static.tongtianta.sitestatic.tongtianta.site/paper_pdf/f950867c-e357-11e... · SHAFIN AND LIU: MULTI-CELL MULTI-USER MASSIVE FD-MIMO: DL PRECODING AND THROUGHPUT ANALYSIS 489 equivalent

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 18, NO. 1, JANUARY 2019 487

Multi-Cell Multi-User Massive FD-MIMO:Downlink Precoding and Throughput Analysis

Rubayet Shafin, Student Member, IEEE, and Lingjia Liu , Senior Member, IEEE

Abstract— In this paper, downlink (DL) precoding and powerallocation strategies are identified for a time-division-duplexmulti-cell multi-user massive full-dimension MIMO network.Utilizing channel reciprocity, DL channel state information feed-back is eliminated and the DL multi-user MIMO precoding islinked to the uplink (UL) direction of arrival (DoA) estimationthrough the estimation of signal parameters via rotational invari-ance technique. Assuming non-orthogonal/non-ideal spreadingsequences of the UL pilots, the performance of the UL DoAestimation is analytically characterized and the characterizedDoA estimation error is incorporated into the corresponding DLprecoding and power allocation strategy. The simulation resultsverify the accuracy of our analytical characterization of theDoA estimation and demonstrate that the introduced multi-userMIMO precoding and power allocation strategy outperforms theexisting zero-forcing-based massive MIMO strategies.

Index Terms— Direction-of-arrival estimation, FD-MIMO,multi-cell, multi-user MIMO, zero-forcing, ESPRIT.

I. INTRODUCTION

MASSIVE-MIMO or large-scale MIMO, has generatedsignificant interest both in academia [1] and indus-

try [2]. Because of the promise of fulfilling future throughputdemand via aggressive spatial multiplexing, massive MIMO isconsidered as one of the key enabling technologies for nextgeneration wireless networks. Due to form factor limitationat the base station (BS), three dimensional (3D) massive-MIMO/Full Dimension MIMO (FD-MIMO) systems havebeen introduced in 3GPP to deploy active antenna elements ina two dimensional (2D) antenna array enabling the exploitationof the degrees of freedom in both azimuth and elevationdomains. Due to the availability of the huge spectrum in themillimeter-wave (mm-wave) band, mm-wave communicationis considered as another enabling technology for future cellularnetworks: 5G and beyond. However, due to its significantlyhigher path loss compared to the microwave channel, it isextremely challenging to establish an effective communicationfor outdoor channels using mm-wave bands. This challenge

Manuscript received January 2, 2018; revised May 19, 2018 andSeptember 10, 2018; accepted November 1, 2018. Date of publicationNovember 29, 2018; date of current version January 8, 2019. This workwas supported by the U.S. National Science Foundation under Grant CNS-1811720, Grant ECCS-1811497, and Grant ECCS-1802710. The associateeditor coordinating the review of this paper and approving it for publicationwas C.-H. Lee. (Corresponding author: Lingjia Liu.)

The authors are with the Bradley Department of Electrical andComputer Engineering, Virginia Tech, Blacksubrg, VA 24061 USA (e-mail:[email protected]).

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

Digital Object Identifier 10.1109/TWC.2018.2882386

can be tackled using beamforming techniques where the basestation serves multiple users with narrower beams. This canbe possible if a large number of antennas are deployed atthe base station in order to realize the narrow beams. As aresult, massive MIMO is a natural counterpart for the mm-wave cellular network.

Since the benefits of massive MIMO or massive FD-MIMOare limited by the accuracy of the downlink (DL) channel stateinformation (CSI) available at the base station, it is criticalfor the BS to obtain corresponding DL CSI information. Ingeneral, the BS can obtain the CSI knowledge through thefollowing: 1) DL CSI feedback where the the CSI informationis fed back from mobile stations (MSs), and 2) DL/UL channelreciprocity where BS estimates the uplink (UL) CSI and infersDL CSI information through channel reciprocity. Note thatDL CSI feedback is heavily used in frequency-division-duplex(FDD) systems where only a few bits of the corresponding DLCSI information are fed back to the BS [3] to achieve a goodtradeoff between DL MIMO performance and UL feedbackoverhead/reliability. To utilize the DL/UL channel reciprocity,the critical point becomes estimating the UL channel at the BS.Based on UL pilots/reference signals sent from MSs, there aregenerally two methods to estimate the UL channel. First is toestimate the corresponding channel transfer function (e.g., ULchannel matrix). Alternatively, UL direction of arrival (DoA)can be estimated at the BS using ESPRIT algorithm [4]. Eventhough the DoA only provides partial information on the ULchannel, it is shown in [5] and [6] that it can be directly linkedto DL MIMO precoding in TDD systems. It is important tonote that the DoA based MIMO precoding strategy has alsobeen introduced to FDD systems demonstrating significantperformance benefits in reality [7].

Despite promising better performance, non-linear precod-ing schemes, such as dirty paper coding (DPC) or vectorperturbation, are not practical for MIMO systems due toits high implementation complexity. In recent years, sim-ple linear processing techniques have been shown to offersignificant performance gains for multi-user massive MIMOscenarios where the base stations employ a large numberof antennas [1]. Hence, most of the prior works in massiveMIMO literature have focused on maximum ratio transmission(MRT) and zero forcing (ZF)-based methods for DL MIMOprecoding [8], [9]. However, for mm-wave mssive FD-MIMOsystems, it is possible to design low-complexity precoderswith better performance than the conventional ZF/MRT-basedprecoders.

1536-1276 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

https://orcid.org/0000-0003-1915-1784

488 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 18, NO. 1, JANUARY 2019

In this paper, we will characterize the optimal/near-optimalDL MIMO precoding and power allocation strategies fora TDD multi-cell multi-user mm-wave massive FD-MIMOnetwork. ESPRIT-based UL DoA estimation scheme will beintroduced and performance of the DoA estimation will beanalytically characterized assuming non-orthogonal spreadingsequences used for UL pilots/reference signals. DL multi-userMIMO precoding and power allocation strategies will be iden-tified based on the UL DoA estimation and their correspondingerror performance. Performance evaluation will be conductedto illustrate benefits of the introduced MIMO precoding andpower allocation strategy over our previous scheme [5] aswell as popular zero forcing (ZF)-based precoding [8], [9].The contribution of this paper can be summarized as thefollowing:

• First, we present a unitary ESPRIT-based uplink DoAestimation method for multi-cell multi-user mm-wavemassive FD-MIMO OFDM network. Unlike majorityof existing work, our scheme considers a more realisticscenario where non-orthogonal spreading sequences areused as UL pilots for both intra-cell and inter-cell users.As a result, due to the non-zero correlation coefficientsamong users’ spreading sequences, the UL DoA/channelestimation is subject to intra-cell interference, inter-cellinterference, and the so called pilot contamination.

• Second, we analytically characterize the mean squareerror (MSE) of unitary ESPRIT-based UL DoA esti-mation for the corresponding multi-cell multi-userFD-MIMO network. Our analytical results show howdifferent perturbation components, namely noise ele-ments, and intra-cell and inter-cell interferences, affectthe UL DoA/channel estimation performance. The MSEhas been related to key physical parameters such asnumber of antennas, BS array geometry, complex pathgains, and correlation coefficients between users’ spread-ing sequences.

• Third, we derive the sum-rate maximizing DL precodingand power allocation strategy for our FD-MIMO system.Furthermore, we perform a large antenna array regimeanalysis for DL precoding and identify the optimumpower allocation under both perfect and imperfect DoAestimation scenarios.

• Finally, we validate our algorithms and analytical resultsthrough extensive simulation. The evaluation resultsdemonstrate that our simulated MSE for different antennanumbers and antenna array geometries match well withthose of analytical expressions for both elevation andazimuth estimation in large SNR regimes. Moreover,we also show that the introduced sum-rate maximizationprecoding strategy outperforms both eigenbeamformingand ZF-based precoding over all SNR regimes.

The rest of the paper is organized as follows. Section IIdescribes the system model and the channel model for theunderlying multi-cell multi-user massive FD-MIMO network.Section III presents the ESPRIT-based UL DoA estimationmethod and the performance characterization for DoA esti-mation. The achievable sum-rate analysis under both perfectand imperfect DoA estimation as well as the optimal MIMO

Fig. 1. Network model.

precoding and power allocation strategies are contained inSection IV. Simulation results are presented in Section V andSection VI concludes the paper.

II. SYSTEM AND CHANNEL MODEL

We consider a multi-cell multi-user MIMO-OFDM systemconsisting of G BSs as depicted in figure 1. Each BS withNr number of antennas supports J number of mobile sta-tions (MSs)–each having Nt number of transmit antennas.After appending cyclic prefix (CP), the resulting time domaintransmit signal at each MS is first passed through a parallel-to-serial converter followed by a digital-to-analog (DAC) con-verter, resulting in the baseband OFDM signal. The basebandsignal is then up-converted and sent through a frequencyselective fading channel, which is assumed to remain time-invariant during an OFDM symbol duration. It is to be notedhere that we assumed same number of antennas at all UEsbecause of better clarity of exposition. However, we want toemphasize here that the algorithm and analysis presented inthis work are not restricted by this assumption. All the resultspresented in this work can be straightforwardly extended tothe scenario where users in the cell have different number ofantennas.

