Aalborg Universitet

Beamforming via large and dense antenna arrays above a clutter

Alrabadi, Osama; Tsakalaki, Elpiniki; Huang, Howard; Pedersen, Gert Frølund

Published in:I E E E Journal on Selected Areas in Communications

DOI (link to publication from Publisher):10.1109/JSAC.2013.130218

Publication date:2013

Document VersionEarly version, also known as pre-print

Link to publication from Aalborg University

Citation for published version (APA):Alrabadi, O., Tsakalaki, E., Huang, H., & Pedersen, G. F. (2013). Beamforming via large and dense antennaarrays above a clutter. I E E E Journal on Selected Areas in Communications, 31(2), 314-325.https://doi.org/10.1109/JSAC.2013.130218

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Beamforming via Large and Dense Antenna Arraysabove a Clutter

Osama N. Alrabadi,Member, IEEE,Elpiniki Tsakalaki,Member, IEEE,Howard Huang,Senior Member, IEEEand Gert F. Pedersen

Abstract— The paper sheds light on the beamforming (BF)performance of large (potentially unconstrained in size) as wellas dense (but physically constrained in size) antenna arrayswhen equipped with arbitrarily many elements. Two operationalmodes are investigated: Single-layer BF and multi-layer BF.In the first mode, a realistic BF criterion namely the averageBF gain is revisited and employed to understand the far-fieldand the near-field effects on the BF performance of large-scaleantennas above a clutter. The diminishing throughput returns in asingle-layer BF mode versus the number of antennas necessitatemulti-layering. In the multi-layer BF mode, the RF coverageis divided into a number of directive non-overlapping sector-beams in a deterministic manner within a multi-user multi-inputmulti-output (MIMO) system. The optimal number of layersthat maximizes the user’s sum-rate given a constrained antennaarray is found as a compromise between the multiplexing gain(associated with the number of sector-beams) and the inter-beaminterference, represented by the side lobe level (SLL).

Index Terms— Beamforming, Capacity, Clustering channels,EM Coupling, MIMO, Large Arrays, High Order Sectorization.

I. I NTRODUCTION

M IMO is a technology that enables multiple paralleldata streams to be communicated by equipping the

transmitter and the receiver with multiple antennas, withoutsacrificing extra bandwidth or transmit power. Thanks to scat-tering environments that make the separation of the transmittedmixture of signals possible by decoding their unique spatialsignatures, the data rate achieved by each individual antennacan be added up so that the multiple antennas act as a data ratemultiplier [1] [2]. Such scattering environments may existinindoor propagation scenarios where signal rays (multipaths)are organized into clusters with wide enough angle spread[3]. However, this is not true when considering a base-station (BS) on top of a clutter as the multipath concentrationseen by a BS in a rural area is only within2 and within5 − 7 in urban environments [4]. Based on this, observingindependent fading channels at the BS dictates the need

Paper received February 1, 2012; revised June 15, 2012. Thispaper ispublished in part in the Loughborough Antennas and Propagation Conference(LAPC) 2012.

Osama N. Alrabadi is with the Antennas, Propagation and RadioNetwork-ing (APNet) group, Department of Electronic Systems, AalborgUniversity,DK-9220 Aalborg, Denmark. (e-mail: ona@es.aau.dk).

Elpiniki Tsakalaki is with the APNet group, Department of Elec-tronic Systems, Aalborg University, DK-9220 Aalborg, Denmark. (e-mail:et@es.aau.dk).

Gert F. Pedersen is with the APNet group, Department of Elec-tronic Systems, Aalborg University, DK-9220 Aalborg, Denmark. (e-mail:gfp@es.aau.dk).

Howard Huang is with Bell Labs, Alcatel Lucent (e-mail:Howard.Huang@alcatel-lucent.com).

Digital Object Identifier XXX/JSAC.2012.XXX.

for an interelement spacing of several wavelengths (almostten times the carrier wavelength), leading to impracticallylarge array architectures. The problem of large arrays can bemitigated by deploying collocated polarized antennas [5],or byusing parasitic antennas [6], i.e., by employing angular ratherthan space diversity. However, having the antenna elementsdecorrelated (be it by distance, polarization or angle) degradesthe BF potential of such antenna arrays as grating lobes, i.e.,multiple main lobes in the array far-field start to appear. Analternative candidate to diversity MIMO systems is to employBF MIMO systems where the BS is equipped with a setof correlated antenna elements (almost of half a wavelengthspacing) leading to a powerful BF system. The motivationbehind such BF communication systems is the fact that thefree-space BF gain (which is the highest possible BF gainin a sector-beam) is almost maintained in low angular spreadchannels [7].

The idea of equipping the BS with BF antennas is scaledup within the ‘massive MIMO’ concept, which proposes toequip the BS with a huge number of antenna elements [8].Theoretical results promise unprecedented capacity gains, RFfrontends complexity reduction as well as remarkable energysavings [9]. The optimal signal precoding in the limitingconditions is found to be matched filtering. From signal spacepoint of view, matched filtering in low angle spread channelstogether with a massive array shapes pencil beams directingtoward the users’ clusters. Massive MIMO is mainly proposedfor time division duplex (TDD) systems as the downlinkBF weights are estimated from the uplink channel. In thispaper, unlike the free-space analysis in [10], we conducta comprehensive analysis regarding the BF performance ofarrays equipped with massive number of antennas, includingboth the far-field scattering environment, i.e., the distributionof the scatterers in the channel, as well as the near-fieldenvironment, i.e., the array electromagnetic (EM) couplingand the power reflections due to impedance mismatch. Thefar-field analysis is simplified by obtaining the diffused sector-beam as a circular convolution of the free-space sector-beamwith the channel angular power spectrum (APS). A properBF criterion is proposed and utilized to evaluate the BFpotential of different array topologies under different prop-agation conditions. On the other hand, the analysis of thenear-field is made tractable by replacing the conventionalarray steering vector with the vector of the active elementresponses, thus taking into account the EM coupling withthe neighboring antenna elements and the coupling with thechassis of the radome. Regarding the TDD mode of operation,simulation results show that the significance of the full channelstate information (CSI) at the transmit BS is small when

communicating in narrow clustering environments comparedto the mere stochastic knowledge. Consequently, instantaneouschannel estimation and pilot contamination requirements canbe relaxed. We also analyze the BF potential when densifyinga constrained physical area with many antennas. It is foundthat, in theory, dense arrays can still have a high BF gainwhile having the simplest matching network. However this istrue when considering only the array matching efficiency butis hard to get in practice because of the higher Ohmic lossesowed to the strong reactive fields [11].

