ieee transactions on vehicular technology, vol. 68, …

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 68, NO. 4, APRIL 2019 3777

Multicell Massive MIMO: Downlink Rate AnalysisWith Linear Processing Under Ricean FadingSi-Nian Jin , Dian-Wu Yue , Senior Member, IEEE, and Ha H. Nguyen , Senior Member, IEEE

Abstract—This paper investigates the downlink (DL) rate ofmulticell massive multiuser multiple-input and multiple-outputsystems over Ricean fading channels that takes into account chan-nel estimation errors. To acquire channel state information at allusers, beamforming training (BT) is examined. Considering bothmaximum-ratio transmission (MRT) and zero forcing, this paperderives closed-form expressions on the lower bound of the achiev-able rates for two cases, with or without BT. With the obtainedexpressions, Bernoulli’s inequality is invoked to find the ranges forthe length of DL pilots such that the sum spectral efficiency of thescheme with BT is superior to that of the scheme without BT, andvice versa. Various power scaling laws concerning DL data and pi-lot transmit powers and uplink pilot transmit power are analyzed.Numerical results corroborate the accuracy of the closed-formexpressions. In particular, the results show that employing BT withMRT processing is only preferred in environments having a highsignal-to-noise ratio, low mobility, and small Ricean K-factors.

Index Terms—Massive MIMO, Ricean fading, downlink trans-mission, multicell, beamforming training, Bernoulli’s inequality.

I. INTRODUCTION

MASSIVE MIMO technology is a promising solution toaddress the very high spectral efficiency requirement

of 5G wireless networks and thus has attracted great interestsin both academia and industry [1]–[6]. The great performancebenefit of massive MIMO can be realized with simplelinear processing methods such as the maximum-ratiocombining/maximum-ratio transmission (MRC/MRT) andzero-forcing (ZF). An important result is given in [2], in whichthe author shows that the effects of intracell interference,small-scale fading and additive noise disappear as the numberof antennas increases without bound. Featuring up to hundredsof antennas, massive MIMO has the ability to drasticallyincrease both spectral and energy efficiencies of wirelesscommunication systems.

In massive MIMO systems, channel state information (CSI)plays an important role for uplink (UL) detection and downlink

Manuscript received April 6, 2018; revised November 1, 2018; acceptedFebruary 18, 2019. Date of publication February 27, 2019; date of current ver-sion April 16, 2019. This work was supported in part by Fundamental ResearchFunds for the Central Universities under Grant 3132016347 and in part by theProvincial Natural Science Foundation of Liaoning under Grant 201602086.The review of this paper was coordinated by Dr. Y. Xin. (Corresponding au-thor: Dian-Wu Yue.)

S.-N. Jin and D.-W. Yue are with the College of Information Scienceand Technology, Dalian Maritime University, Dalian 116026, China (e-mail:,[email protected]; [email protected]).

H. H. Nguyen is with the Department of Electrical and Computer Engineer-ing, University of Saskatchewan, Saskatoon SK S7N 5A9, Canada (e-mail:,[email protected]).

Digital Object Identifier 10.1109/TVT.2019.2902014

(DL) precoding. Due to the fact that time-division duplex (TDD)transmission has the characteristic of channel reciprocity, TDDis suitable for massive MIMO systems. In particular, channelreciprocity allows massive MIMO systems to obtain DL CSIfrom UL pilots sent by the users [2], [7]. The overhead of ULpilots is independent of the number of antennas at the basestation (BS) and is proportional to the number of users. Byusing the information in the unknown data symbols rather thanjust the pilot sequences, the quality of the channel estimates canbe significantly improved [8]–[10]. In particular, the authors in[11] study the performance of channel estimation and achievableUL rate in multi-cell massive MIMO systems based on theleast squares (LS) and minimum mean squared error (MMSE)methods and propose a pilot power allocation (PPA) scheme toimprove channel estimation performance.

For DL transmission in massive MIMO systems, beamform-ing training (BT) is an attractive method for the users to acquireCSI [12]–[16]. The BT method is very practical because it over-comes the problem that the overhead of DL pilots is proportionalto the number of antennas at the BS. Specifically, the BT methodpreprocesses pilot matrices so that the overhead of DL pilots isdetermined only by the number of users on the receiving side.The authors in [12] show that the system performance can be im-proved when a BT scheme is applied to single-cell DL systems.References [13] and [14] then analyze the system performancewhen a BT scheme is employed in multicell massive multiuser(MU) MIMO DL systems. A closed-form lower bound on theachievable DL data rate is derived in [15] for a BT scheme de-signed based on the MMSE precoder. Moreover, reference [16]illustrates that applying a BT scheme under aging channels isnot always beneficial even with the optimal transmission inter-val. In multi-user millimeter wave (mmWave) systems, due tothe fact that high-frequency transmission causes serious propa-gation loss [17], the BT method has been used to achieve beamalignment and obtain high antenna gain [18], [19].

In previous research works on massive MIMO DL systemsemploying BT [12], [13], the fading channels are assumed to beRayleigh distributed. While the Rayleigh fading model is rea-sonable for rich scattering environments, Ricean fading is moregeneral and includes Rayleigh fading as a special case. More-over, Ricean fading is a better model for fading environmentshaving dominant direct line-of-sight (LOS) path [20]–[28]. In[20], the authors study single-cell MIMO DL systems in Riceanfading channels and derive a lower bound on the average signal-to-leakage-and-noise ratio by using the Mullens inequality, andthen use it to analyze the effect of channel mean information

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3778 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 68, NO. 4, APRIL 2019

on the achievable ergodic sum-rate. Under Ricean fadingenvironments, the sum rates in single-cell and multicell massiveMIMO UL systems with MRC are investigated in [21] and[22], respectively. Reference [23] studies a novel statistical-eigenmode space-division multiple-access DL transmissionscheme and derives an exact closed-form expression for theergodic achievable sum-rate in Ricean fading channels. Inaddition, reference [24] considers the Ricean fading channelmodel and examines the impact of the LOS component onthe achievable sum rate and the energy efficiency of a massiveMIMO DL system when a ZF precoder is employed. Given theLOS component in Ricean fading, references [25]–[28] study alinear transmission scheme that takes into account only the LOScomponent and show that such a LOS component-based linearscheme can also achieve very good rate performance in massiveMIMO systems, while significantly reducing the pilot overhead.The reference [13] and [14] only considered the applicationof BT scheme under the environment of multicell massiveMU MIMO systems over Rayleigh fading channels. Riceanfading channels as a more general channel is not considered.However, in the case where BT scheme is not used, the authorsin [22] only considered the performance of multicell massiveMU MIMO systems over Ricean fading channels. Thereare, however, very few works that consider simultaneouslymulticell massive MIMO DL systems with BT and underRicean fading.

To fill this gap, this paper investigates multicell massiveMIMO DL systems with maximum-ratio transmission (MRT)and ZF operating over Ricean fading channels under imperfectCSI. In particular, the paper derives closed-form lower-boundexpressions of the DL achievable rates for two cases: with andwithout BT. Furthermore, different power scaling laws withrespect to DL data and pilot transmit powers and UL pilot trans-mit power are analyzed. The main contributions of the paper aresummarized as follows:

� For MRT and ZF processing, we use Bernoulli’s inequalityto find out the ranges for the length of DL pilots suchthat the sum spectral efficiency of the scheme with BT issuperior to that of the scheme without BT and vice versa.

� It is shown that with MRT and ZF, the DL transmit powerand the UL pilot transmit power can be scaled down toM−a and M−b (a = 1 and b ≥ 0), respectively, whilemaintaining a desirable user rate when the number of an-tennas M at the BS grows unlimited. In addition, whenb > 0 and M → ∞, the effect of pilot contamination canbe eliminated for the above case. For both MRT and ZF,we also find that the DL transmit power and the UL pi-lot transmit power can be scaled down to M−a and M−b

(0 ≤ a < 1 and b > 0), respectively, while the achievablerate approaches infinity when the number of antennas Mat the BS grows unlimited.

� Numerical results indicate that, for environments havinghigh signal-to-noise ratio (SNR), long coherence-time in-tervals and small Ricean K-factors, performance of thesystem employing MRT and BT is better than that withoutBT, while the opposite is true for environments having low

SNR, short coherence-time intervals and high Ricean K-factors. Compared with ZF precoding, MRT precoding ismore suitable for application in the multicell massive MUMIMO systems over Ricean fading channels.

