IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 8, AUGUST 2016 5287
Multi-User Massive MIMO CommunicationSystems Based on Irregular Antenna Arrays
Xiaohu Ge, Senior Member, IEEE, Ran Zi, Student Member, IEEE, Haichao Wang,Jing Zhang, Member, IEEE, and Minho Jo, Member, IEEE
Abstract— In practical mobile communication engineeringapplications, surfaces of antenna array deployment regionsare usually uneven. Therefore, massive multi-input–multi-output (MIMO) communication systems usually transmit wirelesssignals by irregular antenna arrays. To evaluate the performanceof irregular antenna arrays, the matrix correlation coefficient andthe ergodic received gain are defined for massive MIMO commu-nication systems with mutual coupling effects. Furthermore, thelower bound of the ergodic achievable rate, symbol error rate,and average outage probability is first derived for multi-usermassive MIMO communication systems using irregular antennaarrays. Asymptotic results are also derived when the numberof antennae approaches infinity. Numerical results indicate thatthere exists a maximum achievable rate when the number ofantennae keeps increasing in massive MIMO communicationsystems using irregular antenna arrays. Moreover, the irregularantenna array outperforms the regular antenna array in theachievable rate of massive MIMO communication systems whenthe number of antennae is larger than or equal to a giventhreshold.
Index Terms— Massive MIMO, irregular antenna array,mutual coupling, achievable rate.
I. INTRODUCTION
TO MEET the challenge of 1000 times wireless trafficincreasing in 2020 as compared to the wireless traffic
level in 2010, the massive multi-input-multi-output (MIMO)technology is presented as one of the key technologies for thefifth generation (5G) wireless communication systems [1]–[3].
Manuscript received October 10, 2015; revised March 22, 2016; acceptedApril 6, 2016. Date of publication April 21, 2016; date of current versionAugust 10, 2016. This work was supported in part by the National NaturalScience Foundation of China under Grant 61210002, Grant 60872007, andGrant 61271224, in part by the Hubei Provincial Science and Technol-ogy Department under Grant 2013BHE005, in part by the FundamentalResearch Funds for the Central Universities under Grant 2015XJGH011and Grant 2014TS100, in part by the European Union FP7-PEOPLE-IRSESthrough the S2EuNet Project under Grant 247083, in part by the WiNDOWProject under Grant 318992, in part by the CROWN Project underGrant 610524, in part by the National International Scientific and Techno-logical Cooperation Base of Green Communications and Networks underGrant 2015B01008, and in part by the Hubei International Scientific and Tech-nological Cooperation Base of Green Broadband Wireless Communications.The associate editor coordinating the review of this paper and approving it forpublication was S. Jin. (Corresponding authors: Jing Zhang and Minho Jo.)
M. Jo is with the Department of Computer and Information Science, KoreaUniversity, Seoul 136-701, South Korea (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TWC.2016.2555911
Existing studies validated that massive MIMO systems canimprove the spectrum efficiency to 10-20 bit/s/Hz leveland save 10-20 times energy in wireless communicationsystems [4]. However, considering a limited available physicalspace for deployment of large number of antenna elementsin base stations (BSs), the mutual coupling effect amongantenna elements is inevitable for massive MIMO wirelesscommunication systems [5], [6]. Moreover, with hundredsof antennas deployed, new issues of the antenna arraydeployment and architecture may appear [7]. For example,conformal antenna arrays on the surface of buildings mayno longer have uniform antenna spacings because of unevensurfaces of the deployment region. In this case, the antennaarray becomes irregular and then the impact of mutualcoupling on the massive MIMO system is different from thecase with a regular antenna array. Therefore, it is an interestingand practically valuable topic to investigate multi-user massiveMIMO communication systems using irregular antenna arrays.
When antennas are closely deployed in an antenna array,the interaction between two or more antennas, i.e., the mutualcoupling effect, is inevitable and affects coefficients of theantenna array [8]. The mutual coupling effect has beenwidely studied in antenna propagation and signal processingtopics [9], [10]. Based on the theoretical analysis and exper-imental measurement, the performance of antenna array wascompared with or without the mutual coupling effect in [9].It was shown that the mutual coupling significantly affects theperformance of adaptive antenna arrays with either large orsmall inter-element spacing because the steering vector of theantenna array has to be modified both in phases and ampli-tudes [10]. With the MIMO technology emerging in wirelesscommunication systems, the impact of mutual coupling onMIMO systems has been studied [11]–[14]. In a reverberationchamber, measurements and simulation results showed thatthe mutual coupling increases the spatial correlation level andundermines the MIMO channel estimation accuracy as wellas the channel capacity [11]. Considering a fixed-length lineararray that consists of half-wave dipoles, simulation resultsrevealed that the mutual coupling leads to a substantiallylower capacity and reduces degrees of freedom in wirelesschannels [12]. Moreover, analytical results in [13] showed thatin a 2 × 2 MIMO system, the mutual coupling is detrimentalto the subscriber unit (SU) correlation and simultaneouslybeneficial to the channel energy only in the presence ofdirectional scattering conditions and for SU arrays orientedorthogonally to the main direction of arrival with spacings
5288 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 8, AUGUST 2016
between 0.4 and 0.9 wavelengths. Based on the scatteringparameter matrix and power constraint, a closed-form capacityexpression of the MIMO system with mutual coupling wasderived [14]. However, all of the above studies are based onconventional MIMO systems, i.e., antennas at the transmitterand receiver are less than or equal to 8×8. For future massiveMIMO scenarios with hundreds of antennas placed at the BS,the mutual coupling effect needs to be further investigated.
To further improve the transmission rate in 5G wirelesscommunication systems, the massive MIMO technology isenvisaged to satisfy 1000 times wireless traffic increase inthe future decade [15]–[17]. Marzetta revealed that all effectsof uncorrelated noise and fast fading will disappear whenthe number of antennas grows without limit in wirelesscommunication systems [15]. Moreover, massive MIMOsystems could improve the spectrum efficiency by one or twoorders of magnitude and the energy efficiency by three ordersof magnitude for wireless communication systems [16], [18].New precoding and estimation schemes have also beeninvestigated for massive MIMO systems [19], [20]. Motivatedby these results, the impact of mutual coupling on massiveMIMO systems was explored in recently literatures [21]–[23].For massive MIMO systems where dipole antennas are placedin a fixed length linear array, analytical results indicated thatsome ignoring effects, such as mutual coupling effect, givemisleading results in wireless communication systems [21].Based on different antenna elements, such as dipole, patchand dualpolarized patch antennas, it was demonstrated thatthe mutual coupling and spatial correlation have practicallimit on the spectrum efficiency of multi-user massive MIMOsystems [22]. Considering the spatial correlation and mutualcoupling effects on massive MIMO systems, the performanceof linear precoders was analyzed for wireless communicationssystems [23]. However, in all aforementioned studies, onlyregular antenna arrays were considered for massive MIMOsystems with the mutual coupling effect. Considering theaesthetic appearance of the commercial buildings, building aplatform with a large number of regular antennas on the facadewill face confrontations from the building owners. To tacklethe challenge of deploying large number of BS antennas,the antennas elements were integrated into the environments,such as a part of the building facade or signage [7], [24].Moreover, the large number of antennas make it very difficultto maintain uniform antenna spacings in these scenarios. As aconsequence, these antenna arrays are appropriate to be con-sidered as irregular antenna arrays with nonuniform antennaspacings rather than regular antenna arrays with uniformantenna spacings. For irregular antenna arrays, some studieshave been made for conformal antenna arrays where antennaarrays are designed to conform the prescribed shape [25]–[27].Sparse antenna arrays where antenna arrays are configuredto decrease the number of antennas but lead to nonuniformantenna spacings and irregular array shapes have also beenstudied [28]–[31]. However, these antenna arrays were mainlystudied in the field of phased arrays and have never been dis-cussed for massive MIMO communication systems. Motivatedby the above gaps, we investigate multi-user massive MIMOwireless communication systems with irregular antenna arrays
considering the mutual coupling. The contributions andnovelties of this paper are summarized as follows.
1) Considering uneven surfaces of antennas deploymentregions, a massive MIMO communication system withan irregular antenna array is firstly proposed and for-mulated. Moreover, the impact of mutual coupling onirregular antenna arrays is evaluated by metrics of thematrix correlation coefficient and ergodic received gain.
