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Hindawi Publishing CorporationEURASIP Journal on Wireless Communications and NetworkingVolume 2007, Article ID 93421, 9 pagesdoi:10.1155/2007/93421
Research ArticleOptimal Design of Nonuniform Linear Arrays in CellularSystems by Out-of-Cell Interference Minimization
S. Savazzi,1 O. Simeone,2 and U. Spagnolini1
1 Dipartimento di Elettronica e Informazione, Politecnico di Milano, 20133 Milano, Italy2 Center for Wireless Communications and Signal Processing Research (CCSPR), New Jersey Institute of Technology,University Heights, Newark, NJ 07102-1982, USA
Received 13 October 2006; Accepted 11 July 2007
Recommended by Monica Navarro
Optimal design of a linear antenna array with nonuniform interelement spacings is investigated for the uplink of a cellular system.The optimization criterion considered is based on the minimization of the average interference power at the output of a con-ventional beamformer (matched filter) and it is compared to the maximization of the ergodic capacity (throughput). Out-of-cellinterference is modelled as spatially correlated Gaussian noise. The more analytically tractable problem of minimizing the inter-ference power is considered first, and a closed-form expression for this criterion is derived as a function of the antenna spacings.This analysis allows to get insight into the structure of the optimal array for different propagation conditions and cellular layouts.The optimal array deployments obtained according to this criterion are then shown, via numerical optimization, to maximize theergodic capacity for the scenarios considered here. More importantly, it is verified that substantial performance gain with respectto conventionally designed linear antenna arrays (i.e., uniform λ/2 interelement spacing) can be harnessed by a nonuniform opti-mized linear array.
Copyright © 2007 S. Savazzi et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. INTRODUCTION
Antenna arrays have emerged in the last decade as a power-ful technology in order to increase the link or system capac-ity in wireless systems. Basically, the deployment of multipleantennas at either the transmitter or the receiver side of awireless link allows the exploitation of two contrasting ben-efits: diversity and beamforming. Diversity relies on fadinguncorrelation among different antenna elements and pro-vides a powerful means to combat the impairments causedby channel fluctuations. In [1] it has been shown that a sig-nificant increase in system capacity can be achieved by theuse of antenna diversity combined with optimum combin-ing schemes. Independence of fading gains associated to theantennas array can be guaranteed if the scattering environ-ment is rich enough and the antenna elements are sufficientlyspaced apart (at least 5–10 λ, where λ denotes the carrierwavelength) [2]. On the other hand, when fading is highlycorrelated, as for sufficiently small antenna spacings, beam-forming techniques can be employed in order to mitigatethe spatially correlated noise. Interference rejection throughbeamforming is conventionally performed by designing auniform linear array with half wavelength interelement spac-
ings, so as to guarantee that the angle of arrivals can bepotentially estimated free of aliasing. Moreover, beamform-ing is effective in propagation environments where there is astrong line-of-sight component and the system performanceis interference-limited [3].
In this paper, we consider the optimization of a linearnonuniform antenna array for the uplink of a cellular sys-tem. The study of nonuniform linear arrays dates back tothe seventies with the work of Saholos [4] on radiation pat-tern and directivity. In [5] performance of linear and cir-cular arrays with different topologies, number of elements,and propagation models is studied for the uplink of an inter-ference free system so as to optimize the network coverage.The idea of optimizing nonuniform-spaced antenna arraysto enhance the overall throughput of an interference-limitedsystem was firstly proposed in [6]. Therein, for flat fadingchannels, it is shown that unequally spaced arrays outper-form equally spaced array by 1.5–2 dB. Here, different from[6], a more realistic approach that explicitly takes into ac-count the cellular layout (depending on the reuse factor)and the propagation model (that ranges from line-of-sightto richer scattering according to the ring model [7]) is ac-counted for.
2 EURASIP Journal on Wireless Communications and Networking
θ1θ1
θ3
θ2
θ3
θ0Δ12
Δ23
Δ12
(1)
(2)
(3)� = 2 kmθ3 = θ1
� = 2 km(3)
(2)
(1)
Setting A: reuse 3 Setting B: reuse 7
Interferer
User
Figure 1: Two cellular systems with hexagonal cells and trisectorialantennas at the base stations (reuse factor F = 3, setting A, on theright and F = 7, setting B, on the left). The array is equipped withN = 4 antennas. Shaded sectors denote the allowed areas for userand the three interferers belonging to the first ring of interference(dashed lines identify the cell clusters of frequency reuse).
Δ12 Δ23 ΔN
2−1,
N
2
· · · · · ·
Figure 2: Nonuniform symmetric array structure for N even.
