Downlink User Capacity of Massive MIMO Under Pilot ...jeiezhang/document/UserCapacity_TWC.pdf · IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL.14,NO.6,JUNE 2015 3183 Downlink User

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 6, JUNE 2015 3183

Downlink User Capacity of Massive MIMO UnderPilot Contamination

Juei-Chin Shen, Member, IEEE, Jun Zhang, Member, IEEE, and Khaled B. Letaief, Fellow, IEEE

Abstract—Pilot contamination has been regarded as a mainlimiting factor of time division duplexing (TDD) massive multiple-input-multiple-output (Massive MIMO) systems, as it will makethe signal-to-interference-plus-noise ratio (SINR) saturated. How-ever, how pilot contamination will limit the user capacity of down-link Massive MIMO, i.e., the maximum number of users whoseSINR targets can be achieved, has not been addressed. This paperprovides an explicit expression of the Massive MIMO user capac-ity in the pilot-contaminated regime where the number of usersis larger than the pilot sequence length. This capacity expressioncharacterizes a region within which a set of SINR requirementscan be jointly satisfied. The size of this region is fundamentallylimited by the pilot sequence length. Furthermore, the scheme forachieving the user capacity, i.e., the uplink pilot training sequencesand downlink power allocation, has been identified. Specifically,the generalized Welch bound equality sequences are exploitedand it is shown that the power allocated to each user shouldbe proportional to its SINR target. With this capacity-achievingscheme, the SINR requirement of each user can be satisfied andenergy-efficient transmission is achieved in the large-antenna-size(LAS) regime. The comparison with two non-capacity-achievingschemes highlights the superiority of our proposed scheme interms of achieving higher user capacity. Furthermore, for thepractical scenario with a finite number of antennas, the actualantenna size required to achieve a significant percentage of theasymptotic performance has been analytically quantified.

Index Terms—Massive multiple-input-multiple-output (MassiveMIMO), user capacity, pilot contamination, pilot-aided channelestimation, power allocation.

I. INTRODUCTION

MASSIVE MIMO is regarded as an efficient and scalableapproach for multicell multiuser MIMO implementa-

tion, and it can significantly improve both the spectrum effi-ciency and energy efficiency [1]–[3]. By equipping each basestation (BS) with an antenna array whose size is greatly largerthan the number of active user equipments (UEs), the asymp-totic orthogonality among MIMO channels to different UEs isachieved and in turn it makes intra- and inter-cell interferencemore manageable. Hence, using simple linear precoder anddetector can approach the optimal dirty-paper coding capacity

Manuscript received June 27, 2014; revised November 12, 2014; acceptedJanuary 13, 2015. Date of publication February 13, 2015; date of currentversion June 6, 2015. This work is supported by the Hong Kong Research GrantCouncil under Grant No. 610212. The associate editor coordinating the reviewof this paper and approving it for publication was E. A. Jorswieck.

The authors are with the Department of Electronic and Computer Engi-neering, the Hong Kong University of Science and Technology, Kowloon,Hong Kong (e-mail: [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TWC.2015.2403317

[4]. Channel side information (CSI) at BSs plays an importantrole in the exploitation of channel orthogonality. In practice,TDD operation is assumed for CSI acquisition through uplinktraining. The advantage of uplink training is that the requiredlength of pilot training sequences is proportional to the numberof active UEs rather than that of BS antennas. The traininglength is fundamentally limited by the channel coherence time,which can be short due to UEs of high mobility. It has beenshown, nevertheless, in [5] that the effect of using short trainingsequences diminishes in the LAS regime. Specifically, amongthe poorly estimated channels, the asymptotic orthogonality canstill hold.

However, a major problem with TDD Massive MIMO is thatinevitably the same pilot sequences will be reused in multiplecells. The channels to UEs in different cells who share thesame pilot sequence will be collectively learned by BSs. Inother words, the desired channel learned by a BS is contam-inated by undesired channels. Once this contaminated CSI isutilized for transmitting or receiving signals, intercell interfer-ence occurs immediately which limits the achievable SINR.This phenomenon, known as pilot contamination, can not becircumvented simply by increasing the BS antenna size [6], [7].

Several attempts have been made to tackle the problem ofpilot contamination. In [8], a sophisticated precoding methodis proposed to minimize intercell interference due to pilot con-tamination. A more direct approach for pilot decontamination,harnessing second-order channel statistics, can be found in [9].A recent study [10] claims that pilot contamination is due to in-appropriate linear channel estimation. Hence, by using the blindsubspace-based estimation, unpolluted CSI is obtainable. In thecontext of Massive-MIMO-aided OFDM, pilot contaminationcan be eliminated by a novel combination of downlink trainingand scheduled uplink training [11]. However, this method, suf-fering from similar drawbacks as the blind estimation, requiresa substantial training period.

The effect of pilot contamination is usually quantified asSINR saturation due to intercell interference. A number ofstudies have examined this saturation phenomenon and thecorresponding uplink or downlink throughput [6], [12], [13].The former two provide analysis in the LAS regime with a fixednumber of active UEs, while in [13], it analyzes asymptoticSINRs with a fixed ratio of the BS antenna size to the active-UE number. All these studies lead to a similar conclusion thatthe SINR will saturate with an increasing antenna size, makingsystem throughput limited. In [14], several possible definitionsof SINR are provided and discussed.

So far, however, there has been little discussion about theuser capacity of TDD Massive MIMO, which is the maximum

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3184 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 6, JUNE 2015

number of UEs whose SINR requirements can be met for agiven pilot sequence length. This terminology “user capacity,”originally coined for analyzing CDMA systems [15], is em-ployed here for investigating Massive MIMO from a new angle.In this paper, we confine our discussion to the user capacityof single-cell downlink TDD Massive MIMO, where pilotcontamination will occur once the number of UEs is greaterthan the pilot sequence length. Meanwhile, we consider a moregeneral set of pilot sequences whose cross-correlations canrange from −1 to 1. In contrast, in most existing studies ofMassive MIMO, the cross-correlations are restricted to be 1or 0. Our discussion will show that the user capacity can becharacterized by a specific region within which there exists acapacity-achieving pilot sequence and power allocation schemesuch that the SINR requirements are satisfied and the user datacan be energy-efficiently transmitted. This result is significantin the sense that within the specified region we have no worriesabout pilot contamination by using the proposed allocationscheme. Meanwhile, the derivation of this result only involvesa simple least-squares (LS) channel estimator and maximumratio transmission (MRT) precoding. It means that this regionexists without relying on advanced estimation or precodingmethods. Though this region is identified under the single-cellassumption, it sheds light on the possible existence of a similarregion in the general multicell scenario.

