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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 12, DECEMBER 2015 4849

Achievable Downlink Rates of MRC and ZFPrecoders in Massive MIMO With Uplink and

Downlink Pilot ContaminationAmin Khansefid, Student Member, IEEE, and Hlaing Minn, Senior Member, IEEE

Abstract—Massive multi-input multi-output (MIMO) systemsrequire channel state information (CSI) at base stations. In multi-cell environments, adjacent cells using the same spectrum cause apilot contamination problem and degrade CSI quality. Under suchpractical environments, it is important to assess achievable rateof the system for a proper system design. There exist achievablerate bounds of uplink (UL) systems in the literature, but generalclosed-form rate bounds for downlink (DL) systems are not avail-able yet. This paper studies a practical massive MIMO DL systemwith or without pilot-aided coherent detection under the sce-nario of pilot contamination, and derives closed-form approximateachievable DL rate expressions for both maximum ratio combin-ing (MRC) and zero forcing (ZF) precoders. For the case withDL pilot, a corresponding DL effective channel estimator is alsodeveloped. Numerical simulation results corroborate accuracy ofthe closed-form rate expressions and they also show performancecharacteristics of pilot-contaminated massive MIMO DL systemswith MRC and ZF precoders.

Index Terms—Massive MIMO, downlink achievable rate, pilotcontamination, channel estimation, imperfect CSI.

I. INTRODUCTION

M ASSIVE MIMO is a promising technology for nextgeneration wireless systems [1], [2] due to its advan-

tages in enhancing spectrum and energy efficiency [3]–[6].Marzetta shows in [5] that with the use of many antenna ele-ments at the base station (BS), several users can communicatewith BS simultaneously on the same spectrum and time. Withthe use of CSI at BS and by means of asymptotic orthogonalityof independent random vectors of large size, the interferenceamong the simultaneous users is canceled and at the sametime small-scale channel fading effects are averaged out. Theseproperties and potentials of massive MIMO have motivatedsubstantial further research work [7]–[17].

The very large signal vector dimension at a massive MIMOBS favors low complexity algorithms, e.g., linear detectorsbased on MRC, ZF or minimum mean squared error (MMSE)[6]–[8] for the UL and linear pre-coding/beamforming such asMRC and ZF [6], [9], [10] for the DL. Nonlinear detectors[18]–[20] and nonlinear pre-coders such as dirty-paper codingand vector perturbation [21] offer better performance but with

Manuscript received December 17, 2014; revised May 9, 2015 and August 3,2015; accepted September 18, 2015. Date of publication September 28, 2015;date of current version December 15, 2015. The associate editor coordinatingthe review of this paper and approving it for publication was Z. Zhang.

The authors are with the Department of Electrical Engineering,University of Texas at Dallas, Richardson, TX 75080 USA (e-mail:[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/TCOMM.2015.2482965

higher complexity. However, as the number of BS antennas (M)becomes large, linear pre-coding and detection techniques suchas MRC and ZF become near optimal [6].

In assessing performance of various massive MIMO sce-narios, achievable rate expressions or their bounds serve asimportant metrics. In the UL scenario, [7] derives rate boundsfor MRC, ZF and MMSE detectors and analyzes power effi-ciency mainly considering a single cell system. For multicellsystems, UL rate bounds are derived for a ZF receiver in [22]and for MRC and ZF receivers in [12]. The UL rate bounds ofmassive MIMO systems in Ricean fading channels are investi-gated in [13] using MRC and ZF receivers under perfect andimperfect CSI. The accuracies of rate bounds for single cellmassive MIMO UL are evaluated in [23] by deriving upper andlower bounds.

In the DL scenario, the existing works on achievable ratebounds [8]–[11] consider a particular receiver structure whichtreats the channel mean as its effective channel for data detec-tion. Single cell DL systems are addressed in [9], [11]. Ratederivation for both UL and DL of multicell systems in [8] isbased on asymptotic rate analysis when the numbers of basestation antennas and users go to infinity, but our analysis is fora finite number of base station antennas. Another difference is,for downlink transmission we consider both scenarios of withand without downlink pilot. Furthermore, in downlink precod-ing, [8] uses the average total transmit power normalization,while we use per-user downlink power normalization (whichoffers user fairness). Multicell DL scenarios with total transmitpower normalized precoder are considered in [10] and it doesnot consider downlink pilot transmission and downlink effec-tive channel estimation. In addition, there is no closed formrate expression, except for a simple scenario of single user ineach cell with MRC precoding. In our paper, we also considerrate analysis without downlink pilot and with the assumptionof average effective channel gain available at the user. Butwe derive the closed form rate expressions for this case withMRC and ZF precoders. In addition, we also derive closedform rate expressions for systems with downlink pilot for bothMRC and ZF precoders. [24] considers single cell DL transmis-sion for MRC and ZF precoding with average transmit powernormalization and DL pilot.

An important issue in massive MIMO as mentioned in [5] ispilot contamination. The issue arises in multicell environmentswhen users in different cells transmit the same pilot set bywhich the estimated channel is corrupted by channels of usersin other cells. To address this issue, [10] proposes an MMSEprecoding scheme while a review of available techniques is

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4850 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 12, DECEMBER 2015

reported in [4]. Pilot contamination affects both UL and DLperformance, and hence achievable rate expressions under pilotcontamination are important for providing guidance in sys-tem design. Deriving closed-form rate expressions for DL ismore challenging since both UL pilot contamination and DLpilot contamination plague the DL system. To the best of ourknowledge, simple closed-form rate expressions for massiveMIMO DL with an arbitrary number of BS antennas under pilotcontamination are not available in the literature.

In this paper, we study massive MIMO DL transmission withMRC or ZF precoding in a multicell scenario with pilot contam-ination effect. As a common scenario of massive MIMO, weassume time-division duplexing (TDD) which offers channelreciprocity and saves pilot overhead. We consider two sce-narios, namely without DL pilot and with DL pilot. For thelatter scenario, we also develop a corresponding linear MMSE(LMMSE) estimator for the DL effective channel. Our maincontribution is the development of closed-form approximatingexpressions for the achievable rates of the considered multicellmassive MIMO DL system with a finite number of BS antennasunder imperfect CSI and pilot contamination.

The paper is organized as follows. In Section II, we describethe system model, UL channel estimation, and DL transmis-sion. Then in Section III, we drive DL rate bounds for systemswithout DL pilot. In Section IV, DL pilot transmission andDL effective channel estimation are presented, and we derivethe corresponding DL rate bounds and their approximations.Section V discusses numerical results and Section VI providesconclusions.

Notation: Vectors (matrices) are denoted by bold face small(big) letters. The superscripts T , H , and ∗ stand for the trans-pose, conjugate transpose and conjugate of a matrix or vector,respectively. IK is the K × K identity matrix. E{·}, ‖.‖, �{.}and �{.} denote expectation, Euclidean norm, real part andimaginary part, respectively. n ∼ CN(0,C) means n has aprobability density function (pdf) of the zero-mean complexGaussian vector with covariance matrix C. N(0, σ 2) denotesa zero-mean Gaussian pdf with variance σ 2. The δki and �(·)denote the Kronecker delta and the Gamma function, respec-tively. diag{a1, · · · , aK } means a diagonal matrix with diagonalelements {a1, · · · , aK }.

II. SYSTEM MODEL

We consider a single-carrier multi-cell system with L cellswhere each cell has a BS with co-located M antenna ele-ments and K randomly located single antenna users. We assumechannel gains are quasi-static within a frame, and channels ofdifferent users and antennas are independent. Each user expe-riences a frequency-flat fading channel. Let gilkm represent thechannel gain from user k in cell l to the antenna m of the BS incell i (or simply, of BS i). We can write gilkm = √

βilkhilkm

where hilkm ∼ CN(0, 1) is the small scale fading coefficientand βilk is the large scale fading power gain (the same forall channels between user k in cell l and antennas at BS i ,thus the index m is omitted). The overall M × K lowpassequivalent UL channel matrix is denoted by Gil , whose kthcolumn gilk represents the gains of the channels from user k

in cell l to BS i and has a CN(0, βilkIM ) distribution. In thesame way as [8], [10], [12], we assume {βilk, i, l = 1, · · · , L ,k = 1, · · · , K } are known at all BSs, and this assumption isjustified since {βilk} change very slowly compared with {hilkm}and they can be reliably estimated. The same as in the literature,we treat {βilk} as deterministic in our channel estimation.

The above assumption of each base station knowing indi-vidual {βilk} of other cells is in fact not necessary and therequired information for the MMSE channel estimator can beeasily obtained as in [25] which shows almost the same esti-mation performance as the MMSE with perfect knowledge of{βilk}. Thus, information sharing between base stations such asin network MIMO is not necessary. On the other hand, compar-ison of massive MIMO and network MIMO, or combination ofmassive MIMO and network MIMO under different levels ofinformation sharing between base stations is an interesting andseparate research topic.

We also note that the rate analysis in this paper is developedunder equal power allocation among the users. If an appro-priate user power allocation strategy is applied based on theknowledge of large scale fading coefficients, the achievable ratewould be higher. The problem of power allocation for singlecell massive MIMO is studied in [26] while the problem for theconsidered system is a challenging issue which needs a separateinvestigation.

