Fast Beam Tracking for Reconfigurable Intelligent Surface Assisted Mobile mmWave Networks
FFast Beam Tracking for Reconfigurable IntelligentSurface Assisted Mobile mmWave Networks
Xiaowen Tian and Zhi Sun
Abstract —Millimeter wave (mmWave) communications arevulnerable to blockages and node mobility due to the highlydirectional signal beams. The emerging Reconfigurable IntelligentSurfaces (RISs) technique can e ff ectively mitigate the blockageproblem by exploring the non-line-of-sight (NLOS) path, wherethe beam switching is realized by digitally configuring the phasesof RIS elements. To date, most e ff orts have been made in thestationary scenario. However, when considering node mobility,beam tracking algorithms designed specifically for RIS areneeded in order to maintain the NLOS link. In this paper,a fast RIS-based beam tracking algorithm is developed bypartly transforming the large amount of signaling time intothe calculation happens at base station in a mmWave systemwith mobile users. Specifically, the di ff erential form of optimalRIS configuration is exploited as the updating beam trackingparameter to avoid complex channel estimation procedure. TheRIS-based beam tracking problem is then transformed into anoptimization problem whose solution is found by a calculation-based search. Finally, by training on a small set candidate, RIS-based beam tracking is realized. The e ff ectiveness and e ffi ciencyof the proposed RIS-based beam tracking algorithm is evaluatedby simulations. It shows that the proposed algorithm has near-optimal performance with dramatic savings in terms of signalingtime. Index Terms —Beam tracking, Millimeter wave (mmWave)communications, Node mobility, Reconfigurable Intelligent Sur-faces (RISs), RIS beamforming, RIS configuration
I. I ntroduction
Operating at the vast unused millimeter wave spectrum,millimeter wave (mmWave) systems have become a promis-ing technique in the fifth-generation (5G) communicationsto overcome the spectrum congestion problem. Combiningwith beamforming techniques in multiple antenna systems,mmWave signals are of high directionality, i.e., having muchnarrower beams, because of its much shorter wavelengthcomparing with conventional below 6GHz signals [1]. There-fore, despite the advantages, mmWave signals are sensitive toblockage and node mobility at the same time [2].The Reconfigurable Intelligent Surfaces (RISs) [3]- [5],working by digitally configuring the phases of the largenumber of elements on them, o ff er a solution to the problem ofblockage in mmWave systems. It is usually believed that thewireless communication channels are fixed and can only becompensated by the design techniques and signal processing This work was supported by the National Science Foundation (Grant No.1652502).Xiaowen Tian and Zhi Sun are with the Department of Electrical En-gineering, University at Bu ff alo, Bu ff alo, NY 14260 USA (E-mail: xi-aowent@bu ff alo.edu; zhisun@bu ff alo.edu).This work has been submitted to the Elsevier Computer Networks forpossible publication. Copyright may be transferred without notice, after whichthis version may no longer be accessible. procedures at the transceiver sides. However, RISs rise as thekey of the roadmap to smart radio environment because thatthey can be programmed to reflect the electromagnetic wavesto the unnatural directions [6]. By adding an RIS to the point-to-point mmWave communication system, an additional non-light-of-sigh (NLOS) path / channel is o ff ered as is shown inour system model in Fig. 1. The e ff ect of manipulating thewireless communication channels by configuring the phasesof RISs is proved by experiments on prototypes [7]- [10],which provides the foundation of the multiple literatures ondesigning RISs in mmWave systems in order to overcome thechallenge of blockage in mmWave systems.Although adding a RIS into the mmWave systems creates anadditional path and overcome the blockage issue, the supportfor mobile users is also essential, especially with the rapiddevelopments of autonomous vehicle techniques [15], [16].One solution to user mobility in mmWave systems is adoptingbeam tracking algorithms [11]- [17], which automaticallyswitch the paired beams in order to maintain the qualityof communication link above a certain threshold. However,existing beam tracking algorithms in mmWave systems cannot be used directly with RIS-assisted systems. For example,turning on one element of RIS [11] is not able to have largeenough received signal strength to be detected. In addition, thelarge number of RIS elements will deteriorate the performanceof Kalman filter-based tracking algorithms [14], or makethe complexity of beam tracking algorithms intolerable [12].Therefore, beam tracking algorithms designed specifically forRIS-assisted mmWave systems are needed.One natural idea for RIS-based beam tracking algorithmsis to estimate the AP-RIS-UE channel first and configure theRIS elements accordingly. However, many literature on RIS-assisted channel estimation considers the static case, wherethe positions of the three parties remain unchanged [18]- [20].When considering mobility case, where the AP-RIS-UE chan-nel is time-changing, those estimation procedures are morefrequently called, increasing the complexity of beam trackingalgorithms dramatically. Therefore, channel estimation shouldbe avoided when designing RIS-based beam tracking algo-rithms. In [21], channel parameters are updated using extendedKalman filter, which is also utilized in systems without RISs[14]. Drawback of such directly utilization without adaptationis that large antenna array will deteriorate the accuracy ofbeam tracking because of beam misalignment. Novel layoutof the RIS-assisted system is proposed in [22], where the RISis attached on the vehicle. In [22], the unchanging Dopplerfrequency is exploited to estimate the time-changing cascadedchannel. Unfortunately, it is not in mmWave system and itexploits the AP-UE scatter-rich channel, at the same time still1 a r X i v : . [ ee ss . SP ] F e b nvolves multiple training time-slots.In conclusion, RIS-based beam tracking algorithms whenconsidering node mobility in mmWave networks need to bespecifically designed, whose challenges are listed as follows. • RIS is passively reflecting signals. The fact that RISs haveno signal processing