I Introduction
Due to the low hardware cost and energy consumption, analog beamforming is often used in mmWave mobile communications to provide large array gains [1, 2]. However, the beam steered by analog beamforming has small angular spreads. Slight misalignment can cause severe energy loss. Accurate alignment can be achieved by beam training at the expense of large pilot overhead in static or quasistatic scenarios. Nevertheless, this price is unacceptable in fastmoving environments. Therefore, efficient beam tracking is important for serving fast mobility users in mmWave communication.
Some beam tracking methods has been proposed [3, 4, 5]
, utilizing historical observations and estimations to obtain current estimate. Despite this, the analog beamforming vectors are not optimized in those tracking algorithms, resulting in a waste of transmission energy. A beam tracking algorithm is proposed in
[6], trying to optimize the analog beamforming vectors, assuming the channel coefficient is known. In [7], the authors start to jointly track the channel coefficient and beam direction with optimal analog beamforming vectors. The theorems of convergence and optimality are established for joint tracking. However, all these algorithms are based on uniform linear array (ULA) antennas, which can only support onedimensional (1D) beam tracking. While in several mobile scenarios, e.g., unmanned aerial vehicle (UAV) scenarios [8], the beam may also come from different horizontal and vertical directions. Hence, we need to dynamically track the twodimensional (2D) beam direction with 2D phased antenna arrays.This problem is challenging due to the following three reasons: (i) with analog beamforming, we can only obtain part of the system information through one observation. (ii) We need to jointly track channel coefficient and 2D beam direction and the analog beamforming vectors also need to be adjusted. Therefore, it is a dynamic joint optimization problem with sequential analog beamforming vectors and these analog beamforming vectors also need to be optimized. (iii) Compared with 1D beam direction, more analog beamforming vectors are required when tracking 2D beam direction. As a result, the optimization dimension greatly increases.
In this paper, we design a joint beam and channel tracking algorithm for 2D phased antenna arrays to handle the problem above. The main contributions and results are summarized as follows:

This algorithm can achieve the minimum probing overhead for joint beam and channel tracking.

In static scenarios, we get the performance bound, i.e., the minimum CRLB by optimizing the analog beamforming vectors under some constraints. A general way to generate the optimal analog beamforming vectors is proposed with a sequence of parameters. These parameters are proved to be asymptotically optimal in different conditions, e.g., channel coefficients, and path directions, as the number of antennas grows to infinity.

We prove that our algorithm can converge to the minimum CRLB with high probability in static scenarios.