The paper then shifts the analysis from single-layer BF tomulti-layer BF. MIMO in such systems is enabled via beammultiplexing where the users’ signals are mapped onto a set oforthogonal sector-beams thus leading to multi-beam / multi-data stream communication. In the baseband domain this isdone by multiplying the vector of users’ data symbols with aprecoding matrix each column of which shapes a sector-beam(in the signal space) toward a different user. The signal spaceanalysis is made tractable by employing a set of circularlysymmetric sector-beams (i.e., each precoding vector is a cyclicrotation of the other), illuminating a coverage of uniformusers distribution. A multi-layer beamforming is applied onto acircular cylindrical array (measured) prototype thus multiplex-ing B directive sector-beams. The optimal number of sector-beams that maximizes the users’ sum-rate is found as a tradeoff between the multiplexing gain and the SLL (under size-constrained arrays).

Notation: In the following, boldface lower-case and upper-case characters denote vectors and matrices, respectively. Theoperators(·)∗, (·)T, (·)H, (·)−1 designate complex conjugate,transpose, complex conjugate transpose (Hermitian) and ma-trix inverse operators, respectively. The notationIN indicatesan identity matrix of sizeN×N whereas the notionCN (0, σ2)refers to circularly symmetric complex Gaussian distributionwith zero mean andσ2 variance.(·)ij returns thei, j entryof the enclosed matrix and(·)i returns theith element ofthe enclosed vector.‖·‖ denotes the Euclidean norm of theenclosed vector and| · | returns the absolute value.E · isthe expectation operator andC denotes the set of complexnumbers of the specified dimensions. The operators∈ and∼indicate that the (random) variable belongs to a certain setofnumbers and to a certain distribution, respectively.

The rest of the paper is divided as follows: Section IIdescribes different antenna array models to be utilized later forBF and throughput analysis. Section III sheds light on single-layer BF systems whereas Section IV extends the analysis tomulti-layer BF. Finally, Section V suggests some future work.

II. A RRAY SYSTEM MODELS

In this paper we consider different antenna array topologiesranging from a uniform linear array (ULA), a uniform ciculararray (UCA) and uniform cylindrical array (UCyA). The mostcommon and most analyzed geometry is the ULA whichconsists ofN antenna elements placed on a straight line. Weemploy ULA topologies for investigating the BF potential oflarge arrays by increasing the array elements while fixing theinterelement spacing (to a distance at which the EM couplingcan be safely assumed negligible), thus the array length keeps

increasing (potentially size-unconstrained). The steering vectorα(ϕ) ∈ C

N×1 of a ULA of N isotropic elements has theVandermonde structure [12] such that

(α(ϕ))n = exp (jnκd cos(ϕ)) , (1)

wheren ∈ 1, . . . , N − 1, d is the interelement spacing inwavelengths,κ = 2π/λ is the wavenumber,λ is the free-space wavelength andϕ is the azimuthal angle. From signalprecoding point of view, a large number of BF techniques havebeen taken from digital signal processing to shape far-fieldpatterns with small 3dB power beamwidth a.k.a. half-powerbeamwidth (HPBW), or low SLL. These techniques performa type of weighting over the array elements by adoptingproper functions, such as Gaussian, Kaiser-Bessel [14] andBlackman windows [15]. Widely adopted synthesis techniquesfor ULAs are the Dolph-Tschebyscheff [16] and the binomialmethods [17], which consider a nonuniform distribution ofthe excitation amplitudes. Comparing the characteristicsofuniform, Tschebyscheff and binomial arrays, it can be ob-served that the uniform arrays provide the smallest HPBW andthe highest SLL while the binomial arrays yield the largestHPBW and the lowest SLL [17]. In particular, a binomialarray with interelement spacing lower than or equal to halfthe carrier wavelength has no side lobes. The HPBW and theSLL of a Tschebyscheff array lie between those of binomialand uniform arrays. Besides, for a given SLL, the Dolph-Tschebyscheff arrays provide the smallest first null beamwidth(FNBW) and, reciprocally, for a given FNBW, the Dolph-Tschebyscheff arrays provide the lowest possible SLL.

Another commonly employed configuration is the UCA [18]in which the elements are regularly arranged on a circularring. The UCA has significant practical interest and is oftenadopted in radar and sonar systems as well as in cellular BSs.In this paper we employ the UCA mainly for investigating theBF potential of the array when densified with many antennaelements while fixing the array physical area (size-constrainedarray). Our choice on the UCA topology is owed to the UCA’shigh degree of symmetry simplifying the analysis. Therefore,we consider a UCA of radiusr and N half-wavelength thinelectrical dipoles. For properly calculating the EM coupling,the type of the antenna elements comprising the UCA as wellas the elements’ terminations need to be specified. All thedipoles are terminated with the characteristic real impedanceZo (Ω) which is the simplest matching technique. Along theselines we should remark that such simple matching is favoredto sophisticated decoupling and matching networks (DMN) asthe complexity of the DMN grows significantly when dealingwith more than three antennas [19]. Moreover, the insertionlosses introduced by such complex networks are larger than thematching efficiency gains that can be recovered [20]. Finally,the use of DMN, in general, reduces the operative bandwidthof the array [21]; this needs to be taken into account, especiallyfor wideband communication systems.