For illustration purposes, consider the interference sce-narios sketched in Figure 1. Therein, we have two differentsettings characterized by hexagonal cells and different reusefactors (F = 3 for setting A and F = 7 for setting B, frequencyreuse clusters are denoted by dashed lines). The base stationis equipped with a symmetric antenna array1 containing aneven number N of directional antennas (N = 4 in the exam-ple) to cover an angular sector of 120 deg, other BS antennaarray design options are discussed in [8]. Each terminal isprovided with one omnidirectional antenna. On the consid-ered radio resource (e.g., time-slot, frequency band, or or-thogonal code), it is assumed there is only one active user inthe cell, as for TDMA, FDMA, or orthogonal CDMA. Theuser of interest is located in the respective sector accordingto the reuse scheme. The contribution of out-of-cell inter-ferers is modelled as spatially correlated Gaussian noise. InFigure 1, the first ring of interference is denoted by shadedcells. The problem we tackle is that of finding the antennaspacings in vector Δ = [Δ12 Δ23]T (as shown in the example)
1 The symmetric array assumption (as in the array structure of Figure 2)has been made mainly for analytical convenience in order to simplify theoptimization problem. However, it is expected that for a scenario with asymmetric layout of interference (such as setting A), the assumption of asymmetric array does not imply any loss of optimality, while, on the otherhand, for an asymmetric layout (such as setting B), capacity gains couldbe in principle obtained by deploying an asymmetric array.
so as to optimize given performance metrics, as detailed be-low.
Two criteria are considered, namely, the minimization ofthe average interference power at the output of a conven-tional beamformer (matched filter) and the maximizationof the ergodic capacity (throughput). Since in many appli-cations the position of users and interferers is not knowna priori at the time of the antenna deployment or the in-cell/out-cell terminals are mobile, it is of interest to evaluatethe optimal spacings not only for a fixed position of usersand interferers but also by averaging the performance met-ric over the positions of user and interferers within their cells(see Section 2).
Even if the ergodic capacity criterion has to be consideredto be the most appropriate for array design in interference-limited scenario, the interference power minimization is ana-lytically tractable and highlights the justification for unequalspacings. Therefore, the problem of minimizing the inter-ference power is considered first and a closed-form expres-sion for this criterion is derived as a function of the antennaspacings (Section 4). This analysis allows to get insight intothe structure of the optimal array for different propagationconditions and cellular layouts avoiding an extensive numer-ical maximization of the ergodic capacity. The optimal ar-ray deployments obtained according to the two criteria areshown via numerical optimization to coincide for the con-sidered scenarios (Section 5). More importantly, it is veri-fied that substantial performance gain with respect to con-ventionally designed linear antenna arrays (i.e., uniform λ/2interelement spacing) can be harnessed by an optimized ar-ray (up to 2.5 bit/s/Hz for the scenarios in Figure 1).
2. PROBLEM FORMULATION
The signal received by theN antenna array at the base stationserving the user of interest can be written as
y = h0x0 +M∑
i=1
hixi + w, (1)
where h0 is the N × 1 vector describing the channel gainsbetween the user and the N antennas of the base sta-tion, x0 is the signal transmitted by the user, hi and xi arethe corresponding quantities referred to the ith interferer(i = 1, . . . ,M), w is the additive white Gaussian noise withE[wwH] = σ2I. The channel vectors h0 and {hi}Mi=1 are un-correlated among each other and assumed to be zero-meancomplex Gaussian (Rayleigh fading) with spatial correlationR0 = E[h0hH0 ] and {Ri = E[hihHi ]}Mi=1, respectively. Thecorrelation matrices are obtained according to a widely em-ployed geometrical model that assumes the scatterers as dis-tributed along a ring around the terminal, see Figure 3. Thismodel was thoroughly studied in [2, 7] and a brief reviewcan be found in Section 3. According to this model, the spa-tial correlation matrices of the fading channel depend on
(1) the set of N/2 antenna spacings (N is even) Δ =[Δ12 Δ23 · · · ΔN/2,N/2+1]T , where Δi j is the distancebetween the ith and the jth element of the array (the
S. Savazzi et al. 3
Φ0
φ
θ0
θi
Φidi
p
q
Δpq d0
r0
ri
InterfererUser
Figure 3: Propagation model for user and interferers: the scatterersare distributed on a rings of radii ri around the terminals.
array is assumed to be symmetric as shown in Figure 2,extension to an odd number of antennas N is straight-forward);
(2) the relative positions of user and interferers with re-spect to the base station of interest (these latter pa-rameters can be conveniently collected into the vectorη = [ηT0 ηT1 · · · ηTM]T , where, as detailed in Figure 3,vector η0 = [d0 θ0]T parametrizes the geometrical lo-cation of the in-cell user and vectors ηi = [di θi]T (i =1, . . . ,M) describe the location of the interferers (i =1, . . . ,M));
(3) the propagation environment is described by the angu-lar spread of the scattered signal received by the basestation (φ0 for the user and φi (i = 1, . . . ,M) for theinterferers); notice that for ideally φi → 0 all scatter-ers come from a unique direction so that line-of-sight(LOS) channel can be considered. Shadowing can bepossibly modelled as well, see Section 3 for further dis-cussion.
2.1. Interference power minimization
From (1), the instantaneous total interference power at theoutput of a conventional beamforming (matched filter) is [9]
P(
h0,Δ, η) = hH0 Qh0, (2)
where
Q = Q(Δ, η1, . . . ,ηM
) =M∑
i=1
Ri(Δ, ηi
)+ σ2IN (3)
accounts for the spatial correlation matrix of the interferersand for thermal noise with power σ2. Notice that, for clar-ity of notation, we explicitly highlighted that the interfer-ence correlation matrices depend on the terminals’ locations
η and the antenna spacings Δ through nonlinear relation-ships. The first problem we tackle is that of finding the set ofoptimal spacings Δ̂ that minimizes the average (with respectto fading) interference power, P (Δ, η) = Eh0 [P (h0,Δ, η)],that is,
(Problem-1) : Δ̂ = arg minΔ
P (Δ, η) (4)
for a fixed given position η of user and interferers. Problem1 is relevant for fixed system with a known layout at the timeof antenna deployment. Moreover, its solution will bring in-sight into the structure of the optimal array, which can be tosome extent generalized to a mobile scenario. In fact, in mo-bile systems or in case the position of users and interferers isnot known a priori at the time of the antenna deployment,it is more meaningful to minimize the average interferencepower for any arbitrary position of in-cell user (η0) and out-of-cells interferers (η1,η2, . . . ,ηM). Denoting the averagingoperation with respect to users and interferers positions byEη[P (Δ, η)], the second problem (9) can be can be stated as
(Problem-2) : Δ̂ = arg minΔ
Eη[P (Δ, η)
]. (5)
2.2. Ergodic capacity maximization
The instantaneous capacity for the link between the user andthe BS reads [2]
C(
h0,Δ, η) = log2
(1 + hH0 Q−1h0
)[ bit/s/Hz], (6)
and depends on both the antenna spacings Δ and the termi-nals’ locations η. For fast-varying fading channels (comparedto the length of the coded packet) or for delay-insensitiveapplications, the performance of the system from an infor-mation theoretic standpoint is ruled by the ergodic capacityC(Δ,η). The latter is defined as the ensemble average of theinstantaneous capacity over the fading distribution,
C(Δ,η
) = Eh0
[C(
h0,Δ, η)]. (7)
According to the alternative performance criterion hereinproposed, the first problem (4) is recasted as
(Problem 1) : Δ̂ = arg maxΔ
C(Δ,η), (8)
and therefore requires the maximization of the ergodic ca-pacity for a fixed given position η of user and interferers.As before, denoting the averaging operation with respect tousers and interferers positions by Eη[C(Δ,η)], the secondproblem (5) can be modified accordingly:
(Problem 2) : Δ̂ = arg maxΔ
Eη[C(Δ,η)
]. (9)
Different from the interference power minimization ap-proach, in this case, functional dependence of the perfor-mance criterion (7) on the antenna spacings Δ is highly non-linear (see Section 3 for further details) and complicated bythe presence of the inverse matrix Q−1 that relies upon Δ andη. This implies both a large-computational complexity for
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Figure 4: Setting-A: rank-2 approximation of the signal-to-interference ratio SIRr2(Δ,η) (23) versus Δ12/λ and Δ23/λ (a) compared withergodic capacity C(Δ,η) (b) (r = 50 m). Circles denote optimal solutions.
the numerical optimization of (8) and (9), and the impossi-bility to get analytical insight into the properties of the op-timal solution. When the number of antenna array is suffi-ciently small (as in Section 5), optimization can be reason-ably dealt with through an extensive search over the opti-mization domain and without the aid of any sophisticatednumerical algorithm. On the contrary, in case of an arraywith a larger number of antenna elements, more efficient op-timization techniques (e.g., simulated annealing) may be em-ployed to reduce the number of spacings to be explored andthus simplify the optimization process. Below we will prove(by numerical simulations) that the limitations of the aboveoptimization (8)-(9) are mitigated by the criteria (4)-(5) stillpreserving the final result.
3. SPATIAL CORRELATION MODEL
We consider a propagation scenario where each terminal, beit the user or an interferer, is locally surrounded by a largenumber of scatterers. The signals radiated by different scat-terers add independently at the receiving antennas. The scat-terers are distributed on a ring of radius r0 around the ter-minal (ri, i = 1, . . . ,M for the interferers) and the resultingangular spread of the received signal at the base station is de-noted by φ0 � r0/d0 (or φi � ri/di), as in Figure 3. Becauseof the finite angular spreads {φi}Mi=0, the propagation modelappears to be well suited for outdoor channels.
In [7], the spatial correlation matrix of the resultingRayleigh distributed fading process at the base station is com-puted by assuming a parametric distribution of the scatterersalong the ring, namely, the von Mises distribution (variable0 ≤ ϑ < 2π runs over the ring, see Figure 3):
f (ϑ) = 12πI0(κ)
exp[κ cos(ϑ)
]. (10)
By varying parameter κ, the distribution of the scatterersranges from uniform ( f (ϑ) = 1/(2π) for κ = 0) to a Diracdelta around the main direction of the cluster ϑ = 0 (forκ → ∞). Therefore, by appropriately adjusting parameter κand the angular spreads for each user and interferers φi, apropagation environment with a strong line-of-sight com-ponent (φi � 0 and/or κ → ∞) or richer scattering (largerφi with κ small enough) can be modelled. The (normal-ized) spatial correlation matrix has the general expressionfor both user and interferers (for the (p, q)th element withp, q = 1, . . . ,N and i = 0, 1, . . . ,M) [7]:
[Ri(θi,Δ
)]pq = exp
[j2πλΔpq sin
(θi)]
·I0(√κ2 − ((2π/λ)Δpqφi cos
(θi))2)
I0(κ).
