Deep Learning-based Beam Tracking for Millimeter-wave Communications under Mobility
aa r X i v : . [ ee ss . SP ] F e b Deep Learning-based Beam Tracking forMillimeter-wave Communications underMobility
Sun Hong Lim, Sunwoo Kim, Byonghyo Shim, and Jun Won Choi
Abstract
In this paper, we propose a deep learning-based beam tracking method for millimeter-wave (mmWave)communications. Beam tracking is employed for transmitting the known symbols using the soundingbeams and tracking time-varying channels to maintain a reliable communication link. When the poseof a user equipment (UE) device varies rapidly, the mmWave channels also tend to vary fast, whichhinders seamless communication. Thus, models that can capture temporal behavior of mmWave channelscaused by the motion of the device are required, to cope with this problem. Accordingly, we employa deep neural network to analyze the temporal structure and patterns underlying in the time-varyingchannels and the signals acquired by inertial sensors. We propose a model based on long short termmemory (LSTM) that predicts the distribution of the future channel behavior based on a sequence ofinput signals available at the UE. This channel distribution is used to 1) control the sounding beamsadaptively for the future channel state and 2) update the channel estimate through the measurementupdate step under a sequential Bayesian estimation framework. Our experimental results demonstratethat the proposed method achieves a significant performance gain over the conventional beam trackingmethods under various mobility scenarios.
Index Terms
Millimeter-wave communications, beam tracking, mobility, channel estimation, deep learning, deepneural network, LSTM
Sun Hong Lim, Sunwoo Kim, and Jun Won Choi are with Department of Electrical Engineering, Hanyang University, Seoul,Korea, e-mail: [email protected], remero,[email protected] Shim is with Department of Electrical and Computer Engineering, Seoul National University, Seoul, Korea, e-mail:[email protected].
February 22, 2021 DRAFT
Deep Learning-based Beam Tracking forMillimeter-wave Communications underMobility
I. I
NTRODUCTION
Millimeter wave (mmWave) communication has attracted significant attention for achievingthe continuously increasing data throughput requirement of advanced wireless systems [1]–[3]. However, several challenges should be addressed to enable seamless communication overmmWave-band channels. In particular, the received signal power of mmWave communicationsystems experiences significant attenuation. A potential solution is to employ directional transmit(Tx) and receive (Rx) beamforming antennas, which direct highly directional beams in thedesirable directions to enhance the signal power. Such beams are formed by appropriatelyadjusting the phase and amplitude of the signal for each antenna element [4], [5].Consider a base-station (BS) equipped with N b antennas and a user equipment (UE) with N m antennas. In down-link scenarios, the BS uses a beamforming vector to transmit the data symbolsto the UE and the UE applies a combining vector to receive the transmitted data symbols. Thesebeamforming and combining vectors determine the directions of the beams, which should bechosen to maximize the data throughput. The channel state information should be known toboth the BS and UE to determine the directions of the beams. The procedure for acquiring thechannel information using pilot symbols is called beam training [6], [7]. In beam training, thepilot symbols are transmitted using specifically designed Tx and Rx beams. These beams areoften called sounding beams [8]. The BS and the UE use the combinations of M b Tx soundingbeams and M m Rx sounding beams to obtain the channel information. Consequently, the M b · M m pilot symbols are transmitted. Given the absence of prior knowledge about the channel, both M b and M m should be sufficiently large to cover a wide range of directions. Beam tracking techniques have been proposed to reduce the amount of radio resources requiredfor beam training. When the beam tracking is enabled, the BS transmits the pilot symbols usingfewer sounding beams after the initial acquisition step. (see Fig. 1.) The number of sounding
February 22, 2021 DRAFT BS (cid:1) Beamcycling
BeamcontrolUE CSI acquisition CSI tracking CSI trackingBeamcontrol BeamcontrolBeam transmission periods
Fig. 1. Illustration of beam tracking protocol beams can be reduced without significantly by exploiting the temporal channel correlation. TheBS and UE can use the information on the angles of arrival (AoAs) and angles of departure(AoDs) obtained in the previous beam transmissions to direct only a few beams toward thedirections that ensure good channel estimation performance.Two key design issues exist in implementing beam tracking systems. First, both Tx and Rxsounding beams should be determined in response to the time-varying AoAs and AoDs. Note thatsuch beam control should be predictive to steer the sounding beams toward the future channelstate in advance. Second, the UE needs to update the channel estimate by using the receivedpilot symbols and exploiting the temporal channel correlation.Various channel tracking methods have been proposed thus far. In [9], the authors proposeda beam tracking algorithm that exploits the temporal correlation between AoDs and AoAs.However, adapting to rapid channel variations was challenging, as omni-directional trainingbeams were used. In [10]–[14], various types of Kalman filters were employed to track time-varying channels. In [15], an optimal beam training protocol design scheme was derived based onthe partially observable Markov decision process framework. Compressed sensing (CS) recoveryalgorithms [16] were also used to estimate the AoDs and AoAs of multi-path channels in [3],[8], [17] and were extended to utilize the temporal channel correlation in [9], [18], [19]. In [20],[21], channel tracking was formulated as a maximum likelihood estimation problem. Sensor-aided beam tracking methods have been proposed recently [22]–[24]. These methods attemptedto use an inertial measurement unit (IMU) sensor to assist beam alignment and channel tracking
February 22, 2021 DRAFT in mmWave systems. However, modeling different types of data acquired from sensor andcommunication signals to design beam tracking methods is difficult. Thus, applying traditionalmodel-based approaches for sensor-based beam tracking is a significant challenge.Most UEs are hand-held devices. The pose (i.e., location and orientation) of UE devicescan vary based on the motion of their human users. This results in dynamic and instantaneouschannel variations. In practice, this could cause frequent beam tracking failures. This necessitatesthe execution of expensive channel acquisition procedures to recover from failures. Thus, beamtracking algorithms should handle dynamically-varying channels to reduce beam tracking failuresand consequently save resource overhead. However, the performance of most existing beamtracking algorithms is limited because they rely on somewhat simple prior linear models todescribe time-varying channels. In fact, channels often exhibit structured temporal behavior dueto the motion of the UE device. Therefore, channel models that represent such temporal behaviorwell are required.Recently, deep neural networks (DNNs) have received considerable attention owing to theirability to find an abstract representation of high-dimensional data [25]. DNNs can model complexnon-linear relationships using multiple layers of an artificial neural network. DNNs have achievedstate-of-the-art performance in various challenging machine learning tasks. They have beenparticularly effective for applications in which the existing analytical models cannot adequatelydescribe the distribution of the data. Thus, a DNN can be a suitable candidate for modeling thetemporal behavior of mmWave channels caused by the motion of a UE device. Recently, a DNNhas been applied for beam tracking in mmWave systems in [26]–[28].In this paper, we propose an enhanced beam tracking method, that models rapidly-varyingmmWave channels using DNNs. We employ a long short-term memory (LSTM) architecture todescribe the temporal evolution of the AoAs and AoDs, using the information available in aUE device. Specifically, the LSTM predicts the distribution of the AoA and AoD states for thecurrent beam transmission cycle based on the sequence of the previous channel estimates andIMU sensor signals. This distribution is used for two main beam tracking operations. First, thedistribution of the AoAs and AoDs is used to determine the Rx and Tx sounding beams to beused in the current beam transmission cycle. Second, the predicted channel distribution is used asprior information to update the channel estimate. The proposed LSTM-based prediction model isincorporated into a sequential Bayesian estimation framework, in which the channel information
February 22, 2021 DRAFT is updated through a prediction update step and measurement update step in an alternatingmanner. Note that the proposed method uses the LSTM-based prediction model to update thechannel distribution in the prediction update step. This distribution is then used as the priorchannel information in the subsequent measurement update step. We evaluate the performanceof the proposed beam tracking method via computer simulation. Our results demonstrate thatthe proposed method achieves significant performance gains over conventional beam trackingmethods under various high-mobility scenarios.The contributions of this paper are summarized as follows; • Our method uses a DNN to enhance the beam tracking performance in mmWave systems.As compared with widely used simple linear models, the DNN model can capture thecomplex temporal channel behavior caused by the motion of a device, thereby offering anenhanced beam tracking performance. The superiority of the proposed DNN-based beamtracking scheme over the existing methods is confirmed via numerical evaluation. • We incorporate our DNN-based channel model into the sequential Bayesian filtering frame-work. The role of machine learning models is restricted to modeling the temporal behavior ofchannels only and we use the analytical model describing the relation from the transmittedbeam to the measurements in the measurement update step. This is consistent with thedesign principles of respecting established models for well-known physical processes andusing data-driven approaches only where the actual physical process is barely known (e.g.,temporal channel evolution under mobility environment). This approach contrasts with theend-to-end modeling of beam tracking proposed in [26]. • Recently, a DNN-based channel tracking has been proposed in [26]. The method in [26]directly estimates the AoA using the DNN, whereas the proposed method predicts the futuredistribution of AoA. The predicted AoA information is then used to update the channelestimate based on the measurement model. Another key difference from the aforementionedmethod is that the proposed method utilizes various types of signals acquired by motionsensors for beam tracking.II. MM W AVE B EAM T RACKING S YSTEMS
In this section, we describe the mmWave channel model and introduce the widely used beamtracking protocol.
February 22, 2021 DRAFT
A. MmWave Channel Model
Recall that the BS and UE have antenna arrays of sizes N b and N m , respectively. The downlinkchannel from the BS to the UE can be expressed as the matrix H t of size N m × N b , where the ( i, j ) th element of H t represents the channel gain from the j th antenna of the BS to the i thantenna of the UE. The subscript t represents the t th beam transmission period. The channel H t is assumed to be constant within the t th beam training period. The mmWave channel can berepresented in the angular domain as [3], [17] H t = L X l =1 α l,t a ( m ) ( θ ( m ) l,t ) (cid:16) a ( b ) ( θ ( b ) l,t ) (cid:17) H , (1)where L is the total number of paths, α l,t is the channel gain for the l th path, and θ ( b ) l,t and θ ( m ) l,t are the AoD and AoA, respectively. The AoD and AoA are obtained from θ ( b ) l,t = sin( φ ( b ) l,t ) , θ ( m ) l,t = sin( φ ( m ) l,t ) , where φ ( b ) l,t and φ ( m ) l,t ∈ [ − π , π ] are the AoD and AoA in radians, respectively.The steering vectors a ( b ) ( θ ) and a ( m ) ( θ ) are expressed as a ( b ) ( θ ) = 1 √ N b h , e j πdbθλ , e j π dbθλ , · · · , e j π ( Nb − dbθλ i T a ( m ) ( θ ) = 1 √ N m h , e j πdmθλ , e j π dmθλ , · · · , e j π ( Nm − dmθλ i T , where d b and d m are the distances between adjacent antennas for the BS and UE, respectively and λ is the signal wavelength. In practical scenarios, L tends to be small because only a few pathsexhibit dominant energy. Note that the mmWave channel is determined by the set of parameters γ t = [ α ,t , θ ( m )1 ,t , θ ( b )1 ,t , ..., α L,t , θ ( m ) L,t , θ ( b ) L,t ] T . B. Beam Tracking Protocol
Fig. 1 illustrates a typical beam tracking protocol. Without prior knowledge about the channel,the initial channel acquisition is performed using the Tx and Rx sounding beams, whose direc-tions are distributed over a wide range. The beam tracking mode starts once the initial channelacquisition is completed. At the t th beam transmission period, the beam tracking method usesthe channel knowledge to transmit the pilot symbols using significantly fewer sounding beamsdirected at certain desired directions. After beam transmission, the UE updates the channelestimate based on the measurements. These channel estimates are fed back to the BS througha feedback channel or used for data demodulation. This beam tracking procedure is repeated February 22, 2021 DRAFT in each beam transmission cycle. A similar protocol is observed in the 5G standard, where theSS-burst slot and CSI-RS slot are reserved for the initial channel acquisition and beam tracking,respectively [29].
C. mmWave Channel Estimation
At the t th beam transmission, the BS transmits M b · M m pilot symbols to the UE using M b Txbeams and M m Rx beams. Let f t, , ..., f t,M b represent the beamforming vectors used for the Txbeams and w t, , ..., w t,M m represent the combining vectors for the Rx beams. When an analogbeamformer is used, the beamforming and combining vectors are expressed as f t,i = a ( b ) ( µ ( b ) t,i ) and w t,j = a ( m ) ( µ ( m ) t,j ) , respectively, where µ ( b ) t,i and µ ( m ) t,j are the corresponding directions of thesounding beams. The vector received in the t th beam transmission cycle is expressed as y t, ( i − M m + j = w Ht,j H t f t,i s t,i + n t, ( i − M m + j , (2)for ≤ i ≤ M b and ≤ j ≤ M m , where s t,i is the pilot symbol and n t, ( i − M m + j is theadditive noise. Without losing generality, we let s t,i = 1 in the sequel. Note that, for eachTx sounding beam, M m Rx sounding beams are swept, resulting in M m · M u transmissions.Combining the received signals in a vector y t as y t = [ y t, , ..., y t,M b M m ] T and using the angularchannel representation in (1), we obtain y t = vec( W Ht H t F t ) + n t , (3) = vec L X l =1 α l,t W Ht a ( m ) (cid:16) θ ( m ) l,t (cid:17) (cid:16) a ( b ) ( θ ( b ) l,t ) (cid:17) H F t ! + n t , (4)where vec ( · ) is the vectorization operation , n t = [ n t, , ..., n t,M b M m ] T , W t = h w t, ... w t,M m i ,and F t = h f t, ... f t,M b i . We assume that the channel gains α ,t , ..., α L,t vary slowly so that theycan be assumed as being estimated accurately. For specified F t and W t , the channel estimationproblem is equivalent to the determination of the set of parameters γ t = [ γ T ,t , ..., γ TL,t ] T =[[ θ ( m )1 ,t , θ ( b )1 ,t ] , ..., [ θ ( m ) L,t , θ ( b ) L,t ]] T . Accordingly, we formulate the following state-space equation: • State evolution model γ t = A t γ t − + v t , (5) For example, vec = [1 , , , T . February 22, 2021 DRAFT where A t is the auto-regressive parameter and v t is a complex Gaussian vector CN (0 , V t ) . • Measurement model y t = vec L X l =1 α l,t W Ht a ( m ) (cid:16) θ ( m ) l,t (cid:17) (cid:16) a ( b ) ( θ ( b ) l,t ) (cid:17) H F t ! + n t . (6)Owing to the nonlinearity of the state-space equation, we can use nonlinear Bayesian filteringalgorithms. A popular method used in this regard is the extended Kalman filter (EKF)1) Prediction update step ˆ γ t | t − = A t ˆ γ t − | t − P t | t − = A t P t − | t − A Ht + V t , (7)2) Measurement update step K t = P t | t − O Ht (cid:0) O t P t | t − O Ht + σ t I (cid:1) − P t | t = ( I − K t O t ) P t | t − b γ t | t = b γ t | t − + K t (cid:0) y t − q ( b γ t | t − ) (cid:1) , (8)where the vector q ( γ t ) and Jacobian matrix O t are expressed as q ( γ t ) = L X l =1 vec (cid:18) α l,t W Ht a ( m ) (cid:16) θ ( m ) l,t (cid:17) (cid:16) a ( b ) ( θ ( b ) l,t ) (cid:17) H F t (cid:19) O t = ∂q ( γ t ) ∂γ t (cid:12)(cid:12)(cid:12)(cid:12) γ t = ˆ γ t | t − . The expression for O t is provided in Appendix A. As the prior channel model in (5) capturesonly the first-order dynamics of channel variations, EKF often fails to track the complex channeldynamics in the prediction update step, resulting in a large linearization error in the measurementupdate step. III. R EVIEW OF
LSTM M
ODEL
The LSTM is a DNN architecture widely used to analyze time-series data. Fig. 2 depictsthe structure of the LSTM. The LSTM uses recurrent connections to extract features fromsequence data and stores them in a memory called cell state . When unfolded in time, theconnection from the input to the output in the LSTM is deep in time. This enables an efficient
February 22, 2021 DRAFT σσ σ tanh (cid:2) (cid:1) ∘ ∘ tanh ∘
Forget gateInput gate Output gate (cid:3) (cid:1) (cid:4) (cid:1) (cid:4) (cid:1)(cid:2)(cid:3)
Fig. 2. Structure of the basic LSTM model representation of long sequences. The LSTM has been successfully applied to various machine-learning problems, e.g., natural language processing, speech recognition, and machine translation.The LSTM consists of a cell state, and input, output, and forget gates. The input, output, andforget gating functions can control the information flows entering and leaving the cell state.These gating functions are designed to address the vanishing gradient problems , in which thegradient signals attenuate considerably in learning long-term dependency [30]. Whenever theinput x t is fed into the LSTM, the cell state c t at the time step t is updated according to thefollowing recursive equations i t = σ ( W xi x t + W hi h t − + b i ) (9) f t = σ ( W xf x t + W hf h t − + b f ) (10) o t = σ ( W xo x t + W ho h t − + b o ) (11) g t = tanh( W xc x t + W hc h t − + b c ) (12) c t = f t ⊙ c t − + i t ⊙ g t (13) h t = o t ⊙ tanh( c t ) , (14)where • σ ( x ) = e − x : sigmoid function • a ⊙ b : element-wise product February 22, 2021 DRAFT • W xi , W hi , W xf , W hf , W xo , W ho , W xc , W hc : weight matrices for linear transformation • b i , b f , b o , b c : bias vector • i t : input gating vector • f t : forget gating vector • o t : output gating vector • g t : state update vector • h t : output hidden state vector.The output h t of the LSTM contains the feature required to perform the specified task. Thedesired output can be determined from the feature h t through an additional neural network. TheLSTM is trained to minimize the appropriately designed loss function using the back-propagationthrough time (BPTT) algorithm.IV. P ROPOSED D EEP L EARNING - BASED B EAM T RACKING
In this section, we describe the proposed beam tracking method. The structure of the overallsystem is depicted in Fig. 3. The LSTM-based prediction model predicts the distribution ofthe channel state at the t th beam training period based on all the previously available channelestimates and IMU sensor signals. The prediction model produces the mean and covariancematrix of AoD and AoA separately for each path. Note that the parameters of each model areshared among L paths. As shown in Fig. 3, the output of the prediction model is used to steerboth Tx and Rx sounding beams and update the channel estimates based on the received beams. A. LSTM-based Channel Prediction
The input to the LSTM-based prediction model includes • ˆ γ l,t − δ : t − = (cid:2) ˆ γ Tl,t − δ , ..., ˆ γ Tl,t − (cid:3) T : the sequence of the previous δ channel estimates acquiredbefore the t th beam transmission period begins. • s t − δ : t − = [ s Tt − δ , ..., s Tt − ] T : the sequence of the previous sensor signal samples of J types.For example, with J = 3 , the vector s t − i contains the velocity, angular velocity and angularacceleration samples acquired from the IMU sensor. As the sampling frequency of thesesignals can be different from that of the beam transmissions, the sensor signals can beresampled to produce K samples for each beam transmission cycle. Finally, the vector s t − i is filled with KJ signal samples. February 22, 2021 DRAFT0 ⋯ (cid:1) (cid:2) (cid:3) (cid:0) ⋯⋯ (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:2)(cid:9)(cid:4)(cid:10)(cid:6)(cid:2)(cid:8)(cid:2)(cid:3)(cid:11)(cid:4) (cid:12)(cid:1)(cid:13)(cid:14)(cid:15)(cid:16)(cid:9)(cid:4)(cid:2)(cid:17)(cid:18)(cid:6)(cid:2)(cid:17)(cid:19)(cid:20)(cid:11)(cid:19)(cid:5)(cid:3) (cid:1) (cid:2) (cid:4) (cid:21)(cid:2)(cid:9)(cid:8)(cid:22)(cid:5)(cid:3)(cid:11)(cid:6)(cid:5)(cid:23) (cid:1) (cid:2) (cid:4) (cid:21)(cid:2)(cid:9)(cid:8)(cid:11)(cid:6)(cid:9)(cid:3)(cid:4)(cid:8)(cid:19)(cid:4)(cid:4)(cid:19)(cid:5)(cid:3) (cid:22)(cid:1)(cid:24)(cid:13)(cid:6)(cid:9)(cid:20)(cid:25)(cid:19)(cid:3)(cid:26) ⋯ (cid:1) (cid:2) (cid:4) (cid:12)(cid:1)(cid:13)(cid:14)(cid:15)(cid:16)(cid:9)(cid:4)(cid:2)(cid:17)(cid:18)(cid:6)(cid:2)(cid:17)(cid:19)(cid:20)(cid:11)(cid:19)(cid:5)(cid:3) (cid:1) (cid:2) (cid:3) ⋯ (cid:21)(cid:2)(cid:9)(cid:8)(cid:22)(cid:5)(cid:3)(cid:11)(cid:6)(cid:5)(cid:23) (cid:1) (cid:2) (cid:3) ⋯ ⋯ ⋯ ⋯ (cid:27)(cid:13)(cid:27)(cid:13) (cid:2)(cid:3)(cid:4) (cid:1),(cid:3)|(cid:3)(cid:5)(cid:6) , (cid:6) (cid:1),(cid:3)|(cid:3)(cid:5)(cid:6) (cid:7)(cid:2)(cid:3)(cid:4) (cid:6),(cid:3)|(cid:3)(cid:5)(cid:6) , (cid:6) (cid:6),(cid:3)|(cid:3)(cid:5)(cid:6) (cid:7)(cid:2)(cid:3)(cid:4) (cid:6),(cid:3)|(cid:3) , (cid:6) (cid:6),(cid:3)|(cid:3) (cid:7)(cid:2)(cid:3)(cid:4) (cid:1),(cid:3)|(cid:3) , (cid:6) (cid:1),(cid:3)|(cid:3) (cid:7)(cid:3)(cid:4) (cid:6),(cid:3)(cid:5)(cid:6)|(cid:3)(cid:5)(cid:6) (cid:3)(cid:4) (cid:1),(cid:3)(cid:5)(cid:6)|(cid:3)(cid:5)(cid:6) (cid:2)(cid:3)(cid:4) (cid:1),(cid:3)(cid:5)(cid:6)|(cid:3)(cid:5)(cid:6) , (cid:6) (cid:1),(cid:3)(cid:5)(cid:6)|(cid:3)(cid:5)(cid:6) (cid:7)(cid:2)(cid:3)(cid:4) (cid:6),(cid:3)(cid:5)(cid:6)|(cid:3)(cid:5)(cid:6) , (cid:6) (cid:6),(cid:3)(cid:5)(cid:6)|(cid:3)(cid:5)(cid:6) (cid:7) ⋯ (cid:18)(cid:6)(cid:2)(cid:28)(cid:19)(cid:5)(cid:10)(cid:4)(cid:7)(cid:22)(cid:1)(cid:24)(cid:7)(cid:2)(cid:4)(cid:11)(cid:19)(cid:8)(cid:9)(cid:11)(cid:2)(cid:4) Fig. 3. Block diagram of the proposed beam tracking method • C t : the vector that represents contextual information such as the location and activity ofthe UE. Although it does not represent sequential data, contextual information providessupplementary information on the channels.The proposed prediction model aims to determine the distribution p ( γ l,t | ˆ γ l,t − t − δ , s t − t − δ , C t ) for each channel path for the given past channel estimates ˆ γ l,t − δ : t − , sensor signals s t − δ : t − , andcontext information C t . We employ the LSTM to model the dependencies between the inputand the future channel state. The structure of the LSTM-based prediction model is depicted inFig. 4. The signal samples { ˆ γ l,t − δ , s t − δ , C t } , ..., { ˆ γ l,t − , s t − , C t } are encoded separately by the input fully-connected (Fc) layers , i.e., ν l,t − i = Fc( { ˆ γ l,t − i , s t − i , C t } ) , (15)where ν l,t − i is the embedding vector obtained by the Fc layers. The embedding vectors are fedto the LSTM one by one to update the cell state. After δ update of the LSTM, the output h t isfed into the output Fc layers to produce the estimate ˆ γ l,t . That is, ˆ γ l,t = Fc (LSTM ( { ν l,t − δ , ..., ν l,t − } )) . (16)The parameters of the LSTM and Fc layers are determined in the training procedure. Inpractice, the training data could be collected by deploying several reference UEs, that log February 22, 2021 DRAFT1 (cid:1)(cid:2)(cid:3)(cid:4) (cid:1)(cid:2) (cid:1)(cid:2)(cid:3)(cid:4) (cid:1)(cid:2) ⋯ (cid:1)(cid:2)(cid:3)(cid:4) (cid:1)(cid:2) (cid:1)(cid:2) (cid:1),(cid:3)(cid:4)(cid:5) , (cid:4) (cid:3)(cid:4)(cid:5) , (cid:5) (cid:3) (cid:1)(cid:2) (cid:1),(cid:3)(cid:4)(cid:5)(cid:6)(cid:7) , (cid:4) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7) , (cid:5) (cid:3) (cid:1)(cid:2) (cid:1),(cid:3)(cid:4)(cid:7) , (cid:4) (cid:3)(cid:4)(cid:7) , (cid:5) (cid:3) (cid:3)(cid:4)(cid:5)(cid:5) (cid:1) (cid:1),(cid:3) (cid:3)(cid:6)(cid:7)(cid:8)(cid:9) (cid:1)(cid:2) Fig. 4. Structure of the LSTM-based prediction model the channel states and sensor signals in real scenarios. The model is trained to minimize thenegative log-likelihood loss P k γ l,t − ˆ γ l,t k , where the ground truth γ l,t can be obtained directlyfrom the training data. The prediction network is trained using the standard BPTT algorithmwith the ADAM optimization [31]. The model weights are updated over a minibatch of size MINIBATCH . The training starts with the initial learning rate
LEARNING RATE . The learningrate decays by half in each
DECAY EPOCH epochs. Note that the model is trained over a totalof
TOTAL EPOCH epochs.
B. Channel Tracking
The sequential Bayesian estimation framework is widely used to estimate time-varying chan-nels. The Bayesian principle involves updating the distribution of a channel based on all theinformation available at each step. By adopting this principle, we update the mean b γ l,t − | t − and the covariance matrix P l,t − | t − by b γ l,t | t − and P l,t | t − through the prediction update step.Subsequently, the measurement update step updates b γ l,t | t − and P l,t | t − by b γ l,t | t and P l,t | t . In theprediction update step, the LSTM-based prediction model is used to obtain b γ l,t | t − and P l,t | t − .As the LSTM-based prediction model produces the point estimate of the future channel state, weemploy the unscented transformation (UT) [12], [32] to obtain the distribution. First, for given b γ l,t − | t − and P l,t − | t − , we generate P + 1 sigma vectors χ i with the corresponding weights February 22, 2021 DRAFT2 w i , i.e., χ = b γ l,t − | t − (17) χ i = b γ l,t − | t − + (cid:16)q ( L + λ ) P l,t − | t − (cid:17) i i = 1 , ..., P (18) χ i = b γ l,t − | t − − (cid:16)q ( L + λ ) P l,t − | t − (cid:17) i − P i = P + 1 , ..., P (19) w ( m )0 = λ/ ( L + λ ) (20) w ( c )0 = λ/ ( L + λ ) + (1 − α + β ) (21) w ( m ) i = w ( c ) i = 1 / (2( K + λ )) , (22)where λ = α P − P is a scaling parameter, α = 10 − determines the spread of the sigmapoints around b γ t − | t − , and β = 2 reflects the prior distribution of γ l,t . (cid:0)p ( L + λ ) P t − | t − (cid:1) i isthe i th row of the matrix square root. These sigma vectors are passed through the LSTM-basedprediction model, which generates P + 1 outputs ν , ..., ν P . Finally, the updated distribution b γ l,t | t − and P l,t | t − are obtained by the weighted sums b γ l,t | t − = P X i =0 w ( m ) i ν i (23) P l,t | t − = P X i =0 w ( c ) i ( ν i − b γ l,t | t − )( ν i − b γ l,t | t − ) T . (24)It was shown in [32] that the UT yields approximations that are accurate up to at least the secondorder, with the accuracy of the third and higher-order moments determined by the selection of α and β . After the prediction update step is completed, the measurement update step is performedto obtain the statistics b γ l,t | t and P l,t | t from K t = P t | t − O Ht (cid:0) O t P l,t | t − O Ht + σ t I (cid:1) − P l,t | t = ( I − K t O t ) P l,t | t − b γ l,t | t = b γ l,t | t − + K t (cid:0) y t − q ( b γ l,t | t − ) (cid:1) . (25)Note that this measurement update step is equivalent to that of the EKF. C. Predictive Beam Control
The direction of the sounding beams needs to be determined based on the best available channelinformation in the t th beam transmission cycle. Before the sounding beams are transmitted, February 22, 2021 DRAFT3 the best available channel information is the statistics b γ l,t | t − and P l,t | t − obtained using theUT. Based on the Gaussian approximation p ( γ l,t | ˆ γ l,t − t − δ , s t − t − δ , C t ) ≈ N ( b γ l,t | t − , P l,t | t − ) ,we determine the beam angles µ ( b ) t, , ..., µ ( b ) t,M b and µ ( m ) t, , ..., µ ( m ) t,M m , where f t,i = a ( b ) ( µ ( b ) t,i ) and w t,j = a ( m ) ( µ ( m ) t,j ) . The optimal beam angles can be determined by maximizing the expectedchannel estimation performance with respect to the parameters µ ( b ) t, , ..., µ ( b ) t,M b and µ ( m ) t, , ..., µ ( m ) t,M m .An in-depth study on the optimization of the sounding beams has been presented in [3], [15],[33]. In our previous work [33], the problem of sounding beam adaptation was formulated as aminimization of the Cramer-Rao lower bound (CRLB) of the channel estimation error over thecombinations of beam codebook indices. When only two sounding beams are used for Tx andRx, i.e., M b = M m = 2 , the optimal beam directions could be determined by a two-dimensionalsearch over the beam codebook. In this study, adopting the method in [33], the CRLB is derivedfor the given channel distribution N ( b γ l,t | t − , P l,t | t − ) , and the optimal sounding beam anglesare determined. With the setup M b = M m = 2 , we choose the values of the beam angles µ ( b ) t, , µ ( b ) t, , µ ( m ) t, , µ ( m ) t, using two-dimensional search. Note that this beam control algorithm movesthe group of sounding beams toward the future AoD and AoA directions in advance. D. Algorithm Summary
The proposed beam tracking algorithm is summarized in Algorithm 1.V. E
XPERIMENTAL R ESULTS
In this section, we evaluate the performance of the proposed beam tracking algorithm.
A. Simulation Setup1) MmWave System Setup:
In our simulations, we considered 28GHz frequency band com-munications with uniform linear array (ULA) antennas whose adjacent elements are spaced bya half wavelength. We considered the communication between a single BS with N b = 32 Txantennas and a single UE device with N m = 32 Rx antennas. Following the 5G NR standard [29],the symbol duration over which a single beam is transmitted was set to . µs and 14 symbolsare included in each slot of duration µs . In the simulations, sounding beams ( M b = 2 and M m = 2 ) were transmitted every T CSI slots. The periodicity of beam transmission T CSI was setto 160 slots based on the 5G NR standard [29]. We use the beam codebook, which contains 64
February 22, 2021 DRAFT4
Algorithm 1
Proposed beam tracking algorithm At the t th beam transmission cycle Input: b γ ,t − | t − , ..., b γ L,t − | t − and P ,t − | t − , ..., P L,t − | t − Prediction update step: for i = 1 to L ... do Generate P + 1 sigma samples χ , ..., χ P according to b γ l,t − | t − and P l,t − | t − . Generate P + 1 output samples ν , ..., ν P using the LSTM prediction model. Update b γ l,t | t − and P l,t | t − according to (23) and (24). end for Beam adaptation and transmission:
Determine the directions of the sounding beams using the method in [33] and transmit thebeams accordingly.
Measurement update step: for i = 1 to L ... do Update b γ l,t | t and P l,t | t according to (25). end for t ← t + 1 and go back to line 1.beams with equally spaced angles. The symbol slots not used for beam tracking were allocatedfor data transmission. The data symbols in each slot were modulated using binary phase-shiftkeying (BPSK) modulation. The data precoding and combining matrices were obtained from theleft and right singular vectors of the channel matrix associated with the highest singular value,respectively.
2) Mobility Model:
We assume that the BS is located sufficiently far away from the UE andthe AoD does not change considerably in time. On the contrary, owing to the motion of the UE,the AoA varies considerably. Thus, we assume that only AoA varies in time according to thefollowing dynamic model: a l,n = (1 − ρ ) a avg + ρa l,n − + w t (26) θ ( m ) l,n = θ ( m ) l,n − + ∆ t ( a l,n ) , (27) February 22, 2021 DRAFT5 where n is the slot index, a l,n denotes the angular velocity of AoA for the l th path, θ ( m ) l,n denotesthe AoA, a avg denotes the average velocity, ∆ t = 125 us is the symbol duration, ρ = 0 . isthe auto-regressive (AR) parameter, and w t ∼ N (0 , . − ρ )) . Note that the angular velocity a l,n is modeled by the AR process, and the AoA θ ( m ) l,n is generated by accumulating the velocity.We assume that the channel gain and AoD are constant and known. The number of paths L wasset to 3. The AoAs were generated independently for each path. Fig. 5 shows the change in theAoA and angular velocity with different values of a avg . A higher value of a avg leads to fastermotion and consequently more dynamic AoA variations. Note that a avg = 0 . π indicates thatthe UE device rotates in approximately . s .
3) LSTM-based Prediction Model:
We present the detailed configurations of the proposedprediction model. The length of the input sequence for updating the LSTM was set to δ = 3 .Each input vector consists of • The previous channel estimate • K = 4 samples of angular velocity sensor measurements ( rad/s ) • K = 4 samples of angular acceleration sensor measurement ( rad /s ).The IMU sensor measurements were generated by computing the first and second samplederivatives of the AoA and adding Gaussian noise. The signal-to-noise ratio (SNR) was setto 5 dB when testing the prediction model. We assume that the sampling period of the IMUsensor signals is K = 4 times lower than T CSI . The FC layers at the input and output have 16and 32 hidden nodes, respectively. The LSTM uses the stacked cell states of size 32.
4) Training Procedure:
The training configurations are as follows: • MINIBATCH : 64 • INITIAL LEARNING RATE : 0.01 • DECAY EPOCH : 3 • DECAY RATE : 0.1 • TOTAL EPOCH : 30 • OPTIMIZER : Adam optimizer
A total of , , data examples were generated for training and 1,000,000 examples wereused to evaluate the proposed beam tracking method. The training data were generated with arandom SNR uniformly distributed in the range [6 , dB. The SNR of the sensor signals wasalso randomly determined from the same candidate set. February 22, 2021 DRAFT6
Time (ms) -0.0500.050.10.150.20.250.3 A ng l e (r ad ) a avg = 0a avg = 0.2a avg = 0.4 (a) Time (ms) -0.200.20.40.60.811.21.41.61.8 A ngu l a r v e l o c i t y (r ad / s ) a avg = 0a avg = 0.2a avg = 0.4 (b)Fig. 5. Variation of (a) AoA and (b) angular velocity for several values of the parameter a avg . B. Experimental Results
In this section, we compare our method with the following mmWave channel tracking methods:1) Compressive channel tracking [19]: Orthogonal matching pursuit [34] followed by off-gridrefinement was used to track the AoA.2) EKF method [10]: The AoA was estimated using the EKF.3) Least mean square (LMS) method [20]: The AoA was estimated by using the LMS filter.4) LSTM based tracking [26]: The LSTM model directly estimates the AoA. It was trained
February 22, 2021 DRAFT7
SNR -3 -2 -1 BE R Compressive trackingEKF based trackingLMS trackingLSTM based trackingProposed (CSI + IMU)Proposed (CSI only) (a)
SNR -3 -2 -1 BE R Compressive trackingEKF based trackingLMS trackingLSTM based trackingProposed (CSI + IMU)Proposed (CSI only) (b)
SNR -3 -2 -1 BE R Compressive trackingEKF based trackingLMS trackingLSTM based trackingProposed (CSI + IMU)Proposed (CSI only) (c)Fig. 6. BER versus SNR of several channel tracking methods for (a) a avg = 0 . π , (b) a avg = 0 . π , and (c) a avg = 0 . π . using the cosine loss function.5) Proposed (CSI) method: Only previous AoA estimates were used to predict the futurechannel state information (CSI). This method was evaluated to investigate the advantageof using IMU sensors for beam tracking.6) Proposed (CSI+IMU) method: The previous AoA estimates and IMU measurements wereused to predict the future channel distribution.As the compressive channel tracking, LMS, and LSTM-based tracking methods do not producethe distribution of AoA, we used M m = 2 beams closest to the previous AoA estimate as the Rx February 22, 2021 DRAFT8
SNR -14-12-10-8-6-4-2024 M SE Compressive trackingEKF based trackingLMS trackingLSTM based trackingProposed (CSI + IMU)Proposed (CSI only) (a)
SNR -14-12-10-8-6-4-2024 M SE Compressive trackingEKF based trackingLMS trackingLSTM based trackingProposed (CSI + IMU)Proposed (CSI only) (b)
SNR -12-10-8-6-4-2024 M SE Compressive trackingEKF based trackingLMS trackingLSTM based trackingProposed (CSI + IMU)Proposed (CSI only) (c)Fig. 7. Normalized MSE versus SNR of several channel tracking methods for (a) a avg = 0 . π , (b) a avg = 0 . π , and (c) a avg = 0 . π . sounding beams. In contrast, like the proposed method, the EKF method yields the distributionof the AoA, which is used to determine the Rx sounding beams. The normalized mean squareerror (MSE) is defined as M SE = 10 log (cid:13)(cid:13)(cid:13) H t − ˆH t (cid:13)(cid:13)(cid:13) F k H t k F . Fig. 6 shows the bit error rate (BER) performance as a function of SNR. The parameter a avg indicates the extent of mobility of the UE. Fig. 6 (a), (b), and (c) show the performance curvesfor a avg = 0 . π , . π , and . π , respectively. We observe that the proposed (CSI+IMU) method February 22, 2021 DRAFT9 outperforms the existing methods for all the cases considered. When a avg is . π rad/s, theproposed method achieves a performance gain of approximately 1 dB over the EKF method atthe BER of − . As a avg increases, the channel changes more dynamically and the performancegain of the proposed method increases. With a avg = 0 . π , the proposed method achieves a gainof more than 3 dB over other algorithms. Furthremore, with a avg = 0 . π , the performance gainincreases up to more than 10 dB. This indicates that the LSTM-based channel model providesa more accurate model of time-varying AoAs, and thus, superior performance is achieved underhigher mobility. Fig. 