I Introduction
Millimeterwave (mmWave) communication is promising to support the vastly growing data traffic for future wireless systems [1, 2, 3]. In the mmWave band, only several distinctive propagation paths exist, i.e., the lineofsight path and a few relatively strong reflected paths [4, 5]. Therefore, the directional beamforming with large antenna arrays is necessary to provide sufficiently strong received signal power.
To overcome the hardware limitation on the number of radio frequency (RF) chains with large array size and high carrier frequency, analog beamforming with phased antenna arrays was proposed [6, 7, 8, 9, 3]. A phased array can receive the signal that is projected onto a certain spatial subspace, with a cost of requiring much more pilots than the fully digital arrays to find the rare and precious paths. When users move quickly, it is needed to track the dynamic paths and even more pilots are required. Hence, one fundamental challenge is how to accurately track a large number of dynamic paths from many highmobility terminals/reflectors using limited pilots, e.g., in V2V/V2I, highspeed railway, and UAV scenarios [10].
The compressed sensing based algorithms (e.g., [11, 12, 13]) were proposed for phased arrays, which can reduce pilot overhead and make beam direction acquisition faster. However, these algorithms are designed for static or quasistatic scenarios, and will encounter performance deterioration under highmobility scenarios. To cope with highmobility scenarios, the algorithms in [14, 15, 16]
use the prior information to track the dynamic beam directions. However, these solutions do not optimize the tracking scheme with the optimal training beamforming vectors, which leads to poor tracking accuracy.
Since the tracking of a large number of dynamic paths can be decoupled into tracking each path with low pilot overhead, we have proposed a beam tracking algorithm in [17, 18] to optimize both the training beamforming vectors and tracking scheme. However, it assumes known channel coefficients, while both channel coefficient and beam direction might be unknown and timevarying in a real mobile system. In this paper, we further develop a recursive beam and channel tracking (RBCT) algorithm to jointly track the dynamic beam direction and channel coefficient. In static scenarios, the CramérRao lower bound (CRLB) of beam direction is derived, which is a function of the training beamforming vectors. We also obtain the minimum CRLB by optimizing these training beamforming vectors, and establish three theorems to verify that the RBCT algorithm can converge to the minimum CRLB with high probability. Simulations reveal that the RBCT algorithm can achieve much faster tracking speed, lower tracking error, and lower pilot overhead than several existing algorithms.
We use the following notations: is a matrix, is a vector, is a scalar. is the 2norm of . , and are ’s transpose, Hermitian and inverse, respectively. denotes expectation and () obtains the real (imaginary) part. The natural logarithm of is .
Ii System Model
Consider a phased array in Fig. 1, where omnidirectional antennas are placed on a line, with a distance between two neighboring antennas. Each antenna is connected through a phase shifter to the same RF chain. In timeslot , the pilot symbols arrive at the array from an angleofarrival (AoA) . The channel vector is given by
(1) 
where is the sine of the AoA , is the steering vector of the arriving beam, is the wavelength, and is the complex channel coefficient.
To track the beam direction and channel coefficient simultaneously, at least two observations using different beamforming vectors are needed. Hence, we assume that two pilot symbols are applied in each timeslot. To receive the th () pilot symbol, let be the beamforming vector in timeslot , denoted by
(2) 
which is assumed to have the same form as the steering vector. Combining the output signals of the phase shifters yields
(3) 
where is the pilot symbol that is known by the receiver, and is an i.i.d.
circularly symmetric complex Gaussian random variable. Given
and, the conditional probability density function of
is given by(4) 
A beam and channel tracker determines the beamforming matrix
, and provides an estimate
of the channel coefficient and the sine of the AoA. Let be a beam and channel tracking policy. In particular, we consider the set of causal beam and channel tracking policies: The estimate of timeslot and the beamforming matrix of timeslot are determined by using the history of beamforming matrices and channel observations .Iii Joint Beam and Channel Tracking Problem
Our goal is to develop a joint beam and channel tracking algorithm to minimize the beam tracking error. For any timeslot , the joint beam and channel tracking problem is given by
(5)  
s.t. 
where the constraint ensures that is an unbiased estimate of .
Problem (5) is a constrained sequential control and estimation problem that is difficult to solve optimally, where the beamforming matrix is the control action. First, the system is only partially observed through the channel observation . Second, both the beamforming matrix and the estimate need to be optimized: On the one hand, the optimization of is a nonconvex optimization problem of in (2), which is discussed in Section IIIA. On the other hand, as will be discussed in Section V, the optimization of is also nonconvex and has multiple local optimal estimates.
Iiia Cramér Rao Lower Bound of Beam Tracking
(13) 
Now, we try to establish a lower bound of the MSE in (5) in static scenarios, where the ground true of beam direction and channel coefficient is invariant for all timeslot , i.e., . Given the beamforming matrices of the first timeslots, the MSE in (5) is lower bounded by the CRLB as follows [19]:
(6) 
where obtains the matrix element in row and column , and is the Fisher information matrix, i.e., [20]
(7)  
where , , and . By optimizing the beamforming matrices on the RHS of (6), we obtain the minimum CRLB as below:
(8) 
where because the linear additive property of Fisher information matrix [20], the optimal are the same, and from (7), we can get
(9) 
Problem (IIIA) is nonconvex with respect to and , which makes it too hard to obtain the analytical solution. However, we can still use numerical method to find the solution, which yields the optimal beamforming matrix as below:
(10) 
where , and when , is very close to . In Fig. 2, the optimal receiving beam directions are depicted by plotting vs. and , where , and the signaltonoise ratio (SNR) is . It can be observed that is almost the same as and there are two symmetric optimal solutions. Therefore, we will set in the proposed RBCT algorithm in Section IV.
