performance evaluation of cdma reverse links with ... · abstract—reverse link capacity of a...

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Performance Evaluation of CDMA Reverse Links withImperfect Beamforming in a Multicell Environment Using a

Simplified Beamforming Model

Jin Yu, Yu-Dong Yao, A. F. Molisch, Jinyun Zhang

TR2006-126 May 2006

AbstractReverse link capacity of a direct-sequence codedivision multiple-access (DS-CDMA) systemin a multicell environment has been studied recently, and significant capacity improvementsdue to the use of beamforming have been observed. However, system performance withbeamforming will be affected by several impairments, such as direction of arrival estimationerrors, array perturbations, mutual coupling, and signal spatial spreads. In this paper, reverselink performance of CDMA systems with beamforming under these impairments (imperfectbeamforming) is investigated. A simplified beamforming model is developed to evaluate thesystem performance in terms of user capacity, bit-error rates (BER), and outage probabilities.Both signal-to-interference-ratio-based power control and strengthbased power control areconsidered in this paper. The capacity and BER degradations due to different impairmentsare shown, and outage probabilities under different power control schemes are examined.

IEEE Transactions on Vehicular Technology

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 3, MAY 2006 1019

Performance Evaluation of CDMA Reverse LinksWith Imperfect Beamforming in a Multicell

Environment Using a SimplifiedBeamforming Model

Jin Yu, Member, IEEE, Yu-Dong Yao, Senior Member, IEEE, Andreas F. Molisch, Fellow, IEEE, andJinyun Zhang, Senior Member, IEEE

Abstract—Reverse link capacity of a direct-sequence code-division multiple-access (DS-CDMA) system in a multicell en-vironment has been studied recently, and significant capacityimprovements due to the use of beamforming have been observed.However, system performance with beamforming will be affectedby several impairments, such as direction of arrival estimationerrors, array perturbations, mutual coupling, and signal spa-tial spreads. In this paper, reverse link performance of CDMAsystems with beamforming under these impairments (imperfectbeamforming) is investigated. A simplified beamforming model isdeveloped to evaluate the system performance in terms of usercapacity, bit-error rates (BER), and outage probabilities. Bothsignal-to-interference-ratio-based power control and strength-based power control are considered in this paper. The capacityand BER degradations due to different impairments are shown,and outage probabilities under different power control schemesare examined.

Index Terms—Array perturbation, beamforming, code-divisionmultiple access (CDMA), mutual coupling, power control, spatialspread, user capacity.

I. INTRODUCTION

IN THE PAST 15 years, there is an explosive increase inthe number of mobile users. Although second-generation

(2-G) wireless systems, such as the Global System for MobileCommunications (GSM) and IS-95, are successful in manycountries [1], [2], they still cannot meet the requirements ofhigh-speed data and user capacity in high-user-density areas.Code-division multiple access (CDMA) has been chosen as theradio interface technology for third-generation (3-G) systems

Manuscript received April 17, 2004; revised March 4, 2005, June 19,2005, and October 21, 2005. The review of this paper was coordinated byDr. M. Stojanovic.

J. Yu was with the Wireless Information Systems Engineering Laboratory,Department of Electrical and Computer Engineering, Stevens Institute ofTechnology, Hoboken, NJ 07030 USA. He is now with Berkeley VaritronicsSystems, Inc., Metuchen, NJ 08840 USA (e-mail: [email protected]).

Y.-D. Yao is with the Wireless Information Systems Engineering Laboratory,Department of Electrical and Computer Engineering, Stevens Institute ofTechnology, Hoboken, NJ 07030 USA (e-mail: [email protected]).

A. F. Molisch is with Mitsubishi Electric Research Laboratories, Cambridge,MA 02139 USA, and also with the Department of Electroscience, LundUniversity, Lund 221 00, Sweden (e-mail: [email protected]).

J. Zhang is with Mitsubishi Electric Research Laboratories, Cambridge, MA02139 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TVT.2006.874556

[3]. Unlike frequency-division multiple access (FDMA) andtime-division multiple access (TDMA), which are primarilybandwidth- or dimension-limited in capacity, CDMA capacityis interference-limited [4]. Thus, any reduction of the interfer-ence will directly lead to capacity increases. Emerging tech-nologies, such as beamforming and multiuser detections, couldlead to a significant reduction in the interference and result inconsiderable capacity increases [5]–[7]. Capacity estimation isan important element in the performance evaluation of thesenew technologies.

Gilhousen et al. [4] estimated CDMA reverse link user ca-pacity, where strength-based power control [8], [9] is assumedand total other-cell interference I is modeled as Gaussiannoise [4], [10]. The value of I increases with the number ofactive users per cell, N , which results in a decrease of signal-to-interference ratio (SIR). The maximum N can be foundconsidering a target SIR, γ0. Using similar methods employedin [4], Kim and Sung estimated the reverse link user capacity ofSIR-based power-controlled CDMA systems in [11] and [12].User capacity of multicode CDMA systems supporting voiceand data traffic or heterogeneous constant-bit-rate traffic wasanalyzed in [13]. The effects of a Rake receiver and antennadiversity on reverse link user capacity were further investigatedin [14]. Capacity improvements with base-station antenna ar-rays in strength-based power-controlled cellular CDMA sys-tems have been analyzed [15]. CDMA reverse link capacitywith the combined use of antenna arrays and Rake receiversunder SIR-based power control scheme was investigated in[16]. Significant capacity improvement due to the use of beam-forming has been observed in [15] and [16].

In the implementation of beamforming algorithms, the di-rection of arrival (DOA), which is a parameter that is used tosteer the beams toward the desired signals, is usually obtainedthrough estimation algorithms, such as the multiple signalclassification (MUSIC) algorithm and the estimation of signalparameters via rotational invariance techniques (ESPRIT) al-gorithm [17], [18]. Any error in estimating arrival angles willcause the antenna array point away from the desired users andlead to a reduction of the received power of the desired signals.Even with perfect DOA estimations, array perturbations due tothe position errors of antenna elements can also result in mis-matches between exact DOAs and directions of main lobes [17].

0018-9545/$20.00 © 2006 IEEE

1020 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 3, MAY 2006

In addition to DOA estimation errors and array perturbations,the existence of mutual coupling between antenna elementsalso leads to a significant change of beam patterns [19], [20].Imperfect power control in addition to mutual coupling andscattering is also considered in [19]. The obstacles around atransmitter (mobile station, MS), such as buildings, reflect thetransmitted waves and result in multiple paths with different ar-rival angles at a base station (BS; angle spread). A propagationmodel to characterize the angle spread was reviewed in [21].Recently, angle spreads have been measured and reported in[22] and [23]. For rural environments, angular spreads between1◦ and 5◦ have been observed [22]; for urban and hilly terrainenvironments, considerably larger angular spreads as large as20◦, have been found [23]. Angle spreads not only reduce thereceived signal power, as the DOA estimation becomes randomin the interval of arrival angles, but also cause DOA estima-tion uncertainty. Sensitivity of array signal processing/arrayantennas to errors and imperfections has also been describedextensively in [17].

Notice that the CDMA system performance improvementhas been investigated considering perfect beamforming (noimpairments) in most studies [15], [16], [18], whereas im-perfect beamforming considering mutual coupling and spatialspreads (angle spreads) has been investigated for a single-cellCDMA system [19]. This paper differs from previous studiesin two aspects. First, we test a multicell CDMA system withboth in-cell interference and other-cell interference. Second, weinvestigate all beamforming impairments discussed above, i.e.,DOA estimation errors, array perturbations, mutual coupling,and spatial spreads. Additionally, a simplified beamformingmodel is developed in this paper to facilitate the evaluationof CDMA performance with imperfect beamforming. Signalstrength-based power control and SIR-based power controlare considered in the performance evaluation in terms of usercapacity, bit-error rates (BER), and outage probabilities. In thispaper, user capacity is referred to as the number of users that aCDMA system could support under a desired signal-to-noise-plus-interference ratio (SNIR) [6]. This paper is organized asfollows. System models, including beamforming and other-cellinterference, are given in Section II. A simplified beamformingmodel is presented in this section. The impacts of DOA esti-mation errors, array perturbations, mutual coupling, and spatialspreads on the interference statistics are analyzed in Section III.Performance of CDMA reverse link with imperfect beamform-ing is evaluated in Section IV, and numerical results are givenin Section V. Finally, conclusions are drawn in Section VI.

II. SYSTEM MODEL

A. Beamforming

Beamforming has been widely used in wireless systems thatemploy a fixed set of antenna elements in an array. Consideringreceive beamforming in reverse link transmissions, the signalsfrom these antenna elements are combined to form a movablebeam pattern that can be steered to a desired direction thattracks MSs as they move. This allows the antenna system tofocus radio frequency (RF) resources on a particular MS and

Fig. 1. ULA.

minimize the impact of interference [18]–[24]. Although fewantenna elements could be installed at an MS, large antennaarrays can be implemented at a BS. When beamforming isused at the MS, the transmit beam pattern can be adjusted tominimize the interference to unintended receivers (such as BSsin other cells). At a BS, receive beamforming for each desireduser could be implemented independently without affectingthe performance of other links [24]. A uniform linear array(ULA) considering a two-dimensional multicell environment(θ = 90◦) is considered and is shown in Fig. 1 [18]. Thedistance d between the antenna elements is assumed to be 0.5λ,where λ is the carrier wavelength. In the ULA system, a com-bining network connects an array of low-gain antenna elementsand could generate an ideal antenna pattern [18], [25], i.e.,

G(φ, ϕ) =

∣∣∣∣∣ 1MM−1∑m=0

exp {−j(k · pm −mπ sinϕ)}∣∣∣∣∣2

(1)

where M is the number of antenna elements, exp(jmπ sinϕ)is the weighting factor for the mth antenna element, k =2π(cosφ, sinφ)/λ is the phase propagation vector, φ is thearrival angle of signals or interference, pm = (xm, ym) is thevector for the position of the mth element, and ϕ is the scanangle. The beam could be steered to a desired direction byvarying a single parameter ϕ, when ϕ is set to be equal to thearrival angle of the desired signal [18]. In the remaining ofthis paper, we will use the antenna pattern specified in (1) toevaluate the impact of beamforming on the CDMA reverse linkperformance.

B. Other-Cell Interference

A cellular structure is shown in Fig. 2 with a reference cell(BS0) and an interference cell (with BSm). In a CDMA cellularsystem, an MS is power-controlled by a BS in its home cellto ensure that the received SNIR (or power) at the BS is noless than a target value, assuming that SIR-based power control(or strength-based power control) is in use. Considering the jthMS in the mth cell, MSm,j , let the received power at its BS(BSm) be S. Note that imperfect beamforming may result in the

YU et al.: PERFORMANCE EVALUATION OF CDMA REVERSE LINKS WITH IMPERFECT BEAMFORMING 1021

Fig. 2. Cellular structure and reverse link geometry.

change of the gain toward the desired user. The instantaneousreceived power is

S = PT r−µm 10ξm/10Gt(φm,j − π, φ̂m,j − π)Gr(φm,j , φ̂m,j)

(2)

where φ̂m,j is the estimated arrival angle that is used to steerthe beam, Gt(φm,j − π, φ̂m,j − π) and Gr(φm,j , φ̂m,j) aretransmit and receive beamforming gains [16], and φm,j − πand φm,j are the transmit angle and arrival angle from the MSto its home BS, respectively.1 The transmitted power PT isobtained as

PT =S

r−µm 10ξm/10Gt(φm,j − π, φ̂m,j − π)Gr(φm,j , φ̂m,j)

.

(3)

The resulted interference at the BS of the reference cell BS0 is

I = S(rm/r0)µ10(ξ0−ξm)/10

× Gt(ϕm,j − π, φ̂m,j − π)Gr(ϕm,j , φ̂0,0)

Gt(φm,j − π, φ̂m,j − π)Gr(φm,j , φ̂m,j)(4)

where r0 and rm are the distances from MSm,j to BS0 andBSm as shown in Fig. 2, φ0,0 is the arrival angle of thedesired user MS0,0 to BS0 as shown in Fig. 2 and is uniformlydistributed from 0 to 2π, φ̂0,0 is the estimated value of φ0,0, µis a path loss exponent, and ξ0 and ξm describe the shadowingprocesses in the cells of BS0 and BSm. The shadowing

1Note that (2) describes the received power averaged over the small-scalefading. However, recent investigations [26] have shown that even the powerdistribution in the presence of both shadowing and small-scale fading can bewell approximated by a lognormal distribution with a modified shadowingvariance.

processes are assumed to be mutually independent and followa lognormal distribution with standard deviation σ dB andzero mean. Considering all interfering MSs, the total other-cellinterference at BS0 is obtained by integrating the whole cellularcoverage area except the reference cell [12],

∫ ∫(·)dA. The

total other-cell interference is

I =∫ ∫

S(rm/r0)µ10(ξ0−ξm)/10ε(ξ0 − ξm, rm/r0)

× ρGt(ϕm,j − π, φ̂m,j − π)Gr(ϕm,j , φ̂0,0)

Gt(φm,j − π, φ̂m,j − π)Gr(φm,j , φ̂m,j)dA (5)

where

ε(ξ0 − ξm, rm/r0) ={1, if (rm/r0)µ10(ξ0−ξm)/10 ≤ 10, otherwise

ϕm,j = arctan(r1 sinφm + rm sinφm,j

r1 cosφm + rm cosφm,j

)

ρ is the user density per unit area, N is the number of MSsin each cell (ρ = 2N/(3

√3)), assuming that the users are uni-

formly distributed in a cell and the radius of the hexagonal cellis normalized to unity, ε(ξ0 − ξm, rm/r0) is an indicator func-tion to show the cell areas that are excluded in the calculationof I because the MSs in these areas are not power-controlled byBSm but by BS0, φm is the azimuth angle of BSm, and r1 is thedistance between BSm to BS0. In computing the above integral,we simply consider the hexagonal areas of each cell rather thanthe actual coverage area of the BSs [4], [11], [12].

In the remainder of this paper, we consider receive beam-forming only and set the number of transmit antenna elementsto 1, so that Gt(ϕm,j − π, φ̂m,j − π) = Gt(φm,j − π, φ̂m,j −π) = 1. The mean value of the total other-cell interferenceis thus

E[I] = E[S]F (µ, σ)N (6)

with

F (µ, σ) =2

3√3exp

{(σ ln(10)

10

)2}∫ ∫ (

rmr0

)µ

× Φ(

10µ√2σ2

log10

(r0rm

)−√2σ2

ln(10)10

)

× E

[Gr(ϕm,j , φ̂0,0)

Gr(φm,j , φ̂m,j)

]dA (7)

where

Φ(x) =1√2π

x∫−∞

exp{− t2

2

}dt.