A. Uplink Training

Each user in UL mode transmits τu pilot symbols and theneach BS estimates the channels of its users. We assume users ofdifferent cells transmit the same set of pilots at the same time(a typical scenario in massive MIMO) and the pilot reuse factoris one. The pilot sequences of K users are represented by aK × τu matrix �H

u with orthogonality property �Hu �u = τuIK ,

τu ≥ K . The received pilot symbols at BS i are represented byan M × τu matrix Yi as

Yi =L∑

l=1

√quGil�

Hu + Ni (1)

where qu is the UL pilot power and Ni is an M × τu noisematrix with independent and identically distributed (i.i.d.) ele-ments of CN(0, 1). If gilk is to be estimated at BS i , a sufficientstatistic [27] is

zik � 1

τu√

quYiφuk =

L∑l=1

gilk + CN

(0,

1

τuquIM

)(2)

where φuk denotes the kth column of �u. The MMSE estima-tion is given by [27]

gilk � bilkzik = bilk

L∑l=1

gilk + CN

(0,

b2ilk

τuquIM

)(3)

where bilk � βilk∑Ll′=1 βil′k+ 1

τuqu

. We can write gilk = gilk + εilk ,

where the channel estimate gilk and the estimation error εilk are

KHANSEFID AND MINN: ACHIEVABLE DOWNLINK RATES OF MRC AND ZF PRECODERS IN MASSIVE MIMO 4851

independent due to the MMSE property and their Gaussianity.We have gilk ∼ CN(0, σ 2

gilkIM ) and εilk ∼ CN(0, σ 2

εilkIM )with

σ 2gilk

�β2

ilk∑Ll ′=1 βil ′k + 1

τuqu

(4)

σ 2εilk

�βilk

(∑Ll ′=1,l ′ =l βil ′k + 1

τuqu

)∑L

l ′=1 βil ′k + 1τuqu

. (5)

Increasing qu decreases σ 2εilk

but enlarges σ 2gilk

while a larger

βilk or βil ′k increases σ 2εilk

. Note that σ 2gilk

+ σ 2εilk

= βilk . From(5), we see that due to pilot contamination, increasing UL pilotpower cannot reduce the Mean Square Error (MSE) to zero.From (3), we can see

gilk = βilk

βi ikgi ik (6)

and as εilk is independent of gilk , it is also independent of anyfunction of gilk , such as gi ik . Note that BS i only needs to esti-mate {gi ik, k = 1, · · · , K }, and the role of (6) is to facilitate therate analysis by decomposing the inter-cell interference into acorrelated term (due to gilk) and an uncorrelated term (due toεilk) with respect to the desired user’s channel estimate.

B. Downlink Transmission

BS l linearly precodes data of its users through an M × Kprecoding matrix Wl and transmits them. The kth column ofWl is denoted by wlk with ‖wlk‖ = 1. The received signal atuser k in cell i is given by

rik = √p

L∑l=1

gTlikWlsl + nik (7)

where sl = [sl1, · · · , sl K ]T is the complex data vector of usersin cell l with E{sl} = 0 and E{slsH

l } = IK , nik ∼ CN(0, 1) isthe noise term, and p is the transmit data power which alsorepresents the transmit signal-to-noise power ratio (Tx SNR)of each user. So the total transmit power at BS is K p. Weexpand (7) as

rik = √pgT

iikwiksik +K∑

j=1, j =k

√pgT

iikwi j si j

︸︷︷︸intra-cell interference

+L∑

l=1,l =i

K∑j=1

√pgT

likwl j sl j

︸︷︷︸inter-cell interference

+nik . (8)

We define wik � g∗i ik

‖gi ik‖ for MRC precoding and wik �a∗

ik‖aik‖ for ZF precoding where aik is the kth column of Ai �Gi i (GH

ii Gi i )−1 so that aH

ik gi i j = δ jk . The received signals atuser k in cell i with MRC and ZF precoding are respectivelygiven by

rmrcik = √

pgH

iik∥∥gi ik∥∥gi iksik +

K∑j=1, j =k

√p

gHii j∥∥gi i j∥∥gi iksi j

+L∑

l=1,l =i

K∑j=1

√p

gHll j∥∥gll j∥∥gliksl j + nik (9)

r zfik = √

paH

ik

‖aik‖gi iksik +K∑

j=1, j =k

√p

aHi j∥∥ai j∥∥gi iksi j

+L∑

l=1,l =i

K∑j=1

√p

aHl j∥∥al j∥∥gliksl j + nik . (10)

III. DL RATE ANALYSIS WITHOUT DL PILOT

When the users do not estimate their DL channels, followingthe signal model given in (8), the achievable rate lower boundof user k in cell i can be given by [10, Theorem 1]

Rik,np = λnp log2

⎛⎜⎜⎝1 +

∣∣E {√pgTiikwik

}∣∣2E

{∣∣∣ Iik

∣∣∣2}⎞⎟⎟⎠ (11)

where λnp accounts for pilot overhead (will be discussed inRemark 7), the expectation is over all random variables, and

Iik = √p(

gTiikwik−E

{gT

iikwik

})sik+

K∑j=1, j =k

√pgT

iikwi j si j

+L∑

l=1,l =i

K∑j=1

√pgT

likwl j sl j + nik . (12)

Here, users treat the mean effective channel gain as the channelknowledge for data detection. Note that [10] does not providea general closed form expression for (11). In the following, wedrive closed form expressions for DL rate bounds.

Theorem 1: The closed form of the rate bound in (11) withMRC precoding reads as

Rmrcik,np = λnp log2

(1 +

C2Mσ

2gi ik

Imrcik,np

)(13)

where Imrcik,np�VMσ

2gi ik

+σ 2εi ik

+(K−1)βi ik+K∑L

l=1,l =i βlik+1p +(M−1)

∑Ll=1,l =i σ

2glik

=(VM−1)σ 2gi ik

+K∑L

l=1 βlik + 1p +

(M − 1)∑L

l=1,l =i σ2glik

, and CM and VM are given by

CM � � (M + 1/2)

�(M)(14)

VM � M − C2M . (15)

For a moderate or large M , CM ≈ √M and VM ≈ 0.25 [10].

Proof: See Appendix A. �Theorem 2: The closed form of the rate bound in (11) with

ZF precoding reads as


Rzfik,np = λnp log2

(1 +

C2M−K+1σ

2gi ik

Izfik,np

)(16)

where Izfik,np � VM−K+1σ

2gi ik

+ K∑L

l=1 σ2εlik

+ (M − K + 1)∑Ll=1,l =i σ

2glik

+ 1p while CM−K+1 and VM−K+1 are given in

(14) and (15).

Proof: See Appendix B. �Remark 1: As M → ∞ with a fixed K , the rates in (13) and

(16) approach the same rate as

Rmrcik,np, Rzf

ik,np → λnp log2

⎛⎝1 +

σ 2gi ik∑L

l=1,l =i σ2glik

⎞⎠

︸︷︷︸�R∞

ik

. (17)

IV. DL RATE ANALYSIS WITH DL PILOT

From (9) for MRC, the effective DL channel gain isgH

iik‖gi ik‖gi ik , but computing interference power conditioned on theestimate of this complex channel gain is analytically intractable.To overcome this, by replacing gi ik with gi ik + εi ik , we rewrite

the effective channel gain as ‖gi ik‖ + gHiik

‖gi ik‖εi ik , and with obser-vation that the power of the second term is negligible comparedto the power of the first one, we propose to treat ‖gi ik‖ as the

effective channel gain and treat thegH

iik‖gi ik‖εi ik as part of the inter-

ference. This approach is well justified since the power ratio of

these two terms isMσ 2

gi ikσ 2

εi ik

= Mβi ik∑Ll=1,l =i βilk+ 1

τuqu

which is large1

especially for a large M . The same reasoning holds for thedesired signal term in (10) for ZF precoding. So, we definethe effective channel for MRC and ZF precoding scenario asαmrc

ik � ‖gi ik‖ and αzfik � 1

‖aik‖ , respectively.In systems with DL pilot, each user estimates the effective

channel αik and performs coherent detection. With the channelestimate αik and its corresponding error νik (see Lemma 3 and6 for their statistics), we can write αik = αik + νik . Then, thereceived signal at user k in cell i with MRC precoding is

rmrcik = √

p αmrcik sik + I mrc

ik (18)

where the interference plus noise term is

I mrcik � √

pνmrcik sik + √

pgH

iik∥∥gi ik∥∥εi iksik

+K∑

j=1, j =k

√p

gHii j∥∥gi i j∥∥gi iksi j

+L∑

l=1,l =i

K∑j=1

√p

gHll j∥∥gll j∥∥gliksl j + nik . (19)

The corresponding received signal for ZF precoding is

r zfik = √

p αzfik sik + I zf

ik (20)

1e.g., with βi ik = 1, βilk = 0.1 (l = i), qu = 10, L = 7, and M in the rangeof 30–100, the ratio is in the range of 50–164.

where

I zfik � √

pνzfiksik + √

paH

ik

‖aik‖εi iksik +K∑

j=1, j =k

√p

aHi j∥∥ai j∥∥gi iksi j

+L∑

l=1,l =i

K∑j=1

√p

aHl j∥∥al j∥∥gliksl j + nik . (21)

A. Downlink Channel Estimation

To estimate the effective DL channel for coherent detection,BSs in different cells transmit the same set of orthogonal pilotsequences over τd symbol intervals. The pilot set for K usersis collected in a K × τd matrix �H

d which has the orthogonalproperty �H

d �d = τdIK , τd ≥ K . The received pilot vector atuser k in cell i is

yTik = √

qd

L∑l=1

gTlikWl�

Hd + nT

ik (22)

where qd is the DL pilot power. For estimation of αik , asufficient statistic is given by

zik �yT

ikφdk

τd√

qd= gT

iikwik +L∑

l=1,l =i

gTlikwlk+CN

(0,

1

τdqd

)(23)

where φdk is the kth column of �d. Using (23), we developeffective DL channel estimators.