As stated before, the UCA will be employed to investigatethe BF potential of size-constrained arrays, thus the activeelement response rather than the ideal response should beconsidered (because of the non negligible effect of the EMcoupling). The active element response is the radiation pattern

obtained when exciting the corresponding antenna elementwith a unit excitation voltage signal while terminating theother antenna elements with their corresponding matchingimpedances [22]. Consequently, the active element responsetakes into account the EM coupling with the neighboringantenna elements and the coupling with the chassis of theradome. The active element response of the first element canbe expressed as

G1(ϕ) := αT(ϕ)(ZT + ZoIN )−1u1, (2)

whereα(ϕ) ∈ CN×1 is the array steering vector under no EM

coupling, defined from the array topology as

(α(ϕ))n = exp

(−jκr cos(ϕ − (n − 1)

2π

N)

), (3)

where n ∈ 1, . . . , N − 1. In (2), ZT is the mutualimpedance matrix calculated using the analytical expressionsin the Appendix andu1 is a selection vector given by

u1 = [1 0 0 . . . 0︸ ︷︷ ︸N−1

]T. (4)

In general,uk is defined as a vector of all zeros except aunity at thekth position. The active element response of thekth element is given by

(G(ϕ)

)

k

= Gk(ϕ) =

G1

(ϕ − 2π

N(k − 1)

)exp

(−jκr cos

(ϕ − 2π

N(k − 1)

))

(5)

i.e., thekth element response is a rotated and phase-shiftedversion of the first element responseG1(ϕ). On the other hand,the input impedance seen by all elements is equal (by the arraysymmetry). The input impedance can be written in a compactform as

Zin =uT

1ZT (ZT + ZoIN )−1 u1

uT1 (ZT + ZoIN )

−1 u1

. (6)

The input reflection coefficient is expressed as [25]

Γin =Zin −Zo

Zin + Zo, (7)

from which the matching efficiency of each element becomes[25]

ηk = 1 − ΓinΓ∗in ∀ k. (8)

Although ULAs and UCAs are widely adopted geometriesbased upon a regular and symmetrical design, these config-urations are mainly suitable for applications in which themounting support is able to sustain the structure and particularsize or shape constraints are not present. However, in manypractical scenarios, the mounting surface is irregular and/orthe available space is limited. In these situations the arraygeometry must be designed to match the particular require-

−35−30−25

−20

−20

−15

−15

−10

−10

−5

−5

0

0

5

5

10 dBi

10 dBi

90o

60o

30o

0o

−30o

−60o

−90o

−120o

−150o

180o150o

120o

−25−30−35

Fig. 1. UCyA Prototype (left) and the array diagram of the measured activeelement response at 2.45 GHz with a bore-sight gain of6.4 dBi (right).

ments of the support and so a conformal array is needed. Theantenna elements can be disposed on an ellipse, on an arc, orcan be placed according to more complex three-dimensionaltopologies, usually circularly symmetric surfaces, such ascylinders, cones or spheres [26]. We consider a practical arraydesign for BS, namely the UCyA. We study a UCyA prototype(rather than a theoretical model) comprised of a cylindricalradome with a radiusr = 2.17λ. The array is equipped withN = 24 vertical column elements spaced by0.55λ. Eachcolumn is comprised of four patch antennas thus a total of96 patch antennas exist. The array as well as the measuredcolumn (active) response are shown in Fig. 1. The rest of theentries ofG(ϕ) are obtained from the same expression (5),i.e., by physical rotation along with phase-shifting.

III. S INGLE-LAYER BF

A. Signal Model

Assume a BS equipped with an array ofN antenna elementsbeaming a complex data symbolxT toward a single antennauser terminal by exciting the antenna elements with the BFvector w (normalized to a unit power, i.e.,wHw = 1). Usingthe Kronecker separability [27], the signal received by theuserterminal can be expressed as

y =√

ρhHwxT + n =√

ρhHR

1

2

T wxT + n, (9)

wheren ∼ CN (0, 1) is the noise symbol at the receiver side,h ∈ C

N×1 is the complex conjugate of the channel vectorbetween the transmit array and the receive antenna such thath ∼ CN (0, RT). h ∈ C

N×1 is an independent and identicallydistributed channel vector such thath ∼ CN (0, IN ). Thetransmit signal-to-noise ratio (SNR) is denoted byρ whereasRT is transmit correlation matrix defined as

RT :=Eh

hhH√

Eh

‖h‖2

F

. (10)

From a signal space point of view, the transmit correlation

matrix can be alternatively found by obtaining the transmitcovariance matrix as

RC =

∮α(ϕ)αH(ϕ)A(ϕ)dϕ, (11)

whereA(ϕ) is the channel APS. From (11) the correlation(normalized covariance) matrix becomes

(RT)ij =(RC)ij√

(RC)ii (RC)jj

∀ i, j ∈ 1, . . . , N. (12)

The fact that the transmit covariance can be either expressed bythe expectation of the channel fading vectors’ outer product(10) or alternatively by the integration of the array steeringvectors’ outer product (11) is explained in [28]. The steeringvector conventionally represents the responses of antennaelements when each element is alone, i.e., when there is no EMcoupling with the other antenna elements or any neighboringobjects. However this is not the case in real-life antennaimplementations and thus (11) is replaced with

RC =

∮G(ϕ)GH(ϕ)A(ϕ)dϕ, (13)

whereG(ϕ) is a column vector formed by stacking the arrayactive element responses. Thekth active element response isnormalized such that∮

Gk(ϕ)G∗k(ϕ)dϕ = ηk, (14)

whereηk is the efficiency of thekth element. In (9), if weset w to be equal to the eigenvector that corresponds to themaximum eigenvalue ofRT, then the signal model becomes

y =√

ρhxT + n, (15)

where the composite channelhHw is replaced by the fadingcomponenth ∈ C

1×1, h ∼ CN (0, λmax (RT)) andλmax (RT)is the maximum eigenvalue ofRT corresponding to the arraybeamforming gain or the variance ofh. The signal model in(15) can be further written as

y =√

ρλmax (RT) hxT + n, (16)

whereh ∼ CN (0, 1).