The variance of I is found to be

Var[I] ={U(µ, σ)E[S2]− V (µ, σ)E2[S]

}N (8)


where

U(µ, σ) =2

3√3exp

{(σ ln(10)

5

)2}∫ ∫ (

rmr0

)2µ

× Φ(

10µ√2σ2

log10

(r0rm

)−√2σ2

ln(10)5

)

× E

[G2

r(ϕm,j , φ̂0,0)

G2r(φm,j , φ̂m,j)

]dA (9)

and

V (µ, σ) =2

3√3exp

{2(σ ln(10)

10

)2}∫ ∫ (

rmr0

)2µ

× Φ2

(10µ√2σ2

log10

(r0rm

)−√2σ2

ln(10)10

)

× E2

[Gr(ϕm,j , φ̂0,0)

Gr(φm,j , φ̂m,j)

]dA. (10)

C. Simplified Beamforming Modeling: Signal and Interference

The value F (µ, σ), U(µ, σ), and V (µ, σ) can be obtainednumerically [16]. However, the complexity considering theexact beam pattern is very high, especially for the evalua-tion under those impairments mentioned in Section I, due tomultiple integrals. A simple Bernoulli model is introduced in[15] in which a signal is considered to be within a mainlobe(Gr = 1) or out of the mainlobe (Gr = 0), and the half-powerbeamwidth is defined as the beamwidth. This model is easy touse, but it neglects the impact of sidelobes and the effect of anyspecific beam patterns. The work of Spagnolini [27] providesa beamforming model with a triangular pattern to characterizethe beam head. In the following, we develop an accurate, yetsimple, beamforming model to account the impact of sidelobesand the real beam patterns. Assuming that the beamwidth isB (normalized by 2π), the gain of the mainlobe is normalizedto unity, and the gain out of the mainlobe is α. This impliesthat the probability that one interferer is in the mainlobe is B.Considering the Bernoulli distribution, the first and second mo-ments of the antenna gain with respect to all incident angles areB + (1−B)α and B + (1−B)α2, respectively. Consideringthe impact of the impairments of beamforming and normalizingthe beam pattern by the gain at the desired direction, we set thefirst and second moments of the real beam pattern equal to thatin the simplified model and obtain

B + (1−B)α = E

[Gr(ϕ, φ̂)

Gr(ψ, ψ̂)

](11)

and

B + (1−B)α2 = E

[G2

r(ϕ, φ̂)

G2r(ψ, ψ̂)

]. (12)

Solving this equation set, we obtain the parameters for receivebeamforming

α =ΣG −ΥG

ΥG − 1(13)

and

B =ΣG −Υ2

G

ΣG + 1− 2ΥG(14)

where

ΥG = E

[Gr(ϕ, φ̂)

Gr(ψ, ψ̂)

](15)

and

ΣG = E

[G2

r(ϕ, φ̂)

G2r(ψ, ψ̂)

](16)

in which ΥG and ΣG are the antenna gain parameters, φ and ϕare the arrival angles of the desired signal and the interference,φ̂ is an estimate of φ,ψ denotes the arrival angle of the interfererto its own BS, and ψ̂ is an estimation of ψ. Note that the divisionin (15) and (16) is due to power control. When ψ is the arrivalangle of an interferer in another cell (other-cell interference),the power control is performed at another cell. Therefore, ψ isindependent of ϕ, and the denominator and numerator in (15)and (16) are independent of each other. However, when ψ is thearrival angle of an interferer from the same cell, power controlis done at the reference cell and ψ = ϕ. We denote the arraygain parameters for in-cell interference as ΥI

G and ΣIG, and

those for other-cell interference as ΥOG and ΣO

G. The antennagain ratios are averaged with respect to the arrival angles ofthe desired signal and the interference, φ, ϕ, and ψ, whichare assumed to be uniformly distributed in the region from 0to 2π. It is important to point out that a real beam patternis used for evaluating the received power of a desired signalas shown in Fig. 3(a), and the simplified accurate model withparameters B and α is used for interference [Fig. 3(b)]. Whenthe beamforming impairments are considered, ϕ, φ, and ψ willbe related to those random variables (B,α), which are usedto model the impairments. With the simplified/accurate modeldescribed above, we obtain the mean of the interference I

I =NF (µ, σ)E[S]

=N (B + (1−B)α)F0(µ, σ)E[S]

=NΥOGF0(µ, σ)E[S] (17)

where

F0(µ, σ) =2

3√3exp

{(σ ln(10)

10

)2}∫ ∫ (

rmr0

)µ

×Φ(

10µ√2σ2

log10

(r0rm

)−√2σ2

ln(10)10

)dA (18)


Fig. 3. Simplified model for beamforming with arrival angle φ = 30◦.(a) Signal model. (b) Interference model.

and similarly, the variance of the interference I can be simpli-fied as

Var[I] ={U0(µ, σ)ΣO

GE[S2]− V0(µ, σ)E2[S](ΥO

G

)2}N

(19)

where

U0(µ, σ) =2

3√3exp

{(σ ln(10)

5

)2}∫ ∫ (

rmr0

)2µ

×Φ(

10µ√2σ2

log10

(r0rm

)−√2σ2

ln(10)5

)dA (20)

and

V0(µ, σ) =2

3√3exp

{2(σ ln(10)

10

)2}∫ ∫ (

rmr0

)2µ

×Φ2

(10µ√2σ2

log10

(r0rm

)−√2σ2

ln(10)10

)dA (21)

where F0(µ, σ) = 0.6611, U0(µ, σ) = 0.2252, and V0(µ, σ) =0.0451 with µ = 4 and σ = 8 [16]. In the evaluation of reverselink performance of a CDMA system with imperfect beam-forming, the key issue is to find ΣO,I

G and ΥO,IG . We will also

show in Section IV that the numerical results with the use ofthe simplified and accurate model agree very well with thenumerical results without the use of the model.

III. INTERFERENCE STATISTICS

A. DOA Estimation Errors

In this section, the impact of DOA mismatch on the beam-forming parameters (B and α, ΥG and ΣG) is analyzed. Theestimated arrival angle φ̂ is characterized as a random variablewith a uniform distribution or normal distribution [17], [21],[28] as (22), shown at the bottom of the page, where φ isthe accurate arrival angle, ∆2 represents the variance of theestimation errors for both uniform and normal distributions,Bw

is the broadside null-to-null beamwidth, and

κ = erf−1

(0.5Bw√

2∆

)

in which

erf(x) =2√π

x∫0

exp(−t2)dt.

f(φ̂) =

12√

3∆, −

√3∆ ≤ (φ̂− φ) ≤

√3∆, uniform distribution

κ√2π∆

exp{− (φ̂−φ)2

2∆2

}, φ̂ ∈ [−0.5Bw + φ, 0.5Bw + φ], normal distribution

(22)


Therefore, the array gain parameters for the interference fromthe same cell can be found as

ΥIG =Eφ̂,ϕ̂,φ,ϕ

[Gr(ϕ, φ̂)Gr(ϕ, ϕ̂)

]

=1

4π2

2π∫0

2π∫0

∫φ̂

∫ϕ̂

f(φ̂)f(ϕ̂)

(Gr(ϕ, φ̂)Gr(ϕ, ϕ̂)

)dϕ̂ dφ̂ dϕ dφ

(23)

and

ΣIG =Eφ̂,ϕ̂,φ,ϕ

[G2

r(ϕ, φ̂)G2

r(ϕ, ϕ̂)

]

=1

4π2

2π∫0

2π∫0

∫φ̂

∫ϕ̂

f(φ̂)f(ϕ̂)

(Gr(ϕ, φ̂)Gr(ϕ, ϕ̂)

)2

dϕ̂ dφ̂ dϕ dφ.