2) Based on the results of irregular antenna arrays withmutual coupling, the lower bound of the ergodic achiev-able rate, average symbol error rate (SER) and averageoutage probability are derived for multi-user massiveMIMO communication systems. Furthermore, asymp-totic results are also derived when the number ofantennas approaches infinity.
3) Numerical results indicate that there exists a maximumachievable rate for massive MIMO communication sys-tems using irregular antenna arrays. Moreover, the irreg-ular antenna array outperforms the regular antenna arrayin the achievable rate of massive MIMO communicationsystems when the number of antennas is larger than orequal to a given threshold.
The remainder of this paper is outlined as follows. Section IIdescribes a system model for massive MIMO communicationsystems where BS antennas are deployed by an irregularantenna array. In Section III, the impact of mutual couplingon the irregular antenna array is analyzed by the matrix corre-lation coefficient and ergodic received gain. In section IV, thelower bound of the ergodic achievable rate, average SER andaverage outage probability are derived for multi-user massiveMIMO communication systems using irregular antenna arrays.Considering that the number of antennas approaches infinity,asymptotic results are also obtained. Numerical results anddiscussions are presented in Section V. Finally, conclusionsare drawn in Section VI.
II. SYSTEM MODEL
With the massive MIMO technology emerging in 5G mobilecommunication systems, hundreds of antennas have to bedeployed on the BS tower or the surface of a building.However, surfaces used for deploying massive MIMO antennasare not ideally smooth planes in most of the real scenarios.When massive MIMO antennas have to be deployed on unevensurfaces, spatial distances among adjacent antennas are notexpected to be perfectly uniform. In this case, massive MIMOcommunication systems have to be deployed by irregularantenna arrays. Furthermore, the impact of irregular antennaarrays on massive MIMO communication systems need to bereevaluated when the mutual coupling of irregular antennaarrays is considered. Motivated by above challenges, a single-cell multi-user massive MIMO communication system with anirregular antenna array is illustrated in Fig. 1.
In this system model, a BS is located at the cell center andequipped with M antennas which are deployed on an unevensurface. Because of the uneven surface, spatial distancesamong antennas is no longer regular even when antennas areregularly deployed in a two-dimensional plane. To intuitivelyillustrate the spatial distances among the irregular antenna
GE et al.: MULTI-USER MASSIVE MIMO COMMUNICATION SYSTEMS 5289
Fig. 1. Multi-user massive MIMO communication system with irregularantenna array.
array, we project the antenna distances deployed at the unevensurface into a smooth plane meanwhile keeping the spatialdistances between each pair of antennas the same as before, asshown in Fig. 1. What needs to be mentioned is that the mutualcoupling effect depends on the spatial distance among anten-nas, which remains the same through this projection. Hencethe projecting in Fig. 1 does not affect the mutual couplingeffect of irregular antenna array. In the projected plane ofFig. 1, without loss of generality, all antennas are assumedto be covered by a circle centered at � with the radius R.The i -th and j -th antennas of the massive MIMO antennaarray are denoted as Anti and Ant j , i �= j , 1 � i � M ,1 � j � M . Spatial distances between the circle center �and locations of the antennas Anti and Ant j are denoted asdi and d j , respectively. To simplify derivations, all antennasare sorted by the spatial distances between the circle center �and their locations in the circle, i.e., di < d j if i < j .Considering that the number of antennas M and the circlearea are fixed in a given scenario, the distribution of the Mantennas in the circle is assumed to be governed by a binomialpoint process (BPP) [32], [33]. It’s notable that the circle areais an assumed area on the smooth projection plane to cover allantennas. The circle area does not depend on the actual shapeof the antenna deployment regions. Similarly, other randomprocesses can be used for modeling of the irregular antennadistribution according to the specified requirements. K activeuser terminals (UTs) are assumed to be uniformly scattered ina cell and each UT is equipped with an antenna. In this paper,we focus on the uplink transmission of the massive MIMOcommunication system.
The signal vector received at the BS is expressed as
y = √SN RU T Gx + w, (1)
where y ∈ CM×1 is the M × 1 received signal vector,w ∈ CM×1 is the additive white Gaussian noise (AWGN)with zero mean, i.e., w ∼ CN (0, IM), IM is the M × Munit matrix, x ∈ CK×1 is the K ×1 symbol vector transmittedby K UTs. Moreover, the UT transmitting power is normalized
as 1. SN RU T is the transmitting signal-to-noise ratio (SNR)at the UT and values of SN RU T at all UTs are assumed to beequal in this paper. Similar to the finite dimensional physicalchannel and taking into account the mutual coupling effectbetween antennas, the M × K channel matrix G ∈ CM×K isextended as [23], [34]
G = CAHD1/2, (2)
where C is a mutual coupling matrix, A is an array steeringmatrix, H is a small scale fading matrix, and D is the largescale fading matrix. The mutual coupling matrix C and thearray steering matrix A are affected by the irregularity of theantenna array. The detailed modeling of these two matrixeswill be discussed in the next section. The large scale fadingmatrix is a K × K diagonal matrix and is expressed as
D = diag(β1, . . . , βk, . . . , βK ), (3)
the k-th diagonal element of matrix D, i.e., βk , denotes thelarge scale fading factor in the link of the k-th UT and the BS.
The performance of massive MIMO systems depends crit-ically on the propagation environment, properties of antennaarrays and the number of degrees of freedom offered by thephysical channel. The propagation environment offers richscattering if the number of independent incident directionsis large in the angular domain. More precisely, a finite-dimensional channel model is introduced in this paper, wherethe angular domain is divided into P independent incidentdirections with P being a large but finite number [34]. Eachindependent incident direction, corresponding to the azimuthangle φq , φq ∈ [0, 2π ] , q = 1, . . . , P , and the elevationangle θq , θq ∈ [−π/2, π
/2], is associated with an M × 1
array steering vector a(φq , θq
) ∈ CM×1. In this case, all
independent incident directions are associated with an M × Parray steering matrix A which is given by expression (4)
A = [a (φ1, θ1) , . . . , a
(φq , θq
), . . . ,a (φP , θP)
] ∈ M×P.
(4)
Without loss of generality, the BS is assumed to be sur-rounded by a group of scatters and associated with a largebut finite number of P independent incident directions [34].Therefore, despite locations of UTs, the uplink signals arescattered by the scatters around the BS and arrive at the BSfrom the P incident directions. H ∈ CP×K is the P × K smallscale fading matrix and extended as
H = [h1, . . . ,hk , . . . ,hK ] ∈ CP×K, (5a)
hk = [hk,1, . . . , hk,q , . . . hk,P
]T, (5b)
where hk,q is the small scale fading factor in the link of thek-th UT and the BS at the q-th independent incident direction,which is governed by a complex Gaussian distribution withzero mean and unit variance, i.e., hk,q ∼ CN (0,1). C ∈ CM×M
is an M × M mutual coupling matrix which represents themutual coupling effect on the irregular antenna array. Morespecifically, [C]i, j �= 0, i.e., the element at the i -th row andj -th column of C denotes the mutual coupling coefficientbetween the antennas Anti and Ant j in the irregular antennaarray. Considering the mutual coupling between antennas,
5290 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 8, AUGUST 2016
wireless channels in massive MIMO communication systemsare assumed to be correlated in this paper.
III. IRREGULAR ANTENNA ARRAY
WITH MUTUAL COUPLING
To investigate the impact of the mutual coupling on MIMOcommunication systems, some studies have been carried outfor regular antenna arrays [11]–[14], [21]–[23]. However, theimpact of mutual coupling on massive MIMO communica-tion systems with irregular antenna arrays has been rarelyinvestigated. In irregular antenna arrays, the antenna spacingsare no longer uniform and different from those of regularantenna arrays. Because of the irregular and non-uniformantenna spacings, the mutual coupling and spatial correlationof irregular antenna arrays also become different from thoseof regular antenna arrays. Considering the impact of themutual coupling and spatial correlation of antenna arrays onmassive MIMO systems, the achievable rate of massive MIMOsystems is inevitably affected by the antenna array irregularity.In this section, the channel correlation model is firstly derivedfor irregular antenna arrays considering the mutual coupling.Furthermore, the ergodic received gain is defined for evaluat-ing the joint impact of the number of antennas and array sizeon irregular antenna arrays.