ability makes the communicationinvolve three parties. In other words, the configuration ofthe large amount of RIS elements can only be done basedon the received signals, which complicates the RIS-basedbeam tracking algorithms since the algorithms withoutconsidering RIS can not be applied directly. • Finding best beam pairing solution is cumbersome.MmWave signals have narrower beams. Conventionally,e.g., in 11ad protocol, the beam alignment is done byswitching the antenna to omni-directional mode at oneside and remaining narrow beam mode at the other.By exhaustively changing the narrow beam directions atboth sides, a best spatial narrow beam pair which hasthe best communication performance is found. In RIS-assisted systems, such complexity raises from square ofthe number of spatial beams into the cube of it, which isusually unacceptable. • RIS-based beam tracking algorithms need to be e ff ectiveand e ffi cient. Channel parameters other than the completechannel state information (CSI) are to be updated ine ffi cient beam switching algorithms since such procedureis called more frequently in order to maintain the com-munication link quality. On the contrary, conventionalchannel estimation procedure is called only after severesignal loss. Therefore, accuracy of updating the channelparameters without having to estimate the complete CSIis also a big challenge.In this paper, we design the beam tracking algorithm par-ticularly for RIS-assisted mmWave systems considering nodemobility. We propose to avoid the complex channel estima-tion procedure by exploiting the channel di ff erence betweenadjacent user positions to obtain the update information toconfigure the RIS. We also manage to partly transform thetime-consuming signaling time into calculation complexityhappens at the base station. The major contributions of thispaper are summarized as follows. • We propose the system which utilizes RIS to create anadditional NLOS path in order to overcome blockageissue and support node mobility at the same time inmmWave systems. We formulate the beam tracking prob-lem and have the expression of optimal RIS configuration,which will be exploited in di ff erential form, serving as theparameters for updating during beam tracking procedure. • We propose an e ff ective beam tracking algorithm whichtransforms the high time complexity of beam searchingprocedure into calculation complexity at base station side.We transform the RIS beam configuration problem intoan optimization problem by exploiting both the receivedsignal strength (RSS) ratio and received signal angledi ff erence of the optimal RIS configuration status and thecurrent status. By solving the transformed optimizationproblem using a two-dimensional grid-based search, a small-size candidate set of RIS configurations is found.Then by applying this small set downlink training, sig-naling time is saved dramatically. • Simulation studies prove the e ff ectiveness as well as thelow complexity of our proposed RIS-based beam track-ing algorithm in mmWave systems by comparing withconventional exhaustive searching strategy. It is shownthat our proposed algorithm has near-optimal performancewith dramatic savings in terms of signaling time.The remainder of this paper is organized as follows. Therelated work is discussed in Sec. II. Sec. III describes thesystem model when applying RIS into mmWave systemsand formulates the beam tracking problem. The optimal RISconfiguration with respect to channel information is alsoprovided. In Sec. IV, we propose our RIS-based beam trackingalgorithm by obtaining the update rule and proposing thechannel complex gain model. Then we transform the RIS-based beam tracking problem into an optimization problem byexploiting the received signal strength and angles comparing tooptimal configuration case. A search-based calculation returnsa small set candidates and the overall signaling procedureis proposed then. In Sec. V, we explain how the channelis generated when considering a mobile user followed bymultiple simulation studies conducted to verify our proposedalgorithm. Finally, conclusions are drawn in Sec. VI.Notations: Boldface lower-case and upper-case letters indi-cate column vectors and matrices, respectively. ( · ) H denotesthe transpose-conjugate operation. E {·} represents statisticalexpectation. ∠ {·} represents the phase of a complex number. C denotes the set of complex numbers. | A | denotes the deter-minant of matrix A . | a | and | a | are the magnitude and norm ofa scalar a and vector a , respectively.II. R elated W ork A. Beam Tracking Algorithms in mmWave Systems
Beam tracking is needed in mmWave systems because ofits narrow beams and the algorithms can be divided by twocatagories. One is to estimate the channel, the other is to pre-dict the channel. However, most of those algorithms can not bedirectly applied to RIS-assisted systems due to its unique char-acteristics as is mentioned in Sec. I. For example, in [11], theauthors assume omni-directional transmission which is unprac-tical to be directly applied to RIS because it’s impossible toobserve the low-power received signal with only one elementof RIS working. Similar to 802.11ad protocol [12], exhaustivesearching strategy can be utilized in RIS-assisted systemsbut will induce intolerable complexity. When designing RIS-based beam tracking algorithms, decreasing beam alignmentcomplexity by narrowing down the search range is doable.Same idea can be found in [13] without considering RISconfiguration, where the authors exploit the beam patterns ofadjacent positions of user to decrease the search time. Kalmanfilter is useful when designing beam tracking algorithms as in[14] without considering RIS. However, without adapting it inRIS-assisted networks, the accuracy of the algorithm decreaseswith larger number of RIS elements. Learning-based beamtracking algorithms are emerging these days thanks to their2bility in beam prediction [15]- [17]. The machine-learning-based algorithms will require additional information, suchas multiple base stations in [15] and situational awarenessinformation in [16], to train the learning model and executebeam prediction. In addition, the reinforcement-learning-basedalgorithms, such as in [17], also require many times of failuresbefore successfully selecting the next state action.