(24)

For the interference from other cells, ΥOG and ΣO

G can befound as

ΥOG =Eφ̂,φ,ϕ

[Gr(ϕ, φ̂)

]Eψ̂,ψ

[1/Gr(ψ, ψ̂)

]

=1

4π2

2π∫0

2π∫0

∫φ̂

f(φ̂)(Gr(ϕ, φ̂)

)dφ̂ dϕ dφ

× 12π

2π∫0

∫ψ̂

f(ψ̂)

(1

Gr(ψ, ψ̂)

)dψ̂ dψ (25)

and

ΣOG =Eφ̂,φ,ϕ

[G2

r(ϕ, φ̂)]Eψ̂,ψ

[1/G2

r(ψ, ψ̂)]

=1

4π2

2π∫0

2π∫0

∫φ̂

f(φ̂)(G2

r(ϕ, φ̂))dφ̂ dϕ dφ

× 12π

2π∫0

∫ψ̂

f(ψ̂)

(1

G2r(ψ, ψ̂)

)dψ̂dψ (26)

where φ is the actual arrival angles of the desired signal to BS0,φ̂ is an estimation of φ, ϕ and ψ represent the actual arrivalangles of the users (interferers) to the reference BS and its own

BS, ϕ̂ and ψ̂ are the estimates of ϕ and ψ, and ψ is also modeledas a uniform random variable in [0, 2π).

B. Array Perturbations

Assuming there are odd number of antenna elements, M =2D + 1, which are symmetrically distributed along the y-axiswith one at the center. The position of the bth element ispb = (xb, yb) = (0, bλ/2) in the absence of a perturbation.The position errors due to array perturbation are modeled asrandom variables with a normal distribution. The weightingfactor for the bth element is wb = exp{k · pb}, where k is thepropagation vector of the signal. Let pb = xb/λ, qb = yb/λ, andwe have the beam pattern (envelope) as [17]

A =D∑

b=−D

exp {−j2π [(pb cosϕ+ qb sinϕ)− 0.5b sinφ]}

(27)

where pb and qb are independent with E[pb] = 0 and E[qb] =0.5d and Var[pb] = Var[qb] = (σp/λ)2, where σ2

p is the vari-ance of the position errors. Using p and q to represent[p−D · · · p0 · · · pD] and [q−D · · · q0 · · · qD], respectively, thebeam pattern gain is expressed in (28), shown at the bottomof the page. Using u and v to represent [u1 · · ·u2D] and[v1 · · · v2D], the beam pattern gain can be simplified as

G(ϕ, φ,u,v) =1M

{1 +

M−1∑l=1

Gl(ϕ, φ, ul, vl)

}(29)

where

Gl(ϕ, φ, ul, vl) = 2(1− l/M)

× cos {2π(ul cosϕ+ vl sinϕ)− lπ sinφ}

l = a− b, ul = pa − pb, and vl = qa − qb. Therefore, ul andvl are normally distributed with E[ul] = 0, E[vl] = 0.5l, andVar[ul] = Var[vl] = 2(σp/λ)2. If the users are assumed to beuniformly distributed in the cell, ϕ and φ are uniform randomvariables in [0, 2π). The array gain parameters for the interfer-ence from the same cell are obtained as

ΥIG =Eϕ,φ,u,v

[G(ϕ, φ,u,v)G(ϕ,ϕ,u,v)

]

=1

4π2

2π∫0

2π∫0

∫u

∫v

f(u)f(v)[G(ϕ, φ,u,v)G(ϕ,ϕ,u,v)

]du dv dϕ dφ

(30)

G(ϕ, φ,p,q) =|AA∗|M2

=1M2

M +

D∑a=−D

∑a =b

exp {−j2π [((pa − pb) cosϕ+ (qa − qb) sinϕ)− 0.5(a− b) sinφ]}

(28)


and

ΣIG =Eϕ,φ,u,v

[G2(ϕ, φ,u,v)G2(ϕ,ϕ,u,v)

]

=1

4π2

2π∫0

2π∫0

∫u

∫v

f(u)f(v)

×[G2(ϕ, φ,u,v)G2(ϕ,ϕ,u,v)

]du dv dϕ dφ (31)

where f(u) and f(v) represent the probability density functionof u and v. The array gain parameters for the interference fromother cells are obtained as

ΥOG =Eϕ,φ,u,v [G(ϕ, φ,u,v)]Eψ,u,v [1/G(ψ,ψ,u,v)]

=1

4π2

2π∫0

2π∫0

∫u

∫v

f(u)f(v) [G(ϕ, φ,u,v)] du dv dϕ dφ

× 12π

2π∫0

∫u

∫v

f(u)f(v)[

1G(ψ,ψ,u,v)

]du dv dψ

(32)

and

ΣOG =Eϕ,φ,u,v

[G2(ϕ, φ,u,v)

]Eψ,u,v

[1/G2(ψ,ψ,u,v)

]=

14π2

2π∫0

2π∫0

∫u

∫v

f(u)f(v)[G2(ϕ, φ,u,v)

]du dv dϕ dφ

× 12π

2π∫0

∫u

∫v

f(u)f(v)[

1G2(ψ,ψ,u,v)

]du dv d dψ.

(33)

C. Spatial Spreads

Spatial spread implies that the simultaneously arrived signalsmay come from different paths with different arrival angles,and the signal energy is spread in space. Assuming the spreadenergy follows the same distribution as (22) [21], with thenormal distribution limited within [−0.5π + φ, 0.5π + φ] andκ = erf−1(0.5π/

√2∆). Thus, the estimated arrival angle is as-

sumed to follow the same distribution as that of the energy. Theexpected received energy should be averaged considering boththe arrival angle estimations and the spatial spreads. Assumingthe mean value of x as x̄, the array gain parameters ΥI

G and ΣIG

for in-cell interference can be found as

ΥIG =Eφ̂,ϕ̂,ϕ,φ̄,ϕ̄

[Gr(ϕ, φ̂)Gr(ϕ, ϕ̂)

]

=1

4π2

2π∫0

2π∫0

∫φ̂

∫ϕ̂

f(ϕ̂)f(φ̂)

×∫ϕ f(ϕ)Gr(ϕ, φ̂)dϕ∫ϕ f(ϕ)Gr(ϕ, ϕ̂)dϕ

dϕ̂ dφ̂ dϕ̄dφ̄ (34)

and

ΣIG =Eφ̂,ϕ̂,ϕ,φ̄,ϕ̄

[G2

r(ϕ, φ̂)G2

r(ϕ, ϕ̂)

]

=1

4π2

2π∫0

2π∫0

∫φ̂

∫ϕ̂

f(ϕ̂)f(φ̂)

×(∫

ϕ f(ϕ)Gr(ϕ, φ̂)dϕ∫ϕ f(ϕ)Gr(ϕ, ϕ̂)dϕ

)2

dϕ̂ dφ̂ dϕ̄ dφ̄. (35)

The integral with respect to ϕ indicates the accumulation ofthe spatially spread energy, and that with respect to φ̂ and ϕ̂considers the DOA estimations. The antenna gain parametersΥO

G and ΣOG for other-cell interference can be found as

ΥOG =Eφ̂,ϕ,φ̄,ϕ̄

[Gr(ϕ, φ̂)

]Eψ̂,ψ,ψ̄

[1/Gr(ψ, ψ̂)

]