Iv Recursive Beam and Channel Tracking
We propose a twostage algorithm to approach the minimum CRLB in (IIIA), which is given below:
Recursive Beam and Channel Tracking (RBCT):
1) Coarse Beam Sweeping: pilots are used successively (see Fig. 3). The beamforming vector to receive the th observation is set as . Obtain the initial estimate by
(11) 
where , , , the size of determines the estimation resolution, and .
2) Beam and Channel Tracking: In timeslot , two pilots are received at the beginning (see Fig. 3) using beamforming vectors and , given by
(12) 
and the estimate is updated by (13) on the top of the page, where , , , and .
In Stage 1, the exhaustive sweeping is used, and the initial estimate is obtained in (11) by using the orthogonal matching pursuit method (e.g., [13]). This ensures that the initial beam direction is within the mainlobe set, i.e.,
(14) 
In Stage 2, the recursive tracker is motivated by the following maximization likelihood problem:
(15) 
where is subject to (2). We propose a twolayer nested optimization algorithm to find the solution of (15). In the outer layer, we use the stochastic Newton’s method to update the estimate , given by [19]
(16) 
V Asymptotic Optimality Analysis
(20) 
(21) 
(24) 
There are multiple stable points for (13), which correspond to the local optimal estimates for Problem (5) [21]. Hence Problem (5) is nonconvex for the estimate . To study these stable points, we rewrite (13) as follows:
(19) 
where is defined in (20), is defined in (21), with , , , , and .
A stable point should satisfy: 1) , and 2) is a negative definite matrix. Let
(22) 
denote the stable points set at timeslot . Then, we can verify as below:

When , we have Hence, .
Note that except for the real direction , the antenna array gain is quite low at other local optimal stable points in . Hence, one key challenge is how to ensure that the RBCT algorithm converges to the real direction , instead of other local optimal stable points in .
In static beam tracking, where and , we adopt the diminishing stepsizes, given by [19, 21, 22]
(26) 
where and . We use the stochastic approximation and recursive estimation theory [19, 21, 22] to analyze the RBCT algorithm.To support the more general joint beam and channel tracking scenario than [17, 18], three new theorems are developed to resolve the challenge mentioned above:
Theorem 1 (Convergence to Stable Points).
If is given by (26) with any and , then converges to a unique point within with probability one.
Proof.
See the detailed proof in Appendix A. ∎
Hence, for general stepsize parameters and in (26), converges to a stable point in .
Theorem 2 (Convergence to the Real Beam Direction ).
If (i) , (ii) is given by (26) with any , then there exist and such that
(27) 
Proof.
See the detailed proof in Appendix B. ∎
By Theorem 2, if the initial point is in the mainlobe , the probability that does not converge to decades exponentially with respect to . Hence, one can increase the transmit SNR and reduce the stepsize parameter to ensure with high probability.
Theorem 3 (Convergence to with the Minimum MSE).
If (i) is given by (26) with and any , and (ii) , then
(28) 
Proof.
See the detailed proof in Appendix C. ∎
Vi Numerical Results
We compare the RBCT algorithm with three reference algorithms: the compressed sensing algorithm [13], the IEEE 802.11ad algorithm [14], and the beam tracking algorithm [18]. The first two algorithms have the same configuration as that in Section VI of [18]. The third one uses the same training beamforming vectors as the RBCT algorithm, i.e., in each timeslot, it receives two pilots with the beamforming vectors in (12), and the beam direction is tracked by using both observations. Moreover, its channel coefficient is obtained with a least square estimator by using these observations. Consider the system model in Section II with antennas, and the antenna spacing is . The pilot symbol is , and the transmit SNR is set as . To ensure fairness, we assume that 2 pilot symbols are received in each timeslot, hence all the algorithms have the same pilot overhead.
In static scenarios, we set the stepsize as . The real AoA
is randomly generated by a uniform distribution on
in each realization, and the results are averaged over 10000 random realizations. Figure 4 plots the MSE over time. It can be observed that the MSE of the RBCT algorithm converges to the minimum CRLB in (IIIA), which is much smaller than the reference algorithms.In dynamic scenarios, we set the stepsize as a constant value, i.e., . The channel variation is modeled as: The AoA where , denotes the rotation direction, and is a fixed angular velocity. The rotation direction is chosen such that varies within . The channel coefficient is subject to Rician fading with a Kfactor , according to the channel model proposed in [23]. In Fig. 5 and 6, one can observe that the RBCT algorithm can support much higher angular velocities and data rates than other algorithms. According to Fig. 6, the RBCT algorithm can achieve 95% of channel capacity when the angular velocity is 0.19rad (1.09 degrees) per timeslot. If 5 timeslots last for one second, i.e., 10 pilots per second received, then the RBCT algorithm is capable of tracking a mobile moving at an angular velocity of 5.45 degrees per second and achieving over 95% of channel capacity.
Vii Conclusion
We have developed a joint beam and channel tracking algorithm for mmWave phased antenna arrays, and established its convergence and asymptomatic optimality. Our simulation results show that the proposed algorithm can achieve much faster tracking speed, lower beam tracking error, and higher data rate than several stateoftheart algorithms, with the same pilot overhead.
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Appendix A Proof of Theorem 1
Before providing the proof, let us provide some useful definitions. In static beam tracking, where and , recall the recursive procedure (19):
(29) 
where and are given in (20) and (21) separately. From (21), we have
(30) 
where , and is the covariance matrix of calculated by (31). In (31), the step can be obtain as follows:
(31)  