A. Prior Works

The idea of using user capacity as a metric to characterizethe system performance has been explored in [15], [16]. In[15], the discussion of uplink/downlink user capacity is con-fined to CDMA systems, in which the role of the signaturesequences is analogous to that of pilot sequences in TDDMassive MIMO. When the signature/pilot sequence length isless than the number of served UEs, it is described as over-loading in CDMA while as pilot contamination in MassiveMIMO [17]–[19]. However, more emphasis is placed on thesingle-cell scenario for over-loaded CDMA, while the multicellscenario is stressed for pilot-contaminated Massive MIMO. Therecent study of user capacity in multicell CDMA has shiftedits focus to the effects of hard-handoffs instead of over-loadedchannels [20]. On the other hand, the previous investigationof downlink user capacity of multiuser MIMO in [16] is con-ducted without considering the effects of pilot contamination.In addition, the capacity depicted in the same work assumesan infinite number of UEs. With perfect CSI at the BS, up-link/downlink user capacity of multiuser MIMO with identicalSINR constraints is presented in [21]. Basically, the maximumnumber of admissible UEs increases with the BS antenna sizein the downlink, while it is governed by the pilot sequencelength in the uplink. Moreover, the optimal number of UEsfor maximizing the ergodic sum throughput can be identified.Briefly, the foregoing studies of the user capacity of multiuserMIMO have not examined the effects of pilot contaminationwith general SINR constraints. The main purpose of this studyis to develop an understanding of the mentioned effects ondownlink user capacity of Massive MIMO.

B. Contributions

The major contributions of this paper are summarized asfollows:

1) Previous research on Massive MIMO has examined theeffects of pilot contamination on the received SINRs ofindividual users. In contrast, this paper makes the firstattempt to study the impact on a group of UEs and toarticulate conditions under which their SINR targets canbe met simultaneously in the LAS regime. We show that ifthe SINR requirements are located in a prescribed admis-sible region, using our proposed joint pilot sequence andpower allocation can achieve these targets concurrentlyas the BS antenna size grows without limit, i.e., in theLAS regime. We find that the size of the prescribedregion, characterizing the user capacity, is fundamentallylimited by the length of uplink pilot sequences. If cross-correlations of pilot sequences are not well designed, thesize of this region can be further reduced.

2) We advance this research by evaluating the performancein the non-LAS regime, which provides a useful guide topractical BS antenna deployment. Specifically, we char-acterize the average SINR that is actually achieved for agiven number of BS antennas. Based on it, the study ofhow many BS antennas are needed to approach a signifi-cant percentage of SINR performance limits is presented.Generally, the number of the required antennas dependson the cross-correlations of pilot sequences and the SINRrequirements. When there exists a UE with a relativelyhigh SINR requirement, we provide a fresh insight intohow this UE will cause the demand for a larger antennasize. Simulation results will verify the accuracy of thisanalytical investigation, i.e., the prediction of the requiredantenna size.

3) Not only is the user capacity characterized as an ad-missible region of SINR requirements, but we also pro-pose a joint pilot sequence design and transmit powerallocation scheme to achieve this capacity. The detailedcomparisons between our proposed capacity-achievingscheme and other non-capacity-achieving schemes havebeen made. Particularly, the conventional pilot sequencedesign is taken into account in one of the non-capacity-achieving schemes. The results highlight the advantage ofusing the proposed scheme, i.e., the potential of admittingmore UEs into the specified Massive MIMO system. Thisadvantage becomes increasingly apparent when UEs havemore diverse SINR requirements.

C. Organization and Notations

The rest of the paper is as follows. In Section II, we specifythe system model including the uplink training method and thedownlink transmit beamforming scheme. Section III addressesthe issue of admissible users in the pilot-contaminated system.This upper bound is further translated into an admissible regionin which the SINR requirements can be satisfied by usingthe proposed joint scheme of pilot sequence design and trans-mit power allocation. The analysis in the non-LAS regime is

SHEN et al.: DOWNLINK USER CAPACITY OF MASSIVE MIMO UNDER PILOT CONTAMINATION 3185

Fig. 1. A single-cell TDD Massive MIMO network, in which the BS acquiresCSI via uplink training.

provided in the same section. Numerical results are presentedin Section IV, which is followed by the concluding remarks inSection V.

Notations: R: real number, Z: integers, ‖ · ‖p: p-norm, (·)T :transpose, (·)H : Hermitian transpose, ⊗: Kronecker product,◦: Hadamard product, IN : N ×N identity matrix, CN (·, ·):complex normal distribution, E[·]: expectation, tr(·): trace,diag(· · ·): diagonal matrix, �,�: vector inequalities, 0: zerovector, Null(·): null space, card(·): cardinality.

II. SYSTEM MODEL

Consider a single-cell Massive MIMO system, as shown inFig. 1, where a BS equipped with M antennas serves K single-antenna UEs. With TDD operation, the BS acquires downlinkCSI through uplink pilot training. The acquired CSI will thenbe utilized to form MRT precoding vectors for downlink spatialmultiplexing.