=1

4π2

2π∫0

2π∫0

∫φ̂

∫ϕ

f(φ̂)f(ϕ)Gr(ϕ, φ̂)dϕ dφ̂ dϕ̄ dφ̄

× 12π

2π∫0

∫ψ̂

f(ψ̂)1∫

ψ f(ψ)Gr(ψ, ψ̂)dψdψ̂ dψ̄ (36)

and

ΣOG =Eφ̂,ϕ,φ̄,ϕ̄

[G2

r(ϕ, φ̂)]Eψ̂,ψ,ψ̄

[1/G2

r(ψ, ψ̂)]

=1

4π2

2π∫0

2π∫0

∫φ̂

∫ϕ

f(φ̂)f(ϕ)G2r(ϕ, φ̂)dϕ dφ̂ dϕ̄ dφ̄

× 12π

2π∫0

∫ψ̂

f(ψ̂)

(1∫

ψ f(ψ)Gr(ψ, ψ̂)dψ

)2

dψ̂ dψ̄. (37)

D. Mutual Coupling

Antenna array mutual coupling has been found to have asignificant impact on wireless communications. It can affect theestimation of the arrival angles, which results in the disturbanceof the weighting vector in beamforming. Considering thin half-wavelength dipoles, mutual coupling can be characterized by amutual coupling impedance matrix [20], [29], i.e.,

C = (ZT + ZA)(Z + ZT I)−1 (38)

where ZA is the antenna impedance, ZT is the terminatingimpedance, I denotes the identity matrix, and Z is the mutualimpedance matrix. Assuming that the arrival angles are esti-mated correctly, the beam pattern is

A =B∑

m=−B

exp{jmπ sinφ}B∑

n=−B

Cm,n exp{−jπn sinϕ}.

(39)


Therefore, the normalized beamforming gain is

G(ϕ, φ) =|AA∗|M2

. (40)

The antenna gain parameters ΥIG and ΣI

G for the same cellinterference can be found as

ΥIG =

14π2

2π∫0

2π∫0

G(ϕ, φ)G(ϕ,ϕ)

dϕ dφ (41)

and

ΣIG =

14π2

2π∫0

2π∫0

G2(ϕ, φ)G2(ϕ,ϕ)

dϕ dφ (42)

where φ and ϕ are also modeled as uniform random variablesin [0, 2π). The array gain parameters for other-cell interferenceare

ΥOG =

14π2

2π∫0

2π∫0

G(ϕ, φ)dϕ dφ× 12π

2π∫0

1G(ψ,ψ)

dψ (43)

and

ΣOG =

14π2

2π∫0

2π∫0

G2(ϕ, φ)dϕ dφ× 12π

2π∫0

1G2(ψ,ψ)

dψ.

(44)

IV. PERFORMANCE EVALUATION

The performance of a CDMA system with imperfect beam-forming is evaluated in terms of user capacity, error probabili-ties, and outage probabilities. User capacity is referred to as themaximum number of users that a CDMA system could supportunder a target SNIR. In this paper, a closed-form expressionfor the user capacity under the impairments is presented. Ex-pressions for the error probability as a function of the numberof users per cell, antenna gains (ΥI,O

G ,ΣI,OG ), and the CDMA

processing gain are derived, and both the intracell and intercellinterferences are taken into consideration in the evaluation.Expressions of the outage probability for a CDMA systemwith imperfect beamforming are derived considering both thestrength-based power control and SIR-based power control.

A. User Capacity

Considering SIR-based power control in CDMA systems, thereceived SNIR, Eb/I0, should be no less than a target value, γ0,to maintain a required transmission quality. Considering receivebeamforming, we have [16]

Eb

I0≈ GS

23

(∑N−1m=1 SE

[Gr(φ0,m,φ̂0,0)

Gr(φ0,m,φ̂0,m)

]+ I)+ η0W

=GS

23

((N − 1)SΥI

G + I)+ η0W

≥ γ0 (45)

and G is the CDMA processing gain, η0 is the single-sidedwhite noise power spectrum density, and W is the spreadingbandwidth. The factor 2/3 in the denominator is due to theassumption of a square chip pulse. The denominator in (45)includes other-cell interference as well as own-cell interferencedue to other MSs in the reference cell. φ0,m is the azimuth angleof an interfering MS0,m. Gr(φ0,m, φ̂0,0) is the receive beam-forming gain of MS0,0 to the direction of MS0,m. φ0,m andφ0,0 are uniformly distributed in [0, 2π). Considering E[I] > 0and Var(I) > 0, the user capacity can be derived as [16]

N =⌊

ΥIG + 1.5G/γ0

ΥIG +ΥO

GF0(µ, σ)

⌋(46)

where �x� indicates maximum integer no greater than x.

B. Error Probability

Error probability performance of CDMA systems has beenextensively investigated, considering standard Gaussian ap-proximation (SGA), improved Gaussian approximation (IGA),and simplified improved Gaussian approximation (SIGA) [31],[32]. In this subsection, using the simplified beamformingmodel and interference statistics (Sections II and III), we testthe impact of beamforming impairments on the error probabili-ties. The received signals at a BS from C cells are

r(t) =NC∑n=1

Anαnyndn(t− τn)

× cn(t− τn) exp {j(ωn + ςn)}+ n(t) (47)

where An is the nth user’s signal amplitude, dn(t) and cn(t)represent the nth user’s binary data stream and spreadingsequence, respectively, τn indicates that each user has an in-dependent delay due to asynchronous transmissions, ςn is thecarrier phase, αn exp{jωn} is the complex path gain, and n(t)is the additive white Gaussian noise (AWGN) with single-sidedpower spectral density η0. In addition, yn is also an indicatorfunction given by

yn ={1, n ≤ N(rn/r0)µ/210(ξ0−ξn)/20, n > N.

The first user is assumed to be the reference user, and whenbeamforming and power control is considered, the decisionvariable is obtained by correlating the spreading signal, i.e.,

rT =NC∑n=1

Anαnyn√Gr(ϕn, ϕ1)/Gr(ψn, ψn)

× cos(ωn − ω1 + ςn − ς1)In +Nw (48)

where Gr(ϕn, ϕ1) is the receive beam pattern, ϕn representsthe nth user’s azimuth angle, ψn is the angle from the nthuser to its own BS, Nw is the noise component with a normaldistribution N(0, η0T/2), and In is given by [32]

In =

T+τ1∫τ1

dn(t− τn)cn(t− τn)c1(t− τ1)dt


where T is the bit period, and Tc represents the chip period.Assume that the spreading sequences of different users are in-dependent random binary sequences. Considering the strength-based power control (Anαn =

√S) and normalizing rT by

AαnTc, we obtain r′T = rT /(ATc) = G+ Iic + Ioc +N1, inwhich Iic is the normalized interference from other users inthe same cell and Ioc is the cochannel interference from othercells, N1 ∼ N(0, G2/(2γb)), where γb is the bit-energy-to-noise ratio. We have Var(Iic) ≈ ΥI

GGN/3, and

Var(Ioc) =NC∑

n=N+1

E

[y2nGr(ϕn, ϕ1)Gr(ψn, ψn)

]G/3. (49)

Similarly, assuming that all interfering MSs are uniformlydistributed in a cell and the radius of the hexagonal cell isnormalized to unity, Var(Ioc) can be obtained by integratingthe whole cellular coverage area except the reference cell [14],[16], i.e.,

Var(Ioc) = F0(µ, σ)ΥOGNG/3.

Inasmuch as Iic, Ioc, and N1 are independent of each other, themean and variance of r′T can be found as

E [r′T ] = G

and

Var [r′T ] =ΥI

G(N − 1)G3

+ΥO

GF0(µ, σ)NG

3+

G2

2γb.