A. Channel Correlation Model
Since each UT is equipped with an antenna and UTs areassumed to be distributed far away from each other, channelsof different UTs are assumed to be uncorrelated. In thissection, the channel correlation is focused on the side of BSirregular antenna arrays. Based on the channel matrix G in (2),the channel correlation matrix is expressed by [37]
� = 1
KD−1
EH
(GGH
)= CAAH CH , (6)
where D−1 is a normalizing result for the large scale fading,EH (·) is an expectation operator taken over the small scalefading matrix H and the superscript H denotes the conjugatetranspose of a matrix. Considering that the distribution ofspatial distances among M antennas follows a binomial pointprocess for irregular antenna arrays, the probability densityfunction (PDF) of di is expressed as expression (7) [32]
fdi (d) = 2� (M + 1)(R2 − d2
)M−id2i−1
R2M� (i) � (M − i + 1), (7)
where � (x) is a Gamma function. For the antenna Ant j ,the PDF of d j is obtained based on (7) as well. Note thatdi and d j are distances from the origin to Anti and Ant j
measured on the projection plane, respectively. The origin, orthe circle center �, the antenna locations of Anti and Ant j
together form a triangle. The triangle’s interior angle at itsvertex � is denoted as ψi j . The distribution of ψi j is assumedto be governed by a uniform distribution in the range of0 and π . Based on the law of cosines, the distance betweenAnti and Ant j is derived as
di j =√
d2i + d2
j − 2did j cosψi j . (8)
When the type of all antennas is assumed to be the dipoleantenna, the mutual impedance between Anti and Ant j isexpressed as (9a) [8] with
Zi j = ε
4π
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
2
(γ + ln(ς)+
∫ ς
0
cos x − 1
xdx
)
−(γ + ln(μ)+
∫ μ
0
cos x − 1
xdx
)
−(γ + ln(ρ)+
∫ ρ
0
cos x − 1
xdx
)
− j
⎛
⎜⎜⎝
2∫ ς
0
sin x
xdx −
∫ μ
0
sin x
xdx
−∫ ρ
0
sin x
xdx
⎞
⎟⎟⎠
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
, (9a)
ς = 2πdi j
λ, (9b)
μ = 2π
λ
(√di j
2 + l2 + l
), (9c)
ρ = 2π
λ
(√di j
2 + l2 − l
), (9d)
where ε is an impedance of free space, γ is anEuler-Mascheroni constant, l is an antenna length.Furthermore, the mutual impedance matrix ZC of theirregular antenna array is formed with its element Zi j ,i �= j , located at the i -th line and the j -th column. The Mdiagonal elements in ZC represent the self-impedances of theM antennas in the irregular antenna array. Considering allconfiguration parameters of antennas in the irregular antennaarray are equal, all diagonal elements in ZC are denotedas Z0. Based on the mutual impedance matrix ZC, the mutualcoupling matrix C is expressed as [13]
C = (Z0 + Z L) (Z LIM + ZC)−1, (10)
where Z L is the load impedance of each antenna. The M × Mmutual coupling matrix C denotes the coupling of the receivedsignals caused by the antenna array. Based on the resultsin [13], the mutual coupling matrix C can be derived fromthe load, self and mutual impedances of the antenna nodesas shown in (10). Considering all configuration parametersof antennas in the irregular antenna array are same, the loadimpedance of each antenna is assumed to be same.
As for a signal from the q-th incident direction, thecorresponding array steering vector of the irregular antennaarray is expressed as a
(φq , θq
), which is the q-th column of
the steering matrix A. Based on the projected plane in Fig. 1,a polar coordinate system with the origin � is assumed. Theantenna with the largest spatial distance dM to the origin isdenoted as AntM . Moreover, AntM is assumed to be locatedat the polar axis of the polar coordinate system and thecorresponding polar coordinate is denoted as (dM , 0). In thepolar coordinate system, we assume the origin � as the phaseresponse reference point with a zero phase. Consideringthe incident signal with azimuth angle φq and elevationangle θq , the phase response of the point with the coordinate
(dx , ψx ) is exp(− 2πdx
λ α)
in which α = cosψx cosφq sin θq +sinψx sin φq sin θq [8]. Therefore, the phase response of AntM
GE et al.: MULTI-USER MASSIVE MIMO COMMUNICATION SYSTEMS 5291
with the coordinate (dM , 0) is given by
aM(φq , θq
) = exp
(− j
2πdM
λsin θq cosφq
). (11)
For Anti with the spatial distance di in the projected planeof Fig. 1, the corresponding position in the polar coordinatesystem is denoted as (di , ψ). Similarly, given the polar coor-dinate of Anti , its phase response with the origin � as thephase response reference is derived by expression (12)
ai(φq , θq
) = exp
[−2π
λ
(di cosψi cosφq sin θq
+ di sinψi sin φq sin θq)]
= exp
[−2πdi sin θq
λ
(cosψi cosφq
+ sinψi sin φq)]
= exp
[− j
2πdi
λsin θq cos
(φq − ψi
)]. (12)
Based on (11) and (12), all M elements of a(φq , θq
)can be
derived. Furthermore, all elements of array steering matrix Aare derived based on (4), (11) and (12).
When derivation results of the mutual coupling matrix C andarray steering matrix A are substituted into (6), the channelcorrelation matrix of irregular antenna array is derived. As forMIMO systems, the strength of the channel correlation greatlyaffects the performance of wireless communications [11], [38].Moreover, the strength of channel correlation depends on thechannel correlation matrix’s off-diagonal elements in con-ventional MIMO systems with regular antenna arrays [37].For regular antenna arrays, the channel correlation matrix isa Toeplitz matrix, whose off-diagonal elements get smallervalues when the elements’ positions get farther from thematrix’s diagonal. In general, the magnitude of one of theoff-diagonal elements which is the closest to the diagonal ofthe Toeplitz matrix is selected to represent the strength ofthe channel correlation for regular antenna arrays. However,spatial distances among antennas are not identical for irregularantenna arrays. In this case, values of the channel correlationmatrix’s off-diagonal elements do not get smaller when theirpositions get farther from the matrix’s diagonal. Therefore,it is impossible to evaluate the strength of channel cor-relation with only one element of the channel correlationmatrix in irregular antenna arrays. To find a representation ofthe channel correlation strength for irregular antenna arrays,the matrix correlation coefficient η is defined as the ratio of thesum of squared off-diagonal elements to the sum of squareddiagonal elements in the channel correlation matrix, which isexpressed by
η = Tr[��H
]
Tr [� ◦�]− 1, (13)
where Tr is the trace operator for matrixes, ◦ is the Hardmardproduct operator for matrixes. Based on the definition in (13),the value of the matrix correlation coefficient η increases withthe increase of the strength of channel correlation in irregularantenna arrays.
B. Ergodic Received Gain
When the deployment area is fixed for massive MIMOantenna arrays, the number of antennas is inversely pro-portional to the antenna spatial distance in regular antennaarrays. The received SNR of MIMO antenna arrays rises withthe increasing of the received diversity when the number ofantennas is increased. However, the antenna spatial distancedecreases with the increase of the number of antennas in afixed deployment area, which leads to the received SNR todecrease with the strengthening of the channel correlation [23].Hence, the number of antennas and antenna spatial distanceare contradictory parameters for the received SNR of massiveMIMO communication systems in a fixed deployment area.When the deployment area of the irregular antenna array isfixed, it is also a critical challenge to evaluate the impact ofthe number of antennas and antenna spatial distance on thereceived SNR of massive MIMO communication systems withirregular antenna arrays. To easily investigate the impact of thevarying number of antennas and antenna spatial distance to thereceived SNR, the ergodic received gain of irregular antennaarrays is defined as
G (M, R) = E
(SN RBS − SN Rmin
BS
), (14)
where SN RminBS refers to the received SNR when the number
of antennas is configured as its available minimum valueMmin and the circle radius in Fig. 1 is configured as itsavailable minimum value Rmin. Mmin and Rmin are configuredas constants in this paper and SN Rmin
BS can be viewed as areference point for the received SNR. SN RBS refers to thereceived SNR with the number of antennas M and the circleradius R. Comparing with SN RBS , the ergodic received gainin (14) can directly reflect the increment of the received SNRwhen the number of antennas M and the circle radius R keepincreasing from their minimum values. Moreover, the ergodicreceived gain also reflects how the achievable rate changeswith the varying number of antennas and antenna spatialdistances since the achievable rate monotonously increaseswith the increase of received SNR in wireless communicationsystems.
