 ,
n+1/2

Qy0 . By exploiting such an observation, it results that

P SLnD(σx,σy)o_{≈} 1
K2max„
u_{m}_{+ 1}
2,vn+ 12
«
/
∈R
K(−1)m+n
P Q sin[_{P}π(m+1_{2})]sin[_{Q}π(n+1_{2})]+
+PP_{k=0,kl}−1 _{6}_{=0}PQ_{l=0,kl}−1 _{6}_{=0} F(σx,σy)(−1)m+k+n+l
P Q sin[_{P}π(m+k+1_{2})]sin[π_{Q}(n+l+1_{2})]
2
.
(17)

where the mainlobe region,R, is defined analogously to [6] as the visible region where the first
term in (17) exceeds the magnitude of the second one. As regards to the first term, its magnitude
is approximately equal to
K
π2 _{m}_{+}1
2
n+ 1_{2}

and for large values of P and Q. Moreover, the largest coefficients in the second term (i.e.,

m+k+1
2 =±1andn+l+
1
2 =±1) of (17) are bounded by
4max(k,l)∈H0|F
(σx,σy)_{|}
π2 .

Thus, after simple manipulation, it is possible to show thatR extends to the region limited by the following boundary inequality

u_{m+}1
2vn+
1
2 ≤
K
4P Qx0y0max(k,l)∈H0|F(σx,σy)|
. (18)
* - Derivation of* P SLSU P

**in (13)**With reference to discrete version ofR,RD,

RD ,
(m, n)_{∈}Z×Z:
m+1
2 n+
1
2
≤ _{4}_{max} K
(k,l)∈H0|WDF T(k, l)|
(19)

let us consider the following approximation ofP SLopt_{{}_{D}_{}}_{=}_{min}
(σx,σy)
h
P SLnD(σx,σy)oi
P SLopt_{{}D_{}}._{min}_{(σ}
x,σy)
"
max(k,l)∈H0
|F(σx,σy)_{(}_{k, l}_{)}_{|}2
K2 × (20)
×max(m,n)/∈RD
P−1
X
k=0,kl6=0
Q−1
X
l=0,kl6=0
ejφ(_{kl}σx,σy)_{(}_{−}_{1)}m+k+n+l
P Q sin_{π}
P m+k+
1
2
sinh_{Q}π n+l+ 1_{2}i
2

where the complex coefficient_{F}(σx,σy)_{has been expressed in terms of its amplitude,}p_{|}_{F}(σx,σy)(k, l)|2_{,}

and phase,φ(σx,σy)

kl .

It is worth pointing out that, likewise DSs, φ(σx,σy)

kl *is not a-priori known as well as, unlike*

DSs, the termmax(k,l)∈H0

|F(σx,σy)_{(}_{k, l}_{)}_{|}2 _{and they have to be estimated. Towards this end,}

by exploiting the circular correlation property of DF T [22], it is possible to state that

F(σx,σy)(k, l)
2
=DF T
AADS_{w} (p, q) =K_{−}Λ +P QΛδ(k, l) + Ψ(k, l), (21)

and to obtain the following relationship

max(k,l)∈H0
n
F(σx,σy)(k, l)
2o
=K_{−}Λ +max(k,l)∈H0{Ψ(k, l)} (22)
whereΨ(k, l),DF T _{{}ψ(p, q)_{}}beingψ(p, q),PP Q_{r=1}−1−tδ(p_{−}lr,1, q−lr,2).