6 also shows the advantage of using IMU sensors for beam tracking. Theproposed (CSI+IMU) method achieves a performance gain over the proposed (CSI) method,especially in the low SNR range. This appears to be because the channel estimates obtained inthe previous beam transmission cycles would not be reliable in the low SNR range; thus, theIMU sensor signals can compensate the degraded channel estimation. Note that, although both theproposed and EKF methods perform the same measurement update step, the proposed methodachieves a better performance owing to its more accurate prediction results in the predictionupdate step. Note also that, although both our method and the method in [26] employ DNNfor beam tracking, the proposed method performs better by leveraging the underlying domainknowledge in the measurement model.Fig. 7 shows the normalized MSE as a function of SNR for several beam tracking methods.The proposed method achieves a significant performance gain over the existing methods for allthe cases considered. The performance gain of the proposed method also increases with a avg .The proposed method can track rapidly varying channels better by using the DNN and IMUsensor measurements.Fig. 8 illustrates the variation in the MSE performance with the beam transmission period T CSI and the angular velocity a avg when the SNR is set to dB. Fig. 8 (a) shows the plotof MSE versus T CSI when a avg is fixed to . π . As T CSI increases, the sounding beams aretransmitted less frequently, and the beam tracking method experiences larger channel variations.With a small T CSI , the performance of the EKF method is comparable to that of the proposedmethod. However, the performance of the EKF method severely deteriorates with T CSI , andconsequently, the performance gap between these two methods increases rapidly. Fig. 8 (b)shows the plot of MSE versus a avg when T CSI is fixed to 160 slots. The performance of thebeam tracking algorithms degrades as the channel changes more dynamically owing to the fast
February 22, 2021 DRAFT0 T CSI -9-8-7-6-5-4-3-2-101 M SE EKF based trackingLMS trackingLSTM based trackingProposed (CSI+IMU)Proposed (CSI) (a) M SE EKF based trackingLMS trackingLSTM based trackingProposed method (CSI+IMUProposed method (CSI) (b)Fig. 8. Normalized MSE versus (a) T CSI and (b) a avg . motion of the UE. As a avg increases, the EKF method does not perform well because the linearchannel model used in the EKF method does not sufficiently capture the behavior of time-varyingchannels. In contrast, the proposed method successfully models the complex channel behaviorfor reliable beam tracking. February 22, 2021 DRAFT1
VI. C
ONCLUSIONS
In this paper, we proposed a deep learning-based beam tracking method for mmWave com-munication. The proposed beam tracking method was designed to track fast-varying AoD andAoA due the motion of the UE device. We employed the LSTM to model the channel variationand predict the future distribution of the channel state based on the sequence of the previouschannel estimates and IMU sensor measurements. Our method is based on a sequential Bayesianestimation framework, in which the prediction model yields the prior distribution of the channelin the prediction update step, and the predicted distribution is used to update the channel estimatein the measurement update step. Thus, our method is a hybrid approach in that we used bothan LSTM-based channel model and an analytical measurement model for beam tracking. Oursimulation results showed that the proposed method achieved a significant performance gainover the EKF baseline and outperformed the existing beam tracking methods, especially in high-mobility scenarios. VII. A
PPENDIX
A. Derivation of Jacobian Matrix
The Jacobian matrix O t is expressed as O t = " ∂q ( γ t ) ∂θ ( b )1 ,t , ∂q ( γ t ) ∂θ ( m )1 ,t , · · · , ∂q ( γ t ) ∂θ ( b ) L,t , ∂q ( γ t ) ∂θ ( m ) L,t , whose elements in the ( M ( i −
1) + j ) th row are given by ∂q ( M ( i − j ) ∂θ ( b ) l,t = α l,t n t n r k b e (cid:16) k b ( θ ( b ) l,t − ν ( b ) t,i ) (cid:17) − k b N b e (cid:16) k b N b ( θ ( b ) l,t − ν ( b ) t,i ) (cid:17) − k b ( N b − e (cid:16) k b ( N b +1)( θ ( b ) l,t − ν ( b ) t,i ) (cid:17) (cid:16) − e (cid:16) k b ( θ ( b ) l,t − ν ( b ) t,i ) (cid:17) (cid:17) × − e (cid:16) k m N m ( θ ( m ) l,t − ν ( m ) t,j ) (cid:17) − e (cid:16) k m ( θ ( m ) l,t − ν ( m ) t,j ) (cid:17) ∂q ( M ( i − j ) ∂θ ( m ) l,t = α l,t n t n r − e (cid:16) k b N b ( θ ( b ) l,t − ν ( b ) t,i ) (cid:17) − e (cid:16) k b ( θ ( b ) l,t − ν ( b ) t,i ) (cid:17) × k m e (cid:16) k m ( θ ( m ) l,t − ν ( m ) t,j ) (cid:17) − k m N m e (cid:16) k m N m ( θ ( m ) l,t − ν ( m ) t,j ) (cid:17) − k m ( N m − e (cid:16) k m ( N m +1)( θ ( m ) l,t − ν ( m ) t,j ) (cid:17) (cid:16) − e (cid:16) k m ( θ ( m ) l,t − ν ( m ) t,j ) (cid:17) (cid:17) , where k b = − j πd b λ and k m = − j πd m λ . February 22, 2021 DRAFT2 R EFERENCES [1] T. S. Rappaport et al. , “Millimeter wave mobile communications for 5G cellular: It will work!,”
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