I Introduction
Nextgeneration networks continuously adopt newer bands in the higher end of the spectrum, like millimeterwave (mmWave) bands, and even tap into the subterahertz bands for 6G [20, 12, 11]. The adoption of higher spectrum bands helps mitigate the spectrum crunch, but it comes with the challenge of overcoming the associated severe path loss[9, 14]. Deploying Multiple Input Multiple Output (MIMO) communication systems with significant antenna gains alleviates the severe attenuation in the highfrequency bands. However, to reap the potential gain, sharp beams created at the transmitter need to be aligned with those at the receiver. The challenge is that the alignment of such sharp beams requires significant overhead. Even then, the alignment of narrow beams can be shortlived since narrow beams have short coherence time, after which, the chosen transmitter and receiver beams are no longer optimal and realignment is needed. Misalignments can be caused by mobility, among other factors, and such variations rapidly degrade the received signal strength (RSS)[3]. The beam alignment process can be divided into two steps: (i) Initial Beam Alignment, and (ii) Beam Tracking. Next, we will discuss both steps in detail.
Initial beam alignment: constitutes finding the Angle of Arrival (AoA) / Angle of Departure (AoD) at the beginning of a communication session and in the event of beam failure (i.e., when beam alignment is lost, e.g., due to blockage). This process is challenging, especially when analog transceivers are used, because measurements have to be collected one at a time, which elongates this process. Large antenna arrays are particularly susceptible to this issue since their capability to form narrower beams increases the total number of beams to choose from. Beam sweeping, whose computational complexity is , where and denote TX and RX array sizes respectively, is a simple example of initial alignment. Derivatives of beam sweeping with hierarchical structures and lower complexity are adopted in WIFI standards, and 5G NR. Other solutions which take advantage of channel sparsity also exist, and they have been shown to have complexity in order of (for channels with k paths)[19, 15].
Beam Tracking: In fastchanging environments, frequently searching for beam directions is not acceptable due to the massive overhead that would be required. Instead, a more efficient way is to elongate the time between each beam discovery by tracking the already discovered beam(s) for as long as possible. This can be achieved by periodically transmitting pilot symbols to monitor small beam changes. However, those pilot symbols still constitute an overhead that should be minimized to avoid degrading the effective spectral efficiency[17]. A tradeoff presents itself here where more frequent pilots can better track beam directions but at the cost of higher overhead. Less frequent pilots, on the other hand, would result in lower average received SNR due to worse beam alignment, in addition to risking totally losing beam alignment (i.e., beam failure) in which case a repeat of the costly initial beam alignment process would be required.
Our Design: In this work, we propose a beam tracking algorithm that continuously updates antenna beams even though channel measurements are only available sparsely, via the transmission of pilot symbols. In contrast to standard tracking approaches, which update beam directions based on pilots and keep them fixed until the next pilot symbol, our approach models and makes use of gradual beam changes in between pilot symbols, based on an estimate of the variation of the beam directions at the transmitter and the receiver. Our solution, which we refer to as ContinuousDiscrete tracking, leverages the physical nature of beam variation in the spatial domain as a function of time to build a state transition model. Information from this model is fused with measurement data under three different frameworks: (i) an Extended Kalman Filter (EKF) (ii) a Fast Beam Tracking algorithm (iii) a MainLobe approximation. Those three frameworks are discussed in detail in Section V. Our proposed ContinuousDiscrete approach allows our solution to rely on less frequent channel measurements to maintain accurate beam tracking. Decreasing the number of measurements reduces the overhead of reference (i.e., pilot) symbols required to perform such measurements. Our solution (i) improves the quality of beam alignment over stateoftheart solutions, given the same pilot overhead, (ii) elongates the beam tracking time before beam realignment would be needed, and (iii) lowers the associated pilot overhead required to achieve beam tracking performance similar to the stateoftheart solutions.
The main contributions of this work include:

We propose a simple discrete beam tracking algorithm based on a simple approximation of the mainlobe of the array attenuation factor.

We provide three continuousdiscrete tracking algorithms based on different techniques.

We illustrate that continuous variations, accompanying the discrete jumps yield major gains in the average SNR achieved between updates. As a result, at a given desired SNR, the frequency of pilots can be reduced significantly.

We provide an analysis of different antenna arrangements, including linear and planar structures, and study their respective robustness to variations in the beam direction.

We show that, due to shortened beam coherence times associated with narrower beams, larger MIMO arrays do not necessarily lead to improved performance, as they require more frequent beam updates in analog beamforming.