Concerning the real-valued coefficientsΨ(k, l), by applying the Parseval’s theorem [22]

1
P Q
P−1
X
k=0
Q−1
X
l=0
|Ψ (k, l)_{|}2 =
P−1
X
p=0
Q−1
X
q=0
|ψ(p, q)_{|}2 =P Q_{−}1_{−}t,

and noticing thatPP_{p=0}−1PQ_{q=0}−1_{|}ψ(p, q)_{|}2 _{=}_{P Q}_{−}_{1}_{−}_{t}_{and}_{Ψ(0}_{,}_{0) =} _{P Q}_{−}_{1}_{−}_{t}_{, the following}
holds true
P−1
X
k=0,kl6=0
Q−1
X
l=0,kl6=0
|Ψ(k, l)_{|}2 = (t+ 1)(P Q_{−}1_{−}t). (23)
Therefore, since
max(k,l)∈H0{Ψ(k, l)} ≤max(k,l)∈H0{|Ψ(k, l)|} ≤
p
(t+ 1)(P Q_{−}1_{−}t) (24)

and substituting (22) in (20), we obtain
P SLopt_{{}D_{}}._{K} _{−}_{Λ +}p_{(}_{t}_{+ 1)(}_{P Q}_{−}_{1}_{−}_{t}_{)}_{×} _{(25)}
×min(σx,σy)
max(m,n)/∈RD
PP−1
k=0,kl6=0
PQ−1
l=0,kl6=0
ejφ
(σx,σy)
kl (−1)m+k+n+l
P Q sin[_{P}π(m+k+1_{2})]sin[_{Q}π(n+l+1_{2})]
2
K2
.

As regards the phase terms φ(σx,σy)

kl , the analysis is carried out as in [6] in order to give an estimate of the P SL. More specifically, although the phase terms φ(σx,σy)

kl are deterministic quantities, they are dealt with as independent identically distributed (i.i.d.) uniform random variables. Under this assumption, (25) can be expressed as

P SLopt_{{}D_{}}_{.}

K_{−}Λ +p(t+ 1)(P Q_{−}1_{−}t)Γopt

K2 (26)

where for largeP andQ

Γopt _{≈}min(σx,σy){max[Ui; i= 1, ...,Π]} (27)

Ui ,
P∞
k=−∞
P∞
l=−∞ e
jφ(_{kl}σx,σy)
π2_{(}_{k+}1
2)(l+12)
2

,i = 1, ...,Π, being i.i.d. random variables and Πis the
cardinality of RD (≈ P Q). Since the statistics ofΓopt are not available in closed form, Monte
Carlo simulations have been performed to provide an approximation of its mean valueE_{{}Γopt_{}}

E

Γopt _{≈ −}_{0}_{.}_{1 + 1}_{.}_{5}_{log}

10(P Q). (28)

By substituting (28) in (26), the upper boundP SLSU P is finally obtained.

* - Derivation of* P SLIN F

**in (13)**By sampling (12) at (u = _{P d}s_{x}, v = _{Qd}t_{y}), s = 0, ..., P _{−}1, t = 0, ..., Q_{−}1, it can be easily
shown that

P SLopt_{{}D_{} ≥} _{P SL}nD(σx,σy)ok
u= s
P dx,v=Qdyt
= (29)
=
max(s,t)∈H_{0}
˛
˛
˛
˛
˛
PP−1
k=0
PQ−1
l=0 F(σx,σy)(k,l)
sin[π(s+k)]
P sin[π(s_{P}+k)]
sin[π(t+l)]
Q sin[π(t_{Q}+l)]
˛
˛
˛
˛
˛
2
K2 =
max(s,t)∈H_{0}|F(σx,σy)(s,t)|
2
K2

By substituting (22) in (29), and observing that

max(k,l)∈H0{Ψ(k, l)} ≥ −
s
(t+ 1)(P Q_{−}1_{−}t)
P Q_{−}1 (30)
it turns out
P SLopt_{{}D_{} ≥}
K_{−}Λ_{−}q(t+1)(P Q_{P Q}_{−}−_{1}1−t)
K2 ,P SLIN F. (31)
* - Derivation of* P SLmax