B. RIS-based Channel Estimation Algorithms
Since beam tracking is highly related to channel estimationand the participation of RIS will introduce unique challenges,RIS-based channel estimation algorithms are of importance ininspiring RIS-based beam tracking algorithms. Since the pas-sive behaviour and large number of RIS elements make RIS-based channel estimation more complex than in conventionalsystems, many literature focus on saving the signaling time byexploiting the unique channel properties. In [18], compressedsensing and deep learning based algorithm are proposed inorder to obtain the channel state information with RIS systems.In [19], the channel feedback is exploited in order to extractthe essential information needed to configure the RIS. Thecomplexity is transformed from searching exhaustively intoselecting a codeword from a pre-set codebook that representingthe unknown channel. In [20], the authors propose a manifoldoptimization (MO)-based algorithm by exploiting the inher-ent structure of the e ff ective mmWave channel to optimallyconfigure the RIS. In order to decrease the complexity ofobtaining complete CSI, algorithms which update channelparameters are commonly used when considering beam track-ing problem. In [21], the RIS-based beam tracking algorithmis based on hierarchically searching for the auxiliary beamsas the estimated cascaded channel. With estimated channelparameters, the extended Kalman filter is utilized to updatethe estimated channel parameters and the RIS configuration.However, the tracking algorithm will become inaccurate whenthe number of RIS elements is large because of the narrowerbeam misalignment [14]. With time-changing channels whenconsidering node mobility, there are some parameters that arenot changing, which can be exploited to estimate the channel.In [22], the RIS is implemented in the high-speed vehicleto generate slow fading RIS-UE channel. The authors exploitthe Doppler frequency to estimate the cascaded AP-RIS-UEchannel followed by designing the RIS configuration in orderto constructively align with the direct AP-UE channel.In summary, as is mentioned in the challenges of our topic inSec. I, despite the mature studies of beam tracking algorithmsin mmWave systems, they can not be directly applied to RISsystems because of its passive-reflecting characteristic andlarge number of elements. In addition, most of the existingwork about RIS configurations focus on the static scenariosand will require the procedure of channel estimation, which isof high complexity if used in beam tracking scenarios. There-fore, in this paper, we propose the beam tracking algorithmdesigned specifically for RIS-assisted mmWave systems inorder to overcome the issues of blockages and node mobility.By exploiting the signal strengths and angles of adjacent statusreceived signals, our proposed solution can bypass the complex AP Beamformer
UE UE elements RISPosition/state sPosition/state s+1Fixed RIS configuration TrajectoryBlockage
Fig. 1. A mmWave system with a multiple antennas AP, a single antennamobile UE and an N -element third party RIS. The LOS path is blocked andan additional NLOS path is created with the application of RIS. channel estimation procedure and update the beams based onthe di ff erential form of optimal RIS configuration, thus savinglarge amount of signaling time as well as maintaining thecommunication quality above a given threshold.III. S ystem M odel , P roblem F ormulation and O ptimal I nitial C onfiguration In this section, we first describe the mmWave system modelwith a third-party RIS and formulate the RIS beam trackingproblem under node mobility. Then the optimal initial RIS con-figuration obtained by a searching manner is introduced basedon our derivation of optimal RIS configuration represented bychannel angular information.
A. System Model
Consider a mmWave system with a RIS as illustrated inFig. 1, in which the AP is equipped with N t antennas andtransmits the information to a single-antenna UE throughbeamformer f ∈ R N t × . Here we consider the case wherethe line-of-sight (LOS) path is blocked and AP and UE arecommunicating through the RIS, which has N phase shiftersto manipulate the signal’s phases. The transmitted symbolsfirst goes through the AP-RIS channel G ∈ R N × N t , andreflected by the RIS, whose e ff ect is denoted by a diagonalmatrix Θ = diag { e j φ , e j φ , ..., e j φ N } ∈ R N × N . Then the reflectedsignal goes through the RIS-UE channel h ( s ) ∈ R N × , wherethe notation s represents the s -th position / state, and finallyreceived by UE. Therefore, the received signal at position / state s has the following form: y ( s ) DL = ( h ( s ) ) H Θ ( s ) G ( s ) f x ( s ) + n ( s ) . (1)where E { x ( s ) ( x ( s ) ) H } = n ( s ) ∼CN (0 , σ n ) is additive white Gaussian noise. Specifically, G = α a RA ( θ ) a HAP ( φ AP ) , (2)( h ( s ) ) H = β s a HRA ( θ ( s )2 ) . (3)where α and β s are channel coe ffi cients. a AP ( φ AP ), a RA ( θ ) and a RA ( θ ( s )2 ) are steering vectors of the corresponding channels,with φ AP , θ and θ ( s )2 as AoD of AP-RIS channel, AoA of3P-RIS channel, and AoD of RIS-UE channel. The steeringvectors has the form of: a ( θ ) = [1 , e − j π d λ sin θ , ..., e − j π d λ ( N −
1) sin θ ] T . (4) B. Problem Formulation
According to the received signal in (1), the problem ofoptimal RIS configuration has the following form: Θ ∗ = arg max | ( h ( s ) ) H Θ Gf | . (5)Our goal is to propose an RIS beam tracking algorithm in orderto achieve high data rate while keeping the overhead as lowas possible. In order to design the RIS configuration duringnode mobility, an optimal initial configuration is introduced inthe next subsection. C. Optimal Initial Configuration
