IEEE COMMUNICATIONS LETTERS, ACCEPTED FOR PUBLICATION 1

Iterative Joint Beamforming Training with Constant-AmplitudePhased Arrays in Millimeter-Wave Communications

Zhenyu Xiao, Member, IEEE, Lin Bai, Member, IEEE, and Jinho Choi, Senior Member, IEEE

Abstract—In millimeter-wave communications (MMWC), inorder to compensate for high propagation attenuation, phasedarrays are favored to achieve array gain by beamforming, wheretransmitting and receiving antenna arrays need to be jointlytrained to obtain appropriate antenna weight vectors (AWVs).Since the amplitude of each element of the AWV is usuallyconstraint constant to simplify the design of phased arrays inMMWC, the existing singular vector based beamforming trainingscheme cannot be used for such devices. Thus, in this letter,a steering vector based iterative beamforming training scheme,which exploits the directional feature of MMWC channels, isproposed for devices with constant-amplitude phased arrays.Performance evaluations show that the proposed scheme achievesa fast convergence rate as well as a near optimal array gain.

Index Terms—Millimeter wave, beamforming, phased array,beamforming training, 60 GHz.

I. INTRODUCTION

WHILE arousing increasing attentions in both academiaand industry due to abundant frequency spectrum

[1], [2], millimeter-wave communications (MMWC) face thechallenge of high propagation attenuation caused by the highfrequency. To remedy this, phased arrays can be adopted atboth the source and destination devices to exploit array gainsand compensate for the propagation loss [1], [2].

To achieve a sufficiently high array gain, the antenna weightvectors (AWVs) at both ends need to be appropriately set priorto signal transmissions, which is the joint Tx/Rx beamforming.If the channel state information (CSI) is available at bothends, the optimum AWVs can be directly found under well-known performance criteria, e.g., maximizing receiving signal-to-noise ratio (SNR) [1], [2]. Unfortunately, since the numberof antennas is generally large, the channel estimation becomestime-consuming in MMWC. In addition, the computationalcomplexity is high, because the matrix decomposition, e.g., thesingular vector decomposition (SVD), is generally required.Owing to this, the beamforming training approach, whichhas a lower complexity, becomes more attractive to find theAWVs [3]–[5]. Generally, there are two types of joint Tx/Rxbeamforming training schemes. One is switched beamformingtraining based on a fixed codebook [3]. The codebook containsa number of predefined AWVs. During beamforming training,

Manuscript received February 17, 2014. The associate editor coordinatingthe review of this letter and approving it for publication was H.-C. Wu.

This work was partially supported by the National Natural Science Foun-dation of China under Grant Nos. 61201189, 91338106, and 61231011, aswell as GIST College’s 2013 GUP Research Fund.

Z. Xiao and L. Bai are with the School of Electronic and InformationEngineering, Beihang University, Beijing 100191, P.R. China. Z. Xiao is thecorresponding author (e-mail: xiaozy06@gmail.com).

J. Choi is with the School of Information and Communications, GwangjuInstitute of Science and Technology (GIST), Korea.

Digital Object Identifier 10.1109/LCOMM.2014.040214.140351

the AWVs at both ends are examined according to a certainorder, and the pair that achieves the largest SNR is selected.The other one is adaptive beamforming training [4], [5], whichdoes not need a fixed codebook. The desired AWVs at bothends are found via real-time joint iterative training. It is clearthat the switched beamforming training is simpler, while theadaptive one is more flexible.

Most adaptive beamforming training schemes adopt thesame state-of-the-art approach, i.e., finding the best singularvector via iterative training without a priori CSI at both ends[4], [5]. This singular vector based training scheme (SGV)requires that both the amplitudes and phases of the AWVare adjustable. On the other hand, in MMWC, phased arraysare usually implemented with the approach that only phasesof the antenna branches are adjustable; the amplitudes areset constant to simplify the design and reduce the powerconsumption of phased arrays [3], [6], [7]. In fact, even ingeneral multiple-input multiple-output (MIMO) systems [8],antenna branches with constant amplitude (CA) are also anoptimization objective to reduce implementation complexity[9], [10]. In such a case, SGV becomes infeasible due to theCA phased array. The schemes proposed in [9], [10] cannotbe used here either, because these schemes are designed fortransmitting beamforming with full or quantized a priori CSIat only the source device, but for MMWC with CA phasedarrays, joint beamforming is required without a priori CSI atboth the source and destination devices.