B. Proper BF Criterion

The normalization in (12) which is found in many text booksand research articles, e.g. [29] [30],does notreflect the actualBF potential or the directivity of the antenna array as the BFgain λmax (RT) from (12) is merely connected to the numberof the antenna elements. For example, aN -element small-sized as well as aN -element large-sized ULA will both havea free-space BF gain equal ofλmax (RT) = N based on suchnormalization. This is indeed not true as we shall explain inthe lines. In order to resolve BF gain conflict, we review the

concept of the distributed directivityD which measures thearray potential of focusing the power in a certain direction.Assume a free-space sector-beamGtot(ϕ) illuminating a clusterof scatterers with an APS ofA(ϕ), such that the cluster centerof mass is aligned with the beam bore-sight, then the arraydirectivity D is defined from [4] as

D :=

∮Gtot(ϕ)G∗

tot(ϕ)A(ϕ)dϕ12π

∮Gtot(ϕ)G∗

tot(ϕ)dϕ=

1

PT

∮P(ϕ)A(ϕ)dϕ, (17)

whereP(ϕ) = Gtot(ϕ)G∗tot(ϕ) is thepowersector-beam andPT

is the transmit power. Please notice that (17) is a generalizationof the free-space directivity obtained by settingA(ϕ) = δ(ϕ−φmax), i.e., zero angle spread channel (Dirac function) in thedirection of the maximum powerφmax. Moreover, (17) can begeneralized to all angles, thusD(ϕ) is obtained by the circularconvolution ofP(ϕ) with A(ϕ) thus obtaining

D(ϕ) =1

PT

∮P(φ)A(φ − ϕ)dφ. (18)

The distributed directivity is a figure of merit for designingBS antennas [4] analogous to the directivity of point-to-pointapplications. The better the average match of the sector-beamto the distribution of the scatterers, the better the average gainof the multipath components. The average BF gainGav cannow be obtained by multiplying the distributed directivitywiththe array efficiencyηT, i.e.,

Gav = ηTD. (19)

The free-space sector-beamGtot(ϕ) of an array ofN antennaelements is generally expressed as

Gtot(ϕ) = wHG(ϕ). (20)

By plugging (20) into (17) we get

D =

∮ (wHG(ϕ)

) (wHG(ϕ)

)H A(ϕ)dϕ12π

∮(wHG(ϕ)) (wHG(ϕ))

Hdϕ

=wH(∮

G(ϕ)GH(ϕ)A(ϕ)dϕ)

w

wH(

12π

∮G(ϕ)GH(ϕ)dϕ

)w

=wHRCwwHRUw

. (21)

From (21), the distributed directivity in a channel of a givenAPS is maximized when the precoding vector is equal to

wopt = V(RC, RU), (22)

where V(·) is the operator that returns the eigenvectorthat corresponds to the maximum generalized eigenvalueλmax(RC, RU). In order to highlight the importance of theaverage BF gain criterion we consider ULAs ofN isotropicantennas, spaced by a sufficient interelement spacing suchthat the EM coupling is safely assumed negligible whereasthe elements’ efficiencies are set to unity. Fig. 2 shows thebroadside sector-beams obtained by driving two ULAs withwopt in (22). The sector-beam of the ULA withd = 0.5 is

20 40 60 80 100 120 140 160−40

−30

−20

−10

0

10

20

30

ϕ

G av(ϕ

)(d

B)

d = 0.5

d = 10

Fig. 2. The sector-beam when exciting withwopt in (22) a ULA of 100isotropic antenna elements ford = 0.5 andd = 10 spacing.

0 20 40 60 80 1002

4

6

8

10

12

14

16

18

20

22

N

G av

(dB

)

d = 0.5λ

d = 10λ

λmax(RC)

Fig. 3. The average BF gain when exciting withwopt in (22) a ULA of 100isotropic antenna elements ford = 0.5 andd = 10 spacing.

more directive than the ULA withd = 10. This is indeedlogical as the antenna elements of the latter array lie outof the correlation distance from each other thus leading tomultiple main lobes as illustrated in the same figure. Fig. 3showsGav(ϕ = π/2) for both ULAs versus the number ofthe antenna elements, together withλmax (RT) for both arrays(coinciding over one another which is not logical). The figureshows thatλmax (RT) may overestimate or underestimate thetrue BF gain according to the scenario.

Finally, the signal model in (15) can now be accuratelyreplaced with

y =√

ρGav hxT + n. (23)

Gtot(ϕ)

A(ϕ)

Fig. 4. An illustration of the mutual interaction between thesector-beamand the channel statistics (∗ indicates circular convolution).

C. Effect of Channel Statistics

In order to understand the interaction between the sector-beam and the channel statistics represented byA(ϕ) weassume a Gaussian free-space sector-beam with a HPBW ofσG

√2 ln 2 and zero mean angle, thus according to [4] we get

Gtot(ϕ) :=1√

2πσ2G

exp

(− ϕ2

2σ2G

). (24)

We also assume a Gaussian cluster of scatterers with a HPBWof σA

√2 ln 2 and zero mean angle, i.e., aligned with the

sector-beam, thus

A(ϕ) :=1√

2πσ2A

exp

(− ϕ2

2σ2A

). (25)

By convolving (24) with (25) we obtain

D(ϕ) =1

PT

√2π(σ2G + σ2

A

) exp

(− ϕ2

2(σ2G + σ2

A

))

, (26)

i.e., the diffused beam is Gaussian too. The convolution in(26) was performed by first taking the Fourier transform of(24) and (25), multiplying them and then taking the inverseFourier transform of the product. From (26), ifσG << σA,i.e., the sector-beam is too sharp with respect to the cluster,the resultant diffused beam will have angle spread slightlybigger than the cluster spread. On the other hand, ifσG >>σA then the free-space sector-beam is almost identical tothe diffused beam. Fig. 4 illustrates how a Gaussian beamof certain HPBW, illuminating a nearly zero angular spreadcluster is equivalent to a pencil beam illuminating a clusterhaving the same HPBW of the Gaussian beam.

From (26), the (maximum) array directivity is obtainedby settingϕ = 0, thus we getDmax = D(0) ∝ [(σ2

B +σ2A)]−1. Moreover, since the number of antennasN is gen-

0 5 10 15 20−15

−10

−5

0

5

10

15

20

25

30

(σB)−1

[(σ2 A+

σ2 B)]−1dB

σA = 20

σA = 0

Fig. 5. Maximum distributed directivityD(0) ∝ [(σ2

B+ σ2

A)]−1 versus

(σB)−1 ∝ N for different values ofσA.

erally proportional to(σB)−1 we finally obtain Dmax ∝log10

(N2

1+σAN2

), in dB. From this expression we observe

thatDmax ∝ log10 (N) in free-space (by settingσA to zero).On the other hand,Dmax converges to a constant when(σA)increases. In Fig. 5 we plot the maximum directivity versus(σB)−1. The same trends are observed in Fig. 6 where aconcrete ULA setup is established (withd = 0.5) illuminatinga cluster of scatterers of angular spreadσA. The figure showsthatGav deteriorates rapidly when illuminating wider and widerclusters (remember that the precoding in (22) is performedaccording to the channel statistics). On the other hand, Fig. 7shows free-space sector-beams synthesized by exciting ULAsof 100 and1000 isotropic elements,0.5λ spacing withwopt in(22). From the figure we drop the following remarks:

• The peak BF gain for both arrays is almost equal whenσA = 0, namely 24.5 dB and 25.5 dB for the 100-element and the1000-element ULA respectively.