Initial access is an essential step in mmWave communica-tions in order to align the narrow beams of the transceivers.In a mmWave system with RIS, the AP-RIS channel canbe viewed as unchanged since the positions of both AP andRIS are fixed. Therefore, in our paper, we focus on the RISconfiguration under the assumption of f = √ SNR × a AP ( φ AP ) | a AP ( φ AP ) | in the following deduction. Such assumption is reasonable be-cause the AoD can be obtained either by the a 802.11ad beamsearching procedure or by other AoD estimation algorithms.With that being said, we can rewrite the received signal atUE from (1) as: y ( s ) DL = ( β s a HRA ( θ ( s )2 )) Θ ( s ) ( α a RA ( θ ) a HAP ( φ AP )) × √ SNR × a AP ( φ AP ) | a AP ( φ AP ) | x ( s ) + n ( s ) = c αβ s a HRA ( θ ( s )2 ) Θ ( s ) a RA ( θ ) x ( s ) + n ( s ) , (6)where c (cid:44) √ S NR | a AP ( φ AP ) | . Ignoring noise, we can rewritethe objective function in (5) as: { φ , ..., φ N } = arg max | β s a HRA ( θ ( s )2 ) Θ ( s ) a RA ( θ ) | . (7)Replacing the steering vectors with the form of (4), theobjective function (7) can be rewritten as: { φ ∗ , ..., φ ∗ N } = arg max | β s | | N (cid:88) n = e j (cid:16) φ n − π d λ ( n − θ − sin θ ( s )2 ) (cid:17) | . (8)In order to have the maximum value of the summation form,it is obvious to align the phases of RIS elements with thechannel. Therefore, we can have the optimal RIS configurationas: ( φ ( s ) n ) ∗ = , if n = , π d λ (sin θ − sin θ ( s )2 ) , if n = , ( n − φ ( s )2 , if n = , ..., N . (9)Taking a closer look at the optimal RIS configuration (9),the term (sin θ − sin θ ( s )2 ) is unknown to us since we can notextract the exact AoD and AoA information of the channelrelated to RIS. Another observation is that as long as weset the phase of one RIS element, i.e, φ , we can have theoptimal phases for other elements. Inspired by the above two observations, we propose the optimal initial RIS configuration Θ (1) in an exhaustive searching manner, which searches thevalue of φ ∈ (0 ◦ , ◦ ) with a given searching step (1 ◦ insimulation studies).IV. F ast RIS-B ased B eam T racking A lgorithm After obtaining the optimal initial RIS configuration Θ (1) and starting the downlink data transmission with fixed AP-RIS beamformer f , RIS beam tracking algorithm is executed inorder to maintain the link under node mobility. In this section,we first derive the RIS configuration update rule consisting thedi ff erential form of a channel parameter. Then, we propose achannel complex gain model in order to decouple the e ff ect ofRIS misalignment from the observation of a received signalstrength drop. Next, we transform the beam tracking probleminto an optimization problem by exploiting the received sig-nal strength and angle. Using a two-dimensional grid-basedsearch, we find a small-size candidate set of RIS configura-tions. Finally, we explain the whole procedure including thesignaling procedure and the framework of the RIS-based fastbeam tracking algorithm. A. State Transformation and Update Rule
We first define a state during downlink data transmissionas follows: If during time-slots t n , there is | y ( t n ) DL | ≥ γ , where γ is a communication quality threshold, we say these time-slots are in the same state. In the same state / status, theconfiguration of RIS for downlink data transmission remainsthe same. This is reasonable since we don’t have to updatethe RIS configuration to maintain the communication quality.However, if we observe at a time-slot t , there are | y ( t ) DL | < γ and | y ( t − DL | ≥ γ , we say the status has transformed from s into s +
1. We define t as the status transition time-slot . Thisis reasonable since the degradation of RSS (received signalstrength) may indicate that the current RIS configuration cannot maintain the communication quality and an update ofRIS configuration is needed, which calls the beam trackingalgorithm we are going to explain in the following.In order to form our proposed beam tracking algorithm, wewill propose our update rule for the RIS configuration. Recallthe optimal RIS configuration (9), we can have the optimalRIS configuration for status s + φ ( s + n ) ∗ = π d λ ( n − θ − sin θ ( s + ) , n = , ..., N . (10)Combined with (9), we can have the update rule of the optimalRIS configuration as: φ ( s + n = φ ( s ) n − π d λ ( n − w ( s + , n = , ..., N , (11)where w ( s + (cid:44) sin( θ ( s + ) − sin( θ ( s )2 ) denotes the unknown RIS-UE channel update information. B. Time Sequential Signaling and Channel Complex GainModel
Previously, we have defined the status from the view of RSS.In this subsection, we will discuss about the time sequential4 lements RISPhase: A B User position: A->B
Fig. 2. Complex channel gain model. signaling by viewing the received signal during each time-slot.Based on that, we will introduce our own channel complexgain update model while the UE is moving a very shortdistance in a very short time.The position-dependent / time-dependent RIS-UE channel ismodelled as a LOS channel decided by only two parameters,complex channel gain and angle of departure (AoD): h ( t ) = β ( t ) a RA ( θ ( t )2 ) , (12)where the superscript t represents the t -th time-slot. β ( t ) isthe complex channel gain, following Rayleigh distribution. a RA ( θ ( t )2 ) is the steering vector and θ ( t )2 is the AoD.By modelling the time-changing channel as in (12), one ofthe parameters, i.e., θ ( t )2 that describes the angle characteristicof this channel, is easy to obtain by geometrically obtainingthis information. As for the other deciding parameter β ( t ) , wewill describe our principle revealing the relationship between β ( t − and β ( t ) in the following.Since the duration of one time-slot is t = . us accordingto the 11ad standard, which is very small, we make theassumption that the positions of UE at two adjacent time-slotsare very near. Therefore, the complex gain β ( t ) can be viewedas dependent on β ( t − as shown in Fig. 2. The relationship isrevealed as follows: β ( t ) = ρβ ( t − e j θ δ , (13)where θ δ = π r δ λ describes the phase di ff erence of the adjacenttwo channels caused by the di ff erence of mmWave traveldistance, i.e., r δ = r ( t )2 − r ( t − and r represents the distancebetween RIS and UE. ρ is a coe ffi cient describes the path-lossrelationship between these two positions.In the following, I will introduce the path-loss-relatedcomplex channel gain update coe ffi cient. First, I will intro-duce the RSS to be used in the following explanation, i.e., RS S ( t ) = | y ( t ) DL | . We write out the approximation of twoadjacent time-slots’ received signals without considering noiseand with optimal RIS configuration: y ( t − DL = c αβ ( t − a HRA ( θ ( t − ) Θ ( t − a RA ( θ ) , y ( t ) DL = c αβ ( t ) a HRA ( θ ( t )2 ) Θ ( t ) a RA ( θ ) . (14)In this way, we can rule out the e ff ect caused by RIS misalign-ment and only consider the e ff ect of channel changed between two time-slots. By replacing in the optimal configuration as in(9), we have the RSS for both time-slots as: | y ( t − DL | = | c αβ ( t − N | , | y ( t ) DL | = | c αβ ( t ) N | = | c αβ ( t − ρ N | , (15)where (15) is obtained by the definition of our channelcomplex gain model (13). Then we can have the representationof the channel complex gain coe ffi cient ρ as: ρ = | c αβ ( t ) N | | c αβ ( t − N | = | y ( t ) DL | | y ( t − DL | = RS S ( t ) RS S ( t − . (16)Since the RSS is related to path-loss and having the relation ofmultiplication with other coe ffi cients such as transmit