A. Uplink Training

During the uplink training phase, the ith UE transmits itsown pilot sequence si ∈ R

τ×1 with ‖si‖2 = 1. The correlationsTi sj = ρij between different training sequences ranges from−1 to 1. The pilot data over the block-fading channel, syn-chronously received at the BS, can be expressed as

y(τM×1) =

K∑i=1

(p1/2s,i Si

)(β1/2i hi

)+ z, (1)

where ps,i and βi denote respectively the pilot power and thechannel power gain, Si is the τM ×M matrix given by Si =si ⊗ IM , z is the additive white Gaussian noise distributedas CN (0, σ2

zIτM ), and hi(1 ≤ i ≤ K) are independent andidentically distributed (i.i.d.) small-scale fading channels fromUEs to the BS with distribution CN (0, IM ). This channelmodel is commonly assumed in Massive MIMO systems [6],[8], [12]. We further assume that certain uplink power controlis enabled to make the received signals from UEs be of equalpower, i.e., ps,iβi = 1, and thus a fair estimation performance

can be achieved. Without exploiting statistical knowledge of hi,the LS method is adopted to provide the channel estimate

hi =PLS,iy,

=hi +K∑j =i

ρijhj + STi z, (2)

where PLS,i = STi . This estimate indicates how the desire

channel information is polluted by undesired channels in thepilot-contaminated regime (K > τ), where ρij may not be 0for some j = i.

B. Downlink Transmission

Utilizing estimated CSI at the BS, linearly precoded sig-nals are formed and simultaneously transmitted to UEs. Thereceived signal at the ith UE is

ri = β1/2i hH

i

⎛⎝ K∑

j=1

tjxj

⎞⎠+ wi, (3)

where ti is a linear precoding vector, xi denotes uncorrelatedzero-mean data symbols with power E[xH

i xi] = Pi, and wi

is the zero-mean noise with variance σ2w. Let us devise the

ith precoding vector ti = hi/√

M(∑K

j=1 ρ2ij + σ2

z), where thedenominator ensures that ti has unit 2-norm as M growswithout limit. The fact, limM→∞

1M hH

i hi =∑K

j=1 ρ2ij + σ2

z ,can be easily proved by making use of the followingresults [12]

limM→∞

1

MhHi hj =

{0, if i = j,1, if i = j.

(4)

Such asymptotic orthogonality among MIMO channels hasbeen experimentally verified in realistic propagation environ-ments [4].

Consider that channel estimates are available at the BSand only channel statistic E[hH

i ti] is known by the ith UE.Following this, the received signal (3) is recast as

ri = β1/2i E

[hHi ti]xi︸︷︷︸

part1

+β1/2i

(hHi ti − E

[hHi ti])

xi + β1/2i hH

i

⎛⎝ K∑

j =i

tjxj

⎞⎠+ wi

︸︷︷︸part 2

,

(5)

where the second part can be regarded as the effectivenoise which is uncorrelated with the first part. As shown in[8, Theorem 1], the above expansion facilitates defining anergodic achievable rate log2(1 + SINRi,M ), where

SINRi,M =

(E[hHi ti])2

βiPi

Var[hHi ti]βiPi +

∑Kj =i E

∣∣hHi tj

∣∣2 βiPj + σ2w

,

(6)


in which Var[hHi ti] = E|(hH

i ti − E[hHi ti])|2. The following

lemma is a consequence of this definition, describing how theSINR varies with the BS antenna size.

Lemma 1: The SINRi,M defined in (6) can be expressed as

SINRi,M =βiPi

βi

(∑Kj =i ρ

2jiPj +

1M

∑Kj=1 αjPj

)+ 1

M αiσ2w

,

(7)

Proof: Please refer to Appendix A. �In the LAS regime (M → ∞), we obtain the limit form of

the effective SINR

SINRi,∞ =Pi∑K

j =i ρ2jiPj

,

=Pi

tr(sTi SDST si

)− Pi

, (8)

where D = diag(P1, · · · , PK), S = [s1, s2, · · · , sK ], the sym-bol ∞ denotes that the antenna size goes to infinity so thatthe effect of the noise wi can be ignored. This SINR expres-sion suggests that the downlink transmission operates in theinterference-limited regime because of using a large number ofantennas. Moreover, in the pilot-contaminated regime, the inter-ference part,

∑Kj =i ρ

2jiPj , can not be simultaneously eliminated

for every UE as non-orthogonal pilot sequences have to be used.

III. DOWNLINK USER CAPACITY CHARACTERIZATION

The asymptotic SINR expression, identified in the previoussection, will be further employed to examine the user capacityin the LAS regime. In this examination, K > τ will be implic-itly assumed as the pilot-contaminated regime is of interest.Therefore, the following results will show the effects of pilotcontamination on the user capacity, and can be viewed as theperformance limits due to the antenna size growing limitlessly.

A group of UEs is said to be admissible in the specified TDDMassive MIMO system if there exists a feasible pilot sequencematrix S and a power allocation vector p = [P1, · · · , PK ]T � 0such that the SINR requirements, SINRi,∞ ≥ γi for 1 ≤ i ≤K, can be jointly satisfied. A pilot sequence matrix S is claimedto be feasible if S ∈ S = {[s1, s2, · · · , sK ], si ∈ R

τ×1|‖si‖2 =1}. How the number of admissible UEs is related to the pilotlength and SINR requirements will be presented in the proposi-tion below.

Proposition 2: If K UEs are admissible in the TDD MassiveMIMO system, then

K ≤{τ

[K∑i=1

(1 +

1

γi

)]}1/2

. (9)

Proof: Please refer to Appendix B. �It is clear from this proposition that the number of admissible

UEs is fundamentally limited once the length of pilot sequencesand the SINR requirements are given. For instance, if a groupof UEs has high SINR requirements, this group is less likelyto be admissible as (9) may not hold with a given pilot length.On the other hand, (9) is always true for K ≤ τ , which is not

the case under consideration. As the normalized mean-squareerror MSEi,∞ seen by the ith UE is equivalent to 1/SINRi,∞,another explanation can be offered. A lower bound on the sumof MSEi,∞ is given by

K∑i=1

MSEi,∞ =tr(D−1STSDSTS

)−K,

≥ K2

τ−K, (10)

where (31) and (32) are applied in the second inequality. Thisbound can be immediately translated into a bound on K, i.e.,

K ≤ τ(MSE + 1), (11)

where MSE =∑K

i=1 MSEi,∞/K represents the average of thenormalized mean-square errors. Hence, we reach the conclu-sion that the intrinsic bound on K is the result of inevitablemean-square errors caused by pilot contamination.