Paper Organization: Following this introduction, Section II discusses the related work, and we provide a motivating example to utilize the ContinuousDiscrete tracking in Section III. The system model is proposed in Section IV, and section V demonstrates the ContinuousDiscrete beam tracking approaches. Section VI provides an analytical discussion of a complementary problem to the beam tracking, represented by how to choose the pilot period. Section VII shows results comparing the ContinuousDiscrete proposal to its counterparts and evaluation of methods of selecting the Pilot Period, and Section VIII concludes our work.
Ii Related Work
Beam management has received great attention especially in mmWave networks. This has been evident in the design of 5G mmWave bands (Frequency Range 2, i.e., 24.25 to 52.6 GHz) which relies on a beambased air interface, in comparison to the lowfrequency, sectorbased air interface [2]. Beam discovery is one example of beam management problems, which has been tackled using various techniques including (1) Compressed Sensing (CS), which has been investigated for estimating the beam directions (AoA/AoD) [1, 13]
. Thanks to the sparsity of the mmWave channel, we can use CS tools. Solutions based on CS rely on randomized measurements translated into random beamforming vectors. Random beamforming vectors are usually constructed using random phase shifts. The random choice of the phase incorporated in the beamforming and precoding vectors derives irregular shapes of beam patterns, which are sensitive to fluctuations of the received power, and the thermal noise
[16]. Beam discovery techniques also include (2) Beam Sweeping or Exhaustive Search [19] where the transmitter (TX) and receiver (RX) scan all possible beam pairs from predefined sets of TX/RX beams. The channel can then be reconstructed using the accrued knowledge of path gains and their corresponding AoDs and AoAs. In the IEEE 802.11ad/ay WiFi specifications [21, 22], beam sweeping at the TX and at the RX are decoupled as follows: First, the TX utilizes a quasiomnidirectional beam, while the receiver searches the space for the incoming path. Second, the first step is reversed between the transmitter and the receiver. This approach scales down the complexity to instead of . In contrast to beam sweeping, the problem of finding the beam direction is formalized as a MultiArmed Bandit [3]. In comparison to the WIFI standards [21, 22], the approach used in [3] utilizes the contextual information from previous beam alignments is used to minimize pilot overhead.Many algorithms have been proposed for beam tracking, with the motivation of using a small number of overhead pilots used to track the already discovered beams: The EKF is a nonlinear version of the Kalman Filter, where the tracked/predicted object is nonlinearly added to the noise. Therefore, the EKF is a suitable solution for tracking the AoA/AoD where the thermal noise is added to the AoA/AoD in a nonlinear fashion at the antenna elements. Discrete Extended Kalman Filter (EKF) was proposed in[18] as a lowcomplexity tracking algorithm, utilizing a periodic single pilot symbol each time slot to find a new estimate. In [18], the EKF complements a good estimate of an initial estimate of the AoA/AoD, which means they only consider the beam tracking problem without considering the initial beam alignment problem. In[7], a Fast Beam Tracking (FBT) approach was developed to make the MSE converge faster to the CramérRao lower bound (CRLB). In this approach both the initial beam discovery and the beam tracking problems are considered. A coarse beam sweeping is proposed to discover the initial beam direction, and a recursive beam tracking algorithm is proposed to minimizes the MSE of the true AoA. Also, an Auxiliary Particle Filter as another tracking algorithm that deals with the nonlinear noisy function of the AoA is proposed in[8]. In this approach, a group of particles are generated at each channel measurement. An estimate of the AoA is found by averaging the estimate of all generated particles. These approaches update the beams at discrete instances upon fresh measurements and keep the alignment fixed until the next measurement. In contrast, our approach updates the beam directions incrementally in between measurements.
Up to this point, we have introduced some beam tracking algorithms that utilize pilot symbols to perform tracking. On the other hand, other techniques exploit different resources to track the beam direction. For example, a Machine Learning (ML) framework was proposed in
[5] to reduce the dependence of beam tracking on pilot symbols. In this approach, an RGB camera is attached to the Base Station (BS) to capture a group of consecutive images. An object detector is utilized to process the captured images and identify the receiver position. An algorithm then predicts the future position of the receiver and its corresponding beam direction. In this paper, we do not assume the presence of aiding devices like cameras, and we are only concerned with beam tracking techniques that utilize a single pilot symbol arriving at each time slot.Iii Illustrative Example
In this section, we provide a simplified example to demonstrate our proposed ContinuousDiscrete beam tracking solution, and compare it against the traditional discrete beam tracking approaches. The fusion of discrete measurements and continuous AoA updates (which enables continuous beam updates) is referred to as ”ContinuousDiscrete” Beam Tracking.