We can obtain the error probabilities with Gaussian approxima-tion, i.e.,

Pe =Q

(G√

Var [r′T ]

)

=Q

√√√√ G

ΥIG

(N−1)

3 + ΥOGF0(µ,σ)N

3 + G2γb

. (50)

When the number of users N is small, the method using sim-plified improved Gaussian approximation can improve the ac-curacy of the evaluation [31]. Following the derivation of SIGA[31], let ν =

∑NCn=2 Zny

2nGr(ϕn, ϕ1)/Gr(ψn, ψn), where Zn

is a decision statistic as defined in [31], and the error probabilitycan be derived as

Pe =23Q

(G√

Var [r′T ]

)+

16

×

Q G√

Var[r′T ]+√3σν

+Q

G√

Var [r′T ]−√3σν

(51)

where

Q(x) =1√2π

∞∫x

exp(−t2/2)dt

and the variance of ψ is

σ2ν =(N − 1)

{E[Z2n

]ΣI

G − E2[Zn](ΥI

G

)2}+N

{E[Z2n

]U0(µ, σ)ΣO

G −E2[Zn](ΥO

G

)2V0(µ, σ)

}+{(G− 1)(N − 1)(N − 2)

(ΥI

G

)2+ 2(G− 1)N(N − 1)ΥI

GΥOGF0(µ, σ)

}/36

where E[Zn] = G/3 and E[Z2n] = (7G2 + 2G− 2)/40. No-

tice that the error probability derived above is for a strength-based power-controlled CDMA system. The error probabilitiesfor strength-based and SIR-based power-controlled CDMAsystems are different due to different values of γb. In strength-based power-controlled CDMA systems, γb is constant, andtypically, a very large value as the system is designed to beinterference-limited, which is different in an SIR-based power-controlled system. Reverse link capacity of a CDMA systemwith SIR-based power control can be obtained from (46), whichis denoted as N ∗. To calculate the BER in an SIR-basedpower-controlled system, γb, when N ≤ N ∗, can be found asγb = 1.5/(1.5G/γ0 −ΥI

G(N − 1)−NΥOGF0(µ, σ)), and the

error probability is determined through (50) or (51); if thesystem loses power control, N > N ∗, all users transmit withtheir maximum power under Rayleigh fading, and the errorprobability in this case can be found following [30].

C. Outage Probability

Let S be the minimum power level satisfying (45). We get theexpression of S in terms of I , when SNIR satisfies the minimumrequirement γ0, i.e.,

S =I + 1.5η0W

1.5G/γ0 −ΥIG(N − 1)

. (52)

Note that the user capacity N can be found via an iterativemethod [11], [14], in which there are two concatenated iterationloops. In the inner loop, for a given N value, determine E[I]and Var[I] using the following steps.

1) Set E[I] and Var[I] as zeros.2) Calculate E[S] and Var[S] from (52).3) Calculate E[I] and Var[I] from (17) and (19).4) Repeat steps 2 and 3 until the differences between old and

new values of E[I] and Var[I] are less than 1% [14].

Using E[I] and Var[I] obtained above and a specified max-imum transmission power limit, an outage probability (thetransmission power exceeds the power constraint) [14] can becalculated with a fixed N . If the outage probability does notexceeds a required level, the outer loop increases N by 1and enters the inter loop. The iteration loops stop when thecalculated outage probability exceeds the required level. Wethen obtain the user capacity as N − 1. The outage probability


for an SIR-based power-controlled CDMA system consideringpower constraint S∗ is

Pout =1− Pr{0 < S ≤ S∗}

= Pr{0 <

I + 1.5η0W

1.5G/γ0 −ΥIG(N − 1)

≤ S∗, I ≥ 0}. (53)

It can be rewritten as

Pout =1− Pr{0<S ≤ S∗}=1−Pr

{0<I≤S∗(1.5G/γ0−ΥI

G(N−1))−1.5η0W

}.

(54)

Let GSIR=S∗(1.5G/γ0 −ΥIG(N − 1))− 1.5η0W . Inasmuch

as I is assumed to be a Gaussian distribution, we obtain

Pout = 1 +Q

(GSIR − E[I]√

Var[I]

)−Q

(− E[I]√

Var[I]

). (55)

For a strength-based power-controlled CDMA system, S isfixed. The SIR expression (45) is rewritten as

I/S ≤ 1.5Gγ0

− (N − 1)ΥIG − 1.5η0W/S.

Let GS = 1.5G/γ0 − (N − 1)ΥIG − 1.5/η, where η is signal-

to-noise ratio (SNR) and η = S/η0W . Inasmuch as I ≥ 0, weobtain the outage probability

Pout =1− Pr {0 ≤ (I/S) ≤ GS}

=1+Q

(GS − E[I/S]√

Var[I/S]

)−Q

(− E[I/S]√

Var[I/S]

). (56)

V. NUMERIC RESULTS

Throughout this section, we assume the following propa-gation parameters, path loss exponent µ = 4 and shadowingprocess standard deviation σ = 8 dB. The CDMA processinggain G is assumed to be 128, and the required SNIR target γ0

is 5 dB.

A. User Capacity

Fig. 4 presents the impact of the DOA estimation errors onthe reverse link user capacity and illustrates the effect of thenumber of receive antenna elements. Both uniformly and nor-mally distributed DOA errors are considered. In this figure, wefocus on the comparison of the capacity evaluation results usingthe simple/accurate model (Section II-C) and those based ondirect integral computations (7). Good matches of the capacityevaluation results are observed. As shown in the figure, forsome cases, the two evaluation methods give identical results.In some other scenarios, they only differ by one or two usersin the capacity evaluations. In the evaluations presented inFig. 4, the DOA errors follow the distributions specified in(22) with a mean φ and a standard deviation 0.15Bw. Thenormally distributed estimation errors are limited to be within[−0.45Bw + φ, 0.45Bw + φ].

Fig. 5 presents the impact of various impairments on thereverse link user capacity, in which δ, ∆, and σp represent

Fig. 4. Accuracy of using simple model to evaluate user capacity, consideringDOA estimation errors (DE) with ∆ = 0.15Bw (Cmpx: (7), Simp: using thesimplified model).

the standard deviations of the DOA estimation errors, spatialspreads, and perturbation errors, respectively. In consideringthe DOA estimation errors [Fig. 5(a)], the errors are assumedto follow a normal or uniform distribution with a standarddeviation of 0.20Bw. It is seen that user capacity drops approx-imately 30% due to the DOA estimation errors. In Fig. 5(b), theimpact of array perturbations is considered. The perturbationfollows a normal distribution with zero mean and a standarddeviation of 0.05λ, 0.1λ, and 0.2λ. The user capacity decreaseranges from approximately 4% (σp = 0.05λ), 9% (σp = 0.1λ),to 29% (σp = 0.2λ). The impact of spatial spreads is examinedin Fig. 5(c). The spreads follow a uniform or normal distributionwith a standard deviation of 5◦ and 10◦. User capacity decreasesare significant. When the number of antenna elements is equalto 9, the capacity drop is as high as 28% when ∆ = 5◦ and 55%when ∆ = 10◦ (normally distributed spatial spread). The per-formance drop of user capacity increases with an increase of thenumber of antenna elements because of the fact that, whereasthe spatial spread is fixed (5◦ or 10◦), the antenna beamwidthreduces when the number of antenna elements becomes larger.This leads to reduced received signal energy and a more sig-nificant impact due to DOA estimation errors. Fig. 5(d) examsthe impact of mutual coupling, considering a scenario in whichthe terminating impedance matches the antenna impedanceZT = Z∗

A (∗ represents conjugate). It is observed that the usercapacity increases slightly with the presence of mutual coupling(3% with M = 7 or 9). A similar observation (performanceimprovement due to mutual coupling) has also been reportedfor a diversity system [20]. Notice that the impact of DOAestimation errors and spatial spreads is fully taken into accountbecause the real beam pattern is used to characterize the desiredsignal.

