ieee transactions on communications, vol. 62, no. 5,...

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 5, MAY 2014 1541

Retrieving Channel Reciprocity for CoordinatedMulti-Point Transmission with Joint Processing

Liyan Su, Chenyang Yang, Gang Wang, and Ming Lei

Abstract—In time division duplex coordinated multi-pointtransmission with joint processing (CoMP-JP) systems, theuplink-downlink channels are no longer reciprocal. Due to thedifficulty of antenna calibration among the coordinated basestations (BSs), practical downlink channels differ from uplinkchannels by multiplicative ambiguity factors, which lead to severeperformance degradation. In this paper, we propose an inter-BS antenna calibration strategy to facilitate downlink CoMP-JP transmission, whose basic idea is to estimate the ambiguityfactors. To establish the observation equations for estimation, anextra uplink training frame except for the regular uplink anddownlink frames is introduced, and existing signalling frameworkof limited feedback can also be reused. To improve the estimationperformance, we can either select multiple users or employmultiple frames of one user to assist the calibration. Afterestablishing the observation equations respectively with uplinktraining or limited feedback, the weighted least square criterionis used for estimation. We proceed to analyze and compare themean square errors of the estimators, and provide a principleto select the users for assisting calibration. Simulation resultsshow that the channel reciprocity is largely retrieved by theproposed antenna calibration strategy, which provides substantialthroughput gain over the CoMP-JP systems without inter-BScalibration.

Index Terms—Coordinated multi-point transmission, channelreciprocity, antenna calibration.

I. INTRODUCTION

COORDINATED multi-point (CoMP) transmission is aspectral-efficient technique for cellular networks, which

has attracted extensive research efforts recently [1–5]. Whendata and channel information can be shared among multiplebase stations (BSs), CoMP joint processing (CoMP-JP) is ableto fully exploit the potential provided by the network resource[6, 7].

To assist downlink (DL) CoMP-JP multi-user precoding,the global channels of all mobile stations (MSs) should beavailable at the coordinated BSs. The channel informationcan be obtained at the BSs via uplink (UL) training in timedivision duplex (TDD) systems, or by feedback in frequencydivision duplex (FDD) systems. It is widely recognized thatTDD is more applicable for CoMP systems, because large

Manuscript received May 18, 2013; revised October 20, 2013, January 14and March 3, 2014. The editor coordinating the review of this paper andapproving it for publication was B. Clerckx.

L. Su and C. Yang are with the School of Electronics and Information Engi-neering, Beihang University, Beijing China (e-mail: [email protected],[email protected]).

G. Wang and M. Lei are with NEC Laboratories, China (e-mail:{wang gang, lei ming}@nec.cn).

Part of this work was published in IEEE WCNC, 2013. This work wassupported in part by the National 863 Program 2014AA01A703, and by aresearch gift from NEC Laboratories, China.

Digital Object Identifier 10.1109/TCOMM.2014.031014.130367

feedback overhead is required in FDD systems to provide thechannel information at the BSs [8]. Yet this is established onthe assumption of the reciprocity of UL and DL channels.

The global channel of each MS in CoMP systems is aconcatenation of multiple single-cell channels. In TDD sys-tems, the global channel is constructed from multiple per-cellchannels respectively estimated at the BSs by exploiting thechannel reciprocity. However, the UL and DL channels arenot reciprocal in practice due to the imperfect calibration onthe analog gains of radio frequency (RF) chains in multipletransmit and receive antennas [9,10]. In CoMP-JP systems, thenon-ideal channel reciprocity will lead to severe performancedegradation [11,12], because it causes a kind of multiplicativenoise that will hinder the co-phasing of coherent cooperativetransmission. For conciseness, we refer to CoMP-JP as CoMPin the following.

To retrieve the channel reciprocity, antenna calibration isoften employed to compensate the mismatch of the RF chains.Self-calibration is a popular antenna calibration method ap-plied in single-cell systems [9], which adjusts all antennas toachieve the same RF analog gain as that of a reference antenna,and hence yields a scalar ambiguity factor between the UL andDL channel vectors. However, self-calibration among BSs ishard to implement in CoMP systems (because coordinated BSsare not co-located). This leads to multiple ambiguity factorsbetween the UL and DL channels at different coordinatedBSs. Over-the-air calibration is another method applied insingle-cell systems [10, 13], where some users are selectedby the BS to assist the calibration, referred to as calibrationsupporters or simply supporters. When these users are servedas supporters, their UL and DL channels to and from a BS areestimated or quantized to assist antenna calibration, rather thanfor conveying data as a regular user. To reduce the channelestimation errors for providing accurate calibration, the BSshould select a supporter close to the BS. However, in CoMPsystems a user close to one BS will be far from another BS.It is not clear how to calibrate the antennas at different BSswith an acceptable performance.

In this paper, we investigate inter-BS antenna calibrationstrategy for DL CoMP systems, where the ambiguity factorsare estimated from the UL and DL channels under weightedleast squares (WLS) criterion. To provide observation equa-tions for parameter estimation, we consider two approachesto obtain the UL and DL channels, either via UL training orthrough limited feedback. We first propose an inter-BS antennacalibration method by exploiting a regular UL training frame,a regular DL training frame and an extra UL training signalweighted by a ratio. We then present a calibration method

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1542 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 5, MAY 2014

by using the training and feedback frames available in thespecification of Long-term Evolution (LTE) [14, 15], whichare originally designed for providing the channel directioninformation (CDI). To reduce the impact of the channelestimation errors, we use the signals received from multiplesupporters or from multiple frames of a single supporter. Theperformance of the proposed methods is analyzed and theprinciple of selecting supporters is provided.

The rest of this paper is organized as follows. We firstpresent the system mode and formulate the problem in SectionII. In Section III, we propose the strategy to estimate the inter-BS ambiguity factor and address several possible variations.In Section IV, we analyze the performance of the proposedestimators and show the principle to select the supporters.Numerical and simulation results are provided in Section Vand the paper is concluded in Section VI.

II. SYSTEM MODEL AND PROBLEM FORMULATION

Consider a DL cooperative cluster consisting of Nc

BSs each equipped with Nt antennas. These BSs cooper-atively serve multiple single antenna MSs. Denote gD

mb =√αmbh

Dmb ∈ C

1×Nt as the DL channel vector from BSb toMSm, where αmb is the large scale fading gain including pathloss and shadowing, hD

mb is the DL small scale fading channelvector, whose entries are assumed independent and identicallydistributed (i.i.d.). We assume that the BSs are connected witha central unit (CU) via high capacity backhaul links, and theCU and MSs know perfect large scale fading gains. With theper-cell channel information, hD

mb and αmb, b = 1, · · · , Nc,the CU constructs the global channel vector of MSm asgDm = [gD

m1 · · ·gDmNc

]. After gathering the global channelsof all MSs, the CU can compute the multi-cell precoding forDL transmission.

To explain why in TDD systems, the assumption of channelreciprocity does not hold due to the imperfect antenna cali-bration, we take BSb and MSm as an example. As shown inFig. 1, let SB

bj and Y Bbj denote the transmit and receive analog

gains of the jth antenna at BSb, and SMm and YM

m denotethe transmit and receive analog gains of the antenna at MSm.Then the UL channel gUmbj and DL channel gDmbj between thejth antenna of BSb and the antenna of MSm satisfy

gDmbj =SBbjY

Mm

Y Bbj S

Mm

· gUmbj � βmbjgUmbj , (1)

where βmbj is a complex ambiguity factor.As a result, the relationship between the UL and DL per-cell

channel vectors can be written as

gDmb = gU

mb ·Bmb, (2)

where gUmb ∈ C1×Nt is the UL channel vector from MSm to

BSb, Bmb � diag{βmb1 · · ·βmbNt}, and diag{·} stands for adiagonal matrix.

Since without antenna calibration βmbi �= βmbj for i �= j,the UL and DL channels are no longer reciprocal.