In the example shown in Fig. 1, a Transmitter (TX) with a single antenna serves a moving Receiver (RX) with a directional antenna gain. The RX moves on along the shown circular path with a constant velocity . Let us assume that at position 1 the RX has accurately discovered
the AoA, , of the strongest channel path, and has aligned its receive antenna beam towards that path. After that, the RX keeps tracking using periodic pilot symbols that arrive every seconds. Upon the arrival of a pilot symbol, the RX updates the estimate of the AoA, and its slope variation, . Using the slope variation, the RX predicts the change in the AoA, and continuously updates the beamforming vectors to stay tightly coupled with the incoming path.
Recall that RX moves with constant angular velocity. Hence, intuitively, as the prediction errors of the AoA and the slope variations go to zero, the period , which means that pilot symbols are not needed to keep tracking the AoA. That is because perfectly tells us exactly what the value of will be at any future time. On contrary, ”Discrete” Beam Tracking solutions only update the AoA once every pilot symbol arrival. That is, discrete tracking does not update the AoA estimate in the time duration in between pilot arrivals. Also discrete tracking does not rely on AoA rate of change information. Hence, the misalignment between the beam direction and the incoming paths increases as the pilot period increase. This means that, in a Discrete tracking approach the RX needs to minimize the period (increase the pilot overhead) as the beamwidth decreases (array size increase). Our proposed ContinuousDiscrete measurement framework incorporates utilizes extra system information embodied in the rate of change of the AoA. This extra information allows us to predict how the AoA changes even when no measurements are available. This, in turn, allows us to extend the pilot period without significantly degrading the link quality nor risking losing beam alignment.
Now, let us introduce how the AoA changes based on the special scenario assumed in Fig. 1. The RX at position 1 is moving with constant velocity , and the AoA changes continuously based on the following equation:
(1) 
where is the distance from the TX to the RX. So, after a pilot period , the RX will move to position 2, and the new AoA is . The ContinuousDiscrete tracking allows us to update the beam direction continuously based on a prediction of the angular velocity , while Discrete tracking only updates the beam direction upon pilot arrival, (i.e., position 1,2). Fig. 2
, shows the Signal to Noise Ratio (SNR) for the example shown in Fig.
1, under both Discrete and ContinuousDiscrete measurement frameworks. For that experiment, the total tracking time is 100 ms, Km/hr, m, and , and the two values for the pilot period ms. The results from Fig. 2 reveals the weakness of the Discrete tracking approaches, where the instantaneous SNR drops drastically in between pilot symbols. To overcome the drop in the SNR, the pilot period should be decreased, i.e., ms. On the other hand, the prediction of the rate of change of the AoA makes the ContinuousDiscrete framework to stay tightly coupled even when only two pilot symbols are used over the entire tracking duration (i.e., ms). Furthermore, the ContinuousDiscrete approach can stay aligned without any pilot symbols as long as the prediction of the slope variation is available.In this example, we have considered an oversimplified scenario. In a real system, however, the channel environment becomes more complex, requiring more sophisticated mathematical modeling and tracking techniques. Modeling real system complexities will be tackled in the following sections starting with the System Model in Section IV. We will see that errors in the prediction of the slope variation force the ContinuousDiscrete solution to limit the pilot period to alleviate these errors. Yet, ContinuousDiscrete approaches still can do better than the Discrete approach, as measured in two metrics: (1) Low Pilot Overhead, where the pilot period is longer than the period of the Discrete case. (2) SNR, where the ContinuousDiscrete can have higher SNR than the Discrete case for the same pilot period.
Iv System Model
Iva Notation
The lowercase and denote scalar and vector quantities respectively, while the uppercase denotes a matrix. The continuous variation in is denoted by , and is the sampled version of each seconds, where . Moreover, , and denote the conjugate transpose, the real and the imaginary parts respectively. Finally, The standard Q function is denoted by , while its inverse is denoted by .
IvB Channel Model
We consider a Single Input Multiple Output (SIMO) channel model with a single antenna at the transmitter, and antennas at the receiver. The receiver antennas are equally spaced and arranged linearly to form a Uniform Linear Array (ULA). We assume an analog beamforming transceiver architecture, where a single RF chain is utilized, and a phase shifter is applied for each antenna element in the array. We assume a 2D multipath channel model from [4] with a channel impulse response:
(2) 
where is the path index, is the Number of paths, is the complex path gain, is the Doppler shift, is the array steering vector, and is the timevarying AoA, which changes over time due to mobility of the transmitter, receiver and the environment. The steering vector is given by:
(3) 
where , is the antenna spacing normalized by the wavelength, and is the number of array elements. The variation in the AoA is given by:
(4)  
where , and are the initial AoA and the initial rate of change for the AoA, respectively. The variation term denotes the rate of change in AoA. We model the changes in through modeling its rate of change . In a general environment, users can accelerate, decelerate, rotate and may move on curved paths.
