IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. XX, NO. XX, XX XXXX 1

Compressed Beam Alignment with Out-of-BandAssistance in Millimeter Wave Cellular Networks

Jie Zhao, Student Member, IEEE, Xin Wang, Member, IEEE, Harish Viswanathan, Fellow, IEEEArjuna Madanayake, Member, IEEE , and Guangxue Yue, Member, IEEE.

Abstract—Network transmission over millimeter-wave (mmW) bands has a big potential to provide orders of higher bandwidth.However, beamforming is generally needed to compensate for the high path loss. As mmW antennas have a potentially large numberof candidate beamforming directions, to achieve high network throughput, the finding of a high gain direction between a base stationand each mobile in the mmW network may involve a large overhead if training signals are directly sent along all possible directions oraccording to a large volume of codebook. Taking advantage of the block sparse characteristics of the mmW channel and coexistenceof legacy antennas, we propose a comprehensive design for more efficient beam direction finding. Different from existing compressive-sensing-based schemes which just take a random subset of directions to measure, taking advantage of the path clustering feature ofthe mmW channel, we develop a self-adaptive block sparse algorithm which can benefit from preliminary channel estimation duringeach iteration of the problem solving to significantly improve the overall channel estimation accuracy thus the beam alignment gain.We also explore two methods to exploit co-located legacy antennas to provide further guidance for transmission direction finding.Simulation results indicate that our proposed beam alignment scheme outperforms the baseline and peer schemes in terms of thebeamforming gain and training cost. By taking advantage of the block sparse properties of mmW channel, our proposed design is ableto achieve the transmission throughput comparable with the exhaustive direction search at much lower overhead.

Index Terms—millimeter wave, beamforming, beam alignment, directional antenna, compressed sensing.

F

1 INTRODUCTION

Millimeter-wave (mmW or mmWave) communication isreceiving tremendous interest from academia, industryand federal agencies as a promising technique to provideGigabit data rate demanded by the exponential growthof various applications in wireless networks. A key chal-lenge of data transmissions in mmW cellular networks isthe low signal transmission range (Figure 1). Accordingto Frii’s Law, the high frequencies of mmW signals resultin a large isotropic path loss.

Fortunately, the small wavelength of mmW signalsalso enables a large number of antenna elements to beplaced in the same small dimension (e.g. at the basestation, in the skin of a cellphone, or even within achip), which provides a high beamforming gain that cancompensate for the increase in the isotropic path loss.

Although the use of a large antenna array helps tocombat the severe path loss, it also makes it difficultto coordinate network transmissions [1], [2], [3], [4].Taking the initialization and synchronization of basestations (BSs) and mobiles as an example, using only

• J. Zhao and X. Wang are with the Department of Electrical and ComputerEngineering, Stony brook University, Stony Brook, NY 11794 (E-mail:jie.zhao@alumni.stonybrook.edu, x.wang@stonybrook.edu). X. Wang’s re-search has been supported by the NSF under grant ECCS-1731238.H. Viswanathan is with Nokia Bell Labs, Murray Hill, NJ 07974 (E-mail: harish.viswanathan@alcatel-lucent.com). A. Madanayake is with theDepartment of Electrical and Computer Engineering, Florida InternationalUniversity, Miami, FL 33174 (E-mail: amadanay@fiu.edu). G. Yue iswith the College of Mathematical and Information Engineering, JiaxingUniversity, Zhejiang, China (E-mail: guangxueyue@163.com).

omni-directional transmissions of synchronization sig-nals would be problematic in the mmW range: the avai-lability of high gain antennas would bring a discrepancybetween the range at which a cell can be detected (whensignaling messages are transmitted omni-directionallybefore the correct beamforming directions are found)and the range at which reasonable data rates can beachieved (after beamforming is applied). On the otherhand, although a beamed transmission from the basestation provides a larger footprint and allows for hig-her data rate, it is difficult for a mobile to find thebase stations (BSs) initially without knowing the correctbeamforming directions. To address these issues, a cellsearch phase is needed where the base station of a mmWnetwork beams towards different directions to facilitatea mobile to find a direction that maximizes its receivingrate.

The antenna gains of the transmitter (TX) and thereceiver (RX) have significant impacts on the trans-mission quality. Simply transmitting signaling messagesrotationally along each direction would introduce veryhigh delay and cost for finding the optimal beamformingdirection with the maximum gain. An example is whenTX and RX each has 64 beam directions (in practicalmmW networks this number can be even larger), toexhaustively measure every beam pair, 64 × 64 = 212

measurements are required. The finding of an optimalbeam direction may take long time to complete, resultingin a large delay to establish a transmission link. As thechannel conditions are dynamic, the direction findingmay need to be performed frequently, which would sig-

2 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. XX, NO. XX, XX XXXX

nificantly compromise the network capacity. Therefore,it is critical to incorporate an efficient beam searchingor beam alignment scheme into the MAC design toestablish high quality mmW links at low cost.

As lower-frequency legacy radios are likely to bedeployed alongside mmW systems to strengthen widearea signaling control and/or facilitate multi-band com-munications, out-of-band information from legacy bandhas been considered to help reduce the overhead ofestablishing the mmW links [5], [6]. As the co-locatedlow-frequency system and mmW system share similarenvironment, the spatial characteristics of low-frequencychannel and mmW channel have correlation, as confir-med by experimental studies. This however, does notmean the channels at different frequencies have exactlythe same characteristics. The work in [5] simply modelsthe channel with a single path without considering thecharacteristics of mmWave channel [7], [8], [9], while aut-hors in [6] assume paths of legacy channel and mmWavechannel have nearly-exact match. These inaccuracies willlead to mis-alignment of mmWave beams, which willfurther degrade the transmission performance in realisticscenarios. As another source of inaccuracy, the workin [5] exploits model-based estimation to directly inferthe transmission directions on the mmWave channelbased on measurements from the low-frequency channel,which will make the alignment subject to performancedegradation due to difference in channels and failureunder high dynamics.

The measurement from low band can only give arough estimation of transmission direction. To enablehigh performance transmissions, we need to align thebeams with the desired angular resolution. Instead ofexhaustively training all possible beam pairs within arange directed by the low-band training, we are moti-vated to further reduce the number of measurementswith compressed channel estimation. Different from thelegacy low frequency channels, the recent studies [7],[8], [9], show that wireless mmW channels only havea few dominant paths and the paths are often clustered,thus the channels often present sparse characteristics.Some initial studies [10], [11], [12], [13] have been madeto exploit the sparse feature to estimate mmW channelwith compressed sensing [14], [15], but they did not takeinto account the clustering feature of transmission pathsthat differentiates mmW channel from conventional lowfrequency channels [16].

In this paper, we concurrently exploit legacy bandcoarse direction finding and compressed channel estima-tion to enable more accurate beam alignment betweeneach pair of sender and receiver at low cost. The estima-tion of channel allows for finding the optimal fine beamdirection even when it does not fall into the best coarsebeam range of low-frequency channel. Rather than justexploiting the basic compressed sensing technique [7],[8], [9], [17], [18] based on the low rank propertiesof mmW channel, we propose two major techniquesto significantly reduce the training overhead, improve

BS

MobilemmWchannel

Fig. 1. Millimeter-wave cellular network.

channel estimation performance and increase the qualityof beam alignment: 1) We develop a self-adaptive blocksparse reconstruction algorithm which learns from theiterative channel estimation results to further improvethe estimation efficiency; and 2) We design a beam alig-nment algorithm which concurrently exploits the coarsechannel information from low-frequency antennas andblock-based mmW channel estimation to enable high-throughput beamed transmissions. Our design intendsto be flexible. The block-based channel estimation canwork either stand-alone or jointly with out-of-band as-sistance.