1) System with MRC Precoding: In this case, the signal in(23) is rewritten as

zmrcik = αmrc

ik + wmrcik (24)

where αmrcik = ‖gi ik‖ is the effective channel gain and

wmrcik �

gHiik∥∥gi ik∥∥εi ik+

L∑l=1,l =i

gHllk∥∥gllk∥∥glik+CN

(0,

1

τdqd

)(25)

is the interference plus noise term. Note that pilot contamina-tion causes correlation between the precoding matrices of theinterfering BSs and their channels to the desired user, which inturn results in higher interference power in the real domain thanthe imaginary domain. This non-circular symmetric character-istics of inter-cell interference yields different statistics betweenthe real and imaginary parts of wmrc

ik . The following lemmadescribes more details.

Lemma 1: The real part wmrc,reik and the imaginary part

wmrc,imik of wmrc

ik are independent and

E{w

mrc,reik

} = CM

L∑l=1,l =i

√σ 2

glik(26)

E

{w

mrc,imik

}= 0 (27)


var{w

mrc,reik

} = 1

2

L∑l=1

σ 2εlik

+ 1

2τdqd+ VM

L∑l=1,l =i

σ 2glik

(28)

var{w

mrc,imik

}= 1

2

L∑l=1

σ 2εlik

+ 1

2τdqd. (29)

Furthermore, wmrc,imik is Gaussian and w

mrc,reik for a large

value of M approaches Gaussian. We can rewrite (28)as var{wmrc,re

ik } = 12 (σ

2εi ik

+ 1τdqd

)+ ( 12 − VM )

∑Ll=1,l =i σ

2εlik

+VM

∑Ll=1,l =i βlik .

Proof: See Appendix C. �Lemma 1 implies that the interference power var{wmrc,re

ik }decreases with the increase of qu or qd or decrease of βilk . Thenext Lemma describes some statistics of αmrc

ik .Lemma 2: Mean and variance of the DL effective channel

gain αmrcik are respectively given by

E{αmrc

ik

} = E{∥∥gi ik

∥∥} = CM

√σ 2

gi ik(30)

var{αmrc

ik

} = VMσ2gi ik

(31)

and for a large value of M , αmrcik approaches Gaussian.

Proof: See Appendix D. �From Lemma 2, following the approximation of CM ≈√M and VM ≈ 0.25, we can see E{αmrc

ik } increases with√

Mwhile var{αmrc

ik } is approximately fixed. So for a large M ,E{αmrc

ik }√var{αmrc

ik } ≈ 2√

M is large and thus, relative fluctuations of

αmrcik are quite small. In addition, both mean and variance ofαmrc

ik increase with qu.By exploiting Lemma 1, we can estimate αmrc

ik from the realpart zmrc,re

ik of zmrcik where

zmrc,reik = αmrc

ik + wmrc,reik . (32)

Lemma 3: For a system with MRC precoding, the LMMSEestimator αik based on (32) and its MSE ηmrc

ik are respectivelygiven by

αmrcik = cmrc

ik + f mrcik zmrc,re

ik (33)

ηmrcik � E

{∣∣αmrcik − αmrc

ik

∣∣2} = var{αmrc

ik

} · var{w

mrc,reik

}var{αmrc

ik } + var{wmrc,reik } (34)

where

cmrcik �

var{w

mrc,reik

}E{αmrc

ik

}− var{αmrc

ik

}E{w

mrc,reik

}var

{αmrc

ik

}+ var{w

mrc,reik

} (35)

f mrcik �

var{αmrc

ik

}var

{αmrc

ik

}+ var{w

mrc,reik

} . (36)

Denoting the estimation error by νmrcik , we write αmrc

ik = αmrcik +

νmrcik . Then, for a large M , LMMSE estimator becomes MMSE

estimator, and consequently, αmrcik and νmrc

ik are independent,and αmrc

ik ∼ N(μmrcik , σ 2

αmrcik) and νmrc

ik ∼ N(0, ηmrcik )

where

μmrcik � E

{αmrc

ik

}(37)

σ 2αmrc

ik�

(var

{αmrc

ik

})2var

{αmrc

ik

}+ var{w

mrc,reik

} . (38)

Proof: See Appendix E. �The quantities in (34)–(38) are given in Lemma 1 and 2. We

observe that the MSE ηmrcik cannot be reduced to zero by increas-

ing UL and DL pilot powers due to pilot contamination (c.f.Remark 2).

2) System with ZF Precoding: With ZF precoding, thesignal in (23) is rewritten as

zzfik = αzf

ik + wzfik (39)

where αzfik = 1

‖aik‖ is the effective channel gain and

wzfik �

aHik

‖aik‖εi ik +L∑

l=1,l =i

aHlk

‖alk‖glik + CN

(0,

1

τdqd

)(40)

is the interference plus noise term. The next Lemma describesstatistics of αzf

ik .Lemma 4: Mean and variance of the DL effective channel

gain αzfik are given by

E

{αzf

ik

}= E

{1

‖aik‖}

= CM−K+1

√σ 2

gi ik(41)

var{αzf

ik

}= VM−K+1σ

2gi ik

(42)

and αzfik has a Gaussian pdf for a large value of M .

Proof: See Appendix F. �From Lemma 4, following the approximation of CM−K+1 ≈√M − K + 1 and VM−K+1 ≈ 0.25, we can see E{αzf

ik}increases with

√M − K + 1 while var{αzf

ik} is approximatelyfixed.

The real and imaginary parts of wzfik have different statistics

due to the same reason as explained for the MRC case, and thefollowing Lemma describes them.

Lemma 5: The real part wzf,reik and the imaginary part wzf,im

ikof wzf

ik are independent and

E

{w

zf,reik

}= CM−K+1

L∑l=1,l =i

√σ 2

glik(43)

E

{w

zf,imik

}= 0 (44)

var{w

zf,reik

}= 1

2

L∑l=1

σ 2εlik

+ 1

2τdqd+VM−K+1

L∑l=1,l =i

σ 2glik

(45)

var{w

zf,imik

}= 1

2

L∑l=1

σ 2εlik

+ 1

2τdqd. (46)

In addition, wzf,imik is Gaussian and wzf,re

ik for a large value of

M−K approaches Gaussian. We can rewrite (45) as var{wzf,reik }

= 12 (σ

2εi ik

+ 1τdqd

)+ ( 12 − VM−K+1)

∑Ll=1,l =i σ

2εlik

+ VM−K+1∑Ll=1,l =i βlik .


Proof: See Appendix G. �Next, by exploiting Lemma 5, we can estimate αzf

ik from the

real part zzf,reik of zzf

ik where

zzf,reik = αzf

ik + wzf,reik . (47)

Lemma 6: For a system with ZF precoding, the LMMSE DLchannel estimator based on (47) and its MSE are respectivelygiven by

αzfik = czf

ik + f zfik zzf,re

ik (48)

ηzfik � E

{∣∣∣αzfik − αzf

ik

∣∣∣2} =var

{αzf

ik

} · var{w

zf,reik

}var

{αzf

ik

}+ var{w

zf,reik

} (49)

where

czfik �

var{w

zf,reik

}E{αzf

ik

}− var{αzf

ik

}E

{w

zf,reik

}var

{αzf

ik

}+ var{w

zf,reik

} (50)

f zfik �

var{αzf

ik

}var

{αzf

ik

}+ var{w

zf,reik

} . (51)

Denoting the estimation error by νzfik , we write αzf

ik = αzfik +

νzfik . Then, for a large M , LMMSE becomes MMSE estima-

tor and, consequently, αzfik and νzf

ik are independent, and αzfik ∼

N(μzfik, σ

2αik) and νzf

ik ∼ N(0, ηzfik) where

μzfik � E

{αzf

ik

}(52)

σ 2αzf

ik�

(var

{αzf

ik

})2var

{αzf

ik

}+ var{w

zf,reik

} . (53)

Proof: See Appendix E. �The quantities in (49)–(53) are given in Lemma 4 and 5.Remark 2: Due to the pilot contamination, the normalized

MSE of the effective DL channel estimation denoted by ηik �ηik/E{α2

ik} approaches a floor as the UL and DL pilot SNRsincrease. The floors of ηik for the MRC and ZF precoders arerespectively given by

VM

M

12

∑Ll=1 σ

2εlik

+ VM∑L

l=1,l =i σ2glik

12

∑Ll=1 σ

2εlik

+ VM∑L

l=1 σ2glik

(54)

VM−K+1

M − K + 1

12

∑Ll=1 σ

2εlik

+ VM−K+1∑L

l=1,l =i σ2glik

12

∑Ll=1 σ

2εlik

+ VM−K+1∑L

l=1 σ2glik

(55)

where σ 2εlik

� βlik∑L

j=1, j =i βl jk∑Lj=1 βl jk

and σ 2glik

� β2lik∑L

j=1 βl jk. From (54)

and (55), it can be seen by increasing M the floor reduces andasymptotically with increase of M , approaches zero.

Remark 3: αmrcik and �{ gH

iik‖gi ik‖εi ik} are asymptotically inde-

pendent; and even with a small M , they can be considered

as independent for practical purposes. To justify this, weexpand αmrc

ik in (33) by replacing zmrc,reik with (32) and wmrc,re

ikwith the real part of (25). We note αmrc

ik consists of severalterms including these two terms with factors αmrc

ik = ‖gi ik‖and �{ gH

iik‖gi ik‖εi ik}. While the total power of αmrc

ik is summa-tion of powers of individual terms, the power ratio of the two

terms is2Mσ 2

gi ikσ 2

εi ik

= 2Mβi ik1

τuqu+∑L

l=1,l =i βilk, which linearly increases

with M . Even for a small value of M = 10, and typical settingof βi ik = 1, βilk = 0.1, L = 7, qd = 10 dB, that ratio is larger

than 28. Thus, discarding the term with �{ gHiik

‖gi ik‖εi ik} from αmrcik

does not affect the performance for practical purposes, justify-

ing the above statement. Note that αmrcik and �{ gH

iik‖gi ik‖εi ik} are

independent for any M . The same statement applies between

αzfik and �{ aH

ik‖aik‖εi ik}.