In the UL, each MS sends Nt spreading sequences of lengthQ as plots/reference signals: one on each transmit antenna.Accordingly, the Nr × Q frequency-domain received signalfor the k-th subcarrier at the i-th BS can be written as

Zi(k) =G−1�

g=0

J−1�

j=0

�Λjg,iHjg,i(k)Xjg(k) + Wi(k), (1)

where Hjg,i(k) is the Nr×Nt channel matrix for the channelbetween the i-th BS and the j-th MS in the g-th cell at thek-th subcarrier, and Λjg,i is the corresponding large scale fad-ing coefficient which is independent of subcarrier frequency;Xjg(k) is the Nt×Q frequency domain transmit signal fromthe j-th MS in the g-th cell for the k-th subcarrier, and Wi(k)is the corresponding Nr × Q noise matrix. Note that eachrow vector of Xjg(k) is a length-Q spreading sequence. Thechannel transfer function, Hjg,i(k), can be written as

Hjg,i(k) =Ljg,i−1�

�=0

Cjg,i(�)e−j2πk�

Nc , (2)

where Cjg,i(�) is the Nr×Nt channel impulse response (CIR)for the �-th tap of the channel between i-th BS and the j-th MSin the g-th cell. Nc denotes total number of subcarriers. Here,we assume that the channel, which can be represented by an

SHAFIN AND LIU: MULTI-CELL MULTI-USER MASSIVE FD-MIMO: DL PRECODING AND THROUGHPUT ANALYSIS 489

equivalent discrete-time linear channel impulse response, hasa finite number (Ljg,i) of non-zero taps.

Using the geometric channel model for mm-wave frequen-cies, the impulse response for the �-th tap of the channelbetween i-th BS and the j-th MS in the g-th cell can berepresented by [5], [6], [10]

Cjg,i(�) =Pjg,i,�−1�

p=0

αjg,i(�, p)er,jg,i(�, p)eHt,jg,i(�, p), (3)

where αjg,i(�, p), er,jg,i(�, p), and et,jg,i(�, p) are, respec-tively, the channel gain, Nr×1 receive antenna array response,and Nt × 1 transmit antenna array response for the p-thsub-path within the �-th tap of the channel between thei-th BS and the j-th MS in g-th cell; Pjg,i,� is the totalnumber of sub-paths within the �-th tap of the channel; and(·)H denotes Hermitian transpose operation. In the FD-MIMOnetwork of interests, a 1D uniform linear array (ULA) isassumed at each MS. The corresponding transmit antennaarray response can be described using the Vandermondestructure: et,jg,i(�, p) =

�1 ejωjg,i,�,p . . . ej(Nt−1)ωjg,i,�,p

�T,

where ωjg,i,�,p = (2πΔt/λ) cosΩjg,i,�,p, Δt is the spacingbetween the adjacent transmit antenna elements, Ωjg,i,�,p isthe transmit angle (DoD) for p-th sub-path within �-th tapof the channel between i-th base station and the j-th userin g-th cell, and λ is the carrier wavelength. On the otherhand, for FD-MIMO networks the antenna array at the BSis a 2D planar array placed in the X-Z plane, with M1 andM2 antenna elements in vertical and horizontal directions,respectively. Accordingly, the number of total receive antennaelements at the base station is Nr = M1 × M2. Therefore,the receive antenna array response for the p-th sub-path within�-th tap can be expressed as er,jg,i(�, p) = a(vjg,i,�,p) ⊗a(ujg,i,�,p), where ⊗ represents the Kronecker product,and a(ujg,i,�,p) =

�1 ejujg,i,�,p . . . ej(M1−1)ujg,i,�,p

�Tand

a(vjg,i,�,p) =�1 ejvjg,i,�,p . . . ej(M2−1)vjg,i,�,p

�Tcan be

treated as the receive steering vectors in the eleva-tion and azimuth domains, respectively. Here, ujg,i,�,p =2πΔr

λ cos θjg,i,�,p and vjg,i,�,p = 2πΔr

λ sin θjg,i,�,p cosφjg,i,�,pare the two receive spatial frequencies at the BS, Δr is thespacing between adjacent antenna elements in the receiveantenna array, and θjg,i,�,p and φjg,i,�,p are the elevationand azimuth DoAs for the p-th sub-path within �-th tap forthe channel between the i-th BS and the j-th MS in g-thcell, respectively. In this paper, we are not considering usermobility or scheduler impact on the system performance, andwe will address these important issues in our future work.

III. UPLINK CHANNEL CHARACTERIZATION

THROUGH DOA ESTIMATION

In this section, we will present the UL DoA estimationprocedure for the multi-cell multi-user massive FD-MIMOnetwork and characterize assuming non-orthogonal/non-idealspreading sequences and characterize the corresponding esti-mation performance. We choose ESPRIT-based method forDoA estimation over other high resolution DoA estima-tion methods, such as MUSIC, since ESPRIT offers better

resolvability and unbiased estimates with lower variance.Most importantly, ESPRIT provides significant computationaladvantages in terms of faster processing speed, lower storagerequirement and indifference to knowledge of precise arraygeometry.

A. UL DoA Estimation Through Unitary ESPRIT

Let the n-th MS at the i-th cell be the target user whichtries to communicate to the i-th BS. In massive FD-MIMOnetworks, the number of scheduled users may be quite large,and hence due to limited availability of orthogonal spreadingcodes, it may not be possible to assign orthogonal sequencesto all scheduled users. With this in mind, in this work,we assume a more realistic scenario that only the spread-ing sequences used by the same MS are orthogonal whilespreading sequences for different MSs within a cell are non-orthogonal. Furthermore, we assume that the same pool ofspreading sequences are reused across all cells as UL pilotscomplying with the 3GPP LTE/LTE-Advanced standards [11].

Let the correlation among the spreading sequences fromdifferent MSs be denoted as ρ1. Now, for estimating theUL channel of the n-th MS in i-th cell, at the i-th BS,the Nr × Q received signal at the k-th subcarrier, Zi(k),is first correlated with the spreading sequences of n-th MS.Hence, after correlating the received signal with the targetuser’s sequence, from (1), we have

Zi(k)XHni(k)

=G−1�

g=0

J−1�

j=0

�Λjg,iHjg,i(k)Xjg(k)XH

ni(k) + W�i(k), (4)

where W�i(k) = Wi(k)XH

ni(k) is the equivalent noiseelement. Now, we can re-write (4) as

Zi(k)XHni(k) =

�Λni,iHni,i(k) +

J−1�

j=0j �=n

�Λji,iHji,i(k)ρ11Nt

+G−1�

g=0g �=i

J−1�

j=0j �=n

�Λjg,iHjg,i(k)ρ11Nt

+G−1�

g=0g �=i

�Λng,iHng,i(k)+W�

i(k), (5)

where 1Nt denotes an Nt×Nt matrix with each element beingunity. In (5), the first term,

�Λni,iHni,i(k), represents the

target user’s UL channel; first summation term represents theinter-cell interference caused by users in other cells whosespreading sequences are exactly the same as that of target user(pilot contamination); second summation term represents theintra-cell interference; and third summation term representsthe inter-cell interference caused by users in other cells whosespreading sequences are different (non-orthogonal) than that oftarget user. In realistic wireless networks such as LTE/LTE-Advanced networks, there exists a nonzero correlation betweendifferent pilot sequences. For example, Zadoff-Chu sequenceis used to make sure different prime length Zadoff-Chu


sequence has constant cross-correlation [12]. To reflect thispractical constraint as well as provide system design insights,a correlation coefficient, ρ1, in (5) is introduced as a systemdesign parameter to consider the tradeoff between the trainingsequence length and corresponding system performance. It isto be noted that ρ1 = 0 results in the special case where all theusers in a cell are assigned orthogonal codes. Also important tonote that the value of ρ1 depends on the length of scramblingsequences the system designer chooses, which again dependson the coherence time of the channel. This provides us a wayto investigate the impact of channel coherence on the networkperformance: Smaller coherence time will lead to a shorterscrambling sequence resulting in higher values for ρ1.

Because of the large path loss, which is manifested by thelarge-scale fading coefficients, overall gains of the inter-cellinterference channels are relatively small compared to that ofthe target user’s channel. Furthermore, presence of the smallcorrelation coefficients, ρ1, intra-cell interference terms canalso be considered relatively smaller compared to the termfor target user’s channel. Hence, during the ESPRIT-basedparameter estimation phase, we can treat the interference andnoise elements together, and (5) can be written as

Hni,i(k) = Zi(k)XHni(k) =

�Λni,iHni,i(k) + W��

i (k), (6)

where W��i (k) is the equivalent noise-plus-interference matrix.

Now, using (2) and (3), we can write Hni,i(k) as

Hni,i(k)=Lni,i−1�

�=0

Pni,i,�−1�

p=0

αni,i(�, p)er,ni,i(�, p)eHt,ni,i,k(�, p),

(7)

where et,ni,i,k(�, p) = et,ni,i(�, p)e−j2πk�

Nc . In order to jointlyestimate the elevation and azimuth angles of the uplink channelbetween the i-th base station and the n-th user in i-th cell,we can now apply a low-complexity DoA estimation algorithmbased on unitary ESPRIT.