To simplify performance analysis of the ergodic receivedgain in irregular antenna arrays, only one UT is assumed inthis case and the BS has perfect channel state information.Moreover, the maximum ratio combining (MRC) detectorscheme is adopted in the BS. In this case, the BS receivedsignal is expressed as
y1,M RC = √SN RU T β1hH
1 AH CH CAh1x
+ β1/21 hH
1 AH CHw1, (15)
where x is the symbol transmitted by the UT, ω1 is the AWGNwith zero mean over wireless channels. SN RU T is the transmitSNR at the UT and equals to the transmit power of the UT.Furthermore, the received SNR at BS is given by
SN R1,BS,M RC =∣∣√SN RU T β1hH
1 AH CH CAh1x∣∣2
∣∣∣β1/2
1 hH1 AH CHw1
∣∣∣2
= SN RU T β1hH1 AH CH CAh1. (16)
5292 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 8, AUGUST 2016
For the single UT scenario, the minimum received SNRwith the BS irregular antenna array, which is denoted asSN Rmin
1,BS,M RC , is expressed by
SN Rmin1,BS,M RC = SN RU T β1hH
1 AH CH CAh1, (17)
where C and A are the mutual coupling matrix and arraysteering matrix with the minimum number of antennas Mminand the minimum circle radius Rmin, respectively. Based onthe definition of ergodic received gain in (14), the ergodicreceived gain with the single UT is derived by
G (M, R) = E
(SN R1,BS,M RC − SN Rmin
1,BS,M RC
)(18a)
= SN RU T β1 [E (ξ1)− E (ξmin)] , (18b)
with
ξ1 = hH1 AH CH CAh1, (18c)
ξmin = hH1 AH CH CAh1. (18d)
The conventional antenna arrays have been investigatedbased on the regular antenna distance [33], [37], [39]–[41].However, it is difficulty to derive an analytical expression forthe ergodic received gain when the random distributed antennaspatial distances are presented at irregular antenna arrays. As amatter of fact, the channel correlation matrix � = CAAH CH
is modeled based on the random distributed antenna spatialdistances. Comparing with the fast varying small scale fadingmatrix H, � can be viewed as a deterministic matrix just likethe large scale fading matrix D. Therefore, to simplify thederivation in (18), the expectation in (18a) is assumed to betaken over the small scale fading vector, i.e., h1. In addition,all eigenvalues τp, 1 � p � M , of channel correlation matrix� = CAAH CH are assumed to be known. Therefore, thefollowing proposition is derived.
Proposition 1: For the single UT scenario, the BS has theperfect channel state information and adopts the MRC detectorscheme, the ergodic received gain at massive MIMO commu-nication systems with irregular antenna arrays is derived byexpression (19a)
G (M, R)= SN RU T β1
×
⎡
⎢⎢⎢⎣
det(BM,1
)
M∏
i< j
(τ j − τi
)×
⎛
⎝τMM −
M−1∑
p=1
M−1∑
q=1
[B−1
M,1
]
q,pτ
q−1M τM
p
⎞
⎠
−det
(BMmin,1
)
Mmin∏
i< j
(τ j − τi
)
⎛
⎝τMminMmin
−Mmin−1∑
p=1
Mmin−1∑
q=1
×[B−1
Mmin,1
]
q,pτ
q−1Mmin
τMminp
⎞
⎠
⎤
⎥⎥⎥⎦, (19a)
with
BM,1 =⎡
⎢⎣
1 τ1 · · ·...
.... . .
1 τM−1 . . .
τM−21...
τM−2M−1
⎤
⎥⎦, (19b)
BMmin,1 =⎡
⎢⎣
1 τ1 · · ·...
.... . .
1 τMmin−1 . . .
τMmin−21...
τMmin−2Mmin−1
⎤
⎥⎦, (19c)
where τp, 1 � p � Mmin is the eigenvalue of channelcorrelation matrix � = CAAH CH .
Proof: Based on the BS configuration and the single UTscenario, the ergodic received gain is expressed in (18a), (18b)and (18c). When all eigenvalues τp, 1 � p � M , of channelcorrelation matrix � = CAAH CH are assumed to be known,the conditional PDF of ξ1 is derived by expression (20) [38]
fξ1 (x |τ1, . . . , τM )
= det(BM,1
)
M∏
i< j
(τ j − τi
)
×⎛
⎝τM−2M e−x/τM−
M−1∑
p=1
M−1∑
q=1
[B−1
M,1
]
q,pτ
q−1M τM−2
p e−x/τp
⎞
⎠,
(20)
Furthermore, the term of Eh1 (ξ1) is derivedby expression (21), where (a) is obtained becauseξ1 = hH
1 AH CH CAh1 � 0.
Eh1 (ξ1)
=+∞∫
−∞x fξ1 (x |τ1, . . . , τM )dx
(a)= det(BM,1
)
M∏
i< j
(τ j − τi
)
⎛
⎝τM−2M
+∞∫
0
xe−x/τM dx
−M−1∑
p=1
M−1∑
q=1
[B−1
M,1
]
q,pτ
q−1M τM−2
p
+∞∫
0
xe−x/τM dx
⎞
⎠
= det(BM,1
)
M∏
i< j
(τ j − τi
)
(τM
M −⎛
⎝M−1∑
p=1
M−1∑
q=1
[B−1
M,1
]
q,pτ
q−1M τM
p
⎞
⎠,
(21)
Similarly, the term of Eh1 (ξmin) is derived by expres-sion (22),
Eh1 (ξmin) =det
(BMmin,1
)
Mmin∏
i< j
(τ j − τi
)
×⎛
⎝τMminMmin
−Mmin−1∑
p=1
Mmin−1∑
q=1
[B−1
Mmin,1
]
q,pτ
q−1Mmin
τMminp
⎞
⎠, (22)
Substitute (21) and (22) into (18a), the expression of (19)is obtained. Meanwhile, it’s worth mentioning that besidesequation (20) from [38], [42, Th. 1] also can be used to obtainthe PDF of ξ1 and derive the closed form result of the ergodicreceived gain.
GE et al.: MULTI-USER MASSIVE MIMO COMMUNICATION SYSTEMS 5293
Fig. 2. Empirical probability distribution of the eigenvalues of the channelcorrelation matrix � .
C. Numerical Analysis
Based on definitions of the matrix correlation coefficientand ergodic received gain, some performance evaluations canbe numerically analyzed in detail. In the following analysis,default parameters in Fig. 1 are configured: antennas areassumed to be dipole antennas [8], the load impedance ofevery antenna is Z L = 50 Ohms, the self-impedance ofevery antenna is 50 Ohms [21], the carry frequency usedfor wireless communications is 2.5 GHz, the correspondingwavelength is λ = 0.12 meter (m), the number of independentincident directions in propagation environments is configuredas P = 100 [34].
In the performance analysis of conventional regular antennaarrays, the channel correlation matrix is a Toeplitz matrixwhere the eigenvalues of Toeplitz matrix converge to somelimited distributions [37]. However, the channel correlationmatrix of irregular antenna array is not a Toeplitz matrix.In this case, the eigenvalues of the channel correlation matrixis analytically intractable. Therefore, we try to obtain theeigenvalues for irregular antenna arrays empirically by numer-ical simulations. When the circle radius R is configured asλ and 3λ, Fig. 2 shows the empirical distribution of the eigen-values of the channel correlation matrix � with the numberof antennas M = 2, M = 10 and M = 20, respectively.Furthermore, we try to select suitable distributions to bestmatch the simulation results in Fig. 2. The sum of normaldistributions appears to agree well with the simulation results.To simplify the matching results, each eigenvalue of thechannel correlation matrix is approximated as the expectationof each normal distribution. For example, for the case withR = λ and M = 2, the eigenvalues are approximatedas 38.4 and 72.3. When the approximated eigenvalues aresubstituted into (19), the ergodic received gain is obtainedfor the irregular antenna array. It’s worth mentioning that theBPP is used to model a specific antenna deployment scenario,the eigenvalues’ distributions in Fig. 2 are applicable for thespecified scenarios with the BPP antenna distribution.
To illustrate how the irregularity of the antenna array affectsthe performance of massive MIMO systems, the irregularitycoefficient ζ is defined here. As introduced in [43], theparameter ζ is used to model the degree of irregularity of
Fig. 3. Matrix correlation coefficient with respective to the ratio of the circleradius R and the wavelength λ, the number of antennas M and the irregularitycoefficient ζ .
the points’ distribution on a plane. In this paper, we use ζto describe the irregularity of the antenna distribution on theprojection plane. According to the definition in [39], ζ isthe magnitude of the gradient of the sum of the weights,
i.e. ζ =∣∣∣∣∇
(M∑
i=1wi
)∣∣∣∣, in which the weight wi is calcu-
lated as wi = e−(di /1.3)2 , where di is the distance fromthe origin to the antenna Anti and the constant 1.3 is anempirical parameter [43]. ζ is a nonnegative value. For aregular antenna array with identical antenna spacing, the valueof ζ is 0. For an irregular antenna array whose antennadistribution follows the BPP distribution, the value of ζ isapproximately 1.5. Moreover, the larger value of ζ corre-sponds to the higher level of irregularity in the antenna array.In Fig. 3, the matrix correlation coefficient with respective tothe ratio of the circle radius and the wavelength, the numberof antennas, and the irregularity coefficient are illustrated.It can be seen in Fig. 3 that the matrix correlation coefficientdecreases with the increase of the ratio of the circle radiusand the wavelength when the number of antennas and theirregular coefficient of the antenna array are fixed. Whenthe ratio of the circle radius and the wavelength is fixed,the larger number of antennas corresponds to the larger valueof the matrix correlation coefficient. Furthermore, when theratio of the circle radius and the wavelength is small, thesmaller value of the irregularity coefficient corresponds tothe larger correlation coefficient of the array. But with theincreasing of the ratio of the circle radius and the wavelength,the curves with different irregularity coefficients get crossed.When the ratio of the circle radius and the wavelength is large,the smaller value of the irregularity coefficient correspondsto the smaller correlation coefficient of the array. This phe-nomenon shows the irregularity of the antenna array helps todecrease the correlation of antenna arrays when the size ofthe array, i.e. the ratio of the circle radius and the wavelength,is less than a given threshold, but increase the correlationwhen the size of the array is larger than or equal to a giventhreshold. When the array size is less than the cross point,the expectation of the average antenna spacing in the irregular
5294 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 8, AUGUST 2016
Fig. 4. Ergodic received gain with respect to the number of antennas andthe circle radius.
array is larger than the average antenna spacing in the regulararray. But when the array size is larger than the cross point,the expectation of the average antenna spacing in the irregulararray is less than that in the regular array. Because the smalleraverage antenna spacing corresponds to the higher antennacorrelation, the curves of regular antenna arrays and irregularantenna arrays cross each other when the array size increasesa given threshold.
The impact of the number of antennas and the circleradius on the ergodic received gain of irregular antenna arrayswith ζ = 1.5 has been investigated in Fig. 4. It should bementioned that in the rest of this paper, if there isn’t anyspecific notifications, the term “irregular antenna” refers to theirregular antenna array with irregularity coefficient ζ = 1.5.Considering the single UT scenario, default parameters areconfigured as follows: the available minimum radius isRmin = λ, the available minimum number of antennas isMmin = 2, the large scale fading factor is β1 = z
/(l1/
lresist)v
,where z is the random variable of the log-normal distributionwith standard variance σshadow = 8 dB, the distance betweenthe UT and the BS is l1 = 100 m, the protect distancebetween the UT and the BS is lresist = 10 m, and the pathloss coefficient is v = 3.8. When the number of antennasis fixed, the ergodic received gain of irregular antenna arrayincreases with the increase of the circle radius. Numericalresults indicate that the ergodic received gain firstly increaseswith the increase of the number of antennas, but then decreaseswhen the number of antennas at irregular antenna arraysexceeds a given threshold. These results imply that thereexists a maximal value for the ergodic received gain ofirregular antenna arrays. The maximal ergodic received gainvalues are 1.2 × 104, 1.86 × 104, 2.8 × 104 and 3.9 × 104,corresponding to the numbers of antennas as 250, 300, 400and 450, respectively.
IV. MULTI-USER MASSIVE MIMOCOMMUNICATION SYSTEMS
Based on the ergodic received gain of irregular antennaarrays, the lower bound of the ergodic achievable rate, theaverage SER and the average outage probability are presented
for multi-user massive MIMO communication systems withirregular antenna arrays in this section. It’s notable that in thissection we firstly consider the massive MIMO system withhundreds of BS antennas and use a finite-dimensional methodto investigate its performance. Then, the case with an infinitenumber of BS antennas is considered and asymptotic resultsare obtained for massive MIMO system performance metrics.
A. Achievable Rate
Assume that the zero-forcing detector is adopted at the BSto cancel the inter-user interference in the cell in Fig. 1. TheBS detecting matrix is denoted as F = G
(GH G
)−1 ∈ CM×K .Therefore, the received signal at the BS is expressed as
y = FH y
= √SN RU T FH Gx + FH w
= √SN RU T x + FH w. (23)
Considering the BS received signals transmitted from Kactive UTs, y = [
y1, · · · , yk, · · · , yK] ∈ CK×1 is a K × 1
vector. The BS received signal transmitted from the k-th UTis expressed as
yk = √SN RU T xk + FH
k w. (24)
Furthermore, the BS received SNR over the link of the k-thUT is expressed by
SN Rk,BS,Z F = SN RU T∥∥FH
k
∥∥2 = SN RU T[(
GH G)−1
]
kk
. (25)
As a consequence, the achievable rate for the k-th UT isderived by expression (26)
Rk,Z F = E[log2
(1 + SN Rk,BS,Z F
)]
= E
⎡
⎢⎣log2
⎧⎪⎨
⎪⎩1 + SN RU T[(
GH G)−1
]
kk
⎫⎪⎬
⎪⎭
⎤
⎥⎦, (26)
Based on the channel matrix in (2), the closed-form solutionof the lower bound of the ergodic achievable rate for the k-thUT is obtained in Proposition 2.
Proposition 2: For a single-cell multi-user massive MIMOcommunication system with BS irregular antenna arrays andthe zero-forcing detector scheme, the closed-form expressionof the lower bound of the uplink ergodic achievable rate forthe k-th UT is given by expression (27a)�
Rk,Z F
= log2
⎧⎨
⎩1 + SN RU T βk
/⎡
⎣ϒK∑
i=1
K∑
j=2
� ( j − 1)
× D (i, j)(τ
n+ j−2n+i −
n∑
p=1
n∑
q=1
[B−1
M,K
]
q,pτ
q−1n+i τ
n+ j−2p
)
+ϒK∑
i=1
D (i, j)(τ n−1
n+i (ln (τn+i )− γ )
−n∑
p=1
n∑
q=1
[B−1
M,K
]
q,pτ
q−1n+i τ
n−1p
(ln
(τp
) − γ)⎞
⎠
⎤
⎦
⎫⎬
⎭, (27a)
GE et al.: MULTI-USER MASSIVE MIMO COMMUNICATION SYSTEMS 5295
with
ϒ = det(BM,K
)
KM∏
q<p
(τp − τq
) K−1∏
p=1p!, (27b)
BM,K =⎡
⎢⎣
1 τ1 · · ·...
.... . .
1 τM−K . . .
τM−K−11...
τM−K−1M−K
⎤
⎥⎦
=⎡
⎢⎣
1 τ1 · · ·...
.... . .
1 τn . . .
τ n−11...
τ n−1n
⎤
⎥⎦, (27c)
where n = M − K , γ is the Euler-Mascheroni constant,D (i, j) is the cofactor of the element [�]i, j in the K × Kmatrix �, and the element [�]i, j is expressed by expres-sion (27d)
[�]i, j = ( j − 1)!⎛
⎝τ n+ j−1n+i −
n∑
p=1
n∑
q=1
[B−1
M,K
]
p,qτ
p−1n+i τ
n+ j−1q
⎞
⎠.
(27d)
Proof: Substituting (2) into (26) and assuming that eigen-values of the channel correlation matrix are known, the achiev-able rate for the k-th UT is derived by expression (28).