(11)
It is worth mentioning that spatial channel models based ondifferent geometries such as elliptical or disk models [10, 11]may be considered as well by appropriately modifying thespatial correlation (11). Effects of mutual coupling (not ad-dressed in this paper) between the array elements may be in-cluded in our framework too, see [12, 13].
From (11), the spatial correlation matrices Ri of the userand interferers can be written as
Ri(ηi,Δ
) = ρiRi(θi,Δ
)with ρi = K
dαi, (12)
where K is an appropriate constant that accounts for receiv-ing and transmitting antenna gain and the carrier frequency,and α is the path loss exponent. The contribution of shad-owing in (12) will be considered in Section 5.3 as part of anadditional log-normal random scaling term.
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(b)
Figure 5: Setting-A: rank-1 approximation (a) of the signal-to-interference ratio, SIRr1(Δ,η), versus Δ12/λ and Δ23/λ. Dashed lines denotethe optimality conditions (24) obtained by the rank-1 approximation. As a reference, ergodic capacity is shown (b), for an angular spreadapproaching zero.
4. REDUCED-RANK APPROXIMATION FORTHE INTERFERENCE POWER
According to a reduced-rank approximation for the spa-tial correlation matrices of user and interferers Ri for i =0, 1, . . . ,M, in this section, we derive an analytical closedform expression for the interference power (2) to ease theoptimization of the antenna spacings Δ. In Section 4.1, weconsider the case where the angular spread for users and in-terferers φi is small so that a rank-1 approximation of the spa-tial correlation matrices can be used. This first case describesline-of-sight channels. Generalization to channel with richerscattering is given in Section 4.2.
4.1. Rank-1 approximation (line-of-sight channels)
If the angular spread is small for both user and inter-ferers2 (i.e., φi � 1 for i = 0, 1, . . . ,M), the asso-ciated spatial correlation matrices {Ri}Mi=0, can be con-veniently approximated by enforcing a rank-1 constraint.For φi � 1, the following simplification holds in (11):
I0(√κ2 − ((2π/λ)Δpqφi cos(θi))2)/I0(κ) � 1. Therefore, the
spatial correlation matrices (12) can be approximated as (wedrop the functional dependency for simplicity of notation)
Ri � ρ · vivHi , (13)
2 Rank-1 approximation for the out-of-cell interferers is quite accuratewhen considering large reuse factors as the angular spread experiencedby the array is reduced by the increased distance of the out-of-cell inter-ferers.
where
vi(Δ, j) =[
1 exp[− jω
(θi)Δ12] · · · exp
[− jω(θi)Δ1N
]]T
(14)
and ωi(θ) = 2π/λ sin(θi).From (13), the channel vectors for user and interfer-
ers can be written as hi = γ√ρivi, where γ ∼ CN (0, 1).
Therefore, within the rank-1 approximation, the interferencepower reads (the additive noise contribution σ2IN has beendropped since it is immaterial for the optimization problem)
P1(Δ,η) = vH0
( M∑
i=1
ρivivHi
)v0, (15)
therefore, optimal spacings with respect to Problem 1 (4) canbe written as
Δ̂ = arg minΔ
P1(Δ,η), (16)
where the subscript is a reminder of the rank-1 approxima-tion. The advantage of the rank-1 performance criterion (15)is that it allows to derive an explicit expression as a functionof the parameters of interest. In particular, after tedious butstraightforward algebra, we get
P1(Δ,η)
=M∑
i=1
[ρiN + ρi
L∑
j=1
4S(l j , θ0, θi
)+ ρi
C∑
k=1
2S(ck, θ0, θi
)]
,
(17)
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Figure 6: Setting B: rank-2 approximation of the signal-to-interference ratio, SIRr2(Δ,η), (23) versus Δ12/λ and Δ23/λ (a) compared withergodic capacity C(Δ,η) (b) (r = 50 m). Circles denote optimal solutions.
where S(x, θn, θm) = cos[2πx(sin(θn)−sin(θm))], L =((
N2
)−
N/2)/2 is the number of “lateral spacings” li = Δi, j for
i = 1, . . . ,N/2−1, j = i+1, . . . ,N− i, C = N/2 is the numberof “central spacings” ci = Δi,N−i for i = 1, . . . ,N/2.
As a remark, notice that if there exists a set of antennaspacing Δ̃ such that the user vector v0 is orthogonal to the Minterference vectors {vi}Mi=1, then this nulls the interferencepower, P1(Δ,η) = 0, and thus implies that Δ̃ is a solution to(16) (and therefore to (4)).