Proposition 2 provides an upper bound for the user capacity.The next question to answer is whether K UEs are admissibleif the inequality (9) is satisfied, i.e., the achievability issue. Inother words, once the UE number is less than or equal to theupper bound, we wonder if there exists a set of S ∈ S and p �0, fulfilling the SINR requirements. The following section willshow that the answer to this question is positive.

A. Capacity-Achieving Pilot Sequence Design andTransmit Power Allocation

Validating the converse of Proposition 2 requires to identifythe pair of S and p with which we have SINRi,∞ ≥ γi for1 ≤ i ≤ K. The following proposition will specify a conditionunder which this pair, referred to as a valid combination of pilotsequence design and transmit power allocation, can exist.

Proposition 3: If

K∑i=1

(γi

1 + γi

)≤ τ, (12)

then K ≤ {τ [∑K

i=1(1 + 1/γi)]}1/2 and there exists a validcombination of pilot sequence design and transmit power al-location.

Proof: The Cauchy-Schwarz inequality gives

K∑i=1

(1 +

1

γi

)≥ K2∑K

i=1

(γi

1+γi

) ,≥ K2

τ,

which is equivalent to (9) and proves the first part of thestatement. Before going on to the second part, a definition anda lemma to be utilized later are provided.

Definition 4: Given x,y ∈ RN , x majorizes y if

n∑k=1

x[k] ≥n∑

k=1

y[k], for n = 1, · · · , N,


where x[k] and y[k] are respectively the elements of x and y indecreasing order.

Lemma 5 [22, Theorem 9.B.2]: Given x,y ∈ RN , if x ma-

jorizes y and∑N

k=1 x[k] =∑N

k=1 y[k], then there exists a realsymmetric matrix H with diagonal elements y[k] and eigenval-ues x[k].

First consider the case of∑K

i=1[γi/(1 + γi)] = τ .Given that the 1×K vector of eigenvalues e =

[λ1, · · · , λτ , 0 · · · , 0]T majorizes p and∑τ

i=1 λi =∑K

i=1 Pi,following Lemma 5, there exists a real symmetric matrixH = QΛQT , where the vector of diagonal entries of H isequal to p, Λ = diag(λ1, · · · , λτ , 0 · · · , 0), and the orthogonalmatrix Q can be presented as[

VK×τ VK×(K−τ)

].

The approach to constructing Q as well as H is provided in[23, Sec. IV-A]. Define

SΔ= Σ1/2VTD−1/2, (13)

where Σ = diag(λ1, · · · , λτ ). Then, S ∈ S is true as the diago-nal entries of STS = D−1/2HD−1/2 are equal to 1. Moreover,we have

SDST = Σ. (14)

Let us specify that

λ1 = · · · = λτ =

∑Ki=1 Pi

τ, (15)

and

Pi = cγi

1 + γi, for some c > 0. (16)

It can be verified that e majorizes p since for 1 ≤ i ≤ τ ,

λi =c∑K

i=1

(γi

1+γi

)τ

,

= c,

> max {Pk, for 1 ≤ k ≤ K} ,

where the second equality is due to the case under consideration.Next we will check if the SINR requirements are satisfied by

using such pilot sequence design and transmit power allocation.Making use of (14), we have

SINRi,∞ =Pi

tr(sTi Σsi

)− Pi

,

=c γi

1+γi

c− c γi

1+γi

,

= γi, ∀i = 1, · · · ,K,

which means that all SINR requirements are met.Now we turn to the case of

∑Ki=1[γi/(1 + γi)] < τ . As

f(x) = x/(1 + x) is monotonically increasing for x > 0, thereexists a set {γi ≥ γi for 1 ≤ i ≤ K} such that

∑Ki=1[γi/(1 +

γi)] = τ . At the same time, K ≤ {τ [∑K

i=1(1 + 1/γi)]}1/2holds. By exploiting the previous result, we can find a corre-sponding valid set of S ∈ S and p � 0. �

The valid set of S and p used in Proposition 3 is summarizedhere. The matrix S results from the generalized Welch boundequality (GWBE) sequences, satisfying SDST = cIτ for somec ∈ R

+[24]. The transmit power allocated to UEi is given byPi = cγi/(1 + γi) where γi ≥ γi and

∑Ki=1[γi/(1 + γi)] = τ .

In the case of identical SINR requirements, the pilot sequencesin use are called Welch bound equality (WBE) sequences. BothGWBE and WBE sequences have been extensively appliedfor constructing optimal CDMA signature sequences. Thesesequences also find applications in wavelet expansions andGrassmannian frames [25], [26]. In [27], a comparative evalua-tion of several sequence-construction algorithms is presented.

An explanation of the constraint,∑K

i=1[γi/(1 + γi)] ≤ τ , ispresented as follows. When a UE has a high SINR requirement,the pilot sequence assigned to this UE should be orthogonalto others so that it will receive less interference. Overall, onlyτ such assignments are allowed in the system. So the systemcan roughly admit τ UEs with high SINR requirements. Fromanother viewpoint, this constraint indicates that the desired pilotlength is lower bounded for the given K and γi. In addition, theproposed pilot sequence design and transmit power allocationcan be applied to meet the target SINRs using the minimumpilot length. A corollary which follows from Propositions 2 and3 is provided below.

Corollary 6: Given the identical SINR requirement γ, KUEs are admissible in the TDD Massive MIMO system if andonly if

K ≤(1 +

1

γ

)τ. (17)

Unlike (9), the right-hand side of (17) does not depend onK, and thus it provides an explicit upper bound of admissibleUEs. However, this is true only for identical SINR require-ments. In the general case, (9) and (12) do not provide upperbounds of this kind. To have a consistent interpretation of theuser capacity in the general case, we intend to characterizethe user capacity as the admissible region RUC = {γ1∼K ∈R

+|∑K

i=1[γi/(1 + γi)] ≤ τ}. It means that once the SINR re-quirements are located within RUC, the corresponding K UEsare admissible. When it comes to identical SINR requirements,this region maintains the same structure, i.e., RUC = {γ1∼K =γ ∈ R

+|Kγ/(1 + γ) ≤ τ}. Later on, this characterization willbe utilized to evaluate different joint pilot sequence designand power allocation schemes, i.e., different combinations of Sand p.

According to the present analytical results, the followingremarks can be made.