The objective of this paper is not to provide detailed models of motion and integrate them into a cumbersome calculus. Instead, we aim to convey the value of incorporating the variations into beamforming. As a result, we use an approach in which the vagaries of the mobility are summarized in a general stochastic model. In particular, as a representation of , we use
, a zero mean Brownian motion process with variance
, where , and is a zeromean White Gaussian process with a Power Spectral Density (PSD) for all frequencies . A small value of Q means that is almost fixed (and that changes in is steady), and vice versa for large values of Q.Given the sparse nature of the mmWave channel, the received power from all directions is limited to a few sharply defined AoAs [11, 10]. In addition, since our receiver has an analog architecture, it can only form one directional beam towards the strongest path in the channel (i.e., the path with the highest received power). Hence, we can assume that all other paths are highly attenuated since they are weaker and lie outside the main lobe of the receive antenna. Therefore, we omit the subscript for paths and treat attenuated weaker paths as noise.
Using the frame structure in Fig. 3, we have two stages for beam alignment: the first stage consists of pilots required for initial beam alignment, and the second stage applies one pilot each time slot for beam tracking. The second stage can be viewed as a time frame consisting of time slots. Each time slot has a duration seconds, while the total duration of the time frame is seconds (i.e., ).
In this paper, we only focus on the second stage that is concerned with beam tracking. Hence, the initial beam alignment performed at the first stage is out of scope of this work, however, we assume that we have a good initial beam direction. The received pilot sample is given by:
(5)  
where is a fixedknown value assuming that the complex gain changes much slower than the AoA. The amplitude for a known pilot symbol is denoted by , is a circularly symmetric complex Gaussian noise sample, is the beam pointing direction, and is the beamforming vector which is given by:
(6) 
Succinctly, we can rewrite (5) normalized as follows:
(7) 
where , is the SNR including the array gain, and is SNR for each antenna element. For more straightforward dealing with equation 7 in the following sections, we define the signal part by:
(8) 
IvC Uniform Planar Arrays
Up to this point, development in equations 7, 6, 5 and 2 is based on ULAs. Here, we turn our attention to Uniform Planar Arrays (UPA) since they provide higher gain coupled with resilience to channel variations. We assume a square UPA where elements are equally spaced and arranged over the area of a square. The steering and the beamforming vectors are given by:
(9) 
where is the Kronecker product, while and are the azimuth and elevation angles, respectively. We assume AoA variation only in the azimuth direction, and we fix . The beamforming vector is given by . Now, the model for the UPA is based on (9), and .
IvD Problem Statement
Our objective is to minimize the instantaneous MSE between the true AoA and its estimate , which can be stated by:
(10) 
While traditional solutions update at discrete measurement instances only, which create larger MSE as drifts inbetween pilots, we update continuously using an estimate of the AoA rate of change (recall equation (4)). We define the system state as , and let be its estimate, whose error covariance matrix is , where
(11) 
The estimate of which incorporates both the AoA and its rate of change is tackled in the following section.
V ContinuousDiscrete Beam Tracking Approaches
In traditional beam tracking, the estimate of AoA is updated using pilot symbol arrived each time slot and the RX fixes the beam direction to that estimate until the arrival of the next pilot symbol. In contrast, a ContinuousDiscrete beam tracking exploits the prediction of the rate of change of the channel variation to continuously update the beam direction to minimize the instantaneous MSE. Furthermore, the approach presented here hinges on updating the estimate of the AoA not only based on pilot symbols, but also on continuous predictions of the rate of variations of the AoA in between pilots. In other words, we perform continuous beam tracking instead of abrupt adjustments at each pilot.
In this section, we apply the ContinuousDiscrete approach to three different techniques to prove that the proposed approach improves the MSE over discrete baseline solution. The three frameworks are listed as follows:

An Extended Kalman Filter: An extended version of the wellknown Kalman Filter, which is more suitable to a nonlinear system.

A Fast Beam Tracking: The proposed framework in [7], targets converging the MSE faster to the CRLB.