To retrieve the reciprocity of the per-cell channels, selfcalibration [13] can be applied. After perfect self-calibration,all antennas at each BS achieve the same RF analog gain as

that of a reference antenna. Then, the relationship between theUL and DL channel vectors between BSb and MSm becomes

gDmb = βmbg

Umb. (3)

Such a scalar ambiguity factor βmb does not affect thebeamforming direction in single-cell single-user systems andhence has no impact on its performance [16].

However, self calibration designed for single-cell systemsis not applicable to CoMP systems, where the antennas tobe calibrated are not all co-located. On the other hand, evenafter the perfect self-calibration between each BS and eachMS in CoMP systems, the ambiguity factors at different BSsβmb, b = 1, · · · , Nc are not identical. Consequently, the globalUL and DL channels are still not reciprocal, which leadsto severe performance degradation. This suggests that inter-BS antenna calibration is necessary for CoMP systems. Toemphasize the inter-BS antenna calibration and simplify thenotation, we assume perfect self-calibration at each BS in thefollowing design and analysis.

Without loss of generality and for notation simplicity, weonly calibrate two BSs, say, BS1 and BS2. When there aremore BSs in the CoMP system, the CU can select one ofthem as a reference BS and other BSs can calibrate with thereference BS.

After perfect self-calibration, to calibrate BS1 and BS2 weonly need to estimate the following inter-BS ambiguity factor,

μ21 � βm2

βm1=SB2 Y

B1

Y B2 SB

1

, (4)

which only depends on the analog gains at the referenceantennas of the two BSs and therefore the subscript i and jfor antennas are omitted. Further considering (3) and assumingthat the large scale fading gains of UL and DL channels areequal, the factor can be rewritten as,

μ21 =gUm1j

gUm2i

gDm2i

gDm1j

=hUm1j

hUm2i

hDm2i

hDm1j

, i, j = 1, · · · , Nt, (5)

where hUmbj = gUmbj/√αmb is the UL small scale fading

channel from MSm to the jth antenna of BSb.

III. INTER-BS ANTENNA CALIBRATION STRATEGY

Previous analysis shows that we can retrieve the globalchannel reciprocity in CoMP systems by estimating the ambi-guity factor between the reference antennas at any two BSs.To do this, it is shown from (5) that the CU needs to obtainboth the ratios of UL and DL channels between a supporterand the two BSs, which can be obtained in different ways.

In this section, we present two types of methods to estimatethe inter-BS ambiguity factor either via training or via feed-back. We first establish the respective observation equations,and derive the statistics of the observation errors. Then, WLScriterion is used to estimate μ21.

To reduce the impact of the imperfect per-cell channels,we can use the signals received from multiple supporters inone frame or from multiple frames of a single supporter.For conciseness, we present the calibration methods by usingmultiple supporters, and then briefly discuss the methods byusing multiple frames.

SU et al.: RETRIEVING CHANNEL RECIPROCITY FOR COORDINATED MULTI-POINT TRANSMISSION WITH JOINT PROCESSING 1543

m

m

mm

m

M

M

Fig. 1. Illustration of imperfect uplink-downlink channel reciprocity withoutantenna calibration. Each antenna is equipped with a high power amplifier inthe transmitter circuit and a low noise amplifier in the receiver circuit. The ULand DL channels are gUmbj = SM

m cmbj,UY Bb,j and gDmbj = SB

b,jcmbj,DY Mm ,

where cmbj,U and cmbj,D are the UL and DL propagation channels betweenthe antenna pair, which are reciprocal according to electromagnetic theory,i.e. cmbj,U = cmbj,D [9].

A. Inter-BS Antenna Calibration via Training

Suppose that the CU can select several MSs as the support-ers to assist calibration, where MSm located at 1st cell (i.e.,BS1 is the master BS of MSm) is one of them.

To obtain the UL and DL channels between the two BSsand the supporters at the CU, we need two UL and one DLtraining frames.

1) Uplink and Downlink Channel Estimation: In the firstUL frame, the supporters send regular training symbols toassist the BSs for channel estimation, each symbol is

√pULx

with |x|2 = 1 and pUL is the UL transmit power. To avoidinterference among the training signals from multiple MSs,these training symbols are mutually orthogonal. Recall thatafter perfect self-calibration between each BS and each MS,we only need to calibrate the reference antennas of the BSs,say the first antenna. When linear minimum mean squareerror (LMMSE) criterion is used for channel estimation, theUL channel from MSm to the first antenna of BSb can beexpressed by its estimate as

gUmb1 = gUmb1 + eUmb1, b = 1, 2, (6)

where gUmb1 is the UL channel estimated at BSb, eUmb1 ∼CN (0, εUmb) is the estimation error uncorrelated with theestimated channel,1 εUmb = αmbσU/(αmbpUL + σU ), and σUis the variance of receiver noise at each BS.

In the DL frame, the BSs broadcast training symbols withtransmit power pDL. With the LMMSE channel estimator, theDL channel from the first antenna of BSb to MSm can beexpressed as

gDmb1 = gDmb1 + eDmb1, b = 1, 2, (7)

where gDmb1 is the DL channel estimated at MSm, eDmb1 ∼CN (0, εDmb) is the estimation error uncorrelated with theestimated channel, εDmb = αmbσD/(αmbpDL + σD), and σDis the variance of noise at each MS.

1This is valid under the assumption that the user is perfectly synchronized intime and frequency with each BS. Such an assumption is reasonable becausein practice the time and frequency mismatch can be significantly reduced byjudiciously designed synchronization methods, e.g., [17, 18].

2) Observation Equations for Estimating μ21: With theestimated DL channels, MSm calculates the ratio of the smallscale fading channel of its cross link (i.e., the link from BS2

to MSm) to that of its local link (i.e., the link from its masterBS, BS1, to MSm) as follows,√

αm1

αm2

gDm21

gDm11

=

√αm1

αm2

gDm21 + eDm21

gDm11 + eDm11

=

√αm1

αm2

βm2gUm21 + eDm21

βm1gUm11 + eDm11

=

√αm1

αm2

μ21gUm21 + eDm21/βm1

gUm11 + eDm11/βm1

� Am exp(iθm), (8)

where Am and θm are the norm and phase of the ratio,respectively, (3) and (4) are used in the derivation.

In the next UL frame, each supporter, say MSm, sends aweighted training symbol Am exp(iθm) · √pULx instead ofthe regular training symbol, which is referred to as calibrationsounding reference signal (SRS). This does not affect theorthogonality of the training signals of multiple supporters.Then, the received calibration SRS at the reference antenna(i.e., the first antenna) of BS1 is

zm11 =√pULAm exp(iθm)gUm11x+ nU

1 , (9)

where nU1 is the noise at the BS1.

Substituting (8) into (9) and removing x by multiplying itsconjugate, we have

zm11x∗

√pUL

=

√αm1

αm2


gUm11 + eDm11/βm1gUm11 +

uU11√pUL

=

√αm1

αm2


gUm11 + eDm11/βm1

(gUm11 + eDm11/βm1 − eDm11/βm1) +uU11√pUL

=

√αm1

αm2μ21g

Um21 + vm, (10)

where uU11 = nU11x

∗, and vm =√

αm1

αm2

eDm21

βm1−√

αm1

αm2

gDm21

gDm11

eDm11

βm1+

uU11√pUL

.After the CU collects the UL channel estimates obtained

in the first UL frame and the received calibration SRS inthe second UL frame from the BS1, it can establish theobservation equation for estimating the inter-BS ambiguityfactor. Considering (6) and (10), an observation equation forestimating μ21 can be obtained as follows

zm11x∗

√pUL

=

√αm1

αm2μ21g

Um21 + wm, (11)

where wm =√

αm1

αm2μ21e

Um21 + vm is the observation noise.