Notation: Boldface upper-case and lower-case letters denotematrices and column vectors, respectively. The superscripts (·)∗,(·)T and (·)H stand for conjugate, transpose and conjugate-transpose operations, respectively. IN is an N × N identitymatrix and diag {a1, . . . , aN } is a diagonal matrix with diago-nal elements {a1, . . . , aN }. The trace, smallest element, largestelement, expectation, variance and covariance operators are de-noted by tr (·), min{·}, max{·}, E{·}, Var(·) and Cov {X,Y },respectively. [A]mn gives the (m,n)th entry of A. Finally, weuse z ∼ CN (0,A) to denote a circularly symmetric complexGaussian vector z with zero mean and covariance matrix A.

II. SYSTEM MODEL

A. Channel Model

Consider a multicell system with L cells where each cell hasa BS equipped with M equally-spaced, omnidirectional anten-nas (i.e., an uniform linear array (ULA)) serving N randomlylocated single-antenna users. Denote by Gil ∈ CM ×N the chan-nel matrix between the BS in the ith cell and the N users in thelth cell. The matrix Gil can be represented as [1]

Gil = HilD1/2il , (1)

where Dil = diag {βil1, . . . , βiln , . . . , βilN } and βiln repre-sents the large-scale fading effect between the BS in the ithcell and the nth user in the lth cell. In essence, βiln modelsthe geometric attenuation and shadow fading, which is assumedconstant over many coherence time intervals and known a priori[1]. However, the fast fading matrix between the users and theBS in the same cell is modeled to consist of two parts, namelya deterministic component corresponding to the LOS path anda Rayleigh-distributed random component which accounts forthe scattered signals. This assumption is reasonable, since theLOS components exist in channels between the BS and users inthe own cell due to the short range, but, because of the scattersand buildings block, they do not exist any more in channels be-tween the BS and users in different cells. As discussed in [22],Hil ∈ CM ×N can be expressed as

Hil =

⎧⎨

⎩

Hi

[Ω i

Ω i +IN

]1/2+ Hw,ii

[IN

Ω i +IN

]1/2, l = i

Hw,il , l = i(2)

where Ωi = diag {Ki1, . . . , Kin , . . . , KiN } and Kin ≥ 0 is theRicean K-factor that represents the power ratio of the LOS andnon line-of-sight (NLOS) components for the nth user in the ithcell. The entries in matrix Hw,il ∈ CM ×N are independent andidentically distributed (i.i.d.) CN (0, 1) random variables, andHi represents the LOS component of Hii . Each element of Hi

is governed by the steering response of the ULA with a LOSdirection of departure and can be expressed as [21], [22], [29]

[Hi

]

mn= e−j (m−1)(2πd/λ) sin(θi n) , (3)

JIN et al.: MULTICELL MASSIVE MIMO: DOWNLINK RATE ANALYSIS WITH LINEAR PROCESSING UNDER RICEAN FADING 3779

where λ is the carrier wavelength, θin is the departure angle ofthe nth user in the ith cell and d is the antenna spacing. Forconvenience, we also set d = λ/2 and θin ∈ [−π, π] [29].

B. Channel Estimation

Given the channel reciprocity property of a TDD multicellsystem, Gil is usually estimated by utilizing receiving pilot se-quences at the BS. Since the LOS component changes slowly,the BS can estimate the LOS component very accurately withnegligible signaling overhead. As in [22], Hi and Ωi are as-sumed to be known at both the BS and the users. Then, theestimate of Gil is given by [22]

Gil =

⎧⎨

⎩

Gi

[Ω i

Ω i +IN

]1/2+ Gw,ii

[IN

Ω i +IN

]1/2, l = i

Gw,il , l = i(4)

where Gi = HiD1/2ii and Gw,il represents the estimate of

random matrix Gw,il = Hw,ilD1/2il . Assume that, during the

channel estimation phase, users of different cells simultane-ously transmit the same set of UL orthogonal pilot sequencesof τu symbols, which can be stacked in a N × τu matrix√

τu pu Ψ (τu ≥N ), which satisfies ΨΨH = IN (a typical setup inmassive MIMO). In this case, the channel estimate from the ULpilots is contaminated by the users from other cells using thesame pilot. The MMSE estimate of Gw,il is given as [1]

Gw,il =

⎛

⎝L∑

j=1

Gw,ij +1√τupu

Zi

⎞

⎠ Dil , (5)

where Dil = Dil( 1τu pu

IN +∑L

j=1 Dij )−1, pu is the transmit

power of UL pilot and the entries of Zi ∈ CM ×N are i.i.d.CN (0, 1) random variables.

The true and estimated channel matrices are related asGw,il = Gw,il + εw,il , where εw,il is the estimation error ma-trix. Under MMSE estimation, Gw,il is statistically independentof εw,il . Furthermore, the nth column vector of Gw,il and εw,il ,denoted as gw,iln , and εw,iln have the following distributions:gw,iln ∼ CN (0, σ2

gw , i l nIM ) and εw,iln ∼ CN (0, σ2

εw , i l nIM ),

where

σ2gw , i l n

=β2

iln∑L

l ′=1 βil ′n + 1τu pu

, (6)

σ2εw , i l n

=βiln

(∑Ll ′=1,l ′ = l βil ′n + 1

τu pu

)

∑Ll ′=1 βil ′n + 1

τu pu

. (7)

C. Downlink Transmission

In DL transmission, each BS transmits data to its users. Thereceived signal at the nth user in the ith cell is written as [13]

rin =√

pd

L∑

l=1

gTlinWlsl + nin , (8)

where pd is the DL transmitted power, sl = [sl1, . . . , slN ]T is thetransmit signal vector for users in the lth cell with E{sl} = 0and E{slsH

l } = IN , nin ∼ CN (0, 1) is the noise term at thenth user in the ith cell, glin denotes the nth column vector of

Gli and Wl ∈ CM ×N denotes the precoding matrix of the lthcell.

When the BS uses MRT to process the transmitted sig-nal, Wl = αlG∗

ll , where αl satisfies the power constraint ofE{tr(WlWH

l )} = 1, which is given as αl = μl√M

with

μl =1

√∑N

j=1

Kl j βl l j +σ 2g w , l l j

Kl j +1

. (9)

On the other hand, for ZF precoding, Wl = γlAl , whereAl = G∗

ll(GTll G

∗ll)

−1 and γl satisfies the power constraint ofE{tr(WlWH

l )} = 1, which is given as γl =√

M −Nη l

, where ηl =∑N

j=1

[Σ−1

l

]

jj, and [21]

Σl = Λl +TH

l Tl

M, (10)

Λl = diag

{σ2

gw , l l 1

Kl1 + 1, . . . ,

σ2gw , l l N

KlN + 1

}

, (11)

Tl = Gl

[Ωl

Ωl + IN

]1/2

. (12)

The expression in (8) can be expanded as [13]

rin =√

pdaiinnsin +∑

(l,j ) =(i,n)

√pdailnj slj + nin , (13)

where ailnj = αlgTlin g∗

llj and ailnj = γlgTlinalj are the effective

CSI for MRT and ZF precoding with gllj and alj are the jthcolumn vector of Gll and Al , respectively.

D. Achievable Rate of Beamforming Training Scheme

The DL communication in a conventional TDD system con-sists of UL training phase and DL data transmission phase.However, for a system employing BT, an additional phase isallocated to estimate the effective channel. During this phase,all BSs beamform DL pilots synchronously, and each user es-timates ailnj from the received DL pilots. Let Φ ∈ CN ×τd bethe DL pilot matrix, where τd ≥ N is the symbol length of theDL pilot sequences. The pilot matrix is designed such that itsrows are pairwise orthogonal, i.e., ΦΦH = IN . Suppose that Φis reused among BSs in all the cells. Then, the received DL pilotmatrix at the users in the ith cell is given by [13]

Ri =√

τdpd

L∑

l=1

GTliWlΦ + Ni , (14)

where Ni ∈ CN ×τd is the noise matrix with i.i.d. CN (0, 1) en-tries. It is sufficient to use Ri = RiΦH for channel estimation,which can be written as [30]

Ri =√

τdpd

L∑

l=1

GTliWl + Ni , (15)

where Ni = NiΦH . Note that the entries of Ni are i.i.d.CN (0, 1) random variables due to the pairwise orthogonalityof Φ. The DL pilot vector received by the nth user in the ith


cell can be expressed as

rin =√

τdpd

L∑

l=1

gTlinWl + nin =

√τdpd

L∑

l=1

alin + nin ,

(16)

where alin = [ailn1, . . . , ailnN ], and nin is the nth row of Ni .The effective channel alin can be estimated independently

[12]. Based on (16), the MMSE estimation of ailnj is [13]

ailnj = E {ailnj} +Cov {ailnj , rin,j}Cov {rin,j , rin,j} (rin,j − E {rin,j}) ,

(17)

where the jth element of rin is given by

rin,j =L∑

l=1

√τdpdailnj + nin,j (18)

and nin,j is the jth element of nin . Then, the expression in (13)can be rewritten as

rin =√

pdaiinnsin +∑

(l,j ) =(i,n)

√pdailnj slj

+L∑

l=1

N∑

j=1

√pdζilnj slj + nin , (19)

where ζilnj = ailnj − ailnj is the estimation error of the effec-tive channel.

