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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TWC.2016.2594173, IEEETransactions on Wireless Communications

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DoA Estimation and Capacity Analysis for 3DMillimeter Wave Massive-MIMO/FD-MIMO

OFDM SystemsRubayet Shafin, Student Member, IEEE, Lingjia Liu, Senior Member, IEEE, Jianzhong (Charlie) Zhang,

Fellow, IEEE, and Yik-Chung Wu, Senior Member, IEEE

Abstract—With the promise of meeting future capacity de-mands, 3D massive-MIMO/Full Dimension MIMO (FD-MIMO)systems have gained much interest in recent years. Apart fromthe huge spectral efficiency gain, 3D massive-MIMO/FD-MIMOsystems can also lead to significant reduction of latency, simplifiedmultiple access layer, and robustness to interference. However, inorder to completely extract the benefits of the system, accuratechannel state information is critical.

In this paper, a channel estimation method based on directionof arrival (DoA) estimation is presented for 3D millimeter wavemassive-MIMO OFDM systems. To be specific, the DoA isestimated using Estimation of Signal Parameter via Rotational In-variance Technique (ESPRIT) method, and the root mean squareerror (RMSE) of the DoA estimation is analytically characterizedfor the corresponding MIMO-OFDM system. An ergodic capacityanalysis of the system in the presence of DoA estimation erroris also conducted, and an optimum power allocation algorithmis derived. Furthermore, it is shown that the DoA-based channelestimation achieves better performance than traditional linearminimum mean squared error estimation (LMMSE) in termsof ergodic throughput and minimum chordal distance betweenthe subspaces of the downlink precoders obtained from theunderlying channel and the estimated channel.

Index Terms—3D Massive MIMO OFDM Systems, MillimeterWave Communication, Parametric Channel Estimation, 5G.

I. INTRODUCTION

In recent years, wireless communications have experiencedan unprecedented growth-rate of wireless data traffic due tothe rapid introduction of various connected mobile devices andexcessively data-hungry applications run on those devices. Ac-cording to the International Telecommunication Union (ITU)[3], the total number of mobile cellular subscription is fastapproaching the total number of people living on earth, andis expected to be more than 7 billion by the end of 2015,which corresponds to a penetration rate of 97%. Cisco Systemspredicts that Global Mobile data traffic will increase nearly tenfold between 2014 and 2019 reaching 24.3 exabytes per monthin 2019 [4]. In order to fulfill this huge capacity demand,multiple-input-multiple-output (MIMO) [5] systems have beenintroduced as the primary solution to increase the spectral-efficiency of the underlying communication system. However,

R. Shafin and L. Liu are with the Department of Electrical Engineeringand Computer Science, University of Kansas, USA. J. Zhang is with theStandards and Mobility Innovation Lab., Samsung Research America, USA.Y. Wu is with the Department of Electrical and Electronic Engineering, TheUniversity of Hong Kong, Hong Kong. Part of the contents in this paper hasbeen presented in [1] and [2].

The corresponding author: L. Liu.

the presently available MIMO transmissions, allowed by LTE-Advanced systems [6], do not support more than 8 antennaports at the base station.

Recently, a new MIMO paradigm called massive-MIMO,also known as large-scale MIMO, has created much interestboth in academia [7]–[10] and industry [11], with the promiseof meeting future capacity demands by providing increasedspectral-efficiency achieved through aggressive spatial multi-plexing. Considering the form factor limitation at the basestation (BS), instead of placing a large number of antennashorizontally, three dimensional (3D) massive-MIMO systemsemploy those antennas in a two dimensional (2D) antennaarray enabling the exploitation of the degrees of freedomin elevation domain along with those in the azimuth do-main [12]. Accordingly, 3D massive-MIMO is also calledfull-dimension MIMO (FD-MIMO) in 3GPP LTE-Advancedsystems. Apart from the huge potential of providing excellentspatial resolution and array gains, massive-MIMO can alsooffer a significant reduction of latency, a simplified multipleaccess layer, and robustness to interference [13]. With the helpof a large number of antennas, this system can concentratemore energy in a particular direction leading to a dramaticincrease in energy efficiency. Furthermore, massive-MIMOis the key enabling technology for gigabit-per-second datatransmission in millimeter wave (mmW) wireless communi-cations with carrier frequency between 30 and 300 GHz. InmmW communications, it becomes feasible to pack a greaternumber of antennas at the base station. However, the benefitsof Massive MIMO are limited by the accuracy of the channelstate information (CSI) obtained at the transmitter. The CSIis critical for functionalities such as downlink beam-forming,transmit precoding, and user scheduling.

In general, there are two methods to estimate the MIMOchannel. First is the traditional way where the channel trans-fer function is estimated. Channel state information can beobtained by sending some predefined pilot signals/referencesignals and estimating the channel matrix from the receivedsignal. Alternatively, channel estimation can be performedbased on the parametric channel models where directionsof arrival (DoA) and directions of departure (DoD) of theresolvable paths can be estimated [14], [15]. The time divisionduplex (TDD) 3D massive MIMO system can exploit thechannel reciprocity in order to acquire the channel stateinformation (CSI). To be specific, using the parametric model,the base station can estimate the DoAs in the uplink. Then with



the perfect calibration of the uplink-downlink signal chains, inthe downlink the base station can generate the beamformingvectors based on the channel estimation (DoAs) conducted inthe uplink phase.

MIMO technology, together with Orthogonal FrequencyDivision Multiplexing (OFDM), offers efficient ways of in-creasing the spectral efficiency of the cellular system. Thehigh-data-rate wireless transmission schemes, such as OFDM,converts a frequency-selective MIMO channel into a parallelcollection of frequency-flat subchannels, which is beneficialfor detection and channel estimation. It is shown in [16]that unprecedented spectral efficiency and promising systemthroughput can be obtained by combining these two powerfultechnologies– MIMO and OFDM. Because of the sensitivityof MIMO algorithms with respect to the underlying channelmatrix, channel state information is particularly critical inorder to assess the performance of the MIMO-OFDM systems.

In general, channel matrix for a MIMO system can bemodeled in different ways. The parametric channel model isadopted by performing virtual direction-of-arrival (DoA) anddirection-of-departure (DoD) estimations of resolvable paths.It provides a simple geometric interpretation of the scatteringenvironment in characterizing the two key MIMO channelmetrics: ergodic capacity and diversity level [17]. Despite theadvantage of reducing the number of estimation parameters,it is shown in [18] that channel estimation based on DoA andDoD provides the best performance in terms of error bound.

The contribution of the paper can be summarized as follows:• First, based on a parametric channel model, we present

a DoA estimation method for the single user 3D massiveMIMO OFDM system. It has been shown that ESPRITtype algorithms can be elegantly deployed for estimatingthe azimuth as well as the elevation angles efficiently.

• Second, we derive an analytical expression for the rootmean square error (RMSE) of the angle of arrival estima-tion for different channel taps. Our simplified results forthe case of the 3D massive MIMO system show that ifthe angles are drawn independently from any continuousdistribution, the RMSE depends heavily on the numberof antennas, antenna orientation, number of snapshots,and correlation of the transmitted signal. Accordingly,valuable insights into pilot sequence design can be drawnfor the massive MIMO OFDM systems.

• Third, we present a throughput analysis for the 3Dmassive MIMO OFDM system and derive an optimumdownlink precoder for such a system. We show that, forthe parametric channel model assumed in this work, thedownlink precoder can be constituted from the DoAsestimated at the base station. Furthermore, we charac-terize the ergodic throughput under the DoA estimationerror, and identify the optimal transmission and powerallocation strategies. Our results also show that the DoAbased channel estimation method achieves better perfor-mance than the traditional linear minimum mean squaredestimation (LMMSE) in terms of system throughput andminimum chordal distance.

• Finally, we investigate the performance of the elevationand azimuth angle estimation under various antenna

configurations and observe some interesting results. Forexample, for a system with the same 64 antennas, an8 × 8 antenna pattern can yield better performance thana 4 × 16 array in azimuth angle estimation at lowand medium signal-to-noise (SNR) regime. These kindof results can provide significant design intuition fordeveloping practical massive MIMO OFDM system.

The remainder of the paper is organized as follows: thesystem model and an outline of the DoA estimation method isintroduced in Section II. Analytical expression for the RMSEof the DoA estimation is presented in Section III followed bythe capacity and achievable rate analysis in section IV. Thesimulation results are shown in Section V before we drawconclusion in Section VI.