Simulation results show that our algorithm approaches the minimum CRLB quickly in static scenarios. In dynamic scenarios, our algorithm can achieve lower tracking error and faster tracking speed compared with several existing algorithms.
Ii System Model
We consider a mmWave receiver equipped with a planar phased antenna array^{1}^{1}1Note that tracking is needed at both the transmitter and receiver. However, considering the transmitterreceiver reciprocity, the beam and channel tracking of both sides have similar designs. Hence, we focus on beam and channel tracking on the receiver side., as shown in Fig. 1. The planar array consists of antenna elements that are placed in a rectangular area, with a distance () between neighboring antenna elements along axis (axis)^{2}^{2}2To obtain different resolutions in horizontal direction and vertical direction, the antenna numbers along different directions may not be the same, i.e., [9]. To suppress sidelobe, the antennas may be unequally spaced, i.e., [10].. The antenna elements are connected to the same RF chain through different phase shifters. The system is timeslotted. To estimate and track the direction of the incoming beam, the transmitter sends predetermined pilot symbols in each time slot, where is the transmit power of each pilot symbol.
In mmWave channels, only a few paths exist due to the weak scattering effect [1]. Because the angle spread is small and the mmWave system is usually configured with a large number of antennas, the interaction between multipaths is relatively weak. In other words, the incoming beam paths are usually sparse in space, making it possible to track each path independently [11]. Hence, we focus on the method for tracking one path. Different paths can be tracked separately by using the same method.
In timeslot , the direction of the incoming beam path is denoted by (), where is the elevation angle of arrival (AoA) and is the azimuth AoA. The channel vector of this path is
(1) 
where is the complex channel coefficient, is the direction parameter vector determined by (),
(2) 
is the steering vector with , and is the wavelength.
Let be the analog beamforming vector for receiving the th pilot symbol in timeslot , given by
(3) 
where is the direction parameter offset corresponding to . After phase shifting and combining, the observation at the baseband output of RF chain is given by
(4) 
where is an i.i.d.
circularly symmetric complex Gaussian random variable. Define
as the channel parameter vector in timeslot , as the analog beamforming matrix, andas the noise vector. Then the conditional probability density function of the observation vector
is given by(5) 
In timeslot , the receiver needs to choose an analog beamforming matrix and obtain an estimate of the channel parameter vector . From a control system perspective, is the system state, is the estimate of the system state, the analog beamforming matrix is the control action and is a nonlinear noisy observation determined by the system state and control action.
Iii Problem Formulation and Optimal Beamforming Matrix
Iiia Problem Formulation
Let denote a beam and channel tracking scheme. We consider a particular set of causal beam tracking policies: in timeslot , the analog beamforming matrix and estimate are based on the previously used analog beamforming matrix and historical observations . Hence, in th timeslot, the beam and channel tracking problem is formulated as:
(6)  
s.t.  (7)  
where the constraint (7) ensures that
is an unbiased estimation of the channel vector
and the constraints (1)(4) ensure the steering vector form of analog beamforming vectors.Problem (6
) is difficult to solve optimally due to several reasons: (i) it is a constrained partially observed Markov decision process (CPOMDP) that is usually quite difficult to solve. (ii) The analog beamforming matrix
and the estimate need to be optimized. However, both the optimization of and are nonconvex problems.Before giving some theoretical results of problem (6), we will first study the pilot overhead needed for beam and channel tracking in 2D phased antenna arrays.
IiiB How Many Pilots Are Needed?