• On the other side, the SLL of the1000-element ULA ismuch lower than the SLL of the100-element ULA. Basedon this, wireless systems for which the SLL is vital willbenefit by having many antennas, unlike wireless systemsthat merely benefit from the BF gain. This will be thetopic of the next section.

• When the two free-space sector-beams are diffused overa cluster of spreadσA = 8, both sector-beams becomeidentical down to a SLL of−10 dB. Consequently, the1000-element ULA will provide almost similar perfor-mance to the 100-element ULA in channels where themultipath concentration is about8. This assures that theoptimal number of antennas (to be installed at the BS)beyond which we observe diminishing returns should takethe channel statistics into consideration.

D. Effect of Array Efficiency

Till the moment we have been considering potentiallyunconstrained arrays where the array length grows with the

number of the array elements by maintaining the interelementspacing to a minimum ofλ/2 (thus the EM coupling couldbe safely neglected). In this part we consider physicallyconstrained arrays where the interelement spacing gets smallerwhen increasing the number of the array elements. Theelement response when accounting for the EM coupling isgiven in (2) and (5). On the other hand, the average BF gainaccording to (19) is the multiplication of the array efficiency1

and the array distributed directivityD. While D was shown tobe a function of the BF vectorw in (21), it is still not clear howthe BF vector affectsηT. The transmit array efficiency is in turnthe product of the array matching efficiencyηM and the arrayOhmic efficiencyηΩ, i.e.,ηT = ηΩηM . The former is due to themismatch between the RF source and the load (the antenna) aswell as the coupling with the neighboring antennas. The latteris owed to resistive as well as dielectric losses. In this part weassumeηΩ = 1 which is not true when the quality factor of thearray increases. On the other hand,ηM in arrays with negligibleEM coupling is a mere function of the array terminations andremains constant for all excitations. However this is not truewhen the array has strong EM coupling among its antennaelements. In this part we overview the way the array efficiencyis calculated when simultaneously exciting the elements ofanarbitrary array. Given the excitation vectorw the input poweris given by wHw. Part of the input signal reflects back tothe sources, given bySTw whereST is the array scatteringmatrix (normalized to the source impedance). The radiatedpower is the difference between the incident power and thereflected power while the array efficiencyηT = ηM is ratio ofthe radiated power to the input power, i.e.,

ηT :=wHw − wHS

HTSTw

wHw

=wH(IN − S

HTST

)w

wHw=

wHTwwHw

, (27)

whereT := IN −SHTST is the radiation matrix. From (27), the

maximum efficiency is obtained when the precoding vector

wopt = V(T) (28)

is applied, whereV(·) is the operator that returns the eigen-vector that corresponds to the maximum eigenvalueλmax(T).Consequently, the precoding vector that maximizes the averageBF gain is neither the one that maximizes the array directivity(i.e., the one in (22)) nor the one that maximizes the arrayefficiency (i.e., the one in (28)), but a compromise in between.Finding the optimal precoding vector that maximizes theaverage BF gain requires direct search algorithms and getscomplicated when the number of the antennas grow. In Fig. 8we show that the free-space BF gain is quite comparableto the free-space directivity when employing a UCA ofNdipoles andr = λ/2, as modeled in Section II. The dipolesare terminated withZo = 50 Ω while the precoding vectoris the one in (22). The figure thus shows thatthe hit in the

1The array efficiency is different from the elements efficiencyηk. Theformer is the one obtained when the array elements are excited simultaneouslywith w whereas the latter is obtained when the array is excited withuk.

0 20 40 60 80 1004

6

8

10

12

14

16

18

20

22

N

G av

(dB

)

0

5

10

15

20

Fig. 6. The average BF gain when exciting withwopt in (22) a ULA of Nisotropic antenna elements ford = 0.5 andσA = 0, 5, 10, 15, 20.

0 20 40 60 80 100 120 140 160 180−50

−40

−30

−20

−10

0

10

20

30

ϕ

G av(ϕ

)(d

B)

N = 100, σA = 0

N = 1000, σA = 0

N = 100, σA = 8

N = 1000, σA = 8

Fig. 7. Free-space and diffused sector-beams when exciting with wopt in(22) a ULA of 100 and1000 isotropic antenna elements whereasd = 0.5.

average BF gain owed to the impedance mismatch (matchingefficiency) is quite small when driving the array with theexcitations that maximize the array directivity. Consequently,maximizing the average BF gain of large-scale antenna arrayscan be simply done by maximizing the array directivity using(22) while accepting the (small) hit of the non maximummatching efficiency.

E. Discussion

• In single-layer BF systems the performance is governedby the average BF gain. In non-zero angle spread chan-nels the BF gain tends to saturate beyond a threshold thusfurther added antennas seem to bring no gain. However,this is not completely true as there could be otherhardware and complexity-reduction gains e.g. splittingthe power over the antennas relaxes the design of thepower amplifiers and RF filters in the transmit RF chain.

5 10 15 20 25 302

4

6

8

10

12

14

16

18

N

dB

DGav

Fig. 8. Free-space directivity and (average) BF gain when exciting with woptin (22) a UCA ofN isotropic antenna elements andr = 0.5.

Moreover, having more antennas immunizes the arrayagainst element-branch failures [31].