II. SYSTEM MODEL AND DOA ESTIMATION

Consider a MIMO-OFDM system with Nt transmit anten-nas and Nr receive antennas. At each transmitter, the high-rateinformation symbols to be transmitted are grouped into blocksof length Nc. The i-th such block at the jt-th transmitter canbe represented as

xi,jt = [xi,jt(0), xi,jt(1), . . . , xi,jt(Nc − 1)]T , (1)

where xi,jt(k) denotes the k-th information symbol withinthe i-th block at the jt-th transmitter. The IFFT opera-tion yields the OFDM symbol consisting of the sequencesi,jt(0), si,jt(1), . . . , si,jt(Nc − 1), where si,jt(n) corre-sponds to the IFFT samples of the multi-carrier signal at thejt-th transmitter, and can be written as

si,jt(n) =1√Nc

Nc−1∑k=0

xi,jt(k)ej2πkn/Nc , 0 ≤ n ≤ Nc − 1. (2)

The n-th samples at all the transmitters can be expressed as

si(n) = [si,j0(n), si,j1(n), . . . , si,j(Nt−1)(n)]T . (3)

After appending CP, the resulting sequence si,jt(n) is firstpassed through a parallel-to-serial converter followed by adigital-to-analog converter, resulting in the baseband OFDMtransmit signal at the jt-th transmit antenna. The basebandsignal is then upconverted and sent through a frequencyselective fading channel, which is assumed to remain time-invariant during one OFDM symbol duration. We assumethat the channel, which can be represented by an equivalentdiscrete-time linear channel impulse response (CIR), has afinite number (L) of non-zero taps.

In this paper, a clustered channel model is considered,where each scattering cluster is assumed to contribute a singlepropagation path. The channel impulse responses are thenrepresented by a sequence of channel matrices, C(`) for` = 0, 1, . . . , (L− 1). We assume there are a finite number ofresolvable paths between the transmitter and receiver. For thebroadband millimeter wave communication, we can considerthat each channel tap corresponds to each path. The channelimpulse response for the `-th tap is given by

C(`) = α(`)er(`)eHt (`), (4)

where α(`), er(`) and et(`) are, respectively, the channelgain, Nr × 1 receive antenna array response and Nt × 1



transmit antenna array response for the `-th tap; (.)H de-notes Hermitian transpose. It is obvious that the transmitand receive antenna array responses depend on DoD andDoA, respectively. For the transmitter equipped with a uni-form linear array (ULA), the transmit antenna array responsecan be described using the Vandermonde structure: et(`) =[1 ejω` . . . ej(Nt−1)ω`

]T, where ω` = (2π∆/λ) cos Ω`,

∆ is the spacing between the adjacent transmit antennaelements, Ω` is the transmit angle (DoD) for the `-th tap,and λ is the carrier wavelength. The antenna array at thebase station is a planar array placed in the X-Z plane, withM1 and M2 antenna elements in vertical and horizontaldirections, respectively. Accordingly, the number of receiveantenna elements at the base station is Nr = M1M2. Itis to be noted that we are focusing on ULA at the mobilestation in this paper since ULA is the default mobile stationantenna configuration of the FD-MIMO system in 3GPP Rel-14 LTE-Advanced Pro. However, most of our results can bereadily extended for any planer arrays; i.e. most results can begeneralized for uniform rectangular array (URA) or uniformcircular array (UCA). The exact extension of the work to URAand UCA will be the future work of this paper.

Since the antenna elements at the base station are placedin a 2D plane, for each resolvable path, there will be anazimuth DoA and an elevation DoA. Therefore, the re-ceive antenna array response can be expressed as er(`) =a(v`) ⊗ a(u`), where ⊗ represents the Kronecker prod-uct. a(u`) =

[1 eju` . . . ej(M1−1)u`

]Tand a(v`) =[

1 ejv` . . . ej(M2−1)v`]T

can be viewed as the receivesteering vectors of the elevation and azimuth angles, respec-tively. Here, u` = 2πd

λ cos θ` and v` = 2πdλ sin θ` cosφ` are

the two receive spatial frequencies at the base station, d isthe spacing between adjacent antenna elements in the receiveantenna array, and θ` and φ` are the elevation and azimuthDoA, respectively. It is noteworthy here that if instead ofusing ULA, an URA is used at the transmitter side, the uplinktransmit antenna array response can be represented as thekronecker product of steering vectors containing the azimuthand elevation DoD’s, respectively.

The received base-band signal at the jr-th receiver, yi,jr (t)is passed through an analog-to-digital (A/D) converter. Thecyclic prefix is then removed from the output of the A/Dconverter which yields the sequence yi(n) = si(n)~C(n) +wi(n), where ~ represents circular convolution, yi(n) =[yi,j0(n), yi,j1(n), . . . , yi,j(Nr−1)

(n)]T is the vector containingn-th received sample at all the receiver, jr represents the indexfor the receive antenna, and wi(n) is the noise vector. It is tobe noted here that si(n) has a sequence of length Nc, whereas,C(`) has sequence of only length L. Therefore, for facilitatingthe circular convolution in the DFT operation, we append(Nc −L) zero matrices to the sequence of C(`), so that boththe channel impulse response and transmitted IFFT samplevector are of the same sequence-length, Nc. We can arrangethe transmitted time-samples of the OFDM symbol in a tallNtNc × 1 vector, si =

[sTi (0) sTi (1) . . . sTi (Nc − 1)

]T.

For clarity, si can be expressed as si = Fxi, where xi is thecorresponding NtNc × 1 vector containing frequency-domaintransmit symbols for all the subcarriers at all the antennas,and F is the transformation matrix given by F = FH ⊗ Int ,where F is a Nc × Nc DFT matrix. Accordingly, we can

represent the corresponding received Nc time-samples as yi =[yTi (0) yTi (1) . . . yTi (Nc − 1)

]T. We can thereby ex-

press the NrNc×1 received sample vector as yi = Ccsi+wi

where, wi =[wTi (0) wT

i (1) . . . wTi (Nc − 1)

]Tis the

NrNc × 1 AWGN noise vector, and Cc is the NrNc ×NtNcblock-circulant matrix governing the circular convolution [1].Now, after proper rearrangement of the transmit time samplevector, we can write the n-th received time sample vectorby multiplying the transmit time-sample vector with anyarbitrary row (say p-th row, where 1 ≤ p ≤ Nc ) ofthe matrix Cc. To be specific, let sni denote the vector[sTi (Nc − p+ n+ 1), . . . , sTi (Nc − p+ n)

]T. Notice that the

vector sni can be obtained by cyclically shifting the elementsof the vector si defined earlier. Then, the n-th received timesample vector, yi(n) can be obtained by multiplying sni withthe p-th row of the matrix Cc. Therefore, the n-th receivedtime-samples vector can be written as

y(n) = C(p− 1)s(Nc − p+ n+ 1) + C(p− 2)

s(Nc − p+ n+ 2) + . . .+ C(p)s(Nc − p+ n) + w(n) (5)

where C(`) = 0Nr×Nt for L ≤ ` ≤ (Nc − 1). It is tobe noted here that we have dropped the index, i in (5) fornotational convenience. It will be assumed from now on thatall the samples correspond to i-th OFDM symbol. Recall hereagain that the index ` inside the bracket of C(`) representsthe tap index, whereas l inside the bracket of y(l), s(l) andw(l) represents time-sample index. Using (4), and after arearrangement, we can write (5) as

y(n) = AdiagbEtsn + w(n), (6)

where A =[er(p− 1) er(p− 2) . . . er(p)

]can be

viewed as the array steering matrix, Et is a Nc×NtNc blockdiagonal matrix containing the transmit array responses,

Et = blkdiageHt (p− 1), eHt (p− 2), . . . , eHt (p), (7)

where blkdiagQ represents block diagonalization of matrixQ, and b is the 1 × Nc vector containing the the complexfading envelopes. Now, taking V = Nc snapshots of thereceived signal, i.e., one sample per subcarrier, we extend (6):

Y = AdiagbEtS + W, (8)

where Y = [y(0),y(1), . . . ,y(n), . . . ,y(Nc − 1)] is theNr × V received signal matrix at the base-station, S =[s0, s1, . . . , sn, . . . , sNc−1

]is the NtNc×V transmitted signal,

and W is the Nr × V noise matrix.We now introduce a low-complexity DoA estimation algo-

rithm based on ESPRIT to jointly estimate the elevation andazimuth angles. We can write the received signal in (8) as:

Y = AS + W, (9)

where S = diagbEtS can be regarded as the equivalenttransmit signal. The forward-backward averaged received sig-nal can be expressed as:

Yfba =[Y ΠNrY

∗ΠV

]=[AS ΠNrA

∗S∗ΠV

]+[W ΠNrW

∗ΠV

]. (10)

Here, Πp denotes the p × p exchange matrix with ones onits antidiagonal and zeros elsewhere. The dimension of Ais (Nr × Nc), the dimension of diagb is (Nc × Nc), thedimension of Et is (Nc ×NtNc), and the dimension of S is



(NtNc × V ). Hence, the dimension of Yfba is (Nr × 2V ).The subspace decomposition of the signal space of the receivedsignal through singular value decomposition (SVD) then canbe written as:[

AS ΠNrA∗S∗ΠV

]=[Us Un

] [Σs 00 0

] [VHs

VHn

]. (11)

Following our line of work [19], we can apply ESPRIT basedtechniques on the received signal, and have the following shift-invariance relations:

Kx1UsΨx = Kx2Us Ky1UsΨy = Ky2Us, (12)

where Ψx , T−1ΩxT, Ψy , T−1ΩyT, T is the non-singular transformation matrix; Kx1, Kx2, Ky1, and Ky2 arethe selection matrices, and Ωx and Ωx are given by

Ωx , diag

tan(un

2

), tan

(un−1

2

), . . . , tan

(un−Nc+1

2

).

(13)

Ωy , diag

tan(vn

2

), tan

(vn−1

2

), . . . , tan

(vn−Nc+1

2

).

(14)

From (12), we can solve for Ψx and Ψy based on estimatedsignal subspace using least square type of methods. Let theeigenvalues of the Nc×Nc complex matrix Ψx+ jΨy be λ`,for ` = 1, 2, . . . , Nc. Then u` and v` can be calculated from:

u` = 2 tan−1

Re(λ`

)v` = 2 tan−1

Im(λ`

). (15)

Accordingly, 2D DoAs of interest are obtained through simpleparameter transformation.

III. RMSE CHARACTERIZATION

In this section, we present the theoretical analysis of rootmean square error (RMSE) for DoA estimation using thestandard ESPRIT method. Let v` denote the estimated spatialfrequency for `-th tap; the estimation error is then given by∆v` = v` − v`. Similarly, ∆u` = u` − u`. The first orderapproximation of the mean square estimation error of v` forthe standard ESPRIT is given by [20]:

E(∆v`)2 =1

2

(r(v)`

HW∗

matRTnnWT

matr(v)`

−Re

r(v)`

TWmatCnnWT

matr(v)`

), (16)

where

r(v)` = q` ⊗

([(J

(v)1 Us)

+(J(v)2 /ejv` − J

(v)1 )]T

p`

), (17)

Wmat = (Σ−1s VTs )⊗ (UnUH

n ). (18)

Here, J(v)1 and J

(v)2 are the two effective selection matrices

for the first and second subarrays, respectively, for the spatialfrequency v`, T is the transformation matrix as described inSection II, q` is the `-th column of matrix T, pT` is the `-th row of matrix T−1; Rnn and Cnn are the covariance and

complementary covariance matrices of the noise, respectively.Similarly, we have

E(∆u`)2 =1

2

(r(u)`

HW∗

matRTnnWT

matr(u)`

−Re

r(u)`

TWmatCnnWT

matr(u)`

), (19)

where

r(u)` = q` ⊗

([(J

(u)1 Us)

+(J(u)2 /eju` − J

(u)1 )]T

p`

). (20)

Here, J(u)1 and J

(u)2 are the two effective selection matrices

for the first and second subarrays, respectively, for the spatialfrequency u`. At this point, in order to facilitate the derivationof MSE expression, we consider the following Lemma:

Lemma 1. If the elevation and azimuth angles are both drawnindependently from a continuous distribution, the normalizedarray response vectors become orthogonal asymptotically,that is, er(k) ⊥ span er(`) | ∀k 6= ` when the number ofantennas at the base station goes large.

Proof. See Appendix A.

It is to be emphasized here that Lemma-1 holds for anycontinuous distribution. It can be seen that (16) depends on thesingular value decomposition (SVD) of the noiseless receivedsignal, which is not easy to obtain at the base station. In fact, itis extremely difficult to simplify such complicated result in themultiple path case. Fortunately, in the massive MIMO system,it can be significantly simplified because of the orthogonalityof the steering vectors. The simplified result is only related tothe real system parameters such as the number of antennas,number of snapshots, transmit power, and covariance matrixof transmit signal. Specifically, for the massive MIMO system,we have the following theorem:

Theorem 1. For the case of 3D DoA estimation based onuniform rectangular array of M1×M2 elements, the root meansquare errors of the elevation and azimuth angle estimationsare given, respectively, by:

RMSEθ` =σ

π sin(θ`)(M1 − 1)

√R−1SS(`, `)

VM2(21)

RMSEφ` =σ

π sin(θ`)

√R−1SS(`, `)

V√(cot2(θ`) cot2(φ`)

(M1 − 1)2M2+

1

sin2(φ`)(M2 − 1)2M1

)(22)

where σ2 is the noise variance, and RSS(`, `) is the `-thdiagonal element of the covariance matrix of the equivalenttransmit signal.

Proof. See Appendix B.

Now, RSS = ESSH/V , where the expectation is withrespect to the time-samples over different subcarriers ontransmit antennas. RSS can be expressed as:

RSS = diagbEtESSHEHt (diagb)H/V

= diagbEtFEXXHFHEHt (diagb)H/V (23)



where X = F−1S is the frequency domain transmit sig-nal matrix. If we assume, Z = diagbEtF, we canwrite, RSS = ZEXXHZH/V = ZRXXZH/V , whereRXX = EXXH. Therefore, we can write, R−1SS =(V )Z−HR−1

XXZ−1. Plugging the expression for R−1SS in (21)

and (22), we can obtain:

RMSEθ` =σ

π sin(θ`)(M1 − 1)

√(Z−HR−1

XXZ−1)(`, `)

M2(24)

RMSEφ` =σ

π sin(θ`)

√(Z−HR−1

XXZ−1)(`, `)√(

cot2(θ`) cot2(φ`)

(M1 − 1)2M2+

1

sin2(φ`)(M2 − 1)2M1

)(25)

where (Z−HR−1XX

Z−1)(`, `) represents the `-th diagonal ele-ment of the matrix product Z−HR−1

XXZ−1. Based on the proof

of Theorem 1, it is straight forward to obtain the RMSE ofthe spatial frequencies u` and v` as follows:

Corollary 1.1. In massive MIMO system, RMSEs of the spatialfrequencies u` and v` using standard ESPRIT are given by:

RMSEu` =σ

(M1 − 1)

√(Z−HR−1

XXZ−1)(`, `)

M2, (26)

RMSEv` =σ

(M2 − 1)

√(Z−HR−1

XXZ−1)(`, `)

M1. (27)

It is clear from these equations that correlation between thedata on different subcarriers will have adverse effect on theperformance of the DoA estimation, i.e., root mean squarederror increases as the correlation between the data increases.The pilot sequence design will also have a significant impacton the DoA estimation performance through the term RXX.

IV. CAPACITY ANALYSIS

In this section, we investigate the downlink channel capacityof 3D massive MIMO OFDM system. First, we will analyzethe system capacity assuming perfect DoA estimation. Thenwe will look into the system achievable rate under the presenceof DoA estimation errors.

A. Ergodic Capacity with Perfect DoA Estimation:The Nr × Nt uplink channel transfer function at the k-th

subcarrier can be written as

H(k) =

Nc−1∑`=0

C(`)e−j2πk`Nc . (28)

Again, it is assumed there are Nc channel taps, but only L ofthem are non-zero. Using the expression of C(`) from (4),

H(k) =

Nc−1∑`=0

α(`)er(`)eHt (`)e

−j2πk`Nc

=

Nc−1∑`=0

α(`)er(`, k)eHt (`), (29)

where er(`, k) = e−j2πk`Nc er(`). In a compact form:

H(k) = A(k)DBH , (30)

where A(k) = [er(0, k), er(1, k), . . . , er(Nc − 1, k)], D =diagb = diagα(0), α(1), . . . , α(Nc − 1), and B =[et(0), et(1), . . . , et(Nc − 1)], where 1/

√NtB is a unitary

matrix. Accordingly, using the channel reciprocity property,downlink channel at the k-th subcarrier can be represented as

Hdl(k) = [H(k)]T = B∗DAT (k). (31)

So, the Nt× 1 downlink received signal at the k-th subcarriercan be written as

ydl(k) = Hdl(k)xdl(k) + ndl(k)

= B∗DAT (k)xdl(k) + ndl(k), (32)

where xdl(k) is the Nr × 1 downlink transmitted signal, andndl(k) is the Nt × 1 noise vector with Endl(i)ndlH(j) =σ2INtδ(i − j), and E· represents expectation. The mutualinformation at the k-th subcarrier:

Ik = log2 det

[INt +

Hdl(k)QkHdlH(k)

σ2

], (33)

where Qk = Exdl(k)xdlH

(k) is the Nr × Nr covariancematrix of the downlink transmit signal at the k-th subcarrier.Now, the ergodic capacity of the MIMO OFDM system canbe expressed as:

C = E

1

Nc

Nc−1∑k=0

Ik

= E

1

Nc

Nc−1∑k=0

log2 det

[INt +

Hdl(k)QkHdlH(k)

σ2

], (34)

with the total power constraint, Tr(Q) ≤ P , where P is thetotal transmit power, and Q is the covariance matrix of theNrNc × 1 downlink Gaussian input vector, and given by:

Q = diagQkNc−1k=0 . (35)

The design of the taransmit covariance matrix, Qk, andhence, the design of the transmit signal will dictate the powerallocation across the transmit antennas for the k-th tone. Itis to be noted here that in the capacity expression, we haveignored the loss in spectral efficiency due to the presence ofthe cyclic prefix. Let us now consider the following Lemma:

Lemma 2. For a uniform rectangular array (URA) withazimuth and elevation DoAs drawn independently from a con-tinuous distribution, the normalized frequency-domain arrayresponse vectors becomes orthogonal asympototically, thatis, er(i, k) ⊥ spaner(j, k)|∀i 6= j, when the numberof base-station antennas, Nr = M1M2 goes large, whereer(l, k) = 1√

Nrer(l, k).