B. Achievable Downlink Rates

Since the receiver has knowledge of rik and the effectivechannel estimate αik , by starting from mutual information, i.e.,I(sik; rik, αik), and going through the same procedure as men-tioned in [28, Appendix I], we can show the achievable DL rateis lower bounded by

Rik,p = λpE

{log2

(1 + p

∣∣αik∣∣2

E{|rik |2

∣∣αik}− p

∣∣αik∣∣2)}

(56)

where λp accounts for the pilot overhead, and we discuss itlater in Remark 7. The outer expectation is over αik and theexpectation in the denominator is over the random variablesin rik conditioned on αik . Now, let us consider rik in (18)–(21) and compute E{√psik I ∗

ik |αik}. Applying the asymptotic

independence of αik from νik andgH

iik‖gi ik‖εi ik for the system

with MRC precoding and from νik andaH

ik‖aik‖εi ik for the sys-tem with ZF precoding, together with the zero mean prop-erty of individual terms of E{√psik I ∗

ik |αik}, we can concludeE{√psik I ∗

ik |αik} ≈ 0. As a result, the rate expression in (56) isapproximated to

Rik,p ≈ λpE

{log2

(1 + p

∣∣αik∣∣2

E{|Iik |2

∣∣αik})}

. (57)

1) System with MRC Precoding: To find a closed-formexpression for the achievable rate in (57), first we find the con-ditional interference power in the denominator, and then findthe approximate value for the outer expectation.

Lemma 7: The conditional interference power at user k for asystem with MRC precoding is given by

E

{∣∣I mrcik

∣∣2 ∣∣αmrcik

}≈ pκmrc

ik (58)


where

κmrcik � ηmrc

ik + σ 2εi ik

+ (K − 1)θmrcik

+ (K − 1)L∑

l=1,l =i

βlik +L∑

l=1,l =i

ζmrcilk + 1

p, (59)

θmrcik �

∣∣αmrcik

∣∣2M

+ ηmrcik

M+ σ 2

εi ik(60)

ζmrcilk �

(CM

√σ 2

glik+ γmrc

ilk

(αmrc

ik − E{αmrc

ik

})var

{αmrc

ik

})2

+(

1 − γmrcilk

var{αmrc

ik

}+ var{w

mrc,reik

})γmrc

ilk + 1

2σ 2

εlik(61)

γmrcilk � VMσ

2glik

+ 1

2σ 2

εlik. (62)

Proof: See Appendix H. �For a large value of M , due to the Gaussian pdf of αmrc

ikin Lemma 3, |αmrc

ik |2 has a non-central Chi-square pdf with 1degree of freedom (DoF). Its mean and variance are given by

E

{∣∣αmrcik

∣∣2} = (μmrc

ik

)2 + σ 2αmrc

ik(63)

var{∣∣αmrc

ik

∣∣2} = 2σ 4αmrc

ik+ 4σ 2

αmrcik

(μmrc

ik

)2. (64)

From (37) and (38), we can observe that for a large value ofM , σ 2

αmrcik

dose not depend on M but μmrcik increases with

√M .

Thus, both mean and variance of |αmrcik |2 increase linearly with

M . By substituting (58) in (57), the DL achievable rate boundis given by

Rmrcik,p ≈ λpE

{log2

(1 +

∣∣αmrcik

∣∣2κmrc

ik

)}. (65)

Next, we want to find the outer expectation in (65) over the pdfof αmrc

ik . However, the expectation integral does not lead to aclosed-form solution. A common approach adopted in the liter-ature for such a problem is the use of a lower bound by means ofJensen’s inequality, i.e., log2(1 + 1

E{1/X} ) ≤ E{log2(1 + X)}.But such an approach fails in our case because E{ 1

|αmrcik |2 } is

infinity and that lower bound reads as a trivial zero bound. Thus,we use the approximation E{log2(1 + X

Y )} ≈ log2(1 + E{X}E{Y } ).

Following the proof of the lemma in [13, Appendix I], wecan see that if X and Y are two non-negative random vari-ables with properties that 1

E2{Y }var{Y } and 1E2{X+Y }var{X + Y }

approaches zero, then that approximation is asymptoticallytight. Here, X and Y represent |αmrc

ik |2 and κmrcik , respectively.

We can seevar{κmrc

ik }E2{κmrc

ik } → 0 andvar{κmrc

ik +|αmrcik |2}

E2{κmrcik +|αmrc

ik |2} → 0 as M → ∞,

since the two numerators increase with order M while the twodenominators increase with order M2. Hence, the approxima-tion is asymptotically tight as the number of antennas increases.Then, based on this approximation, we obtain an approximaterate Rmrc

ik,p ≈ Rmrcik,p given by

Rmrcik,p � λp log2

⎛⎝1 +

E

{∣∣αmrcik

∣∣2}E{κmrc

ik

}⎞⎠ (66)

where E{|αmrcik |2} is given by (63). As will be shown in simula-

tion results, Rmrcik,p matches Rmrc

ik,p in (65), even for a small valueof M .

This subsection is summarized by the following theorem.Theorem 3: For a system with MRC precoding and orthogo-

nal DL pilot, the approximate DL rate bound is given by

Rmrcik,p = λp log2

⎛⎝1 +

C2Mσ

2gi ik

+ σ 2αmrc

ik

Imrcik,p

⎞⎠ (67)

where Imrcik,p � ηmrc

ik + σ 2εi ik

+ (K − 1)βi ik + K∑L

l=1,l =i βlik +(M − 1)

∑Ll=1,l =i σ

2glik

+ 1p .

Proof: It immediately follows from (66), (63), andstraightforward calculation of E{κmrc

ik } in (59). �Remark 4: In (67), the numerator for a large M is approxi-

mately C2Mσ

2gi ik

≈ Mσ 2gi ik

, which is the same as the numeratorin (13), and the denominators are approximately the same(The denominator for the case with DL pilot is always slightlysmaller than that without DL pilot due to ηmrc

ik < VMσ2gi ik

=var{αmrc

ik } (see (34)). Hence, comparing (67) and (13), andneglecting pilot overhead, we see DL rates for MRC precodingwith DL pilot and without DL pilot are very close for a large M .

2) System with ZF Precoding: First we present the expecta-tion of the interference power conditioned on αzf

ik , and then usean approximation for computing the outer expectation.

Lemma 8: The conditional interference power at user k for asystem with ZF precoding is given by

E

{∣∣∣I zfik

∣∣∣2 ∣∣∣αzfik

}≈ p κzf

ik (68)

where

κzfik � ηzf

ik + Kσ 2εi ik

+ (K − 1)L∑

l=1,l =i

σ 2εlik

+L∑

l=1,l =i

ζ zfilk + 1

p

(69)

with

ζ zfilk �

(CM−K+1

√σ 2

glik+ γ zf

ilk

(αzf

ik − E{αzf

ik

})var

{αzf

ik

})2

+⎛⎝1 − γ zf

ilk

var{αzf

ik

}+ var{w

zf,reik

}⎞⎠ γ zf

ilk + 1

2σ 2

εlik(70)

γ zfilk � VM−K+1σ

2glik

+ 1

2σ 2

εlik(71)

Proof: See Appendix I. �By substituting (68) in (57), the DL achievable rate bound

reads as

Rzfik,p ≈ λpE

{log2

(1 +

∣∣αzfik

∣∣2κzf

ik

)}. (72)

For a large value of M , due to the Gaussian pdf of αzfik in

Lemma 6, |αzfik |2 has a non-central Chi-square pdf with 1 DoF.

Its mean and variance are given by


E

{∣∣∣αzfik

∣∣∣2} =(μzf

ik

)2 + σ 2αzf

ik(73)

var

{∣∣∣αzfik

∣∣∣2} = 2σ 4αzf

ik+ 4σ 2

αzfik

(μzf

ik

)2. (74)

Next finding the outer expectation in (72) over the pdf of αzfik ,

similar to the MRC precoding case, does not lead to a sim-ple closed-form solution and finding a lower bound based onJensen’s inequality fails. Now, in the same way as discussedfor MRC, we find an asymptotically (with respect to M) tightapproximation for (72). Simulation results show this approxi-mation accurately matches with the actual value of expectation.Expression of our approximate rate Rzf

ik,p ≈ Rzfik,p reads as

Rzfik,p � λp log2

⎛⎝1 +

E

{∣∣αzfik

∣∣2}E{κzf

ik

}⎞⎠ (75)

where E{|αzfik |2} is given by (73). This subsection is summarized

in the following theorem.Theorem 4: For a DL system with ZF precoding and orthog-

onal DL pilot, the approximate achievable rate bound isgiven by

Rzfik,p = λp log2

⎛⎝1 +

C2M−K+1σ

2gi ik

+ σ 2αzf

ik

Izfik,p

⎞⎠ (76)

where Izfik,p � ηzf

ik + K∑L

l=1 σ2εlik

+ (M − K + 1)∑L

l=1,l =i

σ 2glik

+ 1p .

Proof: It immediately follows from (75), (73), andstraightforward calculation of E{κzf

ik} in (69). �Remark 5: In (76), the numerator for a large M is approx-

imately C2M−K+1σ

2gi ik

, which is the same as the numerator in(16), and the denominator of (76) is always smaller than thedenominator of (16) due to ηzf

ik < VM−K+1σ2gi ik

= var{αzfik} (see

(49)). However, the difference between the denominators isnegligible. So, the DL rates for ZF precoding with DL pilotand without DL pilot are very close for a large M .