B. Error Probabilities

The error probabilities of a CDMA system with imperfectbeamforming due to the impairments are evaluated using (50),and the corresponding numerical results are shown in Fig. 6


Fig. 5. User capacity of CDMA systems with beamforming under (a) DOA estimation errors, (b) array perturbations, (c) spatial spreads, and (d) mutual coupling.

Fig. 6. Error probabilities of CDMA systems with M = 7, imperfect beam-forming, and strength-based power control (DE: DOA estimation errors, AP:array perturbations, SS: spatial spreads).

with M = 7 and γb = 30 dB (an interference-limited environ-ment). Although noticeable error probability increases are seendue to DOA estimation errors, array perturbations, and spatialspreads, the error performance improves slightly in the presenceof mutual coupling.

C. Outage Probabilities

Strength-based power control expects a constant receivedpower level regardless of the interference. The total other-cell

interference, which is modeled as a random variable, results in anonzero outage probability even when the CDMA system has asmall number of users. However, SIR-based power control triesto maintain SIR above a target value. Thus, if SIR-based powercontrol operates normally in a system, the required power levelsare within a reasonable range, and the outage probability isclose to 0. When the system loses power control, the systembecomes unstable and every user tries to meet the requirementof SIR by transmitting its maximum allowed power, and theoutage probability will suddenly jump to a large value. FromFig. 7, we observe that, for a given outage probability, e.g.,0.01, CDMA systems with SIR-based power control have largercapacity than those with strength-based power control, whereS∗/η0W is assumed to be 60 dB in setting a signal powerconstraint.

VI. CONCLUSION

The impact of DOA estimation errors, spatial spreads, arrayperturbations, and mutual coupling on reverse link performanceof a multicell CDMA system is investigated in this paper. Asimple beamforming model is developed, which is shown to beaccurate in performance evaluations. User capacity, error prob-abilities, and outage probabilities are derived to characterizethe CDMA performance. Noticeable performance degradationsare observed due to the impairments such as DOA estimationerrors, array perturbations, and spatial spreads. However, aminimal performance improvement is seen in the presence ofmutual coupling. The performance degradations due to thebeamforming impairments are illustrated. Notice that errors


Fig. 7. Outage probabilities of CDMA systems with M = 7 and imperfect beamforming under different power control schemes and impairments. (a) Arrayperturbations. (b) Spatial spreads with uniform distribution. (c) Spatial spreads with normal distribution. (d) DOA errors with uniform distributions. (e) DOAerrors with normal distributions. (f) Mutual coupling (SIR-PC: SIR-based power control, S-PC: strength-based power control).

due to array perturbations (antenna mispositioning) can beeliminated or reduced by appropriate calibration of the antennaarray. However, the errors due to misestimation of the DOAand due to spatial spreads cannot be reduced through antennaarray calibration.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers fortheir valuable comments.

REFERENCES

[1] J. E. Padgett, C. G. Gunther, and T. Hattori, “Overview of wire-less personal communications,” IEEE Commun. Mag. (Special Issue onWireless Personal Communications), vol. 33, no. 1, pp. 28–41, Jan.1995.

[2] A. F. Molisch, Wireless Communications. Hoboken, NJ: Wiley, 2005.[3] M. Zeng, A. Annamalai, and V. K. Bhargava, “Recent advances in

cellular wireless communications,” IEEE Commun. Mag. (Special Issueon Wideband CDMA), vol. 36, no. 9, pp. 128–138, Sep. 1999.

[4] K. S. Gilhousen et al., “On the capacity of a cellular CDMA sys-tem,” IEEE Trans. Veh. Technol., vol. 40, no. 2, pp. 303–312, May1991.

[5] S. Verdu, Multiuser Detection. Cambridge, U.K.: Cambridge Univ.Press, 1998.

[6] D. N. C. Tse and S. V. Hanly, “Linear multiuser receivers: Effectiveinterference, effective bandwidth and user capacity,” IEEE Trans. Inf.Theory, vol. 45, no. 2, pp. 641–657, Mar. 1999.

[7] U. Madhow and M. Honig, “MMSE interference suppression for direct-sequence spread-spectrum CDMA,” IEEE Trans. Commun., vol. 42,no. 12, pp. 3178–3188, Dec. 1994.

[8] M. Xiao, N. B. Shroff, and E. K. P. Chong, “Resource management inpower-controlled cellular wireless systems,” J. Wireless Commun. MobileComput., vol. 1, no. 2, pp. 185–199, Apr. 2001.

[9] A. El-Osery and C. Abdallah, “Distributed power control in CDMA cel-lular systems,” IEEE Antennas Propag. Mag., vol. 42, no. 4, pp. 152–159,Aug. 2000.

[10] A. J. Viterbi, A. M. Viterbi, and E. Zehavi, “Other-cell interference in cel-lular power-controlled CDMA,” IEEE Trans. Commun., vol. 42, no. 2–4,pp. 1501–2504, Feb.–Apr. 1994.

[11] D. K. Kim and D. K. Sung, “Capacity estimation for a multicodeCDMA system with SIR-based power control,” IEEE Trans. Veh. Tech-nol., vol. 50, no. 3, pp. 701–710, May 2001.

[12] ——, “Capacity estimation for an SIR-based power-controlled CDMAsystem supporting on-off traffic,” IEEE Trans. Veh. Technol., vol. 50,no. 3, pp. 1094–1101, Jul. 2000.

[13] S. J. Lee, H. W. Lee, and D. K. Sung, “Capacity of single-code and mul-ticode DS-CDMA systems accommodating multiclass services,” IEEETrans. Veh. Technol., vol. 48, no. 2, pp. 376–384, Mar. 1999.

[14] D. K. Kim and F. Adachi, “Theoretical analysis of reverse link capacityof an SIR-based power-controlled cellular CDMA system in a multipathfading environment,” IEEE Trans. Veh. Technol., vol. 50, no. 2, pp. 452–464, Mar. 2001.

[15] A. F. Naguib, A. Paulraj, and T. Kailath, “Capacity improvement withbase-station antenna arrays in cellular CDMA,” IEEE Trans. Veh. Tech-nol., vol. 43, no. 3, pp. 691–698, Aug. 1994.

[16] J. Yu and Y. D. Yao, “Reverse link capacity of SIR-based power-controlledCDMA systems with antenna arrays,” J. Wireless Commun. Mobile Com-put., vol. 3, no. 6, pp. 759–772, Sep. 2003.

[17] H. L. Van Trees, Optimum Array Processing, Part IV of Detection, Esti-mation, and Modulation Theory. New York: Wiley, 2002.


[18] J. C. Liberti and T. S. Rappaport, Smart Antennas for Wireless Commu-nications: IS-95 and Third Generation CDMA Applications. EnglewoodCliffs, NJ: Prentice-Hall, 1999.

[19] A. M. Wyglinski and S. D. Blostein, “On uplink CDMA cell capacity:Mutual coupling and scattering effects on beamforming,” IEEE Trans.Veh. Technol., vol. 52, no. 2, pp. 289–304, Mar. 2003.