The norms of the ambiguity factors βmb can be modeled asthe log-uniformly distributed random variables [19], therefore,E{|μ21|2} = E{ 1

|βmb|2 }2 � μ2 > 1. Then, the variance of theobservation noise can be derived as

εT,m � E{|wm|2} =αm1

αm2μ2εUm2 +

αm1

αm2μεDm2

+1− εDm2/αm2

1− εDm1/αm1μεDm1 +

σUpUL

. (12)


After the CU received Ms calibration SRSs, it can obtainMs observation equations, which is written in a vector formas follows

zT = uT · μ21 +wT , (13)

where zT = [z111x∗, · · · , zMs11x

∗]T /√pUL, uT =

[√

α11

α12gU121, · · · ,

√αMs1

αMs2gUMs21

]T ,

wT = [w1, · · · , wMs ]T is the observation noise with zero

mean and covariance matrix

RT = diag{εT,1, · · · , εT,Ms}. (14)

With (13), which is established from one DL training andtwo different UL training frames of Ms supporters, μ21 canbe estimated with WLS shown later. We refer to this methodas SRS based multi-supporter calibration (S-MSC).

Remark 1: The calibration SRSs can also be sent by thesame supporter during Ms DL and 2Ms UL frames. Thecorresponding method is referred to as SRS based multi-framecalibration (S-MFC).

Remark 2: The calibration SRS will change the transmitpowers of the UL training signals. To avoid this, MSm needsto send a normalized calibration SRS, i.e., exp(iθm) ·√pULx.As will be shown in the simulations later, such a normalizationleads to a minor performance loss in calibration. By replacinga regular SRS by a normalized calibration SRS, which can alsobe used to estimate the UL channels, the calibration strategydoes not need additional overhead.

B. Inter-BS Antenna Calibration via Limited Feedback

To obtain the UL and DL channels between two BSs anda supporter at the CU, we can also employ one UL trainingframe, one DL training frame and one feedback frame. In thisway, the signalling framework in LTE-A specification [14]can be reused for supporting the inter-BS antenna calibration,where the UL training is served as the SRS, the DL trainingand the succeeded UL feedback aim at reporting CDI to theBS for precoding. In particular, BS1 obtaining gU

mb and MSm

obtaining gDmb with the regular UL and DL training shown in

III-A1, and the MS conveys the DL CDI to the BS using theregular limited feedback, with which the observation equationsfor estimating μ12 can be established

In [12], another inter-BS antenna calibration method basedon limited feedback was proposed, where each supporterestimates the channels from the reference antennas at the twoBSs to itself, then combines the two channel coefficients intoa vector and feeds back the quantized vector to the CU. Thedifference of the method presented in this subsection from thatin [12] lies in which information is quantized for feedback.

1) Limited Feedback for CDI: After each DL channel isestimated at MSm, which is hD

mb = gDmb/

√αmb, its direction

is quantized and fed back to the BS originally used to assistDL beamforming [8]. We consider the per-cell codebook basedlimited feedback for CoMP systems, where MSm employssingle-cell random vector quantization (RVQ) codebooks toseparately quantize its per-cell channels hD

mb, b = 1, 2, whichis tractable for analysis. We assume that the instantaneousnorms of per-cell channels ‖hD

mb‖, b = 1, 2 are perfectly fed

back. We consider independent codeword selection [20]. Then,MSm quantizes its per-cell CDIs as follows,

hDFmb = argmax

ci∈C‖hD

mbcHi ‖, b = 1, 2, (15)

where C is the per-cell codebook, which consists of unit normcodewords ci ∈ C1×Nt , i = 1, · · · , 2BC , BC is the numberof bits to quantize the per-cell CDI and

hDmb = hD

mb/‖hDmb‖ (16)

is the per-cell CDI.With the independent codeword selection, phase ambiguity

(PA) ejφmb exists, which is a phase rotation between each per-cell CDI and each per-cell codeword. When the PA differenceωm2 is fed back with BP bits using scalar quantization, thenegative impact of the PA on CoMP systems can be recovered[8], where the PA differences is ωmb � φmb − φm1, b = 1, 2,and ωm1 = 0.

After MSm quantized the per-cell CDIs and the PA differ-ence, it feeds back the indices of the selected codewords to itslocal BS. Each BS reconstructs the quantized version of theper-channel of MSm and sends to the CU, which is

gDFmb =

√αmb‖hD

mb‖hDFmb e

jωmb , b = 1, 2, (17)

where ωm2 is the quantized PA difference.2) Observation Equations for Estimating μ21: Define

sin2 θmb = 1 − ‖hDFmb (h

Dmb)

H‖, which is the per-cell CDIquantization error. Then, the per-cell CDI can be expressed as

hDmb = cos θmbe

jφmbhDFmb + sin θmbsmb, (18)

where smb ∈ C1×Nt is a unit norm vector isotropicallydistributed in the null space of hDF

mb [21].Considering (16), (17), (18) and (7) , the DL per-cell

channels can be expressed as

gDmb =

√αmb‖hD

mb‖hDmb + eDmb

=√αmb‖hD

mb‖(cos θmbejφmbhDF

mb + sin θmbsmb)

= ejφmb1DmbgDFmb + eFmb, (19)

where Dmb = cos θmbejΔωmb is a multiplicative error that

comes from the quantization errors of the per-cell CDIsand the PA difference Δωmb = ωmb − ωmb, and eFmb =√αmb‖hD

mb‖ sin θmbsmb + eDmb is an additive error that iscaused by the quantization error of the per-cell CDIs and DLchannel estimation error.

After perfect self-calibration at each individual BS, from(5) the relationship between the UL and DL channels can beexpressed as

gDm2igUm1i = μ21g

Dm1ig

Um2i, i = 1, · · · , Nt, (20)

which can be written in a matrix form as follows

gDm2diag{gU

m1} = μ21gDm1diag{gU

m2}, (21)

where diag{gUmb} = diag{gUmb1, · · · , gUmbNt

}, b = 1, 2.Substituting (6) and (19) into (21),Nt observation equations

for estimating μ21 can be written in a matrix form as

gDFm2 diag{gU

m1} =Dm1

Dm2gDFm1 diag{gU

m2} · μ21 +wm, (22)


where wm = 1Dm2

(μ21gDFm1 diag{eUm2}+μ21e

Fm1diag{gU

m2}−gDFm2 diag{eUm1} − eFm2diag{gU

m1}) is the observation noise.Because the elements of gDF

mb , gUmb, e

Umb and eFmb are i.i.d.,

the elements of wm are also i.i.d.. Therefore, the covariancematrix of the observation noise is

Rm � E{wHmwm} = εF,mINt , (23)

where INt is Nt ×Nt identity matrix.After the CU collects the UL channel estimates and DL

CDIs fed back from Ms supporters, it can estimate the inter-BS ambiguity factor with MsNt observation equations, whichcan be written in a matrix form as follows

zF = DFuF · μ21 +wF , (24)

where zF = [gF12diag{gU

11} · · ·gFMs2

diag{gUMs1

}]H , DF =

diag{D11

D12INt · · · DMs1

DMs2INt},

uF = [gDF11 diag{gU

12} · · ·gDFMs1

diag{gUMs2

}]H , wF =[w1 · · ·wMs ]

H , and the covariance matrix of the observationnoise is

RF = diag{R1 · · ·RMs}. (25)

Then, by using WLS, μ21 can be estimated. We refer to thismethod as CDI based MSC (C-MSC).

Remark 3: The CU can also collect the UL channel es-timates and DL CDIs fed back from one supporter in Ms

UL frames, such a method is referred to as CDI based MFC(C-MFC). In this case, differential quantization [22] can beapplied to quantize the CDIs in the multiple frames, suchthat fewer bits is required to achieve the same quantizationaccuracy as non-differential quantization.

Remark 4: The multiple supporters can also feed backthe ratio Am exp(iθm) in (8) after quantizing with a scalarcodebook instead of sending the weighted training symbol,in order to establish the observation equations. The corre-sponding method is referred to as ratio based MSC (R-MSC).Alternatively, the observation equations can be establishedby allowing one supporter to feed back the ratio in multi-ple frames using differential quantization. The correspondingmethod is referred to as ratio based MFC (R-MFC).