antennagain and receive antenna gain, we can have the relationshipof channel complex gain coe ffi cient ρ and path-loss PL as: ρ = PL ( t − PL ( t ) . (17)Then let’s look at the representation of path-loss [23]: PL ( r )[ dB ] = log ( 4 π r λ ) + log ( rr ) , (18)where r represents the distance between AP and UE and r is a reference distance. Then we know that the path-loss inmultiplication relationship has the following form: PL = ( 4 π r λ ) . (19)Then, according to (17), we have the definition of channelcomplex gain coe ffi cient as: ρ = r ( t − r ( t ) = r + r ( t − r + r ( t )2 , (20)where r is the distance between AP and RIS. By viewingcomplex gain definition (13) and equation (20), we findthat this coe ffi cient is related to the distance between UEand AP, which means this coe ffi cient is determined only bythe characteristic of channel itself other than the signalingprocedures. And since it is a ratio, we can extend this equationinto the relationship of two adjacent status: ρ = (cid:114) RS S ( s + RS S ( s ) = r + r ( s )2 r + r ( s + , (21)which is going to be used in our proposed beam trackingalgorithm.In addition, since we are representing the complex channelgain coe ffi cient as two RSS ratio, we can always generatethe time-changing RIS-UE channel h according to (12) afterobtaining the AoD θ ( t )2 , given the initial RIS-UE channelparameters β (1) and θ (1)2 , together with the update principleof the complex gain β as in (13). C. Fast RIS-based Beam Tracking Algorithm
Recall the definition of di ff erent status during downlinkdata transmission phase in Sec. IV-A, beam tracking algorithmtakes place after status transition. Therefore, let’s focus on thetransition time-slot t as in Sec. IV-A, where it’s in status s + Θ ( s ) . We can have the approximationas: y ( s ) DL = c αβ ( s ) a HRA ( θ ( s )2 ) Θ ( s ) a RA ( θ ) , y ( t ) DL = c αβ ( s + a HRA ( θ ( s + ) Θ ( s ) a RA ( θ ) , (22)Observing these two equations, it can be found that thechannel angle information for the next status, i.e., θ ( s + can berepresented by the parameters at status s , since y ( t ) DL is utilizing Θ ( s ) which is also utilized in y ( s ) DL . As for the complex channelcoe ffi cient β , we have the relationship of two adjacent statusas (21), stated in Sec. IV-B. For simplicity and without losinggenerality, we use time-slot t = s , and time-slot t to represent next status s +
1, where t is thestatus transition time-slot. And we assume that when t = | y (1) DL | = | y ( s ) DL | = | c αβ ( s ) N | . (23)As for time-slot t , as the definition of status transition time-slot, we have | y ( t − DL | ≥ γ and | y ( t ) DL | < γ . Since the RISconfiguration has not been updated as shown in (22), thedecrease in RSS is caused by two factors: one is path-loss,the other is the RIS misalignment. However, these two factorsare coupled but only the information of RIS misalignmentis needed for us to update the RIS configuration. Therefore,in this subsection, we will introduce our proposed RIS beamtracking algorithm aiming at ruling out the e ff ect of path-lossand finding the RIS optimal configuration for next status.Rewrite the status transition time-slot received signal basedon (22), by replacing Θ ( s ) with (9), we have: y ( t ) DL = y ( s + DL = c αβ s + N (cid:88) n = e j [ π d λ ( n − w ( s + ] , (24)and the corresponding RSS as: | y ( s + DL | = | c α | | β ( s + | | N ( s + | , (25)where we have the definition: N ( s + (cid:44) N (cid:88) n = e j [ π d λ ( n − w ( s + ] , (26)where w ( s + is defined in (11). Since N ( s + is a geometricsequence, we have: N ( s + = N , if w ( s + = , e j [ 2 π d λ w ( s + N ] − e j [ 2 π d λ w ( s + − , if w ( s + (cid:44) . (27)Given both RSSs of current status (23) and next status (25),we can have the RSS ratio as: RS S ( s + RS S ( s ) = | y ( s + DL | | y ( s ) DL | = ρ | N ( s + | N = ( r ( s ) r ( s + ) | N ( s + | N = η ( s + . (28)Observing the above equation, we can see that the drop of RSSis represented by two factors: path-loss and misalignment ofRIS configuration.In order to obtain two unknown parameters, we will need
30 35 40 45 50 55 60 A m p li t ude (a) Amplitude of N ( s +
30 35 40 45 50 55 60 -4-3-2-101234 A ng l e (b) Angle of N ( s + Fig. 3. Properties of | N ( s + | and ∠ ( N ( s + ). an extra equation along with (28). Thus, we turn to examinethe angle feature of the two received signals by observing thereceived signal formation as in (24). Similarly as we obtainthe relationship of RSS ratio, we have the di ff erential angle oftwo adjacent status as: ∠ ( y ( s + DL ) − ∠ ( y ( s ) DL ) = ∠ ( β ( s + ) + ∠ ( N ( s + ) − ∠ ( β ( s ) ) − ∠ ( N ( s ) ) = ∠ ( β ( s + ) − ∠ ( β ( s ) ) + ∠ ( N ( s + ) = θ ( s + δ + ∠ ( N ( s + ) = πλ ( r ( s + − r ( s ) ) + ∠ ( N ( s + ) = ξ ( s + (29)According to (27), we have: ∠ ( N ( s + ) = , if w ( s + = , ∠ ( e j [ 2 π d λ w ( s + N ] − e j [ 2 π d λ w ( s + − ) , if w ( s + (cid:44) . (30)Observing the above equation, we also see that the twouncertain parameters, i.e., path-loss and RIS misalignment, arepossible to be decoupled.Since we have the relationships of RSS ratio and di ff erentialangle of two adjacent status, the problem of finding the optimalRIS update configuration can be solved by decoupling thee ff ects of path-loss and RIS misalignment based on (28) and(29). Particularly, recall the update rule (11), we have thefollowing optimization problem based on the fact that weconsidered the above deduction without noise: { ( r ( s + ) ∗ , ( N ( s + ) ∗ } = arg min errorI + errorIIs . t . | ( r ( s ) r ( s + ) | N ( s + | N − η ( s + | = errorI (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) πλ | r ( s + − r ( s ) | + ∠ ( N ( s + ) − ξ ( s + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = errorII . (31) D. Two-Dimensional Search-Based Solution
In this subsection, we will illustrate the solution of theoptimization problem (31). We propose a searching based solu-tion by calculating both errors of possible uncertain parameterpairs, i.e., errorI and errorII. Such idea is inspired by theobservation of the characteristics of | N ( s + | and ∠ ( N ( s + ) asshown in Fig. 3. The above figures are obtained when we set θ = ◦ . It can be seen that as long as we keep the searchingangle of θ within a certain range, we can always obtain arather large outcome of | N ( s + | . Also, the angle characteristicof it can be viewed as linear within the same θ changing6 lgorithm 1 Two-Dimensional Search-Based Solution
Input:
A series of θ ( s + centered at θ ( s )2 with searching rangeas ± . ◦ . Output:
Best N sol pairs of N ( s + and r ( s + . Calculate corresponding series of N ( s + according to (27)based on a series of θ ( s + . Calculate corresponding series of r ( s + cal according to (32). for each r ( s + cal do Obtain series of r ( s + centered at r ( s + cal with searchingrange as ± . m . end for for each pair of N ( s + and r ( s + do Calculate errorI + errorII in (31). end for Select best N sol pairs of N ( s + and r ( s + that has minimumerror summation. FB FB FB
DLUL RIS Beam Tracking
Transmissin with Small set RIS beam training Transmission withtimetimeFeedbackoptimal Feedbackbeam update request Feedbackoptimal Exhaustive Searching
RIS Beam Training
Fig. 4. The overall signaling procedure with our proposed fast beam trackingalgorithm. range. Based on these observations and the intuition that theAoD changing happens within a certain range in respect to theoptimal θ at time sequence t =
1, i.e., when RIS configurationis optimal at status s , we have our search-based solution toproblem (31) as described below.We first set the θ ( s + searching range as ± . ◦ centeredat the optimal θ ( s )2 , and calculate the corresponding N ( s + according to (27). In this case, the center of | N ( s + | is when w ( s + =
0, i.e., we start the searching procedure from status s when RIS configuration is optimal. Then according to thefirst constraint equation in (31), and the calculated series of N ( s + , we can calculate the corresponding r ( s + cal as: r ( s + cal = r ( s ) | N ( s + | (cid:112) η ( s + N . (32)Next, we set the r ( s + searching range as ± . m centered atthe calculated series of r ( s + cal . With multiple N ( s + and r ( s + pairs, we can calculate the summation of the correspondingtwo errors according to (31). Then we select the best severalpairs of N ( s + and r ( s + as the best several solutions toproblem (31). The algorithm for solving this problem isillustrated in Algorithm 1. E. Overall Signaling Procedure and Framework
To clearly demonstrate the whole working process of ourproposed RIS-based fast beam tracking algorithm, we illustratethe overall signaling procedure between a fixed RIS and a
Start1. Obtaining , based on 2. Solve the optimizationproblem for
N_sol best RSS < YInitially configure RIS bysweeping, find DL data transmissionwith N 1 1DL RIS configurationtraining with onecandidate Tested allUse which hashighest SNR toconfigure YN
Fig. 5. The framework of our proposed beam tracking algorithm. mobile UE, including both RIS beam training phase and RISbeam tracking phase in Fig. 4. We also illustrate the frameworkof the whole process in Fig. 5.Initially, the configuration of RIS is done by sweepingthe all possible phases, which corresponds to the RIS beamtraining phase as in Fig. 4. This phase takes the exhaustivesearching strategy thus will cost a large amount of signalingtime. After obtaining the optimal initial RIS configuration Θ (1) ,it goes into downlink data transmission phase where the RISbeam tracking algorithm is working. During such phase, theuser will observe the RSS at each time-slot and compare it witha certain threshold γ . Once the RSS is below the threshold, theuser will feedback a signal indicating that RIS configurationupdate is needed. At the same time, the two parameters of RSSratios and received signal di ff erential angle are also fedback toAP in order to solve for the optimization problem accordingto Algorithm 1. Notice that such complex procedure is doneby calculating at AP side, which saves a large amount ofsignaling time. After solving for N sol best possible solutions N ( s + for updating RIS configuration, a small set downlinktraining is processed. Notice that such time is rather short sincethe complexity is transformed into the calculation complexityhappened at AP. After testing all candidate solutions, the onewhich can has the highest SNR will be utilized as the updateinformation to configure Θ ( s + .V. S imulation E valuations In this section, we will first describe how to generatethe ground true channel with user mobility according to ourchannel complex gain model as in Sec. IV-B followed byextensive evaluations based on it.
A. User Trajectory Generation and Simulation Parameters
In this subsection, we will showcase how to generatethe user trajectory and the corresponding parameters in oursimulation studies as shown in Fig. 6. We generate our groundtrue channel as described below:Firstly, we want to generate the ground true channel for path AB as illustrated in Fig. 6, where a human is walking along7 lements RISRIS configurationA User Trajectory: A->B->CB C Fig. 6. User trajectory in simulation study. a straight line starting from point A to point B. To decidethe channels for path AB , we only need to know the startingposition, the direction of walking and the distance of path AB .We decide the starting point A with initial AoD at RIS set as θ (1)2 and the initial distance between RIS and UE as r (1)2 . Thenwe assume the angle between RIS-to-A and path AB is setas ψ A , which indicates the direction the user is walking. Thetime duration of walking along path AB is set to t , indicatingthe distance of path | AB | = vt t .There are four steps to generate the ground true channel h for path AB . • Step 1 : Decide the position of starting point A. In thisstep, we initialize A by assigning r (1)2 and ψ A = ◦ ,with the initial AoD as θ (1)2 = ◦ , meaning that path AB is parallel to the RIS. • Step 2 : Decide the number of time-slots when UE movesalong path AB . In this step, we calculate the number oftime-slots n . We first calculate the distance UE movesduring one time-slot t = . us by vt , where v repre-sents the speed of UE. Then we assign the distance ofpath AB as | AB | . The number of time-slots on this pathis calculated by n = (cid:100)| AB | / ( vt ) (cid:101) , then t = n meaningeach transmission time-slot. • Step 3 : Decide the complex channel gain. The complexchannel gain β (1) is generated following Rayleigh distri-bution. • Step 4 : Generate the channel h along path AB accordingto geometrical relationship. Given the distance of AP-RISas r , the update rule of RIS-UE distance r is as follows: r ( t )2 = (cid:113) r + ( vt ∗ t ) − r ( vt ∗ t ) cos ( ψ A ) . (33)Then following the update rule of complex gain (13) andcoe ffi cient (20), we can get the complex channel gain foreach time-slot. To update the AoD θ , we use the equationbelow: cos ( θ ( t )2 − θ (1)2 ) = r + ( r ( t )2 ) − ( vt ∗ t ) r r ( t )2 . (34)Then the channel at time-slot t can be obtained accordingto (12).When the UE is changing direction during the walk, the
20 25 30 35 40 45 50 55 60 I n s t an t aneou s R a t e Proposed (a) Proposed algorithm.
20 25 30 35 40 45 50 55 60 I n s t an t aneou s R a t e Exhaustive, res = 1 ° (b) Exhaustive search, res = ◦ .
20 25 30 35 40 45 50 55 60 I n s t an t aneou s R a t e Exhaustive, res = 5 ° (c) Exhaustive search, res = ◦ .
20 25 30 35 40 45 50 55 60 I n s t an t aneou s R a t e Exhaustive, res = 10 ° (d) Exhaustive search, res = ◦ .Fig. 7. Instantaneous rate (bit / s / Hz) versus user position in spatial angle θ (degree) (Distance of AP-RIS r = m , distance of RIS-UE r = m , userwalking speed v = . m / s ). same procedure can be done to generate the channel for thenew path, e.g., path BC as is shown in Fig. 6. Specifically, weneed the position of starting point B , which is also the endingpoint of path AB and such information can be easily acquired.In the following subsections, we will illustrate the sim-ulation results of the proposed RIS configuration algorithmunder node mobility in mmWave systems based on the channelgenerated in this subsection.Consider a mmWave system including an AP equipped with N t =
16 Uniform Linear Antennas (ULA), whose spacing is d = λ , a single antenna mobile UE, and a RIS equipped with N =
64 elements, which is able to achieve continuous phaseshifting. The AoA at RIS side is assumed to be θ = ◦ . Forsimplicity, the noise variances are set to 1 and transmissionpower is SNR = N ( s + and r ( s + is set as ± . ◦ and ± . m . The threshold while calling RIS beam trackingalgorithm is set as γ = . γ exh = .
5. Third is the oraclecase, where RIS will always find the optimal configurationafter the RSS is below the threshold of γ = . Spatial angle C u m u l a t i v e A v e r age R a t e OracleProposedExhaustive, res = 1 ° Exhaustive, res = 5 ° Exhaustive, res = 10 ° Fig. 8. Cumulative average rate (bit / Hz) versus user position in spatial angle θ (degree) (Number of transmit antennas N t =
16, number of RIS elements N =
64, SNR = γ = .
9, exhaustivesearching idea threshold γ exh = .
5, exhaustive searching resolution 1 ◦ , 5 ◦ ,10 ◦ , and walking speed v = . m / s ). B. E ff ect of Noise on Instantaneous Rate We first look at the performance of instantaneous ratecomparing both our proposed algorithm and the exhaustivesearching strategy in Figs. 7. In our simulation, we considerthe instantaneous rate as 0 when DL data transmission is nothappening. Such time-slots including DL signaling time andUL feedback time. From the figure, we can see that whenconsidering noise, the instantaneous rates are fluctuating withtheir trends clear. By observing such trend, we can see thatboth our proposed algorithm and the exhaustive search ideawith 1 ◦ resolution can obtain near-optimal RIS configurationalmost every time-slot, thus maintain the rate at a rather highlevel most of the time. The fact that coarser resolution willdeteriorate instantaneous rate can be verified by comparingFigs. 7(b)-7(d). However, at the same time, finer resolutionwill induce higher signaling time during DL training phase.Therefore, there’s a trade-o ff between instantaneous rate andsignaling time. The idea of our proposed beam trackingalgorithm is to shorten the signaling time while maintainingthe instantaneous rate above a certain threshold. C. Cumulative Average Rate
In order to evaluate the overall performance of the three RISconfiguration ideas, we propose to use cumulative average rateto reflect the trade-o ff of the signaling time and instantaneousrate. The cumulative average rate is calculated by the summa-tion of instantaneous rates over total number of time-slots andhas the following update rule: R ( t + cum = (cid:80) t + i = R ( i ) ins t + = tR ( t ) cum + R ( t + ins t + , (35)where R cum stands for the cumulative average rate and R ins stands for the instantaneous rate. The superscript stands forthe corresponding time-slot.
20 25 30 35 40 45 50 55 60
Spatial angle C u m u l a t i v e A v e r age R a t e Oracle, v=0.6Proposed, v=0.6Proposed, v=1.2Proposed, v=1.8Exhaustive, res = 5 ° , v=0.6 Fig. 9. Cumulative average rate (bit / Hz) versus user position in spatial angle θ (degree) (Number of transmit antennas N t =
16, number of RIS elements N =
64, SNR = γ = .
9, exhaustivesearching idea threshold γ exh = .
5, and walking speed v = . m / s , v = . m / s and v = . m / s ). We compare our proposed beam tracking algorithm withthe oracle one, serving as the upper bound together with theexhaustive searching strategy with three di ff erent resolutionsettings in Fig. 8. Di ff erent from Figs. 7, which reflect in-stantaneous rate and will always have certain positions thatcan reach the highest achievable instantaneous rate, cumulativeaverage rates for di ff erent algorithms has gaps. It can be seenthat the oracle idea has the highest performance since it has nocost on signaling time. Our proposed beam tracking algorithmhas near-optimal performance since we tried to transform thetime cost into calculations happen inside the BS, significantlysaving signaling time comparing with conventional exhaustivesearch ideas. In addition, we found that the performance ofexhaustive search idea with 5 ◦ resolution outperforms whenwith 1 ◦ and 10 ◦ resolutions. This is because that with finerresolution, the accuracy of the updated RIS configuration ishigher, which explains why it outperforms when res = ◦ .However, with finer resolution, the signaling time for eachsearch will get longer, resulting in lower cumulative averagerate since the number of time-slots when R ( t ) ins = = ◦ .According to Fig. 8, cumulative average rate can e ff ectivelyreflect the trade-o ff of signaling time and instantaneous rateand is suitable when evaluting the overall performance. D. E ff ect of User Velocity Then we turn to evaluate the e ff ect of user velocity on ourproposed beam tracking algorithm. We compare our algorithmwith velocities as v = . m / s , v = . m / s , v = . m / s ,respectively, along with the upper bound and the conventionalexhaustive searching idea with 5 ◦ resolution. It can be seenfrom Fig. 9 that with lower user velocity, the proposedalgorithm has higher cumulative average rate. Intuitively, withlower user velocity, the angle changed during a same amountof time will be smaller, meaning that one particular configura-9 Spatial angle C u m u l a t i v e A v e r age R a t e Oracle, =0.9Proposed, =0.9Proposed, =0.8Proposed, =0.5Exhaustive, res = 5 ° , =0.5 Fig. 10. Cumulative average rate (bit / Hz) versus user position in spatial angle θ (degree) (Number of transmit antennas N t =
16, number of RIS elements N =
64, SNR = γ = . γ = . γ = .
5, exhaustive searching idea threshold γ exh = .
5, and walking speed v = . m / s ). tion of RIS will have longer time having the communicationquality above the same threshold. Therefore, the frequencyof implementing the beam tracking algorithm will be lower,thus can save signaling time and achieves higher cumulativeaverage rate. E. E ff ect of Threshold Next we examine the e ff ect of threshold on our proposedbeam tracking algorithm. We compare our algorithm withthresholds set as γ = . γ = . γ = .
5, respectively,along with upper bound and the exhaustive searching idea with5 ◦ resolution. As is shown in Fig. 10, with higher threshold,which means higher communication quality demand, the per-formance of cumulative average rate will be higher. Comparingwith the exhaustive strategy with the same threshold when γ = .
5, our proposed algorithm still outperforms it, whichverifies that our proposed beam tracking algorithm saves muchmore signaling time than the conventional idea.
F. E ff ect of Candidate Set Size The signaling time of our proposed algorithm depends onthe size of candidate solution for the optimization problem(31). Intuitively, with larger size of the candidate set, thechance of selecting the suitable RIS configuration is higher atthe cost of occupying longer signaling time. However, if thesize of such set is too small, it is possible that the solution for(31) will not include the suitable RIS configuration. Therefore,keeping the size of such set as small as possible whilemaintaining a rather small rate di ff erence with the oracle ideawill get us higher cumulative average rate.In order to reflect the superior of our proposed algorithmin saving signaling time, in Fig. 11, we examine the e ff ect ofthe size of candidate set for the optimization problem (31).We evaluate the average error rate for the proposed algorithm, A v e r age E rr o r R a t e Size of candidate set
Proposed
Fig. 11. Average error rate (bit / s / Hz) versus size of candidate set for downlinktraining phase (Number of transmit antennas N t =
16, number of RIS elements N =
64, SNR = γ = .
9, and walkingspeed v = . m / s ). TABLE IP ercentage of time - slots below corresponding thresholds . Prop. Exh, res = ◦ Exh, res = ◦ Exh, res = ◦ .
32% 4 .
53% 0 .
54% 0 . .
5% 7 .
47% 0 .
94% 0 . .
8% 11 .
89% 1 .
54% 0 . ff erence of instantaneous rate betweenthe oracle idea, i.e., | R ( t ) ins − R ( t ) Orc | . It can be seen that the errorof average rate converges very quickly, with near-zero errorwhen the size is larger than 5, which verifies that our proposedalgorithm can dramatically save signaling time, comparingwith conventional exhaustive search idea. However, in orderto make sure the optimal solution is always obtained insidethe found candidate set, we set the size of the candidate setas 7 in all other simulation studies. G. Complexity Comparison
Finally, we turn to evaluate the complexity of our pro-posed beam tracking algorithm (Prop.) comparing with theconventional exhaustive searching idea (Exh.) with di ff erentresolutions. In Tab. I, we summarized the percentage of time-slots below corresponding thresholds for both algorithms. Weimplement this table with 3 channel realizations. The data inthe first row is generated under the channel condition thatthe distances of AP-RIS and RIS-UE are both r = r = m with user velocity as v = . m / s . The second row is whenthe distances are set as both r = r = m with user velocityas v = . m / s , and the third row is when the distances areset as both r = r = m with user velocity as v = . m / s .We calculate the number of time-slots below correspondingthresholds under three cases: 1) Number of time-slots in DL10 ABLE IIN umber of beam tracking procedure called
Prop. Exh, res = ◦ Exh, res = ◦ Exh, res = ◦ = ◦ , 72 in exh, res = ◦ , and 36 in exh, res = ◦ ;2) Number of time-slots when DL data transmission is belowthe thresholds: 1 for both ideas; 3) Number of time-slots whenconducting UL feedback: 1 for indicating the start of the beamtracking algorithm and 1 for indicating its end for both ideas.From the table, we can see that the proposed algorithm hasa percentage of time-slots below the threshold γ = . .
32% while this percentage for exhaustive searchingidea with resolution 1 ◦ is 4 . ff erent resolution. Therefore,we can conclude that, in case of comparable instantaneousrate, our proposed algorithm has over 14 times signaling timesaved than the conventional exhaustive searching idea.In addition, when we compare di ff erent rows, it is verifiedthat with higher user velocity, the percentage of signaling timeused for beam tracking will get higher. Such observation alsoverifies the conclusion drawn for Fig. 10.We also illustrate the number of beam tracking procedurecalled for both ideas in Tab. II. As is shown in the table, amore frequent calling of the beam tracking procedure results ina higher achievable rate. This is because that the e ff ectivenessof RIS configuration will be deteriorated by the mobility ofthe user and a more frequent calling of the beam trackingprocedure will adjust the RIS configuration more frequently,thus maintains the instantaneous rate at a rather high level.VI. C onclusion In this paper, we propose a fast RIS-based beam track-ing algorithm in a mmWave system. We first exploit thedi ff erential form of optimal RIS configuration and serve itas the updating beam tracking parameter to avoid complexchannel estimation procedure. Then we transform the RIS-based beam tracking problem into an optimization problemsolved by a two-dimensional grid-based search. Finally, witha small set candidate solutions, the downlink signaling timecan be dramatically saved. Simulation studies showed that theproposed algorithm outperforms exhaustive searching idea andhas near-optimal performance.R eferences [1] Z. Pi and F. Khan, “An introduction to millimeter-wave mobile broad-band systems,” IEEE Commun. Mag. , vol. 49, no. 6, pp. 101-107, June2011. [2] T. Rappaport, et al. , “Millimeter wave mobile communications for 5Gcellular: It will work!”
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