4.2. Rank-a (a > 1) approximation
In a richer scattering environment, the conditions on the an-gular spread φi � 1 that justify the use of rank-1 approx-imation can not be considered to hold. Therefore, a rank-aapproximation with a > 1 should be employed (in general)for the spatial correlation matrix of both user and interferers:
Ri �a∑
k=1
ρ(k)i · v(k)
i v(k)Hi (18)
for i = 0, 1, . . . ,M. The set of vectors {v(k)i }ak=1 in (18) is re-
quired to be linearly independent. In this paper, we limit theanalysis to the case a = 2, which will be shown in Section 5to account for a wide range of practical environments. The
expression of vectors v(k)i from (11) with respect to the an-
tenna spacings is not trivial as for the rank-1 case. However,in analogy with (14), we could set
v(k)i =
[1 exp
[− jω(k)
i Δ12
]· · · exp
[− jω(k)
i Δ1N
]]T,
(19)
where the wavenumbers ωi = [ω(1)i ,ω(2)
i ] for user and inter-ferers have to be determined according to different criteria.In order to be consistent with the rank-1 case considered in
the previous section, here we minimize the Frobenius norm
of approximation error matrix ‖Ri −∑a
k=1 ρ(k)i · v(k)
i v(k)Hi ‖2
with respect to ω = [ω(1)i ,ω(2)
i ] vector and ρ = [ρ(1)i , ρ(2)
i ]vectors. For instance, for a uniform distribution of the scat-terers along the ring (i.e., κ = 0), it can be easily proved thatthe optimal rank-2 approximation (for i = 0, . . . ,M) resultsin
ω(1)i = ωi
(θi)
+ ϕi, ω(2)i = ωi
(θi)− ϕi, (20)
where ϕi = 2π/λ · φi cos(θi) and ρ(1)i = ρ(2)
i = ρi/2.As for the rank-1 case in (17), after some alge-
braic manipulations, the performance criterion P2(Δ,η) =Eh0 [hH0 Qh0] admits an explicit expression in terms of the pa-rameters of interest:
P2(Δ,η) =M∑
i=1
[ρiN + ρi
L∑
j=1
4S(l j , θ0, θi
) · Ti(l j)
+ ρi
C∑
k=1
2S(ck, θ0, θi
)Ti(ck)]
,
(21)
where Ti(x) = cos(ϕ0x) cos(ϕix); notice that in practicalenvironments, the angular spread for the in-cell user, ϕ0,is larger than the out-of-cell interferers angular spreads,ϕ1, . . . ,ϕM (see Section 5). Therefore, the optimization prob-lem (4) can be stated as
Δ̂ = arg minΔ
P2(Δ,η). (22)
5. NUMERICAL RESULTS
In this section, numerical results related to the layouts inFigure 1 (N = 4, M = 3, F = 3 for setting A and F = 7for setting B with a cell diameter � = 2 km) are presented.Both the interference power minimization problems (4), (5)
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Figure 7: Setting-B: rank-1 approximation (a) of the signal-to-interference ratio SIRr1(Δ,η) versus Δ12/λ and Δ23/λ. Circular marker de-notes the optimal solution (24) obtained by the rank-1 approximation. As a reference, ergodic capacity is shown (b), for an angular spreadapproaching zero.
and the ergodic capacity optimization problems (8), (9) forProblems 1 and 2, respectively, are considered and com-pared for various propagation environments. For Problem1, user and interferers are located at the center of their re-spective allowed sectors (η, as in Figure 1), instead, for Prob-lem 2 average system performances are computed over the al-lowed positions (herein uniformly distributed) of users andinterferers.
Exploiting the rank-a-based approximation (rank-1 andrank-2 approximations in (17) and (21), resp.), the inter-ference power (for fixed user and interferers position η asfor Problem 1, or averaged over terminal positions as forProblem 2) is minimized with respect to the array spac-ings and the resulting optimal solutions are compared tothose obtained through maximization of ergodic capacity.Herein, we show that the proposed approach based on in-terference power minimization is reliable in evaluating theoptimal spacings that also maximize the ergodic capacityof the system. Since the number of antenna array is lim-ited to N = 4, ergodic capacity optimization can be car-ried out through an extensive search over the optimizationdomain.
The channels of user and interferers are assumed to becharacterized by the same scatterer radius ri = r (for therank-2 case) and r → 0 (for the rank-1 case) with κ = 0.Furthermore, the path loss exponent is α = 3.5. The signal-to-background noise ratio (for the ergodic capacity simu-lations) is set to Nρ0/σ2 = 20 dB. For the sake of visual-ization, the rank-a approximation of the interference poweris visualized (in dB scale) as the signal-to-interference ratio(SIR):
SIRra(Δ,η) =[
ρ0
Pa(Δ,η)
]
dB. (23)
5.1. Setting A (F = 3)
Assuming at first fixed position η for user and interfer-ers (Problem 1), Figure 4(b) shows the exact ergodic capac-ity C(Δ,η) for r = 50 m (and thus the angular spread isφ0 = 5.75 deg, φ1 = φ3 = 0.87 deg, φ2 = 0.82 deg) andFigure 4(a) shows the rank-2 SIR approximation SIRr2(Δ,η)(23) versus Δ12 and Δ23 for setting A. According to both op-timization criteria, the optimal array has external spacingΔ̂12 � 1.26 λ and internal spacing Δ̂23 � 3.6 λ. It is interestingto compare this result with the case of a line-of-sight channelthat is shown in Figure 5. In this latter scenario, the optimalspacings are easily found by solving the rank-1 approximateproblem (16) as (k = 0, 1, . . .)