C. WLS Estimation

When the large scale fading gains and the receiver noisevariances at each BS and each MS are available at the CU, itis shown from the definition of εUmb and εDmb in (6) and (7),(12) and (14) that RT is known. Similarly, it is shown fromthe definition of eFmb in (19), wm in (22) and wF

m in (24) thatRF is known.

Note that there is a multiplicative error DF in (24) that isunknown at the CU. Without any knowledge of DF , the CUcan simply ignore the error by regarding DF = IMsNt . Thisonly leads to a minor performance loss, as will be shownin simulations later. Then with the observation equations in(13) or (24) and the corresponding statistics of the observationnoises, WLS criterion can be applied to estimate the inter-BSambiguity factor, which gives rise to

μ21 =uHR−1

uHR−1u· z. (26)

For S-MSC, μ21 = μT21, u = uT , R = RT and z = zT .

For C-MSC, μ21 = μF21, u = uF , R = RF and z = zF .

For all calibration methods using multiple frames, the WLSestimation reduces to the least square estimation, since theobservation noises are identical for multiple frames.

Considering that R is a diagonal matrix, the complexity ofthe matrix inverse is negligible. As a result, the CU only needsto calculate several additional complex multiplications for theinter-BS calibration. It means that the proposed methods havealmost the same complexity as the self-calibration.

D. Procedure of Establishing Observation Equations

For the reader’s convenience, the procedures for the two pre-sented calibration methods based on SRS and CDI feedback aswell as several possible variations to assist the CU establishingthe observation equations are summarized as follows.

◦ S-MSC: 1) each of the multiple supporters sends aregular UL training frame, 2) each BS broadcasts aregular DL training frame, and 3) each supporter sendsa calibration SRS.

◦ S-MFC: 1) one supporter sends a regular UL trainingframe, 2) each BS sends a regular DL training frame,3) the supporter sends the calibration SRS in next ULframe, and 4) the above steps repeat multiple times.

◦ C-MSC: 1) each of the multiple supporters sends aregular UL training frame, 2) each BS broadcasts aregular DL training frame, and 3) each MS feeds backthe quantized DL per-cell CDIs and the PA difference.

◦ C-MFC: 1) one supporter sends a regular UL trainingframe, 2) each BS sends a regular DL training frame, 3)the supporter feeds back the quantized per-cell CDI andPA difference, and 4) the above three steps repeat mul-tiple times and the CDIs and PA differences in multipleframes are quantized with differential codebooks.

◦ R-MSC: 1) each of the multiple supporters sends aregular UL training frame, 2) each BS broadcasts aregular DL training frame, and 3) each supporter feedsback the ratio of its DL per-cell channels with a scalarcodebook.

◦ R-MFC: 1) one supporter sends a regular UL trainingframe, 2) each BS sends a regular DL training frame,and 3) the supporter feeds back the quantized ratio ofits DL per-cell channels, and 4) the above three stepsrepeat multiple times and the ratios in multiple framesare quantized with differential codebook.

IV. PERFORMANCE ANALYSIS

In this section, we compare the mean square error (MSE)of the inter-BS ambiguity factor estimation achieved by usingthe calibration SRS and by reusing the CDI feedback. We alsocompare the MSEs achieved respectively by using multiplesupporters and multiple frames. Based on which, we showhow the supporting MSs should be selected.


A. Comparison of the S-MSC, S-MFC and C-MSC

From (13) and (26), the MSE of the inter-BS ambiguityfactor estimated with S-MSC or S-MFC is

MSE(μT21)

= EuT ,wT {|μ21 − μT21|2}

(a)= EuT {(uH

T RT−1uT )

−1}(b)= Eg{(

Ms∑m=1

αm1|gUm21|2αm2εT,m

)−1}

= Eg{(Ms∑m=1

αm1(αm2 − εUm2)

αm2εT,m

|gUm21|2αm2 − εUm2

)−1}, (27)

where (a) is derived after substituting (13) and (26) by usingthe independence between uT and wT , (b) is obtained bysubstituting the definition of uT in (13) and RT in (14).

From (11) it is not hard to show that the first term of (27),

γT,m � αm1(αm2 − εUm2)

αm2εT,m(28)

is the average observation signal-to-noise ratio (SNR), andXT,m � gU

m21√αm2−εUm2

is the normalized channel estimate with

unit variance. Then, (27) can be equivalently expressed as

MSE(μT21) = EX{(

Ms∑m=1

γT,m|XT,m|2)−1}. (29)

Analogous to the derivations for S-MSC and S-MFC, from(24) and (26) the MSE of the inter-BS ambiguity factorestimated with C-MSC can be derived as

MSE(μF21) = EuF ,wF {|μ21 − μF

21|2}(c)= E{|u

HF RF

−1wF

uHF RF

−1uF

|2+|1−uHF RF

−1DFuF

uHF RF

−1uF

|2|μ21|2},(30)

where (c) is derived by substituting (24) and (26). The secondterm of the MSE comes from treating DF as IMsNt in theWLS estimator.

The following proposition provides the MSE lower boundsof the S-MSC, S-MFC and C-MSC, considering that inpractice the number of supporters or frames, Ms, will be finite.

Prop. 1: When Ms → ∞, the MSEs of S-MSC and S-MFCconverge to zero, i.e.,

limMs→∞

MSE(μT21) = 0, (31)

and the MSE of C-MSC converges to a positive constant, i.e.,

limMs→∞

MSE(μF21) = ψ2 + μ2(1 − Φ · sin(2

−BP π)

2−BP π)2, (32)

where ψ � |E{(gDFm1 diag{gU

m2}ε−1F,mwH

m)}NtE{γF,m|XF,m,n|2} |, Φ �

E{γF,m|XF,m,n|2 cos θm1cos θm2

}E{γF,m|XF,m,n|2} depends on the number of bits

used to quantize the per-cell CDI and the averageobservation SNR, γF,m � (αm1−εDm1)(αm2−εUm2)

εF,mis the

average observation SNR of each scalar equation in (22), andXF,m,n � gF

m1n gUm2n√

(αm1−εDm1)(αm2−εUm2).

Proof: See Appendix A.

Note that ψ, Φ and sin(2−BP π)

2−BP πreflect the impact of the

quantization error of the per-cell CDIs and PA difference. Theproposition suggests that the calibration performance of the C-MSC will be limited by the number of bits for feedback, evenwhen we are able to employ a large number of supportersfor the calibration. Considering that the number of bits forfeedback can not be large in practice, the C-MSC will preformmuch worse than S-MSC and S-MFC, as will be shown fromthe simulations later. Therefore, in the sequel, we will nolonger discuss the C-MSC.

Now we proceed to compare the performance of the S-MSCand S-MFC. (29) can be rewritten in a matrix form as follows

MSE(μT21) = EX{(XH ΓX)−1}, (33)

where Γ � diag{γT,1 · · · γT,Ms}, X = [XT,1 · · ·XT,Ms ]H is

a random vector of complex Gaussian distribution with zeromean and covariance matrix ΦX, and the diagonal elementsof ΦX are equal to 1 that are the variances of XT,m, m =1, · · · ,Ms.

The channels of multiple MSs are usually independent, butthe channels in multiple frames of one MS may be correlatedin general. As a result, when using the S-MSC, ΦX = IMs .When using the S-MFC, the off-diagonal elements of ΦX maynot be equal to zero due to the correlation of the channels inuT defined in (13).

For a fair comparison, we consider that the average observa-tion SNRs of the observation equations for different calibrationmethods are all equal to γ, i.e., Γ = γ · IMs . Then, we canrewrite (27) as

MSE(μT21) = EX{(γXHX)−1}. (34)

Because ΦX = IMs when using the S-MSC, and the covari-ance matrix is not an identity matrix when using the S-MFC,the following proposition implies that the S-MSC outperformsthe S-MFC.