Remark 6: As M → ∞, DL rates for MRC and ZF precod-ing in (67) and (76) saturate to the same value as Rmrc

ik,p, Rzfik,p →

λp R∞ik , where R∞

ik is defined in (17).Remark 7: The pilot overhead for DL rate expressions can

be defined based on DL pilot only or both DL and UL pilotssince UL pilot is used for both UL data detection and DL trans-mit beamforming. In the second definition, UL pilot overheadcan be divided between UL and DL according to the proportionof UL and DL frame lengths; with the same frame length of

T symbols, the overhead factors read as λudp = T −τd− τu

2T and

λudnp = T − τu

2T . In the first definition, the overhead factors are

given by λdnp = 1 and λd

p = T −τdT . If we use the minimum length

for orthogonal pilot sequences, K replaces τu and τd in theabove overhead factors.

Remark 8: Comparing DL rates for systems without DLpilot in (13) and (16), we observe that MRC precoding has ahigher rate if M is smaller than a threshold given by

Mth, np �VMσ

2gi ik

+ 1p + K

∑Ll=1 βlik∑L

l=1 σ2glik

− 1. (77)

Similarly, from the DL rates in (67) and (76), the threshold forsystems with DL pilots is obtained as

Mth, p �ηik + 1

p + K∑L

l=1 βlik − σ 2αik∑L

l=1 σ2glik

+ 2σ 2αik

σ 2gi ik

− 1 (78)

where in deriving the threshold we have used the approxima-tions σ 2

αik= σ 2

αmrcik

≈ σ 2αzf

ikand ηik = ηmrc

ik ≈ ηzfik, These thresh-

olds are for each user, and they can be useful in developingadaptive beam-forming for each user.

V. NUMERICAL RESULTS AND DISCUSSIONS

We use a typical multicell structure with L = 7 cells andfrequency reuse factor of 1. There are K = 10 users in eachcell, and the numbers of pilot symbols for UL and DL areτu = τd = K , and there are T = 200 symbols in each frame.We assume the UL transmit pilot SNR is qu = 10dB (unlessmentioned otherwise) and the DL pilot and data SNRs are thesame p = qd. We evaluate both settings of fixed {βilk} and ran-dom {βilk}. In the fixed setting, we set βi ik = 1 and βilk = a,∀l = i . In the random setting, users in each cell are uniformlylocated in the disk with radius 1000 m around BS not closerthan d0 = 100 m to BS, and the distance of user k in cell lto BS i is denoted by dilk . So, {βilk} are independently gen-erated by βilk = ψ/(

dilkd0)ν where ν = 3.8 and 10 log10(ψ) ∼

N(0, σ 2shadow, dB)with σshadow, dB = 8. We use the first definition

of pilot overhead mentioned in Remark 7 in all the simulationsas the difference between the two definitions is just a scaling.The exception is in Fig. 6 where we use both of the definitionsto investigate sum-rate versus the number of users. In the figurelegends, “NP” refers to the system without DL pilot, “sim” rep-resents Monte Carlo simulation results of non-closed-form ratebounds while “ana” denotes our proposed analytical closed-form rates. For systems with DL pilot, the rate bounds for MRCand ZF and their corresponding approximate closed-form ratesare (65), (72), (67) and (76), respectively. For systems withoutDL pilot, the rate bound is given in (11) and the closed formrate bounds for MRC and ZF are given in (13) and (16).

First, we compare our closed-form DL rates with the MonteCarlo based actual DL rate bounds for systems with and with-out DL pilots. Fig. 1 shows the sum rate versus Tx SNR forM = 100 and a = 0.2 while Fig. 2 presents the sum rate ver-sus the number of BS antennas M for p = qd = 10 dB anda = 0.2. The closed-form results match with the simulation-based rates for all cases. After verifying the accuracy of theclosed-form expressions, we study performance characteristicsunder various conditions. Fig. 1 illustrates rate saturation of allschemes as the DL Tx SNR increases. This implies an interfer-ence dominated operating condition and suggests a low SNRregion for the operation of the massive MIMO systems in mul-ticell environments. We have also evaluated for other settingswith M = 30 and/or a = 0.1; except the obvious increase of


Fig. 1. The DL sum rate performance versus the Tx SNR for systems with andwithout DL pilot (M = 100, a = 0.2, qu = 10dB).

Fig. 2. The effect of the number of BS antennas on the DL sum rates (p =qd = 10 dB, a = 0.2).

the saturated rate for a larger M or a smaller a, the results showthe same trend as in Fig. 1 and hence they are omitted.

From Fig. 2, we observe that MRC scheme performs bet-ter (worse) than ZF scheme if the number of BS antennasis small (large). This can be explained as follows. The rateexpression has a numerator factor (beam-forming power gain)of M − K + 1 for ZF and M for MRC which yields a substan-tial rate difference between the two schemes at a small M buta negligible rate difference at a large M . The denominator inthe rate expression is due to interference plus noise, and the ZFscheme has better interference suppression capability (smallerdenominator) than the MRC scheme. Thus, at a small M , bet-ter beam-forming gain of MRC outperforms better interferencesuppression capability of ZF, yielding performance advantageof MRC. But as M increases, the beam-forming gains of thetwo schemes become similar and the better interference sup-pression of ZF yields performance advantage of ZF. Note that

Fig. 3. The impact of the cross-cell interference level a on the sum rates(p = qd = qu = 10 dB).

as M → ∞, the user channels become orthogonal and henceMRC achieves the same interference suppression capability asZF, thus yielding the same asymptotic rate (given in (17)).

Next, Fig. 3 elaborates effects of the cross-cell large-scalechannel power gain a under the settings of p = qd = 10 dB,M = 100 or 30, and K = 2 and 10. We observe that as aincreases, the sum rates of the two transmission schemesdecrease due to their imperfect interference suppression. ZFoutperforms MRC at a low cross-cell interference level (i.e.,a ≤ 0.2 for the system with M = 100, and a ≤ 0.15 forM = 30), but both of the schemes show almost the same per-formance as the cross-cell interference level increases. This canbe explained as follows. At a low cross-cell interference level,the effect of pilot contamination is insignificant and the ZFscheme can maintain most of its interference suppression capa-bility. However, as the cross-cell interference level increases,the pilot contamination becomes significant which in turn voidsthe better interference suppression capability of the ZF scheme.

Another observation from Fig. 3 is for a small a, the ratewith DL pilot is larger than that without DL pilot for ZF, butfor MRC the same holds if additionally K is very small. For alarge a, the system without DL pilot yields a larger rate thanthat with DL pilot as can be observed in Fig. 1 and Fig. 2. Thereasons are as follows. First, the advantage of the system with-out DL pilot comes from zero DL pilot overhead; at a largea (interference-dominated scenario), rates of the two systemsexcluding pilot overhead are similar and the overhead cost ofthe system with DL pilot yields its smaller rate (including over-head cost). Second, when a is small, the pilot contamination andintercell interference are small, and the rate is mainly influencedby the intra-cell interference. The main difference in contribut-ing the intra-cell interference is the variance of the effective DLchannel gain for the system without DL pilot versus the MSEof the effective DL channel estimate for the system with DLpilot. When a is small, the former is much larger than the lat-ter, thus giving rate advantage for the system with DL pilot and


Fig. 4. Per-user rate versus the frame length (with a = 0.01,M = 100, andp = qd = 10dB).

Fig. 5. DL sum rate versus DL Tx SNR (= qu = p) under various settings ofUL pilot power (with M = 100 and a = 0.2).

ZF precoding. For MRC, non-orthogonality of user channelscauses additional multi-user interference which is proportionalto K , thus requiring an additional condition of a small K forthe system with DL pilot to outperform the system without DLpilot.

Fig. 4 shows the effects of the frame length in compar-ing per-user rates between the system with and without DLpilot, for K = 10 and K = 5 with a = 0.01,M = 100, andp = qd = 10dB. As we expect, a longer frame length yields alarger rate for systems with DL pilot since the cost of the addi-tional overhead due to DL pilot reduces. The gain of the systemwith DL pilot over that without DL pilot is larger for ZF thanMRC, and for a smaller number of users. This can be explainedby the same reasons used for Fig. 3 in the above paragraph.

In the above results, we use a fixed UL pilot SNR qu =10 dB. In Fig. 5, we present how different UL pilot SNR set-tings impact the DL rate bounds for systems with a = 0.2

Fig. 6. DL sum rate versus the number of users K (DL Tx SNR (= qd = p) of10 dB, M = 100, and a = 0.2).

and DL pilot. Both MRC and ZF precoders show the sametrend. All the curves for different qu show DL rate saturation.For convenience of discussion, let us loosely term the satura-tion SNR as the DL Tx SNR above which the rate saturationoccurs. Then we observe that the UL pilot SNR qu much lowerthan the saturation SNR, e.g., qu = 0 dB, yields a drop of theDL rate ceiling. However, qu above the saturation SNR, e.g.,qu = 20 dB, gives very little improvement of the DL rate ceil-ing. These results imply that both UL and DL pilot Tx SNRshould be set around the DL saturation SNR to avoid rate lossor power-inefficient rate saturation.