**in (13)**With reference to (20), let us assume that theADS D_{at hand is known. Thus,}

Ω_{{}D_{}},_{max}_{(k,l)}_{∈H}
0
F(σx,σy)(k, l)
2
(32)

is now a known quantity. By substituting (32) in (20), we obtain

P SLopt_{{}D_{}}._{Ω}_{{}D_{}}_{min}_{(σ}
x,σy)
max(m,n)/∈RD
PP−1
k=0,kl6=0
PQ−1
l=0,kl6=0
ejφ
(σx,σy)
kl (−1)m+k+n+l
P Q sin[π
P(m+k+12)]sin[Qπ(n+l+12)]
K2
2
.
(33)

As regards to the phase terms φ(σx,σy)

kl , let us consider the same procedure used for deriving

P SLSU P and let us rewrite (33) as follows

P SLopt_{{}D_{}}. Ω{D}Γ

opt

K2 (34)

whereΓopt_{is successively approximated with its mean value (28) to obtain}_{P SL}
max.

* - Derivation of* P SLmin

**in (13)**Analogously to the derivation ofP SLmax, a lower bound forP SLopt whenDis known can be obtained starting from (29) and employing (32):

P SLopt_{{}D_{} ≥} max(s,t)∈H0
F(σx,σy)(s, t)
2
K2 =
Ω_{{}D_{}}
K2 ,P SLmin. (35)

**References**

[1] *C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997.*

[2] *Y. T. Lo, “A mathematical theory of antenna arrays with randomly spaced elements,” IEEE*

*Trans. Antennas Propag., vol. 12, no. 3, pp. 257-268, May 1964.*

[3] B. Steinberg, “The peak sidelobe of the phased array having randomly located elements,”

*IEEE Trans. Antennas Propag., vol. 20, no. 2, pp. 129-136, Mar. 1972.*

[4] D. G. Leeper, “Thinned periodic antenna arrays with improved peak sidelobe level con-trol,” U.S. Patent 4071848, Jan. 31, 1978.

[5] *R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag.,*
vol. 42, no. 7, pp. 993-999, Jul. 1994.

[6] D. G. Leeper, “Isophoric arrays - massively thinned phased arrays with well-controlled
*sidelobes,” IEEE Trans. Antennas Propag., vol. 47, no. 12, pp. 1825-1835, Dec 1999.*

[7] S. Caorsi, A. Lommi, A. Massa, and M. Pastorino, “Peak sidelobe reduction with a hybrid
*approach based on GAs and difference sets,” IEEE Trans. Antennas Propag., vol. 52, no.*
4, pp. 1116-1121, Apr. 2004.

[8] *R. L. Haupt and D. H. Werner, Genetic algorithms in electromagnetics. Hoboken, NJ:*
Wiley, 2007.

[9] B. Steinberg, “Comparison between the peak sidelobe of the random array and
*algorith-mically designed aperiodic arrays,” IEEE Trans. Antennas Propag., vol. 21, no. 3, pp.*
366-370, May 1973.

[10] S. Holm, B. Elgetun, and G. Dahl, “Properties of the beampattern of weight- and
*layout-optimized sparse arrays,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 44, no.*
5, pp. 983-991, Sep. 1997.

[11] T. G. Spence, D. H. Werner, “Thinning of aperiodic antenna arrays for low side-lobe lev-els and broadband operation using genetic algorithms,” IEEE Antennas and Propagation Society International Symposium 2006, pp. 2059-2062, 9-14 July 2006.