Δ̂12 = (2k + 1)Ψ(θ1)
with any Δ̂23 ≥ 0 (24a)
or Δ̂12 + Δ̂23 = (2k + 1)Ψ(θ1), (24b)
where Ψ(θ1) = λ/(2 sin(θ1)) � 0.6 λ as θ1 = θ2 = 52 deg.Conditions (24) guarantee that the channel vector of the useris orthogonal to the channel vectors of the first and third in-terferers (the second is aligned so that mitigation of its inter-ference is not feasible). Moreover, the optimal spacings forthe line-of-sight scenario (24) form a grid (see Figure 5(a))that contains the optimal spacings for the previous case inFigure 4 with larger angular spread. Notice that, for everypractical purpose, the solutions to the ergodic capacity maxi-mization (Figure 5(b)) are well approximated by SIRr1(Δ,η)maximization in (23). As a remark, we might observe thatwith line-of-sight channels, there is no advantage of deploy-ing more than two antennas (Δ̂12 = 0 or Δ̂23 = 0 satisfythe optimality conditions (24)) to exploit the interferencereduction capability of the array. Instead, for larger angu-lar spread than the line-of-sight case, we can conclude that
8 EURASIP Journal on Wireless Communications and Networking
14.5
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IRr 2
](d
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odic
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(b)
Figure 8: Setting B: rank 2 approximation of the signal-to-interference ratio Eη[SIRr2(Δ,η)] (a) and ergodic capacity Eη[C(Δ,η)] (b) aver-aged with respect to the position of user and interferers within the corresponding sectors for Δ12 = Δ23.
(i) large enough spacings have to be preferred to accommo-date diversity; (ii) contrary to the line-of-sight case, there isgreat advantage of deploying more than two antennas (ap-proximately 5-6 bit/s/Hz) whereas the benefits of deployingmore than three antennas are not as relevant (0.6 bit/s/Hzfor an optimally designed three-element array with uniformspacing 3.6 λ); (iii) compared to the λ/2-uniformly spaced ar-ray, optimizing the antenna spacings leads to a performancegain of approximately 2.5 bit/s/Hz.
Let us now turn to the solution of Problem 2 (9). In thiscase, the optimal set of spacings Δ̂ should guarantee the bestperformance on average with respect to the positions of userand interferers within the corresponding sectors. It turns outthat the optimal spacings are Δ̂12 = Δ̂23 � 1.9 λ for both op-timization criteria (not shown here), and the (average) per-formance gain with respect to the conventional adaptive ar-rays with Δ̂12 = Δ̂23 = λ/2 has decreased to approximately0.5 bit/s/Hz. This conclusion is substantially different for sce-nario B as discussed below.
5.2. Setting B (F = 7)
For Problem 1, the exact ergodic capacity C(Δ,η) for r =50 m (and angular spread φ0 = 5.75 deg, φ1 = 0.34 deg,φ2 = 0.56 deg, and φ3 = 0.58 deg) and the rank-2 approx-imation SIRr2(Δ,η) (23) are shown versus Δ12 and Δ23, inFigure 6, for setting B. In this case, the optimal linear min-imum length array consists, as obtained by both optimiza-tion criteria, by uniform 2.2 λ spaced antennas. Optimal de-sign of linear minimal length array leads to a 2.5 bit/s/Hzcapacity gain with respect to the capacity achieved throughan array provided with four uniformly λ/2 spaced antennas.Similarly as before, we compare this result with the case of aline-of-sight channel (Figure 7(a)), where the optimal spac-ings, solution to the rank-1 approximate problem (16), are
Δ̂12 = Ψ(θ3) � 0.7 λ (external spacing) and Δ̂23 = 3Ψ(θ3) �2.2 λ (internal spacing) θ3 = 43.5 deg. In this case, the solu-tions (confirmed by the ergodic capacity maximization, seeFigure 7(b)) guarantee that the channel vector of the user isorthogonal to the channel vector of the third (predominant)interferer (the second is almost aligned so that mitigation ofits interference is not feasible, the first one has a minor im-pact on the overall performances). As pointed out before,a larger angular spread than the line-of-sight case require-larger spacings to exploit diversity.