1) The valid combination of pilot sequence design andpower allocation is also referred to as the capacity-achieving scheme or alternatively the GWBE scheme. Itmeans that any K UEs having the SINR requirementswithin RUC can be admitted by using this scheme. Inthe next section, it will be shown that other non-capacity-achieving schemes can not guarantee this.


2) When using the capacity-achieving pilot sequence andpower allocation, the converse of Proposition 3 can beshown to be true.

3) Generally, the region specified by (9) and∑K

i=1[γi/(1 +γi)] > τ has not been characterized, in which the exis-tence of a valid set of S and p is unknown. However,when all the SINR requirements are the same, (9) turnsinto

∑Ki=1[γ/(1 + γ)] ≤ τ , making this uncharacterized

region empty.4) Following the previous remark, we can not remove the

possibility that there exists a better scheme which leads toan admissible region larger than RUC. For example, whenGWBE sequences are employed, there might exist somesuperior power allocation scheme other than the pro-posed one.

B. Comparison With Non-Capacity-Achieving Schemes

The superiority of the proposed capacity-achieving schemeover other existing schemes will be presented in this section.Let us first define two non-capacity-achieving schemes asfollows.

1) The WBE Scheme: Pilot sequences in use are the WBEsequences, having properties: SST = K

τ Iτ , and ρ2ij =(K − τ)/[(K − 1)τ ] for i = j[28]. The transmit powerPi allocated to the ith UE is given by cγi/(1 + γi) forsome c > 0.

2) The Finite Orthogonal Sequence (FOS) Scheme: Givena pilot sequence length τ , only τ orthogonal pilot se-quences will be repeatedly used in the pilot-contaminatedregime. Assume that K = qτ + r where q, r ∈ Z and 0 ≤r < τ . Each pilot sequence si is used by a collection Ei ofUEs. Let card(Ei) = q + 1 for 1 ≤ i ≤ r, card(Ei) = qfor r + 1 ≤ i ≤ τ , and Ei ∩ Ej = ∅ for i = j. This pilotdesign is commonly adopted in TDD Massive MIMO.The power allocation is the same as the one in the WBEscheme.

The main difference of these two schemes from the GWBEscheme lies in the pilot sequence design. Therefore, the com-parisons made below largely point out the benefit of using theGWBE sequences. Note that the power allocation in the WBEscheme is adopted for ease of comparison. It is possible tohave an alternative scheme such as Pi = cγi. However, howto identify the optimal power allocation corresponding to theWBE sequences remains an open question.

The following lemmas will show the potential reduction ofthe user capacity when the WBE and FOS schemes are appliedto the case of general SINR constraints.

Lemma 7: The general SINR requirements γi are satisfiedby using the WBE scheme if and only if

K∑i=1

(γi

1 + γi

)≤ min

{τ, κ− (κ− 1)

(γmax

1 + γmax

)},

(18)

where κ = (K−1)τ(K−τ) and γmax = max{γi, 1 ≤ i ≤ K}.

Proof: Please refer to Appendix C. �

Lemma 8: The general SINR requirements γi are satisfiedby using the FOS scheme if and only if∑

k∈Ei

(γk

1 + γk

)≤ 1, for 1 ≤ i ≤ τ, (19)

and

K∑i=1

(γi

1 + γi

)≤ τ. (20)

Proof: Similar to the proof of Lemma 7. �In Lemma 7, the second term in the curly brackets becomes

less than or equal to τ as γmax ≥ τ/(K − τ). When γmax � 1,(18) turns into

∑Ki=1[γi/(1 + γi)] ≤ 1 + ε ∀ ε > 0, which im-

plies potentially fewer admissible UEs as the allowable com-binations of γ1 ∼ γK become much less. From Lemma 8,it follows that the FOS scheme imposes more constraints,i.e., (19), on the admissibility of UEs. Hence, compared withthe GWBE scheme, the FOS scheme potentially can admitfewer UEs. More comparisons, taking account of the admis-sible region, among these three schemes will be provided inSection IV.

C. How Many BS Antennas Are Required?

So far this paper has focused on the LAS regime, and theprevious discussion applies provided the BS antenna size goesto infinity. This section moves on to examine the performancein the non-LAS regime. With the aid of Lemma 1, we are ableto answer the question “How many BS antennas are requiredto achieve a significant fraction of the ultimate SINRi,∞?” Thelemma below presents the required antenna sizes for three jointpilot sequence design and transmit power allocation schemes.

Lemma 9: To have SINRi,M/SINRi,∞ ≥ νi, where νi ∈(0, 1), for the GWBE scheme, the WBE scheme, and the FOSscheme, we need antenna sizes

MGWBE,i ≥βi

∑Kj=1(αjPj) + αiσ

2w[

βi

∑Kj =i

(ρ2jiPj

)] (1−νν

) , (21)

MWBE,i ≥(K − 1)

(K + τσ2

z

) (βiPtotal + σ2

w

)(K − τ)βi(Ptotal − Pi)

(1−νν

) , (22)

MFOS,i ≥

[card(Ek) + σ2

z

] (βi

∑j∈Ek

Pj + σ2w

)βi

(∑j∈Ek\{i} Pj

) (1−νν

) , (23)

respectively, where Ptotal =∑K

i=1 Pi, and i ∈ Ek is assumedfor the FOS scheme.

Proof: (21) results from Lemma 1. (22) and (23) arebasically the same as (21) except that the cross-correlations ρjiare replaced by the actual values. �

There are important implications of this result. With a higherν, the number of the required BS antennas becomes largerwhichever scheme is applied. In addition, the BS antenna sizehas to be significantly large in order to achieve 100ν% of thelimit SINRi,∞ when the transmit power allocated to a UE is


relatively higher. For example, if the ratio Pi/Ptotal approaches1 in the WBE scheme, then the corresponding MWBE increaseswithout limit. This observation suggests that if the BS antennasize is limited, UEs with similar SINR requirements, to whomsimilar powers are assigned, should be clustered together andbe served cluster by cluster. Some supplementary remarks areadded below.

1) If there is a shortfall in the transmission power budgetPtotal, leading to βiPtotal � σ2

w, then the required an-tenna size to achieve a significant fraction of SINRi,∞will become very large. This conclusion can be drawnfrom Lemma 9.