[12] *A. Trucco and V. Murino, “Stochastic optimization of linear sparse arrays,” IEEE J. Ocean*

*Eng., vol. 24, no. 3, pp. 291-299, Jul. 1999.*

[13] *A. Trucco, “Thinning and weighting of large planar arrays by simulated annealing,” IEEE*

*Trans. Ultrason., Ferroelectr., Freq. Control, vol. 46, no. 2, pp. 347-355, Mar. 1999.*
[14] *L. E. Kopilovich, “Square array antennas based on hadamard difference sets,” IEEE Trans.*

*Antennas Propag., vol. 56, no. 1, pp. 263-266, Jan. 2008.*

[15] *La Jolla Cyclic Difference Set Repository (http://www.ccrwest.org/diffsets.html).*

[16] C. Ding, T. Helleseth, and K. Y. Lam, “Several classes of binary sequences with three-level
*autocorrelation,” IEEE Trans. Inf. Theory, vol. 45, no. 7, pp. 2606-2612, Nov. 1999.*

[17] K. T. Arasu, C. Ding, T. Helleseth, P. V. Kumar, and H. M. Martinsen, “Almost difference
*sets and their sequences with optimal autocorrelation,” IEEE Trans. Inf. Theory, vol. 47,*
no. 7, pp. 2934-2943, Nov 2001.

[18] Y. Zhang, J. G. Lei, and S. P. Zhang, “A new family of almost difference sets and some
*necessary conditions,” IEEE Trans. Inf. Theory, vol. 52, no. 5, pp. 2052-2061, May 2006.*

[20] G. Oliveri, M. Donelli, and A. Massa, “Linear array thinning exploiting almost difference
*sets," IEEE Trans. Antennas Propag., revised.*

[21] *M. I. Kargapolov and J. I. Merzljako, Fundamentals of the Theory of Groups. New York:*
Springer-Verlag, 1979.

[22] *J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and*

*Applications, 3nd ed. London: Prentice Hall, 1996.*

[23] M. Donelli, A. Martini, and A. Massa, “A hybrid approach based on PSO and Hadamard
*difference sets for the synthesis of square thinned arrays,” IEEE Trans. Antennas Propag.*
*(in press).*

**FIGURE CAPTIONS**

• **Figure 1. Autocorrelation functions and associated binary sequences of the**ADSs in Tab.
*I: (a)(d)*D

1*, (b)(e)*D2*, and (c)(f )*D3.

• **Figure 2. Plot of the normalized array factor derived from** D

i =D

(σx=0, σy=0)

i and
asso-ciated_{|}_{F}(k, l)_{|}*values: (a)*i= 1*and (b)*i= 2*.*

• **Figure 3.** P SL values of the ADS-based planar arrays derived from the sequences

D(σx, σy)

3 , σx = 0, ..., P − 1,σy = 0, ..., Q −1 *(a) and* P SL *bounds (b). Number of*
elements:P _{×}Q= 7_{×}11- Aperture size: 3λ_{×}5λ.

• * Figure 4. Numerical Validation. Plots of the*P SLbounds ofADS-based planar arrays,
the estimator of theP SLof random (RND) and random lattice (RNL) arrays, and values
of the P SLof DS-based finite arrays versus η whenν = 0.5

*and (a)*P Q = 102

_{, (b)}P Q= 104_{, (c)}_{P Q}_{= 10}6_{.}

• * Figure 5. Numerical Validation. Plots of the*P SLbounds ofADS-based planar arrays
and the estimators of the P SL of random (RND) and random lattice (RNL) arrays
versus the array aperture, P Q, whenη = 0.5

*and (a)*ν = 0.3

*, (b)*ν = 0.4

*, (c)*ν = 0.5,

*(d)*ν = 0.6.

• * Figure 6. Numerical Validation. Plots of* ∆ADS versus the array aperture, P Q, when

η = 0.5and in correspondence with different thinning values [ν _{∈}[0, 1]].