Rk,Z F = EH
⎡
⎢⎣log2
⎧⎪⎨
⎪⎩1 + SN RU T[(
GH G)−1
]
kk
⎫⎪⎬
⎪⎭
⎤
⎥⎦
� log2
⎧⎪⎨
⎪⎩1 + SN RU T
EH
[(GH G
)−1]
kk
⎫⎪⎬
⎪⎭
= log2
⎡
⎣1 + SN RU T Kβk
EH
{trace
[(HH AH CH CAH
)−1]}
⎤
⎦
= log2
{
1 + SN RU T Kβk
/
E
(K∑
i=1
ξ−1i
)}
, (28)
where ξi , 1 � i � K , is the i -th ordered eigenvalueof matrix HH AH CH CAH. Let fξ (x |τ1, . . . , τM ) denotethe conditional marginal PDF of the unordered eigenvaluesof matrix HH AH CH CAH, and based on results in [38],fξ (x |τ1, . . . , τM ) is expressed by expression (29)
fξ (x |τ1, . . . , τM )
= ϒ
K∑
i=1
K∑
j=1
x j−1 D (i, j)
×
⎛
⎜⎜⎝
τ n−1n+i e−x/τn+i −
n∑
p=1
n∑
q=1
[B−1
M,K
]
q,pτ
q−1n+i τ
n−1p e−x/τp
⎞
⎟⎟⎠. (29)
Therefore, the term E
(K∑
i=1ξ−1
i
)in (28) is derived by
expression (30)
E
(K∑
i=1
ξ−1i
)
= K
+∞∫
−∞
1
xfξ (x |τ1, . . . , τM ) dx
= ϒK
⎧⎨
⎩
K∑
i=1
K∑
j=2
� ( j − 1) D (i, j)
×⎡
⎣τ n+ j−2n+i −
n∑
p=1
n∑
q=1
[B−1
M,K
]
q,pτ
q−1n+i τ
n+ j−2p
⎤
⎦
+K∑
i=1
D (i, j)[τ n−1
n+i (ln (τn+i )− γ )
−n∑
p=1
n∑
q=1
[B−1
M,K
]
q,pτ
q−1n+i τ
n−1p
(ln
(τp
) − γ)⎤
⎦
⎫⎬
⎭.
(30)
Substituting (30) into (28), (28) is derived as
Rk,Z F
� log2
⎧⎨
⎩1 + SN RU T βk
/⎡
⎣ϒK∑
i=1
K∑
j=2
� ( j − 1) × D (i, j)
×(τ
n+ j−2n+i −
n∑
p=1
n∑
q=1
[B−1
M,K
]
q,pτ
q−1n+i τ
n+ j−2p
)
+ϒK∑
i=1
D (i, j)(τ n−1
n+i (ln (τn+i )− γ )
−n∑
p=1
n∑
q=1
[B−1
M,K
]
q,pτ
q−1n+i τ
n−1p
(ln
(τp
) − γ)⎞
⎠
⎤
⎦
⎫⎬
⎭, (31)
and the lower bound of the ergodic achievable rate of the kthUT is just at the right side of the sign of inequality.
Hence, Proposition 2 gets proved. When all UTs are con-sidered in the cell, the lower bound of the uplink ergodic
achievable sum rate is derived by�
RBS =K∑
k=1
�
Rk,Z F . From
Proposition 2, it can be intuitively found that the transmitSNR at the UT, the large scale fading coefficient and thenumber of UTs directly affect the lower bound of the ergodicachievable rate for the kth UT. With the higher transmit SNR atthe UT or the higher path loss, the lower bound of the ergodicachievable rate is logarithmically increased or decreased,respectively. Moreover, the increasing of the number of UTsdeteriorates the lower bound of the ergodic achievable rate.Because the eigenvalues of the matrix HH AH CH CAH areinvolved within (27) in a complicated form, it’s hard tointuitively estimate the impact from the small scale fad-ing coefficient, the mutual coupling effects and the array
5296 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 8, AUGUST 2016
steering matrix. In the following numerical results will beprovided to illustrate the variation trend of the lower boundof the ergodic achievable rate with the changing of theseparameters.
B. Symbol Error Rate
Considering the zero-forcing detector adopted at the BS, theaverage received SER of multi-user massive MIMO communi-cation systems with irregular antenna arrays is expressed [44]
SE RZ F = 1
K
K∑
k=1
E
[ωk Q
(√2�k SN Rk,BS,Z F
)], (32)
where Q (·) is the Gaussian Q function while ωk and �k aremodulation-specific constants. For the quadrature phase shiftkeying (QPSK) modulation, modulation-specific constants areconfigured as ωk = 2 and �k = 0.5. Assuming that alleigenvalues of channel correlation matrix � = CAAH CH areknown, Proposition 3 is obtained.
Proposition 3: For a single-cell multi-user massive MIMOcommunication system with BS irregular antenna arrays andthe zero-forcing detector scheme, the average received SER ofmulti-user massive MIMO communication systems is given byas expression (33), shown at the bottom of this page.
Proof: Substitute (25) into (32), the average received SERof multi-user massive MIMO communication systems withirregular antenna arrays is derived by expression (34)
SE RZ F = 1
K
K∑
k=1
ωk
2× EH
⎡
⎢⎣er f c
⎛
⎜⎝
√√√√
�k SN RU T[(GH G
)−1]
kk
⎞
⎟⎠
⎤
⎥⎦
= 1
K
K∑
k=1
ωk
2EH
[er f c
(√�k SN RU T βkξ
)], (34)
where er f c (·) is the complementary error function whichcan be expressed by Meijer’s G-function as er f c
(√x) =
1π G2,0
1,2
(x
∣∣∣∣10, 1
/2
)[45, eq. (8.4.14.2)]. Furthermore, the
average received SER of multi-user massive MIMO commu-nication systems with irregular antenna arrays is rewritten byexpression (35)
SE RZ F
= 1
K
K∑
k=1
[ωk
2
∫ ∞
0
(1
π× G2,0
1,2
(
�k SN RU T βkx
∣∣∣∣∣
1
0, 1/
2
)
× fξ (x |τ1, . . . , τM ) dx)]. (35)
Substitute (29) into (35), the average received SER of multi-user massive MIMO communication systems with irregular
antenna arrays in (33) is completed by replacing the integralexpression in [46, eq. (7.831)]. From (33) it can be intuitivelyfound that the average received SER decreases with theincreasing of the number of UTs even after equalization. Butsimilar to (27), the eigenvalues of the matrix HH AH CH CAHare still complexly involved within the expression of theaverage received SER. Therefore, the impact of the smallscale fading, mutual coupling and spatial correlation on theaverage received SER is difficult to be intuitively estimated.In the following it will be shown that the impact of the smallscale fading will be neglected after assuming the numberof antennas growing without bound. Numerical results willillustrate how the mutual coupling and spatial correlationinfluence the average received SER.
C. Outage Probability
The outage probability is one of the most important met-rics for wireless communication systems. Assuming the SNRthreshold is given by SN Rth, the average outage probabilityof multi-user massive MIMO communication systems withirregular antenna arrays is defined as [47]
Pout = 1
K
K∑
k=1
Pr(SN Rk,BS,Z F � SN Rth
). (36)
Based on the scenario illustrated in Fig. 1, Proposition 4 isobtained as following.
Proposition 4: For a single-cell multi-user massive MIMOcommunication system with BS irregular antenna arrays andthe zero-forcing detector scheme, the average outage probabil-ity of multi-user massive MIMO communication systems withis given by expression (37)
Pout = 1
K
K∑
k=1
ϒ
K∑
i=1
K∑
j=1
D (i, j)
×
⎛
⎜⎜⎝
ϑ (τn+i , j, k)
−n∑
p=1
n∑
q=1
[B−1
M,K
]
q,pτ
q−1n+i ϑ
(τp, j
)
⎞
⎟⎟⎠, (37a)
with
ϑ (x, y, k) = (y − 1)!xn+y−1 − exp
(− SN Rth
SN RU T βkx
)
×y−1∑
s=0
(y − 1)!s!