[20] T. Svantesson, “Antennas and propagation from a signal processing per-spective,” Ph.D. dissertation, Dept. Signals Syst., Chalmers Univ. Tech-nol., Gothenburg, Sweden, 2001.

[21] R. B. Ertel, P. Cardieri, K. W. Sowerby, T. S. Rappaport, and J. H.Reed, “Overview of spatial channel models for antenna array commu-nication systems,” IEEE Pers. Commun., vol. 5, no. 1, pp. 10–22, Feb.1998.

[22] P. Pajusco, “Experimental characterization of DOA at the base stationin rural and urban area,” in Proc. IEEE Veh. Technol. Conf., May 1998,pp. 18–21.

[23] M. Toeltsch, J. Laurila, K. Kalliola, A. F. Molisch, P. Vainikainen,and E. Bonek, “Statistical characterization of urban spatial radio chan-nels,” IEEE J. Sel. Areas Commun., vol. 20, no. 3, pp. 539–549, Apr.2002.

[24] F. Rashid-Farrokhi, K. J. R. Liu, and L. Tassiulas, “Transmit beamformingand power control for cellular wireless systems,” IEEE J. Sel. AreasCommun., vol. 16, no. 8, pp. 1437–1450, Oct. 1998.

[25] S. Haykin, Adaptive Filter Theory, 3rd ed. Englewood Cliffs, NJ:Prentice-Hall, 1986.

[26] J. Wu, N. B. Mehta, and J. Zhang, “A flexible lognormal sum approxima-tion method,” in Proc. IEEE GLOBECOM, Nov. 2005, pp. 3413–3417.

[27] U. Spagnolini, “A simplified model for probability of error in DS CDMAsystems with adaptive antenna arrays,” in Proc. IEEE ICC Conf., 2001,pp. 2271–2275.

[28] D. Aszetly, “On antenna arrays in mobile communication systems: Fastfading and GSM base station receiver algorithm,” Ph.D. dissertation,Royal Inst. Technol., Stockholm, Sweden, Mar. 1996.

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

[30] J. Cheng and N. C. Beaulieu, “Accurate DS-CDMA bit-error probabilitycalculation in Rayleigh fading,” IEEE Trans. Wireless Commun., vol. 1,no. 1, pp. 3–14, Jan. 2002.

[31] J. M. Holtzman, “A simple, accurate method to calculate spread-spectrummultiple-access error probabilities,” IEEE Trans. Commun., vol. 40, no. 3,pp. 461–464, Mar. 1992.

[32] N. Kong and L. B. Milstein, “Error probability of multicell CDMA overfrequency selective fading channels with power control error,” IEEETrans. Commun., vol. 47, no. 4, pp. 608–617, Apr. 1999.

Jin Yu (S’01–M’05) received the B.S. degree fromWuhan University, Wuhan, China, in 1998, the M.S.degree from the University of Mississippi, Oxford, in2001, and the Ph.D. degree from Stevens Institute ofTechnology, Hoboken, NJ, in 2005, all in electricalengineering. During his study at the University ofMississippi, his research work focused on the com-putational electromagnetics and microwave circuits.At Stevens Institute of Technology, he had beendoing research in the applications of antenna arraysin wireless communications and in secure chaotic

spread spectrum systems in the Wireless Information Systems EngineeringLaboratory.

He is currently with Berkeley Varitronics Systems, Inc., Metuchen, NJ,as a Wireless Engineer. His research interest areas include signal processingfor wireless communications, secure/covert communications, code-divisionmultiple access, and smart antennas.

Dr. Yu was a recipient of the Peskin Award from Stevens Institute ofTechnology in 2005.

Yu-Dong Yao (S’88–M’88–SM’94) received theB.Eng. and M.Eng. degrees from Nanjing Universityof Posts and Telecommunications, Nanjing, China,in 1982 and 1985, respectively, and the Ph.D. degreefrom Southeast University, Nanjing, in 1988, all inelectrical engineering.

From 1989 and 1990, he was at Carleton Univer-sity, Ottawa, ON, Canada, as a Research Associateworking on mobile radio communications. From1990 to 1994, he was with Spar Aerospace Ltd.,Montreal, QC, Canada, where he was involved in

research on satellite communications. From 1994 to 2000, he was with Qual-comm Inc., San Diego, CA, where he participated in research and developmentin wireless code-division multiple-access (CDMA) systems. He joined StevensInstitute of Technology, Hoboken, NJ, in 2000, where he is an AssociateProfessor in the Department of Electrical and Computer Engineering and aDirector of the Wireless Information Systems Engineering Laboratory. He isthe holder of one Chinese patent and nine U.S. patents. His research interestsinclude wireless communications and networks, spread spectrum and CDMA,and digital signal processing for wireless systems.

Dr. Yao is an Associate Editor of the IEEE COMMUNICATIONS LETTERS

and the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY and an Editorfor IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. He was a GuestEditor for a special issue on wireless networks for the International Journal ofCommunication Systems.

Andreas F. Molisch (S’89–M’95–SM’00–F’05) re-ceived the Dipl. Ing., Dr. Techn., and Habilitation de-grees from the Technical University Vienna, Vienna,Austria, in 1990, 1994, and 1999, respectively.

From 1991 to 2000, he was with the TU Vienna,where he became an Associate Professor in 1999.From 2000 to 2002, he was with the Wireless Sys-tems Research Department at AT&T (Bell) Labora-tories Research, Middletown, NJ. Since then, he hasbeen a Senior Principal Member of Technical Staffwith Mitsubishi Electric Research Labs, Cambridge,

MA. He is also Professor and Chairholder for radio systems at Lund University,Lund, Sweden. He has done research in the areas of surface acoustic wavefilters, radiative transfer in atomic vapors, atomic line filters, smart antennas,and wideband systems. He has authored, coauthored, or edited four books[among them the recent textbookWireless Communications (New York: Wiley-IEEE Press, 2005)], 11 book chapters, some 85 journal papers, and numerousconference contributions. His current research interests are multiple-input–multiple-output (MIMO) systems, measurement and modeling of mobile radiochannels, and ultra wideband (UWB).

Dr. Molisch is an Editor of the IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS, Coeditor of a recent special issue on MIMO and smartantennas in the Journal of Wireless Communications and Mobile Computing,and Coeditor of an upcoming special issue on UWB in the IEEE JOURNAL

ON SELECTED AREAS IN COMMUNICATIONS. He has been a Member ofnumerous technical program committees (TPCs), Vice Chair of the TPCof VTC 2005 spring, and will be the General Chair of ICUWB 2006. Hehas participated in the European research initiatives “COST 231,” “COST259,” and “COST273,” where he was the Chairman of the MIMO channelworking group and is the Chairman of Commission C (signals and systems) ofInternational Union of Radio Scientists. He was a recipient of several awards.

Jinyun Zhang (S’87–M’90–SM’04) received thePh.D. degree in electrical engineering from Univer-sity of Ottawa, Ottawa, ON, Canada, in 1991.

She is currently the Group Manager and aSenior Principal Technical Staff of digital communi-cations and networking group at Mitsubishi ElectricResearch Laboratories (MERL) Cambridge, MA.She is leading many new wireless communicationsand networking projects that include ultra wide-band, ZigBee, wireless sensor networks, multipleinput–multiple output, high-speed wireless local area

networks, and next-generation mobile communications. Prior to joining MERL,she worked for Nortel Networks for more than ten years, where she heldengineering and management positions in the areas of digital signal processingvery large scale integration design, advanced wireless technology development,wireless communications, and optical networks.

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