• * Figure 7. Numerical Validation - Planar Array* Dopt

_{3}

_{[Number of elements:}

_{P}

_{×}

_{Q}

_{=}

7_{×}11- Aperture size: 3λ_{×}5λ*]. Plots of the*P SL*bounds versus (a) the array aperture*

P Q[ν = 0.4805,η = 0.4736*] and (b)*η[P Q= 77,ν = 0.4805]. Plot of the normalized
*array factor (c) generated from the*Dopt

3 *-based array arrangement (d).*

• * Figure 8. Numerical Validation - Planar Array* Dopt

4 [Number of elements: P × Q =

23_{×}23- Aperture size: 11λ_{×}11λ*].* P SLvalues of theADS-based arrays derived from
the sequencesD(σ_{4} x, σy)_{,}_{σ}_{x} _{= 0}_{, ..., P}_{−}_{1}_{,}_{σ}_{y} _{= 0}_{, ..., Q}_{−}_{1}_{(a). Plots of the}_{P SL}_{bounds}

*versus (b)*η[P Q= 529,ν = 0.5*]. Plot of the normalized array factor (c) generated from*
theDopt

4 *-based array arrangement (d).*

• * Figure 9. Numerical Validation - Planar Array* Dopt

5 [Number of elements: P × Q =

73_{×}23- Aperture size: 36λ_{×}36λ*].* P SLvalues of theADS-based arrays derived from
the sequencesD(σx, σy)

5 ,σx = 0, ..., P−1,σy = 0, ..., Q−1*(a). Plots of the*P SLbounds
*versus (b)* η [P Q = 5329, ν = 0.5*]. Plot of the normalized array factor (c) generated*
from theDopt

5 *-based array arrangement (d).*

• * Figure 10. Numerical Validation - Planar Array*Dopt

6 [Number of elements: P ×Q =

199_{×}199 - Aperture size: 99λ_{×}99λ*].* P SLvalues of the ADS-based arrays derived
from the sequencesD(σx, σy)

6 ,σx = 0, ..., P −1,σy = 0, ..., Q−1*(a). Plots of the*P SL
*bounds versus (b)* η [P Q = 39601, ν = 0.5*]. Plot of the normalized array factor (c)*
generated from theDopt

6 *-based array arrangement (d).*

• * Figure 11. Numerical Validation - Non-Isotropic elements.* P SLvalues of the ADS
-based arrays generated from the sequences D(σx, σy)

4 ,σx = 0, ..., P −1,σy = 0, ..., Q−1
*(a). Normalized array patterns of the arrays generated from the sequences (b)* Dopt

4 =

D(7,5)_{4} _{with isotropic elements,}D(5,20)_{4} _{with isotropic (c) and directive elements (d).}

• * Figure 12. Comparative Assessment. Plots of the*P SLbounds versus the array aperture,

P Q, whenη = 0.5*and for (a)*ν = 0.54*[5], (b)*ν = 0.507 *[7], (c)*ν = 0.48*[23], (d)*

ν = 0.44[23].

**TABLE CAPTIONS**

A_{w}ADS (p,q)
K
Λ+1
Λ
0
1
2
3
p
0
1 2
3
4 5
6
q
6
8
10
12
14
16
*(a)* *(d)*
A_{w}ADS (p,q) _{K}
Λ+1
Λ
0
1
2
3
4
5
6
p
0
1
2
3
4
5
6
q
12
14
16
18
20
22
24
26
*(b)* *(e)*
A_{w}ADS_{ (p,q)} _{K}
Λ+1
Λ
0
1
2
3
4
5
6
p
0
2
4
6
8
10
q
15
20
25
30
35
40
*(c)* *(f )*

*(a)*

*(b)*

-15
-13
-11
-9
PSL(D
3
(
σx
,
σy
) ) - Normalized value [dB]
0
1
2
3
4
5
6
0
2
4
6
8
10
σx
σy
*(a)*
-20
-15
-10
-5
77
0 10 20 30 40 50 60 70
Normalized value [dB]
Qσ_{x}+σ_{y}
PSL_{opt}
PSL(D_{3}(σx,σy)_{)}
PSL_{INF}
PSL_{SUP}
PSL_{min}
PSL_{max}
*(b)*