Prop. 2: The MSE of the inter-BS ambiguity factor esti-mate achieves its minimum if and only if ΦX = I.

Proof: See Appendix B.

B. Selection of Calibration Supporters

From (33) we can see that the MSE depends on the numberof supporters or frames Ms, the covariance matrix ΦX andthe average observation SNR γT,m. This suggests that thefollowing means can be applied to improve the calibrationperformance, since the CU can gather all the required in-formation to establish the observation equations and thenestimate the ambiguity factors. 1) The CU can use S-MSCby selecting more supporters when there are large numberof active MSs. Otherwise, the CU can employ the S-MFCwith large number of frames from one supporter. Recall thatthe supporter assisting S-MSC or S-MFC acts differently assummarized in the calibration procedure in section III.D. 2)The CU selects appropriate supporters to increase the averageobservation SNR.

From the definition of γT,m in (28), we can see that itdepends on the location of the supporters. In the followingproposition, we show how the supporters should be selectedto optimize the calibration performance.


We first define two notions to be used. When MSm islocated at the “exact cell-edge”, we mean all the large scalefading gains of the user are equal, i.e., α1 = · · · = αNc .2

For a MS with distance r from its master BS, its averageDL receive SNR is called cell-edge SNR, where r is the cellradius. To reflect the impact of the inter-cluster interference,the interference power can be included in the noise power.

Prop. 3: When a system has high cell-edge SNR, theoptimal location of the supporters that maximizes the averageobservation SNR is the exact cell-edge.

Proof: Upon substituting (12), the average observation SNRcan be obtained as

γT,m =αm1(αm2 − εUm2)

αm1(μ2εUm2 + μεDm2) + αm2(1−εDm2/αm2

1−εDm1/αm1μεDm1 +

σU

pUL),

(35)where μ is defined before (12).

When the cell-edge SNR is high, αmbpUL σU . From thedefinition of εUmb after (6), we can see that its denominatoris dominated by αmbpUL. Thus, we have εUmb ≈ σU/pUL.For the same reason, εDmb ≈ σD/pDL. Then, (35) can beapproximated as

γT,m ≈ (μ2σUpULαm2

+μσD

pDLαm2+

μσDpDLαm1

+σU

pULαm1)−1

=1

4· 4

(γUm2

μ2 )−1+ (γDm2

μ )−1+ (γDm1

μ )−1+ (γUm1)−1, (36)

which is a harmonic mean multiplied by a constant, where γUmb

and γDmb are the UL and DL SNRs of the received signals atBSb and at MSm, respectively.

Note that μ > 1, and in practical systems, the SNR oflocal link (i.e., the link from the master BS of MSm, BS1, toMSm) is higher than that of cross link (i.e., the link from BS2

to MSm), because the local link have higher channel gain.As a result, the MSE is dominated by the minimum term,i.e.,γUm2/μ

2 or γDm2/μ, where γUm2 and γDm2 are the UL andDL SNRs of the cross link, respectively. It implies that inorder to reduce the observation errors, cell-edge MSs shouldbe selected to assist the inter-BS calibration. As shown insimulations later, this conclusion is still valid when the cell-edge SNR is not high.

V. NUMERICAL AND SIMULATION RESULTS

In this section we validate previous analytical analysisand evaluate the performance of several inter-BS antennacalibration methods via simulations.

A. Simulation Setting

In the simulation, we consider hexagon cells, the radius ofeach cell r = 250 m. Each BS transmits with a maximal powerof 46 dBm, and each MS transmits with a maximal power of23 dBm. The path loss exponent is 3.76, and the minimumBS-MS distance is 35 m. These simulation parameters arefrom [23]. Sectorized antenna patterns and shadowing are not

2Note that this is a mathematical definition to facilitate analysis. In practice,when considering the path loss, shadowing and sector antenna power gains,such a user may rarely appear.

considered. The receiver noise floor of BS is -116.4 dBm, andthat of MS is -97.5 dBm [24].

Unless otherwise specified, we consider the following set-ting in all of the simulations.

◦ Three neighboring BSs are cooperated. All the MSsare randomly distributed in a “x dB edge region”, e.g.,the value of αmbm/

∑b�=bm

αmb is less than x dB forMSm. When the distance between MS and its local BSis d, its DL SNR is SNRc + 37.6 log10(r/d), whereSNRc is the ratio of average receive power to noisevariance for a user exactly with distance r from the BS,denoted as cell-edge SNR, which is used as a controlparameter to change the bias points of the simulations.We model the inter-cluster interference as white Gaus-sian noise, which is the worst case interference andresults in pessimistic performance [25]. Consequently,inter-cluster interference is implicitly reflected in SNRc.We set SNRc as 10 dB, corresponding to a particularchannel and deployment scenario. The simulation resultsare obtained from 100 small scale channel realizationswith i.i.d. Rayleigh fading and 200 random locations ofthe MSs.

◦ We consider a realistic scenario where the self-calibration within each BS is imperfect, such that βmbj

defined in (1) is not equal to βmb in (3). Differentvalues of βmbj/βmb are assumed to be independent,their norms are modeled as a log-uniformly randomvariable distributed within [−1, 1] dB, and their phasesare modeled as a random variable uniformly distributedin [−10◦, 10◦] [19]. The ambiguity factors of the ref-erence antenna in different BSs, βmb, are independent,whose norms are modeled as a log-uniformly randomvariable distributed within [−3, 3] dB, and the phasesare modeled as random variable a uniformly distributedwithin [−180◦, 180◦].

◦ For all the methods using multiple supporters, the CUrandomly selects 30 MSs as the supporters. For all themethods using multiple frames, we consider a realistictime-varying channel model, Jakes Model, which iswidely applied [26]. The duration of each UL or DLframe is 10 ms. For the R-MSC, the norm and phaseof the ratio Am exp(iθm) are fed back with 4 bits,respectively. For the C-MSC, Grassmannian line packing(GLP) codebook is used to quantize the per-cell CDIs,which is optimal for i.i.d. channels [27]. Each per-cellCDI and each PA difference are fed back with 4 bits,respectively.

B. Validation of the Analytical Results

To validate the asymptotic results in proposition 1 and theapproximated results in proposition 3, in Fig. 2 we comparethe simulated MSE with finite number of supporters underrealistic cell-edge SNR with the numerical results. Since inthe analysis RVQ codebook is considered, in this simulationwe also use this codebook. From Fig. 2(a) we can see that thesimulated MSE of the C-MSC converges to the asymptoticresult (obtained from (32)) faster than that of the S-MSC(whose asymptotic result is obtained from (31)). This is


5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Number of calibration supporters

MS

E o

f int

er−

BS

am

bigu

ity fa

ctor

Simulated result of C−MSCAsymptotic result of C−MSCSimulated result of C−MSCwith knowen errorSimulated result of S−MSCAsymptotic result of S−MSC

(a) Simulation and asymptotic results of the MSEs of the S-MSC andC-MSC. The supporters are located at exact cell-edge.

10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Cell−edge SNR (dB)

MS

E o

f int

er−

BS

am

bigu

ity fa

ctor

Simulated resultApproximate result

exact cell−edge

10 dB cell−edge region

(b) Simulation and approximated results of the MSE of the S-MSC.

Fig. 2. Simulation and numerical results of the MSEs of S-MSC and C-MSC.

because the number of scalar observation equations of C-MSCis NtMs, while that of S-MSC is Ms. However, the MSE ofC-MSC is much higher than the MSE of S-MSC, since thenumber of bits for feedback is limited. To show the impact ofsimply setting the unknown multiplicative error DF as identitymatrix, we provide the simulation result when DF is known.We can see that the knowledge of DF only improves theperformance slightly. The approximated results in Fig. 2(b)is obtained from (29), where the observation SNR γT,m iscalculated from (36). We can see that the simulation resultsconverge to the approximated results at high cell-edge SNR.