In this letter, a steering vector based joint beamformingtraining scheme (STV), which exploits the directional featureof MMWC channels, is proposed. Performance comparisonsshow that for line-of-sight (LOS) channels, both STV andSGV have fast convergence rates and achieve the optimal arraygain. On the other hand, for non-LOS (NLOS) channels, STVachieves a faster convergence rate at the cost of a slightlylower array gain than SGV, which can also achieve the optimalarray gain. In conclusion, STV achieves a fast convergencerate and a near optimal array gain under both LOS and NLOSchannels, which highlights its applicability in practice.

II. SYSTEM AND CHANNEL MODELS

Without loss of generality, we consider a MMWC systemwith half-wave spaced uniform linear arrays (ULAs) of M andN elements at the source and destination devices, respectively.The ULAs are phased arrays where only the phase can becontrolled. A single RF chain is tied to the ULA at each of thesource and destination devices. According to the measurementresults of channels for MMWC in [1], [11], only reflectioncontributes to generating multipath components (MPCs), whilescattering and diffraction effects are negligible due to theextremely small wave length of MMWC. Thus, the MPCs

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2 IEEE COMMUNICATIONS LETTERS, ACCEPTED FOR PUBLICATION

Algorithm 1 The SGV Scheme1) Initialize:

Pick an initial transmitting AWV t at the sourcedevice. This AWV may be chosen either randomlyor with the approach specified in Section V.

2) Iteration: Iterate the following process ε times, and thenstop.

Keep sending with the same transmitting AWV tat the source device over N slots. Meanwhile, useidentity matrix IN as the receiving AWVs at thedestination device, i.e., the nth column of IN asthe receiving AWV at the nth slot. Consequently,we receiving a vector r = IHNHt+ IHNnr = Ht+nr, where nr is the noise vector. Normalize r.Keep sending with the same transmitting AWVr at the destination device over M slots. Mean-while, use identity matrix IM as the receivingAWVs at the source device. Consequently, wereceiving a new vector t = IHMHHr + IHMnt =HHr+nt, where nt is the noise vector. Normalizet.

3) Result:t is the AWV at the source device, and r is theAWV at the destination device.

in MMWC have a directional feature, i.e., different MPCshave different physical transmitting steering angles φt� andreceiving steering angles φr�. Consequently, the channel modelis expressed as [12], [13]

H =√NM

∑L−1

�=0λ�g�h

H� , (1)

where L is the number of multipath components, (·)His the conjugate transpose operation, λ� are the chan-nel coefficients, and g� and h� are the receiving andtransmitting steering vectors [12], [13] that are givenby g� = {ejπ(n−1)Ωr�/

√N}n=1,2,...,N and h� =

{ejπ(m−1)Ωt�/√M}m=1,2,...,M , respectively. Note that Ωt�

and Ωr� represent the transmitting and receiving cosine anglesof the �th MPC, respectively [13], i.e., Ωt� = cos(φt�) andΩr� = cos(φr�).

Given the transmitting AWV t and the receiving AWV r,where ‖t‖ = ‖r‖ = 1, the received signal y is given byy = rHHts + rHn, where s is the transmitted symbol, n isthe noise vector. The target of beamforming training is to findappropriate transmitting and receiving AWVs to obtain a highreceiving SNR, which is given by γ = |rHHt|2/σ2, whereσ2 is the noise power.

III. SINGULAR-VECTOR BASED SCHEME

Let us introduce the SGV scheme first. It is known that theoptimal AWVs to maximize γ is the principal singular vectorsof the channel matrix H [4], [5]1. Denote the SVD of H as

H = UΣVH =∑K

k=1ρkukv

Hk , (2)

1The SVD on H gives a set of orthogonal transmitting and receiving AWVpairs, as well as the energies projected to these AWV pairs.

where U and V are unitary matrices with column vectors(singular vectors) uk and vk, respectively, Σ is an N × Mrectangular diagonal matrix with nonnegative real values ρkon the diagonal, i.e., ρ1 ≥ ρ2 ≥ ... ≥ ρK ≥ 0 and K =min({M,N}). The optimal AWVs are t = v1 and r = u1.