High frequency channels, especially millimeter-wave chan-nels, usually have fewer number of scattering clusters [13].In this work, we focus on the simple case where eachscattering cluster contributes a single propagation path. Thisis a reasonable assumption for the analysis of FD-MIMOsystems [5], [14], [15]. Hence for the clarity of exposition,and notational convenience, we can drop the subpath index,p, from αjg,i(�, p), er,jg,i,k(�, p), and eHt,jg,i(�, p). However,our results also hold for multiple subpaths scenario due tothe fact that ESPRIT can be used to distinguish subpaths aslong as the spatial resolvability of the array is higher thanthe angular spread between two subpaths [16]. It is to benoted here that in this work, instead of Standard ESPRIT,we utilize Unitary ESPRIT [17] for DoA estimation, whichprovides superior estimation performance for the case wherethe sub-paths within the same clusters are highly correlated.Moreover, because of Forward-Backward Averaging (FBA),Unitary ESPRIT can still estimate the corresponding DoAs oftwo sub-paths which are completely correlated or coherent.It is also noteworthy here that it is unlikely to have more thantwo completely coherent sub-paths in the mm-wave propaga-tion channel based on 3GPP mm-wave channel model [18] and

the seminal work in [19]. Therefore, our introduced algorithmis applicable for most general mm-wave channels. However,for the very special case where more than two sub-paths arecompletely coherent, and all such sub-path DoAs are requiredto be estimated, the spatial smoothing technique can be appliedin conjunction with FBA to de-correlate the correspondingsignals [20]. However, this is out of the scope of currentmanuscript and we will consider this special case in our futurework. Now, (7) can be written as

Hni,i(k) = Ani,iDni,iBHni,i(k), (8)

where Ani,i =�er,ni,i(0) . . . er,ni,i(Lni,i − 1)

�,

Dni,i = diag�αni,i(0) . . . αni,i(Lni,i − 1)

�, and

Bni,i(k) =�et,ni,i,k(0) . . . et,ni,i,k(Lni,i − 1)

�. Hence,

from (6), the channel matrix, Hni,i(k), can be written as

Hni,i(k) =�

Λni,iAni,iDni,iBHni,i(k) + W��

i (k). (9)

By converting all the complex matrices to the real matrices,Unitary ESPRIT performs the computations in real, instead ofcomplex, numbers from beginning to the end of the algorithm,and hence, reduces the computational complexity significantly.Since we are only interested in estimating UL DoAs, the noisychannel from (9) can be expressed as

Hni,i(k) = Ani,iSni,i(k) + W��i (k), (10)

where Sni,i(k) =�

Λni,iDni,iBHni,i(k). In order to perform

unitary ESPRIT, we need to use forward-backward averagingon the received signal:

Hfbani,i(k) =

�Hni,i(k) ΠNrH

∗ni,i(k)ΠNt

�

=�Ani,iSni,i(k) ΠNrA

∗ni,iS

∗ni,i(k)ΠNt

�

+�W��

i (k) ΠNrW��i∗(k)ΠNt

�, (11)

where A∗ represents complex conjugate of A, and Πp denotesthe p × p exchange matrix with ones on its antidiagonaland zeros elsewhere. The subspace decomposition of thesignal space of the received signal through singular valuedecomposition then can be written as:�Ani,iSni,i(k) ΠNrA

∗ni,iS

∗ni,i(k)ΠV

�

=�Usigni,i Unoise

ni,i

� �Ρsigni,i 00 0

Vsigni,i

H

Vnoiseni,i

H

�. (12)

From this step onward, we can now follow our line of work [5]in order to apply ESPRIT-based techniques on (12). Hence,the details are not repeated here due to page limitation.

B. RMSE Characterization

The theoretical performance of the unitary ESPRIT-basedUL DoA estimation can be characterized where the root meansquared error (RMSE) of the estimation is served as theperformance metric. Let vni,i,� denote the estimated spatialfrequency for �-th tap of the target user’s channel, i.e, the chan-nel between the i-th BS and the n-th MS in the i-th cell; theestimation error is then given by Δvni,i,� = vni,i,� − vni,i,�.Similarly, Δuni,i,� = uni,i,� − uni,i,�.


It has been shown in [21] that the unitary transformationdoes not affect the MSE of the ESPRIT methods; however,the statistics of the noise and the signal subspace are changeddue to the forward and backward averaging performed in(11). To be specific, the covariance and complementary covari-ance matrices for the equivalent noise-plus-interference matrixW��

i (k) in (5) become, respectively [21]:

R(fba)i (k) =

�Ri(k) 0

0 ΠNrNtR∗i (k)ΠNrNt

;

C(fba)i (k) =

�0 Ri(k)ΠNrNt

ΠNrNtR∗i (k) 0

,

(13)

where Ri(k) = Eα,β,φ,ψ

�vec {W��

i (k)} vec {W��i (k)}H

,

and the expectation, Eα,β,φ,ψ, is taken with respect to differentchannel realizations (i.e., w.r.t. channel gains, DoA’s– bothazimuth and elevation– and DoD’s of the interference chan-nels). Now the expression of the covariance matrix, Ri(k),can be simplified using the following lemma:

Lemma 1: The covariance matrix, Ri(k), of the equivalentnoise-plus-interference matrix, W��

i , is given by:

Ri(k) = Ri,1(k) + Ri,2(k) + Ri,3(k) + Ri,4(k), (14)

where Ri,4(k) = σ2INrNt , where σ2 is the noise variance,and

Ri,1(k)

= Eα,β,φ,ψ

⎡

⎢⎢⎣G−1�

g=0g �=i

��Λng,i

�2

Rng,i(k)

⎤

⎥⎥⎦, (15)

Ri,2(k)

= Eα,β,φ,ψ

⎡

⎢⎢⎣ρ21

J−1�

j=0j �=n

��Λji,i

�2

X t,rRji,i(k)X t,r

⎤

⎥⎥⎦, (16)

Ri,3(k)

= Eα,β,φ,ψ

⎡

⎢⎢⎣ρ21

G−1�

g=0g �=i

J−1�

j=0j �=n

��Λjg,i

�2

X t,rRjg,i(k)X t,r

⎤

⎥⎥⎦,

(17)

where X t,r = (1Nt ⊗ INr), and Rpq,r(k) =Ppq,r(k)PH

pq,r(k), where Ppq,r(k) =�B∗pq,r(k) ⊗ Apq,r

�vec {Dpq,r}.

Proof Sketch: Lemma 1 can be proved using the proper-ties of matrix vectorization, and with the assumption of theindependence of channel gains. �

It is noteworthy here that in Lemma 1, Ri,1(k), Ri,2(k),Ri,3(k), and Ri,4(k) correspond, respectively, to the effectsof pilot contamination, intra-cell interference, inter-cell inter-ference, and noise element of the noise-plus-interference sig-nal. Now, the first order approximation of the mean squareestimation error of vni,i,� for the Unitary ESPRIT is given

by [21]:

E

�(�vni,i,�)

2

=12

�r(v)H

ni,i,� · W∗ni,i,mat · R(fba)T

i · WTni,i,matr

(v)ni,i,�

−Re�r(v)T

ni,i,� · Wni,i,mat · C(fba)i ·WT

ni,i,mat · r(v)ni,i,�

�,

(18)

where

r(v)ni,i,� =q�⊗

��(J(v)

1 Usigni,i)

+(J(v)2 /ejvni,i,�−J(v)

1 )�T

p�

�,

(19)

Wni,i,mat = (Ρsig−1

ni,i VsigT

ni,i ) ⊗ (Unoiseni,i UnoiseH

ni,i ). (20)

Here, Jv,1 = [IM2−1 0] ⊗ IM1 and Jv,2 = [0 IM2−1] ⊗ IM1

are the selection matrices for the first and second subarrays,respectively, for the spatial frequency vni,i,�; T is the transfor-mation matrix, q� is the �-th column of matrix Tni,i, pT� is the�-th row of matrix T−1

ni,i; Rfbai and Cfba

i are the covarianceand complementary covariance matrices of the noise-plus-interference, respectively. Now, let us consider the followinglemma:

Lemma 2: Covariance and complementary covariancematrices of the forward-backward averaged signal can bedecomposed as

R(fba)i (k) =R(fba)

i,1 (k)+R(fba)i,2 (k)+R(fba)

i,3 (k)+R(fba)i,4 (k),

(21)

C(fba)i (k) =C(fba)

i,1 (k)+C(fba)i,2 (k)+C(fba)

i,3 (k)+C(fba)i,4 (k),

(22)

where

R(fba)i,m (k) =

�Ri,m(k) 0

0 ΠNrNtR∗i,m(k)ΠNrNt

;

C(fba)i,m (k) =

�0 Ri,m(k)ΠNrNt

ΠNrNtR∗i,m(k) 0

,

(23)

for m = 1, . . . , 4, where Ri,m(k)’s are given by Lemma 1.Proof Sketch: This Lemma can be proved by substituting

(21) into (13), and by utilizing the definitions of Ri,m(k)sfrom Lemma 1. �

Using Lemma 2, we can separately investigate the effectsof different elements of noise-plus-interference signal on theDoA estimation performance, and hence, can write (18) as

E

�(�vni,i,�)

2

=4�

m=1E

�(�vni,i,�)

2

mwhere

E

�(�vni,i,�)

2

m

=12

�r(v)H


i,m ·WTni,i,matr

(v)ni,i,�

−Re�r(v)T

ni,i,� · Wni,i,mat · C(fba)i,m · WT


�.

(24)

Now, (24) depends on the singular value decomposition (SVD)of the noiseless received signal, which can be difficult toobtain at the BS. However, for massive MIMO systems, thisbecomes possible due to the fact that the steering vectors are


orthogonal. We consider the following Lemma [5] to facilitatethe derivation of the MSE expression for massive MIMOsystems:

Lemma 3: If the elevation and azimuth anglesare both drawn independently from a continuousdistribution, the normalized array response vectorsbecome orthogonal asymptotically, that is, er,jg,i(m) ⊥span {er,j�g�,i�(n) | ∀(j, g, i, m) �= (j�, g�, i�, n)} when thenumber of antennas at the base station goes large, whereer,jg,i(m) = 1√

Nrer,jg,i(m).

Using this property, we can analytically characterize theeffect of each individual perturbation element on the DoAestimation performance. To be specific, the MSE of UL DoAestimation due to pilot contamination is given by the followingTheorem:

Theorem 1: For the massive MIMO network, the MSE ofthe unitary ESPRIT-based UL DoA estimation due to pilotcontamination is given by . . .