In Fig. 3 we show the MSE and phase error (defined asE{| arg(μ21) − arg(μ21)|}, arg(z) stand for the phase ofz) of the inter-BS ambiguity factor estimated with the S-

0 5 10 15 20 25 3010

−2

10−1

100


MS

E o

f int

er−

BS

am

bigu

ity fa

ctor

Exact cell−edge3 dB cell−edge region10 dB cell−edge region

SNRc = 10 dB

SNRc = 15 dB

(a) MSE of the inter-BS ambiguity factor estimation

0 5 10 15 20 25 300

5

10

15

20

25

30

35


Pha

se E

rror

( o )

Exact cell−edge3 dB cell−edge region10 dB cell−edge region

SNRc = 10 dB

SNRc = 15 dB

(b) Phase error of the inter-BS ambiguity factor estimation

Fig. 3. MSE and phase error of the inter-BS ambiguity factor estimated withthe S-MSC.

MSC. We can see that the supporters located at the exactcell-edge provides the best performance, which agrees withthe conclusion in proposition 2. We can also see that theconclusions obtained for MSE analysis are also valid for phaseerror, but the phase error of the estimation is not sensitive tothe location of the supporters.

C. Performance Comparison of the Calibration Methods

In Fig. 4, we compare the MSEs of three calibration meth-ods using multiple supporters when the number of supportersMs or the number of feedback bits of each user in the C-MSC and R-MSC increases. In R-MSC, the number of bitsfor quantizing the norm and phase of the ratio are identical(e.g., 8 bits for a user means 4 bits for the norm and 4 bitsfor the phase). In C-MSC, the number of bits for quantizing


0 10 20 30 4010

−3

10−2

10−1

100

Number of calbration supporters

MS

E o

f int

er−

BS

am

bigu

ity fa

ctor

SNRc = 10 dB

SNRc = 15 dB

SNRc = 20 dB

C−MSCR−MSC

S−MSC

(a) MSE versus Ms, the number offeedback bits per-user is 8 bits in R-MSC, and 12 bits in C-MSC.

4 6 8 10 1210

−3

10−2

10−1

100

Number of feedback bits

MS

E o

f int

er−

BS

am

bigu

ity fa

ctor

SNRc = 10 dB

SNRc = 15 dB

SNRc = 20 dB

C−MSC

R−MSC

(b) MSE versus the number of feed-back bits per-user, Ms = 30

Fig. 4. Comparison of several calibration methods using multiple supporters,which are located at exact cell-edge.

each per-cell CDI and PA difference feedback are identical(e.g., 9 bits for a users means 3 bits for each per-cell CDIand 3 bits for the PA difference). As shown in Fig. 4(a), theMSE of the proposed S-MSC reduces rapidly with the numberof supporters, while the MSEs of the other two methods areseverely limited by the quantization error. To achieve the sameMSE, the S-MSC needs much less supporters than the R-MSC.As shown in Fig. 4(b), the MSEs achieved by the R-MSCand C-MSC reduce with the increase of the number of bitsfor feedback. The R-MSC outperforms the C-MSC. However,when each user employs 12 bits, the R-MSC is still muchworse than the S-MSC. In this case, overall 12Ms = 360 bitsare feedback, which is indeed a large number. When morebits are used, a MSE floor will appear for both C-MSC andR-MSC, which is not shown in the figure.

In Fig. 5 we compare the MSEs of the inter-BS ambiguityfactor estimated with the S-MSC and the S-MFC, given thesame number of observation functions (i.e., the number ofsupporters in MSC, and the repeated times of the multiplesteps in S-MFC). For the S-MFC, a supporter is located at theexact cell edge and moves at different speeds of 0 km/h, 3km/h and 10 km/h, where the carrier frequency is 2 GHz. Wecan see that the MSE of the S-MFC decreases when the speedof the supporter increases. This is because the coherence timedecreases with the MS’s speed, and the covariance matrix ΦX

approaches to IMs when the coherence time is close to zero.Furthermore, the MSE of the S-MSC decreases much fasterthan that of the S-MFC. We can also see that even when thesupporters are located within a 10 dB edge region, the S-MSCstill outperforms the S-MFC with a low speed supporter.

In Fig. 6 we compare the MSEs achieved by severalcalibration methods based on limited feedback, given thesame number of observation functions (i.e., the number offeedback frames in C-MFC and R-MFC). Unless explainedin the legend, time-invariant channel is considered and thesupporters are located in the exact cell edge. When using the

5 10 15 20 25 30 35 4010

−2

10−1

100

Number of scalar observation equations

MS

E o

f int

er−

BS

am

bigu

ity fa

ctor

Time−invariant channel3km/h MS speed10km/h MS speedSupporters at 10 dB cell−edge regionSupporters at exact cell−edge

S−MSC

S−MFC

Fig. 5. Comparison of the calibration methods using multiple supporters andmultiple frames.

5 10 15 20 25 30 35 4010

−2

10−1

100

Number of matrix observation equations

MS

E o

f int

er−

BS

am

bigu

ity fa

ctor

C−MFC with GLP C−MSC with GLPC−MFC with DifferentialCodebook, 3km/h MS speedC−MFC with Differential CodebookR−MFC with scalar quantizationR−MFC with Differential CodebookR−MFC with DifferentialCodebook, 3km/h MS speedR−MSC with scalar quantization

Fig. 6. Comparison of the calibration methods based on limited feedback.

GLP and differential codebook [22], each per-cell CDI and PAdifference are fed back with 4 bits, respectively. When usingthe scalar quantization and scalar differential quantization,the norm and phase of the ratio are fed back with 4 bits,respectively. In the scalar quantization, uniform quantizationof the norm (from -20 dB to 20 dB) and phase (from −πto π) is considered. In the scalar differential quantization,we employ a method similar to that presented in [28]. Fromthe figure we can see that using differential quantization forthe calibration methods with multiple frames can improve theperformance, and the gain for R-MFC is more significant.When the supporter moves with 3 km/h, the MSE achieved byR-MFC further reduces, but is still higher than the R-MSC.This result seems inconsistent with the intuition that differ-


ential codebook achieves the highest feedback accuracy whenthe MS does not move. Yet this intuition is only true whenwe quantize CDI, while in R-MFC the ratio Am exp(iθm) isquantized, which does not have a direct relation with channelvariation. Moreover, we have shown in Fig. 5 that MFC has abetter performance when MS moves faster, which dominatesthe MSE of the ambiguity factor estimate. On the other hand,the MSE does not decrease with the number of observationsfor C-MFC with differential codebook even when the MS isstationary. Such an unexpected result is caused by the selecteddifferential codebook. Other type of differential codebooksmay not show such a saturation and may provide a lowerMSE. A comprehensive comparison of differential codebooksoptimized for inter-BS antenna calibration is for further study.Comparing Fig. 5 and Fig. 6, we can find that the R-MSC isinferior to the S-MSC.

D. Impact of the Inter-BS Calibration on DL Achievable Rate

To show the impact of the inter-BS calibration on the DLtransmission performance, we simulate the average achievabledata rate of each user, where three cooperative BSs eachequipped with four antennas cooperatively serve three MSs.We consider zero-forcing (ZF) precoding both for CoMP andnon-CoMP systems. In CoMP systems, ZF precoding consistsof ZF beamforming and an optimal power allocation thatmaximizes the sum rate under the per-BS power constraint(PBPC) [29]. The DL achievable rate is computed by Shannoncapacity formula.

In Fig. 7 we show the average per-user rate when thefollowing methods are applied. All the MSs and calibrationsupporters are randomly distributed in a 3 dB edge region.Note that the receive SNRs of these supporters may be higherthan the cell-edge SNR (the x-axis) because they are randomlylocated.

1) Perfect calibration: the CU knows error-free inter-BSambiguity factors μi1, i = 2, 3.

2) Perfect phase calibration: the CU knows error-freephases of μij , i.e., arg(μij).