2) If SINRi,MGWBE,i ≥ νiSINRi,∞ ≥ γi holds for any indexi, then using M ≥ max{MGWBE,i, 1 ≤ i ≤ K} shouldguarantee that all downlink SINR targets are satisfiedwhen the GWBE scheme is applied. A similar argumentalso holds for the other two schemes.

3) Knowing the achievable rates that are guaranteed is in-trinsically important. Hence, a sensible alternative to thecomputation in Lemma 9 is to evaluate the requirednumber of antennas to achieve a fraction of the ergodicrate log2(1 + SINRi,∞).

IV. NUMERICAL RESULTS

In this section, comparisons between the capacity-achievingand non-capacity-achieving schemes are made. The first setof analyses examine the impact of different schemes on thesize of the admissible region of SINR requirements. Next,we investigate how target SINRs are asymptotically achievedand how many BS antennas are used to attain the desiredperformance.

A. Admissible Region Characterization in the LAS Regime

To verify the results in Proposition 3 and in Lemmas 7 and 8,we consider a pilot-contaminated Massive MIMO system withK = 6 and τ = 3. By fixing certain SINR requirements {γ4 =γ5 = γ6 = 1}, we look into admissible regions of the remainingSINR requirements given by

RGWBE =

{γ1∼3 ∈ R

+|3∑

i=1

[γi/(1 + γi)] ≤ 3/2

}, (24)

RWBE =RGWBE

∩{

3∑j=1

γ1∼3 ∈ R+|

3∑j=1

[γj/(1 + γj)]

≤ (7/2− 4γi/(1+γi)), for 1 ≤ i ≤ 3

}, (25)

and

RFOS = RGWBE ∩{γ1∼3 ∈ R

+| γi ≤ 1, for 1 ≤ i ≤ 3},

(26)

for the GWBE, WBE, and FOS schemes. Note that it is implic-itly assumed that E1 = {UE1,UE4}, E2 = {UE2,UE5}, and

Fig. 2. Upper boundaries of admissible regions for the GWBE, WBE, andFOS schemes.

Fig. 3. Achievable SINR γ versus the factor ζ for the GWBE, WBE, and FOSschemes given a fixed SINR-requirement pattern, that is {γ1 = γ2 = γ3 =γ, γ4 = γ5 = γ6 = ζγ}.

E3 = {UE3,UE6} for the FOS scheme. The upper boundariesof these regions in the positive orthant are plotted in Fig. 2.For the GWBE scheme, an extra restriction γ3 = min{γ3, 5} isplaced as the admissible γ3 can go to infinity. It can be observedthat the boundary surface of RGWBE lies well above those ofRWBE and RFOS. This implies that RGWBE contains moreadmissible points than RWBE and RFOS, so more general SINRconstraints γ1∼3 can be met in the GWBE scheme.

Fig. 3 presents the achievable SINR for different schemeswhen the pattern of the SINR requirements is specified. TheUE grouping, assumed for the FOS scheme, is given by E1 ={UE1,UE2}, E2 = {UE3,UE4}, and E3 = {UE5,UE6}. Thefactor ζ adjusts a subset of the SINR requirements, makingoverall requirements of 6 UEs more or less diverse. As shownin this figure, there is a clear trend of decreasing the achievableSINR with increasing ζ. Additionally, it is obvious that theGWBE scheme outperforms the other two in terms of achievinghigher SINRs for ζ ranging from 0.1 to 1. For ζ < 1, the advan-tage of the GWBE scheme remains significant over the FOSscheme while it becomes negligible over the WBE scheme.Compared to the FOS scheme, the GWBE scheme is more


Fig. 4. Achievable SINR versus the number K of UEs for the GWBE, WBE,and FOS schemes given a fixed SINR-requirement pattern, that is {γ1 = γ2 =γ3 = γ, γ4 = · · · = γK = γ/2}.

Fig. 5. Number of admissible UEs versus pilot sequence length for theGWBE, WBE, and FOS schemes, given a fixed SINR-requirement pattern, thatis {γ1∼l = 1/3, γ(l+1)∼2l = 1, γ(2l+1)∼3l = 3}.

favorable when we have diverse SINR requirements, i.e, whenζ approaches 0.1 or 10.

To explore the effects of having different numbers of UEs,Fig. 4 plots the achievable SINR versus the number of UEswhen fixing τ = 3. For the FOS scheme, the grouping amongUEs for any given K is assumed to be optimal in the senseof maximizing the achievable SINR. It can be observed thatincreasing K, making pilot contamination more serious, willdecrease achievable SINRs for all three schemes. Our pro-posed GWBE scheme, however, attains relatively higher SINRscompared with the WBE and FOS schemes. Interestingly, theWBE scheme does not always outperform the FOS scheme forK < 7, but does so for K ≥ 7. This highlights that the GWBEscheme exhibits a consistent superiority over the FOS schemecompared with the WBE scheme.

By specifying the SINR-requirement pattern of K = 3l UEs,how many UEs are admissible for a given pilot length isdepicted in Fig. 5. It can be observed that the number ofadmissible UEs scales almost linearly with the pilot lengthwhatever scheme is adopted. This linear relationship directly

Fig. 6. Achieved SINR1,M for a given number of BS antennas when usingdifferent joint pilot sequence and power allocation schemes.

demonstrates how the user capacity is limited by the pilotlength. Also shown in the same figure, the GWBE scheme,without doubt, substantially outperforms the other two schemesin terms of admitting more UEs. In addition, two non-capacity-achieving schemes exhibit comparable user capacities espe-cially with short pilot lengths.

B. Performance Comparison in the Non-LAS Regime

The results discussed in Section III-C will be numericallyevaluated in this section. The system of interest still operates inthe pilot-contaminated regime with K = 6 > τ = 3. The noisepowers, σ2

z and σ2w, at the BS and UEs, and all channel power

gains βi are set to 1. Three UE sets for the FOS scheme are thesame as those specified in the previous section.