Next, Fig. 6 shows the sum rate with respect to the numberof users K in each cell for systems with and without DL pilotbased on the closed form results. Since K affects the overhead,we consider both definitions of overhead factors mentioned inRemark 7. The results show that the sum rate is nonlinearlyrelated to K and there exists an optimal number of users to max-imize the sum rate for all cases except the MRC scheme withoutDL pilot under the overhead definition of λd

np. The reason canbe explained by the following factors as K increases: (i) theincreasing spatial multiplexing gain, (ii) the increasing interfer-ence level and (iii) the (potentially) increasing pilot overhead.For the exception case, the third factor is nulled, the first factorlinearly increases the sum rate, and the second factor decreasesthe sum rate from inside the log2 function. Thus, the sum rateincreases with K . For the MRC scheme without DL pilot underthe overhead definition of λud

np, the roles of the first and secondfactors are the same, but the third factor reduces the sum ratelinearly, thus yielding existence of an optimal value of K . Forall of the ZF cases, the roles of the first and third factors arethe same as the corresponding MRC cases. The second factoris less for ZF than MRC due to the interference suppressioncapability of ZF. But an additional factor for ZF, namely thedecreasing beam-forming gain of ZF as K increases, results inan existence of an optimal K for all the ZF cases and their sumrates decay faster than the MRC counterparts as K increasesbeyond its optimal point.


Fig. 7. The DL performance cross-over point between MRC and ZF transmis-sion schemes with DL training in terms of the number of BS antennas M underdifferent conditions of Tx SNR (with p = qd) and cross-cell large-scale channelpower gain a.

Fig. 8. Preferred regions of system parameters for ZF and MRC precoders(without DL pilot, p = qd = 10dB).

Next, we present in Fig. 7 how different conditions can influ-ence Mth, the threshold point of M for choosing the precoder.We observe a larger Mth at a larger a or a smaller DL Tx SNR.Both conditions increase the interference plus noise level forboth precoders which in turns dilutes the interference suppres-sion advantage of ZF scheme. Thus, a larger beam-forming gain(a larger M) is required for ZF scheme to outperform MRCscheme.

We further investigate the selection of the precoding schemebased on Mth,np as mentioned in Remark 8. As the thresholdsfor systems with and without DL pilot are quite similar, wejust show the results for systems without DL pilot in Fig. 8.The considered discrete points in the parameter space of aand K over which MRC outperforms ZF scheme are shownwith cross marks for M = 30 and M = 100. We observe that

Fig. 9. Average sum rate under random {βilk } versus the Tx SNR (with M =100 and K = 10).

Fig. 10. Average sum rate under random {βilk } versus M for different K(p = qd = 10dB) (left) and for different DL Tx SNR (K = 10) (right).

for a fixed M , smaller K or/and smaller a yield performanceadvantage of ZF over MRC since under those conditions ZFhas much better interference suppression capability and similarbeamforming gain if compared to MRC scheme.

Next, we evaluate the sum rate performance under randomlarge-scale fading coefficients. The results are obtained by aver-aging the sum rates over 1000 large-scale fading coefficients.Fig. 9 presents the average sum rate versus DL Tx SNR whichhas the same trend as the sum rate for fixed large-scale fad-ing coefficients (see Fig. 1) but with an SNR shift. The reasonfor the SNR shift can be ascribed to the larger path loss andsmaller values of log-normal shadowing coefficients than thefixed coefficient scenario. Fig. 10 shows the average sum rateversus the number of antennas for different values of K in theleft and different values of DL Tx SNR in the right. The resultsare in the same trend as in Fig. 7 except the crossover points ofthe rate curves between ZF and MRC are shifted to the left.


The reason is due to the larger path loss and smaller valuesof log-normal shadowing coefficients than the fixed coefficientscenario, and it is also consistent with the characteristics of asmaller a in Fig. 7.

VI. CONCLUSIONS

We have studied a multicell massive MIMO DL system withpilot contamination for MRC and ZF precoders at the base sta-tion for two DL scenarios of with and without a pilot-aidedcoherent detector at users. We have developed an LMMSEDL channel estimator by incorporating the non-circularly-symmetric characteristics of the effective DL channels. We havederived closed-form approximate achievable rate expressionsfor both of the DL scenarios. Numerical results illustrate thatthe derived rate expressions are very accurate. Our investiga-tion shows that the considered massive MIMO DL system isinterference-dominated in medium and high SNR regions dueto imperfect CSI and pilot contamination, and the system is bestdeployed in a low SNR region. The system with DL pilot mayhave advantage over the system without DL pilot under scenar-ios with small cross-cell interference level, small pilot overheadwith respect to the frame length, and small number of users.We have also presented criteria for choosing between MRC andZF precoder, which depends on the number of BS antennas,the number of users, the cross-cell interference level, and theDL transmit SNR. In most of the scenarios investigated, thereexists an optimal number of users to maximize the sum rate,which can be easily found from our closed form rate expres-sions. Thus, our closed form rate analysis offers insights andguidance for an efficient massive MIMO system design.

APPENDIX APROOF OF THEOREM 1

First we present a Lemma which we use frequently in thisappendix and others.

Lemma 9: If x ∼ CN(0, σ 2x IM ) and y ∼ CN(0, σ 2

y IM ) are

independent, then xH

‖x‖y ∼ CN(0, σ 2y ).

Proof: Define z � xH

‖x‖y. Then, by conditioning on x, wesee z|x is a weighted sum of i.i.d. complex Gaussian ran-dom variables with pdf CN(0, σ 2

y ) where the powers of theweights sum up to 1. Thus, the pdf of z|x denoted by fz|x(z)is CN(0, σ 2

y ) which is independent of x. Consequently, the pdf

of z denoted by fz(z) equals fz|x(z) = CN(0, σ 2y ). �

When users do not estimate their channels, we decompose

the received signal in (9) as rmrcik = E{√p

gHiik

‖gi ik‖gi ik}sik + I mrcik ,

where I mrcik = ∑5

j=1 I mrcik, j and we define these different interfer-

ence components as I mrcik,1 � √

p(gH

iik‖gi ik‖gi ik − E{ gH

iik‖gi ik‖gi ik})sik ,

I mrcik,2 �

∑Kj=1, j =k

√p

gHii j

‖gi i j ‖gi iksi j , I mrcik,3 �

∑Ll=1,l =i

∑Kj=1, j =k

√p

gHll j

‖gll j ‖gliksl j , I mrcik,4 �

∑Ll=1,l =i

√p

gHllk

‖gllk‖glikslk , and I mrcik,5

� nik . The rate expression is given in (11) and the remain-ing part is finding powers of the terms in (11). Firstfor the signal term in the numerator, by replacing gi ik

with gi ik + εi ik , we find E{√pgH

iik‖gi ik‖gi ik} = √

p E{‖gi ik‖} =√

p√σ 2

gi ikCM , where we have used the fact that ‖gi ik‖ has a

Chi distribution. Then, from independence of user’s data, wecan see E{| I mrc

ik |2} = ∑5j=1 E{| I mrc

ik, j |2}. Also, E{| I mrcik,1|2} = p

var{ gHiik

‖gi ik‖gi ik} = p (var{‖gi ik‖} + σ 2εi ik)= p(VMσ

2gi ik

+ σ 2εi ik),

where we have replaced gi ik with gi ik + εi ik and used theindependence between gi ik and εi ik and Lemma 9. For com-puting power of the other interference terms, due to inde-pendence between gi ik and gi i j ( j = k), from Lemma 9 we

havegH

ii j

‖gi i j ‖gi ik ∼ CN(0, βi ik). Similarly, we havegH

ll j

‖gll j ‖glik ∼

CN(0, βlik), j = k. But for the terms in Iik,4, gllk and glik arecorrelated due to pilot contamination. To proceed further, wereplace glik with glik + εlik and then glik with βlik

βllkgllk from (6).

As a result, we havegH

llk‖gllk‖glik ∼ (

βlikβllk)‖gllk‖ + CN(0, σ 2

εlik).

Then we obtain E{| gHllk

‖gllk‖glik |2}=(βlikβllk)2 Mσ 2

gllk+σ 2

εlik=Mσ 2

glik+

σ 2εlik

= (M − 1)σ 2glik

+ βlik , where we have used the fact that

‖gllk‖2 has a Chi square distribution with 2M DoF, and σ 2glik

+σ 2

εlik= βlik . Now, we can compute the powers of the remaining

interference terms as E{| I mrcik,2|2} = p(K−1)βi ik , E{| I mrc

ik,3|2} =p(K−1)

∑Ll=1,l =i βlik , and E{| I mrc

ik,4|2} = p∑L

l=1,l =i (M−1)

σ 2glik

+ βlik . By substituting the signal power and the interfer-ence power in (11), we arrive at (13).

APPENDIX BPROOF OF THEOREM 2

When users do not estimate their channels, we decompose

the received signal in (9) as r zfikE{√p

aHik‖aik‖gi ik}sik + I zf

ik ,

where I zfik = ∑5

j=1 I zfik, j and we define these different interfer-

ence components as I zfik,1 � √

p(aH

ik‖aik‖gi ik − E{ aHik‖aik‖gi ik})sik ,

I zfik,2 �

∑Kj=1, j =k

√p

aHi j

‖ai j ‖gi iksi j , I zfik,3 �

∑Ll=1,l =i

∑Kj=1, j =k

√p

aHl j

‖al j ‖gliksl j , I zfik,4 �

∑Ll=1,l =i

√p

aHlk‖alk‖glikslk , and I zf

ik,5

� nik . The rate expression is given in (11) and the remainingpart is finding powers of the terms in (11). From Sec. II-B,we recall for ZF precoding aH

ik gi i j = δ jk . First for the signalterm in the numerator, by replacing gi ik with gi ik + εi ik , we

find E{√paH

ik‖aik‖gi ik} = √p E{ 1

‖aik‖ } = √p√σ 2

gi ikCM−K+1,

where we have used Lemma 4. Then, from independenceof user’s data, we can see E{| I zf

ik |2} = ∑5j=1 E{| I zf

ik, j |2}.Also, E{| Iik,1|2}=p var{ aH

ik‖aik‖gi ik}=p (var{‖ 1aik

‖}+σ 2εi ik)=

p(

VM−K+1σ2gi ik

+ σ 2εi ik

), where we have replaced gi ik with

gi ik + εi ik and used the independence between gi ik and εi ik

in conjunction with Lemma 9, and for computing variance ofthe first term we have used Lemma 4. For computing powerof the other interference terms, from Lemma 9, we have

aHi j

‖ai j ‖gi ik = aHi j

‖ai j ‖εi ik ∼ CN(0, σ 2εi ik), j = k. Similarly, we have


aHl j

‖al j ‖glik ∼ CN(0, σ 2εlik), j = k. But for the terms in I zf

ik,4,aH

lk‖alk‖glik = βlikβllk

1‖alk‖ + aH

lk‖alk‖εlik , where we have replaced glik

with glik + εlik and then glik with βlikβllk

gllk from (6). So we

obtain E{| aHlk‖alk‖glik |2} = (M − K + 1)σ 2

glik+ σ 2

εlik, where we

have used Lemma 10. Now, we can compute the powers of theremaining interference terms as E{| I zf

ik,2|2} = p(K − 1)σ 2εi ik

,

E{| I zfik,3|2} = p(K − 1)

∑Ll=1,l =i σ

2εlik

, and E{| I zfik,4|2} = p∑L

l=1,l =i (M − K + 1)σ 2glik

+ σ 2εlik

. By substituting the signalpower and the interference power in (11), we arrive at (16).