3) S-MSC: the supporters send the calibration SRS withoutnormalization, Am exp(iθm) · x.

4) S-MSC with normalized SRS: the supporters send thecalibration SRS with normalization, exp(iθm) · x.

5) R-MSC: each of the supporters feed back 4 bits foruniformly quantizing the norm and 4 bits for the phaseof Am exp(iθm) · x.

6) C-MSC: each of the supporters feed back 4 bits for eachper-cell CDI and 4 bits for each PA difference.

7) FDD systems: 4 bits are used for quantizing each per-cell CDI and for quantizing each phase ambiguity (PA)difference, respectively. Then the quantized global CDIis reconstructed at the CU and used for DL precoding[20].

8) Perfect self-calibration: CoMP with perfect self-calibration but without inter-BS calibration.

9) Non-CoMP: three BSs transmit to the users in their owncells without cooperation.

10) Method in [12]: the inter-BS calibration method pro-posed in [12].

To show how the MSE of the ambiguity factor estimationtranslates to the rate loss of CoMP systems, we providesome typical MSE values of several schemes. It is shownthat the achievable rate gap between the perfect calibrationand the perfect self-calibration-only method is large, which iscaused by the inter-BS ambiguity factors. Without the inter-BS calibration, CoMP will become inferior to non-CoMP.The S-MSC without normalized SRS performs closely tothe perfect calibration, and the S-MSC with normalized SRSperforms close to the perfect phase calibration. When theR-MSC is used, the performance is always worse than thatof the S-MSC without normalized SRS, but is a little bitbetter than the S-MSC with normalized SRS for high cell-edge SNR. By comparing the performance between the perfectcalibration and the perfect phase calibration, we can see thatthe performance loss is dominated by the phase of μij . Thisis because only the multi-user interference (MUI) depends onthe norm of ambiguity factor, but both the signal and MUIdepend on the phase ambiguity. As anticipated, the C-MSCperforms the worst among all the inter-BS calibration methods.In FDD systems, the antennas do not need to calibrate, butthe performance is destroyed by the quantization errors ofCDI and PA. Comparing Fig. 7(a) and Fig. 7(b), we can seethat the S-MSC outperforms the R-MSC, both outperform themethod in [12].

In LTE systems, only finite number of orthogonal SRS areavailable. In Fig. 8, we show the impact of the non-orthogonalSRS on the average per-user achievable rate, where all theMSs and calibration supporters are randomly distributed in a3 or 10 dB cell edge region or whole cell, and the trainingsequences for the supporters and MSs in different cells arenonorthogonal as generated in [30]. It is shown that when theMSs are located in the cell edge region, the non-orthogonalSRS leads to minor performance degradation. This is becauseduring the UL training the interference from other cells aremuch lower than the receiver noise.

In summary, the evaluation results suggest that the S-MSC is the best calibration method when there are multiplesupporters to be selected in practice. If only one MS is able toassist the calibration, the S-MFC will perform well when thechannel of the MS is in fast fading. If the feedback overheadis not a concern, the R-MSC performs close to the S-MSCat high cell-edge SNR. Finally, if the training and feedbackframes in LTE specification are not allowed to revise forsupporting the inter-BS calibration, the C-MSC is a solutionbut the performance is not good.

VI. CONCLUSION

In this paper, we studied inter-BS antenna calibration strat-egy to alleviate the performance degradation caused by thenon-ideal uplink-downlink channel reciprocity in DL TDDCoMP systems. The strategy can be implemented either withmultiple supporters or with multiple uplink frames from asingle supporter, and either with a calibration SRS or withfeedback. We analyzed the performance of typical methods,and provided the principle to select the users assisting calibra-tion. Both analytical and simulation results showed that cell-edge users should be selected to assist the inter-BS antennacalibration. In general, the calibration with multiple supporters


5 7 9 11 13 151.5

2

2.5

3

3.5

4

4.5

5

5.5

6


Ave

rage

Per

−U

ser

Rat

e (b

ps/H

z)

Perfect calibrationS−MSCPhase perfect calibrationS−MSC with normalize SRSNon−CoMPPerfect self calibration

(a) Performance of training based calibrations

5 7 9 11 13 151.5

2

2.5

3

3.5

4

4.5

5

5.5

6


Ave

rage

Per

−U

ser

Rat

e (b

ps/H

z)

Perfect calibrationR−MSCMethod in [12]C−MSCFDD systemsPerfect self calibration

(b) Performance of quantization based calibrations

Fig. 7. Average per-user achievable data rate with different antennacalibration methods. When cell-edge SNR is 10 dB, the MSEs of S-MSC,R-MSC and C-MSC are 5× 10−2, 9× 10−2 and 1.1 respectively.

outperforms the calibration with multiple frames from onelow-mobility supporter. The proposed method of sendingcalibration SRS performs the best with a few supporters. Theaverage data rate achieved by each user increases significantlyby using the proposed inter-BS calibration method, which isclose to that with perfect calibration.

APPENDIX APROOF OF PROP.1

When the number of supporters Ms → ∞, the MSE ofS-MSC or S-MFC is

limMs→∞

MSE(μT21) = lim

Ms→∞1

MsEX{( 1

Ms

Ms∑m=1

γT,m|XT,m|2)−1}.(A.1)

3dB 10dB Whole2

3

4

5

6

7

8

Edge Region

Ave

rage

Per

−U

ser

Rat

e (b

ps/H

z)

SNRc = 5 dB

3dB 10dB Whole2

3

4

5

6

7

8

Edge Region

Ave

rage

Per

−U

ser

Rat

e (b

ps/H

z)

SNRc = 15 dB

Non−OrthogonalOrthogonal

Fig. 8. Average per-user achievable data rate with orthogonal or non-orthogonal SRS.

Because γT,m|XT,m|2,m = 1, · · · ,Ms are independent, fromthe law of large numbers we obtain

limMs→∞

1

Ms

Ms∑m=1

γT,m|XT,m|2 = EX{γT,m|XT,m|2}, (A.2)

which is is a constant not depending on Ms.Substituting (A.2) into (A.1) and further considering (23),

we have

limMs→∞

MSE(μT21) = lim

Ms→∞(EX{γT,m|XT,m|2})−1

Ms= 0.

This proves (31). In the sequel, we prove (32) for the C-MSC.From the definition of uF in (24) and RF in (25), we have

uHF RF

−1uF =

Ms∑m=1

(γF,m

Nt∑n=1

|XF,m,n|2). (A.3)

Substituting (A.3) into the first and second terms in (30) andfurther considering (23), we obtain

E{|uHF RF

−1wF

uHF RF

−1uF

|2}

= E{|∑Ms

m=1(gDFm1 diag{gU

m2}ε−1F,mwH

m)∑Ms

m=1(γF,m

∑Nt

n=1 |XF,m,n|2)|2}

= E{|1

Ms

∑Ms

m=1(gDFm1 diag{gU

m2}ε−1F,mwH

m)

1Ms

∑Ms

m=1(γF,m

∑Nt

n=1 |XF,m,n|2)|2},(A.4)

and

E{|1− uHF RF

−1DFuF

uHF RF

−1uF

|2|μ21|2}

= E{|1−∑Ms

m=1(γF,mDm1

Dm2

∑Nt

n=1 |XF,m,n|2)∑Ms

m=1(γF,m

∑Nt

n=1 |XF,m,n|2)|2|μ21|2}

(b)= E{|1−

1Ms

∑Ms

m=1(γF,mcos θm1

cos θm2

∑Nt

n=1 |XF,m,n|2e−jΔωm2)

1Ms

∑Ms

m=1(γF,m

∑Nt

n=1 |XF,m,n|2)|2

·|μ21|2}, (A.5)


where (b) is obtained by substituting Dmb = cos θmbejΔωmb .