MainLobe Approximation: A novel approach used to track the beam direction based on an approximation of the amplitude of mainlobe proposed in Section VC1.
Following this, we provide a detailed discussion of the three proposed approaches.
Va Approach 1: Extended Kalman Filter
The Klaman Filter is a popular tool used in tracking problems, which motivates its use in beam tracking. However, recall from equation 7 that the measurement is a nonlinear function of the AoA (). This makes the traditional Kalman Filter unsuitable for this system. Instead, the EKF is the nonlinear version of the KF, which is more suitable for the considered nonlinear system in this paper. The EKF can be described by two equations: (1) State, which incorporates the continuous dynamics in the AoA (), and its rate of change (). Hence, we can write the state dynamics of the ContinuousDiscrete Extended Kalman Filter (EKF) as follows:
(12)  
and (2) Measurement/Observation, which represents the channel measurements using known pilot symbols, and it is given by equation 7. The derivation of the state dynamics of the EKF can be found in [6], but we only provide a sketch as follows: The main idea of the EKF derivation is the linearization of the signal part () of equation 7. The following procedures are the same as that of the standard Kalman filter which can be found in [6].
The standard recursion equations (see [6]) are divided into two stages: (1) Prediction, and (2) Update. The prediction stage is given by:
(13) 
(14) 
where is the prediction of the time derivative of the state vector , is the prediction of the time derivative of , and is the process covariance that drives the state vector, and is assumed to be bounded by . The estimate during the prediction stage is the solution of the differential equation in (13). The update stage, which amends the iteration of the standard Kalman gain (), is given by:
(15) 
(16) 
(17) 
where , and is the estimate and the error covariance matrix at iteration given measurements. Finally, is the gradient of the signal part () and estimated at .
An overview of the EKF procedure can be described in Fig. 4. First, we assume our knowledge of the estimate of the initial AoA . Then, the state is updated based on equation 17 if a measurement is available. Otherwise, we repeat the prediction stage times, i.e., we make a prediction every seconds. When the tracking period is over, we start searching for a fresh estimate of the AoA, similar to estimating . In this paper, our main focus is beam tracking, and our solution complements any beam discovery (initial AoA estimation). Hence the latter is not included in this work. Further, optimizing the tracking period length is left as future work.
VB Approach 2: Fast Beam Tracking
The FBT was proposed in [7] as a low complexity algorithm that makes the MSE of the true AoA converge faster to the CRLB. In [7], they considered both the beam discovery and the beam tracking problems. In this section, we are only concerned with extending the discrete FBT (beam tracking only) to a ContinuousDiscrete Fast Beam Tracking (FBT) solution. The key idea of the ContinuousDiscrete is to predict the slope variation of the AoA, so we need a prediction of the slope variation besides the estimate of the AoA itself using FBT. We take advantage of the prediction stage in the EKF to continuously predict the AoA and its slope between two pilot symbols. Two stages describe the recursion of the FBT: (1) the prediction stage using equation (13), which continuously predict the estimate of the AoA and its rate of change found in the update stage. (2) the update stage is explained by the following two steps:
VB1 Discrete Fast Beam Tracking
The first step is to update the AoA using the FBT [7], which can be described as follows:
(18) 
where , is the current and the previous estimates of and respectively, and is the step size. The estimate is found by taking the inverse of the sine function. To accommodate for differences from the model of [7], we recalculate the step size as follows:
(19) 
It is easily to find the nominator and the denominator (Fisher Information) of by the following:
(20)  
Also,
(21) 
Hence, we have the step size as:
(22) 
VB2 Slope Variation Update
The FBT does not consider updating the slope variation of the AoA (), we utilize the EKF in the background of FBT to update , which is needed for the prediction stage.
VC Approach 3: MainLobe Tracking Algorithm
In EKF, and FBT, the slope variation estimate was found using an EKF framework. Here, we introduce a ContinuousDiscrete MainLobe Tracking Algorithm (ML) based on the proposed Algorithm 1 to update the estimate of the AoA. The estimate of the rate of change () is updated using an MMSE of the difference between two consecutive slope variation instants (). The proposed approach not only proves the validity of the ContinuousDiscrete over all discrete baseline but also it provides a tractable analysis needed for the upcoming Section VI. The ML is described by two stages as in the EKF, and FBT; the prediction stage using equation 13, and update stage can be divided into two separate steps: (i) updating the AoA based on a Discrete MainLobe Tracking (ML), and (ii) updating the slope variation.