APPENDIX CPROOF OF LEMMA 1

Following (6), we can write glik as

glik = glik + εlik = βlik

βllkgllk + εlik . (79)

Then substituting (79) into (25), we have

wmrcik =

L∑l=1

gHllk

‖gllk‖εlik +L∑

l=1,l =i

βlik

βllk‖gllk‖ + CN

(0,

1

τdqd

).

(80)

Due to the independence between gllk and εlik for l =1, · · · , L , and the Gaussian distribution of εlik , from Lemma 9

we havegH

llk‖gllk‖εlik ∼ CN(0, σ 2

εlik). Different terms in (80) are

independent, and due to circularly symmetric distribution ofcomplex random variables in (80), the real and imaginary partsare independent. Thus, we have

wmrc,imik ∼ N

(0,

1

2

L∑l=1

σ 2εlik

+ 1

2τdqd

). (81)

The ‖gllk‖ has a Chi pdf with 2M DoF, E{‖gllk‖} = CM

√σ 2

gllk

and var{‖gllk‖ } = VMσ2gllk

. In addition for a large M , its pdfapproaches a Gaussian pdf [29]. Then, for a large M , we have

wmrc,reik ∼ N

(E{wmrc,re

ik }, var{wmrc,reik }) (82)

where E{wmrc,reik } and var{wmrc,re

ik } are given in (86) and (28).

APPENDIX DPROOF OF LEMMA 2

As αmrcik = ‖gi ik‖ follows a Chi distribution with 2M DoF,

its mean and variance are straightly given by (30) and (31) andits pdf approaches a Gaussian pdf [29].

APPENDIX EPROOF OF LEMMA 3

Following the independence between αik andwreik , and apply-

ing the LMMSE principle [27], the estimator of αik can bederived as

αik = E {αik} + var {αik}(zre

ik − E {αik} − E{wre

ik

})var {αik} + var

{wre

ik

} . (83)

The MSE of the above estimation reads as

ηik � E

{∣∣αik − αik∣∣2} = var{αik} · var

{wre

ik

}var{αik} + var

{wre

ik

} . (84)

So by substituting the mean and variance terms, we arrive at theresults in Lemmas 3 and 6.

Due to asymptotically (with respect to M) Gaussian distribu-tion of αik and wik , for a large M , LMMSE estimator becomesMMSE estimator and αik has Gaussian distribution, and αik andνik become independent.

APPENDIX FPROOF OF LEMMA 4

For the proof of Lemma 4 and 5, we use the followingLemma.

Lemma 10: For independent random vectors {xi }Ki=1 with

xi ∼ CN(0, σ 2i IM ), define X as an M × K random matrix

(M ≥ K ) with its i th column being xi and yi as i th column

of X(XH X

)−1. Then E{ 1

‖yi ‖2 } = σ 2i (M − K + 1). In addition,

the pdf of 1‖yi ‖ is given by

f 1‖yi ‖(t) = 2 t2(M−K )+1e

− t2

σ2i(

σ 2i

)M−K+1�(M − K + 1)

, t > 0. (85)

Proof: We have ‖yi‖2 = [(XH X)−1]i i . We can writeX = H�, where H is an M × K matrix of i.i.d. CN(0, 1)elements, and � = diag{σ1, · · · , σK }. So [(XH X)−1]i i =σ−2

i [(HH H)−1]i i . Let vi � ([(HH H)−1]i i

)−1. Then vi has

a Gamma distribution with pdf fvi (v) = vM−K e−v

�(M−K+1) , v > 0,

[30], and E{vi } = (M − K + 1). As 1‖yi ‖2 = σ 2

i vi , we have

E{ 1‖yi ‖2 } = σ 2

i E{vi } = σ 2i (M − K + 1). Next, from 1

‖yi ‖ =σi

√vi and the pdf of vi , we can straightly obtain the pdf of 1

‖yi ‖which turns out to be a Chi pdf. �

By following Lemma 10 and replacing {xi }Ki=1 with

{gllk}Kk=1, the pdf of 1

‖alk‖ is given by a Chi pdf as

f 1‖alk‖

(t) = 2 t2(M−K )+1e− t2

σ2gllk(

σ 2gllk

)M−K+1�(M − K + 1)

, t > 0. (86)

Next, by direct calculation, the mean and variance of 1‖alk‖ read

as in (41) and (42).

APPENDIX GPROOF OF LEMMA 5

By substituting (79) into (40) and simplifying the expression,we have

wzfik =

L∑l=1

aHlk

‖alk‖εlik +L∑

l=1,l =i

βlik

βllk

1

‖alk‖ + CN

(0,

1

τdqd

).

(87)


Fig. 11. Comparison of the approximate and actual pdfs of αmrcik (upper) and

αzfik (lower) with M = 20 antennas, K = 10 users, and pilot powers qu =

10 dB.

alk depends on Gll due to its definition but it is indepen-dent from εlik , and due to the Gaussian distribution of εlik ,

from Lemma 9 we haveaH

lk‖alk‖εlik ∼ CN(0, σ 2εlik) and all dif-

ferent terms are independent. Then the same as in the proof ofLemma 1, the real and imaginary parts of wzf

ik are independent,and we have

wzf,imik ∼ N

(0,

1

2

L∑l=1

σ 2εlik

+ 1

2τdqd

). (88)

Next, using (41) and (42), we have E{wzf,reik } and var{wzf,re

ik }as given in (43) and (45), respectively. In addition, for a largevalue of M − K + 1, 1

‖alk‖ approaches a Gaussian randomvariable [29]. So, for a large M − K + 1, we have

wzf,reik ∼ N

(E

{w

zf,reik

}, var{wzf,re

ik }). (89)

Remark 9: Although the Gaussian approximations inLemma (2) and (4) are described for a large M , we notice in ourinvestigation that they also hold for a small M . To illustrate this,we present in Fig. 11 the pdfs of the actual (Chi) pdf and theapproximate (Gaussian) pdf of αik for a system with M = 20antennas, K = 10 users and pilot power of qu = 10 dB. Thepdfs show a close match even for a small M .

APPENDIX HPROOF OF LEMMA 7

We can decompose the interference term in (19) into sixcomponents as

I mrcik = I mrc

ik,1 + I mrcik,2 + I mrc

ik,3 + I mrcik,4 + I mrc

ik,5 + nik (90)

where I mrcik,1 � √

pνmrcik sik , I mrc

ik,2 � √p

gHiik

‖gi ik‖εi iksik , I mrcik,3 �∑K

j=1, j =k√

pgH

ii j

‖gi i j ‖gi iksi j , I mrcik,4 �

∑Ll=1,l =i

∑Kj=1, j =k

√p

gHll j

‖gll j ‖gliksl j , and I mrcik,5 �

∑Ll=1,l =i

√p

gHllk

‖gllk‖glikslk .

From Lemma 3 and Remark 3, νmrcik and

gHiik

‖gi ik‖εi ik are

asymptotically independent from αik . Also νmrcik and

gHiik

‖gi ik‖εi ik

are asymptotically independent. As a result, we haveE{(I mrc

ik,1)(Imrcik,2)

∗|αik} ≈ 0. The other pairs of interference termsin (90) are also uncorrelated due to uncorrelated {sl j }. As aresult, based on (90), we can write the conditional averageinterference power as

E

{∣∣I mrcik


}≈

5∑j=1

E

{∣∣∣I mrcik, j

∣∣∣2 ∣∣αmrcik

}+ 1. (91)

In the following, we find the conditional power of each interfer-ence component.