Proof. See Appendix C.

From Lemma 2, the matrix√

1/NrA(k) can be regardedas unitary for the large antenna system. Therefore, downlinkchannel representation in (31) can essentially be thought of assingular value decomposition for the massive MIMO system.In order to simplify the analysis, we can assume that themobile station has perfect knowledge about the DoD so thatoptimal received processing can be used at the mobile station.



Observing (32), the optimum precoding matrix, therefore, canbe expressed as

Vopt(k) =1

NrA∗(k)

=1

Nr[e∗r(0, k), e∗r(1, k), . . . , e∗r(Nc − 1, k)]

=1

Nr

[ej2πk(0)Nc e∗r(0), e

j2πk(1)Nc e∗r(1), . . . , e

j2πk(Nc−1)Nc e∗r(Nc − 1)

]=

1

Nr

[ej2πk(0)Nc a∗(v0)⊗ a∗(u0), e

j2πk(1)Nc a∗(v1)⊗ a∗(u1),

. . . , ej2πk(Nc−1)

Nc a∗(vNc−1)⊗ a∗(uNc−1)

]. (36)

From (36), it is apparent that the optimum precoding matrixis essentially composed of the array steering vectors, whichagain depend on the DoA estimation at the base station.Let sdl(k) be the Nc × 1 information signal vector forthe k-th subcarrier. Therefore, using the optimum precodingmatrix, we have xdl(k) = (1/Nr)A

∗(k)sdl(k). Plugging thisinto (33) , for the perfect DoA estimation case, we canobtain (37). It is not difficult to show that D∗BTB∗D =Ntdiag[|α(0)|2, |α(1)|2, . . . , |α(Nc− 1)|2]. Using Lemma 2 ,we have AT (k)A∗(k)/Nr = INc . Assuming Gaussian inputinformation signal, sdl(k)sdl

H(k) will also be diagonal, i.e.,

sdl(k)sdlH

(k) = diag[|sdl0 (k)|2, . . . , |sdlNc−1(k)|2]= diag[p0(k), . . . , pNc−1(k)], where sdl` (k) is the `-th infor-mation symbol on the k-th subcarrier, and p`(k) = |sdl` (k)|2is the power to be allocated on the corresponding channel.Finally, using Hadamard inequality, we can write (37) as

Ik = log2

∏`

(1 +

Nt|α(`)|2p`(k)

σ2

)

=

Nc−1∑`=0

log2 (1 + γ`p`(k)) , (38)

where γ` = Nt|α(`)|2/σ2. It is to be noted here that thematrix inside the log2 det is diagonal, and hence Hadamard in-equality follows with equality. Accordingly, the optimal powerallocation is the well-known water-filling solution which canbe expressed as p`(k) = [µ`(k) − 1/γ`]

♦, where [x]♦ is thefunction with [x]♦ = 0 when x < 0, and [x]♦ = x whenx ≥ 0, and µ`(k) is the corresponding Lagrange multiplier.

We now briefly discuss about the optimum receive process-ing for the perfect DoA estimation scenario. From (32), theNt × 1 downlink received signal can be expressed as

ydl(k) = B∗DAT (k)Vopt(k)sdl(k) + ndl(k)

=1

NrB∗DAT (k)A∗(k)sdl(k) + ndl(k)

= B∗Dsdl(k) + ndl(k)

= [e∗t (0), e∗t (1), . . . , e∗t (Nc − 1)]

diagα(0), α(1), . . . , α(Nc − 1)sdl(k) + ndl(k). (39)

Let the received signal vector corresponding to i-th informa-tion symbol on the k-th subcarrier be represented as ydli (k),and the corresponding noise vector as ndli (k). Then,

ydli (k) = e∗t (i)α(i)sdli (k) + ndli (k) = g′isdli (k) + ndli (k), (40)

where g′

i = e∗t (i)α(i). Accordingly, for the receive processing,the maximum ratio combining (MRC) vector, w

′

i can bewritten as w

′

i = g′

i/|g′

i|. Multiplying both sides of (40) by

w′

i

H, we obtain the MRC detected signal ydli (k) corresponding

to the i-th information symbol:

ydli (k) = w′i

Hydli (k) =

g′i

H

|g′i|g′isdli (k) + ndli (k)

= |g′i|sdli (k) + ndli (k) = |α(i)||e∗t (i)|sdli (k) + ndli (k)

=√Nt|α(i)|sdli (k) + ndli (k), (41)

where ndli (k) = w′i

Hndli (k) =

g′i

H

|g′i|ndli (k) is the noise element

corresponding to i-th information symbol on k-th subcarrier.

B. System Achievable Rate under the presence of DoA Esti-mation Errors:

In this section, we study the effects of DoA estimation erroron the achievable rate of the massive-MIMO OFDM system.Since base station does not have perfect DoA estimation,the precoding vectors, in the presence of DoA estimationerror, will be in the form (1/Nr)er(`, k), where er(`, k) =

e−j2πk(`)

Nc a(v` + ∆v`) ⊗ a(u` + ∆u`), and ∆u` and ∆v`represent the DoA estimation errors for the `-th tap. Forthis imperfect channel estimation case, let sdl(k) denote theNc× 1 information symbol vector for the k-th subcarrier. Theoptimum precoding matrix, in the presence of DoA estimationerror, can be written as Vopt(k) = (1/Nr)A

∗(k), whereA∗(k) = [er(0, k), er(1, k), . . . , er(Nc − 1, k)]. Accordingly,similar to (33), the expected mutual information or achievablerate of the k-th subcarrier, Ik, can be represented as (42),where the expectation is with respect to the DoA estimationerror. Here, we are treating the terms inside the expectationas random variable, where, for each channel realization, theDoA estimation error will be random. Hence, from (42), wecan have (43), where the term AT (k)A∗(k) can be written as

AT (k)A∗(k)

=

eTr (0, k)e∗r(0, k) . . . eTr (0, k)e∗r(Nc − 1, k)eTr (1, k)e∗r(0, k) . . . eTr (1, k)e∗r(Nc − 1, k)

.... . .

...eTr (Nc − 1, k)e∗r(0, k) . . . eTr (Nc − 1, k)e∗r(Nc − 1, k)

.(44)

Now, let us consider the following lemma:

Lemma 3. For the 3D massive MIMO OFDM systems, thenormalized vectors er(i, k) = 1/

√Nrer(i, k) and ˆer(m, k) =

1/√Nrer(m, k), ∀i 6= m becomes orthonormal asymptoti-

cally as the number of antenna, Nr →∞ .

Proof. See Appendix C.

Using Lemma 3, we can write AT (k)A∗(k) =diag[eTr (0, k)e∗r(0, k), eTr (1, k)e∗r(1, k), . . . , eTr (Nc −1, k)e∗r(Nc − 1, k)] . With Gaussian information signal,sdl(k)(sdl(k))H = diag[p0(k), p1(k), . . . , pNc−1(k)], wherep`(k) is the power to be allocated for the `-th informationsymbol on the k-th subcarrier. We can clearly see thatAT (k)A∗(k)sdl(k)(sdl(k))HAT (k)A∗(k)D∗BTB∗Dis a diagonal matrix with `-th diagonal element being



Ik = log2 det

[INt +

B∗DAT (k)A∗(k)sdl(k)sdlH

(k)AT (k)A∗(k)D∗BT

N2r σ

2

]

= log2 det

[INc +

AT (k)A∗(k)sdl(k)sdlH

(k)AT (k)A∗(k)D∗BTB∗D

N2r σ

2

]

= log2 det

[INc +

AT (k)A∗(k)sdl(k)sdlH

(k)AT (k)A∗(k)D∗BTB∗D

N2r σ

2

]. (37)

Ik = E

log2 det

[INt +

B∗DAT (k)Vopt(k)sdl(k)(sdl(k))HVoptH (k)A∗(k)D∗BT

σ2

]. (42)

Ik = E

log2 det

[INt +

B∗DAT (k)A∗(k)sdl(k)(sdl(k))HAT (k)A∗(k)D∗BT

N2r σ

2

]

= E

log2 det

[INc +

AT (k)A∗(k)sdl(k)(sdl(k))HAT (k)A∗(k)D∗BTB∗D

N2r σ

2

]. (43)

Nt|eTr (`, k)e∗r(`, k)|2|α(`, k)|2p`(k). Now, invokingHadamard’s Inequality, we can write (43) as

Ik = E

log2

∏`

(1 +

Nt|eTr (`, k)e∗r(`, k)|2|α(`)|2p`(k)

N2r σ2

)

= E

Nc−1∑`=0

log2

(1 +

Nt|eTr (`, k)e∗r(`, k)|2|α(`)|2p`(k)

N2r σ2

)

= E

Nc−1∑`=0

log2

(1 + γ`|eTr (`, k)e∗r(`, k)|2p`(k)

), (45)

where γ` = Nt|α(`)|2/(N2r σ

2). Using method of Lagrangianmultiplier, the optimal power allocation for the k-th subcarrier:

p`(k) =

[µ`(k)− 1

γ`|eTr (`, k)e∗r(`, k)|2

]♦, (46)

where µ`(k) is the corresponding Lagrange multiplier, and ♦indicates the water-filling algorithms. The expected transmitpower for the `-th tap can be expressed as