From (19), when BT is employed, a lower bound onthe DL achievable rate of the nth user in the ith cell forMRT or ZF precoding is given in (20), shown at the bot-tom of this page [13], and the expressions of E{|ζilnj |2}(i, l = 1, . . . , L;n, j = 1, . . . , N) are given in (104) and (141)for MRT and ZF precoding, respectively.

Finally, the DL sum spectral efficiency of the ith cell with BTis expressed as

C(A)i =

T − τu − τd

T

N∑

n=1

R(A)in , (21)

where A ∈ {MRT,ZF} corresponds to MRT or ZF processing,and T is the coherence time of the channel. The computation ofC

(A)i is examined next.With the technique from [13, Lemma 2], the lower bound in

(20) can be approximated by (22), shown at the bottom of this

page. Given the closed-form expression of R(A)in in (22), the sum

spectral efficiency in the ith cell for MRT and ZF precoding withBT scheme can be approximated as:

C(A)i =

T − τu − τd

T

N∑

n=1

R(A)in . (23)

For comparison, we now study the conventional DL schemewithout BT, i.e., each user detects the signal based only on thestatistical CSI [31]. To this end, rewrite (13) as

rin =√

pdE {aiinn} sin +√

pd (aiinn − E {aiinn}) sin

+∑

(l,j ) =(i,n)

√pdailnj slj + nin . (24)

Then the lower bound on the achievable rate (without BT) ofthe nth user in the ith cell for MRT or ZF precoding is given in(25), shown at the bottom of this page. The result in (25) canthen be used to compute the DL sum spectral efficiency of theith cell for MRT and ZF precoding without BT scheme as:

C(A)i =

T − τu

T

N∑

n=1

R(A)in . (26)

III. DOWNLINK RATE ANALYSIS FOR MRT PROCESSING

A. Achievable Rates of MRT Processing: With and Without BT

By performing the expectation of each term in (22) and (25),we obtain Theorem 1 and Theorem 2 for MRT processing withand without BT, respectively.

Theorem 1: In a multicell MU massive MIMO system withBT and MRT precoding, the lower bound on the DL achievablerate of the nth user in the ith cell can be given as

R(MRT)in = log2

(

1 +Δ1 + Δ5

∑Nj=1 Δij + Δ2 + Δ3 − Δ5 + 1

pd

)

,

(27)

R(A)in = E

⎧⎨

⎩log2

⎛

⎝1 +pd |aiinn |2

pd

∑(l,j ) =(i,n) |ailnj |2 + pd

∑Ll=1

∑Nj=1 E

{|ζilnj |2

}+ 1

⎞

⎠

⎫⎬

⎭(20)

R(A)in = log2

⎛

⎝1 +pdE

{|aiinn |2

}

pd

∑(l,j ) =(i,n) E

{|ailnj |2

}+ pd

∑Ll=1

∑Nj=1 E

{|ζilnj |2

}+ 1

⎞

⎠ (22)

R(A)in = log2

⎛

⎝1 +pd |E {aiinn}|2

pd

∑(l,j ) =(i,n) E

{|ailnj |2

}+ pdVar {aiinn} + 1

⎞

⎠ (25)


where

Δ1 =Mμ2

i

(Kinβiin + σ2

gw , i i n

)2

(Kin + 1)2 , (28)

Δij =μ2

i βiin

(Kijβiij + (Kin + 1) σ2

gw , i i j

)

(Kin + 1) (Kij + 1), (29)

Δ2 =N∑

j=1,j =n

μ2i βiinβiijKinKijΔ2

inj

M (Kin + 1) (Kij + 1), (30)

Δinj =sin

(M π

2 [sin (θin ) − sin (θij )])

sin(

π2 [sin (θin ) − sin (θij )]

) , (31)

Δ3 =L∑

l=1,l =i

Mμ2l σ

2gw , l l n

σ2gw , l i n

Kln + 1+ βlin , (32)

Δ4 =L∑

l=1,l =i

μ2l βlin

(Klnβlln + σ2

gw , l l n

)

Kln + 1, (33)

Δ5 =Δ2

in

Δin + Δ4 + 1τd pd

. (34)

Proof: See Appendix A. �Through Appendix A, we can know that Δ1 means the part

of useful signal,∑N

j=1 Δij and Δ2 represent intra-cell inter-ference, Δ3 represents inter-cell interference and Δ5 representsthe performance gain brought by the BT scheme for MRT pro-cessing. In particular, the first part of Δ3 is the effect of pilotcontamination.

With MRT processing, the authors in [32] derive the effec-tive signal-to-interference-plus-noise ratio (SINR) and max-minpower control under LOS, and carry out comparison of theasymptotic properties of the correlation of channel vectors inLOS and Rayleigh fading as M increases. Real-world propa-gation environment is likely to fall somewhere between LoSand Rayleigh fading environment. Ricean fading channel is oneof representative channel in actual environment. Therefore, theresults of Theorem 1 can be considered a more general resultfor the paper [32].

Following the same steps in the derivation of R(MRT)in , a

closed-form expression of R(MRT)in (without BT) can be obtained

in [28]. In order to facilitate comparison with the performanceof MRT with BT, it is stated in Theorem 2.

Theorem 2: In a multicell MU massive MIMO system with-out BT, the lower bound on the DL achievable rate of the nthuser in the ith cell for MRT precoding can be given as [28]

R(MRT)in = log2

(

1 +Δ1

∑Nj=1 Δij + Δ2 + Δ3 + 1

pd

)

. (35)

Substituting (27) into (23) and (35) into (26) yields the DLsum spectral efficiency, denoted C

(MRT)i and C

(MRT)i , for MRT

processing with and without BT, respectively. To facilitate thecomparison between the two systems (with and without BT),the next two corollaries establish the sufficient conditionsthat the length of DL pilot sequences should satisfy so

that C(MRT)i ≥ C

(MRT)i and C

(MRT)i ≥ C

(MRT)i , respectively.

These corollaries are obtained by using Bernoulli’s inequality.Corollary 1: For MRT precoding, it is true that C

(MRT)i ≥

C(MRT)i if τd satisfies

max{τud,MRT , N

} ≤ τd ≤ T − τu , (36)

where

τud,MRT = max

n=1,...,N

{pd (T − τu ) (Δ1 + Δ6) Δ2

in − Δ1Δ6

pd

(Δ6Δ2

in + Δ1Δ6Δin + Δ1Δ4Δ6)

}

,

(37)

Δ6 =N∑

j=1

Δij + Δ2 + Δ3 +1pd

. (38)

Proof: See Appendix B. �Corollary 2: For MRT precoding, it is true that C

(MRT)i ≥

C(MRT)i if τd satisfies

N ≤ τd ≤ min{τ ld,MRT , T − τu

}, (39)

where

τ ld,MRT = min

n=1,...,N

{pd (T − τu ) Δ2

in − Δ1

pd

(Δ2

in + Δ1Δin + Δ1Δ4)

}

. (40)

Proof: See Appendix C. �Furthermore, it follows from Corollary 1 and Corollary 2

that there exists a value of τd that leads to C(MRT)i = C

(MRT)i ,

which is stated in the following corollary.Corollary 3: For MRT precoding, when N ≤ τ l

d,MRT <τud,MRT ≤ T − τu , both schemes, with and without BT, have

the same DL sum spectral efficiency for a value of τd in therange τ l

d,MRT ≤ τd ≤ τud,MRT .

B. Power Scaling Laws With MRT Processing

In general, power scaling is an important characteristic of anymassive MIMO system since it indicates how the deployment oflarge-scale (massive) antenna array helps to scale down transmitpower while maintaining system’s target rate [28], [33]. Basedon C

(MRT)i and C

(MRT)i , this subsection provides asymptotic

rate expressions of MRT precoding with and without BT indifferent power scaling laws.