For the first component, using the result of Lemma 3 thatfor a large M the estimation error νmrc

ik and the estimate αmrcik

approach to be independent, we obtain

E

{∣∣I mrcik,1


}= p E

{∣∣νmrcik


}≈ p ηmrc

ik . (92)

For the second component, from Lemma 9 we use the result

thatgH

iik‖gi ik‖εi ik ∼ CN(0, σ 2

εi ik) and obtain

E

{∣∣I mrcik,2


}=p E

⎧⎨⎩∣∣∣∣∣ gH

iik

‖gi ik‖εi ik

∣∣∣∣∣2∣∣αmrc

ik

⎫⎬⎭≈pσ 2

εi ik. (93)

For the third component, due to uncorrelated {si j , j = 1, · · · ,K }, we have

E

{∣∣I mrcik,3


}=

K∑j=1, j =k

pE

⎧⎨⎩∣∣∣∣∣ gH

ii j

‖gi i j‖gi ik

∣∣∣∣∣2 ∣∣αmrc

ik

⎫⎬⎭⎫⎬⎭ .(94)

In the following, we compute E{| gHii j

‖gi i j ‖gi ik |2|αmrcik }. Note that

gHii j

‖gi i j ‖gi ik and αmrcik are dependent. By substituting gi ik with

gi ik + εi ik , we have

gHii j

‖gi i j‖gi ik = gHii j

‖gi i j‖ gi ik + gHii j

‖gi i j‖εi ik

= gHii j

‖gi i j‖gH

iik

‖gi ik‖ αmrcik + gH

ii j

‖gi i j‖gH

iik

‖gi ik‖νmrcik

+ gHii j

‖gi i j‖εi ik (95)

where we have used ‖gi ik‖ = αmrcik = αmrc

ik + νmrcik in (95). The

third term in (95), i.e.,gH

ii j

‖gi i j ‖εi ik , is independent fromgH

ii j

‖gi i j ‖because gi i j and εi ik are independent and εi ik has a circularlysymmetric distribution [31, Lemma 4]. In addition, conditioned


on αmrcik , the three terms in (95) are mutually asymptotically

uncorrelated (see Remark 3, and Lemma 3), and we can write(95) as

E

⎧⎨⎩∣∣∣∣∣ gH

ii j

‖gi i j‖gi ik

∣∣∣∣∣2 ∣∣αmrc

ik

⎫⎬⎭ ≈ E

⎧⎨⎩∣∣∣∣∣ gH

ii j

‖gi i j‖gH

iik

‖gi ik‖ αmrcik

∣∣∣∣∣2 ∣∣αmrc

ik

⎫⎬⎭

+E

⎧⎨⎩∣∣∣∣∣ gH

ii j

‖gi i j‖gH

iik

‖gi ik‖νmrcik

∣∣∣∣∣2∣∣αmrc

ik

⎫⎬⎭+E

⎧⎨⎩∣∣∣∣∣ gH

ii j

‖gi i j‖εi ik

∣∣∣∣∣2∣∣αmrc

ik

⎫⎬⎭ .

(96)

To proceed further, we introduce the following Lemma.Lemma 11: If x ∼ CN(0, σ 2

x IM ) and y ∼ CN(0, σ 2y IM ),

then z � | xH

‖x‖y

‖y‖ |2 has a beta distribution with parameter 1

and M − 1, and its pdf is fz(u) = (M − 1)(1 − u)M−2 foru ∈ [0, 1] with E{z} = 1

M .

Proof: See [32]. �From Lemma 11 with x = gi i j and y = gi ik , we obtain

E{| gHii j

‖gi i j ‖gH

iik‖gi ik‖ |2} = 1

M . Then we have

E

⎧⎨⎩∣∣∣∣∣ gH

ii j

‖gi i j‖gi ik

∣∣∣∣∣2 ∣∣αmrc

ik

⎫⎬⎭ ≈ |αmrc

ik |2M

+ ηmrcik

M+ σ 2

εi ik(97)

and by substituting (97) in (94), we obtain

E

{∣∣I mrcik,3


}≈ p

K − 1

M

∣∣αmrcik

∣∣2 + pK − 1

Mηmrc

ik

+ p(K − 1)σ 2εi ik. (98)

Next, in computing E{|I mrcik,4|2|αmrc

ik }, we use the fact thatgH

ll j

‖gll j ‖glik is independent from αmrcik . In addition, due to the inde-

pendence between gll j and glik for j = k, from Lemma 9 we

havegH

ll j

‖gll j ‖glik ∼ CN(0, βlik). Then we obtain

E

{∣∣I mrcik,4


}= p(K − 1)

L∑l=1,l =i

βlik . (99)

But, in computing E{|I mrcik,5|2|αmrc

ik }, gHllk

‖gllk‖glik is dependent on

αmrcik . In particular, when we expand αmrc

ik in (33) by replacing

wmrc,reik with the real part of (25), we see �{ gH

llk‖gllk‖glik |2} and αmrc

ik

are dependent while �{ gHllk

‖gllk‖glik |2} and αmrcik are independent.

Next, considering expression of αmrcik and using the asymptot-

ically Gaussian distribution of a Chi random variable, we can

find E{(�{ gHll j

‖gll j ‖glik})2|αmrcik }, a conditional second moment,

from the sum of square of the conditional mean [27, eqn. 14.6]and conditional variance [27, eqn. 14.8]. After algebraic manip-

ulations and simplification, E{| gHll j

‖gll j ‖glik |2|αmrcik } can be found

as ζmrcilk as given in (61).

With the above result, we obtain

E

{∣∣I mrcik,5


}= p

L∑l=1,l =i

ζmrcilk . (100)

Finally, by substituting (92), (93), (98), (99), (100) into (91),we complete the proof.

APPENDIX IPROOF OF LEMMA 8

We can decompose the interference term in (21) into sixcomponents as

I zfik = I zf

ik,1 + I zfik,2 + I zf

ik,3 + I zfik,4 + I zf

ik,5 + nik (101)

where I zfik,1 � √

pνzfiksik , I zf

ik,2�√

paH

ik‖aik‖εi iksik , I zfik,3�∑K

j=1, j =k√

paH

i j‖ai j ‖gi iksi j , I zf

ik,4�∑L

l=1,l =i∑K

j=1, j =k√

paH

l j‖al j ‖

gliksl j , and I zfik,5 �

∑Ll=1,l =i

√p

aHlk‖alk‖glikslk . In the same way

of proving Lemma 7, we have

E

{∣∣∣I zfik

∣∣∣2 ∣∣∣αzfik

}≈

5∑j=1

E

{∣∣∣I zfik, j

∣∣∣2 ∣∣∣αzfik

}+ 1. (102)

For the first and second components, similar to the proof ofLemma 7, we have

E

{∣∣∣I zfik,1

∣∣∣2 ∣∣∣αzfik

}= p E

{∣∣∣νzfik

∣∣∣2 ∣∣∣αzfik

}≈ p ηzf

ik (103)

E

{∣∣∣I zfik,2

∣∣∣2 ∣∣∣αzfik

}= pE

⎧⎨⎩∣∣∣∣∣ aH

ik

‖aik‖εi ik

∣∣∣∣∣2 ∣∣∣αzf

ik

⎫⎬⎭ ≈ pσ 2

εi ik. (104)

Next, as ai j and εi ik are independent, from Lemma 9 we

haveaH

i j‖ai j ‖εi ik ∼ CN(0, σ 2

εi ik), and together with Remark 3, we

obtain

E

{∣∣∣I zfik,3

∣∣∣2 ∣∣∣αzfik

}≈ p(K − 1)σ 2

εi ik. (105)

For the fourth component, al j and αik are independent. Weuse glik = glik + εlik and replace glik with βlik

βllkgllk . Then, for

the fourth component, from Lemma 9 we haveaH

l j‖al j ‖glik =

aHl j

‖al j ‖εlik ∼ CN(0, σ 2εlik). As a result, we obtain

E

{∣∣∣I zfik,4

∣∣∣2 ∣∣∣αzfik

}= p(K − 1)

L∑l=1,l =i

σ 2εlik. (106)

For the fifth component, expanding αzfik in (48) by replacing

wzf,reik with the real part of (40), we find �{ aH

lk‖alk‖glik} and αzfik

are dependent. With the same approach of Lemma 7, we can

find the conditional second moment E{| aHlk‖alk‖glik |2|αzf

ik} = ζ zfilk ,

where ζ zfilk is given in (70) which yields

E

{∣∣∣I zfik,5

∣∣∣2 ∣∣∣αzfik

}= p

L∑l=1,l =i

ζ zfilk . (107)

By substituting (103), (104), (105), (106) and (107) into (102),we complete the proof.


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Amin Khansefid (S’06) received the B.S. degreesin electrical engineering and mathematics, both fromIsfahan University of Technology, Isfahan, Iran, in2005, and the M.S. degree in electrical engineeringfrom the University of Tehran, Tehran, Iran, in 2008.He is currently pursuing the Ph.D. degree in electri-cal engineering at the University of Texas at Dallas,Richardson, TX, USA. He was an RF and ElectronicDesigner with Isfahan University of Technology from2008 to 2010, and a Research Associate with theDepartment of Electrical and Computer Engineering,

University of Tehran from 2010 to 2012. His research interests include commu-nication systems and signal processing.

Hlaing Minn (S’99–M’01–SM’07) received the B.E.degree in electrical engineering (electronics) fromYangon Institute of Technology, Yangon, Myanmar,in 1995, the M.Eng. degree in telecommunicationsfrom Asian Institute of Technology, Pathumthani,Thailand, in 1997, and the Ph.D. degree in electricalengineering from the University of Victoria, Victoria,BC, Canada, in 2001. From January to August 2002,he was a Postdoctoral Fellow with the University ofVictoria. Since September 2002, he has been with theErik Jonsson School of Engineering and Computer

Science, University of Texas at Dallas, Richardson, TX, USA, where he iscurrently a Full Professor. His research interests include wireless communica-tions, statistical signal processing, signal design, cross-layer design, cognitiveradios, dynamic spectrum access and sharing, energy efficient wireless systems,next-generation wireless technologies, and wireless healthcare applications. Heis an Editor of the IEEE TRANSACTIONS ON COMMUNICATIONS and theInternational Journal of Communications and Networks (Wiley). He has servedas a Technical Program Co-Chair for the Wireless Communications Symposiumof the IEEE Global Communications Conference (GLOBECOM 2014) and theWireless Access Track of the IEEE Vehicular Technology Conference (VTC2009”Fall), and a Technical Program Committee Member for several IEEEconferences.

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