• From throughput point of view, the logarithmic growthof the capacity with the number of BS antennas is notattractive when operating in a single-layer BF mode.A topology that combines the benefits of diversity aswell as BF arrays is illustrated in Fig. 9. The idea is tosynthesize a super-array comprised of a set of subarrays.Each subarrayserves as a single directive virtual antenna.The virtual antennas are decorrelated from each otherby distance (could alternatively be by polarization). Thesubarray preserves most of the BF benefits by having amoderate number of antenna elements within the corre-lation distance from each other. On the other hand, thesuper-array preserves the benefits of diversity systems byhaving the virtual antennas decorrelated. Fig. 10 showsthe throughput potential of two subarrays having a totalnumber of antennasN equally divided between the twosubarrays, as well as the throughput potential when theNantennas are collocated in one UCA. The RF channel isRayleigh fading, Laplacian clustering with angle spreadof 5 and a mean angle aligned with the broadside axis ofsymmetry of the two subarrays. The figure shows that40collocated antennas provide the same spectral efficiency(or capacity) as2 (subarrays)× 4 (antenna/subarray)= 8antennas.

• Last but not least, in order to investigate the(in)significance of full CSI on the throughput potentialwhen transmitting under different channel statistics, weconsider a UCA of20 and 100 dipole elements withr = 0.5, illuminating a Laplacian cluster of angle spreadσA. Fig. 11 shows the capacity versus theσA at SNR= 10dB, when partial CSI (denoted by P) is available at thetransmit BS, i.e., the UCA is beamforming accordingto the transmit covariance matrix, and when full CSI(denoted by F) is available, i.e., the UCA is time-reversing the uplink channel [32]. The figure shows that

Fig. 9. Super-array topology at a BS above a clutter.

4 8 12 16 20 24 28 32 36 405

5.5

6

6.5

7

7.5

8

N

C(b

psH

z−1)

Two Subarrays

One Array

Fig. 10. Capacity performance of a single array ofN collocated antennas anda super-array comprised of two subarrays, each subarray hasN/2 antennas.

the significance of full CSI knowledge decreases asσA

decreases which is an important issue when dealing withchannel estimation limits (or pilot contamination issues).

IV. M ULTI -LAYER BF

Till now we have been investigating the BF potential ofantenna arrays in a single user scenario thus the term singe-layer BF. However, future wireless systems will allow manyusers to share the spatial resources within space divisionmultiplexing (SDM). SDM is a technology that allows one BSto serve a number of users in a point-to-multi-point fashion.The key difference between SDM and point-to-point MIMOstems from the independent decoding of the users receivedsignals, thus users’ unintended signals are seen as colorednoise. The fact that every user knows his own channel butnot the others’ necessitates careful processing at the BS forminimizing the inter-user interference which otherwise rulesout any MIMO multiplexing gain. Generally, this requires afeedback channel from the end users back to the BS which inturn applies a transmit filter like a conventional zero forcing

5 10 15 20 25 305

5.5

6

6.5

7

7.5

8

8.5

9

9.5

σA

C(b

psH

z−1)

N = 20, P

N = 20, F

N = 100, P

N = 100, F

Fig. 11. Capacity performance versus the angle spread of the cluster whenpartial (P) and full (F) CSI is available at the BS. The performance is shownfor two ULAs of 20 and100 isotropic antenna elements.

precoding matrix. From a beam-space point of view and underlow angle spread channels, e.g., users’ channels seen by a BSon top of a clutter,the transmit filter maps the users’ signalsonto a basis of directive beams toward the desired users andnulls toward the interfering onesi.e., the geometrical and thephysical (actual) directions of the channel coincide. A simpleSDM technology that utilizes the BF potential of BS arraysis the high order sectorization (HOS) [7] [33]. HOS mapsthe users’ signals onto apredefined basis of directive sector-beams with low inter-beam interference, regardlessof theusers’ instantaneous channels. The approach is found practicalas it requires the minimum feedback from the users to theBS though some assumptions should be made on the usersdistribution within the RF coverage. In the following partswe show how the sector-beams can be designed to maximizethe system throughput while maintaining full RF azimuthalcoverage.

A. Signal Model

We consider a scenario where a set of circularly symmetricsector-beams (see Fig. 12 in [7]) serve a number of users(equal to the number of beams) uniformly distributed withinthe RF coverage. The precoded signal is given byx = Wx,whereW ∈ C

N×B = [w1 w2 . . . wB] is the precoding matrix,B is the number of BF layers (sector-beams) andwk is thekthprecoding vector (cyclic rotation ofw1). x = [x1 x2 . . . xB]

T ∈C

B×1 is the vector of the complex data symbols to betransmitted. Assuming single antenna users and uniform powerdistribution across the sector-beams, the signal receivedby thefirst user is given by

y1 =

√ρ

BhH1 w1x1 +

√ρ

BB∑

k=2

hHk wkxk + n1

=

√ρGav(φ1)

B h1x1 +

√ρ

BB∑

k=2

√Gav(φk)hkxk + n1

(29)

−150 −100 −50 0 50 100 150−15

−10

−5

0

5

10

15

ϕ

G av(ϕ

)(d

B)

σA = 4

σA = 8

σA = 12

σA = 16

Fig. 12. Effective sector-beam resulting by the convolution the free-spacepower sector-beam with the channel APS.

−150 −100 −50 0 50 100 150−80

−70

−60

−50

−40

−30

−20

−10

0

ϕ

G tot(

ϕ)

(dB

)

B = 3

B = 4

B = 6

B = 8

B = 12

B = 24

Fig. 13. Free-space sector-beams synthesized using (31) andthe UCyA inFig. 1.

wherehk ∈ CN (0, 1), k ∈ 1, . . . ,B andφk is the directionof the kth user.