The rest of this paper is organized as follows. Afterbriefly reviewing related work in Section 2, we pre-sent compressed sensing preliminaries in Section 3. Wethen describe the system model and our motivationin Section 4. Block sparse mmW channel estimationis introduced in Section 5, followed by Section 6 thatpresents our channel reconstruction algorithm and beamalignment design. We analyze the simulation results inSection 7. The paper concludes in Section 8.

2 RELATED WORK

In millimeter-wave (mmW or mmWave) networks, ahigh beamforming gain is needed to compensate for thelarge path loss and occlusions in the mmW spectrumrange [19]. This requires a joint beamforming (BF)scheme in the MAC protocol to select the best transmis-sion and reception beam directions according to a metricsuch as signal-to-noise ratio (SNR) [20].

IEEE 802.15.3c [21] and IEEE 802.11ad [22] standardsare proposed to enable operation in the 60GHz mm-Wave band. Multi-level codebook is suggested in bothprotocols to facilitate the training of beams at hierar-chical resolutions of beamwidth. Some other codebook-based beamforming methods are also proposed in [23],[24], where the beam training overhead highly relieson the codebook design thus search space. Althoughcodebook-based schemes reduce the beam search space,the overhead for uplink feedback of beam conditionsand selection would be big when the beamwidth issmall and there are a large number of directions tosearch. Another drawback of multi-resolution codebookschemes is that the optimal fine beam direction may befiltered out during high-level coarse direction estimation,as fine beam direction may not exactly align with coarsebeam direction. There are also no discussions in existingschemes on how to reduce the large overhead when the

ZHAO ET AL.: COMPRESSED BEAM ALIGNMENT WITH OUT-OF-BAND ASSISTANCE IN MILLIMETER WAVE CELLULAR NETWORKS 3

codebook volume is huge, which is often the case inmmW band.

Some existing works attempt to reduce the beamfor-ming overhead by intelligently reducing the search cope(e.g. select only part of the candidate beams to train).In [25], Hur et al. propose the use of outdoor millimeterwave communications for backhaul networking to con-nect cells with the core network and mobile access withina cell between the base station and mobiles. To over-come the outdoor impairments found in millimeter wavepropagation, this paper studies beamforming using largearrays. The authors propose an efficient beam alignmenttechnique using adaptive subspace sampling and hier-archical beam codebooks. To perform initial directionalcell search in mmW cellular networks, the mobile andbase station must jointly search over a potentially largeangular directional space to locate a suitable path toinitiate communication. Barati et al. in [26] propose adirectional cell search procedure where a base stationperiodically transmits synchronization signals in rand-omly varying directions, and derive detectors for bothanalog beamforming and digital beamforming. An effi-cient beam switching technique for the emerging 60GHzwireless personal area networks is proposed in [27],where Li et al. formulate a global optimization problemto look for the best beam-pair for data transmissionsand adopt a numerical approach to implement the beamsearching strategy with divide and conquer used in smallregions. Rather than just reducing the beam search space,we also take advantage of the mmW channel feature andthe information of trained beams to estimate the mmWchannel and guide for better beam selection.

Channel estimation for mmW beamforming is inves-tigated in [28], where Singh et al. investigate the fea-sibility of employing multiple antenna arrays to obtaindiversity/multiplexing gains in mmW systems. The aut-hors exploit the sparse multipath property of the mmWchannel and propose to reduce the complexity involvedin jointly optimizing the beamforming directions acrossmultiple arrays by focusing on a small set of candidatedirections. Differently in our work, we not only takeadvantage of the sparsity of mmW channel, but alsoemploy a useful tool, compressed sensing, to fully extractuseful information from mmW channels. In [29], Olfatet al. propose to estimate the channel for a frequency-selective millimeter-wave communication system with aminimum number of pilots. A learning-based schemeis taken to find the optimal precoding and combiningvectors for transmitting and receiving pilot signals in theface of channel dynamics, where the learning requiresthe previous knowledge of the channel.

Compressed sensing [14], [15] (CS) has been exploitedfor channel estimation in [10], [11], [12], [13], [30].Although these studies exploit the sparse feature ofmmW channel, they did not consider the clusteringfeature of transmission paths, and the path clusteringis a distinct characteristic of mmW channel comparedto conventional low-frequency channels [16]. In this

work, rather than just exploiting the basic low rankproperties [7], [8], [9] of the mmW channel based oncompressed sensing techniques, we propose a block-sparse channel estimation algorithm that takes full ad-vantage of the path clustering to significantly improvethe channel estimation efficiency. In addition, we alsoinvestigate the possibility of exploiting the informationfrom co-located legacy antennas (e.g. from 3G cellularnetworks) to further improve the performances of mmWsystems. One major difference of our design from ot-her methodologies on out-of-band assistance for mmWnetworks [5], [6] is that we take the legacy band infor-mation as an optional procedure that can jointly workwith mmW channel estimation to enable low cost andhigh accuracy alignment at the desired beam resolution.These literature studies usually rely on the out-of-bandassistance to operate stand-alone and have strong as-sumptions on the mmWave channel, such as having onlysingle dominating path [5] or having paths matched withthose on the legacy channel almost perfectly [6]. Theseassumptions make the beam alignment more prone toinaccuracy in practical scenarios. In addition, model-based estimation [6] based on coarse measurements fromlow frequency band can not well capture the channel dy-namics at mmWave band to find the fine beam directionswith the highest gains.

User motion can make the beamforming issue morechallenging in mmW networks. Based on the observationthat 60 GHz channel profiles at nearby locations arehighly-correlated, Zhou et al. in [31] propose a beam-forecast scheme to reconstruct the channel profile andpredict new optimal beams. The beam prediction duringmobility case is built upon the initial channel estima-tion and direction finding, and channel scanning willbe called for again to realign beams as the predictionerrors accumulate over time. Complementary to thework in [31], we propose an efficient scheme to find theoptimal beam direction without pre-channel knowledge.

To summarize our differences from existing literature,in this work, we propose to design an intelligent andefficient beam alignment scheme to enable the quick fin-ding of a good transmission direction between a BS anda mobile in mmW cellular networks. Rather than simplyand exhaustively measuring all possible beam directionsor searching in a large codebook, we can measure asmall portion of the beam pairs to achieve comparableperformance, thus saving resources like time and power.Specifically, to enable beam alignment with lower trai-ning overhead and higher beamforming performances,we propose an efficient CS-based mmW channel estima-tion methodology that takes full advantage of the blocksparse features (due to path clustering) of mmW chan-nel, and the channel estimated serves as a guidance todiscover the optimal beam pairs. We further improve thebeam alignment performance by gathering informationfrom co-located legacy antennas. Some important issueswe consider include: (a) Why does a mmW channel haveblock-sparse properties? (b) How to take advantage of

4 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. XX, NO. XX, XX XXXX

the block-sparse properties of mmW channel to performmore efficient channel estimation in order to guide thebeam alignment? (c) How to comprehensively designbeam alignment schemes to achieve a better networkperformances? (d) How to exploit the information fromlegacy antennas?

To answer these questions, we’ll first present the sy-stem model in the next section.

3 COMPRESSED SENSING PRELIMINARIES

In this work, we will exploit the use of the emergingcompressed sensing (CS) technique for more efficientbeam alignment in mmWave cellular networks. We firstintroduce some backgrounds on compressed sensing.

The main idea of compressed sensing (CS) is to takeadvantage of the sparsity within the signal to signi-ficantly reduce the sampling rate. An N -dimensionalsignal d is considered to be K-sparse in a domain (alsocalled a dictionary matrix) Ψ ∈ CN×N if there exists anN -dimensional vector x ∈ RN×1 so that d = Ψx andx has at most K non-zero entries (K � N ). The CStheory suggests that d can be fully reconstructed froma sufficient number M (M ≥ cK log

(NK

), where c is a

fairly small constant) of linear measurements.If one performs linear measurements of the signal d

with a measurement matrix Φ, then one can considerthe obtained linear measurements y, possibly affectedby noise as:

y = ΦΨx + n = Ax + n, (1)

where the measurements are y ∈ RM×1, the sparse vec-tor x ∈ RN×1, the additive noise n ∈ RM×1, the sensingmatrix A ∈ RM×N , and M < N . A is essentially theproduct of the measurement matrix and the dictionarymatrix: A = ΦΨ, where Φ ∈ CM×N , Ψ ∈ CN×N . Wenotice that different from the notations above, a smallnumber existing works call Φ the sensing matrix. Inorder to avoid inconsistency and misunderstanding, wewill consistently regard Φ as the measurement matrix,and A = ΦΨ the sensing matrix.

Obviously, the number of measurements is smallerthan the number of variables in Equation 1, and itindicates this is an under-determined system. Donoho,Candes, Romberg, and Tao show in [32] that the under-determined equation system can be solved as:

min ‖x‖l1 (2)s.t. ‖Φd− y‖l2 ≤ ε (3)

d = Ψx (4)

where the parameter ε is the bound of the error causedby noise n, lp means the lp-norm (p = 1, 2, ...). Thesolution can also be expressed as:

x = argminu: ‖y−Au‖l2≤ε

‖u‖l1 . (5)

The signal d = Ψx can then be recovered as d = Ψx.

DAC ADC

TX RX

Beamforming

selection v Beamforming

selection u

s y2

1 1

2

M N

noise

noise

noise

H

Fig. 2. Beam alignment between TX and RX.

Compressed Beam Selection

Compressed Beam Training

Block-sparse Channel Reconstruction

Compressed Beam Alignment

Beam Selection Range Beam Range Quality

Legacy Band Assistance (if available)

Fig. 3. Framework Overview.

The form of the optimization problem in (5) is knownas LASSO [33] or BPDN [34] and also some other varia-tions such as the Dantzig selector [35]. In addition to theconvex optimization approach, such as l1 minimization[36], to reconstruction in compressed sensing, there existseveral iterative/greedy algorithms such as IHT [37] andCosamp [38]. Such convex or greedy approaches aregenerally called reconstruction algorithms.

4 SYSTEM MODEL AND MOTIVATION

In order to perform the beam alignment in mmWavecellular networks, a challenge is that the base stationsand mobiles need to search for an optimal transmissiondirection for each transmission pair over a large numberof possible beam directions. To frame the problem andthe design of our beam-alignment algorithm, we firstintroduce the basic system model considered in thispaper.

4.1 Framework Overview

As shown in Figure 3, our proposed framework consistsof two major components: (1) Compressed Beam Align-ment and (2) Legacy Band Assistance. The two compo-nents can jointly work together. Compressed beam alig-nment can operate stand-alone if there is no assistancefrom the legacy band. Our low-cost beam alignment isachieved by reducing the beam search overhead withCompressed Beam Selection and Training, and guidedthrough channel information obtained from Block-SparseChannel Estimation. Different from the literature work,

ZHAO ET AL.: COMPRESSED BEAM ALIGNMENT WITH OUT-OF-BAND ASSISTANCE IN MILLIMETER WAVE CELLULAR NETWORKS 5

block sparsity of the mmW channel is exploited inthe modeling and reconstruction of the channel, whichimproves the accuracy of channel estimation and thusenhances the performance of beam alignment. The im-pacts of Legacy Band Assistance on Compressed BeamAlignment lie in two aspects: (a) beam selection rangesuggestion and (b) beam range quality suggestion, wherethe former helps narrow the angular range of beam se-arch thus reducing the overhead and the channel recon-struction in the latter takes advantage of the suggestedbeam range qualities to achieve higher accuracy.

4.2 NotationsFor a matrix A, we use AT to denote its transpose.A∗, AH , A−1 are its conjugate, Hermitian transpose(conjugate transpose) and inverse, respectively. For avector a, diag(a) is a diagonal matrix whose diagonalelements are entries from a. AB is the matrix productof A and B. A ◦B is the Hadamard product (element-wise product) of A and B. A ⊗ B is the Kroneckerproduct of A and B. A ∗ B is the Khatri-Rao productof A and B. diag(a) is the diagonal matrix of vector awhose diagonal elements of diag(a) are entries from a.vec(A) is vectorized vector of matrix A, and the vecoperator creates a column vector from a matrix A bystacking the column vectors of A = [a1,a2, ...,an] belowone another.

4.3 Channel Model and Downlink TransmissionWe will now explain the model using the downlinktransmission as an example, where the RX denotes themobile and the TX means the base station (BS). Fora sender and receiver pair, assume there are Nrx RXantennas and Ntx TX antennas.

The signal transmitted from TX is represented as

xNtx×1 = uNtx×1 ◦ sNtx×1 (6)

where u is the BF weight vector for the TX antennasand s is the training signal sent from the TX antennas.The multiplication ◦ means the operation between thevectors is element-wise.

The signal received by the RX antennas before thereceiver beamforming is

yNrx×1 = HNrx×NtxxNtx×1 + eNrx×1 (7)

where H is the channel matrix (its element Hn,m is thechannel coefficient from the TX antenna m to the RXantenna n), e is AWGN at the RX antennas.

In [16], statistical models of mmW channels arederived from real-world measurements at 28 and 73GHz in New York City (NYC). The mmW channel isfound to be sparse and can be modeled as a smallnumber of clusters each consisting of a small numberof subpaths. For ease of presentation, we consider onlythe azimuth and neglect the elevation in this paper,implying that all scattering happens in the azimuth andthat the TX and RX conduct the horizontal beamforming

only. Implementations that facilitate both horizontal andvertical beamforming can be built on top of our design.While our proposed design can be used for any kindof antenna arrays, without loss of generality, we adoptuniform linear arrays (ULAs) in this work. Assuming ammW channel is composed of K clusters within eachthere are L subpaths, then the mmW channel matrix weconsider can be expressed as:

H =

K∑

k=1

L∑

`=1

ak` ·Drx(θrxk` ) ·DH

tx(θtxk`), (8)

where ak` is the complex path gain for a path ` (` =1, 2, ..., L) in the cluster k (k = 1, 2, ...,K), with k` jointlycorresponding to the `-th sub-path in the k-th cluster. Forthe sake of consistency, in this work, we use the termspath and sub-path interchangeably. θtxk` and θrxk` denotethe angle of departure (AoD) and the angle of arrival(AoA) for the corresponding path, which throughout thispaper indicates the horizontal scattering.

Dtx(θtxk`), the TX antenna’s directional response co-

lumn vector (Ntx × 1 dimension) for the sub-path at theangle of departure θtxk`, is expressed as:

Dtx(θtxk`)

=[D(1)(θtxk`), D

(2)(θtxk`), ..., D(m)(θtxk`), ..., D

(Ntx)(θtxk`)]

=[1, ej·1·w

txk` , ej·2·w

txk` , ..., ej·(Ntx−1)·wtx

k`

]T,

(9)

where D(m)(θtxk`) is from antenna basics, the spatialfrequency wtxk` can be written in terms of AoDs, aswtxk` = 2πdt

λ sin θtxk`. dt is the distances between twoadjacent antenna elements in the ULAs in the TX. λ = c

fis wavelength in meters. f is the carrier frequency of thesignal in Hz, c is the speed of light (3× 108 meters/sec).

Similarly, Drx(θrxk` ), the RX antenna’s directional re-

sponse column vector (Nrx × 1 dimension) for the pathat an angle of arrival θrxk` , is expressed as

Drx(θrxk` )

=[D(1)(θrxk` ), D

(2)(θrxk` ), ..., D(n)(θtxk`), ..., D

(Nrx)(θtxk`)]

=[1, ej·1·w

rxk` , ej·2·w

rxk` , ..., ej·(Nrx−1)·wrx

k`

]T,

(10)

where D(n)(θtxk`) is from antenna basics, the spatial fre-quency wrxk` can be written in terms of AoAs, as wrxk` =2πdrλ sin θrxk` . dr is the distances between two adjacent

antenna elements in the ULAs in the RX.We now use a concatenated column vector a (1×KL)

to denote the complex path gains. Then

a = [a11, a12, ..., a1L︸ ︷︷ ︸cluster 1

, a21, a22, ..., a2L︸ ︷︷ ︸cluster 2

, ..., aK1, aK2, ..., aKL︸ ︷︷ ︸cluster K

]T .

(11)

Note a is concatenated in a manner that the first Lelements are for the first cluster, and the next L elements

6 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. XX, NO. XX, XX XXXX

are for the second cluster and so on, and there areKL elements in a. The major task of mmW channelestimation in our work is to estimate a efficiently.

In order to achieve a high receiving gain, a beamfor-ming vector will be applied at RX. The beamformingoutput at the RX antenna array will be

r = vHy + e = vHHu ◦ s + e, (12)

where e is the received noise at the RX. The beamformingprocess can be presented as in the Figure 2.

We have introduced the model for the downlink trans-mission, where TX is BS and RX is MS. The uplinkcommunication can be easily derived according to ourpresentation above by making MS the TX and BS theRX.

4.4 Motivation and ProblemA grand challenge in mmW communications is to

efficiently find the optimal beamforming directions fordata transmissions in the mmW networks. This requiresdetermining the optimal weights for both the transmitterand the receiver of each transmission pair in the mmWnetwork. To compensate for the path loss in the highfrequency mmWave channel, TX and RX potentially needa large number of antennas to achieve a high beamfor-ming gain, which unfortunately also greatly expands thesearch space of TX and RX antenna directions.

As an example, when TX and RX each has 64 beamdirections (in practical mmW networks, this number canbe even larger), to exhaustively measure every beampair, 64 × 64 = 212 measurements are required. Simplytransmitting signaling messages rotationally along eachdirection as suggested by the existing standard [21]would introduce very high delay and cost for findingthe optimal beamforming direction with the maximumgain. Therefore, an essentially important question formmW band beamforming is: how to reduce the searchspace of TX and RX beam directions in mmW cellularnetworks while ensuring the high beamforming gainthus transmission rate?

Existing work such as [10], [11], [12], [13] appliedcompressed sensing (CS) to alleviate this training over-head by randomly training a small subset of beam pairsand then estimating the mmW channel to facilitate beamalignment. However, these studies did not fully exploitthe path clustering effect of mmW channels and theresulted block sparsity of the gain coefficients of mmWpaths.

Moreover, the small angular spread in the mmW chan-nel indicates that if some directions in the low-frequencychannel are dominant, it is also likely that an overlappedmmW channel has a good gain in surrounding angularspace. Although transmission paths in the mmW chan-nel and the low-frequency channel may not be exactlyidentical, the information in the low-frequency channelcan give a rough information on which angular range ofdirections is more likely to have better path gains in the

mmW channel. In the existence of some co-located legacyantennas (e.g. from 3G cellular networks) around mmWantennas, more efficient beam alignment schemes can bedesigned by taking advantage of the information fromlegacy band. Different from the literature on CS-basedmmW channel estimation and beamforming, to supportmore efficient beam alignment, we aim to answer thefollowing questions in this work:

(a) How to efficiently exploit the block sparse featureof mmW channels to better formulate the mmW channelestimation problem?

(b) How to develop an efficient block sparse CS re-construction algorithm to achieve more accurate mmWchannel estimation?

(c) How can the information from co-located legacyantennas provide more useful guidance for mmW beamalignment?

In response of (a), Section 5 presents a block-sparsity-based mmW channel estimation problem. The solutionsfor (b) and (c) are presented in Section 6. We proposea self-adaptive weighted algorithm to iteratively learnthe weights of channel blocks to increase the CS recon-struction performance. We also develop two methodo-logies which exploit legacy antennas to facilitate beamalignment, one aiming to help select better beams tobe trained and the other aiming to further improve thereconstruction performance of mmW channels.5 SPARSE MMWAVE CHANNEL ESTIMATIONFor each pair of transmitter and receiver in the mmWnetwork, in order to find the best transmission/receptiondirection, there is a need to estimate the channel betweenthem. This is generally facilitated with the sending ofthe training signals from the transmitter to the receiver.However, if the channel measurement is performed in astraight-forward way, it would require the transmissionsalong a large number of directions to find more accuratechannel information. In this section, we first discuss thechannel features, and then explain how the emergingcompressed sensing technique can be applied to reducethe channel measurement overhead.

5.1 Sparse Channel EstimationFrom matrix basics we know that the channel matrix Hfrom (8) can be equivalently written as follows:

H = DR diag(a)DHT (13)

where diag(a) is a diagonal matrix whose diagonalelements are those from a in order. The matrices DT

and DR contain the TX and RX array response vectors,respectively, in the following form:

DT = [Dtx(θtx11),Dtx(θ

tx12), ...,Dtx(θ

tx1L),

Dtx(θtx21), ...,Dtx(θ

tx2L), ...,

Dtx(θtxK1), ...,Dtx(θ

txKL)],

(14)

DR = [Drx(θrx11 ),Drx(θ

rx12 ), ...,Drx(θ

rx1L),

Drx(θrx21 ), ...,Drx(θ

rx2L), ...,

Drx(θrxK1), ...,Drx(θ

rxKL)],

(15)

ZHAO ET AL.: COMPRESSED BEAM ALIGNMENT WITH OUT-OF-BAND ASSISTANCE IN MILLIMETER WAVE CELLULAR NETWORKS 7

The dimensions of matrices DR, diag(a) and DT areNrx ×KL, KL×KL and Ntx ×KL, respectively, whichcorresponds to the fact that H both (8) and (13) is anNrx ×Ntx matrix.

If we take measurements by transmitting the trainingsignals along P directions and letting the receiver toestimate the signals from Q directions, which is achievedthrough the use of P TX beamforming (BF) vectors (up,p = 1, 2, ..., P ) and Q RX beamforming vectors (vq ,q = 1, 2, ..., Q), we have the matrix form

RQ×P = VHHU ◦ S + E, (16)

where an element in the location (q, p) in R denotes theBF output when TX and RX use beamforming weights upand vq , respectively, S and E are respectively the trainingsignals and noise in the matrix form, and

VNrx×Q = [v1,v2, ...,vq, ...,vQ], (17)

UNtx×P = [u1,u2, ...,up, ...,uP ]. (18)

With the training signals transmitted at the power A,we have

RQ×P =√AVHHU + E (19)

We can vectorize R,

r = vec(R) =√Avec(VHHU) + vec(E)

Theorem 1 [39]==========

√A(UT ⊗VH)vec(H) + vec(E)

Proposition 1 [40]============

√A(UT ⊗VH)Ψa + vec(E)

= ΦΨa + vec(E) = Aa + vec(E),(20)

where Ψ = D∗T ∗DR (Khatri-Rao product), Φ =√A(UT⊗

VH) (Kronecker product) is the measurement matrix(determined by the TX and RX beamforming trainingdirections), Ψ is the basis matrix that will be introducedin Proposition 1. In the derivation, we have used The-orem 1 [39] and Proposition 1 [40]. For the integrity ofpresentation, we still provide the proofs below.

Theorem 1. vec(AXB) = (BT ⊗A)vec(X).

Proof. Let B = [b1,b2, ...,bn] (of size m × n) and X =[x1,x2, ...,xm]. Then, the k-th column of AXB is

(AXB):,k = AXbk = A

m∑

i=1

xibi,k

= [b1,kA, b2,kA, ..., bm,kA] [x,x2, ...,xm]T

︸ ︷︷ ︸vec(X)

= ([b1,k, b2,k, ..., bm,k]⊗A)vec(X)

(21)

Stacking the columns together

vec(AXB) = [(AXB):,1,AXB):,2, ..,AXB):,n, ]T

=[bT1 ⊗A,bT2 ⊗A, ...,bTn ⊗A

]Tvec(X)

= (BT ⊗A)vec(X)(22)

�

Proposition 1. vec(H) = Ψa, where Ψ = D∗R∗DT (Khatri-Rao product).

Proof. From (13) we know we can estimate channel asfollows:

H = DR diag(a)DHT (23)

From Theorem 1 we know,

vec(H) = (D∗T ⊗DR)vec(diag(a))= (D∗T ⊗DR)Ja

(a)=== (D∗T ∗DR)a = Ψa

(24)

where Ψ = D∗T ∗ DR (Khatri-Rao product) defines abasis domain where we can map the channel to. J isa K2L2 × KL selection matrix, which selects diagnoalelements from diag(a) to form a. (a) is derived from therelationship between Kronecker product and Khatri-Raoproduct.

�

To facilitate the channel estimation, we can discretizethe channel using an angular grid with the size of Gtx×Grx. Then the spatial frequencies wtxk` (related to AoDs)and wrxk` (related to AoAs) in (9) and (10) can be takenfrom the uniform grid of Gtx and Grx points:

DT →

1 1 · · · 1

1 ej2π1

Gtx · · · ej2π(Gtx−1)

Gtx

...... · · ·

...

1 ej2π1

Gtx(Ntx−1) · · · ej2π

(Gtx−1)Gtx

(Ntx−1)

.

(25)

DR →

1 1 · · · 1

1 ej2π1

Grx · · · ej2π(Grx−1)

Grx

...... · · ·

...

1 ej2π1

Grx(Nrx−1) · · · ej2π

(Grx−1)Grx

(Nrx−1)

.

(26)From Equation 24, the channel can be estimated as a

vector of the dimension GtxGrx × 1 (vec(H)). In orderto differentiate the estimated channel and the actualchannel a, we refer the estimated a as x. For easeof presentation, we will also refer the channel as theestimated path gains of x.

Replacing the vector a in the equation 20 with x, wehave the compressed sensing form r = Ax + e, wherer is the measurement results, A is the sensing matrix,x is a sparse vector to be reconstructed, and e is thenoise. After finding a with the estimation of x, H can beestimated from (23).Gtx and Grx determine the angular discretization

levels of the channel, which impact the accuracy of

8 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. XX, NO. XX, XX XXXX

estimating the angle of departure (AoD) and angle ofarrival(AoA) (whose values are continuous in reality oneach transmission path). They also impact the dimensi-ons of the basis domain (or the size of the reconstructiondictionary) in the channel reconstruction. The largerthe values of Gtx and Grx, the finer the resolution ofthe channel is discretized at, the more computationalcomplexity for the channel reconstruction, and the closerthe estimated AoD grid and AoA grid are to their actualangles. Gtx and Grx can be chosen according to thedesired estimation accuracy, conditions of the equipmentsuch as antenna hardware precision and the complexityin the channel estimation when the dimension is large.If Gtx = Ntx and Grx = Nrx, we have

Ψ = IDFT ∗Ntx∗ IDFTNrx

, (27)

where IDFTN denotes an N -dimensional IDFT matrix.

5.2 Sparse Virtual Channel Recovery

The sparseness of mmW channel can be exploited toreduce the number of beamforming pairs to train. Toenable the optimal beam alignment, the virtual mmWchannel can be more efficiently reconstructed with thetraining signals sent over a small number of directions.From (20), we know x (whose elements are the path gaincoefficients being vectorized) can be recovered using CStheory, with the following CS components:

1) Basis matrix: Ψ = D∗R ∗DT (Khatri-Rao product);2) Measurement matrix: Φ =

√A(UT ⊗VH) (Kronec-

ker product);3) Sensing matrix: A = ΦΨ =

√A(UT ⊗VH)D∗R ∗DT ;

The channel can be estimated by solving the followingoptimization problem from Equation (20):

min ‖x‖1, (28)subject to r = Ax + e, (29)

where ‖·‖1 denotes the `1-norm, a is the noise.To be consistent with the compressed sensing no-

tations introduced earlier, we now use the followingnotations:

min ‖x‖1, (30)subject to y = Ax + n, (31)

where ‖·‖1 denotes the `1-norm. Note the measurementsr is denoted as y now.

After recovering x, the virtual channel H can be esti-mated as in Equation (13). Existing recovery algorithmsof the compressed sensing, such as l1 minimization[36] and Cosamp [38], can be applied to recover x.However, from the compressed sensing theory, we knowthat directly recovering vector x from PQ number ofmeasurements in y may not be accurate if PQ is not bigenough.

6 BLOCK SPARSE CHANNEL RECON-STRUCTION AND BEAM-ALIGNMENT DESIGN

In the basic channel model we consider, there are Kclusters for transmission paths, each containing L paths.Kd out of K clusters are dominating, i.e., concentratingmost of the signal power. The path clustering bringsthe block sparsity characteristics to the coefficient vectora in (24) for the gain of the path estimated. That is,not only that a is sparse with only a small fractionof elements non-zero, but also the non-zero elementsare clustered. In simulations, we adopt the statisticalmmW channel model developed based on real-worldmeasurements in New York City [16] and also select thechannel parameters based on empirical values suggestedby the NYC model.

In order to improve the recovery performance, wepropose to exploit the block property of the sparse vectorx in the reconstruction problem in (30), which translatesto the path gain vector a in Section 5, with the followingdesign goals:

1) Reducing the number of measurements requiredfor the recovery, i.e., reducing the number of trai-ning beam pairs ( P and Q in (16));

2) Improving the channel reconstruction accuracy fora given number of measurements.

6.1 Block Sparse Property of MmW Channel

In CS-based channel estimation, the vector of the pathgain a of mmW channel is what we are most interestedin, where only a small fraction of the elements are non-zero. One of the effects of the path clustering featureof mmW channels is that it results in assembled non-zero elements in the path gain vector (Observation 1)and this additional block sparsity information can befurther exploited in CS reconstruction to improve thereconstruction performance.

Observation 1. The vector of the path gain a of mmWchannel presents block sparsity due to the path clustering.

Remarks: In [16], spatial statistical models of mmWchannels are derived from real-world measurements at28 and 73 GHz in New York City. It indicates thatin micro-cell level, receivers in typical measurementlocations experience a small number of path clusters,two to three being dominant. Moreover, within eachpath cluster, the angular spread is relatively small. Thecovariance matrix of the mmW channel is low-rank inthe sense that the paths are clustering into relativelysmall and narrow beam clusters. The authors in [16]also studied the distribution of energy fraction in spatialdirections and the results show that for 28GHz NYCchannel, 3 dimensions of spatial directions can capture95% of the channel energy for a 4 × 4 uniform planararray (which has a dimension of 16).