Corollary 4: Let pd = pd/Ma and pu = pu/Mb , where 0 ≤

a < 1, b = 0, and pd and pu are fixed. When M → ∞, the sumspectral efficiencies achieved by MRT with and without BT tendto

C(MRT)i → T − τu − τd

T

N∑

n=1

log2

(

1 +Δ7

Δ8

)

, (41)

and

C(MRT)i → T − τu

T

N∑

n=1

log2

(

1 +Δ7

Δ8

)

, (42)


where

Δ7 =μ2

i

(Kinβiin + σ2

gw , i i n

)2

(Kin + 1)2 , (43)

Δ8 =L∑

l=1,l =i

μ2l σ

2gw , l l n

σ2gw , l i n

Kln + 1. (44)

Corollary 4 implies that when the number of BS antennasgrows without bound, the effects of channel estimation error,noise and interference from users that use different pilots disap-pear, and pilot contamination which is denoted as Δ8 becomesthe only remaining interference for MRT precoding in this powerscaling case. It can also be seen that both C

(MRT)i and C

(MRT)i

are increasing functions of Kin , an intuitively satisfying result.Corollary 5: Let pd = pd/M

a and pu = pu/Mb , wherea = 1, b = 0, and pd and pu are fixed. When M → ∞, thesum spectral efficiencies achieved by MRT with and withoutBT tend to


T

N∑

n=1

log2

(

1 +pdΔ7

pdΔ8 + 1

)

, (45)

and


T

N∑

n=1

log2

(

1 +pdΔ7

pdΔ8 + 1

)

. (46)

Comparing Corollary 4 and Corollary 5 shows that thesum spectral efficiencies in Corollary 4 are greater than thecorresponding (with or without BT) spectral efficiencies inCorollary 5, respectively. This is a direct consequence of thefact that the DL power is scaled down more in Corollary 5 thanin Corollary 4.

Corollary 6: Let pd = pd/Ma and pu = pu/Mb , where

a = 1, b > 0 (including b ≥ a = 1), and pd and pu are fixed.When M → ∞, the sum spectral efficiencies achieved by MRTwith and without BT tend to

C(MRT)i →

T − τu − τd

T

N∑

n=1

log2

⎛

⎝1 +pdK

2inβ2

iin

(1 + Kin )2 ∑Nj=1

Ki j βi i j

Ki j +1

⎞

⎠,

(47)

and


T

N∑

n=1

log2

⎛

⎝1+pdK

2inβ2

iin

(1 + Kin )2 ∑Nj=1

Ki j βi i j

Ki j +1

⎞

⎠.

(48)

Corollary 5 and Corollary 6 reveal that the impact of powerreduction of UL pilot power on the sum spectral efficiencydepends on the Ricean K-factor. Comparing Corollary 5 andCorollary 6, we also find that when b > 0, the system perfor-mance are only affected by Ricean K-factor and large scalefading coefficient, the interference from other cells does notimpact the DL sum spectral efficiencies, which means that theeffect of pilot contamination can be eliminated for both schemes(with and without BT).

Corollary 7: Let pd = pd/Ma and pu = pu/Mb , where 0 ≤

a < 1, b > 0, pd and pu are fixed. When M → ∞ and Kin > 0,the sum spectral efficiencies C

(MRT)i and C

(MRT)i go to ∞.

Corollary 7 show that, as M → ∞, the effect of interferenceand noise will not be able to limit system performance and therate performance of system approaches infinity for MRT pro-cessing with and without BT in this power scaling law. Com-paring Corollary 4, we can see that when b > 0, the systemof asymptotic performance has been improved. This is becauseas M → ∞ and pu → 0, the influence of pilot contaminationis transformed into the influence of independent noise and theeffect of pilot contamination gradually disappears (Δ8 → 0).Furthermore, enlarging the large-scale antenna arrays at BS hasa much stronger effect on the system performance as comparedwith decreasing UL pilot and DL transmit powers.

Corollary 8: Let Kin → ∞, i = 1, 2, . . . , L; n = 1, 2,. . . , N . Then the sum spectral efficiencies of MRT precodingwith and without BT converge to


T

N∑

n=1

log2

(

1 +M 2β2

iin

Δ9 + Δ10

)

, (49)

and


T

N∑

n=1

log2

(

1 +M 2β2

iin

Δ9 + Δ10

)

, (50)

where

Δ9 =N∑

j=1,j =n

βiinβiijΔ2inj , (51)

Δ10 = M

N∑

j=1

βiij

⎛

⎝L∑

l=1,l =i

βlin +1pd

⎞

⎠ . (52)

We can see that as the Ricean K-factor increases withoutbound, the sum spectral efficiencies with and without BT ap-proach constant values. When M → ∞, (49) and (50) will ap-proach infinity. This shows that when both the Ricean K-factorand the number of BS antennas approach infinity, the effect ofall interference and noise disappear for the two schemes (withand without BT).

IV. DOWNLINK RATE ANALYSIS OF ZF PROCESSING

The analysis and presentation in this section for ZF processingparallel those in Section III for MRT processing.

A. Achievable Rates of ZF Processing: With and Without BT

Based on (22) and (25), we can obtain the DL achievable rateof ZF processing with and without BT as stated in Theorem 3and Theorem 4.

Theorem 3: In a multicell MU massive MIMO system withBT and ZF precoding, the lower bound on the DL achievable


rate of the nth user in the ith cell can be given as

R(ZF)in =

log2

(

1 +Θ1 + Θ3

∑Nj=1 Θj +

∑Ll=1;l =i

∑Nj=1 Θlj + Θ2 − Θ3 + 1

pd

)

,

(53)

where Σl is given in (10) and

Θj =γ2

i σ2εw , i i n

[Σ−1

i

]

jj

(M − N) (Kin + 1), (54)

Θlj = γ2l

⎛

⎝σ2

εw , l i n

[Σ−1

l

]

jj

M − N+

σ2gw , l i n

(Klj + 1)

M(Kljβllj + σ2

gw , l l j

)

⎞

⎠ ,

(55)

Θ1 = γ2i , (56)

Θ2 =L∑

l=1;l =i

γ2l σ

2gw , l l n

σ2gw , l i n

(Kln + 1)(Klnβlln + σ2

gw , l l n

)2 , (57)

Θ3 =Θ2

n

Θn +∑L

l=1;l =i Θln + 1τd pd

. (58)

Proof: See Appendix D. �Through Appendix D, it is observed that Θ1 denotes the

part of useful signal,∑N

j=1 Θj represent intra-cell interfer-

ence,∑L

l=1;l =i

∑Nj=1 Θlj and Θ2 which is the effect of pilot

contamination represent inter-cell interference, and Θ3 repre-sents the performance gain brought by the BT scheme for ZFprocessing.

Theorem 4: In a multicell MU massive MIMO system with-out BT, the lower bound on the DL achievable rate of the nthuser in the ith cell for ZF precoding can be given as

R(ZF)in =

log2

(

1 +Θ1

∑Nj=1 Θj +

∑Ll=1;l =i

∑Nj=1 Θlj + Θ2 + 1

pd

)

.

(59)

Substituting (53) into (23) and (59) into (26), we can obtainthe DL sum spectral efficiencies, C

(ZF)i and C

(ZF)i , for ZF pro-

cessing with and without BT. Furthermore, following similarderivations in Appendix B and Appendix C for MRT precoding,we can obtain the following corollaries regarding the length ofDL pilot sequences for ZF precoding.

Corollary 9: For ZF precoding, it is true that C(ZF)i ≥ C

(ZF)i

if τd satisfies

max{τud,ZF , N

} ≤ τd ≤ T − τu , (60)

where

τud,ZF = max

n=1,...,N

{pd (T − τu ) (Θ1 + Θ5) Θ2

n − Θ1Θ5

pd (Θ5Θ2n + Θ1Θ5Θn + Θ1Θ4Θ5)

}

,

(61)

Θ4 =L∑

l=1;l =i

Θln , (62)

Θ5 =N∑

j=1

Θj +L∑

l=1;l =i

N∑

j=1

Θlj + Θ2 +1pd

. (63)

Corollary 10: For ZF precoding, it is true that C(ZF)i ≥

C(ZF)i if τd satisfies

N ≤ τd ≤ min{τ ld,ZF , T − τu

}, (64)

where

τ ld,ZF = min

n=1,...,N

{pd (T − τu ) Θ2

n − Θ1

pd (Θ2n + Θ1Θn + Θ1Θ4)

}

. (65)

Based on Corollary 9 and Corollary 10, we also can finda value of τd that leads to C

(ZF)i = C

(ZF)i and it is stated in

Corollary 11.Corollary 11: For ZF precoding, when N ≤ τ l

d,ZF <τud,ZF ≤ T − τu , both schemes, with and without BT, have the

same DL sum spectral efficiency for a value of τd in the rangeτ ld,ZF ≤ τd ≤ τu

d,ZF .