We first consider a set of SINR requirements γ1 ∼ γ6, givenby {5/2, 3/2, 2/3, 2/3, 3/8, 2/9}. This SINR set is located inthe admissible region for the GWBE scheme, while outsidethose for the WBE scheme and FOS scheme. The transmitpower allocated to the ith UE is Pi = cγi/(1 + γi) for the WBEscheme and FOS scheme and Pi = cγi/(1 + γi) for the GWBEscheme, where c = 10 and γ1∼6 = {3, 2, 1, 1, 1/2, 1/3}. Fig. 6presents the SINRi,M of UE1 achieved by different schemesfor a given antenna size M . It can be seen that the simulated(dashed, dotted, dash-dot) curves due to the average over100 000 channel realizations fit well with the analytical (circle-marker) values obtained from (7). Hence, the result in Lemma 1is verified. Another observation is that the SINR1,M curves ofthe WBE scheme and FOS scheme can not approach the targetSINR by increasing the antenna size. It is because that bothSINR1,∞ = 1.92 for the WBE scheme and SINR1,∞ = 1.19for the FOS scheme are less than γ1. In contrast, the GWBEscheme, having SINR1,∞ = 3 greater than γ1, can achieve thetarget SINR with a finite number of BS antennas.

Fig. 7 shows the SINRi,M curves of different UEs for theGWBE scheme and WBE scheme. We can see that the curvesof SINR3,M and SINR6,M reach their own target SINRs re-spectively for the WBE scheme. This result means that as theset γ1 ∼ γ6 falls outside the admissible region for the WBEscheme, only partial SINR requirements can be satisfied by this


Fig. 7. Average received SINRs at UEs versus the number of BS antennas forthe GWBE, WBE, and FOS schemes.

scheme. For UEs with lower SINR constraints, such as UE3 andUE6, it can be observed that the difference between the resultsdue to two schemes becomes less significant. Next, the GWBEscheme is taken as an example for demonstrating the accuracyof results in Lemma 9. The SINR limits for this scheme aregiven by SINRi,∞ = γi > γi for 1 ≤ i ≤ 6. If 83.3% of thelimit SINR1,∞ is achieved, the corresponding SINR require-ments γ1 will be met. Based on (22), the required antennas sizeis expected to be MGWBE ≥ 186 for UE1. Similarly, we needto achieve 66.6% of the limit SINR3,∞, resulting in MGWBE ≥38 for UE3. This analytical inference is verified in Fig. 7,where the simulated SINRi,M and target γi curves intersectat the points around M = 196 for UE1 and M = 38 for UE3.From then onwards, the simulated SINRi,M is greater than thetarget SINR.

V. CONCLUSION

This paper has investigated the user capacity of down-link TDD Massive MIMO systems in the pilot-contaminatedregime. The necessary condition for admitting a group of UEswith general SINR requirements has been provided. It hasshown an intrinsic capacity upper bound due to the limitedlength of pilot sequences. Meanwhile, the capacity-achievingGWBE scheme, which can achieve the identified user capacityand satisfy the SINR requirements, has been proposed and com-pared with the non-capacity-achieving WBE and FOS schemes.The results of this study indicate that the capacity-achievingGWBE scheme is superior in terms of enhancing higher usercapacity.

The current investigation has only examined the single-cell scenario. More study is needed to better understand theuser capacity in the multicell scenario. Future research mightexplore the capacity region under intra- and inter-cell pilotcontamination, the corresponding capacity-achieving pilot se-quence design and transmit power allocation, and an efficientmethod for performing intercell resource allocation.

APPENDIX APROOF OF LEMMA 1

First, we have∣∣∣hHi hj

∣∣∣2=

(K∑

m=1

ρjmhHi hm + hH

i SHj z

)H

×(

K∑n=1

ρjnhHi hn + hH

i SHj z

),

=

K∑m=1

K∑n=1

ρjmρjnhHmhih

Hi hn +

K∑m=1

ρjmhHmhih

Hi SH

j z

+

K∑n=1

ρjnzHSjhih

Hi hn + zHSjhih

Hi SH

j z.

Then,

E

[∣∣∣hHi hj

∣∣∣2]

=K∑

m=1

K∑n=1

ρjmρjnE[hHmhih

Hi hn

]+ E

[zHSjhih

Hi SH

j z],

= ρ2jiE[hHi hih

Hi hi

]+

K∑m=n=i

ρ2jmE[hHmhih

Hi hm

]

+

K∑m =n

ρjmρjnE[hHmhih

Hi hn

]+E

[tr(Sjhih

Hi SH

j zzH)]

,

= ρ2ji(M2 +M) +

K∑m =i

ρ2jm [tr (RhiRhm

)]

+ tr(Rhi

SHj RzSj

),

= ρ2ji(M2 +M) +M

K∑m =i

ρ2jm +Mσ2z ,

= M2ρ2ji +Mαj , (27)

where αj =∑K

m=1 ρ2jm + σ2

z , Rhiand Rz respectively denote

the covariance matrices of hi and z, and E[hHi hih

Hi hi] =

(M2 +M) because of hHi hi being gamma-distributed

Γ(M, 1).By definition,

SINRi,M

=

(E[hHi hi])

2βiPi

Mαi

βi

(E|(hH

ihi−E[hH

ihi])|2Pi

Mαi+∑K

j =i

E|hHihj|2Pj

Mαj

)+ σ2

w

,

=M2

MαiβiPi

βi

[Mαi

MαiPi +

∑Kj =i

MMαi

(Mρ2ji + αj

)Pj

]+ σ2

w

,

=βiPi

βi

[αi

M Pi +∑K

j =i1M

(Mρ2ji + αj

)Pj

]+ αi

M σ2w

,

=βiPi

βi

(∑Kj =i ρ

2jiPj +

∑Kj=1

αj

M Pj

)+ αi

M σ2w

, (28)

where E[hHi hi] = M , and Var[hH

i hi] = Mαi.


APPENDIX BPROOF OF PROPOSITION 2

Making use of (8), we haveK∑i=1

1 + SINRi,∞SINRi,∞

=

K∑i=1

1

Pitr(sTi SDST si

),

=tr(D−1STSDSTS),

=tr(D−1/2GsDGsD

−1/2), (29)

where

GsΔ=STS,

=

⎡⎢⎢⎢⎢⎣

1 ρ12 ρ13 · · · ρ1Kρ12 1 ρ23 · · · ρ2Kρ13 ρ23 1 · · · ρ3K

......