(SN Rth
SN RU T βk
)s
xn+y−s−3,
(37b)
SE RZ F = 1
K
K∑
k=1
ωkϒ
2√π
K∑
i=1
K∑
j=1
D (i, j)
[
τn− j−1n+i G2,1
2,2
(
�k SN RU T βkτn+i
∣∣∣∣∣1 − j, 1
0, 1/
2
)
−n∑
p=1
n∑
q=1
[B−1
M,K
]
q,pτ
q−1n+i τ
n− j−1p G2,1
2,2
(
�k SN RU T βkτn+i
∣∣∣∣∣1 − j, 1
0, 1/
2
)]. (33)
GE et al.: MULTI-USER MASSIVE MIMO COMMUNICATION SYSTEMS 5297
Proof: Substitute (25) into (36), the average outageprobability of multi-user massive MIMO communicationsystems is derived by
Pout = 1
K
K∑
k=1
Pr
⎛
⎜⎝
SN RU T[(GH G
)−1]
kk
� SN Rth
⎞
⎟⎠
= 1
K
K∑
k=1
Pr (SN RU T βkξ � SN Rth)
= 1
K
K∑
k=1
Pr
(ξ � SN Rth
SN RU T βk
). (38)
Assuming that all eigenvalues of channel correlation matrix� = CAAH CH are known, (38) is rewritten by
Pout = 1
K
K∑
k=1
∫ SN RthSN RUT βk
0fξ (x |τ1, . . . , τM ) dx. (39)
Substitute (29) into (38), the average outage probabilityof multi-user massive MIMO communication systemsin (36) is completed by replacing the integral expression in[46, eq. (3.351.8)]. For the estimation of the system outageperformance, the setting of the SNR threshold is very criticalin practical systems. Equation (37) implies that the averageoutage probability monotonously increases with the increasof the SNR threshold. When the number of UTs is increased,the average outage probability also increases. Numericalresults will be provided in the next section to show how thenumber of BS antennas and antenna array size affect theaverage outage probability.
D. Asymptotic Analysis
To further investigate asymptotic results on multi-user mas-sive MIMO communication systems with irregular antennaarrays, we then assume the scenario where the number ofantennas M grows without bound, i.e. M → ∞. In such ascenario, the following expressions will hold [48]
pH Rp − 1
Mtrace (R) −−−−→
M→∞ 0, (40a)
pH Rq −−−−→M→∞ 0, (40b)
where matrix R ∈ CM×M has uniformly bounded spectralnorm with respect to M . And p,q ∈ C
M×1 are two indepen-dent vectors with distributions p,q ∼ CN (0, 1
M IM ). q and pare also independent from R. With the above expressions, wehave the following proposition for the received SNR at the BSwhen the zero-forcing detector is adopted.
Proposition 5: For a single cell multi-user massive MIMOcommunication system with BS irregular antenna arrays andthe zero-forcing detector, when the number of antennas at theBS approaches infinity, i.e., M → ∞, the received SNR overthe link of the kth UT is expressed as
SN Rk,BS,Z F −−−−→M→∞ SN RU T βk trace
(AH CH CA
). (41)
Proof: The BS received SNR over the link of the kth UTis expressed by (25). Multiply the numerator and denominatorby M , we get
SN Rk,BS,Z F = M × SN RU T[( 1M GH G
)−1]
kk
. (42)
For the channel matrix multiplication term 1M GH G, we have
expression (43)
1
MGH G = 1
MDH HH AH CH CAHD
= 1
M
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎣
β1/21 hH
1
...
β1/21 hH
k
...
β1/21 hH
K
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎦
AH CH CA
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎣
β1/21 h1
...
β1/21 hk
...
β1/21 hK
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎦
T
. (43)
It can be seen that the entry at the i th row and j th column
of 1M GH G is expressed by
√βiβ j
M hHi AH CH CAh j . If i �= j ,
applying (40), we have√βiβ j
M hHi AH CH CAh j −−−−→
M→∞ 0.
If i = j , we have βiM hH
i AH CH CAhi −−−−→M→∞
βiM trace
(AH CH CA
). In other words, when the number
of antennas at the BS approaches infinity, 1M GH G
converges to a diagonal matrix whose i th diagonal entry isβiM trace
(AH CH CA
). Therefore, the received SNR over the
link of the kth UT is derived by expression (44)
SN Rk,BS,Z F = M × SN RU T[( 1M GH G
)−1]
kk
−−−−→M→∞
M × SN RU TM
βi trace(AH CH CA)
= SN RU T βk trace(
AH CH CA). (44)
So Proposition 5 is proved.Proposition 5 illustrates that when the number of antennas
at the BS approaches infinity, the received SNR only dependson the UT transmit power, large scale fading and channelcorrelation, despite of the small scale fading and noise. Thisresult is very concise and coincides with the precious studieson massive MIMO [15], [16].
With the derived asymptotic results of the received SNR,the asymptotic result for the achievable sum rate, whichis previously investigated in Proposition 2, is given byexpression (45)
RBS =K∑
k=1
Rk,Z F −−−−→M→∞
K∑
k=1
log2
×[1 + SN RU T βk trace
(AH CH CA
)]. (45)
For the average SER investigated in Proposition 3, i.e.,substituting (41) into (32), its asymptotic result when thenumber of BS antennas grows without a bound is given byexpression (46), as shown at the bottom of the next page.
It is worth nothing that if the MRC detector is employedat the BS, the received SNR over the link of the kth UT can
5298 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 8, AUGUST 2016
be expressed as expression (47), as shown at the bottom ofthis page.
When the number of antennas grows without a bound,similar to the proof of Proposition 5, the numerator of (47)converges to SN RU T
∣∣βk trace
(AH CH CA
)∣∣2 and the denom-inator of (47) converges to βk trace
(AH CH CA
). Therefore,
the received SNR over the link of the kth UT goes toSN RU T βk trace
(AH CH CA
)when the number of antennas
approaches infinity. It can be found that the MRC detector andZF detector have identical asymptotical results when the num-ber of antennas approaches infinity. With the asymptotic resultof the received SNR when the MRC detector is employed,the asymptotic result for the ergodic received gain, which isinvestigated in Proposition 1, is obtained as
(M, R) = SN RU T βk trace(
AH CH CA)
− SN RU T βk trace(
AH CH CA). (48)
V. SIMULATION RESULTS AND DISCUSSIONS
Based on the proposed models of massive MIMO com-munication systems, the effect of mutual coupling on themassive MIMO communication systems with irregular andregular antenna arrays is analyzed by numerical simulations.In the following, some default parameters and assumptions arespecified. The type of all antennas at the BS is assumed to bethe same, and the load impedance and self-impedance of eachantenna are assumed to be 50 Ohms [23]. The number of UTsin a cell is K = 10 and the large scale fading factor βk ismodeled as βk = z
/(lk/
lresist)v , which is similar to β1 except
with lk being a uniformly distributed random variable rangingfrom 10 m to 150 m [34], [35]. The transmitting SNR at eachUT is assumed to be 15 dB [21]. Since the BS is assumed tobe associated with a large but finite number of independentincident directions, the number of incident directions P areassumed to be 100 [23], [36].
Fig. 5 illustrates the uplink ergodic achievable sum ratewith respect to the number of antennas M , the circle radius Rand the irregularity coefficient ζ of antenna arrays. The linescorrespond to the lower bound of the uplink ergodic achievable
sum rate�
RBS . The square points correspond to the asymptoticresults of the achievable sum rate obtained in (45). When thecircle radius is fixed, numerical results demonstrate that thereexists a maximum of the uplink ergodic achievable sum ratewith the increasing number of antennas. The uplink ergodic
Fig. 5. The achievable sum rate with respect to the number of antennas, thecircle radius and the irregularity coefficient.
achievable sum rate increases with the increase of the numberof antennas before achieving the maximum. After the numberof antennas exceeds a given threshold, the uplink ergodicachievable sum rate becomes to decrease. When the numberof antennas is fixed, the uplink ergodic achievable sum rateincreases with the increase of the circle radius. Furthermore, itcan be seen that the smaller value of the irregularity coefficientcorresponds to the larger value of the achievable sum rate whenthe number of antennas is less than a given threshold. Butwhen the number of antennas increases, curves with differentirregularity coefficients get crossed. When the number ofantennas is larger than a given threshold, the smaller valueof the irregularity coefficient corresponds to the smaller valueof the achievable sum rate. These results indicate that theirregular antenna array has contributed to improve the uplinkergodic achievable sum rate when the number of antennasis larger than or equal to a given threshold. In addition, theasymptotic results well match the lower bound of the uplinkergodic achievable sum rate in Fig. 5, especially when thenumber of antennas is large.