B. Power Scaling Laws With ZF Processing

Based on C(ZF)i and C

(ZF)i , Corollary 12 to Corollary 16

establish the power scaling laws of ZF scheme with and withoutBT in different case.


0 ≤ a < 1, b = 0, and pd and pu are fixed. When M → ∞,the sum spectral efficiencies achieved by ZF with and withoutBT tend to

C(ZF)i → T − τu − τd

T

N∑

n=1

log2

(

1 +1

Θ6

)

, (66)

and

C(ZF)i → T − τu

T

N∑

n=1

log2

(

1 +1

Θ6

)

, (67)

where

Θηl=

N∑

j=1

Klj + 1Kljβllj + σ2

gw , l l j

, (68)

Θ6 =L∑

l=1;l =i

Θηiσ2

gw , l l nσ2

gw , l i n(Kln + 1)

Θηl

(Klnβlln + σ2

gw , l l n

)2 . (69)

It follows from Corollary 12 that, when M → ∞, it can beobserved that Θ6 originated from Θ2 is the only factor limitingsystem performance. It indicate that the effects of channel esti-mation error, noise and interference from users that use differentpilots disappear, and pilot contamination becomes the only fac-tor that affects the system performance for ZF precoding withand without BT.



a = 1, b = 0, and pd and pu are fixed. When M → ∞, thesum spectral efficiencies achieved by ZF with and without BTtend to


T

N∑

n=1

log2

(

1 +pd

pdΘ6 + Θηi

)

, (70)

and


T

N∑

n=1

log2

(

1 +pd

pdΘ6 + Θηi

)

. (71)

Compared with Corollary 12, we can find that, when a = 1,noise also become a factor affecting the system except the effectof pilot contamination for ZF processing with and without BT.


a = 1, b > 0 (including b ≥ a = 1), and pd and pu are fixed.When M → ∞, the sum spectral efficiencies achieved by ZFwith and without BT tend to


T

N∑

n=1

log2

(

1 +pd

Θ7

)

, (72)

and


T

N∑

n=1

log2

(

1 +pd

Θ7

)

. (73)

where

Θ7 =N∑

j=1

Kij + 1Kijβiij

. (74)

For ZF processing over Ricean fading, Corollary 13 andCorollary 14 reveal that, as the number of antennas goes toinfinity, the DL transmit and pilot power can be scaled downby M−1 to maintain a given DL rate performance for the twoschemes (with and without BT). When b > 0, Θ7 consists ofRicean K-factor and large scale fading coefficient, the interfer-ence from other cells does not influence the lower bound onthe DL achievable rate, which indicates that the effect of pilotcontamination can be eliminated and noise becomes the onlyfactor limiting system performance. It is pointed out that simi-lar results are obtained in Corollary 5 and Corollary 6 for MRTprecoding.


0 ≤ a < 1, b > 0, pd and pu are fixed. When M → ∞ andKin > 0, the sum spectral efficiencies C

(ZF)i and C

(ZF)i go

to ∞.The above power scaling law implies that, as the number

of antennas goes to infinity, all interference and noise will notaffect the system performance of ZF processing for the twoschemes (with and without BT). By comparing Corollary 12and Corollary 15, it can be seen that when b > 0, the system ofZF processing leads to better asymptotic performance. This isbecause as M → ∞ and pu → 0, the influence of pilot contam-ination can be gradually eliminated (Θ6 → 0) and the influenceof independent noise becomes more dominant in this powerscaling law. In particular, the gain from the increasing numberof BS antennas is much greater than the impact of decreasing

UL pilot and DL transmit power. For the system of MRT pre-coding with and without BT, we can obtain similar results inCorollary 4 and Corollary 7.

Corollary 16: Let Kin → ∞, i = 1, 2, . . . , L; n = 1, 2,. . . , N . Then the sum spectral efficiencies of ZF precoding withand without BT converge to


T

N∑

n=1

log2

(

1 +pdΘγi

pdΘ8 + 1

)

, (75)

and


T

N∑

n=1

log2

(

1 +pdΘγi

pdΘ8 + 1

)

, (76)

where

Θllj =[GH

l Gl

M

]−1

jj

, (77)

Θγl=

M − N∑N

j=1 Θllj

, (78)

Θ8 =L∑

l=1;l =i

N∑

j=1

Θγl

(σ2

εw , l i nΘllj

M − N+

σ2gw , l i n

Mβllj

)

. (79)

Corollary 16 shows that, as the Ricean K-factor go to ∞,the sum spectral efficiencies of ZF processing with and withoutBT approach constant values. When M → ∞, (75) and (76)will approach infinity. This indicates that when both the RiceanK-factor and the number of BS antennas approach infinity, allinterference and noise do not limit the system’s rate performancefor ZF precoding with and without BT. These are similar toresults obtained in Corollary 8 for MRT processing.

Before closing this section, it is pointed out that the powerscaling laws for MRT and ZF processing with and without BTstudied in Sections III-B and IV-B are all under the case ofequal DL data and pilot transmit powers and UL pilot transmitpower. However, it is known that proper power allocation andpilot design can improve the spectral efficiency, mitigate pilotcontamination and provide good service for everyone in multi-cell massive MIMO systems. In particular, in massive MIMO,power allocation can break the limitations from the assumptionof equal transmit power among user and BS, and contributemuch to harvest all the benefits brought by the large antennaarrays [34]–[36]. In [34], joint power control is considered forquality-of-service (QoS) when noncooperative beamforming isused, which results in a low overhead to exchange limited CSIbetween BS. For the joint power control, the bounds on SINRin terms of slow-fading coefficients are derived and a better per-formance is achieved. This paper [35] proposed a new methodto jointly optimize the power allocation and user association inmulti-cell Massive MIMO systems. The DL non-coherent jointtransmission was designed to minimize the total transmit powerconsumption while satisfying QoS constraints. Research on thepilot assignment and power allocation into a unified optimiza-tion framework have attracted strong interests in recent years[36], but these issues are beyond the scope of this paper and leftfor further studies.


V. NUMERICAL RESULTS

In this section, simulation results are presented and com-pared with the theoretical results obtained in the previous sec-tion. The network is simulated with L = 7 base stations anda frequency reuse factor of 1. This means that a given cell isinterfered by its six neighboring cells. The radius of each cellis set to be rc = 1000 meters. There are N = 5 users that aredistributed randomly and uniformly in each cell with the exclu-sion of a central disk of radius rh = 100 meters. The randomnature of the user’s location will bring different large scale fad-ing coefficient for each user and base station. The large scalefading coefficient is the important parameter affecting systemperformance. The large scale fading coefficient βiln is gener-ated by using the formula βiln = γ/( ri l n

rh)υ , where γ is the

log-normal random variable with standard deviation σ = 8 dB,υ = 4 is the path loss exponent, and riln is the distance betweenthe nth user in the lth cell and the BS in the ith cell. All ofthe large-scale fading coefficients are chosen as given in [37]with the intercell interference factor β = 1 and are shown as(80) for l1, l2, l3, l4, l5, l6, l7 ∈ {1, 2, . . . , L}. For convenience,we assume that users in all cells have the same Ricean K-factor,denoted as K. For the system of MRT with BT, the simulationresults and the analytical results are obtained by (21), (23) and(27), respectively. For the system of MRT without BT, the sim-ulation results and the analytical results are obtained by (25),(26) and (35), respectively. For the system of ZF with BT, thesimulation results and the analytical results are obtained by (21),(23) and (53), respectively. For the system of ZF without BT,the simulation results and the analytical results are obtained by(25), (26) and (59), respectively.

Dl1l1 = 10−3diag [29.028, 0.319, 908.275, 286.484, 0.525]

Dl1l2 = 10−3diag [0.0135, 0.1112, 0.0210, 0.5061, 1.2007]

Dl1l3 = 10−3diag [1.6199, 0.0028, 0.4856, 0.1218, 0.0108]

Dl1l4 = 10−3diag [0.0684, 0.0130, 0.0236, 0.0692, 0.2540]

Dl1l5 = 10−3diag [2.5533, 0.0189, 0.9230, 0.0112, 0.0089]

Dl1l6 = 10−3diag [0.1040, 0.1345, 0.0234, 1.3590, 0.3573]

Dl1l7 = 10−3diag [1.5309, 0.3749, 0.3129, 0.0647, 0.0210](80)

Fig. 1 compares the DL sum spectral efficiencies of MRTand ZF with and without BT in Ricean fading for two differentRicean K-factors, K = −3 dB and K = 10 dB. For this figure,we set M = 100, pu = 10 dB, τu = τd = N and T = 200. Ascan be seen, the performance gaps between the simulation re-sults and analytical results are small. As the DL SNR increases,the performance of MRT precoding is better than that of ZFprecoding, and the system of MRT with BT gradually showsperformance advantage over the system of MRT without BT,and the performance of ZF with and without BT are basicallythe same. This shows that the scheme of MRT with BT has thegreatest performance advantage in the multicell massive MUMIMO systems over Ricean fading channels. As K increases,the DL rate grows noticeably for MRT and ZF precoding. This

Fig. 1. Effect of SNR on the DL sum spectral efficiency: M = 100, pu =10 dB, τu = τd = N and T = 200.

Fig. 2. Effect of coherence interval on the DL sum spectral efficiency: M =100, τu = τd = N and pu = pd = 10 dB.

also suggests that the DL rates under Ricean fading are higherthan that under Rayleigh fading for MRT and ZF precoding. Dueto the tightness between the simulation and theoretical results,we will only report the theoretical results in the remaining ofthis section.

Fig. 2 plots the sum spectral efficiency versus the length ofthe coherence interval for K = −3 dB and K = 10 dB. In thisexample, we assume M = 100, τu = τd = N and pu = pd =10 dB. As the length of the coherence interval increases, we canfind that the length of the coherence interval has less influenceon the scheme of ZF with and without BT. Compared to MRTprocessing, the performance of the ZF precoding is relativelypoorer. This also shows that MRT, which has lower processingcomplexity, is more suitable for multicell massive MU MIMOsystems over Ricean fading channels. We note that the perfor-mance of MRT with BT is better than the performance of MRTwithout BT for long coherence interval (i.e., in low-mobility en-vironments). This is expected since with large coherence time,the length of DL pilot sequences is relatively small compared


Fig. 3. Effect of the number of BS antennas on the DL sum spectral efficiencyfor MRT precoding: K = −3 dB, T = 200, τu = τd = N and pu = pd =10 dB.

to the coherence time. It can also be seen that as K increases,the scheme of MRT with BT has advantage over the schemeof MRT without BT only for larger coherence time. These ob-servations suggest that the scheme of MRT with BT is moresuitable for channel environments with low-mobility and smallRicean K-factor.

We now study two typical power scaling cases as follows:Case 1: pd = pd

M and pu = pu ; Case 2: pd = pd

M and pu = pu

M . Inthis example, we assume K = −3 dB, T = 200, τu = τd = Nand pu = pd = 10 dB. The DL sum spectral efficiency for MRTwith and without BT in these two cases are plotted in Fig. 3. Itis seen that in Case 1 when M grows large, both sum spectralefficiencies, with and without BT, increase gradually until ap-proaching their corresponding constant asymptotic values pre-sented in Corollary 5. In contrast, in Case 2 both sum spectralefficiencies with and without BT decrease gradually until ap-proaching their corresponding constant asymptotic values pre-sented in Corollary 6. We can also see that as the number of BSantennas increases, the performance without BT gradually ex-ceeds that with BT for the two cases. For the scheme of ZF withand without BT, similar results of power scaling were obtainedbut not included here due to space limitation.

Fig. 4 shows how the sum spectral efficiencies vary withRicean K-factor when M = 100, T = 200, τu = τd = N andpu = pd = 10 dB. It can be seen from Fig. 4 that, as the RiceanK-factor increases, the sum spectral efficiencies for MRT andZF increase gradually toward the asymptotic results given inCorollary 8 and Corollary 16 for the systems with and withoutBT. This indicates that having a larger Ricean K-factor leadsto better rate performance. We can also see that the system ofMRT with BT performs better than the system of MRT withoutBT when the Ricean K-factor is smaller, and vice versa.

We assume that the randomness is due to random user loca-tions and large scale fading realizations. Furthermore, we chooseM = 100, pu = pd = 20 dB, τu = τd = N , K = 10 dB andT = 200. Fig. 5 illustrates the cumulative distributions func-tion (CDF) of the spectral efficiencies per user for MRT and ZF

Fig. 4. Effect of Ricean K -factor on the DL sum spectral efficiency: M =100, T = 200, τu = τd = N and pu = pd = 10 dB.

Fig. 5. Cumulative distribution function of the spectral efficiency per user forMRT and ZF processing with and without BT scheme: M = 100, pu = pd =20 dB, τu = τd = N , K = 10 dB, and T = 200.

processing with and without BT scheme. The main observationis that all of the user will (statistically) obtain higher spectralefficiency when the MRT processing is utilized at the BS in thisexample, since the CDF curves with MRT processing are to theright of the corresponding curves with ZF processing. We seethat ZF processing performs much worse than MRT processingeven for the 95% likely throughput. The spectral efficiency peruser of ZF processing with and without BT is more concen-trated around its mean compared to the MRT processing withand without BT. Furthermore, we can see that, for MRT process-ing, the scheme with BT is better than the scheme without BT,while for ZF processing, depending on the large scale fading,the performance with and without BT are basically the same.In this example, it is also shown that MRT processing with BTprovides a large gain under multicell MU MIMO systems overRicean fading channels.

Figs. 6 and 7 plot the sum spectral efficiencies versus thelength of DL pilot sequences for MRT and ZF processing,respectively. For these two figures, we set M = 50, τu = N ,


Fig. 6. Effect of the lengths of DL pilots on the DL sum spectral efficienciesfor MRT processing: M = 50, τu = N , pu = 0 dB, pd = 20 dB, T = 1000,K = −10 dB, Dii = IN , and Di l = aIN , ∀l = i with a = 0.1.

Fig. 7. Effect of the lengths of DL pilots on the DL sum spectral efficienciesfor ZF processing: M = 50, τu = N , pu = 0 dB, pd = 20 dB, T = 1000,K = −10 dB, Dii = IN , and Di l = aIN , ∀l = i with a = 0.1.

pu = 0 dB, pd = 20 dB, T = 1000 and K = −10 dB, and as-sume that Dii = IN and Dil = aIN for any l = i with a = 0.1.For MRT and ZF precoding, it can be seen that the performancewith BT is better than that without BT for short lengths ofDL pilot sequences and vice versa. Compared with ZF, MRTneeds longer DL pilot sequences to reach the performance atthe cross-over point. It can also be seen that the cross-overpoint for two schemes (with and without BT) is in the rangeof τ l

d,A ≤ τd ≤ τud,A (A ∈ {MRT,ZF}). When τd ≤ τ l

d,A , theperformance of the scheme with BT is better than that withoutBT, while the opposite is true when τd ≥ τu

d,A . These simula-tion results verify Corollary 1 to Corollary 3 and Corollary 9 toCorollary 11.

By observing Fig. 1, Fig. 2, Fig. 4 and Fig. 5, we can findthat the sum spectral efficiency of MRT processing is muchhigher than the sum spectral efficiency of ZF processing forthe scheme with and without BT. In Figs. 6 and 7, it can be

seen that the performance of ZF processing with and withoutBT is better than the performance of MRT processing with andwithout BT. Through these simulation comparisons for differentparameters (including DL transmit and pilot power, the lengthof the coherence interval, Ricean K-factor and the large-scalefading coefficients, etc), we find that the cause of this fact ismainly related to the large-scale fading coefficients.

VI. CONCLUSION

In this paper, we have investigated the multicell massive MU-MIMO DL systems operating in Ricean fading environments un-der imperfect CSI. Because the channel estimation overhead ofBT is small and does not depend on the number of BS antennas,it is attractive for massive MU-MIMO systems operating overRicean fading channels. We analyzed the spectral efficienciesof MRT and ZF processing with and without BT. Analysis andsimulation results showed that the considered massive MIMODL systems of MRT processing with BT have the biggest per-formance advantage over the systems which apply the MRTprocessing without BT, or the ZF processing with and withoutBT when the length of DL pilot sequences is small comparedto the coherence time interval and when the Ricean K-factor issmall. In addition, for MRT or ZF precoding, we gave a rangeof performance cross-over point for the scheme with and with-out BT. Simulation results also verified the correctness of theperformance cross-over points.

APPENDIX A

For MRT precoding, from the definition of ailnj =αlgT

lin g∗llj , (i, l = 1, . . . , L;n, j = 1, . . . , N), we analyze

E{|ailnj |2} and E{|ζilnj |2} under the following four cases:Case 1: l = i and j = n; Case 2: l = i and j = n; Case 3: l = iand j = n; Case 4: l = i and j = n.

For convenience, Case 1 and Case 2 are considered together.For l = i and j = n, we have [22]

biinn = E {aiinn} =

√Mμi

(Kinβiin + σ2

gw , i i n

)

Kin + 1. (81)

For l = i and j = n, we obtain [22]

bilnn = E {ailnn} =

√Mμlσ

2gw , l l n

βlin√Kln + 1βlln

. (82)

From (81) and (82), when j = n, we have

E {rin,n} =√

τdpd

L∑

l=1

E {ailnn} =√

τdpd

L∑

l=1

bilnn . (83)


E{aiinn r∗in,n

}=

√τdpdciinn +

√τdpd

L∑

l =i

biinn b∗ilnn , (84)


where

ciinn =μ2

i Kinβiinσ2gw , i i n

(Kin + 1)2 +μ2

i βiin

(Kinβiin + σ2

gw , i i n

)

(Kin + 1)2

+μ2

i M(Kinβiin + σ2

gw , i i n

)2

(Kin + 1)2 . (85)


E{ailnn r∗in,n

}=

√τdpdcilnn +

√τdpd

L∑

l ′ = l

bilnn b∗il ′nn , (86)

where

cilnn =μ2

l

[βlin

(Klnβlln + σ2

gw , l l n

)+ Mσ2

gw , l l nσ2

gw , l i n

]

Kln + 1.

(87)

Based on (84) and (86), when j = n, we obtain

E{|rin,n |2

}= τdpd

L∑

l=1

cilnn + τdpd

L∑

l=1

L∑

l ′ = l

bilnn b∗il ′nn + 1.

(88)

Substituting (81), (83) and (84) into (89), when l = i andj = n, gives

diinn = Cov {aiinn , rin,n}

=

√τdpdμ

2i

(Kinβiinσ2

gw , i i n+ βiin

(Kinβiin + σ2

gw , i i n

))

(Kin + 1)2 .

(89)

Substituting (82), (83) and (86) into (90), when l = i andj = n, yields

dilnn = Cov{ailnn , rin,n}

=√

τdpdμ2l βlin (Klnβlln + σ2

gw , l l n)

Kln + 1. (90)

Substituting (83) and (88) into (91), when j = n, we get

ein,n = Cov {rin,n , rin,n} =L∑

l=1

√τdpddilnn + 1. (91)

Similarly, Case 3 and Case 4 are considered together next.For l = i and j = n, we have [22]

biinj = E {aiinj}

=

√KinKijβiinβiijμiΔinj ej ((M−1)π/2)[sin(θi n)−sin(θi j)]

√M (Kin + 1) (Kij + 1)

,

(92)

where Δinj is defined in (31). For l = i and j = n, we obtain

bilnj = E {ailnj} = 0. (93)

From (92) and (93), when j = n, we obtain

E {rin,j} =√

τdpd

L∑

l=1

E {ailnj} =√

τdpdbiinj . (94)

For l = i and j = n, we get [22]

E{aiinj r

∗in,j

}=

√τdpdE

{aiinj a

∗iinj

}=

√τdpdciinj , (95)

where

ciinj =μ2

i

((Kin + 1) βiinσ2

gw , i i j+ Kijβiinβiij

)

(Kin + 1) (Kij + 1)

+μ2

i βiinβiijKinKijΔ2inj

M (Kin + 1) (Kij + 1). (96)

For l = i and j = n, we have [22]

E{ailnj r

∗in,j

}=

√τdpdE

{ailnj a

∗ilnj

}=

√τdpdcilnj , (97)

where

cilnj =μ2

l βlin

(Kljβllj + σ2

gw , l l j

)

Klj + 1. (98)


E{|rin,j |2

}= τdpd

L∑

l=1

cilnj + 1. (99)

Substituting (92), (94) and (95) into (100), when l = i andj = n, we get

diinj = Cov {aiinj , rin,j}

=√

τdpdμ2i

((Kin + 1) βiinσ2

gw , i i j+ Kijβiinβiij

(Kin + 1) (Kij + 1)

)

.

(100)

Substituting (93), (94) and (97) into (101), when l = i andj = n, we obtain

dilnj = Cov {ailnj , rin,j} =√

τdpdcilnj . (101)

Substituting (94) and (99) into (102), when j = n, yields

ein,j = Cov {rin,j , rin,j} =L∑

l=1

√τdpddilnj + 1. (102)

Based on (81)–(102) for four cases, if i, l = 1, . . . , L andn, j = 1, . . . , N , we have

E{|ailnj |2

}= E

{∣∣∣∣bilnj +

dilnj

ein,j(rin,j − E {rin,j})

∣∣∣∣

2}

= |bilnj |2 +d2

ilnj

ein,j. (103)

E{|ζilnj |2

}= E

{|ailnj − ailnj |2

}=

dilnj√τdpd

− d2ilnj

ein,j.

(104)

Substituting (103) and (104) into (22), we finally obtain (27).

APPENDIX B

If C(MRT)i ≥ C

(MRT)i , we have

N∑

n=1

log2

(

1 +Δ1

Δ6

)T −τ u

T −τ u −τ d

≥N∑

n=1

log2

(

1 +Δ1 + Δ5

Δ6 − Δ5

)

,

(105)


where Δ1, Δ5 and Δ6 are defined in (28), (34) and (38), respec-tively. The expression in (105) can be rewritten as

N∏

n=1

(

1 +Δ1

Δ6

) T −τ uT −τ u −τ d ≥

N∏

n=1

(

1 +Δ1 + Δ5

Δ6 − Δ5

)

. (106)

By using Bernoulli’s inequality, i.e., (1 + x)r ≥ 1 + rx (x >−1 and r ≥ 1), when N ≤ τd ≤ T − τu , (106) can be trans-formed into

N∏

n=1

(

1 +T − τu

T − τu − τd

Δ1

Δ6

)

≥N∏

n=1

(

1 +Δ1 + Δ5

Δ6 − Δ5

)

. (107)

Thus, if (107) is satisfied, we have C(MRT)i ≥ C

(MRT)i . In

order to satisfy (107), for n = 1, . . . , N , we must have

T − τu

T − τu − τd(Δ1Δ6 − Δ1Δ5) ≥ Δ1Δ6 + Δ5Δ6. (108)

The inequality in (108) can be simplified to

τd ≥ maxn=1,...,N

{pd (T − τu ) (Δ1 + Δ6) Δ2

in − Δ1Δ6

pd

(Δ6Δ2

in + Δ1Δ6Δin + Δ1Δ4Δ6)

}

.

(109)

Taking (109) and N ≤ τd ≤ T − τu into consideration, wefinally arrive at Corollary 1.

APPENDIX C

The inequality C(MRT)i ≥ C

(MRT)i implies that

N∑

n=1

log2

(

1 +Δ1 + Δ5

Δ6 − Δ5

) T −τ u −τ dT −τ u ≥

N∑

n=1

log2

(

1 +Δ1

Δ6

)

,

(110)

where Δ1, Δ5 and Δ6 are defined in (28), (34) and (38), respec-tively. The inequality in (110) can be rewritten as

N∏

n=1

(Δ1 + Δ6

Δ6 − Δ5

)T −τu −τd

≥N∏

n=1

(Δ1 + Δ6

Δ6

)T −τu

, (111)

which can be transformed intoN∏

n=1

(

1 +Δ5

Δ6 − Δ5

) T −τ uτ d ≥

N∏

n=1

(

1 +Δ1 + Δ5

Δ6 − Δ5

)

. (112)

Again, by using Bernoulli’s inequality, i.e., (1 + x)r ≥ 1 +rx (x > −1 and r ≥ 1), when N ≤ τd ≤ T − τu , (112) can betransformed into

N∏

n=1

(

1 +T − τu

τd

Δ5

Δ6 − Δ5

)

≥N∏

n=1

(

1 +Δ1 + Δ5

Δ6 − Δ5

)

.

(113)

In order to satisfy (113), for n = 1, . . . , N , we should have

T − τu

τdΔ5 ≥ Δ1 + Δ5. (114)

Simplifying (114) leads to

τd ≤ minn=1,...,N

{pd (T − τu ) Δ2

in − Δ1

pd

(Δ2

in + Δ1Δin + Δ1Δ4)

}

. (115)

Finally, taking (115) and N ≤ τd ≤ T − τu into considera-tion arrives to Corollary 2.

APPENDIX D

For ZF precoding, define ailnj = γlgTlinalj , (i, l = 1, . . . ,

L;n, j = 1, . . . , N) and also analyze E{|ailnj |2} andE{|ζilnj |2} under the following four cases: Case 1: l = i andj = n; Case 2: l = i and j = n; Case 3: l = i and j = n; Case 4:l = i and j = n.

Similar to MRT precoding, Case 1 and Case 2 are consideredtogether. For l = i and j = n, we have

biinn = E {aiinn} = γi. (116)

When M N , we can use the law of large numbers to obtainthe following approximation:

(GT

ll G∗ll

)−1≈ 1

MDll , (117)

where Dll is a N × N diagonal matrix whose (n, n)th elementis

[Dll

]

nn=

Kln + 1Klnβlln + σ2

gw , l l n

. (118)

From (117), when l = i and j = n, we obtain

bilnn = E {ailnn} ≈γl

[Dll

]

nnE{gT

lin g∗lln

}

M

=γlσ

2gw , l l n

βlin

√Kln + 1

Klnβ2lln + βllnσ2

gw , l l n

. (119)

From (116) and (119), when j = n, we have

E {rin,n} =√

τdpd

L∑

l=1

E {ailnn} =√

τdpd

L∑

l=1

bilnn .

(120)

For l = i and j = n, we obtain

E{aiinn r∗in,n

}=

√τdpdciinn +

√τdpd

L∑

l =i

biinn b∗ilnn ,

(121)

where [21]

ciinn = γ2i

(

1 +σ2

εw , i i n

[Σ−1

i

]

nn

(M − N) (Kin + 1)

)

. (122)

From (117), when l = i and j = n, we obtain [21]

E{ailnn r∗in,n

}=

√τdpdcilnn +

√τdpd

L∑

l ′ = l

bilnn b∗il ′nn ,

(123)

where

cilnn =γ2

l σ2εw , l i n

[Σ−1

l

]

nn

M − N+

γ2l σ

2gw , l i n

(Kln + 1)

M(Klnβlln + σ2

gw , l l n

)

+γ2

l σ2gw , l l n

σ2gw , l i n

(Kln + 1)(Klnβlln + σ2

gw , l l n

)2 . (124)


E{|rin,n |2

}= τdpd

L∑

l=1

cilnn + τdpd

L∑

l=1

L∑

l ′ = l

bilnn b∗il ′nn + 1.

(125)


Substituting (116), (120) and (121) into (126), when l = iand j = n, gives

diinn = Cov {aiinn , rin,n} =√

τd pd γ 2i σ 2

εw , i i n[Σ−1

i ]n n

(M −N )(Ki n +1) . (126)

Substituting (119), (120) and (123) into (127), when l = iand j = n, yields

dilnn = Cov {ailnn , rin,n}

=√

τdpdγ2l

⎛

⎝σ2

εw , l i n

[Σ−1

l

]

nn

M − N+

σ2gw , l i n

(Kln + 1)

M(Klnβlln + σ2

gw , l l n

)

⎞

⎠ .

(127)

Substituting (120) and (125) into (128), when j = n, we get

ein,n = Cov {rin,n , rin,n} =L∑

l=1

√τdpddilnn + 1. (128)

Similarly, Case 3 and Case 4 are considered together next.For l = i and j = n, we have

biinj = E {aiinj} = 0. (129)

From (117), when l = i and j = n, we obtain

bilnj = E {ailnj} = 0. (130)

From (129) and (130), when j = n, we obtain

E {rin,j} =√

τdpd

L∑

l=1

E {ailnj} = 0. (131)

For l = i and j = n, we get [21]

E{aiinj r

∗in,j

}=

√τdpdE

{aiinj a

∗iinj

}=

√τdpdciinj ,

(132)

where

ciinj =γ2

i σ2εw , i i n

[Σ−1

i

]

jj

(M − N) (Kin + 1). (133)

From (117), when l = i and j = n, we have [21]

E{ailnj r

∗in,j

}=

√τdpdE

{ailnj a

∗ilnj

}=

√τdpdcilnj ,

(134)

where

cilnj = γ2l

⎛

⎝σ2

εw , l i n

[Σ−1

l

]

jj

M − N+

σ2gw , l i n

(Klj + 1)

M(Kljβllj + σ2

gw , l l j

)

⎞

⎠ .

(135)


E{|rin,j |2

}= τdpd

L∑

l=1cilnj + 1. (136)

Substituting (129), (131) and (132) into (137), when l = iand j = n, we get

diinj = Cov {aiinj , rin,j} =√

τdpdciinj . (137)

Substituting (130), (131) and (134) into (138), when l = iand j = n, we obtain

dilnj = Cov {ailnj , rin,j} =√

τdpdcilnj . (138)

Substituting (131) and (136) into (139), when j = n, yields

ein,j = Cov {rin,j , rin,j} =L∑

l=1τdpdcilnj + 1. (139)

Similarly, based on (116)–(139) for the four cases of ZFprecoding, if i, l = 1, . . . , L and n, j = 1, . . . , N , we have

E{|ailnj |2

}= E

{∣∣∣∣bilnj +

dilnj

ein,j(rin,j − E {rin,j})

∣∣∣∣

2}

= |bilnj |2 +d2

ilnj

ein,j. (140)

E{|ζilnj |2

}= E

{|ailnj − ailnj |2

}=

dilnj√τdpd

− d2ilnj

ein,j.

(141)

Substituting (140) and (141) into (22), we finally obtain (53).

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Si-Nian Jin received the B.S. degree in com-munication engineering from Shandong University,Weihai, China, in 2014, and the M.S. degree in in-formation and communication engineering, in 2017,from Dalian Maritime University, Dalian, China,where he is currently working toward the Ph.D. de-gree in information and communication engineering.His research interests include massive MIMO sys-tems and cooperative communications.

Dian-Wu Yue received the B.S. and M.S. degrees inmathematics from Nankai University, Tianjin, China,in 1986 and 1989, respectively, and the Ph.D. de-gree in communications and information engineeringfrom the Beijing University of Posts and Telecom-munications, Beijing, China, in 1997. From 1989 to1993, he was a Research Assistant of applied math-ematics with the Dalian University of Technology,Dalian, China. From 1996 to 2003, he was an Asso-ciate Professor of communications and informationengineering with the Nanjing University of Posts and

Telecommunications, Nanjing, China. Since December 2003, he has been a fullProfessor of communications and information engineering with Dalian Mar-itime University, Dalian. During 2000–2001, he was a Visiting Scholar with theUniversity of Manitoba, Winnipeg, MB, Canada. During 2001–2002, he was aPostdoctoral Fellow with the University of Waterloo, Waterloo, ON, Canada.During November 2017–December 2017, he was a Visiting Professor with theUniversity of Saskatchewan, Saskatoon, SK, Canada. His current research inter-ests include massive MIMO systems, millimeter wave MIMO communications,and cooperative relaying communications.

Ha H. Nguyen (M’01–SM’05) received the B.Eng.degree from the Hanoi University of Technology(HUT), Hanoi, Vietnam, in 1995, the M.Eng. de-gree from the Asian Institute of Technology (AIT),Bangkok, Thailand, in 1997, and the Ph.D. degreefrom the University of Manitoba, Winnipeg, MB,Canada, in 2001, all in electrical engineering. Hejoined the Department of Electrical and ComputerEngineering, University of Saskatchewan, Saskatoon,SK, Canada, in 2001, and became a full Profes-sor in 2007. He currently holds the position of

NSERC/Cisco Industrial Research Chair in Low-Power Wireless Access forSensor Networks. His research interests fall into broad areas of communica-tion theory, wireless communications, and statistical signal processing. He wasan Associate Editor for the IEEE TRANSACTIONS ON WIRELESS COMMUNICA-TIONS and IEEE WIRELESS COMMUNICATIONS LETTERS during 2007–2011 and2011–2016, respectively. He currently serves as an Associate Editor for theIEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. He was a Co-Chair forthe Multiple Antenna Systems and Space-Time Processing Track, IEEE Vehic-ular Technology Conferences (Fall 2010, Ottawa, ON, Canada and Fall 2012,Quebec, QC, Canada), Lead Co-Chair for the Wireless Access Track, IEEE Ve-hicular Technology Conferences (Fall 2014, Vancouver, BC, Canada), LeadCo-Chair for the Multiple Antenna Systems and Cooperative Communica-tions Track, IEEE Vehicular Technology Conference (Fall 2016, Montreal, QC,Canada), and Technical Program Co-Chair for Canadian Workshop on Infor-mation Theory (2015, St. John’s, NL, Canada). He is a co-author, with EdShwedyk, of the textbook “A First Course in Digital Communications” (Cam-bridge University Press). He is a Fellow of the Engineering Institute of Canada(EIC) and a registered member of the Association of Professional Engineersand Geoscientists of Saskatchewan (APEGS).

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