VC1 Discrete MainLobe Tracking Algorithm
In this section, we propose a Discrete MainLobe Tracking Algorithm, which is consistent with the definition of beam coherence time from [17]. This model is built over the assumption that pilot symbol arrives each within the mainlobe (i.e, within the beam coherence time). This approach is based on the following approximation of the received signal part for ULAs:
(23) 
A comparison between the true value of the amplitude of the received signal, and the approximation given in (23) can be shown in Fig. 5. We can notice from the figure that the main beamlobe of the true value almost the same as the approximation , and the sidelobes can be neglected especially for a large array size.
The basic idea of the proposed algorithm is that; the value of the signal part : is a function of the true value , and the previous estimate . If we have an estimate of the signal part from a given measurement, and using the approximation formula in equation 23 we can solve it to find the current estimate. We assume the signal and noise parts are zeromean Gaussian with variances, , and respectively. Also, we assume the signal and noise parts are orthogonal i.e. . Hence, the Linear Minimum Mean Estimate (LMMSE) of is found as follows:
(24)  
The current estimate is found by solving the approximation given in equation 23, and the estimate given in equation 24. An overview of the proposed algorithm is presented in Algorithm 1, which can be described as follows: first, we equalize the estimate from (24) with the approximation formula in (23), then take for both sides. Also, we are going to change the symbol in equations 24 and 23 with since we looking for an estimate.
(25) 
VC2 Slope Variation Update
The current estimate of the slope variation () is found as an update of as follows:
(26) 
where , and we assume initial estimate . As shown from equation 26, the current estimate depends on the previous estimate and an update term. The update term is found as follows; first, we use a first order approximation of the signal part in the measurement equation by:
(27) 
where , and by assuming good estimate (i.e., ,) we can have , and . In that case, the imaginary part of the measurement equation is given by:
(28) 
where , and for a given , the MMSE of (i.e., ) is given by:
(29)  
Vi Performance Evaluation
The main goal of beam tracking, is to elongate the link reestablishment time ; the time at which we lost the tracking of the AoA, and we need to search for the new best incoming path. Recall the frame structure in Fig. 3, where we need to send a frame of time slots with a fixed rate. In Section VIIA, we show that large array size is not able to cope with channel variation for a given pilot period (Time slot) . Hence, we need to choose the pilot period based on array size beside other factors we will discuss later. In the following, we propose two different ways to choose the pilot period
; (i) comparable to the beam coherence and (ii) to maintain specific outage probability and fixed rate.
Via Beam Coherence Time
Intuitively, the pilot period has to be comparable to the beam coherence time to cope with the channel variation. From [17], the beam coherence time is defined as the time at which the power received at perfect alignment time drop by at time .
(30) 
Since we assume perfect alignment at time , then the received power . For ULA, the received power at time while ignoring the noise power is given by:
(31)  
Now, we try to visualize the picture of a simple example to grasp the whole idea. As shown in Fig. 6, we assume the receiver is perfectly aligned with the incoming path at the first position. The receiver is moving with speed and a fixed beam direction equals to the AoA at the first position, i.e. . Hence, after motion for a distance there will be a misalignment, and the received power will be dropped to . Our model relates the channel variation to the angular variation , and Fig. 6 shows how we relate the linear speed to that angular variation as:
(32)  
where is the distance from source of the path to the receiver and now we can say that the angular variation is bounded by speed over the distance as : . This is the typical situation for the Discrete tracking approach, and we are seeking the pilot period to be within the beam coherence time . For a ContinuousDiscrete tracking algorithm, the beam direction changes dynamically based on estimate of the slope variation , then to find the pilot duration we need to take care of slope variation estimate in addition to the AoA variations. In this case, the notion of beam coherence time does not hold for the ContinuousDiscrete algorithms, and we introduce the notion of the Beam Locking Time . This time represents the duration at which the power will drop by while the receiver continuously update the estimate of the AoA and the beamforming. In order to find a consistent formula of the pilot period, we had to assume it is the time to have an average drop instead of since both sides of equation 30 are random. Now, we introduce how to choose the pilot period for a Discrete and ContinuousDiscrete beam tracking using Theorems 1.
Theorem 1.
For the channel model with angular variation given by equation 4, the link reestablishment interval is achievable if the pilot period is given by:
(33) 
where is the average drop in the received power, and , the lower value of the better estimate of the slope variations and the gets larger. If , then the beam locking time is limited to the beam coherence time which is used in the Discrete beam tracking to choose the pilot period . Proof of theorem 1 can be found in Appendix A, and B.