Ep`(k) =

[µ`(k)− 1

γÈ|eTr (`, k)e∗r(`, k)|2

]♦. (47)

Now,

E[|eTr (`, k)e∗r(`, k)|2] = E[|eHr (`, k)er(`, k)|2]

= E[|[a(v` + ∆v`)⊗ a(u` + ∆u`)]

H [a(v`)⊗ a(u`)]|2]

= E[|[aH(v` + ∆v`)⊗ aH(u` + ∆u`)][a(v`)⊗ a(u`)]|2

]= E

[|[aH(v` + ∆v`)a(v`)]⊗ [aH(u` + ∆u`)a(u`)]|2

]= E

[|[aH(v` + ∆v`)a(v`)][a

H(u` + ∆u`)a(u`)]|2]

= E[|aH(v` + ∆v`)a(v`)|2]E[|aH(u` + ∆u`)a(u`)|2

], (48)

where the last equation follows based on the assumption that∆v` and ∆u` are independent. Now, we can write

∣∣∣aH(v` + ∆v`)a(v`)∣∣∣2

=∣∣∣1 + ejvè−j(v`+∆v`) + . . .+ ej(M1−1)vè−j(M1−1)(v`+∆v`)

∣∣∣2=∣∣∣1 + e−j∆v` + (e−j∆v`)2 + . . .+ (e−j∆v`)(M1−1)

∣∣∣2=

∣∣∣∣1− e−jM1∆v`

1− e−j∆v`

∣∣∣∣2 =

∣∣∣∣1− cos(M1∆v`) + j sin(M1∆v`)

1− cos(∆v`) + j sin(∆v`)

∣∣∣∣2=

1− cos(M1∆v`)

1− cos(∆v`)

=1−

(1− (M1∆v`)

2

2!+ (M1∆v`)

4

4!+ . . .+∞

)1−

(1− (∆v`)

2

2!+ (∆v`)

4

4!+ . . .+∞

)(a)≈

M21 (∆v`)

2

2− M4

1 (∆v`)4

24

(∆v`)2

2− (∆v`)

4

24

=M2

1

(1− M2

1 (∆v`)2

12

)1− (∆v`)

2

12

(b)≈ M2

1

(1− M2

1 (∆v`)2

12

), (49)

where (a) results from truncating the higher order terms, and(b) results by noticing that (∆v`)

2/12 is negligible comparedto unity. Therefore,

E[|aH(v` + ∆v`)a(v`)|2] = M21

(1−

M21E[(∆v`)

2]

12

). (50)

Following exactly same procedure, we can also show that

E[|aH(u` + ∆u`)a(u`)|2] = M22

(1−

M22E[(∆u`)

2]

12

). (51)



Therefore, from (48), we have1

E[|eTr (`, k)e∗r(`, k)|2

=1

M21M

22

(1−

M21E[(∆v`)

2]

12

)−1(1−

M22E[(∆u`)

2]

12

)−1

≈ 1

M21M

22

(1 +

M21E[(∆v`)

2]

12

)(1 +

M22E[(∆u`)

2]

12

).

(52)

Hence, from (47):

Ep`(k) =

[µ`(k)− 1

γ`M21M

22

(1 +

M21E[(∆v`)

2]

12

)(

1 +M2

2E[(∆u`)

2]

12

)]♦. (53)

From (53), it can be observed that in the absence of DoAestimation error, the proposed power allocation algorithmbecomes identical to traditional water-filling solution.

V. SIMULATION RESULTS

In this section, we evaluate the RMSE of the ESPRIT-based DoA estimation for 3D millimeter wave massive MIMOsystems. Furthermore, we will evaluate the system achievablerate under the presence of DoA estimation errors and presenta performance comparison between our proposed power allo-cation algorithm and the traditional water-filling solution. Toevaluate the performance of the DoA estimation, we assumethere are 4 resolvable paths, which is a typical number forthe outdoor millimeter-wave communication systems at both28GHz and 73GHz [21]. Number of subcarriers of the OFDMsystem is 32, and the length of cyclic prefix is 4. The antennaspacing for both the received and transmit antennas is assumedto be 0.5λ. The number of transmit antennas is set to be 8. Theelevation and azimuth DoAs are chosen randomly from eitherthe uniform distribution: U [−180, 180], the exponential dis-tribution: Exp(1/50), or the Gaussian distribution: N (50, 20).

In our work, we invoke the far field assumption, and thewavefront impinging on the antenna array was assumed tobe planer. However, using models such as Costa model, itwould not be difficult to extend our results to incorporatethe spherical wavefront. It is also noteworthy here that thetransmission medium assumed in this paper is isotropic andlinear. Normally distributed random numbers were used forgenerating frequency domain data for simulation. The numberof snapshots is taken to be 32. The success of ESPRIT typehigh resolution DoA estimation method depends on the fullrank condition of the data covariance matrix. However, ifappropriate preprocessing schemes such as forward-backwardaveraging or spatial smoothing can be applied, the data covari-ance matrix can be ensured to be of full rank and non-singulareven when all the data signals are correlated. This is the mainreason why we performed the forward-backward averaging inour DoA estimation process in Section II to deal with thepotential issue of not having enough independent realizationsof the random wavefield. Therefore, even though we usedindependent data signals in our simulation evaluation, our

method/simulation procedure is expected to handle the non-singular covariance matrix case smoothly. The main require-ment for forward-backward averaging to be valid is that theproperties of the process under consideration be approximatelythe same independent of the orientation of space axis and thatthe samples be taken in a geometry that is also reversible.The antenna array that we are using at the base station fulfillsthis condition since it has a centro-symmetric structure. Forthe forward-backward averaging, the data are first taken fromthe 2D rectangular array, and then these data are vectorized.Finally, an extended data matrix is used for forward-backwardaveraging which is defined in Equation (10). Through forward-backward averaging, the data is effectively doubled to improvethe estimation performance. Finally, the total available transmitpower is assumed to be unity, and the SNR is defined as theratio of the received signal power to the noise power, i.e.SNR = 10 log10

(1/σ2

).

The performance of the estimation of elevation and azimuthangles for an 8× 8 antenna array with uniform DoA distribu-tion is shown in Figure 1 and Figure 2, respectively, where theanalytical results for RMSEs are compared with the empiricalones. It can be observed that as SNR increases, the empiricalresults match the analytical results asymptotically. We can

−5 0 5 10 15 20 2510

−2

10−1

100

101

SNR (dB)

RM

SE

for θ

Est

imat

ion

(Deg

rees

)

Empirical ResultsAnalytical Results

Figure 1: Elevation angle estimation for 8× 8 array(Uniform DoA distribution).

investigate the impact of various antenna configurations onthe estimation performance and obtain some interesting designintuitions. For example, for a 3D massive MIMO system withthe same total 64 antenna elements, the RMSE for 4 × 16antenna array is shown in Figure 3 and Figure 4, while theelevation and azimuth angle estimation for a 16× 4 array areshown in Figure 5 and Figure 6, respectively. Comparing thesefigures with Figure 1 and 2, we observe that as the numberof antennas in the elevation domain increases, the elevationangle estimation performance improves. However, the RMSEof azimuth estimation of an 8× 8 array shown in Figure 2 iseven better than that of the 4 × 16 array shown in Figure 4,especially at low and medium SNR regime. This is somewhatsurprising because 4 × 16 array has more antenna elementsin the azimuth domain compared with the 8 × 8 array. Thereason behind this is that the azimuth estimation is actually



−5 0 5 10 15 20 2510

−2

10−1

100

101

SNR (dB)

RM

SE

for φ

Est

imat

ion

(Deg

rees

)


Figure 2: Azimuth angle estimation for 8× 8 array(Uniform DoA distribution).

−5 0 5 10 15 20 2510

−2

10−1

100

101

SNR (dB)

RM

SE

for θ

Est

imat

ion

(Deg

rees

)


Figure 3: Elevation angle estimation for 4×16 array(Uniform DoA distribution).

−5 0 5 10 15 20 2510

−2

10−1

100

101

SNR (dB)

RM

SE

for φ

Est

imat

ion

(Deg

rees

)



coupled with the elevation estimation performance, and whenthe elevation estimation performance decreases, azimuth esti-mation performance also deteriorates even though the number

of horizontal antennas is increased. On the other hand, theelevation angle estimation performance does not depend onthe azimuth angle estimation performance. For the case of

−5 0 5 10 15 20 2510

−2

10−1

100

101

SNR (dB)

RM

SE

for θ

Est

imat

ion

(Deg

rees

)



−5 0 5 10 15 20 2510

−2

10−1

100

101

SNR (dB)

RM

SE

for φ

Est

imat

ion

(Deg

rees

)



16 × 16 antenna array, the RMSE estimation performance isshown in Figure 7 and 8. As expected, the 16 × 16 arrayoutperforms the 8 × 8, 4 × 16 and 16 × 4 arrays in bothelevation and azimuth angle estimation. It is to be emphasizedhere that these DoA estimation results hold for any continuousdistribution, and is not specific to uniform distribution only.The results for 8 × 8 and 16 × 16 antenna arrays where theDoA’s are drawn from Gaussian (N (50, 20)) and exponentialdistribution (Exp(1/50)), are shown in Figures 9 to 12. Fromthese results, and also comparing with Figures 1,2,7, and 8, wecan clearly observe that for the same antenna configuration,the DoA estimation performance is very similar irrespective ofthe underlying distribution from which the DoA’s are drawn.This, in fact, validates the results in Lemmas 1 to 3.

Figures that plot the RMSE of elevation and azimuth anglesestimation as a function of the number of antennas are shown



−5 0 5 10 15 20 2510

−2

10−1

100

101

SNR (dB)

RM

SE

for θ

Est

imat

ion

(Deg

rees

)



−5 0 5 10 15 20 2510

−2

10−1

100

101

SNR (dB)

RM

SE

for φ

Est

imat

ion

(Deg

rees

)


Figure 8: Azimuth angle estimation for 16×16 array(Uniform DoA distribution).

−5 0 5 10 15 20 2510

−2

10−1

100

101

SNR (dB)

RM

SE

for θ

Est

imat

ion

(Deg

rees

)

Empirical (Gaussian)Analytical (Gaussian)Empirical (Exponential)Analytical (Exponential)

Figure 9: Elevation angle estimation for 8× 8 array.

in Figures 13 and 14 for uniform, Gaussian, and exponentialDoA distributions, and SNR = 15 and 20 dB. Dashed linesrepresent the empirical results while solid lines represent theanalytical results. The square array, where M1 = M2, is

−5 0 5 10 15 20 2510

−2

10−1

100

101

SNR (dB)

RM

SE

for φ

Est

imat

ion

(Deg

rees

)


Figure 10: Azimuth angle estimation for 8×8 array.

−5 0 5 10 15 20 2510

−2

10−1

100

101

SNR (dB)

RM

SE

for θ

Est

imat

ion

(Deg

rees

)


Figure 11: Elevation angle estimation for 16 × 16array.

−5 0 5 10 15 20 2510

−2

10−1

100

101

SNR (dB)

RM

SE

for φ

Est

imat

ion

(Deg

rees

)


Figure 12: Azimuth angle estimation for 16 × 16array.

assumed for the evaluation. It can be observed that the DoAestimation performance improves as the number of antennasincreases. Furthermore, in all cases, the empirical results are



very close to the analytical results irrespective of the DoAdistribution.

6 8 10 12 14 160

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

M1=M

2

RM

SE

for θ

Est

imat

ion

(Deg

rees

)

Emp.(15dB) UniformAna.(15dB) UniformEmp.(20dB) UniformAna.(20dB) UniformEmp.(15dB) GaussianAna.(15dB) GaussianEmp.(20dB) GaussianAna.(20dB) GaussianEmp.(15dB) ExponentialAna.(15dB) ExponentialEmp.(20dB) ExponentialAna.(20dB) Exponential

15 dB

20 dB

Figure 13: Elevation angle estimation for differentnumber of antennas.

6 8 10 12 14 160

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

M1=M

2

RM

SE

for φ

Est

imat

ion

(Deg

rees

)

Emp.(15dB) UniformAna.(15 dB) UniformEmp.(20dB) UniformAna.(20 dB) UniformEmp.(15dB) GaussianAna.(15dB) GaussianEmp.(20dB) GaussianAna.(20dB) GaussianEmp.(15 dB) ExponentialAna.(15dB) ExponentialEmp.(20dB) ExponentialAna.(20dB) Exponential

15 dB

20 dB

Figure 14: Azimuth angle estimation for differentnumber of antennas.

Figure 15 provides a throughput comparison of the systemin the presence of the DoA estimation error for uniform DoAdistribution. To be specific, three schemes are compared inFigure 15: the first one utilizes the introduced DoA estimationfor channel acquisition and conducts the corresponding water-filling power allocation; the second one utilizes the introducedDoA estimation for channel acquisition and conducts theintroduced power allocation strategy in Equation (53); thethird one employs the linear mean squared error (LMMSE)channel estimation to acquire the channel transfer functionsand conducts the corresponding water-filling power allocation.In performing LMMSE channel estimation, no apriori knowl-edge of second order statistics, i.e. channel correlation matrix,is assumed to be available at the base station. Instead, eachcolumn of the Nr × Nt channel matrix for each subcarrieris estimated by sending the Nt × 1 transmit signal vector ofthe form [0, . . . , 0, 1, 0, . . . , 0] with the l-th element of thevector being 1 which is used to estimate the l-th column ofthe channel matrix. Therefore, in LMMSE channel estimationit takes at least Nt OFDM symbols to estimate the Nr ×Nt

channel matrix. The comparison in Figure 15 is performedfor an 8 × 8 antenna array, and for uniform DoA distribu-tion. It can be observed that in the low and medium SNRregime, the proposed power allocation algorithm outperformsthe traditional water-filling scheme. However, in the high-SNR regime, both algorithms perform almost identically. Thisis because in the high-SNR regime, the DoA estimation isvery accurate with very small estimation errors. As shownin Section IV-B, our introduced power allocation strategy willconverge to the traditional water-filling one as DoA estimationerror decreases. Furthermore, from Figure 15, it is obviousthat both of the parametric-based methods outperform theLMMSE-based channel estimation in terms of the through-put of the underlying 3D massive MIMO/FD-MIMO OFDMsystem. This is mainly because in parametric-based estimationmethods we exploit the channel structure for the underlying 3DMIMO channel, which significantly improves the estimationperformance. Chordal distance between the subspace spanned

-5 0 5 10 15 20 25 30 35

SNR (dB)

10 1

10 2

10 3

10 4

10 5

10 6

10 7

10 8

Erg

odic

Thr

ough

put i

n b/

s

Traditional Water-filling AlgorithmProposed AlgorithmLMMSE

Figure 15: Ergodic Throughput under Traditionaland Proposed Power Allocation Algorithms.

−5 0 5 10 15 20 2510

−4

10−3

10−2

10−1

100

101

SNR (dB)

Cho

rdal

Dis

tanc

e

Chordal Distance From DoAChordal Distance From MMSE

Figure 16: Chordal Distance Comparison.

by different sets of precoding vectors is another metric thatis widely used for evaluating the underlying MIMO precod-ing strategies. In Figure 16, we present a chordal distancecomparison between the Rank-4 precoding matrix obtainedfrom the DoA estimation and that obtained from the LMMSE-



based method. To be specific, let UH denote the matrixcontaining four left-singular vectors corresponding to the fournonzero singular values obtained from SVD of the underlyinguplink channel, H. Note that UH essentially contains optimaldownlink MIMO precoders. Let UDoA denote the estimatedsteering matrix; and UMMSE denote the matrix attained bytaking the four dominant singular vectors obtained from theSVD of the channel estimated using the LMMSE method.Now the chordal distance between the subspace spanned bythe optimal precoding matrix UH , and UDoA is defined asγDoA = 1/

√2E||UHUH

H −UDoAUHDoA||F

. Similarly,

γMMSE = 1/√

2E||UHUH

H −UMMSEUHMMSE ||F

. It is

important to note that chordal distance is a measure of close-ness of subspaces spanned by the corresponding basis vectors.As we can see from Figure 16, the chordal distance betweenthe precoders obtained from DoA-based channel estimationand that from the ideal channel is much smaller. This meansthat DoA-based precoding is much more effective than theMIMO precoding based on LMMSE channel estimation.

VI. CONCLUSION

In this paper, we presented DoA-based channel estimationstrategies as well as novel power allocation algorithms for 3Dmassive-MIMO/FD-MIMO OFDM systems. To be specific,we derived the analytical expression for the RMSE of ESPRIT-type DoA estimation algorithms. The performance of elevationand azimuth angle estimation of various antenna configurationshave been investigated. The capacity and achievable rate of3D massive MIMO/FD-MIMO downlink systems are charac-terized, and the optimal precoding matrix is related to DoAvectors. The optimal power allocation under the presence ofDoA estimation error is also introduced.

DoA estimation evaluation showed that the empirical per-formance of DoA estimation matches very well with that ofthe analytical performance. Furthermore, the antenna arrayconfiguration plays a vital role in determining the perfor-mance of the 3D massive-MIMO/FD-MIMO OFDM systems:the performance of azimuth angle estimation is tightly cou-pled with that of elevation angle estimation. The correlationamong data on different subcarriers has an adverse effecton the DoA estimation performance. An ergodic throughputevaluation suggested that the DoA-based channel estima-tion achieves much higher throughput than traditional non-parametric channel estimations such as LMMSE, while ourintroduced power allocation strategy outperforms the tradi-tional water-filling scheme. These observations may providesignificant insights and intuitions toward designing realistic 3Dmassive-MIMO/FD-MIMO OFDM systems for 5G networks.

APPENDIX APROOF OF LEMMA 1

Proof. For any i 6= m,

[er(i)]H [er(m)] =

1

Nr[a(vi)⊗ a(ui)]

H[a(vm)⊗ a(um)]

=1

Nr[1, e−jui , . . . , e−j(M1−1)ui , . . . , e−j(M2−1)vi ,

e−j(M2−1)vie−jui , . . . , e−j(M2−1)vie−j(M1−1)ui ]

[1, ejum , . . . , ej(M1−1)um , . . . , ej(M2−1)vm ,

ej(M2−1)vmejum , . . . , ej(M2−1)vmej(M1−1)um ]T

=1

Nr[1 + e−j(ui−um)+, . . . ,+e−j(M1−1)(ui−um)+, . . . ,

+ e−j(M2−1)(vi−vm) + e−j(M2−1)(vi−vm)e−j(ui−um)+, . . . ,

, . . . ,+e−j(M2−1)(vi−vm)e−j(M1−1)(ui−um)]

=1

Nr

M2−1∑n=0

M1−1∑p=0

[e−j(vi−vm)]n[e−j(ui−um)]p (54)

Since both azimuth and elevation DoA’s are drawn inde-pendently from a continuous distribution, (vi − vm 6= 0)with probability = 1; therefore e−j(vi−vm) 6= 1. Similarly,e−j(ui−um) 6= 1. We can write,

limNr→∞

∣∣[er(i, k)]H [er(j, k)]∣∣

= limNr→∞

1

Nr

∣∣∣∣∣M2−1∑n=0

M1−1∑p=0

[e−j(vi−vm)]n[e−j(ui−um)]p

∣∣∣∣∣a= limNr→∞

1

Nr

∣∣∣∣1− e−j(ui−um)M1

1− e−j(ui−um)

∣∣∣∣ ∣∣∣∣1− e−j(vi−vm)M2

1− e−j(vi−vm)

∣∣∣∣≤ limNr→∞

1

Nr

∣∣∣∣ 2

1− e−j(ui−um)

∣∣∣∣ ∣∣∣∣ 2

1− e−j(vi−vm)

∣∣∣∣ = 0 (55)

where, (a) follows by noticing that both∑M2−1n=0 [e−j(vi−vm)]n

and∑M1−1p=0 [e−j(ui−um)]p form geometric series with the

ratios e−j(vi−vm) and e−j(ui−um), respectively.

APPENDIX BPROOF OF THEOREM 1

Proof. In the case of standard ESPRIT and for circularlysymmetric white noise, Cnn = 0 [20]. So, we can write (16)and (19) as

E(∆v`)2 =1

2(r

(v)`

HW∗

matRTnnWT

matr(v)` ), (56)

E(∆u`)2 =1

2(r

(u)`

HW∗

matRTnnWT

matr(u)` ). (57)

Let us now denote

β` = VsΣ−1s q`, (58)

αv,` = (pT` (J(v)1 Us)

+(J(v)2 /ejv` − J

(v)1 )(UnUH

n ))T , (59)

αu,` = (pT` (J(u)1 Us)

+(J(u)2 /eju` − J

(u)1 )(UnUH

n ))T . (60)

So, we can have

WTmatr

(v)` =

((Σ−1

s VTs )⊗ (UnUH

n ))T

(q` ⊗

([(J

(v)1 Us)

+(J(v)2 /ejv` − J

(v)1 )]T

p`

))=(VsΣ

−1s q`

)⊗(pT` (J

(v)1 Us)

+(J

(v)2 /ejv` − J

(v)1

)(UnUH

n

))T= β` ⊗αv,`. (61)



Similarly, WTmatr

(u)` = β`⊗αu,`. The MSE in (56) and (57)

then can be expressed as

E(∆v`)2 =1

2((β` ⊗αv,`)

HRTnn(β` ⊗αv,l)), (62)

E(∆u`)2 =1

2((β` ⊗αu,`)

HRTnn(β` ⊗αu,`)). (63)

It can be easily verified that αv,l can be written as

αTv,l = cT`

((Jv,2A

)+Jv,2 −

(Jv,1A

)+Jv,1

); where c` =

[0, . . . , 1, . . . 0]T is the column selection vector with `-thelement being one, and other elements being zero, Jv,1 =IM1⊗ [IM2−1 0] and Jv,2 = IM1

⊗ [0 IM2−1] are theselection matrices. The pseudo inverse of the selected signalcan be significantly simplified as follows:(

Jv,1A)+

=

((Jv,1A

)H(Jv,1A)

)−1 (Jv,1A

)H

=1

(M2 − 1)M1

(Jv,1A

)H (Jv,1A

)(M2 − 1)M1

−1 (

Jv,1A)H

(a)=

1

(M2 − 1)M1

(Jv,1A

)H,

(64)

where (a) holds due to Lemma 1. Similarly, we have(Jv,2A

)+

=1

(M2 − 1)M1

(Jv,2A

)H. (65)

In the same way, αu,l can be written as αTu,l =

cT`

((Ju,2A

)+Ju,2 −

(Ju,1A

)+Ju,1

); where Ju,1 =

[IM1−1 0] ⊗ IM2and Ju,2 = [0 IM1−1] ⊗ IM2

are theselection matrices. Therefore, we can have(

Ju,1A)+

=1

(M1 − 1)M2

(Ju,1A

)H, (66)

(Ju,2A

)+

=1

(M1 − 1)M2

(Ju,2A

)H. (67)

Now,

Jv,1 =

1 0 . . . 00 1 . . . 0...

.... . .

...0 0 . . . 1

M1×M1

⊗

1 0 . . . 0 00 1 . . . 0 0...

.... . .

......

0 0 . . . 1 0

M2−1×M2

(68)

The `-th row of(Jv,1A

)Hthus can be written as[

1, e−ju` , . . . , e−j(M2−2)u` , . . . , e−j(M2−1)vè−j(M1−1)u`].

Accordingly, `-th row of(Jv,1A

)HJv,1 can be written as

cT`

((Jv,1A

)HJv,1

)=[1, e−ju` , . . . , e−j(M2−2)u` , . . . , e−j(M2−1)vè−j(M1−2)u` , 0

].

(69)

Moreover, Jv,2 can be represented as

Jv,2 =

1 0 . . . 00 1 . . . 0...

.... . .

...0 0 . . . 1

M1×M1

⊗

0 1 0 . . . 00 0 1 . . . 0...

......

. . ....

0 0 0 . . . 1

M2−1×M2

(70)

The `-th row of(Jv,2A

)Hthen can be written as[

e−ju` , e−j2u` , . . . , e−j(M2−1)u` , . . . , e−j(M2−1)vè−j(M1−1)u`],

and the `-th row of(Jv,2A

)HJv,1 can be expressed as

cT`

((Jv,2A

)HJv,2

)=[0, e−ju` , . . . , e−j(M2−1)u` , . . . , e−j(M2−1)vè−j(M1−1)u` , 0

].

(71)

Combining (69) and (71), we can obtain

cT`

((Jv,2A

)HJv,2 −

(Jv,1A

)HJv,1

)=[−1, 0, . . . , 0, e−j((M2−1)u`), . . . , e−j((M1−1)u`), 0,

, . . . , e−j((M2−1)v`+(M1−1)u`)]. (72)

Accordingly, we obtain ||αv,`||2 = 2/(M2−1)2M1. Followingthe same procedure, we can also have ||αu,`||2 = 2/(M1 −1)2M2. With a view to deriving the expression for β`, wefirst rewrite the singular value decomposition of the forward-backward averaged noiseless received signal as:[

AdiagbEtS ΠNrA∗diagbE∗t S∗ΠV

]=[AdiagbEtS AΛdiagbE∗t S∗ΠV

]= Adiagb

[EtS ΛE∗t S

∗ΠV

], (73)

where

Λ =diag[e−j((M1−1)un+(M2−1)vn), . . . ,

, . . . , e−j((M1−1)u(n−Nc+1)+(M2−1)v(n−Nc+1))]. (74)

Comparing with (11), and based on Lemma 1, we haveUs = 1/

√NrA, Σs =

√2NrV diagb, and VH

s =1/√

2V[EtS ΛE∗t S

∗ΠV

]. The vector, β` is given by β` =

VsΣ−1s UH

s Ac` [22]. It can be shown that the unitary trrans-formation does not affect the MSE of the ESPRIT method [23].However, the statistics of the noise and signal subspace arechanged when preprocessing like forward-backward averagingis applied. Now, the noise covariance matrix can be written as:

Rnn = EvecWvecWH. (75)

However, if the noise is assumed to be circularly symmetricand white Gaussian, we have Rnn = σ2INrV . Therefore

(β` ⊗αv,`)HRT

nn(β` ⊗αv,`) = σ2(β` ⊗αv,`)H(β` ⊗αv,`)

= σ2(βH` β`)⊗ (αHv,`αv,`). (76)

Since β` is the `-th column of VsΣ−1s UH

s A, ||β`||2 is the `-thdiagonal element of AHUsΣ

−2s UH

s A, which we can write as||β`||2 = R−1SS(`, `)/V [22], [24], where RSS(`, `) is the `-th



diagonal element of the equivalent transmit signal covariancematrix. Plugging the value of ||β`||2 and ||αv,`||2 in (76):

(β` ⊗αv,`)HRT

nn(β` ⊗αv,`) = σ2 R−1SS(`, `)

V

2

(M2 − 1)2M1.

(77)

Similarly, we also have

(β` ⊗αu,`)HRT

nn(β` ⊗αu,`) = σ2 R−1SS(`, `)

V

2

(M1 − 1)2M2.

(78)

Plugging the expression from (77) into (62) we get

E(∆v`)2 =R−1SS(`, `)

V

σ2

(M2 − 1)2M1. (79)

Similarly, by plugging the expression from (78) into (63):

E(∆u`)2 =R−1SS(`, `)

V

σ2

(M1 − 1)2M2. (80)

Based on Jacobian matrix, we have:

E

(4θ`)2 = E

(4u`)2 1

π2 sin2(θ`), (81)

E

(4φ`)2 =E

(4u`)2 cot2(θ`) cot2(φ`)

π2 sin2(θ`)

+E

(4v`)2π2 sin2(θ`) sin2(φ`)

. (82)

Recognizing that RMSEθ` =√

E(4θ`)2 and RMSEφ` =√E(4φ`)2, by substituting (79) and (80) into (81) and (82),

respectively, we obtain the desired results.

APPENDIX CPROOF OF LEMMA 2


[er(i, k)]H [er(m, k)] =1

Nrej 2πkiNc eHr (i)e

−j 2πkmNc er(m)

=1

Nrej2πk(i−m)

Nc [a(vi)⊗ a(ui)]H [a(vm)⊗ a(um)]

=1

Nrej2πk(i−m)

Nc [1, e−jui , . . . , e−j(M1−1)ui , . . . , e−j(M2−1)vi ,

e−j(M2−1)vie−jui , . . . , e−j(M2−1)vie−j(M1−1)ui ]

[1, ejum , . . . , ej(M1−1)um , . . . , ej(M2−1)vm ,

ej(M2−1)vmejum , . . . , ej(M2−1)vmej(M1−1)um ]T

=1

Nrej2πk(i−m)

Nc [1 + e−j(ui−um)+, . . . ,+e−j(M1−1)(ui−um)

+, . . . ,+e−j(M2−1)(vi−vm) + e−j(M2−1)(vi−vm)e−j(ui−um)

+, . . . ,+e−j(M2−1)(vi−vm)e−j(M1−1)(ui−um)]

=1

Nrej2πk(i−m)

Nc

M2−1∑n=0

M1−1∑p=0

[e−j(vi−vm)]n[e−j(ui−um)]p (83)

Since both azimuth and elevation DoA’s are drawn inde-pendently from a continuous distribution, (vi − vm 6= 0)

with probability = 1; therefore e−j(vi−vm) 6= 1. Similarly,e−j(ui−um) 6= 1 with probability 1. We can write,

limNr→∞

∣∣∣[er(i, k)]H [er(j, k)]∣∣∣

=

∣∣∣∣ej 2πk(i−m)Nc

∣∣∣∣ limNr→∞

1

Nr

∣∣∣∣∣M2−1∑n=0

M1−1∑p=0

[e−j(vi−vm)]n[e−j(ui−um)]p

∣∣∣∣∣a=

∣∣∣∣ej 2πk(i−m)Nc


1

Nr

∣∣∣∣1− e−j(ui−um)M1

1− e−j(ui−um)

∣∣∣∣ ∣∣∣∣1− e−j(vi−vm)M2

1− e−j(vi−vm)

∣∣∣∣≤∣∣∣∣ej 2πk(i−m)

Nc


1

Nr

∣∣∣∣ 2

1− e−j(ui−um)

∣∣∣∣ ∣∣∣∣ 2

1− e−j(vi−vm)

∣∣∣∣ = 0

(84)

where, (a) follows by noticing that both∑M2−1n=0 [e−j(vi−vm)]n

and∑M1−1p=0 [e−j(ui−um)]p form geometric series with the

ratios e−j(vi−vm) and e−j(ui−um), respectively.

APPENDIX DPROOF OF LEMMA 3


[êr(i, k)]H [er(m, k)] =1

NreHr (i, k)er(m, k)

=1

Nrej2πk(i−m)

Nc [a(vi + ∆vi)⊗ a(ui + ∆ui)]H [a(vm)⊗ a(um)]

=1

Nrej2πk(i−m)

Nc

[1, e−j(ui+∆ui), . . . , e−j(M1−1)(ui+∆ui), . . . ,

e−j(M2−1)(vi+∆vi), e−j(M2−1)(vi+∆vi)e−j(ui+∆ui), . . . ,

e−j(M2−1)(vi+∆vi)e−j(M1−1)(ui+∆ui)]

[1, ejum , . . . , ej(M1−1)um , . . . , ej(M2−1)vm ,

ej(M2−1)vmejum , . . . , ej(M2−1)vmej(M1−1)um]T

=1

Nrej2πk(i−m)

Nc

[1 + e−j(ui+∆ui−um)+, . . . ,

+ e−j(M1−1)(ui+∆ui−um)+, . . . ,+e−j(M2−1)(vi+∆vi−vm)

+ e−j(M2−1)(vi+∆vi−vm)e−j(ui+∆ui−um)+, . . . ,

+e−j(M2−1)(vi+∆vi−vm)e−j(M1−1)(ui+∆ui−um)]

=1

Nrej2πk(i−m)

Nc

M2−1∑n=0

M1−1∑p=0

[e−j(vi+∆vi−vm)]n[e−j(ui+∆ui−um)]p

(85)

Now, using the same procedure of the proof of Lemma 2, wecan show that

limNr→∞

∣∣[êr(i, k)]H [er(m, k)]∣∣ = 0. (86)

Hence, êr(i, k) and er(m, k), ∀i 6= m, becomes orthonormalas Nr goes large.

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[2] R. Shafin, L. Liu, and J. C. Zhang, “DoA Estimation and CapacityAnalysis for 3D Massive-MIMO/FD-MIMO OFDM System,” in IEEEGlobalSIP, Dec. 2015, pp. 181–184.

[3] World Telecommunication/ICT Indicators Database 2015 InternationalTelecommunications Union (ITU), Jun. 2015.

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Rubayet Shafin received his B.Sc. degree in Elec-trical and Electronic Engineering from BangladeshUniversity of Engineering and Technology, Dhaka,Bangladesh, in 2013. Currently, he is working to-ward the Ph.D. degree in Electrical Engineeringat the University of Kansas, USA. His researchinterests include MIMO systems, cellular networks,and mmWave Communications.

Lingjia Liu received his B.S. degree in ElectronicEngineering from Shanghai Jiao Tong Universityand Ph.D. degree in Electrical and Computer En-gineering from Texas A&M University. Prior tojoining the faculty in the EECS at KU, he spentmore than three years at Samsung Research Americawhere he received the Global Samsung Best PaperAward in 2008 and 2010. He was leading Samsung’sefforts on downlink multiuser MIMO, coordinatedmultipoint (CoMP), and heterogeneous networks(HetNet) in LTE/LTE-Advanced standards.

Dr. Liu is currently an Editor for the IEEE Trans. Wireless Commun., anEditor for the IEEE Trans. Commun. and Associate Editor for the EURASIP J.on Wireless Commun. and Netw. and Wiley’s Intl J. on Commun. Systems. Heis co-editor of several special issues. His general research interests mainly liein emerging technologies for 5G cellular networks including massive MIMO,massive MTC communications, and mmWave communications.

Jianzhong (Charlie) Zhang is a Vice President(VP) and the head of Standards and Mobility Inno-vation Lab with Samsung Research America, wherehe leads research and standards for 5G cellularsystems and next generation multimedia networks.He received his Ph.D. degree from University ofWisconsin, Madison. From August 2009 to August2013, he served as the Vice Chairman of the 3GPPRAN1 working group and led development of LTEand LTE-Advanced technologies such as 3D channelmodeling, UL-MIMO and CoMP, Carrier Aggrega-

tion for TD-LTE, etc. Before joining Samsung, he was with Motorola from2006 to 2007 working on 3GPP HSPA standards, and with Nokia ResearchCenter from 2001 to 2006 working on IEEE 802.16e (WiMAX) standard andEDGE/CDMA receiver algorithms. Charlie is a Fellow of IEEE.

Yik-Chung Wu received the B.Eng. (EEE) degreein 1998 and the M.Phil. degree in 2001 from theUniversity of Hong Kong (HKU). He received theCroucher Foundation scholarship in 2002 to studyPh.D. degree at Texas A&M University, CollegeStation, and graduated in 2005. From August 2005to August 2006, he was with the Thomson Cor-porate Research, Princeton, NJ, as a Member ofTechnical Staff. Since September 2006, he has beenwith HKU, currently as an Associate Professor. Hewas a visiting scholar at Princeton University, in

summers of 2011 and 2015. His research interests are in general areasof signal processing, machine learning and communication systems, and inparticular distributed signal processing and robust optimization theories withapplications to communication systems and smart grid. Dr. Wu served asan Editor for IEEE Communications Letters, is currently an Editor forIEEE Transactions on Communications and Journal of Communications andNetworks.

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