.... . .

...ρ1K ρ2K ρ3K · · · 1

⎤⎥⎥⎥⎥⎦ . (30)

Also, we can expand the trace in (29) and obtain its lower boundas below

tr(D−1/2GsDGsD

−1/2)

= K +K∑i=1

K∑j>i=1

(Pi

Pj+

Pj

Pi

)ρ2ij ,

≥ K +

K∑i=1

K∑j>i=1

2ρ2ij ,

= tr (GsGs) , (31)

where the inequality is due to (Pi/Pj + Pj/Pi) ≥ 2. The Grammatrix Gs has an eigendecomposition UDGUT , where U is aunitary matrix and DG = diag(d1, · · · , dK) with d1 ∼ dτ > 0,dτ+1 ∼ dK = 0, and

∑τi=1 di = K. Then, we have

tr(GsGs) = tr(UD2

GUT),

=

τ∑i=1

d2i ,

≥ 1

τ

(τ∑

i=1

di

)2

=K2

τ. (32)

As

K∑i=1

1 + SINRi,∞SINRi,∞

≤K∑i=1

1 + γiγi

, (33)

the desired inequality follows.

APPENDIX CPROOF OF LEMMA 7

The SINR requirement

Pi∑Kj =i ρ

2jiPj

=Pi∑K

j=11κPj − 1

κPi

,

=c γi

1+γi

cκ

(∑Kj=1

γj

1+γj

)− c

κγi

1+γi

≥ γi,

can be recast as

K∑j=1

γj1 + γj

≤ κ− (κ− 1)

(γi

1 + γi

). (34)

Let us sum up (34) over i = 1, · · · ,K, i.e.,

K∑i=1

K∑j=1

γj1 + γj

=K

K∑j=1

γj1 + γj

,

≤K∑i=1

[κ− (κ− 1)

(γi

1 + γi

)],

=Kκ− (κ− 1)

(K∑i=1

γi1 + γi

),

which gives

K∑j=1

γj1 + γj

≤ Kκ

K + κ− 1= τ. (35)

The desired result follows from (34) and (35).

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Juei-Chin Shen (S’10–M’14) received the Ph.D. de-gree in communication engineering from Universityof Manchester, U.K., in 2013. In the same year, hejoined the Department of Electronic and ComputerEngineering, Hong Kong University of Science andTechnology (HKUST), as a Research Associate. Heis a co-recipient of IEEE PIMRC 2014 Best Pa-per Award. His research interests include statisticalsignal processing, cognitive radio, spectrum sens-ing, compressed sensing, massive MIMO systems,millimeter-wave communications, and random ma-

trix theory.

Jun Zhang (S’06–M’10) received the B.Eng. de-gree in electronic engineering from the Universityof Science and Technology of China, in 2004, theM.Phil. degree in information engineering from theChinese University of Hong Kong, in 2006, andthe Ph.D. degree in Electrical and Computer En-gineering from the University of Texas at Austin,in 2009. He is currently a Research Assistant Pro-fessor in the Department of Electronic and Com-puter Engineering, Hong Kong University of Scienceand Technology (HKUST). He co-authored the book

Fundamentals of LTE (Prentice-Hall, 2010). He received the 2014 Best PaperAward for the EURASIP Journal on Advances in Signal Processing, and thePIMRC 2014 Best Paper Award. He served as a MAC track co-chair forIEEE WCNC 2011. His research interests include wireless communicationsand networking, green communications, and signal processing.

Khaled B. Letaief (S’85–M’86–SM’97–F’03) re-ceived the B.S. degree (with distinction), and theM.S. and Ph.D. degrees, all in electrical engineering,from Purdue University in 1984, 1986 and 1990,respectively.

From January 1985 and as a Graduate Instructorat Purdue, he taught courses in communications andelectronics. From 1990 to 1993, he was a facultymember at the University of Melbourne, Australia.Since 1993, he has been with the Hong Kong Uni-versity of Science and Technology (HKUST) where

he is currently Chair Professor and the Dean of Engineering, with expertise inwireless communications and networks. In these areas, he has over 500 journaland conference papers and has given invited keynote talks as well as courses allover the world. He has 13 patents including 11 US patents.

Dr. Letaief serves as a consultant for different organizations and isthe founding Editor-in-Chief of the IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS. He has served on the editorial board of other pres-tigious journals including the IEEE JOURNAL ON SELECTED AREAS IN

COMMUNICATIONS—WIRELESS SERIES (as Editor-in-Chief). He has beeninvolved in organizing a number of major international conferences. Theseinclude WCNC’07 in Hong Kong; ICC’08 in Beijing; ICC’10 in Cape Town;TTM’11 in Hong Kong; and ICCC’12 in Beijing.

Professor Letaief has been a dedicated teacher committed to excellence inteaching and scholarship. He received the Mangoon Teaching Award fromPurdue University in 1990; the HKUST Engineering Teaching ExcellenceAward; and the Michael Gale Medal for Distinguished Teaching (HighestUniversity-wide Teaching Award at HKUST). He is also the recipient of manyother distinguished awards including the 2007 IEEE Communications SocietyPublications Exemplary Award; the 2009 IEEE Marconi Prize Award in Wire-less Communications; the 2010 Purdue University Outstanding Electrical andComputer Engineer Award; the 2011 IEEE Communications Society HaroldSobol Award; the 2011 IEEE Wireless Communications Technical CommitteeRecognition Award; and 11 IEEE Best Paper Awards.

Dr. Letaief is a Fellow of HKIE. He has served as an elected member ofthe IEEE Communications Society (ComSoc) Board of Governors, as an IEEEDistinguished lecturer, IEEE ComSoc Treasurer, and IEEE ComSoc Vice-President for Conferences.

He is currently serving as the IEEE ComSoc Vice-President for TechnicalActivities as a member of the IEEE Product Services and Publications Board,and is a member of the IEEE Fellow Committee. He is also recognized byThomson Reuters as an ISI Highly Cited Researcher.

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