According to [7], two pilots in each timeslot are sufficient to jointly track the channel coefficient and 1D beam direction. When tracking the horizontal and vertical beam direction simultaneously, four pilots are feasible by separately using two pilots to track each dimension of the 2D beam direction. However, with four pilots, the channel coefficient is updated twice in each timeslot, possibly leading to redundancy. Hence, we can jointly track channel coefficient and 2D beam direction to further reduce pilot overhead.
When tracking the channel parameters jointly, four real variables (i.e., the real part and imaginary part of channel coefficient and the two direction parameters ) need to be estimated. Then the following lemma is proposed to help determine the smallest :
Lemma 1.
If the analog beamforming vectors are steering vectors, i.e., , then at least observations are needed to estimate real variables in timeslot .
Proof.
See Appendix A. ∎
Lemma 1 tells us at least three observations are required in each timeslot to estimate four real variables. Hence, the smallest pilot number in each timeslot is , i.e., the analog beamforming matrix .
IiiC Lower Bound of Tracking Error
The huge challenge to solve problem (6) optimally makes it hard to complete in just one paper. Therefore, we perform some theoretical analysis for static scenarios as the first step in this paper.
Consider the problem of tracking a static beam, where for all timeslots. The CramérRao lower bound theory gives the lower bound of the unbiased estimation error according to [12]. Based on this, we introduce the following lemma to obtain the lower bound of tracking error:
Lemma 2.
The MSE of channel vector in (6) is lower bounded as follows:
(8)  
where and the Fisher information matrix is given by
(9)  
with , , and .
Proof.
See Appendix B. ∎
The CRLB in (8) is a function of the analog beamforming matrices . It is hard to optimize so many beamforming matrices simultaneously. Suppose that . Then we can get the minimum CRLB under this constriant, given by
(10) 
Solving problem (IIIC) yields the optimal analog beamforming matrix :
(11) 
where denote the optimal direction parameter offsets. Hence, let and we can obtain the minimum CRLB by (IIIC).
IiiD Asymptotically Optimal Analog Beamforming Matrix
Let us consider the optimal analog beamforming matrix . In (IIIC), three 2D direction parameter offsets need to be optimized. It is hard to get analytical results for such a sixdimensional nonconvex problem. Numerical search is a feasible way to handle the problem. However, these optimal offsets may be related to some system parameters, e.g., channel coefficient , direction parameter vector x and antenna array size . Once these system parameters change, numerical search has to be reconducted, leading to high complexity. To overcome this challenge, we explore the properties of and obtain the following lemma:
Lemma 3.
The optimal direction parameter offsets have the following three properties:
1) are invariant to the channel coefficient ;
2) are invariant to the direction parameter vector x;
3) converge to constant values as :
Proof.
See Appendix C. ∎
Lemma 3 reveals that are only related to array size . Hence, the numerical search complexity can be reduced to one for a particular array size . Even if may change for different array sizes, we can adopt to take the place of as long as and are sufficiently large. Therefore, the numerical search times are reduced to one.
By numerical search in the main lobe of the direction parameter vector:
(12) 
we can obtain the asymptotically optimal direction parameter offsets in TABLE I and Fig. 2.
With these offsets, a general way to generate the asymptotically optimal analog beamforming matrix is obtained to achieve the minimum CRLB as below:
(13) 
By adopting to smaller size antenna arrays, we compare the minimum CRLB and the CRLB corresponding to in TABLE I.
As illustrated in Fig. 3, when antenna number , we can approach the minimum CRLB with a relative error less than by using . Therefore, with , the minimum CRLB is obtained for different beam directions, different channel coefficients and different antenna numbers when .
Iv Asymptotically Optimal Joint Beam and Channel Tracking
Iva Joint Beam and Channel Tracking
The proposed tracking algorithm is similar to that in [7]. The main difference is that we need pilots to estimate the initial direction parameter offsets and three analog beamforming vectors to track the timevarying beam direction.
Joint Beam and Channel Tracking:

[leftmargin=*]

Coarse Beam Sweeping: As shown in Fig. 4, pilots are received successively. The analog beamforming vector corresponding to the observation is . The initial estimate is obtained by:
(14) where , is the codebook size with and , , , and .

Beam and channel tracking: In timeslot , three pilots are received by using analog beamforming vectors given below:
(15) where and are given by TABLE I. The estimate is updated by
(16) where , , and . Here, is the step size and will be specified later.
IvB Asymptotic Optimality Analysis
In the tracking procedure (16), there exist multiple stable points and these stable points correspond to the local optimal points for our proposed algorithm. To study these stable points, we rewrite (16) as (17):
(17) 
where is defined as follows:
(18) 
and is given by
(19) 
A stable point of satisfies two conditions: 1) ; 2) is negative definite. Hence, we define the stable points set in timeslot as : .
The channel parameter is a stable point: when ,
1) in (IVB). Hence, ;
2) by derivation, where is a identity matrix. Thus,
is negative definite.
Therefore, is a stable point.
Other stable points in correspond to the local optimal points of the beam and channel tracking problem, which are without the main lobe . Except for the channel parameter vector , the antenna array gain of other stable points in is quite low, resulting in low tracking accuracy. Therefore, one key challenge is to ensure that the tracking algorithm converges to rather than other stable points.
In static scenarios, where , the corresponding theorems are developed to study the convergence of our algorithm. We adopt the diminishing stepsize in (20), given by [14, 15, 16]
(20) 
where and .
Theorem 1 (Convergence to a Unique Stable Point).
If is given by (20) with and , then converges to a unique stable point with probability one.
Proof.
See Appendix D. ∎
Therefore, for the general stepsize in (20), converges to a unique stable point.
Theorem 2 (Convergence to Direction parameter vector x).
If (i) the initial estimate of x is within the main lobe, i.e., , and (ii) is given by (20) with , then there exist some and such that
(21) 
Proof.
See Appendix E. ∎
At the coarse beam sweeping stage of our proposed algorithm, the initial estimation within main lobe in (12) can be obtained with high probability. Under the condition , Theorem 2 tells us the probability of is related to . Hence, we can reduce the stepsize and increase the transmit SNR to make sure that with probability one.
Theorem 3 (Convergence to x with minimum CRLB).
If (i) and (ii) is given by (20) with and any , then is asymptotically Gaussian and
(22) 
Proof.
See Appendix F. ∎
V Numerical Results
We compare the proposed algorithm with four other algorithms: the compressed sensing algorithm in [5], the IEEE 802.11ad algorithm in [17]
, the extended Kalman filter (EKF) method in
[18] and the joint beam and channel tracking algorithm in [7] (using two pilots to track each dimension of the 2D beam direction). In each timeslot, three pilots are transmitted for all the algorithms to ensure fairness. When adopting the joint beam and channel tracking algorithm by using four pilots, we use a buffer to store the received pilots and update the estimate when receiving four new pilots. Based on the model in Section II, the parameters are set as: , the antenna spacing , the codebook size , the pilot symbol = 1, and the transmit .In static scenarios, the AoA (,) as defined in Section II is chosen evenly and randomly in , . The channel coefficient is set as a constant . The stepsize is set as . Simulation results are averaged over 1000 random system realizations. Fig. 5
indicates that the channel vector MSE of our proposed algorithm approaches the minimum CRLB quickly and achieves much lower tracking error than other algorithms.
In dynamic scenarios, the AoA (,) as defined in Section II is modeled as a random walk process, i.e., , ; . The initial AoA values are chosen evenly and randomly in , . The channel coefficient is modeled as Rician fading with a Kfactor =15dB, according to the channel model in [19]. As for the stepsize , we adopt the constant stepsize. Numerical results show that when , the joint beam and channel tracking algorithm can track beams with higher velocity. Therefore, the stepsize is set as a constant . Fig. 6
indicates the proposed algorithm can achieve higher tracking accuracy than the other four algorithms. In addition, if we set a tolerance error , e.g., , then our algorithm can support higher angular velocities.
Vi Future Work Remarks
In this paper, we have developed a joint beam and channel tracking algorithm for 2D phased antenna arrays. A general sequence of optimal analog beamforming parameters is obtained to achieve the minimum CRLB. The work is a first step to beam and channel tracking with 2D phased antenna arrays. In our future work, we will focus on the following aspects: i) establishing the corresponding theorems in dynamic scenarios; ii) jointly tracking multiple paths; iii) tracking at both the transmitter and receiver.
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Appendix A Proof of Lemma 1
Since the effect of noise can be reduced to zero by multiple observations, we ignore the observation noise in the proof for the sake of simplicity.
If the analog beamforming vectors are steering vectors, i.e., , where denotes the th direction parameter offset, then we get the complex observation equation for the th observation:
(23) 
which contains two real equations, i.e., an amplitude equation and a phase angle equation. From (23), we can obtain the phase angle equation:
(24) 
Then the relationship between the phase angles of two different observations and is given by
where and are determined by the direction parameter offsets and unrelated to the channel parameter vector .
Hence, the phase angles of any two observations and are correlated. After observations, we can obtain independent amplitude equations and only 1 independent phase angle equation, which are independent real equations in total.
When estimating real variables, at least independent real equations are required. Therefore, at least observations are needed to obtain independent real equations and estimate real variables, which completes the proof.
Appendix B Proof of Lemma 2
In problem (6), the constraint (7) ensures that is an unbiased estimation of . In static scenarios, where , we consider each element of the channel vector h. Given , we have since . According to section 3.8 of [12], if a function is an unbiased estimation of , i.e., , then we can obtain that
(25) 
where is the corresponding Fisher information matrix.
The partial derivative of is given as follows:
(26) 
Hence, Lemma 2 is proved.
Appendix C Proof of Lemma 3
Lemma 3 is proved in three steps:
Step 1: We prove that are unrelated to channel coefficient .
The basic method is block matrix inversion: the Fisher information matrix in (9) is divided into four matrices as follows:
(28) 
where , , are defined as: