Tracking Sparse mmWave Channel under Time Varying Multipath Scatterers
TTracking Sparse mmWave Channel under TimeVarying Multipath Scatterers (Invited Paper)
Veljko Boljanovic, Han Yan, and Danijela Cabric
Electrical and Computer Engineering Department, University of California, Los AngelesEmail: { vboljanovic, yhaddint } @ucla.edu, [email protected] Abstract —Due to severe signal attenuation at millimeter-wave(mmWave) frequencies large antenna arrays are required atboth base station and user equipment to achieve necessarybeamfoming gain and compensate for the signal power loss.The initial access and beamforming algorithms are typicallydesigned assuming sparsity of mmWave channels, resultingfrom a very few significant multipath clusters, and consideringfixed locations of terminals and scatterers. Channel trackingalgorithms have been proposed to account for channel variationsdue to user mobility. Existing works did not consider mobilityof the scatterers, which adds new challenges and opportunitiesinto a channel tracking problem. In this work, we consider amore realistic assumption of mobile scatterers and their impacton channel tracking algorithms. We propose a novel channeltracking algorithm that takes into account the dynamics ofcluster evolution, and adaptively tracks channel parameterswith the objective to reduce training overhead. We also proposea simple implementation of aperiodic tracking to accommo-date tracking to different channel variations. We analyze theperformance of the proposed tracking algorithm under highlydynamic channels, and compare it to existing channel trackingalgorithms with respect to tracking accuracy, achievable rate,and required training overhead, when aperiodic and periodictrackings are used.
I. I
NTRODUCTION
Communication over millimeter-wave (mmWave) fre-quency bands is considered as one of the key featuresin the fifth generation New Radio (5G-NR) standard [1].Although mmWave frequencies offer abundant spectrum andhence high data rates, their use comes at a cost of lessfavorable propagation conditions [2]. Both the transmitterand the receiver are required to use large antenna arrayswith up-to-date channel state information (CSI) to achieve thebeamforming gain and to compensate for severe propagationloss. Conventionally, the CSI is obtained through periodicchannel estimation, whereas a number of advanced channeltracking techniques have been proposed recently to reduceinevitable training overhead. It is commonly assumed thatmmWave wireless channel is sparse, meaning that there area few significant signal paths coming to the receiver. Thus,the problem of acquiring the CSI for narrowband communi-cations reduces to estimation and tracking of few parametersthat describe those paths - angles of departure (AoD), anglesof arrival (AoA), and path gains. Previous works on channeltracking can be roughly divided into two groups based
This work is supported by NSF under grant 1718742. on assumed spatial consistency model in the channel. Thefirst group does not consider channel geometry and oftenassumes that AoD, AoD, and path gains change according tothe certain statistical models, e.g., Gauss-Markov processes,while the other line of works takes geometry into accountand models changes of channel parameters accordingly.A number of papers on channel tracking assuming statisti-cal models was published recently [3]–[5]. In [3], authorsapply rotation matrices onto old beamforming vectors tocreate new ones for tracking. Moreover, [3] considers hybridanalog-digital beamforming and reduces overhead by usingmultiple RF chains for channel sounding. In [4], [5], channeltracking using Kalman filter is investigated. This approachprovides continuous channel tracking, but it is sensitive tonoise and estimation error accumulates fast. Recent worksalso consider channel geometry and model parameter changesin a deterministic way [6], [7]. The work [6] studies wide-band channel tracking using geometry-based spatial channelmodel (GSCM) which assumes static scatterers and constantangles of departure [8], while [7] uses a ray-tracing tool toaccurately model mobility effects and spatial consistency.Recent mmWave measurement campaign in [9] showedthat a significant part of received signal energy comes asa reflection from mobile objects in the channel, includingcars and pedestrians, hence, these objects can be consideredas mobile scatterers. In urban environment with densely de-ployed base stations, distances between the base station, themobile stations, and scatterers are small, so mobile scattererscan cause significant AoD/AoA changes even for static users.Due to fairly predictive movements of scatterers withoutsudden direction changes, AoD and AoA will tend to changein one direction with high probability. However, AoD andAoA can experience changes in the other direction due to un-expected array orientation changes, e.g., due to rotations. Inthis work, we propose a new channel tracking algorithm thattakes advantage of predictive AoD/AoA changes in GSCMand remains robust to potential changes in the other direc-tion. Similar to [10], the proposed algorithm initially usescompressed sensing (CS) approach with pseudo-random se-quences for beamformer precoding and combining, but laterit exploits knowledge of the previous parameter estimates tofurther reduce channel tracking overhead. Unlike beam track-ing in the Third Generation Partnership Project New Radio(3GPP-NR) Standard [11], where reduction in overhead is a r X i v : . [ ee ss . SP ] D ec T frame Dataphase Dataphase DataphaseChannel tracking
Initial channelestimation M cs M pcs Fig. 1. The frame structure used in this work. achieved by steering certain number of beams around theprevious estimates, the proposed algorithm achieves overheadreduction by using projections of pseudo-random sequencesaround the previous estimates. We also propose a simpleimplementation of aperiodic channel tracking, and use it inconjunction with the proposed algorithm and algorithms from[10] and [11] to compare their AoD/AoA tracking accuracy,achievable rate with aperiodic channel tracking, and overheadwhen aperiodic channel tracking is used. In addition, wecompare these algorithms in terms of achievable rate whenperiodic tracking with maximum overhead is used.The rest of the paper is organized as follows. In Section II,we introduce the channel and system models. Section III de-scribes the proposed algorithm and a simple implementationof aperiodic tracking. In Section IV, we compare the pro-posed algorithm with existing channel tracking algorithms.Finally, conclusions are drawn in Section V.
Notations:
Scalars, vectors, and matrices are denoted bynon-bold, bold lower-case, and bold upper-case letters, re-spectively, e.g., h , h and H . Transpose, conjugate transpose,and pseudo-inverse are denoted by ( . ) T , ( . ) H , ( . ) + , respec-tively. The l -norm of a vector h is denoted by || h || . Operatordiag ( A ) aligns diagonal elements of A in a vector.II. C HANNEL AND S YSTEM M ODELS
We consider a narrowband mmWave MIMO system withuniform linear arrays (ULA) consisting of N BS and N MS antenna elements at the base station (BS) and the mobilestation (MS), respectively. There are L distinct single-raysignal paths that are coming to the mobile station in thedownlink (DL). We focus on the azimuth plane, and thechannel matrix H is H = (cid:114) N BS N MS L L (cid:88) l =1 g l a MS ( φ l ) a HBS ( θ l ) (1)where g l = e jψ l , θ l ∈ [ − π , π ] , and φ l ∈ [ − π , π ] , representthe complex path gains, AoDs, and AoAs, respectively. Phase ψ l in the complex gain is a uniform random variable in theinterval [ − π, π ) . The vectors a BS ( θ l ) ∈ C N BS and a MS ( φ l ) ∈ C N MS are the spatial responses corresponding to the BS andthe MS, and they are defined as a BS ( θ l ) = [1 , e − jπ sin( θ l ) , ..., e − j ( N BS − π sin( θ l ) ] T √ N BS , (2) a MS ( φ l ) = [1 , e − jπ sin( φ l ) , ..., e − j ( N MS − π sin( φ l ) ] T √ N MS . (3) For the sake of simplicity, we can assume that L = 1 , i.e.,there is one Non-Line-of-Sight (NLoS) path coming to thereceiver from a mobile scatterer. Therefore, we can removesummation and index l from (1) and use notation H ( g, θ, φ ) to indicate that H depends on parameters g , θ , and φ .We consider DL communication with the frame structuredepicted in Fig. 1. Each time frame is of length T frame andit consists of 10000 slots. At the beginning of the frame,channel parameter estimates ˆ g , ˆ θ , and ˆ φ are acquired throughthe initial channel estimation, fed back to the BS during ashort period reserved for control messages , and then usedfor beam steering in data communication phase. It is assumedthat the channel does not change during the initial channelestimation and tracking periods. However, due to high userand scatterer mobilities, channel parameters g , θ , and φ cansignificantly change during the data communication phase,therefore, beam tracking and data communication slots mustbe interleaved, as depicted in Fig. 1, to maintain the requiredlink budget. After each channel tracking period, channelparameter estimates are fed back to the BS, and beamsteeringvectors are updated for the following data phase. Channeltracking can be done in a periodic or aperiodic manner[13], i.e., period T can be fixed or time-varying withinthe frame. In this work, we propose and explain a simpleimplementation of aperiodic tracking in Sec. III-C.We focus on phase-only analog beamforming where boththe BS and the MS have only one RF chain, and theycan steer beams in Q BS and Q MS angles from sets A BS = {− π + ( Q BS − i ) πQ BS | i = 1 , , ..., Q BS } and A MS = {− π +( Q MS − i ) πQ MS | i = 1 , , ..., Q MS } , respectively. At k -th timeslot, the BS uses precoding vector f k ∈ C N BS and the MS usescombining vector w k ∈ C N MS for beam steering. Elementsof f k have magnitudes √ N BS , and elements of w k havemagnitudes √ N MS . Note that scaling factor √ N BS N MS in (1)compensates for unit beamforming gains while the channelstill remains normalized. Assuming perfect synchronizationand unit pilot symbol, the received signal at the k -th slot canbe expressed as y k = w H k Hf k + n k , (4)where n k ∈ CN (0 , σ n ) is post-beamforming noise.III. P ROPOSED T RACKING A LGORITHM
We propose a two-step tracking algorithm based onpseudo-random precoding and combining vectors. The firststep is the initial channel estimation based on CS approach[10], while the second step includes a novel channel trackingalgorithm that utilizes information about previous estimatesof channel parameters to further reduce overhead.
A. Initial Channel Estimation
We exploit the sparsity of mmWave channel and use CSapproach for initial parameter estimation. In the k -th slot We assume non-standalone (NSA) mmWave system architecture withLong Term Evolution (LTE) link between the BS and the MS [12] usedfor control information exchange for a short period of time after channelestimation/tracking. a) (b)
Fig. 2. Beam pattern comparison. (a) Normalized quasi-omnidirectionalbeam pattern generated by a random vector. (b) Normalized beam patternof projection of a random vector onto column space of F with ˆ θ t − = 0 ◦ of the initial channel estimation, the BS uses precodingvector f k and the MS uses combining vector w k , whereelements of both vectors are drawn randomly from the set {± ± j } . For normalization purposes, vectors f k and w k aremultiplied by √ N BS and √ N MS , respectively. An example ofquasi-omnidirectional beam pattern generated by f k or w k ispresented in Fig. 2(a). By using superscript t in g t , θ t , and φ t to denote estimation/tracking period, the received signalafter M cs slots (measurements) at t = 1 is y = diag ( W Hcs H ( g , θ , φ ) F cs ) + n , (5)where F cs = [ f , ..., f M cs ] , W cs = [ w , ..., w M cs ] , and n ∈ C M cs is a Gaussian random vector with CN ( , σ n I M cs ) .By defining z ( θ, φ ) = diag ( W H H ( θ, φ ) F ) , the expressionbecomes y = g z ( θ , φ ) + n . (6)The maximum likelihood (ML) estimates of parameters canbe found as ˆ g , ˆ θ , ˆ φ = argmin g,θ,φ (cid:107) y − g z ( θ, φ ) (cid:107) . (7)Using least squares (LS), we estimate the path gain g as ˆ g = y H z ( θ, φ ) (cid:107) z ( θ, φ ) (cid:107) , (8)for any angles θ and φ . If (8) is substituted in (7), the MLangle estimates are calculated as ˆ θ , ˆ φ = argmax θ ∈A BS ,φ ∈A MS y H z ( θ, φ ) (cid:107) z ( θ, φ ) (cid:107) . (9)The ML estimation in (9) assumes that the MS have knowl-edge of F cs , which is obtained just once during the firstcontrol information exchange and then reused in all timeframes. The total number of measurements M cs needed forreliable parameter estimation in the CS scales linearly withthe number of paths L [10]. In this work, we assume thatestimation is reliable if the root mean square error (RMSE)of AoD/AoA estimation vanishes at dB signal-to-noise ratio(SNR), and our simulations show that M cs = 45 is enoughfor reliable estimation when N BS = N MS = 32 . B. Channel Tracking: Projected Compressed Sensing (PCS)
The proposed algorithm relies on the previous angle esti-mates ˆ θ t − and ˆ φ t − from tracking period t − to trackchannel parameters because the current angles θ t and φ t are close to the previous ones with high probability. Since (a) (b) , , Fig. 3. Scanned beamspace. (a) Scanned beamspace at the BS withsgn (ˆ θ t − − ˆ θ t − ) = 1 , i.e., with counter-clockwise angular change. (b)Scanned beamspace at the BS with sgn (ˆ θ t − − ˆ θ t − ) = − , i.e., withclockwise angular change. ˆ θ t − and ˆ φ t − are known, there is no need to use quasi-omnidirectional beam patterns for tracking, and beamspacearound ˆ θ t − and ˆ φ t − can be scanned instead. The trackingprocedures at the BS and the MS are identical, and wedescribe just the former for the sake of brevity.We first take a pseudo-random vector f from the initialchannel estimation. Using the previous estimate ˆ θ t − wecreate four matrices F , F , F , and F , in the waydescribed in (10) and (11). The angular shift of πN BS makes thecolumns of these matrices approximately orthogonal, sincefor large antenna arrays a BS ( α ) H a BS ( α + πN BS ) ≈ for any α ∈ [ − π, π ) . The additional angular shifts for F , F , and F , are defined as δ F = sgn (ˆ θ t − − ˆ θ t − ) πN BS , δ F = sgn (ˆ θ t − − ˆ θ t − ) πN BS , and δ F = sgn (ˆ θ t − − ˆ θ t − ) πN BS ,respectively. The function sgn ( x ) is a sign function of x , andit takes value if x ≥ , and − otherwise.The vector f is projected onto column spaces of all fourmatrices, i.e. four new vectors are obtained as follows f = F F +1 f , f = F F +2 f , f = F F +3 f , f = F F +4 f . (12)Since we focus on phase-only analog beamforming, magni-tude of each element in f , f , f , and f must be scaled to √ N BS , which also ensures that all vectors have unit norm.An example of the beam pattern of f is presented Fig. 2(b).Due to nearly orthogonal columns of F , most of the signalenergy is in three distinct beam lobes. The beam patternsfor f , f , and f , look similar, but are rotated for δ F , δ F , and δ F , respectively. Note that these rotations dependon sgn ( x ) function, i.e., on estimated direction of angularchanges based on the last two tracking periods. Illustrationsof scanned beamspace with sgn (ˆ θ t − − ˆ θ t − ) = 1 andsgn (ˆ θ t − − ˆ θ t − ) = − are provided in Fig. 3(a) andFig. 3(b). The rotations provide asymmetric scanning aroundthe last estimate ˆ θ t − by ensuring that an additional part ofbeam space in the estimated direction of angular change (thepart colored in red) is scanned.The precoding matrix for slots (measurements) is con-structed as F pcs = [ f , f , f , f ] . After a similar procedure,the matrix W pcs is obtained, and the received signal at t -thestimation/tracking period is y = diag ( W Hpcs H ( g t , θ t , φ t ) F pcs ) + n , (13)where n ∈ CN ( , σ n I ) . However, due to pseudo-randomness in f and imperfect orthogonality among columns = (cid:20) a BS (ˆ θ t − ) , a BS (cid:18) ˆ θ t − + 2 πN BS (cid:19) , a BS (cid:18) ˆ θ t − − πN BS (cid:19)(cid:21) (10) F i = (cid:20) a BS (ˆ θ t − + δ F i ) , a BS (cid:18) ˆ θ t − + 2 πN BS + δ F i (cid:19) , a BS (cid:18) ˆ θ t − − πN BS + δ F i (cid:19)(cid:21) , i = 2 , , (11)of F i , i = 1 , , , , beam patterns of projections can haveless than three distinct lobes, which causes gaps in thescanned beamspace. In addition, most of the signal energy isdistributed in up to three directions and full beamforminggain is not achieved. Therefore, more than one pseudo-random sequence must be projected to increase diversity.Our simulations show that projections of five sequences areenough to achieve vanishing AoD/AoA RMSE at SNR =0 dB. In other words, reliable estimation requires precodingmatrix F pcs ∈ C N BS × M pcs with M pcs = 20 precoding vectors,i.e., F pcs = (cid:2) f , f , f , f , f , f , f , f , ..., f , f , f , f (cid:3) ,where the superscripts denote pseudo-random sequences andthe subscripts denote projections. Finally, the channel param-eter estimates ˆ g t , ˆ θ t , and ˆ φ t , can be found from the receivedsignal y ∈ C M pcs using the ML approach described in (6)-(9). Note that the BS does not have to share informationabout F pcs used in the next tracking period with the MS,since F pcs can be created using the AoD estimate ˆ θ t and fiveout of M cs pseudo-random precoding sequences used in theinitial channel estimation. C. Implementation of Aperiodic Tracking
The 3GPP-NR Standard supports the followingset of possible tracking periodicities: S T = { , , , , , , , } slots [11].In this work, we propose a simple technique for aperiodictracking where period T does not remain fixed over a periodof one frame, but it rather changes adaptively by takingdifferent values from S T . We first define the maximumAoD/AoA change γ max that can be tolerated as a half of dB beamwidth. For example, for large antenna arrayswith elements, this value is approximately γ max = 2 . ◦ .After the initial beam training, the first period T shouldtake smaller values from S T , e.g., one of the first fourvalues, to avoid channel changes much greater than γ max .In each following tracking instance, period T t , t = 2 , ... is computed according to the ratio between γ max and themaximum estimated angular change ∆ t T t Z + = (cid:24) γ max ∆ t T t − (cid:25) (14)where ∆ t = max {| ˆ θ t − ˆ θ t − | , | ˆ φ t − ˆ φ t − |} . The function (cid:100) x (cid:101) rounds x up to the closest integer. Since T t Z + in (14)can be any positive interger, it has to be quantized to one ofthe values from S T . We define a range around each value We scale periodicities by 14 since the notion of slot in [11] is differentand it has 14 symbols. from S T using the midpoints between adjacent values, andwe quantize T t Z + in the following way T t = , if ≤ T t Z + < , if ≤ T t Z + < ... , if ≤ T t Z + . (15)If the previous estimates of AoD and AoA are the same astheir current estimates, i.e., if ∆ t = 0 , then T t = 2 T t − assuming that T t − < . Information about the currentperiod T t is fed back to the BS as a part of controlinformation exchange. Hence, both the BS and the MS knowwhen the next tracking is going to be triggered.IV. N UMERICAL E VALUATION
In this section, we numerically evaluate the proposed chan-nel tracking algorithm and compare it to two existing channeltracking algorithms including beam-sweeping from 3GPP-NR Standard [11] and compressed sensing based tracking[10]. Although the 3GPP-NR Standard sets the maximumnumber of beam pairs for beam-sweeping to 64, the initialchannel estimation with large antenna arrays and analogbeamforming requires even higher number of beam pairsdue to small beam width . We exclude the initial channelestimation and compare these algorithms based on theirchannel tracking performance in order to make requiredoverheads comparable.The 3GPP-NR Standard does not specify the maximumnumber of directions that is scanned at the BS/MS duringthe channel tracking, and in [11] the authors consider 4directions around the previous angle estimate as an option.However, our simulations show that the tracking performanceis significantly boosted if an additional direction along theprevious estimate is scanned, thus, we use M bs = 25 beampairs for beam-sweeping with resolution of 32 possibleangles at both the BS and the MS. As discussed in Sec. III, forvanishing AoD/AoA RMSE at SNR = 0 dB, the compressedsensing approach and the projected compressed sensing re-quire M cs = 45 and M pcs = 20 slots, respectively.We consider a communication system with N BS = 32 and N MS = 32 antennas at the BS and the MS, respec-tively. Both the BS and the MS have angular resolutionsof Q BS = Q MS = 256 angles. Since there is L = 1 path,we assume without loss of generality that the path gain g = e jψ preserves unit magnitude and randomly changes thephase ψ during the data communication phase. We considertwo piece-wise linear models of AoD/AoA changes, and we The previous works usually assumed N BS N MS beam pairs. The number of beam pairs is equal to the number of slots in this work. t [ms] A ng l e [ deg r ee ] AoA change AoD change Beam-sweeping CS PCS (a) The first model of AoD/AoA changes. t [ms] A ng l e [ deg r ee ] AoA change AoD change Beam sweeping CS PCS (b) The second model of AoD/AoA changes.Fig. 4. The AoD/AoA change models used in this work, along with timeinstances when algorithms perform aperiodic tracking. extend them by adding Gaussian random variables in bothmodels. In the first model, AoD θ k and AoA φ k at k -th slotare given by θ k = θ k − + ∆ θ , (16) φ k = φ k − + ∆ φ , (17)where ∆ θ = 10 ◦ , ∆ φ = 10 ◦ , Θ ∈ N (0 , − ) , and Φ ∈N (0 , − ) . In the second model, the angles at k -th slot aredescribed as in (16)-(17), but with ∆ θ = 5 ◦ and ∆ φ definedas ∆ φ = ◦ , if ≤ k < − ◦ , if ≤ k < ◦ , if ≤ k ≤ . (18)Note that we allow AoD/AoA changes across all timeslots just to ensure equal angle changes for all algorithmswhen aperiodic tracking is used. When channel tracking isperformed, angles are assume to be fixed, i.e., we use θ k − and φ k − corresponding to the last data communication slot.Described AoD/AoA change models are depicted in Fig. 4along with time instances when different algorithms triggeraperiodic tracking at SNR = 0 dB. We set T = 560 and γ max = 2 . ◦ slots for all algorithms, and then adaptivelychange T t for t = 2 , , ... , as described in Sec. III-C. Forthe first model, period T t should remain constant within theframe due to nearly linear angle changes. This happens forthe CS-based and PCS-based algorithms with high probabil-ity, while beam-sweeping fails to properly trigger channeltracking. Due to randomness in the model, channel trackingtime instances can be offset or differently distributed withinthe frame for all three algorithms. The second model requiresfrequent channel tracking for fast changes, and the frequencyreduces as the angle changes become slower. Again, the -15 -10 -5 0 5 10 15 SNR [dB] -1 A o A R M SE [ deg r ee ] Beam-sweeping Compressed sensing Projected CS
Fig. 5. AoA RMSE of the three algorithms for different SNR levels.
CS-based and the PCS-based algorithms perform better thanbeam-sweeping, but their channel tracking time instances canslightly vary due to randomness.Better tracking triggers of the CS-based and the PCS-based algorithms come from their better AoD/AoA trackingaccuracy as compared to that of beam-sweeping. In Fig. 5,we compare algorithms in terms of AoA tracking accuracyand express the result in terms of RMSE of AoA estimationfor 300 Monte Carlo (MC) runs at different SNR levels. Theresult for the AoD RMSE is similar, and we omit to include itfor brevity. The PCS-based algorithm has lower AoA RMSEthan the CS-based algorithm in low SNR regime, but bothRMSEs experience floor due to finite angular resolution Q MS .Beam-sweeping retains high AoA RMSE regardless of theSNR level due to small number of scanned directions andcoarse angular resolution.In Fig. 6, we compare spectral efficiencies (SE) of thealgorithms as functions of time, averaged over 300 MC runsat SNR = 0 dB. We also find average required overheads o bs , o cs , and o pcs , when aperiodic tracking is used in conjuctionwith beam-sweeping, the CS-based algorithm, and the PCS-based algorithm, respectively. We assume that ˆ θ k = θ k = 12 ◦ and ˆ φ k = φ k = 15 ◦ for k = 0 , i.e., perfect AoD/AoAestimates are available at the beginning of the frame. With thefirst model of angular changes, the PCS-based algorithm hashigher spectral efficiency than the CS-based algorithm, andboth of them are better than beam-sweeping which suffersfrom coarse angular resolution. Beam-sweeping algorithmwaits until the angles significantly change, and then it updatesAoD and AoA. Multiple notches in spectral efficiency of theCS-based and the proposed algorithms come from the factthat in a number of MC runs channel tracking periods canbe offset, as discussed earlier. The PCS-based and the CS-based algorithms have similar spectral efficiencies when thesecond model of angular changes is used. Their deep notchat the beginning is a result of choosing T = 560 slots whichcan be too long for fast angular changes. Nevertheless, bothalgorithms can recover and adapt T t for the future channeltracking. Beam-sweeping is unable to successfully track fastAoD/AoA changes, which results in low spectral efficiency.In order to highlight advantage of the proposed algorithm,we compare average spectral efficiencies within a frame atSNR = 0 dB when periodic tracking is used and maximumoverhead is predefined. For given maximum overhead o max ,we determine the maximum number of channel trackingperiods within a frame for each algorithm, i.e., we find R bs , R cs , and R pcs , and then we choose correspondingtracking periodicities T bs , T cs , and T pcs from the set S T . t [ms] SE [ b / s / H z ] Beam-sweeping Compressed sensing Projected CS o cs =2.16%o pcs =1%o bs =1.39% (a) Spectral efficiency with the first model of AoD/AoA changes. t [ms] SE [ b / s / H z ] Beam-sweeping Compressed sensing Projected CS o cs =4.32%o pcs =1.88% o bs =2.85% (b) Spectral efficiency with the second model of AoD/AoA changes.Fig. 6. Average spectral efficiencies during a period of one time frame. We calculate time offset before the first channel tracking as T offx = (cid:24) T − ( R x − T x (cid:25) , where x = { bs , cs , pcs } . In Fig. 7,we show results for the first model of AoD/AoA changesand observe that the proposed algorithm achieves the highestaverage SE regardless of given maximum overhead. Theperformance with aperiodic tracking tracking is included inthe figure, and we observe that aperiodic tracking tendsto optimize performance by increasing the average spectralefficiency and decreasing required overhead at the same time.In Fig. 8, we show effective spectral efficiency (ESE)for the first model of AoD/AoA changes, and we note thatsimilar results are obtained for the second model. The ESEis calculated for each algorithm in the following wayESE ( N ) = max ( SE ,o max ) (1 − N o max ) SE , (19)where N is the number of users and ( SE , o max ) is a 2-tuplethat represent calculated pairs from Fig. 7. The PCS-basedalgorithm achieves the highest ESE regardless of the numberof users in the network. It is worth noting that we consideredan extreme case when all mobile stations require channeltracking at different time instances and that the CS-basedalgorithm would perform better if multiple users were trackedat the same time. V. C ONCLUSIONS
We proposed a new channel tracking scheme that utilizesknowledge of the previous AoD/AoA estimates to asym-metrically scan beamspace by projecting pseudo-random se-quences onto it. A simple implementation of aperiodic chan-nel tracking was proposed and proved to provide high spectralefficiency with low overhead. Our results showed that thePCS-based algorithm is advantageous over beam-sweepingand the CS-based algorithm in terms of AoD/AoA accuracy,achievable spectral efficiency, and required overhead. In thefuture work, we will design a tracking algorithm that predicts o max [%] A v e r age SE [ b / s / H z ] Beam-sweeping Compressed sensing Projected CS
Aperiodic AperiodicAperiodic
Fig. 7. Average SE within a frame for the first model of AoD/AoA whenperiodic tracking with different maximum overheads is used.
10 20 30 40 50 60
Number of users N
ESE [ b / s / H z ] Beam-sweeping Compressed sensing Projected CS
Fig. 8. Average ESE within a frame for the first model of AoD/AoA changesand different number of users N . the next AoDs and AoAs and thus significantly reduces thetracking overhead. R EFERENCES[1] 3GPP, “NR - Overall description, Stage-2, TS 38.300,” Dec. 2017.[2] T. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. Wong,J. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobilecommunications for 5G cellular: It will work!”
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