B. Sum-Rate Analysis

Assume every user is immersed in a cluster with a center ofmass being uniformly distributed within the angular domainof his sector-beam, thenφj becomes a random variable ofuniform distribution withinΦ. Assuming every single beamserves one user and by the fact that the users are assumed tobe uniformly distributed within the RF coverage, the users’sum-rate will then be equal to the average rate (of one user)scaled by the number of sector-beams as follows:

RΣ = Eh,φ

B log2

(1 + SINR(h1, φ1)

)(30a)

= Eh,φ

B log2

(1 +

ρB |h1|2

1 + ρBΣB

k=2|hk|2)

(30b)

≤ Eh,φ

B log2

(1 +

ρB |h1|2

ρBΣB

k=2|hk|2)

(30c)

≤ Eh,φ

B log2

(1 +

Gav(φ1)|h1|2ℓΣB

k=2|hj |2

)(30d)

≤ Eφ

B log2

(1 + Eh

Gav(φ1)|h1|2ℓΣB

k=2|hj |2

)(30e)

= Eφ

B log2

(1 +

Gav(φ1)

ℓ (B − 2)

)(30f)

≤ B log2

(1 +

Gav

ℓ (B − 2)

), (30g)

where SINR is the signal to interference and noise ratio,Gav

is the average ofGav over the random variableφ, finally ℓis the minimum value ofGav(ϕ). From (30b) to (30c) by thefact thatlog2(.) is a monotonic function. This approximationapproaches equality in the high SNR regime. From (30d) to(30e) by the fact thatℓ ≤ Gav(φ). From (30e) to (30f) by

the fact that |h1|2

ΣBk=1

|hj |2is an F-distribution with a mean 1

B−2 ,

Finally from (30f) to (30g) by Jensen’s inequality and theconcavity of thelog2(.) function.

C. Sector-Beam Design

Classical filter-design techniques like Prolate [34], Taylor[35], Chebychev [16] etc, can not be directly applied on theproposed UCyA (as the element response may not be factoredout) and thus adaptive beamforming techniques are adopted.We set the width of the sector-beam toΦ = 2π

B so that theBsector-beams cover the full azimuth plane. The distribution ofthe users is assumed uniform, however other user distributionscan be considered resulting in some amplitude tapering withinthe side-lobes (beyond the scope of this paper). The sector-beam is designed by minimizing the maximum (infinite norm)response within the complementary region ofΦ denoted byΦ,while maintaining a unit response (with some toleranceτ ) atthe desired directionφ1 , i.e.,

minimize(over w1, ϕ∈Φ)

max(wH

1 G(ϕ)GH(ϕ)w1

)

subject to real(wH

1 G(φ1))≤ 1 + 10

τ10

real(wH

1 G(φ1))≥ 1 − 10

τ10 (31)

(31) is a convex optimization problem that is efficientlyhandled by the software CVX [36]. The choice of the max-imum norm is known to provide the optimal solution bycreating side-lobes with equal ripple (thus a constant signalto interference ratio is maintained), while the factorτ ensuresthe problem not getting over constrained. Having obtainedw1,the circular rotation ofw1 by k taps shifts the direction of thedesired response by2π

k.

• The fact that we optimize the free-space sector-beamin (31) is justified by the slight beam diffusion over

the narrow users’ clusters. Under a non-zero channelangle spread, the diffused sector-beam will have a smallerBF gain (as explained in Section III) whereas the deepnulls observed in the free-space sector-beam are smeared.Fig. 12 illustrates this by showing the angular convolutionof a free-space sector-beam (obtained using (31) and theUCyA in Fig. 1), with a Laplacian APS having anglespread of4, 8, 12.

• In order to investigate the tightness of (30g) with respectto (30a), we consider the UCyA in Fig. 1 and optimizeits sector-beam using (31). The restB − 1 precodingvectors are obtained by cyclic rotations ofw1 by B ∈3, 4, 6, 8, 12, 24 rotations. Fig. 15 shows the sum-ratefor SNRs of 10, 20 and 30 dB as well as sum-rateupperbound. Although (30g) is somehow loose even athigh SNRs, it still provides some useful insights bypreserving the same trend of (30a).

• In case the antenna system is constrained, e.g., by sizeor the number of the antenna elements, then the mini-mum of the side-lobe levelℓ increases by decreasingΦ,or increasing the number of sector-beams (fundamentaltrade off [37] as shown in Fig. 13), leading to sum-ratedeterioration beyond a threshold number of sector-beams.

• As Gav decreases by increasing the channel spread, thesum-rate consequently degrades as shown in Fig. 14.Notice that in case the channel angle spread becomesrelatively wide (e.g. indoor access point surrounded bymany scatterers), the multi sector-beams have no mul-tiplexing gain anymore (above a single sector-beam) asthe inter-beam interference of the diffused sector-beamsbecomes intolerable.

• If the ratio Gav/ℓ is made independent of the numberof beams, for example by keep increasing the array sizeand the number of the antenna elements (unconstrainedantenna system), then the sum-rate upperbound convergesasymptotically toGav/(ℓ ln(2)). Thus, the higher the ratioGav/ℓ, the higher the system capacity.

Based on the aforementioned remarks, HOS provides consid-erable performance gains in channels with low angle spread,making such a technique a viable candidate for outdoorcellular systems where feedback is costly and the narrowconcentration of multipath is satisfied.

V. FUTURE WORK

In the future, we aim at conducting some channel mea-surement campaigns using both large as well as dense an-tenna arrays, for better understanding the interaction of thecomplex RF propagation channel with large-scale antennasystems. An example of the antenna arrays we intend toinvestigate is the uniform rectangular array shown in Fig. 16,comprised of 64 short monopoles above a small finite a groundoperating at5 − 6 GHz. Moreover, since the explosion inthe antenna dimension may lead to implosion in the RFhardware dimension, an important issue that we will addressisthe implementation of reduced-complexity BF-MIMO systemarchitectures. In Fig. 17 we propose a hybrid analogue/digital2-layers BF system. In such systems, the number of the RFchains scales withB rather thanN . However, each antenna

3 4 6 8 12 240

20

40

60

80

100

120

B

RΣ

Upper

bound

(bits/

s/H

z)

σA = 0

5

10

15

20

100

Fig. 14. Users’ sum-rate upperbound versus the number of sector-beams atdifferent channel angle spreads. The channel APS is assumed Laplacian witha mean angle that is uniformly distributed within2π

B.

3 4 6 8 12 24240

10

20

30

40

50

60

B

RΣ

(bpsH

z−1)

SNR=0 dB

SNR=10 dB

SNR=100 dB

Upperbound

Fig. 15. Users’ sum-rate and its upperbound versus the number of sector-beams at different SNRs. The channel APS is assumed Laplacian with a spreadof 20 and a mean angle that is uniformly distributed within2π

B.

in such architecture is connected toB paths of analoguegain controllers and phase-shifters. The RF complexity canbe further simplified exploiting the concept of HOS wheredeterministic multi-layer BF approach requires static phase-shifters and gain controls (i.e. static BF network).