-25
-20
-15
-10
-5
0
0 0.2 0.4 0.6 0.8 1
Normalized value [dB]
η
PQ=102, ν=0.5
PSLRND (β=0.99)
PSL_{RNL} (β=0.99)
PSLINF
PSL_{SUP}
PSL_{DS}
*(a)*
-45
-40
-35
-30
-25
-20
0 0.2 0.4 0.6 0.8 1
Normalized value [dB]
η
PQ=104, ν=0.5
PSLRND (β=0.99)
PSLRNL (β=0.99)
PSL_{INF}
PSL_{SUP}
PSL_{DS}
*(b)*
-65
-60
-55
-50
-45
-40
-35
0 0.2 0.4 0.6 0.8 1
Normalized value [dB]
η
PQ=106, ν=0.5
PSL_{RND} (β=0.99)
PSLRNL (β=0.99)
PSL_{INF}
PSL_{SUP}
PSLDS
*(c)*

-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
102 103 104 105
Normalized value [dB]
PQ
ν=0.3, η=0.5
PSL_{INF}
PSL_{SUP}
PSLRND (β=0.99)
PSL_{RNL} (β=0.99)
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
102 103 104 105
Normalized value [dB]
PQ
ν=0.4, η=0.5
PSL_{INF}
PSL_{SUP}
PSLRND (β=0.99)
PSL_{RNL} (β=0.99)
*(a)* *(b)*
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
102 103 104 105
Normalized value [dB]
PQ
ν=0.5, η=0.5
D4
opt
D_{5}opt
D6opt
PSL_{INF}
PSLSUP
PSLRND (β=0.99)
PSLRNL (β=0.99)
PSLmin
PSL_{opt}
PSLmax
-55
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
102 103 104 105
Normalized value [dB]
PQ
ν=0.6, η=0.5
PSL_{INF}
PSL_{SUP}
PSLRND (β=0.99)
PSL_{RNL} (β=0.99)
*(c)* *(d)*
**F**
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-B
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2
6

10
11
12
13
14
15
16
102 103 104 105 106
Normalized value [dB]
PQ
η=0.5
∆_{ADS}, ν=0.2
∆_{ADS}, ν=0.3
∆_{ADS}, ν=0.4
∆_{ADS}, ν=0.5
∆_{ADS}, ν=0.6
∆_{ADS}, ν=0.7
∆_{ADS}, ν=0.8

-40
-35
-30
-25
-20
-15
-10
-5
0
10 102 103 104
Normalized value [dB]
PQ
ν=0.4805, η=0.4736
PSL_{INF}
PSL_{SUP}
PSL_{RND} (β=0.99)
PSL_{RNL} (β=0.99)
PSL_{min}
PSL_{max}
PSL_{opt}
-25
-20
-15
-10
-5
0
0 0.2 0.4 0.6 0.8 1
Normalized value [dB]
η
PQ=77, ν=0.4805
PSLRND (β=0.99)
PSL_{RNL} (β=0.99)
PSLINF
PSL_{SUP}
PSL_{DS}
PSL_{min}
PSLmax
PSL_{opt}
*(a)* *(b)*
*(c)* *(d)*
**F**
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2
8

-30
-25
-20
-15
529
0 50 100 150 200 250 300 350 400 450 500
Normalized value [dB]
Q σx+σy
PSL_{opt}

PSL(D_{4}(σx,σy)_{)} PSL_{INF} PSL_{SUP} PSL_{min} PSL_{max}

-35
-30
-25
-20
-15
-10
-5
0
0 0.2 0.4 0.6 0.8 1
Normalized value [dB]
η
PQ=529, ν=0.5
PSLRND (β=0.99)
PSLRNL (β=0.99)
PSLINF
PSL_{SUP}
PSL_{DS}
PSLmin
PSL_{max}
PSL_{opt}
*(a)* *(b)*
*(c)* *(d)*
**F**
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2
9

-40
-35
-30
-25
-20
5329
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Normalized value [dB]
Q σx+σy
PSL_{opt}