Eβ,φ,φ{(Δvni,i,�)2}1

=1

8|αni,i(�)|2N2t Λni,i(M2 − 1)2M2

1

×G−1�

g=0g �=i

Λng,iXng,i

⎛

⎝Lng,i−1�

m=0

|αng,i(m)|2⎞

⎠

×�Yng,i + Y �

ng,i − 2��ejΦYng,i

�, (25)

where Φ = ((M1 − 1)uni,i,� + (M2 − 1)vni,i,�), and Xng,i

and Yng,i are given by

Xng,i = Eψ

��1 + e−j(ωni,i,�−ωng,i,m)

+ . . . + e−j(Nt−1)(ωni,i,�−ωng,i,m)��

2

, (26)

Yng,i = Eβ,φ

��1 + ej(uni,i,�−ung,i,m)

+ . . . + ej(M1−1)(uni,i,�−ung,i,m)�

×�ej(M2−1)(vni,i,�−vng,i,m) − 1

��2

, (27)

and Y �ng,i and Yng,i are given by

Y �ng,i =Eβ,φ

��ej(M1−1)ung,i,m + ejuni,i,�ej(M1−2)ung,i,m

+ . . . + ej(M1−1)uni,i,�

�

×�ej(M2−1)vni,i,� − ej(M2−1)vng,i,m

��2

, (28)

Yng,i =Eβ,φ

��e−j(M1−1)ung,i,m +e−juni,i,�e−j(M1−2)ung,i,m

+ . . . + e−j(M1−1)uni,i,�

�

×�ej(M2−1)(vng,i,m−vni,i,�) − 1

�

×�1 + . . . + e(M1−1)(ung,i,m−uni,i,�)

�

×�e−j(M2−1)vni,i,� − e−j(M2−1)vng,i,m

�(29)

for m = 0, . . . Lng,i−1, and Eψ and Eβ,φ denote, respectively,expectations with respect to DoD and DoAs of the interferencechannel.

Proof: See Appendix A. �Remark 1: Based on Jacobian, MSEs of the elevation and

azimuth angles can be obtained from the MSEs of the spatialfrequencies as follows [22]

Eβ,φ,φ{(Δθ�)2} = Eβ,φ,φ{(Δu�)2} 1π2 sin2(θ�)

, (30)

Eβ,φ,φ{(Δθ�)2} =Eβ,φ,φ{(Δu�)2} cot2(θ�) cot2(φ�)

π2 sin2(θ�)

+Eβ,φ,φ{(Δv�)2}

π2 sin2(θ�) sin2(φ�). (31)

Remark 2: The expectation expressed in (25) of Theorem 1is NOT taken with respect to time, rather, it is taken withrespect to the DoAs or DoDs of interference users due torandom locations of those interference users.

Similarly, the effect of intra-cell interference, inter-cellinterference, and noise elements on the MSE performanceare characterized in Theorem 2, Theorem 3, and Theorem 4,respectively:

Theorem 2: For the massive MIMO network, the MSE ofthe unitary ESPRIT-based UL DoA estimation due to intra-cell interference is given by

Eβ,φ,φ

�(�vni,i,�)

2

2

=ρ21|X ��

ni,i,�|28|αni,i(�)|2N2

t Λni,i(M2 − 1)2M21

×

⎡

⎢⎢⎣J−1�

j=0j �=n

Λji,iXji,i

⎛

⎝Lji,i−1�

m=0

|αji,i(m)|2⎞

⎠

×�Yji,i + Y �

ji,i − 2��ejΦYji,i

�⎤

⎥⎥⎦, (32)

where X ��ni,i,� =

Nt−1�m=0

ejmωni,i,� .

Proof: See Appendix B. �Theorem 3: For the massive MIMO network, the MSE of

the unitary ESPRIT-based UL DoA estimation due to inter-cell interference is given by

Eβ,φ,φ

�(�vni,i,�)

2

3

=ρ21|X ��

ni,i,�|28|αni,i(�)|2N2

t Λni,i(M2 − 1)2M21

×G−1�

g=0g �=i

J−1�

j=0j �=n

Λjg,iXjg,i

⎛

⎝Ljg,i−1�

m=0

|αjg,i(m)|2⎞

⎠

×�Yjg,i + Y �

jg,i − 2��ejΦYjg,i

�. (33)

Proof: The proof is similar to the proof ofTheorem 2. �


Theorem 4: For the massive MIMO network, the MSE ofthe unitary ESPRIT-based UL DoA estimation due to noiseelement is given by

E

�(�vni,i,�)

2

4=

σ2

2|αni,i(�)|2NtΛni,i(M2 − 1)2M1,

where σ2 is the noise variance.Proof: This theorem can be proved following the line of

proof for Theorem-1 in [5]. �Similarly, we can also obtain the MSE expressions for ele-vation spatial frequency, E

�(�uni,i,�)

2

. Accordingly, basedon Jacobian matrices, we can characterize MSE expressionsfor UL elevation and azimuth DoAs from the MSEs of thespatial frequencies.

Remark 3: From Theorem 2 and Theorem 3 we can observethat non-zero correlation among the spreading sequences ofdifferent MSs does cause intra- and inter-cell interferencefor UL DoA estimation, and the corresponding MSEs of theestimation are directly affected by the correlation coefficient,ρ1. On the other hand, as we can see from Theorem 1, the MSEdue to pilot contamination is not dependent on the correlationcoefficient.

Furthermore, these four theorems suggest that our originalwork in [5] may yield strictly suboptimal solutions sincein that work we only consider DoA estimation error dueto noise elements. This observation will be verified throughperformance evaluation in Section V.

C. Complexity Analysis

In this subsection, we discuss the computationalcoomplexity of unitary ESPRIT-based DoA estimationprocedure as presented in Section III-A. We first summarizethe computational complexity of some basic operationsin terms of floating point operations (FLOPS). It requires2(n − 1)mp FLOPS for computing product of two matricesof sizes (m × n) and n × p. For taking inverse of a positivedefinite matrix of size (n × n) requires (n3 + n2 + n)floating point operations; number of FLOPS required fortaking SVD of an m × n matrix is (4m2n + 22n3), andcomplexity for finding eigenvalues of an n × n matrix isn3. Now, we can describe the computational complexityof each step of our algorithm presented. For complexityanalysis, we assume all the channels have L resolvablepaths. Now, correlating the received signal with trainingsymbol matrix in (4) requires Ca = 2(Q − 1)NrNt

number of FLOPS. Taking forward-backward averagingin (11) requires Cb = 2NrNt(Nr + Nt − 2) FLOPS.Number of FLOPS required for taking SVD of theforward-backward averaged received signal in (12) isCc = (8N2

rNt + 176N3t ). Now, for solving shift-invariance

equations for elevation and azimuth spatial frequencies,the number of FLOPS required are, respectively, Cd =2[M2(M1−1)−1]L2+[L3+L2+L]+[2(L−1)M2L(M1−1)],and Ce = 2[M1(M2 − 1) − 1]L2 + [L3 + L2 + L] + [2(L −1)M1L(M2 − 1)]. Finally, for calculating the eigenvalues oftwo shift-invariance operator matrices requires Cf = 2L3

number of FLOPS. Hence, total computational complexity

of our ESPRIT-based DoA estimation method can bewritten as CESPRIT = Ca + Cb + Cc + Cd + Ce + Cf .Next, for comparison. we compute the computationalcomplexity of MUSIC algorithm. For computing thecovariance matrix of the received signal, the number ofFLOPS required is Da = (Q + 1)N2

r . Next, computingthe SVD of the covariance matrix requires Db = 26N3

r

FLOPS. Let Ng denote the number of grids for candidateDoA search. Hence, total number of FLOPS requiredfor extracting the eigenvectors corresponding to noisesubspace is Dc = Ng ([2Nr(Nr − L)] + [2Nr − 3]).Hence computational cost for MUSIC algorithm isCMUSIC = Da + Db + Dc.

IV. DOWNLINK PRECODING AND ACHIEVABLE

RATE ANALYSIS

A. Optimum Precoding for Sum-Rate Maximization

In the DL, at the i-th BS, the Ns × 1 informationsymbol vector intended for the n-th MS in the i-th cellon the k-th subcarrier can be expressed as sdlni[k] =�sdlni,0[k], . . . , sdlni,Ns−1[k]

�, where sdlni,p[k] is the p-th infor-

mation symbol intended for the n-th MS. Accordingly,the Nr × 1 downlink frequency domain transmit signal fromthe i-th BS can be written as

xdli [k] =J−1�

j=0

xdlji[k] =J−1�

j=0

Vji[k]sdlji[k], (34)

where xdlji [k] = Vji[k]sdlji[k], and Vji[k] is the Nr × Ns

precoding matrix for the j-th MS in the i-th cell on the k-th subcarrier. Now, the Nt× 1 received signal at the n-th MSin the i-th cell on the k-th subcarrier, ydlni[k], can be writtenas

ydlni[k] =G−1�

g=0

�Λni,gHdl

ni,g[k]xdlg [k] + ndlni[k]

=G−1�

g=0

J−1�

j=0

�Λni,gHdl

ni,g[k]Vjg[k]sdljg[k] + ndlni[k]

=�

Λni,iHdlni,i[k]Vni[k]sdlni[k]

+J−1�

j=0

j �=n

�Λni,iHdl

ni,i[k]Vji[k]sdlji[k]

+G−1�

g=0

g �=i

J−1�

j=0

j �=n

�Λni,gHdl

ni,g[k]Vjg [k]sdljg[k] + ndlni[k],

(35)

where Hdlni,g[k] is the Nt×Nr downlink channel between the

g-th BS and the n-th MS in i-th cell on the k-th subcarrier,and ndlni[k] is the corresponding Nt × 1 noise vector atthe receiver with E{ndlni[m]ndlni[n]} = σ2INtδ(m − n). In(35), the first term is the desired signal, while the secondand third terms represent the intra- and inter-cell interfer-ences, respectively. Now, the rate for n-th MS in i-th cell is


given by

Ini[k]

= log2 det

⎛

⎜⎝I + Λni,iHdlni,i[k]Vni[k]VH

ni[k]HdlH

ni,i[k]

×⎛

⎝�

(n,i) �=(j,g)

Λni,gHdlni,g[k]Vjg[k]VH

jg [k]HdlH

ni,g[k]+σ2I

⎞

⎠−1⎞

⎟⎠ .

(36)

Accordingly, the sum-rate maximization (SRM) problem canbe expressed as

max{Vji[k]}

J−1�

j=0

Iji[k]

s.t.J−1�

j=0

Tr�Vji[k]Vji[k]H

� ≤ Pt, (37)

where Pt is the total power available at the BS for eachsucarrier. In general, it is challenging to solve the problemin (37) since it is highly non-convex. Alternatively, sum-MSE(mean square error) minimization is another popular utilitymaximization problem for DL multi-user MIMO systems. LetTni be the DL receive processing matrix for the n-th MS.The estimated received symbol vector can then be writtenas sdlni[k] = TH

niydlni[k]. Now, n-th MS’s MSE matrix can be

defined as

Eni[k]

= E��

sdlni[k] − sdlni[k]� �

sdlni[k] − sdlni[k]�H�

=�I −�Λni,iTH

niHdlni,i[k]Vni[k]

�

×�I−�Λni,iTH

niHdlni,i[k]Vni[k]

�H

+�

(n,i) �=(j,g)

Λni,gTHniH

dlni,g[k]Vjg[k]VH

jg [k]HdlH

ni,g[k]Tni

+ σ2THniRni. (38)

Accordingly, the sum-MSE minimization problem can bedefined as

min{Vji[k]}

J−1�

j=0

�ji[k]

s.t.

J−1�

j=0

Tr�Vji[k]Vji[k]H

� ≤ Pt, (39)

where �ni[k] = Tr{Eni[k]}. The relationship between theproblems in (37) and (39) can be established by the followinglemma [23]:

Lemma 4: The sum-rate maximization problem in (37) andthe sum-MSE minimization problem in (39) are equivalent inthe sense that the optimal solutions, {Vji[k]}J−1

j=0 , for bothproblems are identical.

In this work, we assume that no coordination is avail-able among BSs, which is a typical scenario in TDD-basedFD-MIMO networks. Hence, problem in (39) can bewritten as

min{Vji[k]}

��THi Hdl

i,i[k]Vi[k] − I��2F

s.t.

J−1�

j=0

Tr�Vji[k]Vji[k]H

� ≤ Pt, (40)

where THi = blkdiag{TH

0i,TH1i, . . . ,T

H(J−1)i, },

Vi[k] =�V0i[k],V1i[k], . . . ,V(J−1)i[k]

�, and

Hdli,i[k] =

��Λ0i,iHdlT

0i,i[k], . . . ,�

Λ(J−1)i,iHdlT

(J−1)i,i[k]�T

.Now, using channel reciprocity property, the downlinkchannel can be written in terms of the uplink channel:

Hdlni,i[k] = HT

ni,i[k]

= B∗ni,i[k]Dni,iAT

ni,i

= B∗ni,iDni,iAT

ni,i[k], (41)

where Ani,i[k] =�er,ni,i,k(0) . . . er,ni,i,k(Lni,i − 1)

�,

where er,ni,i,k(�) = er,ni,i(�)e−j2πk�

Nc , and Bni,i =�et,ni,i(0) . . . et,ni,i(Lni,i − 1)

�. Assuming each MS will

only use its own DL CSI for receive processing, we haveTHn,iH

dlni,i[k] = Dni,iAT

ni,i. Accordingly, the problem in (40)can be expressed as

min{Vji[k]}

��Di,iATi,i[k]Vi[k] − I

��2F

s.t.

J−1�

j=0

Tr�Vji[k]Vji[k]H

� ≤ Pt, (42)

where Di,i = blkdiag{D0i,i, D1i,i, . . . , D(J−1)i,i, },Ai,i[k] =

�A0i,i[k], A1i,i[k], . . . , A(J−1)i,i[k]

�, and

Dni,i =�

Λni,iDni,i accounts for both the large andsmall scale fading effect. The solution to this problem isgiven by the following theorem:

Theorem 5: Let Di,iATi,i[k] = Ui,i[Λi,i,0]WH

i,i

be the SVD of the effective channel, Di,iATi,i[k],

where Λi,i = diag{λ0i,i, λ1i,i, . . . , λ(JNs−1)i,i}.Then the optimal precoding matrix problem in (42)is given by Vi[k] = Wi,i[Ξi,i,0]T UH

i,i, whereΞi,i = diag{ξ0i,i, ξ1i,i, . . . , ξ(JNs−1)i,i}, and ξmi,i =λmi,i/(λ2

mi,i + η), with the smallest η ≥ 0 satisfying�JNs−1m=0 |ξmi,i|2 ≤ Pt.

Proof: See Appendix C. �From theorem 5, it can be seen that the optimal precoder

that minimizes the sum-MSE, and hence maximizes the sum-rate can be constructed from the estimated UL DoAs as wellas the path gains. In this paper, we assume that the BS hasperfect knowledge of the path gains. However, path gains canalso be estimated using maximum likelihood (ML) methodonce the DoAs have been estimated. Our work on this aspectcan be found in [24].

B. Large-Antenna System Analysis

In this section, we present the achievable rate analysisand simplified precoding strategy for massive FD-MIMO


systems. Our discussions in this sub-section are based onasymptotic analysis. This can be viewed as the special case ofSection IV-A where the number of antennas at the base stationgoes large asymptotically.

1) Achievable Rate Under Perfect Channel Estimation: Inthis case, (35) can be written as

ydlni[k] = B∗ni,iDni,iAT

ni,i[k]Vni[k]sdlni[k]

+J−1�

j=0j �=n

B∗ni,iDni,iAT

ni,i[k]Vji[k]sdlji[k] + n�ni[k],

(43)

where

n�ni[k] =

G−1�

g=0g �=i

J−1�

j=0

�Λni,gHdl

ni,g[k]Vjg [k]sdljg[k] + ndlni[k]

(44)

is the equivalent noise-plus-inter-cell-interference vector. Asthe number of antennas grows large, the right singular matrix,Wi,i, in Theorem 5 can be approximated as the DoA matrix,A∗i,i. In other words, for massive FD-MIMO systems, eigen

directions align with the directions of arrivals, which is alsovalidated in [5] and [6]. From Lemma 3, the array steeringvectors for different MSs become orthogonal as the number ofantennas grows large, i.e., we have (1/Nr)AH

ji,i[k]Aj�i,i[k] →0 as Nr → ∞ for all j �= j�. Hence for the massive MIMOsystems, beamforming in the DoA directions nullifies the intra-cell interferences. Therefore, the optimum eigen-beamformerunder perfect DoA estimation:

Veigni [k] =

1Nr

A∗ni,i[k], (45)

and accordingly, the received signal in (43) can be written as

ydlni[k] = B∗ni,iDni,isdlni[k] + n�

ni[k]. (46)

Now, the signal in (46), under the optimal receive processing,results in

ydlni[k] = Dni,isdlni[k] + n�ni[k], (47)

where ydlni[k] =�BTni,iB

∗ni,i

�−1BTni,iy

dlni[k], and n�

ni[k] =�BTni,iB

∗ni,i

�−1BTni,in

�ni[k]. Accordingly, the achievable rate

for the n-th user in i-th cell, Ini[k], can be expressed as in(48), shown at the top of the next page, where, BH

ni,i =�BTni,iB

∗ni,i

�−1BTni,i, and Qdl

ni[k] = E{sdlni[k]sdlH

ni [k]} is thecovariance matrix of the transmit symbol vector from the i-thBS intended for the n-th MS on the k-th subcarrier. Now,(48) can succinctly be written as

Ini[k]=log2 det�ILni,i +Dni,iQdl

ni[k]DHni,iR

�−1

ni [k]�, (49)

where inter-cell interference-plus-noise covariance matrix,R

�ni[k], is defined in (50), shown at the top of next page.

Let us now consider the following lemma:Lemma 5: Assuming all BSs apply the same precoding

strategy, equivalent inter-cell interference-plus-noise covari-ance matrix, R

�ni[k], is approximated by

R�ni[k] ≈ (ζni + σ2)ILni,i , (51)

where ζni = J(G − 1)E{Λni,gpjg,�[k]|αni,g(�)|2}, wherepjg,�[k] is the power allocated on the �-th symbol for j-thuser in g-th cell on the k-th subcarrier.

Proof: This lemma can be proved by substituting (41)in (50), and by utilizing the orthogonality property fromLemma 3. Details are omitted due to page limitation. �

Accordingly, (49) results in

Ini[k] = log2 det�ILni,i +

1ζni + σ2

Dni,iQdlni[k]DH

ni,i

�.

(52)

Assuming Gaussian input signal, Qdlni[k] =

E{sdlni[k]sdlH

ni [k]} = diag{pni,0[k], . . . , pni,L−1[k]}, wherepni,�[k] is the power to be allocated on the �-th informationsymbol on the k-th subcarrier for the target user. Now, usingHadamard inequality, (52) can be rewritten as

Ini[k] = log2 Π�

�1 +

Λni,i|αni,i(�)|2pni,�[k]ζni + σ2

�

=Lni,i−1�

�=0

log2 (1 + γni,�pni,�[k]), (53)

where γni,� = Λni,i|αni,i(�)|2/(ζni + σ2). Accordingly,the optimal power allocation under perfect DoA estima-tion is the well-known water-filling solution which can beexpressed as

pni,�[k] = [μni,�[k] − 1/γni,�]♦, (54)

where [x]♦ denotes a function with [x]♦ = 0 when x < 0,and [x]♦ = x when x > 0, and μni,�[k] is the correspondingLagrange multiplier.

2) System Achievable Rate Under DoA Estimation Errors:In this case, since the BS does not have perfect DoA estima-tion, the array steering matrix for the n-th MS in i-th cell,in the presence of DoA estimation error, can be expressed inthe form of

Âni,i[k] =�er,ni,i,k(0) . . . er,ni,i,k(Lni,i − 1)

�,

where er,ni,i,k(�) = e−j2πk�

Nc a(vni,i,� + Δvni,i,�)⊗ a(uni,i,� +Δuni,i,�), and Δuni,i,� and Δvni,i,� represent the DoA estima-tion errors in the azimuth and elevation spatial frequencies forthe �-th path of the channel between the i-th BS and the n-thMS in i-th cell. Now, let us consider the following lemma:

Lemma 6: For the massive FD-MIMO OFDM sys-tem, the normalized steering vectors er,jg,i,k(�) =1/

√Nrer,jg,i,k(�) and êr,j�g�,i,k(��) = 1/

√Nrer,j�g�,i�,k(��),

∀{j, g, i, �} �= {j�, g�, i�, ��}, becomes orthonormal asymptoti-cally as the number of antenna, Nr → ∞ .

Proof: A similar lemma is proved in [5], and henceomitted here. �

Using Lemma 6, we have 1Nr

ATni,i[k] Â∗

jg,i[k] = 0,

∀{n, i} �= {j, g}, and 1Nr

ATni,i[k] Â∗

ni,i[k] = 1Nr

diag{er,ni,i,k(0)er,ni,i,k(0), . . . , er,ni,i,k(Lni,i−1)er,ni,i,k(Lni,i−1)} for {n, i} = {j, g}. Hence, for massive MIMO systems,we can express the optimum eigen-beamformer under imper-fect DoA estimation as

Veigni [k] =

1Nr

Â∗ni,i[k]. (55)


Ini[k] = log2 det

⎛

⎜⎜⎜⎜⎜⎝ILni,i +

Dni,iQdlni[k]DH

ni,i

G−1�g=0g �=i

J−1�j=0

Λni,gBHni,iH

dlni,g[k]Vjg [k]Qdl

jg[k]VHjg[k]HdlH

ni,g[k]Bni,i + σ2I

⎞

⎟⎟⎟⎟⎟⎠. (48)

R�ni[k] =

G−1�

g=0g �=i

J−1�

j=0

Λni,gBHni,iH

dlni,g[k]Vjg [k]Qdl

jg[k]VHjg [k]HdlH

ni,g[k]Bni,i + σ2I. (50)

Ini[k] = E

⎧⎨

⎩

Lni,i−1�

�=0

log�1 + γni,� |er,ni,i,k(�)er,ni,i,k(�)|2 pni,�[k]

�⎫⎬

⎭. (56)

E{pni,�[k]} =

μni,�[k] − 1

γni,�M21 M2

2

(1 +

M21E�(Δvni,i,�)2

�

12

)(1 +

M22 E�(Δuni,i,�)2

�

12

)�♦. (58)

Accordingly, achievable rate, in the presence of UL DoAestimation error, can be written as in (56), shown at thetop of this page, where pni,�[k] denotes the power to beallocated in the presence of DoA estimation error, γni,� =Λni,i|αni,i(�)|2/(N2

r (σ2 + ζni)), and the expectation is takenwith respect to estimation error. Using method of Lagrangianmultiplier, the optimal expected power allocation on �-thinformation symbol for the n-th MS in the i-th cell on thek-th subcarrier is given by

E{pni,�[k]}

=

μni,�[k] − 1

γni,�E{|eTr,ni,i,k(�)e∗r,ni,i,k(�)|2}

�♦, (57)

where μni,�(k) is the corresponding Lagrange multiplier.Finally, (57) can be simplified [5] as in (58), shown at thetop of this page.

It can be observed that in the absence of DoA estimationerror, the optimal power allocation algorithm in (58) convergesto water filling solution in (54). It is also to be noted here thatboth power allocations in (54) and (58) take into account theeffects of inter-cell interference, unlike the single-user eigen-beamforming presented in [5].

C. Precoding Complexity Analysis

In this subsection, we briefly discuss the computationalcomplexity of the proposed DoA-based precoding strategy aspresented in Theorem 5. Similarly to the Section III-C, forcomputational complexity analysis, here we again assume thatall the channels have L resolvable paths. Now, for formingthe effective channels, Di,iAT

i,i[k], the number of FLOPSrequired is Ea = 2(JL−1)JLNr. Taking SVD of the effectivechannel requires Eb = 4J2L2Nr + 22N3

r FLOPS. Now, forconstructing the final precoder, number of FLOPS requiredis Ec = [2(Nr − 1)NrJL] + 2(JL − 1)NrJL. Hence, totalnumber of FLOPS required for our DoA-based precoder canbe written as CDoA = Ea + Eb + Ec. Next, for comparison,we calculate the computational cost for conventional Block

Diagonalization-based precoding. Let Lj denote the rank ofthe matrix [HdlT

0i,i, . . . ,HdlT

(j−1)i,i,HdlT

(j+1)i,i, . . . ,HdlT

(J−1)i,i]T .

For complexity analysis, we assume that Lj = L; ∀j. Hence,following the steps of conventional block diagonalizationprecoding, total number of FLOPS required for BD is CBD =J([4(J − 1)2N2

t Nr + 22N3r ] + [2(Nr − 1)Nt(Nr − L)] +

[4N2t (Nr − L) + 22(Nr − L)3]).

V. PERFORMANCE EVALUATION

In this section, we evaluate the ESPRIT-based UL DoA esti-mation for multi-cell multi-user massive FD-MIMO OFDMnetworks through simulation. For simulation evaluation,we consider seven hexagonal cells with MSs uniformly dis-tributed in each cell. Without loss of generality, we assumethat number of co-scheduled MSs in each cell is 10. AnM1×M2 (antenna elements in elevation direction, and antennaelements in the azimuth direction) rectangular antenna array isassumed at the BS, whereas the mobile device has a uniformlinear array. Different MSs are using non-orthogonal spreadingsequences as UL pilots, and the same pool of sequences isreused in all seven cells. Therefore, in the UL, the targetBS is subject to intra-cell interference as well as interferencefrom MSs in six other neighboring cells for the purpose ofDoA estimation. Cell radius is set to be 1000 meters. Thesystem is assumed to operate at the mm-wave band with28 GHz carrier frequency. 4 dominant clusters are assumedfor each UL channel from the MS to the BS, and each clustercontributes one resolvable path. The antenna spacing for boththe received and transmit antenna arrays is assumed to be 0.5λ.The number of transmit antennas at each MS is set to be 8.In this paper, we invoke the far field assumption, and thewavefront impinging on the antenna array is assumed to beplaner. The transmission medium is assumed to be isotropicand linear.

The estimation performance of elevation and azimuth anglesfor 8 × 8, 4 × 16, and 16 × 4 antenna arrays are shownin Fig. 2 and Fig. 3, respectively, where the RMSE of theDoA estimation has been used as the performance metric,


Fig. 2. Elevation angle estimation for 64 antennas.

Fig. 3. Azimuth angle estimation for 64 antennas.

and the correlation coefficient of spreading sequences, ρ1,is chosen to be 0.1. As the figure suggests, the analyticalresults of DoA estimation match well with that of empiricalresults asymptotically with SNR. Furthermore, antenna arraygeometry has a significant impact on estimation performance.Fig. 2 clearly suggests that the 16× 4 antenna array performsbetter than 8 × 8 and 4 × 16 arrays in elevation angleestimation. However, 8×8 array configuration may outperformthe 4×16 configuration in azimuth angle estimation as shownin Fig. 3. This is quite counter-intuitive since the 4 × 16array has more elements in the azimuth domain. The reasonmainly comes from the fact that the azimuth DoA estimationis actually coupled with elevation DoA estimation. For the4×16 array, the performance of the elevation DoA estimationmay be so bad that it affects the azimuth DoA estimationperformance. This dependence is manifested through Jaco-bians (see Remark 1), which, in fact, results from the under-lying physics/ coordinate system of the 3D MIMO model.On the other hand, elevation estimation is not dependenton azimuth estimation, and hence, 16 × 4 array geometry

Fig. 4. Elevation angle estimation for 256 antennas.

Fig. 5. Azimuth angle estimation for 256 antennas.

still outperforms 8 × 8 array in elevation angle estimation.These observations can provide important design intuitions forFD-MIMO networks that adopt subspace-based channel esti-mation methods. The elevation and azimuth angle estimationresults for 16×16, 8×32, and 32×8 antenna arrays are shownin Fig. 4 and Fig. 5, respectively. Comparing the results withthose presented in Fig. 2 and Fig. 3, we can observe that asthe total number of antennas increases, the DoA estimationaccuracy accordingly increases, which is also evident fromour analytical results.

In Fig. 6, the average achievable sum-rates for differentprecoding strategies are compared for multi-cell multi-user massive FD-MIMO networks. Five schemes are com-pared: the introduced scheme presented in Theorem 5; theblock-diagonalization based zero forcing (BD-ZF) precodingmethod [8], [9] assuming full CSI at the BS; and three eigen-beamforming schemes based on the large antenna systemanalysis. To be specific, Scheme A is the single-user eigen-beamforming introduced in [5]. This scheme uses eigen-beamformer in (55), and applies the modified water-filling


Fig. 6. Average achievable sum-rate comparison.

power allocation presented in [5] taking into account the DoAestimation error due to the noise. However, Scheme A doesn’tconsider the effects of intra/inter-cell interference into powerallocation. In Scheme B, (55) is used as the beamformer andthe traditional water-filling in (54) is used as power allocationassuming ideal DoA estimation. Scheme C uses the samebeamformer as Scheme A and B, however, it utilizes the powerallocation in (58) considering the DoA estimation error dueto intra/inter-cell interference of the network. Fig. 6 clearlysuggests that the scheme introduced in Theorem 5 achievesbest performance among all precoding strategies over theentire SNR regime of interests. Even assuming full CSI atthe BS, the BD-ZF scheme performs worst in the medium tohigh SNR regime. This suggests that BD-ZF based precodingstrategy may yield strictly suboptimal performance for massiveFD-MIMO networks. It is to be noted here that even thoughBD is using full channel state information, the performancegain of DoA-based method over BD method is coming fromthe fact that DoA-based method utilizes the structure of theunderlying channel, whereas BD method does not take intoaccount the underlying structure of the MIMO channel.

For the three eigen-beamforming schemes based on the largeantenna system analysis, we have the following observation:Scheme C outperforms both Schemes A and B over the entireSNR regime since Scheme C considers the comprehensivecharacterization of the DoA estimation for power allocationas discussed in Remark 1. Scheme A performs better thanScheme B at low SNRs indicating the importance of incor-porating the DoA estimation error. Since DoA estimationerror decreases as SNR increases, both Scheme A and Bapproach Scheme C asymptotically. This is because the powerallocation in (58) converges to water-filling solution in (54)with increasing SNR.

In Fig. 7 we compare computational complexity of ourESPRIT-based DoA estimation method with the widely usedMUSIC algorithm, for a square BS antenna array (i.e.,M1 = M2). Number of transmit antennas is 8, and number ofpaths in the channel is 4. For MUSIC algorithm, number ofgrids for candidate DoA search is 360 which is a typical num-

Fig. 7. Computational complexity comparison for DoA estimation algo-rithms.

Fig. 8. Computational complexity comparison for precoding methods.

ber. We can observe that as the number of antennas increases,the complexity of the MUSIC algorithm increases much fastercompared to the complexity of ESPRIT algorithm. Hence,for massive MIMO system, MUSIC-based DoA estimationwill incur significantly more computational burden than theESPRIT method. In Fig. 8, we compare the computationalcomplexity of the our proposed DoA-based precoding scheme(Theorem 5) is compared with the complexity of traditionalBlock Diagonalization (BD) precoding. We can observe that asfor low and mid-size arrays, the complexity of both algorithmsare similar. However, as the number of antennas increases theDoA-based precoder outperforms the BD method in term ofthe computational complexity.

VI. CONCLUSION

Accurate DL CSI is critical for massive MIMO to realize thepromised throughput gain. In this paper, we introduced optimalDL MIMO precoding and power allocation strategies formulti-cell multi-user massive FD-MIMO networks based onUL DoA estimation at the BS. The UL DoA estimation error


for such a network has been analytically characterized andhas been incorporated into the proposed MIMO precoding andpower allocation strategy. Simulation results suggested that theproposed strategy outperforms existing BD-ZF based MIMOprecoding strategies which requires full CSI at the BS. Thiswork sheds a light on system design for massive FD-MIMOcommunications which is critical for 5G and Beyond 5Gcellular networks.

APPENDIX APROOF OF THEOREM 1

Effect of pilot contamination on the MSE of DoA estimationis given by

E

�(�vni,i,�)

2

1

=12

�r(v)H


i,1 ·WTni,i,mat · r(v)

ni,i,�

−Re�r(v)T

ni,i,� · Wni,i,mat · C(fba)i,1 · WT


�.

(59)

Let us now denote

βni,i,� = Vsigni,iΡ

sig−1

ni,i q�, (60)

αv,ni,i,� =�pT��J(v)

1 Usigni,i

�+ �J(v)

2 /ejvni,i,� − J(v)1

�

×�Unoiseni,i UnoiseH

ni,i

��T. (61)

Using (19) and (20), we have WTni,i,matr

(v)ni,i,� = βni,i,� ⊗

αv,ni,i,�. The MSE in (59) becomes

E

�(�vni,i,�)

2

1

=12

�(βni,i,� ⊗ αv,ni,i,�)

H · R(fba)Ti,1 · (βni,i,� ⊗ αv,ni,i,�)

−Re�(βni,i,� ⊗ αv,ni,i,�)

T ·C(fba)i,1 (βni,i,� ⊗ αv,ni,i,�)

�.

(62)

It can be easily verified that αv,ni,i,� can be written as

αTv,ni,i,�

= cT�

��Jv,2Ani,i

�+

Jv,2 −�Jv,1Ani,i

�+

Jv,1

�,

=1

(M2 − 1)M1

×�− 1,−e−juni,i,� , . . . ,−e−j(M1−1)uni,i,� ,

0, . . . , 0, e−j(M2−1)vni,i,� , e−j((M2−1)vni,i,�+uni,i,�), . . . ,

e−j((M2−1)vni,i,�+(M1−1)uni,i,�)

. (63)

Next, in order to obtain the expression for βni,i,�, we need toperform the SVD of the perturbation-free signal in (11):�

Λni,i�Ani,iDni,iBH

ni,i(k) ΠNrA∗ni,iD

∗ni,iB

Tni,i(k)ΠNt

�

= Ani,idiag*bni,i

+ �BHni,i(k) Γni,iBT

ni,i(k)ΠNt

�,

where

Γni,i = diag��

e−j((M1−1)uni,i,0+(M2−1)vni,i,0), . . . ,

e−j((M1−1)uni,i,Lni,i−1+(M2−1)vni,i,Lni,i−1)�

,

BHni,i(k) = diag

�ejφ

�ni,i,0 , . . . , e

jφ�ni,i,Lni,i−1

BHni,i(k),

and

bni,i =�bni,i,0, . . . , bni,i,Lni,i−1

�,

where bni,i,� and φ�ni,i,� are the amplitude and the phase of

the channel gain αni,i(�), respectively. Accordingly, based onLemma 3, we can obtain

Usigni,i = 1/

�NrAni,i,

Ρsigni,i =

�2NrNt

�Λni,idiag

*bni,i

+,

and

VsigH

ni,i = 1/�

2Nt

�BHni,i(k) Γni,iBT

ni,i(k)ΠNt

�.

In [25], the vector βni,i,� is given as βni,i,� =Vsigni,iΡ

sig−1

ni,i UsigH

ni,i Ani,ic�. Now substituting here the expres-sions of Usig

ni,i, Ρsigni,i, and Vsig

ni,i, we obtain:

βni,i,� =1

(bni,i,�)�

Λni,i√

2Nt

Vsigni,ic�. (64)

Hence, the expression (βni,i,� ⊗ αv,ni,i,�) in (62) can bewritten as equation (65), shown at the top of next page. Now,using equation (23) and (65), the first term in (62) can bewritten as (66), shown at the top of next page. Now, using(15), and after some simplification, we have�eHt,ni,i,k(�) ⊗ αH

v,ni,i,�

�RTi,1 (et,ni,i,k(�) ⊗ αv,ni,i,�)

=1

(M2−1)2M21

G−1�

g=0g �=i

��Λng,i

�2

Xng,iYng,i

L�

m=1

|αng,i(m)|2,

(67)

where Xng,i and Yng,i are given by

Xng,i = Eψ

��1 + e−j(ωni,i,�−ωng,i,m)

+ . . . e−j(Nt−1)(ωni,i,�−ωng,i,m)��

2

, (68)

Yng,i = Eβ,φ

��1 + ej(uni,i,�−ung,i,m)

+ . . . + ej(M1−1)(uni,i,�−ung,i,m)�

× �ejvni,i,�e−jvng,i,m − 1� ��

2

, (69)

for m = 0, . . . Lng,i−1, and Eψ and Eβ,φ denote, respectively,expectations with respect to DoD and DoAs. Now, similarlyto (67), we also have�eTt,ni,i,k(�)ΠNt ⊗ αH

v,ni,i,�

�ΠNrNtR

Hi,1ΠNrNt

× �ΠNte∗t,ni,i,k(�) ⊗ αv,ni,i,�

�

=1

(M2−1)2M21

G−1�

g=0g �=i

��Λng,i

�2

Xng,iY�ng,i

L�

m=1

|αng,i(m)|2,

(70)


β� ⊗ αv,� =1

(bni,i,�)2Nt

�Λni,i

e−jφ

�ni,i,�et,ni,i,k(�)

ejφ�ni,i,�ej((M1−1)uni,i,�+(M2−1)vni,i,�)ΠNte

∗t,ni,i,k(�)

�⊗ αv,�

=1

(bni,i,�)2Nt

�Λni,i

e−jφ

�ni,i,�et,ni,i,k(�) ⊗ αv,ni,i,�

ejφ�ni,i,�ej((M1−1)uni,i,�+(M2−1)vni,i,�)ΠNte

∗t,ni,i,k(�) ⊗ αv,ni,i,�

�. (65)

(βni,i,� ⊗ αv,ni,i,�)H · R(fba)T

i,1 · (βni,i,� ⊗ αv,ni,i,�)

=1

b2ni,i,�4N2

t Λni,i

��eHt,ni,i,k(�) ⊗ αH

v,ni,i,�

�RTi,1 (et,ni,i,k(�) ⊗ αv,ni,i,�)

+�eTt,ni,i,k(�)ΠNt ⊗ αH

v,ni,i,�

�ΠNrNtR

Hi,1ΠNrNt

�ΠNte


��. (66)

where

Y �ng,i = Eβ,φ

��ej(M1−1)ung,i,m + ejuni,i,�ej(M1−2)ung,i,m

+ . . . + ej(M1−1)uni,i,�

�

×�ej(M2−1)vni,i,� − ej(M2−1)vng,i,m

��2

, (71)

for m = 0, . . . Lng,i − 1. Now, using (67) and (70), we canwrite (66) as


i,1 · (βni,i,� ⊗ αv,ni,i,�)

=1

b2ni,i,�4N2

t Λni,i1

(M2 − 1)2M21

×G−1�

g=0g �=i

��Λng,i

�2

Xng,i

Lng,i−1�

m=0

|αng,i(m)|2 �Yng,i+Y �ng,i

�.

(72)

Similarly, we can also have

(βni,i,� ⊗ αv,ni,i,�)T ·C(fba)

i,1 (βni,i,� ⊗ αv,ni,i,�)

=1

b2ni,i,�2N2

t Λni,i1

(M2 − 1)2M21

ejΦ

×G−1�

g=0g �=i

��Λng,i

�2

Xng,iYng,i

Lng,i−1�

m=0

|αng,i(m)|2, (73)

where Φ = ((M1 − 1)uni,i,� + (M2 − 1)vni,i,�), and Yng,i isgiven in (29). Finally, plug the expressions from (72) and (73)into (62), and the proof is finished.

APPENDIX BPROOF OF THEOREM 2

Similarly to (62), MSE due to intra-cell interference can bewritten as

E

�(�vni,i,�)

2

2

=12

�(βni,i,� ⊗ αv,ni,i,�)

H · R(fba)Ti,2 · (βni,i,� ⊗ αv,ni,i,�)

−Re�(βni,i,� ⊗ αv,ni,i,�)

T ·C(fba)i,2 (βni,i,� ⊗ αv,ni,i,�)

�.

(74)

Using (23), for m = 2, the first term in (74) can beexpressed as equation (75), shown at the top of next page.Now, using (16), and after some simplifications, we can writethe first term in (75) as (76), also shown at the top of nextpage.

It can be shown that

Eα��

eHt,ni,i,k(�)1NtBji,i(k)

⊗αTv,ni,i,�ΠNrA

∗ji,i

�vec {Dji,i}∗

×vec{Dji,i}T�BHji,i(k)1Ntet,ni,i,k(�) ⊗ AT

ji,iαv,ni,i,�

��

=��X ��

ni,i,�

��2Lji,i−1�

m=0

�|αji,i(m)|2 |Xji,i(m)|2 αT

v,ni,i,�ΠNr

× e∗ji,i(m)eTji,i(m)αv,ni,i,�

�, (77)

where X ��ni,i,� =

Nt−1�r=0

ejrωni,i,� , and Xji,i(m) =

Nt−1�r=0

ejr(ωni,i,�−ωji,i,m). After some tedious but straight for-

ward calculations, we have

Eβ,φ�αTv,ni,i,�ΠNre

∗ji,i(m)eTji,i(m)αv,ni,i,�

�

=1

(M2 − 1)2 M21

Yji,i. (78)

Now, using (77) and (78), we can simplify (76) as follows:

�eHt,ni,i,k(�)ΠNt ⊗ αT

v,ni,i,�

�ΠNrNtR

∗i,2

× (et,ni,i,k(�) ⊗ αv,ni,i,�)

= ρ21

1(M2 − 1)2M2

1

|X ��ni,i,�|2

×

⎡

⎢⎢⎣J−1�

j=0j �=n

��Λji,i

�2

Xji,iYji,i

⎛

⎝Lji,i−1�

m=0

|αji,i(m)|2⎞

⎠

⎤

⎥⎥⎦.

(79)


(βni,i,� ⊗ αv,ni,i,�)T · C(fba)


=1

b2ni,i,�4N2

t Λni,iejΦ��

eHt,ni,i,k(�)ΠNt ⊗ αTv,ni,i,�

�ΠNrNtR

∗i,2 (et,ni,i,k(�) ⊗ αv,ni,i,�)

+�eTt,ni,i,k(�) ⊗ αT

v,ni,i,�

�Ri,2ΠNrNt

�ΠNte


��. (75)

�eHt,ni,i,k(�)ΠNt ⊗ αT

v,ni,i,�

�ΠNrNtR

∗i,2 (et,ni,i,k(�) ⊗ αv,ni,i,�)

= ρ21Eα,β,φ,ψ

⎡

⎢⎢⎣J−1�

j=0j �=n

��Λji,i

�2 �eHt,ni,i,k(�)1NtBji,i(k) ⊗ αT

v,ni,i,�ΠNrA∗ji,i

�vec{Dji,i}∗

×vec{Dji,i}T�BHji,i(k)1Ntet,ni,i,k(�) ⊗ AT

ji,iαv,ni,i,�

�

⎤

⎥⎥⎦ . (76)

Similarly, we can simplify the second term in (75) as follows:�eHt,ni,i,k(�)ΠNt ⊗ αT

v,ni,i,�

�ΠNrNtR

∗i,2

×(et,ni,i,k(�) ⊗ αv,ni,i,�)

= ρ21

1(M2 − 1)2M2

1

|X ��ni,i,�|2

×

⎡

⎢⎢⎣J−1�

j=0j �=n

��Λji,i

�2

Xji,iYji,i

⎛

⎝Lji,i−1�

m=0

|αji,i(m)|2⎞

⎠

⎤

⎥⎥⎦.

(80)

Now, plugging the expressions from (79) and (80) into (75),we obtain

(βni,i,� ⊗ αv,ni,i,�)T · C(fba)


=1

b2ni,i,�4N2

t Λni,iρ21

1(M2 − 1)2M2

1

|X ��ni,i,�|2ejΦ

×

⎡

⎢⎢⎣J−1�

j=0j �=n

��Λji,i

�2

⎛

⎝Lji,i−1�

m=0

|αji,i(m)|2⎞

⎠Xji,i

�2Yji,i

�⎤

⎥⎥⎦ .

(81)

Following similar procedure, we can also obtain


i,2 · (βni,i,� ⊗ αv,ni,i,�)

=1

b2ni,i,�4N2

t Λni,iρ21

1(M2 − 1)2M2

1

|X ��ni,i,�|2

×

⎡

⎢⎢⎣J−1�

j=0j �=n

��Λji,i

�2

⎛

⎝Lji,i−1�

m=0

|αji,i(m)|2⎞

⎠

× Xji,i

�Yji,i + Y �

ji,i

�

⎤

⎥⎥⎦ . (82)

Now, plugging the expressions from (81) and (82) into (74),we obtain the desired result.

APPENDIX CPROOF OF THEOREM 5

The problem in (42) is a convex quadratic optimizationproblem, and can be solved using Lagrangian method. TheLagrange function for (42) can be written as

L (Vi[k], μik)= Tr

*RHi Hi,i[k]Vi[k]VH

i [k]Ri − RHi Hi,i[k]Vi[k]

−VHi [k]Ri + I

++ μik

�Tr{Vi[k]VH

i [k]} − Pt�,

(83)

where μik is the corresponding Lagrange multiplier. Now,taking the derivative of the Lagrange function w.r.t. Vi[k] andsetting the derivative equal to zero, we can obtain the desiredresult.

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Rubayet Shafin received the B.S. degree in electri-cal and electronic engineering from the BangladeshUniversity of Engineering and Technology, Dhaka,Bangladesh, in 2013, and the M.S. degree in elec-trical engineering from the University of Kansasin 2017. He is currently pursuing the Ph.D. degree inelectrical engineering at Virginia Tech. His researchinterests include MIMO systems, mm-wave com-munications, and the application of machine learn-ing/deep learning in wireless communication. Hewas a recipient of the 2018 IEEE TAOS Best Paper

Award and the 2018 IEEE TCGCC Best Conference Paper Award.

Lingjia Liu (SM’15) received the B.S. degree inelectronic engineering from Shanghai Jiao TongUniversity and the Ph.D. degree in electrical andcomputer engineering from Texas A&M University.He is currently an Associate Professor in the BradleyDepartment of Electrical and Computer Engineering,Virginia Tech. He is also the Associate Director forAffiliate Relations at Wireless@Virginia Tech. Priorto that, he was an Associate Professor in the EECSDepartment, University of Kansas (KU). From 2008and 2011, he was with the Standards & Mobility

Innovation Lab., Samsung Research America, where he received the GlobalSamsung Best Paper Award in 2008 and 2010. He was leading Samsung’sefforts on multiuser MIMO, CoMP, and HetNets in 3GPP LTE/LTE-advancedstandards.

His general research interests mainly lie in emerging technologies for5G cellular networks including machine learning for wireless networks,massive MIMO, massive machine-type communications, and mm-wave com-munications. He received the Air Force Summer Faculty Fellowship from2013 to 2017, the Miller Scholarship at KU in 2014, the Miller ProfessionalDevelopment Award for Distinguished Research at KU in 2015, the 2016 IEEEGLOBECOM Best Paper Award, the 2018 IEEE ISQED Best Paper Award,the 2018 IEEE TCGCC Best Conference Paper Award, and the 2018 IEEETAOS Best Paper Award.

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