In the common case that H is unavailable, iterative beam-forming training can be adopted to find the optimal AWVs.According to [4], [5], H2m Δ

= (HHH)m =∑K

k=1 ρ2mk vkv

Hk ,

which can be obtained by m iterative trainings utilizing thereciprocal feature of the channel. When m is large, H2m ≈ρ2m1 v1v

H1 . Thus, the optimal transmitting and receiving AWVs

can be obtained by normalizing H2mt and H × H2mt,respectively.

The SGV scheme is described in Algorithm 1. The iterationnumber ε depends on practical channel response, which will beshown in Section V. It is clear that although the SGV schemeis effective, it is required that both the amplitudes and phasesof AWV are adjustable, which cannot be satisfied when CAphased arrays are used, where only phases are adjustable.

IV. STEERING-VECTOR BASED SCHEME

In fact, the SGV scheme is a general one suitable for anarbitrary channel H. It does not use the specific feature ofMMWC channels. In MMWC, the channel has a directionalfeature, i.e., H can be naturally expressed as in (1), whichis similar to the expression in (2). The difference is thatin (1), the vectors {g�} and {h�} are CA steering vectors,not orthogonal bases, but in (2), {uk} and {vk} are strictnon-CA orthogonal bases. Nevertheless, according to [13],|gH

mgn| and |hHmhn| are approximately equal to zero given that

|Ωrm−Ωrn| ≥ 1/N and |Ωtm−Ωtn| ≥ 1/M , respectively, i.e.,the receiving and transmitting angles can be resolved by thearrays, which is the common case in MMWC. Consequently,as a suboptimal approach, the steering vectors of the strongestMPC can be adopted as the transmitting and receiving AWVsat the source and destination devices, which leads to theproposed STV scheme. The advantage of STV is that theelements of the steering vector have a constant envelope,which is suitable for the devices with CA phased arrays.Moreover, although the transmitting and receiving angles arerequired to be resolved by the arrays in the following analysis,the STV scheme can work even when there exists angles thatcannot be resolved, because two or more MPCs associatedwith sufficiently close angles that cannot be resolved actuallybuild a single equivalent MPC.

Assuming that H is available in advance, the back-ground of STV is presented as follows. Using the di-rectional feature of MMWC channels, we have H2m ≈∑L

�=1 |√MNλ�|2mh�h

H� , for a positive integer m. Suppose

the kth MPC is the strongest one. For � �= k, |λ�|2m/|λk|2mexponentially decreases. This means that the contribution tothe matrix product H2m from the the other L − 1 MPCsexponentially decreases, compared with the strongest one.Therefore, we have lim

m→∞H2m = |√MNλk|2mhkhHk . Thus,

for given a sufficiently large m and an arbitrary initial trans-mitting AWV t, we have

H2mt = |√MNλk|2mhkh

Hk t =

(|√MNλk|2mhH

k t)hk,

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XIAO et al.: ITERATIVE JOINT BEAMFORMING TRAINING WITH CONSTANT-AMPLITUDE PHASED ARRAYS IN MILLIMETER-WAVE COMMUNICATIONS 3

Send with Receive with :

Send with

Signature estimation:

Receive with :

Source device Destination device

Signature estimation:

H

H HrN N

H HrN N

r F Ht F n

1/2rexp(j ( ))

NN 1/2

g ut e esexp(jexp(je F r

at o :( ))NN(

rrr e

H H HtM M

H H HtM M

t F H r F n

1/2texp(j ( ))

MM 1/2e p(j

g(exp(je F t( ))

MM(

ttt e

HH

NF

MF

Fig. 1. Process of the proposed STV scheme.

which is hk multiplied by a complex coefficient. It is notedthat hk is a constant-envelope steering vector. Hence, thedesired transmitting AWV can be obtained by the signatureestimation2 where et = exp(j�(H2mt))/

√M is to be esti-

mated. Here, �(x) represents the angle vector of x in radian.In fact, the signature estimation can be carried out by theentry-wise normalization on H2mt.