Clearly, the power will drop in the discrete beam tracking algorithm faster than the continuousdiscrete tracking algorithms. This is because continuousdiscrete tracking continuously varies the beamforming based on the prediction of the AoA variation and gain from discrete pilot updates. While the discrete approaches only update the AoA upon the arrival of pilot symbols.
ViB Outage Probability
For the situation where the transmitter is aware of the channel distribution, this motivates the transmitter to choose the pilot period that sustain a specific outage probability for a fixed rate . In that case, the pilot period is chosen based on Theorem 2.
Theorem 2.
For the channel model with angular variation given be equation 4, the link reestablishment interval is achievable with outage probability , and a fixed rate if the pilot period is given by:
(34) 
similarly, , and for , then is valid as the pilot period for the Discrete beam tracking algorithm, otherwise, is valid for the ContinuousDiscrete beam tracking algorithm.
Proof.
From Appendix C, we get the outage probability for a ContinuousDiscrete tracking approach is given by equation 59:
(35)  
for a given , the corresponding spectral efficiency is . Alternatively, we can replace in equation 35 by . After that, we divide both sides by , then take the inverse of the Q function. Finally, equation 34 is found directly by taking to the left side.
Similarly, The Discrete case where is derived in similar procedures as equation 34 by starting from the CDF for the Discrete case which is given by equation 56.
∎
ViC Pilot Overhead Reduction
The key point of utilizing the spatial variations inbetween two pilot symbols is to reduce pilot overhead by extending the pilot symbols duration. Now, we show how a ContinuousDiscrete tracking algorithms reduce pilot overheads as follows:
ViC1 Beam Coherence and Locking Time
(36)  
ViC2 Outage Probability Time
(37)  
Hence, utilizing the ContinuousDiscrete algorithms results in overhead reductions by .
ViD Effective Achievable Rate
Now, we are looking to represent the overhead reduction by the Effective achievable rate by excluding the time of pilots training. The effective rate is given by:
(38)  
where is the achievable rate, will be replaced by , , (D), and (CD) for each case of choosing the pilot period. Also, is the training time for a single pilot symbol, and is the beam sweeping time, which is needed to find the angle of arrival after losing the tracking. Considering the sweeping time, we utilize the same formula proposed by [17] based on the hierarchical beam code book in [19] and follows the IEEE 802.15.3c guidance. The formula was given as function of the beamwidth, which can be shown as function of the array size as:
(39) 
where is the number of levels in the hierarchical beam codebook given in [19].
Vii Numerical Results
Viia Performance of Beam Tracking Algorithms
In this section, we evaluate and compare the performance of EKF and FBT algorithms under both the ContinuousDiscrete and the Discrete frameworks, while the ML and ML are left to the evaluation in Section VIIB. For each experiment, we assume perfect knowledge of the initial AoA at , i.e., . We simulate as a Brownian motion and as in (4), and we let the algorithms run for a total tracking time of 100 ms and average the output over 5000 runs. Our performance metrics are: (1) the MSE of the estimated AoA, and (2) the average received SNR. Except when stated otherwise, we let , ULA and we fix dB. We evaluate the performance under two array orientations, i.e., ULA and UPA.
Different ULA Array Size: Fig. 7 depicts a sample path of the progression of the simulated true value of the AoA as well as its tracking performance using both EKF and EKF algorithms when ULAs of sizes 16 and 64 are employed. The pilot period is fixed at . For the ULA of size 16, both EKF and EKF frameworks provide accurate tracking performance for the entire tracking duration. Only a closer inspection reveals the superiority of EKF as shown in the zoomedin part. However, as the array size gets larger, the beamwidth gets narrower, making it harder to track the beam. Consequently, for ULAs with 64 antenna elements, we see a clear difference in favor of EKF where it is able to track the beam for a significantly longer time duration (200% longer), compared to EKF with discrete updates. In general, as the number of antenna elements increases, the performance gap between EKF and EKF increases in favor of our EKD solution as long as the variation in the AoA over is less than the variation over the beam coherence time.
In Fig. 8, the MSE for the continuousdiscrete and the discrete approaches for is shown. Here, we also provide results for the FBT and FBT algorithms. These results show that the continuousdiscrete framework significantly improves performance over the baseline discrete solutions. We also see that EKF has the lowest MSE, i.e., best tracking performance, among all other solutions. Interestingly, EKF performs even better for the larger array size, despite the overall difficulty in tracking sharper beams. On the other hand, the performance of EKF degrades by increasing the array size (especially when AoA variations become faster over the tracking time). This happens since the larger array size has shorter beam coherence time, and EKF becomes unable to cope with the fast channel variation. The Effect of the array size over FBT and FBT is similar to that of EKF and EKF. Yet, EKF and EKF outperform the Fasttracking approach (especially when the channel variations become faster). EKF achieves and lower MSE than EKF, for 16 and 64 array sizes, respectively.