Note that ωm1 � φm1 − φm1 = 0 and ωm1 = Δωm1 = 0.Again, from the law of large numbers we have

limMs→∞

1

Ms

Ms∑m=1

(gDFm1 diag{gU

m2}ε−1F,mwH

m)

= E{gDFm1 diag{gU

m2}ε−1F,mwH

m}, (A.6)

and

limMs→∞

1

Ms

Ms∑m=1

(γF,mcos θm1

cos θm2

Nt∑n=1

|XF,m,n|2e−jΔωm2)

= NtE{γF,mcos θm1

cos θm2|XF,m,n|2e−jΔωmb2}, (A.7)

and

limMs→∞

1

Ms

Ms∑m=1

(γF,m

Nt∑n=1

|XF,m,n|2)

= NtE{γF,m|XF,m,n|2}. (A.8)

Substituting (A.6), (A.7), (A.8) and E{|μ21|2} = μ2 into(30), the asymptotic MSE of the C-MSC is

limMs→∞

MSE(μF21)

= ψ2+μ2|1− E{γF,mcos θm1

cos θm2|XF,m,n|2e−jΔωmb2}

E{γF,m|XF,m,n|2} |2,(A.9)

where ψ = |E{(gDFm1 diag{gU

m2}ε−1F,mwH

m)}NtE{(γF,m|XF,m,n|2)} |.

Considering the definition of PA difference and the factthat e−jΔωmb2 is independent from the quantization of per-cell channels, we have

E{γF,mcos θm1

cos θm2|XF,m,n|2e−jΔωm2}

= E{γF,mcos θm1

cos θm2|XF,m,n|2}E{e−jΔωm2}. (A.10)

When uniform quantization is applied to quantize the PAdifference, the quantization errors are modeled as randomvariables uniformly distributed in [−2−BP π, 2−BP π]. Then,we have

E{e−jΔωm2} =

∫ 2−BP π

−2−BP π

e−jΔωm2

2× 2−BP πdΔωm2

=sin(2−BP π)

2−BP π. (A.11)

Substituting (A.10) and (A.11) into (A.9), we immediatelyobtain (32).

APPENDIX BPROOF OF PROP.3

From (34) the MSE of the inter-BS ambiguity factor esti-mate is

MSE(μT21) = EX{(γXHX)−1}, (B.1)

where E{XXH} = ΦX, and Tr(ΦX) =Ms.Because ΦX is a Hermitian matrix, its eigen-decomposition

is ΦX = UΛUH , where U is a unitary matrix, Λ =diag([λ1 · · ·λMs ]) is a diagonal matrix, and λ1 ≥ · · · ≥λMs =Ms −

∑Ms−1m=1 λm.

Denote ξ = Λ−1/2UHX. Then, ξ is a complex Gaussianrandom vector with zero mean and covariance matrix IMs , theelements of ξ are independent, and |ξm|2 follows exponentialdistribution with parameter 1, whose probability density func-tion is f(x) = exp(−x), x ≥ 0.

After these matrix transformations, we can rewrite (B.1) as

MSE(μT21)

=1

γE{(

Ms∑m=1

λm|ξm|2)−1}

=1

γ

∫∫ +∞

0

exp(−∑Ms

m=1 xm)∑Ms

m=1 λmxmdx1 · · · dxMs . (B.2)

We can prove that MSE(μT21) is an increasing function of λ1

by finding the partial derivative as

∂MSE(μT21)

∂λ1

=1

γ

∫∫ +∞

0

xMs − x1

(∑Ms

m=1 λmxm)2exp(−

Ms∑m=1

xm)dx1 · · · dxMs

=1

γ

∫∫x1>xMs

xMs − x1

(∑Ms

m=1 λmxm)2exp(−

Ms∑m=1


+1

γ

∫∫x1<xMs

xMs − x1

(∑Ms

m=1 λmxm)2exp(−

Ms∑m=1


(a)=

1

γ

∫∫x1>xMs

xMs − x1

(∑Ms

m=1 λmxm)2exp(−

Ms∑m=1


− 1

γ

∫∫x′1>x′

Ms

(x′Ms− x′1) exp(−

∑Ms

m=1 xm)

(λMsx′1+

∑Ms−1m=2 λmxm+λ1x′Ms

)2dx′1 · · · dx′Ms

=1

γ

∫∫x1>xMs

(− xMs − x1

(λMsx1 +∑Ms−1

m=2 λmxm + λ1xMs)2

+xMs − x1

(∑Ms

m=1 λmxm)2) exp(−

Ms∑m=1

xm)dx1 · · · dxMs , (B.3)

where (a) is a coordinate transformation by setting x′1 = xMs

and x′Ms= x1. When x1 > xMs and λ1 > λMs , λ1x1 +

λMsxMs > λMsx1 + λ1xMs , therefore xMs−x1

(∑Ms

m=1 λmxm)2>

xMs−x1

(λMsx1+∑Ms−1

m=2 λmxm+λ1xMs )2

, and the term inside the inte-

grals is positive. Then we obtain ∂MSE(μT21)

∂λ1> 0.

It indicates that the decrease of the largest eigenvalue of ΦX

will lead to the decrease of the MSE. Therefore, the MSEachieves its minimum if and only if all the eigenvalues areequal, i.e., when the covariance matrix ΦX = I.

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Liyan Su received his B. Eng degree in the Schoolof Advanced Engineering from Beihang University(BUAA), Beijing, China in 2010. He is now pursu-ing a Ph.D degree in the School of Electronics andInformation Engineering from BUAA. His researchinterests lie in the area of energy efficient andspectral efficient multi-cell MIMO systems.

Chenyang Yang (SM’08) received the M.S.E andPh.D. degrees in electrical engineering from Bei-hang University (formerly Beijing University ofAeronautics and Astronautics), Beijing, China, in1989 and 1997, respectively. She is currently a FullProfessor with the School of Electronics and Infor-mation Engineering, Beihang University. Her recentresearch interests include green radio, interferencecoordination and cooperative transmission for largescale networks. Prof. Yang was the Chair of theIEEE Communications Society Beijing chapter from

2008 to 2012. She has served as Technical Program Committee Memberfor many IEEE conferences. She currently serves as an Associate editor forIEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, a guest editorof IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, anAssociate Editor-in-Chief of the Chinese Journal of Communications, andan Associate editor-in chief of the Chinese Journal of Signal Processing.She was nominated as an Outstanding Young Professor of Beijing in 1995and was supported by the First Teaching and Research Award Program forOutstanding Young Teachers of Higher Education Institutions by Ministry ofEducation (P.R.C. “TRAPOYT”) during 1999-2004.

Gang Wang received the Ph.D. degree from Ts-inghua University in 2006. He joined NEC Lab-oratories China in 2006 and worked on wirelessnetworking. From 2009, he started research on 3GPPLTE standardization. He visited UC Davis in 2011and worked with Prof. Zhi Ding on CoMP. Heis now a senior researcher at NEC LaboratoriesChina. His research interest includes MIMO, CoMP,dynamic TDD system and coverage enhancement.

Ming Lei received the B.Eng. degree from theSoutheast University in 1998 and the Ph.D. degreefrom BUPT (Beijing University of Posts & Telecom-munications) in 2003, all in Electrical Engineering.From April 2003 to February 2008, he was a re-search scientist with the National Institute of Infor-mation and Communications Technology (NICT),Japan, where he contributed to Japan’s nationalprojects on 4G mobile communications (MIRAIprojects) and IEEE standardization of 60-GHz multi-gigabit WPAN (IEEE 802.15.3c). From March 2008

to May 2009, he was a project lead of Intel Corporation, where he contributedto the standardization of WiGig (60-GHz WPAN), IEEE 802.11ad (60-GHzWLAN) and IEEE 802.16m (mobile WiMAX). Since May 2009, he has beenwith NEC Laboratories China, NEC Corporation, as the department headmanaging the wireless research and standardization projects on 4G cellularmobile communications (LTE and LTE-Advanced), 60-GHz, mobile backhaul,etc. Dr. Ming Lei was elected to IEEE Senior Member in 2009.

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