APPENDIX

An approached expression of the dipoles self-impedanceis given by the impedance of ann wavelength long isolateddipole [23]:

ZTii = 30 [ln (2πnγ) − CI (2πn) + jSI(2πn)] , (32)

whereγ is the Euler constant (γ = 0.5772156649...) and SIand CI are the sine and cosine integral functions defined asfollows

SI(z) =

∫ z

0

sin(t)t

dt

CI (z) =

∫ z

∞

cos(t)t

dt = γ + ln z+

∫ z

0

cos(t) − 1

tdt .

(33)

The coupling-impedance termsZTij = Rij + jXij , i 6= j

Fig. 16. An array of 64 monopoles above a finite ground, operating at5−6GHz.

Gain Controller Phase Shifter

Combiner

RF Frontend

Antenna

Fig. 17. Hybrid analogue-digital two-layers BF system.

are calculated using the induced electromotive force method[24]

Rij = +ηf

4π[2CI (u0) − CI (u1) − CI (u2)] ,

Xij = − ηf

4π[2SI(u0) − SI(u1) − SI(u2)] , (34)

where u0 = κd, u1 = κ(√

d2ij + L2 + L

), u2 =

κ(√

d2ij + L2 − L

), ηf = 120π is the free-space impedance,

dij is the spacing between theith and thejth adjacent dipolesandL = λ/2 is the dipole length.

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Osama N. Alrabadi (S’08–M’11) is an industrialpostdoctoral fellow at the Antennas, Propagationand radio Networking (APNet) group, department ofElectronic Systems, Aalborg University, Denmark.He was born in 1979 in Jordan, where he receivedthe Diploma of Electrical Engineering from the Uni-versity of Jordan in 2002. He received the M.Sc. inInformation and Telecommunications from AthensInformation Technology, Greece in 2007 with thehighest honor, and the Ph.D. degree from the Aal-borg University in 2011. In summer 2010, he was

an intern at Bell Laboratories (Alcatel-Lucent), Holmdel, New Jersey, wheredeveloped beamforming techniques for cellular networks. Currently he isworking on the ”Smart Antenna Frontend (SAFE)” project, in cooperationwith the antenna company Molex, the tuner company Wispry (California,US) and Intel Mobile Communications. His research interests include designand modeling of small antenna arrays, array processing as wellas MIMO andcognitive radio transceiver architectures. Dr. Osama N. Alrabadi is a co-authorof numerous journal and conference papers and holds several patents. He isan IEEE member and a member of the Jordan Engineering Associationsince2002.

Elpiniki Tsakalaki is a postdoctoral fellow at theAntennas, Propagation and radio Networking (AP-Net) group at the department of Electronic Sys-tems,Aalborg University, Denmark, from where shereceived her Ph.D. degree in Wireless Communica-tions in 2012. With a scholarship from the HellenicFederation of Enterprises (SEV), she received in2009 the M.Sc. in Information Networking from theInformation Networking Institute, Carnegie MellonUniversity, Pittsburgh with the highest honor. In2008, she received the Diploma in Electrical and

Computer Engineering (Dipl.-Ing.) from the National Technical University ofAthens, Greece. From 2009 until 2012 she worked in the research project”Cognitive Radio Oriented Wireless Networks (CROWN)” funded under theFET-Open scheme within the Seventh Framework Programme (FP7) oftheEuropean Commission. During the summer of 2011 she was an intern atNokia Siemens Networks, Madrid, Spain, developing beamforming designs foractive base station antenna systems. Currently, she is working on the project”MIMO wireless communications for closely spaced antennas” granted bythe Danish Council for Independent Research, Technology and ProductionSciences (FTP). Her research interests are in the combined area of antennadesign and signal processing including reconfigurable and compact antennaarrays, diversity and MIMO systems, and cognitive radio communication. Sheis a member of the IEEE and a member of the Technical Chamber of Greece.

Howard Huang was born in Houston, Texas in1969. He received a BS in electrical engineeringfrom Rice University in 1991 and a Ph.D. in electri-cal engineering from Princeton University in 1995.Since then, he has been a researcher at Bell Labs(Alcatel-Lucent) in Holmdel, New Jersey, currentlyas a Distinguished Member of Technical Staff inthe Wireless Communication Theory Department.His interests are in communication theory, multipleantenna (MIMO) techniques, and the system designof wireless networks. He served as a guest editor

of two issues of the IEEE Journal of Selected Areas in Communications-one on next-generation MIMO wireless networks and another on cooperativecommunications in MIMO cellular networks. He has taught at ColumbiaUniversity and is a co-author of the book MIMO Communication for CellularNetworks (published by Springer 2011). Dr. Huang holds overa dozen patentsand is a Senior Member of the IEEE.

Gert Frølund Pedersen was born in 1965, mar-ried to Henriette and has 7 children. He receivedthe B.Sc. E. E. degree, with honor, in electricalengineering from College of Technology in Dublin,Ireland, and the M.Sc. E. E. degree and Ph.D. fromAalborg University in 1993 and 2003. He has beenwith Aalborg University since 1993 where he is nowfull Professor heading the Antenna, Propagation andNetworking LAB with 36 researchers. Further he isalso the head of the doctoral school on WirelessCommunications with around 100 Ph.D. enrolled

students. His research has focused on radio communication formobile termi-nals especially small antennas, diversity systems, propagation and biologicaleffects and he has published more than 175 peer reviewed papers and holds28 patents. He has also worked as consultant for developmentsof more than100 antennas for mobile terminals including the first internalantenna formobile phones in 1994 with lowest SAR, first internal triple-band antennain 1998 with low SAR and high TRP and TIS, and lately various multiantenna systems rated as the most efficient in the market. He has workedmost of the time on joint university and industry projects and has receivedmore than 12 million USD in direct research funding. Latest he is the projectleader of the SAFE project with a total budget of 8 million USD investigatingtunable front-ends including tunable antennas for the future multiband mobilephones. He has been one of the pioneers in establishing Over-The-Air (OTA)measurement systems. The measurement technique is now well established formobile terminals with single antennas and he has been chairingthe variousCOST groups (swg2.2 of COST 259, 273, 2100 and currently ICT1004) withliaison to 3GPP for OTA testing of MIMO terminals.