PSL(D_{5}(σx,σy)_{)} PSL_{INF} PSL_{SUP} PSL_{min} PSL_{max}

-40
-35
-30
-25
-20
-15
0 0.2 0.4 0.6 0.8 1
Normalized value [dB]
η
PQ=5329, ν=0.5
PSL_{RND} (β=0.99)
PSL_{RNL} (β=0.99)
PSLINF
PSL_{SUP}
PSL_{DS}
PSLmin
PSLmax
PSLopt
*(a)* *(b)*
*(c)* *(d)*
**F**
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3
0

-50
-45
-40
-35
-30
39601
0 5000 10000 15000 20000 25000 30000 35000
Normalized value [dB]
Q σx+σy
PSL_{opt}

PSL(D_{6}(σx,σy)_{)} PSL_{INF} PSL_{SUP} PSL_{min} PSL_{max}

-50
-45
-40
-35
-30
-25
-20
0 0.2 0.4 0.6 0.8 1
Normalized value [dB]
η
PQ=39601, ν=0.5
PSL_{RND} (β=0.99)
PSLRNL (β=0.99)
PSLINF
PSL_{SUP}
PSL_{DS}
PSL_{min}
PSL_{max}
PSLopt
*(a)* *(b)*
*(c)* *(d)*
**F**
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**0**
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D
S
-B
as
ed
G
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id
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fo
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3
1

-25
-20
-15
0 135 166 265 410 441 529
Normalized value [dB]
Q σx+σy
PSL(D4
(σx,σy)_{), Isotropic Elements} _{PSL(D}
4
(σx,σy)_{), Directive Elements}

PSLopt, Isotropic Elements

PSLopt, Directive Elements

*(a)* *(b)*
**F**
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**1**
**1**
**-G**
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3
2

-25
-20
-15
-10
-5
0
0 0.2 0.4 0.6 0.8 1
Normalized value [dB]
η
PQ=200, ν=0.54
PSLRND (β=0.99)
PSLRNL (β=0.99)
[Haupt, 1994]
PSLINF
PSL_{SUP}
PSL_{DS}
-25
-20
-15
-10
-5
0
0 0.2 0.4 0.6 0.8 1
Normalized value [dB]
η
PQ=63, ν=0.507
PSLRND (β=0.99)
PSL_{RNL} (β=0.99)
[Caorsi, 2004]
PSLINF
PSL_{SUP}
PSL_{DS}
*(a)* *(b)*
-25
-20
-15
-10
-5
0
0 0.2 0.4 0.6 0.8 1
Normalized value [dB]
η
PQ=144, ν=0.48
PSL_{RND} (β=0.99)
PSL_{RNL} (β=0.99)
[Donelli, 2009] - HSPSO
PSLINF
PSL_{SUP}
PSL_{DS}
-30
-25
-20
-15
-10
-5
0 0.2 0.4 0.6 0.8 1
Normalized value [dB]
η
PQ=576, ν=0.44
PSL_{RND} (β=0.99)
PSLRNL (β=0.99)
[Donelli, 2009] - HSPSO
PSLINF
PSL_{SUP}
PSL_{DS}
*(c)* *(d)*
**F**
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**1**
**2**
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“A
D
S
-B
as
ed
G
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fo
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...”
3
3

ADS P Q K Λ t ν , K
P Q η ,
t
P Q−1 Reference
D
1 4 7 15 7 6 ≈0.5357 ≈0.22 [18]
D
2 7 7 25 12 24 ≈0.5102 0.5 [18]
D_{3} _{7} _{11} _{37} _{17} _{36} _{≈}_{0}_{.}_{4805} _{≈}_{0}_{.}_{4736} _{[18]}
D
4 23 23 265 132 264 ≈0.5 0.5 [19]
D
5 73 73 2665 1332 2664 ≈0.5 0.5 [19]
D
6 199 199 19801 9900 19800 ≈0.5 0.5 [19]