Impact of the UT SNR and the number of BS antennas onthe lower bound of the uplink ergodic achievable sum rateof multi-user massive MIMO communication systems withand without mutual coupling is investigated in Fig. 6. Whenthe number of antennas at the BS is fixed, the lower boundof the uplink ergodic achievable sum rate increases with the
SE RZ F −−−−→M→∞ = 1
K
K∑
k=1
ωk
2er f c
(√�k SN RU T βk trace
(AH CH CA
))
= 1
2πK
K∑
k=1
ωk G2,01,2
(�k SN RU T βk trace
(AH CH CA
) ∣∣∣∣
10, 1
/2
). (46)
SN Rk,BS,M RC = SN RU T∣∣βkhH
k AH CH CAhk∣∣2
SN RU T
K∑
i=1,i �=k
∣∣∣β1/2
k hHk AH CH CAhiβ
1/2i
∣∣∣2 + βkhH
k AH CH CAhk
. (47)
GE et al.: MULTI-USER MASSIVE MIMO COMMUNICATION SYSTEMS 5299
Fig. 6. Impact of the UT SNR and the number of BS antennas on theachievable sum rate of multi-user massive MIMO communication systemswith and without mutual coupling.
Fig. 7. The average SER with respect to the UT SNR, the number ofBS antennas and the circle radius of the massive MIMO communicationsystem with irregular antenna arrays. Both cases with limited and infinitenumber of antennas are illustrated.
increase of the UT SNR. When the UT SNR is fixed, the lowerbound of the uplink ergodic achievable sum rate increases withthe increase of the number of BS antennas. Moreover, theresults of 10000 times Monte-Carlo simulation on the uplinkergodic achievable sum rate are illustrated. Both numerical andMonte-Carlo simulation results demonstrate that the uplinkergodic achievable sum rate with mutual coupling is less thanthe uplink ergodic achievable sum rate without mutual cou-pling for multi-user massive MIMO communication systemswith irregular antenna arrays.
Without loss of generality, the QPSK modulation scheme isadopted for numerical simulations in Fig. 7. The modulationconstants are configured as ωk = 2 and �k = 0.5. Impact ofthe UT SNR, the number of BS antennas M and the circleradius R on the average SER of multi-user massive MIMOcommunication systems with irregular antenna arrays is eval-uated in Fig. 7. The solid and dashed lines corresponds to theSER with limited number of antennas under different circleradii. And the square points correspond to the asymptotic
Fig. 8. The average outage probability with respect to the UT SNR,the number of BS antennas and the circle radius of the massive MIMOcommunication system with irregular antenna arrays.
results obtained in (46). When the number of BS antennas andthe circle radius are fixed, the average SER decreases with theincrease of the UT SNR. When the UT SNR and the circleradius are fixed, the average SER decreases with the increaseof the number of BS antennas. When the UT SNR and thenumber of BS antennas are fixed, the average SER decreaseswith the increase of the circle radius.
The average outage probability with respect to the UT SNR,the number of BS antennas and the circle radius is analyzedin Fig. 8. Without loss of generality, the SNR threshold isconfigured as SN Rth = −3 dB. When the number of BSantennas and the circle radius are fixed, the average outageprobability of massive MIMO communication systems withirregular antenna arrays decreases with the increase of theUT SNR. When the UT SNR and the circle radius are fixed, theaverage outage probability of massive MIMO communicationsystems with irregular antenna arrays decreases with theincrease of the number of BS antennas. When the UT SNRand the number of BS antennas are fixed, the average outageprobability of massive MIMO communication systems withirregular antenna arrays decreases with the increase of thecircle radius.
VI. CONCLUSION
In this paper, multi-user massive MIMO communicationsystems with irregular antenna arrays and mutual couplingeffects have been investigated. In real antenna deploymentscenarios, antenna spatial distances of massive MIMO antennaarrays are usually irregular. Considering engineering require-ments from real scenarios, the effect of the mutual coupling onthe irregular antenna array is firstly analyzed by the channelcorrelation model and ergodic received gain. Furthermore,the lower bound of the ergodic achievable rate, the averageSER and the average outage probability of multi-user massiveMIMO communication systems with irregular antenna arraysare proposed. Numerical results indicate that there exists amaximum for the achievable rate considering different num-bers of antennas for massive MIMO communication systems.Compared with the regular antenna array, the irregular antenna
5300 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 8, AUGUST 2016
array has contributed to improve the achievable rate whenthe number of antennas is larger than or equal to a specificthreshold. Our results provide some guidelines for the massiveMIMO antenna deployment in real scenarios. For the futurestudy, we will try to investigate multi-cell multi-user massiveMIMO communication systems with irregular antenna arrays.
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GE et al.: MULTI-USER MASSIVE MIMO COMMUNICATION SYSTEMS 5301
Xiaohu Ge (M’09–SM’11) received thePh.D. degree in communication and informationengineering from the Huazhong University ofScience and Technology (HUST), China, in 2003.He was a Researcher with Ajou University,Korea, and the Politecnico di Torino, Italy,from 2004 to 2005. He has been with HUSTsince 2005, where he is currently a Full Professorwith the School of Electronic Information andCommunications. He is an Adjunct Professorwith the Faculty of Engineering and Information
Technology, University of Technology Sydney, Australia. He has authoredover 100 papers in refereed journals and conference proceedings andholds about 15 patents in China. His research interests are in the areaof mobile communications, traffic modeling in wireless networks, greencommunications, and interference modeling in wireless communications.He received the Best Paper Awards from the IEEE Globecom 2010.
Dr. Ge is a Senior Member of the China Institute of Communicationsand a member of the National Natural Science Foundation of China andthe Chinese Ministry of Science and Technology Peer Review College.He has been actively involved in organizing over ten international conferencessince 2005. He served as the General Chair of the 2015 IEEE InternationalConference on Green Computing and Communications. He serves as anAssociate Editor of the IEEE ACCESS, the Wireless Communicationsand Mobile Computing Journal (Wiley), and the International Journal ofCommunication Systems (Wiley). Moreover, he served as the Guest Editorof the IEEE Communications Magazine Special Issue on 5G WirelessCommunication Systems.
Ran Zi (S’14) received the B.E. degree in communi-cation engineering and the M.S. degree in electronicsand communication engineering from the HuazhongUniversity of Science and Technology, Wuhan,China, in 2011 and 2013, respectively, where he iscurrently pursuing the Ph.D. degree. His researchinterests include MIMO systems, millimeter-wavecommunications, and multiple access technologies.
Haichao Wang received the bachelor’s degree inelectronic science and technology from the WuhanUniversity of Technology, Wuhan, China, in 2013.He is currently pursuing the master’s degree withthe Huazhong University of Science and Technology,Wuhan. His research interests include the mutualcoupling effect in antenna arrays and optimizationof the number of RF chains in antenna arrays.
Jing Zhang (M’13) received the M.S. and Ph.D.degrees in electronics and information engineeringfrom the Huazhong University of Science and Tech-nology (HUST), Wuhan, China, in 2002 and 2010,respectively. He is currently an Associate Professorwith HUST. From 2014 to 2015, he was a Visit-ing Researcher with Friedrich-Alexander-UniversityErlangen-Nuremberg, Germany. He has authoredabout 20 papers in refereed journals and conferenceproceedings. He has done research in the areas ofmultiple-input multiple-output, CoMP, beamform-
ing, and next-generation mobile communications. His current research inter-ests include cellular systems, green communications, channel estimation, andsystem performance analysis.
Minho Jo (M’07) received the B.A. degree from theDepartment of Industrial Engineering, Chosun Uni-versity, South Korea, in 1984, and the Ph.D. degreefrom the Department of Industrial and Systems Engi-neering, Lehigh University, USA, in 1994. He is cur-rently a Professor with the Department of Computerand Information Science, Korea University, SejongCity, South Korea. He is one of the founders ofthe Samsung Electronics LCD Division. Areas ofhis current interests include LTE-unlicensed, Inter-net of Things, cognitive networks, HetNets in 5G,
green (energy efficient) wireless communications, mobile cloud computing,5G wireless communications, optimization and probability in networks, net-work security, software defined networks, and massive MIMO. He receivedthe Headong Outstanding Scholar Prize with price money of U.S. $20 000in 2011. He is the Founder and Editor-in-Chief of the KSII Transactionson Internet and Information Systems [SCI (ISI) and SCOPUS indexed,respectively]. He is an Editor of the IEEE Wireless Communications, and anAssociate Editor of the IEEE INTERNET OF THINGS JOURNAL, Security andCommunication Networks, Ad-Hoc & Sensor Wireless Networks, and WirelessCommunications and Mobile Computing. He is the Vice President of theInstitute of Electronics and Information Engineers and was the Vice Presidentof the Korea Information Processing Society.