ZHAO ET AL.: COMPRESSED BEAM ALIGNMENT WITH OUT-OF-BAND ASSISTANCE IN MILLIMETER WAVE CELLULAR NETWORKS 9

6.2 Block-Sparse Channel State ReconstructionIn our framework, the channel states to be recovered byCS (path gains) exhibit the characteristics of the blocksparsity, as shown in Figure 4, which motivates us touse the block sparse properties to help reconstruct thechannel signal for better beam-alignment.

6.2.1 Weighted RecoveryTo exploit the block structure, instead of solving an`1 optimization problem in (2), we instead solve thefollowing problem:

minn∑

i=1

‖Xi‖2

subject to Ax = y,x = [X1,X2, ...,Xn], (32)

where Xi = x(i−1)d+1:id, , as shown in Figure 4.In [41], an algorithm has been proposed to solve

the problem in (32). Instead of equally treating all thechannel blocks, we would like to take advantage of thechannel estimation and learning after each iteration ofproblem solving to further increase the reconstructionquality. We give a block with higher channel estimatedenergy a lower weight so its recovered gain value is lessrestricted. We solve the following problem in our design:

minn∑

i=1

wi‖Xi‖2

subject to Ax = y,x = [X1,X2, ...,Xn], (33)

where wi is a weighted factor for block Xi.

6.2.2 Channel State Estimation with Co-located LegacyAntennasA major challenge in estimating the channels of themmW band is its narrow angular spread of signal andthus the need of performing the channel measurementsin a large number of directions. At the BS end andmobile end of mmW cellular networks, it is likely thatthere are some co-located legacy antennas (e.g. from3G cellular networks), which operate in much lower-frequency band than mmW, e.g., L Band: 1 to 2 GHzor Ultra High Frequency: 300MHz to 3GHz (for easeof presentation we will refer to it interchangeably aslow-frequency band, legacy band or low-band). Sincethe number of antennas in the legacy band is typicallysmall, these co-located TX and RX antennas can performthe direction search in a low cost fashion, i.e., even anexhaustive search will not bring unbearable overhead.As the mmW interface and co-located legacy interfaceshare similar spatial environment, which leads to strongcorrelation between the mmW channel and the legacychannel, it would be helpful if we could exploit thechannel information from the low-frequency channelsto guide more intelligent beam direction finding in themmW band.

Due to the small angular spread in the mmW channel,if some clusters (blocks in sparsity) in the low-band

x1

xid−d+1xid−d+2

xid

xnd−d+2

Xi

x2

xd

xnd−d+1

Xn

xnd

X1

y

...

...

...

}}}

...

...

A1 AnAi

A1 — columns 1, 2, . . . , d

Ai — columns id− d+ 1, id− d+ 2, . . . , id

An — columns nd− d+ 1, nd− d+ 2, . . . , nd

= . . .

y =Ax =∑ni=1AiXi

. . .

Fig. 4. Block sparse model.

are dominant, it is likely that an overlapped mmWchannel has a good channel gain. Although signal pathsin the mmW channel and the low-frequency channelmay not be exactly the same, the information on thestrong channel in the low-frequency band can give arough information on which antenna direction range ismore likely to have better antenna gains in the mmWchannel. To provide a guide for more efficient mmWband beam training and channel estimation, we pro-pose to exploit the legacy band information with twotechniques: (1) legacy-band-assisted beam selection and(2) legacy-band-assisted channel reconstruction.

Legacy-band-assisted beam selection: We propose to usethe low-band information to efficiently compress theinitial range of beam selection for beamforming training.From the low-frequency channel information, one caneasily find the best low-band direction (beam) pair.For mmW band, instead of training within the wholeangular range at high cost, we propose to only randomlyselect beams within the best direction range detectedin low-band training. This will significantly reduce thetraining overhead and speed up the training process.On the other hand, given the same number of beamdirections to train, the beams selected with the guidanceof low-band results are likely to have better quality thanthose randomly selected from the whole angular space,thus improving the channel estimation and beamformingperformance.

After selected beams are trained, channel recon-struction is called for to estimate the channel, wherelegacy band information can be further exploited.

Legacy-band-assisted channel reconstruction: We proposeto use the direction-range quality from the low-bandas the initial weights wi in our proposed beam align-ment algorithm, and order the directions by their beamalignment gains. If a low-band direction overlapped bya mmW direction is detected to have a good channelquality, the weight of the corresponding mmW channelblock is set to a smaller value in Problem (33).

Our weighted-block-sparse recovery algorithm is fa-cilitated with the low-band information, as shown inAlgorithm 1. The main idea is to assign a weight to

10 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. XX, NO. XX, XX XXXX

each block and update the weights through iterativelysolving (33). For the blocks that are not likely to containmany non-zero elements, weights are increased, so thatthe values of these blocks can be reduced when solvingthe minimization problem in (33). For the blocks thatare likely to be non-zero, weights are reduced to relaxthose blocks. Our algorithm enhances the channel recon-struction performance from two aspects: (a) error/noisereduction by disregarding data from likely-zero blocksas in Algorithm 1; (b) accuracy improvement by upda-ting block weights through each iteration to concentratethe residual information to the likely-nonzero blocks toincrease the signal recovery quality.

From Algorithm 1 we can see that it learns from theiterations and updates the weights by self-adaptation,and this contributes to the fast convergence of the algo-rithms. The computational complexity of Algorithm 1 isO(n3), where n = N/d is the block sparsity level (thenumber of blocks), d is the size of a block, and N isthe size of path gain vector x that we try to recover inthis algorithm. In the conventional CS recovery basedon min-`1 [36], a typical convex optimization approachwithout considering block effect will incur a computa-tional complexity of O(N3) >> O(n3). Obviously, ourblock-based CS recovery has much lower complexity,thus can be completed with much shorter time. Ourperformance studies also demonstrate that our methodcan better capture the clustering features of mmWavechannels to achieve higher recovery accuracy.

6.3 Beam Alignment after Channel Estimation

Optimizing the beamforming gain is a critical goal ofbeam alignment, since a larger beamforming gain cantranslate to larger receiving power and better signal-to-noise ratio thus achievable transmission rate. Assumethere are I and J possible directions at TX and RX, re-spectively, after obtaining the channel estimation result,the optimal beam pair (uopt,vopt) can be determinedfrom Algorithm 2. The optimal transmission and recep-tion directions are the ones that can maximize the beam-forming gain. Instead of blind or purely random beamtraining, the estimated channel information is appliedin our scheme to guide the finding of the optimal beamdirection.

7 SIMULATIONS AND RESULTS

In this section, we will perform simulations to show theeffectiveness and efficiency of the proposed design.

We will compare the performances of the followingschemes:• CODEBOOK: Multi-resolution codebook-based

beam searching as in IEEE 802.11ad [22] withoutCS channel reconstruction.

• CS: Use a greedy CS reconstruction algorithm suchas [38] without block-sparsity to recover the chan-nel.

Algorithm 1 Reconstruction of block sparse signalsRequire:

Initialization.Measured vector y, size of blocks d, block sparsitylevel n and measurement matrix A.

Ensure:1: Low-band exhaustive search.

Each block Xi is given a weight wi based on the co-located low-land antenna search result.

2: Solve the following optimization problem usingsemi-definite programming

minx

n∑

i=1

wi‖Xi‖2

subject to Ax = y,x = [X1,X2, ...,Xn]. (34)

3: Reduce noise by updating y.Sort Xi such that ‖Xj1‖2 ≥ ‖Xj2‖2 ≥ ...‖Xjn‖2.Update A to be the submatrix of A containingcolumns of first (n− 1)d rows of A that correspondsto the blocks j1, ..., jn−1.Update y← Ax (Disregard the weakest block infor-mation from measurements.)

4: Update weights wi for each blockCalculate x = A−1y = [X1,X2, ...,Xn].For each block Xi, count the number of x entries thatare above a predetermined threshold as mi.Update the weight of each block, wi as follows:wi ←

∑ni=1mi

mi.

5: IterationIf the termination condition (e.g. allowed maximalnumber of iterations) is not met, go back to Step 2.Otherwise algorithm terminates.

6: OutputOutput the recovered signal x = [X1,X2, ...,Xn].

Algorithm 2 Beam AlignmentRequire: mmW channel estimation result x

Initialization.Ensure:

1: RX constructs channel H from x.2: RX determines the best beam pair as follows and

transmit the best beam pair to TX:(uopt,vopt) = argmaxu1,u2,...,uI ,v1,...,vJ

‖vHHu‖1 (vjis j-th RX beam direction).

3: TX and RX align their beams according to directionpair (uopt,vopt).

• BLOCK: Use an unweighted block-sparse-based re-construction algorithm [41] to recover channel.

• PROPOSED1: Use proposed weighted block-sparse-based reconstruction algorithm to recover channel,but without low-band block information (legacyband assistance). Block weights are initialized to thesame value for all blocks.

• PROPOSED2: Use proposed weighted block-sparse-

ZHAO ET AL.: COMPRESSED BEAM ALIGNMENT WITH OUT-OF-BAND ASSISTANCE IN MILLIMETER WAVE CELLULAR NETWORKS 11

based reconstruction algorithm with legacy-band-assisted reconstruction (but without legacy-band-assisted beam selection) to recover the channel.Block weights are initialized according to the legacy-band information.

• PROPOSED3: PROPOSED2 with legacy-band-assisted beam selection.

7.1 Simulation Settings

In the simulations, we consider a 28GHz mmW cellularnetwork, and assume the TX has 64 λ/2 uniform lineararrays, and the RX has 16 λ/2 uniform linear arrays. Wealso assume the numbers of colocated low-band TX andRX antennas are both 4. The mmW channel is generatedbased on the model discussed in Section 4. We alsoselect the channel parameters based on empirical valuesbased on real-world measurements in [16], with defaultKd = 2 dominating clusters, L = 16 transmission pathsin each cluster. We assume the transmission bandwidthis 0.5 GHz.

To evaluate the effectiveness of different beam-alignment schemes, we compare their losses in link gainfrom the optimal one with different ratios of directionssearched. The optimal beam direction is found throughthe exhaustive search of all possible beam pairs at highoverhead. A bigger loss reflects a larger reduction ofthe transmission rate from the optimal. To reduce thesearch range, CODEBOOK exploits a multi-level code-book. After each coarse resolution of beam search, onlythe beam pairs within the range detected to have the bestquality will be further trained. Although CS, BLOCK andour proposed schemes all train a subsect of beam pairsand exploit the channel information to guide the beamtraining, our proposed schemes further take advantageof the block sparse properties of mmWave channel andthe assistance from the legacy band to improve the beamalignment efficiency. We expect our scheme can achievecomparable performances as the optimal one with thesearching cost lower than that of other schemes.

The first metric we will use to evaluate the perfor-mance of a beam pair is the SNR degradation comparedwith the SNR value obtained at the optimal beam pair.We define the optimal SNR as Ropt = R(uopt,vopt) andthe actual SNR obtained from the estimated best beampair (u,v) as R(u,v). Then the SNR loss for this beampair in decibels is defined as:

Loss(dB) = −10 log10[R(u,v)

Ropt

]. (35)

The smaller the Loss (Loss is larger than 0 in definition),the better the beam pair selected.

The second metric we will evaluate is the Search Rate,which is defined as the number of measured subset ofbeam pairs (PQ) normalized to all the possible beampairs NtxNrx, that is:

Search Rate = PQ/(NtxNrx). (36)

The third metric we adopt is the actual data transmis-sion rate achieved by communicating over the beam pairsuggested by beam alignment.

10 15 20 250

5

10

15

20

Search Effectiveness

Search Rate (%)

Loss (

dB

)

CODEBOOK

CS

BLOCKPROPOSED1

PROPOSED2

PROPOSED3

Fig. 5. Search effectiveness: Loss.

10 15 20 250

1

2

3

4

5Effect of search rate on data transmission rate

Search Rate (%)

Data

tra

nsm

issio

n r

ate

(G

b/s

)

CODEBOOK

CS

BLOCKPROPOSED1

PROPOSED2

PROPOSED3

Fig. 6. Search effectiveness: data transmission rate.

7.2 Search EffectivenessWe use two metrics to evaluate the search effectivenessof beam alignment: Loss and data transmission rate.

Figures 5 and 6 show how beam alignment qualitychanges with different search rates for various beamalignment schemes. In Figure 5, among all the schemescompared, CODEBOOK is the most inefficient one withthe lowest SNR loss. It may filter out the optimal beamdirection in its coarse-level search and end up withthe selection of some sub-optimal ones. In contrast,CS, BLOCK and our proposed schemes can rely onthe estimated channel information to infer the optimalbeam direction, even if the optimal one hasn’t beentrained earlier or falls out of the best coarse beamrange suggested by the legacy channel. We also seethat BLOCK performs better than the conventional CSrecovery algorithm by considering the block propertiesof the sparse channel. At the same search rate, our threeproposed schemes always outperform CS and BLOCKwith lower SNR Loss. PROPOSED2 is able to achieve asmall Loss of 1.8dB when only measuring one fourthof all the possible beam pairs. In comparison withCODEBOOK, and PROPOSED1, PROPOSED2 achievesa Loss reduction by 69.44% and 78.42%, respectively.Compared to CS, the Loss reduction of PROPOSED2 is

12 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. XX, NO. XX, XX XXXX

Fig. 7. Cost efficiency: Loss for fixed search rate.

Fig. 8. Cost efficiency: required rate for a given Loss.

59.34%, whereas BLOCK and PROPOSED1 reduce theloss by 14.14% and 42.42%, respectively. Our proposedschemes not only take advantage of the sparse propertiesof mmW channel, but also exploit its block feature andlearning from channel estimation in the previous roundsof iteration to further improve the channel estimationaccuracy. In addition, PROPOSED2 outperforms PRO-POSED1 by making use of the information from thelow-frequency channels to facilitate the beam alignment.We also observe an additional improvement of about20.83% for PROPOSED3, compared with PROPOSED2.This improvement is mainly contributed by the guidanceof legacy band for the efficient selection of beams totrain, which further increases the accuracy of channelreconstruction. Compared with the scheme using CSdirectly, the loss reduction from the use of PROPOSED3is about 66.35% in the case of 25% sampling rate, andthe loss reduction is generally above 38% for all testedcases.

In Figure 6, as expected, the transmission rates incre-ase with the search rate. Our proposed schemes achievea drastic enhancement in the data transmission ratesfrom other methods. Compared with CODEBOOK, at thesearch rate 25%, PROPOSED1, PROPOSED2 and PRO-POSED3 obtain an improvement of data rate by 224%,255% and 267%, respectively. Compared with CS, theimprovements are 30.73%, 43.06% and 48.17%, whereasthe enhancements over BLOCK are 18.63%, 29.82% and34.46%.

2 3 4 5 60

2

4

6

8

10

12

14Effect of channel cluster sparsity

Number of power concentrating clusters

Loss (

dB

)

CODEBOOKCSBLOCKPROPOSED1PROPOSED2PROPOSED3

Fig. 9. Effect of channel clusters on Loss.

2 3 4 5 61

1.5

2

2.5

3

3.5

4

4.5

5Effect of channel cluster sparsity

Number of power concentrating clusters

Data

tra

nsm

issio

n r

ate

(G

b/s

)

CODEBOOKCSBLOCKPROPOSED1PROPOSED2PROPOSED3

Fig. 10. Effect of channel clusters on transmission rate.

7.3 Cost Efficiency

For a given number of Search Rate (this usually happensin a resource-limited network, where TX and RX mayonly be able to measure a limited number of beam pairs),the estimated best beam pair quality determines theeffectiveness of a beam alignment scheme. For a givenSNR Loss, a certain Search Rate requirement will needto be met. The required Search Rate in this case indicatesthe cost efficiency of the scheme. When more beam pairsare searched, the overhead will become higher (e.g. time,energy, computational complexity).

Figure 7 shows the advantages of our proposedschemes over CS and BLOCK. For a given number ofbeam pairs trained, our proposed schemes experiencemuch smaller Loss than CS and BLOCK. In Figure 8, wecompare the search rate needed by different schemes toachieve the same beam alignment quality. Our schemesrequire a smaller number of training signal transmissionsas compared to CS and BLOCK. At a Loss of 10dBachieved by CS, the search rate of our PROPOSED3 isonly 59% that of CODEBOOK, 65.48% that of CS, 74.32%that of BLOCK. Compared to the exhaustive search, ourrate is only 13.75%. The cost reduction can be hugeunder the circumstance that the number of possiblebeam directions is large.

7.4 Effect of Channel Clusters

The number of clusters that concentrate most of the sig-nal power in the channel impacts the block sparse level

ZHAO ET AL.: COMPRESSED BEAM ALIGNMENT WITH OUT-OF-BAND ASSISTANCE IN MILLIMETER WAVE CELLULAR NETWORKS 13

in the estimated path gain vector. The fewer the numberof power concentrating clusters, the sparser the channel.By varying the number of dominating clusters Kd whilekeeping the total number of paths KL unchanged, wefurther investigate the effect of channel clustering on theachieved Loss when the Search Rate is kept the same at25%. The results are shown in Figure 9.

In Figure 9, the cluster number has a significant im-pact on the channel estimation thus beam alignmentperformances. When the number of clusters increases(i.e., the channel becomes less sparse), the performancesof all schemes degrade with a larger Loss, as 25% SearchRate is not enough. Among all the schemes, CODEBOOKis impacted the most, because more clusters introducemore uncertainty. The optimal beam direction has a hig-her chance of being filtered out during the coarse-leveltraining. The performances of other schemes change lessdramatically, because they are able to estimate the actualchannel information to guide the beam alignment. Whenthere are 6 channel clusters, compared to CODEBOOK,CS and BLOCK have a loss reduction of 46.93% and54.43%, respectively, whereas Proposed1, Proposed2 andProposed 3 reduce the loss at 69.44%, 78.42% and 82.14%,respectively. Again, from PROPOSED1 to PROPOSED2then to PROPOSED3, we see consistently increase ofimprovement. Better exploiting the block sparsity, chan-nel knowledge and low-frequency band assistance, ourschemes can more effectively select the beams to train.

Figure 10 depicts the effects of clusters on the datatransmission rate. We observe that when the channelbecomes less sparse, the users will experience lower datatransmission rate as a result of the degraded qualityof the beam pair found in beam alignment. Comparedto CODEBOOK, CS, BLOCK, PROPOSED1, PROPOSED2and PROPOSED3 achieve rate improvements of 148%,173%, 224%, 255% and 267%, respectively.

8 CONCLUSION

This paper presents an efficient beam alignment schemefor the transmitter and receiver to jointly decide thebeam directions to combat the large path loss inthe mmW cellular networks. Unlike the conventionalscheme which searches through all the possible beampairs or a large volume of codebook at the cost of severedelay and overhead, to enable efficient beam alignment,our algorithm takes advantage of the mmW channelsparsity to enable efficient beam direction matching. Rat-her than just randomly selecting a small subset of beamdirections to train as done in existing CS-based channelmeasurement, to further reduce the beam alignmentoverhead and improve the gain, we exploit the blocksparse properties and learning of iterative channel esti-mation to more accurately estimate the mmW channel atlower cost. We also exploit the assistance from co-locatedlow-frequency antennas to guide the beam selection fortraining and further increase the channel reconstructionaccuracy thus beam alignment quality. Simulation results

demonstrate the significant advantages of our design inthe search effectiveness and cost efficiency.

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Jie Zhao received the Ph.D. degree in Electrical Engineering from theState University of New York at Stony Brook, New York, USA, and theB.S. degree in Telecommunications Engineering from Huazhong Univer-sity of Science and Technology, Wuhan, China. His research interestsinclude millimeter-wave communications, cognitive radio networks, aswell as networked sensing and detection.

Xin Wang received the B.S. and M.S. degrees in telecommunicationsengineering and wireless communications engineering respectively fromBeijing University of Posts and Telecommunications, Beijing, China, andthe Ph.D. degree in electrical and computer engineering from ColumbiaUniversity, New York, NY. She is currently an Associate Professor inthe Department of Electrical and Computer Engineering of the StateUniversity of New York at Stony Brook, Stony Brook, NY. Before joiningStony Brook, she was a Member of Technical Staff in the area of mobileand wireless networking at Bell Labs Research, Lucent Technologies,New Jersey, and an Assistant Professor in the Department of Com-puter Science and Engineering of the State University of New Yorkat Buffalo, Buffalo, NY. Her research interests include algorithm andprotocol design in wireless networks and communications, mobile anddistributed computing, as well as networked sensing and detection.She has served in executive committee and technical committee ofnumerous conferences and funding review panels, and serves as theassociate editor of IEEE Transactions on Mobile Computing. Dr. Wangachieved the NSF career award in 2005, and ONR challenge award in2010.

Harish Viswanathan is a CTO Partner in the Corp CTO organization.As a CTO Partner he advises the Corporate CTO on TechnologyStrategy through in-depth analysis of emerging technology and marketneeds. Harish Viswanathan joined Bell Labs in 1997 and has workedon multiple antenna technology for cellular wireless networks, mobilenetwork architecture, and M2M. He received the B. Tech. degree fromthe Department of Electrical Engineering, Indian Institute of Technology,Chennai, India and the M.S. and Ph.D. degrees from the School ofElectrical Engineering, Cornell University, Ithaca, NY. He holds morethan 50 patents and has published more than 100 papers. He is a Fellowof the IEEE and a Bell Labs Fellow.

Arjuna Madanayake is an Associate Professor at Florida InternationalUniversity (FIU) in Miami, Florida. He completed his Ph.D. and M.Sc.degrees, both in electrical engineering, from the University of Calgary,in Alberta, Canada. He obtained a B.Sc with Specialization in Electronicand Telecommunication Engineering from the University of Moratuwa,Sri Lanka, in 2002. Dr. Madanayake was a tenured faculty memberat the University of Akron, in Akron, Ohio, between 2010-2018 beforejoining the faculty at FIU in August 2018. His research interests are inarray signal processing, circuits, systems, electronics, fast algorithmsand computer architecture.

Guangxue Yue is a Professor in the College of Mathematical andInformation Engineering, Jiaxing University, Jiaxing, Zhejiang, China.He received the M.S. and Ph.D. degrees from Hunan University in2004 and 2012, respectively. His main research interests include cloudyconvergence and collaborative services, wireless mesh networks andmobile cloud computing.