In addition, we have

H×H2mt =(λk

√MN |λk

√MN |2mhH

k t)gk. (3)

Thus, the desired receiving AWV can be obtained by thesignature estimation of er = exp(j�(H×H2mt))/

√N .

It is clear that given full CSI, the AWVs steering alongthe strongest MPC in both ends can be obtained. In practicalMMWC, however, H is basically unavailable in both ends;thus we propose the joint iterative beamforming training pro-cess of STV, which is shown in Fig. 1, and the correspondingalgorithm is described in Algorithm 2. The iteration numberε depends on practical channel response. According to thesimulation results in Section V, ε = 2 or 3 can basicallyguarantee convergence.

It is noted that STV is tailored for MMWC devices withCA phased arrays based on SGV. Thus, STV and SGV havecommon features, e.g., both schemes need iteration. However,their mathematical fundamentals are different. SGV is to findthe principal singular vectors of the channel matrix H, whichis optimal and applicable for arbitrary channels, while STVis to find the CA steering vectors of the strongest MPC byexploiting the directional feature, which is sub-optimal andonly feasible under MMWC channels. Thus, in each iteration,STV requires the signature estimation, which is to estimatethe CA steering vector of the strongest MPC. Meanwhile, inorder to make STV feasible for CA phased arrays, it adopts theDFT matrices in transmitting and receiving training sequences,because the entries of them have a constant envelope.

V. PERFORMANCE EVALUATION

In this section we evaluate the performances of STV,including array gain and convergence rate, and compare themwith those of SGV via simulations. In all the simulations,the channel is normalized as E(

∑L�=1 |λ�|2) = 1. The

transmitting SNR is thus γt = 1/σ2, and the array gainbecomes the ratio of receiving SNR to transmitting SNR,i.e., η = γ/γt = |rHHt|2. The initial transmitting AWVsin the two schemes are selected under the principle that its

2There are other approaches to obtain the desired AWV here. The presentedone is a simple one in implementation.

Algorithm 2 The STV Scheme1) Initialize:

Pick an initial transmitting AWV t at the sourcedevice. This AWV may be chosen either randomlyor with the approach specified in Section V.

2) Iteration: Iterate the following process ε times, and thenstop.

Keep sending with the same transmitting AWVt at the source device over N slots. Meanwhile,use discrete Fourier Transform (DFT) matrix FN

as the receiving AWVs at the destination device,i.e., the nth column of FN as the receiving AWVat the nth slot. Note that IN cannot be usedfor the receiving AWVs here, due to its non-constant-envelope entries, but other unitary ma-trices with constant-envelope entries are feasible.Consequently, we receiving a vector r = FH

NHt+FH

Nnr, where nr is the noise vector. Estimate thesignature er as er = exp(j�(FNr))/

√N and

assign er to r.Keep sending with the same transmitting AWVr at the destination device over M slots. Mean-while, use DFT matrix FM as the receivingAWVs at the source device. Consequently, we re-ceiving a new vector t = FH

MHHr+FHMnt, where

nt is the noise vector. Estimate the signature etas et = exp(j�(FM t))/

√M and assign et to t.

3) Result:t is the AWV at the source device, and r is theAWV at the destination device.

power evenly projects on the M basis vectors of the receivingmatrices at the source, i.e., IM and FM , respectively. Thus, theinitial transmitting AWV for SGV is 1M/

√M , while that for

STV is a normalized constant-amplitude-zero-autocorrelation(CAZAC) sequence with length M .

The array gain is empirically found using the ratio ofthe average receiving SNR to the average transmitting SNRover 103 realizations of channels. Furthermore, the SVDupper bound is obtained by averaging the squares of theprincipal singular values of these channel realizations. Channelrealizations are generated under the Rician and Rayleighfading models for the LOS and NLOS channels, respec-tively. For the LOS channel, the power of the LOS MPC is|λ1|2 = 0.7692, and the average powers of the NLOS MPCsare E({|λ�|2}�=2,3,4) = [0.0769 0.0769 0.0769]. For theNLOS channel, E({|λ�|2}�=1,2,3,4) = [0.25 0.25 0.25 0.25].The transmitting and receiving steering angles are randomlygenerated within [0 2π) in each realization.

The left and middle figures in Fig. 2 show the achievedarray gains of SGV and STV under the LOS and NLOSchannels, respectively, with different numbers of iterations,where M = N = 16. The right figure in Fig. 2 showsthe comparison of convergence rate between SGV and STVunder the LOS and NLOS channels with a high transmittingSNR, i.e., 25 dB, in the cases of M = N = 16 andM = N = 32, respectively. From the left and right figures,

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4 IEEE COMMUNICATIONS LETTERS, ACCEPTED FOR PUBLICATION

0 5 10 15 20 25 3014

16

18

20

22

24

SNR (dB)

Arr

ay G

ain

(dB

)

STV, ε=1

STV, ε=2

STV, ε=3

SGV, ε=1

SGV, ε=2

SGV, ε=3SVD Bound

0 5 10 15 20 25 3012

14

16

18

20

22

SNR (dB)

Arr

ay G

ain

(dB

)

STV, ε=1

STV, ε=2

STV, ε=3

SGV, ε=1

SGV, ε=2

SGV, ε=3SVD Bound

0 2 4 6 820

22

24

26

28

30

32

Iteration Number ε

Arr

ay G

ain

(dB

)

STVSGVSVDB

LOS, M=N=32

NLOS, M=N=32

LOS, M=N=16

NLOS, M=N=16

Fig. 2. Comparison of array gain between SGV and STV under the LOS (left) and NLOS (middle) channels with different numbers of iterations, whereM = N = 16, and that of convergence rate (right), where SVDB represents SVD bound.

it is found that under the LOS channel, both schemes achievefast convergence rates and approach the optimal array gain,i.e., the SVD upper bound. From the middle and right figures,it is observed that under the NLOS channel, both the twoschemes have slower convergence rates, and STV achievesa faster convergence rate at the cost of a slightly lower arraygain than SGV that also approaches the SVD upper bound. It isnoted that, although not shown in these figures, similar resultsare observed with a smaller or larger number of antennas.

Explanations for these observations are as follows. Underthe LOS channel, there is one and only one strong MPC, andthe steering vectors of this MPC are almost the optimal AWVs.Thus, STV can achieve the optimal array gain. But under theNLOS channel, there are several MPCs with different steeringangles (or steering vectors) and the STV scheme obtains oneof them as an AWV, which is not optimal. Hence, STV cannotachieve the optimal array gain in such a case. On the otherhand, since the SGV is based on the principal singular vector,it can surely achieve the SVD upper bound once convergencehas been achieved. Besides, the fact that STV achieves a fasterconvergence rate in NLOS channel indicates that the signatureestimation in each iteration of STV is more robust againstnoise, while the AWV estimation of SGV is more sensitive tonoise.

In brief, although STV is tailored for MMWC devices withCA phased arrays, where SGV is infeasible, it has comparableperformances to SGV in terms of the convergence rate andarray gain, under both the LOS and NLOS channels. Onthe other hand, it is noted that a single iteration consumesM + N training slots, which may significantly degrade thesystem efficiency, especially when the number of antennasis large. Hence, even if there is no CA constraint, i.e., bothphase and amplitude are adjustable and thus SGV is feasible,STV may still be favored in the case that the iteration numberis constrained to be 1 or 2 to save training time, because itachieves a higher array gain according to the right figure ofFig. 2.

VI. CONCLUSIONS

Since the existing SGV scheme cannot be used in MMWCwith CA phased arrays, the STV scheme has been proposedin this study, which effectively exploits the directional featureof MMWC channels. Performance comparisons showed that

under LOS channel, both the schemes achieve fast conver-gence rates and achieve the optimal array gain; under NLOS

channel, STV achieves a faster convergence rate at the cost ofa slightly lower array gain than SGV that can still approachthe optimal array gain. In summary, while the proposed STVscheme is well-suited to MMWC with CA phased arrays, it hascomparable performances to SGV in terms of the convergencerate and array gain under both the LOS and NLOS channels.

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