As for Problem 2 (9), in Figure 8, we compare the ana-lytical rank-2 approximation Eη[SIRr2(Δ,η)] averaged overthe position of users and interferers with the exact aver-aged ergodic capacity for a uniform-spaced antenna array.The minimal length optimal solutions turn out again to beΔ̂12 = Δ̂23 � 2.2 λ. Moreover, we can conclude that in thisinterference layout the capacity gain with respect to the ca-pacity achieved through an array provided with four uni-formly λ/2 spaced antennas is 2.5 bit/s/Hz.
5.3. Impact of nonequal power interferingdue to shadowing effects
In this section, we investigate the impact of nonequal in-terfering powers caused by shadowing on the optimal an-tenna spacings. This amounts to include in the spatial cor-relation model (12) a log-normal variable for both user andinterferers as ρi = (K/di
α) · 10Gi/10 and Gi ∼ N (0, σ2Gi
) fori = 0, 1, . . . ,M. All shadowing variables {Gi}Mi=0 affect receiv-ing power levels and are assumed to be independent. Figure 9shows the ergodic capacity averaged over the shadowing pro-cesses for setting B and r = 50 m (as in Figure 6), when thestandard deviation of the fading processes are σG0 = 3 dB forthe user (e.g., as for imperfect power control) and σGi = 8 dB(i = 1, . . . ,M) for the interferers. By comparing Figure 9 with
S. Savazzi et al. 9
0
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Δ12 λ
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odic
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city
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Δ23
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Figure 9: Setting B: ergodic capacity C(Δ,η) averaged with respectto the distribution of shadowing (r = 50 m). Circle denotes the op-timal solution.
Figure 6, we see that the overall effect of shadowing is that ofreducing the ergodic capacity but not to modify the optimalantenna spacings; similar results can be attained by analyzingthe interference power (not shown here).
6. CONCLUSION
In this paper, we tackled the problem of optimal design oflinear arrays in a cellular systems under the assumption ofGaussian interference. Two design problems are considered:maximization of the ergodic capacity (through numericalsimulations) and minimization of the interference power atthe output of the matched filter (by developing a closed formapproximation of the performance criterion), for fixed andvariable positions of user and interferers. The optimal ar-ray deployments obtained according to the two criteria areshown via numerical optimization to coincide for the con-sidered scenarios. The analysis has been validated by studyingtwo scenarios modelling cellular systems with different reusefactors. It is concluded that the advantages of an optimizedantenna array as compared to a standard design depend onboth the interference layout (i.e., reuse factor) and the prop-agation environment. For instance, for an hexagonal cellularsystem with reuse factor 7, the gain can be on average as highas 2.5 bit/s/Hz. As a final remark, it should be highlightedthat optimizing the antenna array spacings in such a way toimprove the quality of communication (by minimizing theinterference power) may render the antenna array unsuitablefor other applications where some features of the propaga-tion are of interest, such as localization of transmitters basedon the estimation of direction of arrivals.
REFERENCES
[1] J. H. Winters, J. Salz, and R. D. Gitlin, “The impact of antennadiversity on the capacity of wireless communication systems,”
IEEE Transactions on Communications, vol. 42, no. 234, pp.1740–1751, 1994.
[2] G. J. Foschini and M. J. Gans, “On limits of wireless commu-nications in a fading environment when using multiple an-tennas,” Wireless Personal Communications, vol. 6, no. 3, pp.311–335, 1998.
[3] F. Rashid-Farrokhi, K. J. R. Liu, and L. Tassiulas, “Transmitbeamforming and power control for cellular wireless systems,”IEEE Journal on Selected Areas in Communications, vol. 16,no. 8, pp. 1437–1449, 1998.
[4] J. Saholos, “A solution of the general nonuniformly spaced an-tenna array,” Proceedings of the IEEE, vol. 62, no. 9, pp. 1292–1294, 1974.
[5] J.-W. Liang and A. J. Paulraj, “On optimizing base station an-tenna array topology for coverage extension in cellular radionetworks,” in Proceedings of the IEEE 45th Vehicular TechnologyConference (VTC ’95), vol. 2, pp. 866–870, Chicago, Ill, USA,July 1995.
[6] R. Jana and S. Dey, “3G wireless capacity optimization forwidely spaced antenna arrays,” IEEE Personal Communica-tions, vol. 7, no. 6, pp. 32–35, 2000.
[7] A. Abdi and M. Kaveh, “A space-time correlation model formultielement antenna systems in mobile fading channels,”IEEE Journal on Selected Areas in Communications, vol. 20,no. 3, pp. 550–560, 2002.
[8] P. Zetterberg, “On Base Station antenna array structures fordownlink capacity enhancement in cellular mobile radio,”Tech. Rep. IR-S3-SB-9622, Department of Signals, Sensors& Systems Signal Processing, Royal Institute of Technology,Stockholm, Sweden, August 1996.
[9] H. L. Van Trees, Optimum Array Processing, Wiley-Intersci-ence, New York, NY, USA, 2002.
[10] R. B. Ertel, P. Cardieri, K. W. Sowerby, T. S. Rappaport, and J.H. Reed, “Overview of spatial channel models for antenna ar-ray communication systems,” IEEE Personal Communications,vol. 5, no. 1, pp. 10–22, 1998.
[11] T. Fulghum and K. Molnar, “The Jakes fading model incorpo-rating angular spread for a disk of scatterers,” in Proceedingsof the 48th IEEE Vehicular Technology Conference (VTC ’98),vol. 1, pp. 489–493, Ottawa, Ont, Canada, May 1998.
[12] I. Gupta and A. Ksienski, “Effect of mutual coupling on theperformance of adaptive arrays,” IEEE Transactions on Anten-nas and Propagation, vol. 31, no. 5, pp. 785–791, 1983.
[13] N. Maleki, E. Karami, and M. Shiva, “Optimization of antennaarray structures in mobile handsets,” IEEE Transactions on Ve-hicular Technology, vol. 54, no. 4, pp. 1346–1351, 2005.
EURASIP JOURNAL ON ADVANCES IN SIGNAL PROCESSING
Special Issue on
Wireless Cooperative Networks
Call for PapersMotivations
Cooperative networks are gaining increasing interest fromICT society due to the capability of improving the perfor-mance of communication systems as well as providing a fer-tile environment for the development of context-aware ser-vices.
Cooperative communications and networking is a newcommunication paradigm involving both transmission anddistributed processing promising significant increase of ca-pacity and diversity gain in wireless networks, by counteract-ing faded channels with cooperative diversity.
On one hand, the integration of long-range and short-range wireless communication networks (e.g., infrastruc-tured networks such as 3G, wireless ad hoc networks, andwireless sensor networks) improves the performance interms of both area coverage and quality of service (indeedrepresenting a form of diversity reflected in a greater capacityand number of potential users). On the other hand, the coop-eration among heterogeneous nodes, as in the case of wire-less sensor networks, allows a distributed space-time signalprocessing supporting, with a reduced complexity or energyconsumption per node, environmental monitoring, localiza-tion techniques, distributed measurements, and so forth.
The relevance of this topic is also reflected by numeroussessions in current international conferences on the field aswell as by the increasing number of national and interna-tional projects worldwide financed on these aspects.
List of topics
This issue tries to collect cutting-edge research achievementsin this area. We solicit papers that present original and un-published work on topics including, but not limited to, thefollowing:
• Physical layer models, for example, channel models(statistics, fading, MIMO, feedback)
• Device constraints (power, energy, multiple access,synchronization) and resource management
• Distributed processing and resource managementfor cooperative networks, for example, distributed
compression in wireless sensor networks, channel andnetwork codes design
• Performance metrics, for example, capacity, cost, out-age, delay, energy, scaling laws
• Cross-layer issues, for example, PHY/MAC/NET in-teractions, joint source-channel coding, separationtheorems
• Multiterminal information theory• Multihop communications• Integration of wireless heterogeneous (long- and
short-range) systems
Authors should follow the EURASIP JASP manuscriptformat described at http://www.hindawi.com/journals/asp/.Prospective authors should submit an electronic copy of theircomplete manuscript through the EURASIP JASP Manu-script Tracking System at http://mts.hindawi.com/, accord-ing to the following timetable:
Manuscript Due November 1, 2007
First Round of Reviews February 1, 2008
Publication Date May 1, 2008
GUEST EDITORS:
Andrea Conti, Engineering Department in Ferrara(ENDIF), University of Ferrara, 44100 Ferrara, Italy;[email protected]
Jiangzhou Wang, Department of Electronics, University ofKent, Canterbury, Kent CT2 7NT, UK; [email protected]
Hyundong Shin, Department of Radio CommunicationEngineering, School of Electronics and Information, KyungHee University, Gueonggi-Do 449-701, South Korea, Korea;[email protected]
Ramesh Annavajjala, ArrayComm Inc., San Jose,CA 95131-1014, USA; [email protected]
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Smart Antennas—State of the Art brings together the broad expertise of 41 European experts in smart
antennas. They provide a comprehensive review and an extensive analysis of the recent progress and new results generated during the last years in almost all fields of smart antennas and MIMO (multiple-input multiple-output) transmission. The following represents a summarized table of contents.
Receiver: space-time processing, antenna combining, reduced rank processing, robust beamforming, subspace methods, synchronization, equalization, multiuser detection, iterative methods
Channel: propagation, measurements and sounding, modelling, channel estimation, direction-of-arrival estimation, subscriber location estimation
Transmitter: space-time block coding, channel side information, unified design of linear transceivers, ill-conditioned channels, MIMO-MAC strategies
Network Theory: channel capacity, network capacity, multihop networks
Technology: antenna design, transceivers, demonstrators and testbeds, future air interfaces
Applications and Systems: 3G system and link level aspects, MIMO HSDPA, MIMO-WLAN/UMTS implementation issues
This book serves as a reference for scientists and engineers who need to be aware of the leading edge research in multiple-antenna communication, an essential technology for emerging broadband wireless systems.
Smart Antennas—State of the ArtEdited by: Thomas Kaiser, André Bourdoux, Holger Boche, Javier Rodríguez Fonollosa, Jørgen Bach Andersen, and Wolfgang Utschick
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