Different SNR values: Fig. 9 shows the MSE under different SNR values; 10 and 20 dB. Here, we fix the array size to 64 and the pilot period to . As expected, decreasing the SNR degrades the performance for all approaches. EKF, however, still outperforms all other algorithms due to its ability to predict the rate of change of the AoA between pilot symbols. The superiority of EKF is very dominant that even at 10 dB, it still has comparable performance to FBT and EKF with 20 dB. The MSE for all approaches is tiny at the beginning of the tracking period since the variation of AoA is negligible. As the tracking time advances, the variation in AoA increases, and the MSE for all approaches increases.
Different Pilot Frequencies: Measurement frequency is an important design parameter. More frequent pilots ensure better beam tracking, but it also wastes more transmission opportunities. Here, we shed more light on the effect of tracking performance as the measurement frequency changes, where we plot the MSE for a medium AoA variation at for a 64element ULA, as shown in Fig. 10. First, we observe that for , the performance of both EKF and EKF are almost identical at the measurement update instances. However, the MSE of EKF fluctuates more aggressively inbetween measurements. This is due to the AoA estimate of the EKF being kept constant between measurements despite the continuous AoA changes over time. Second, accounting for the nature of variation between measurements gives an advantage to EKF and allows it to use less frequent measurements while keeping the MSE comparable to EKF. This can be seen when comparing the MSE of EKF with to the MSE of EKF with . Finally, at the same , FBT achieves up to lower MSE compared to FBT. Also, FBT performs almost identically at both and , showing the effectiveness of continuous AoA updates.
In Fig. 11, we present the average received SNR for different . Averaging is over the whole tracking time and 5000 runs. Here, we also normalize , with respect to the array size. This normalization helps isolate the tracking performance as a function of the beamwidth only. Next in Fig. 12, we will remove this normalization and study the overall performance with sharp beams, by accounting for the beam gain, as well. We notice that average SNR is degraded by increasing the array size since increasing the array size decreases the beam coherence time. In order to have good performance over large array sizes, we need to utilize smaller . The EKF outperforms the EKF by 1 dB and up to 4 dB for small and large array sizes, respectively, while the FBT has an advantage by 4 dB and can reach 7 dB. Increasing the pilot period degrades the average SNR for all approaches, yet EKF still outperforms EKF by 2 dB for . Also, FBT has superiority over FBT by 5 dB for .
Fig. 12 shows the average SNR for 16 dB per each antenna element. We observe that increasing the array size increases the average received SNR to a certain point, before it drops slowly for larger sizes. This can be explained by arguing that the array gain compensates for small mismatches in beam alignment, despite the difficulty imposed by the larger array size. However, as arrays get larger, beams get sharper, and alignment fails more often, leading to a drop in the average SNR.
Uniform Planar Arrays: A UPA has better spatial properties than ULA in terms of beam coherence time. Hence, a UPA can utilize longer and still obtain better performance than ULA. We notice from Fig. 13 that UPA scenarios, which are denoted by EKFP and FBTP have better average SNR for , especially for large array sizes and fast channel variations.
ViiB Choosing the pilot period
Here, we verify our discussion of choosing the Pilot Period using the proposed tracking algorithms ML, and ML since these algorithms are modeled on the concept of beam coherence. The assumptions that we assumed in Section VIIA still hold. In addition, we assume , , the frame length ms, and SNR/antenna element 8 dB. Finally, the average drop in the SNR at time , the outage probability , and a fixed rate as a factor of the maximum rate . Our performance metrics are: (1) the CDF of the received SNR and the achievable rate , and (2) the effective achievable rate.
First, we numerically investigate the pilot overhead reduction, as shown in Table I. We compare the overhead reduction for different array sizes by choosing the pilot period either by the beam coherence or outage probability definitions. The overhead reduction increases with the array size, and this comes from the fact that for a small array size, the beam is wide, and the ML can cope with channel variation with a pilot period comparable to ML. Yet, for a large array size, the beam coherence time becomes much smaller, and the pilot period becomes smaller for a ML, which gives the advantage to the ML. Increasing the rate to , imposes the tracking algorithms to sustain higher received SNR, which can be maintained by decreasing the pilot period. The ML algorithm can sustain a higher rate with a very comparable pilot period used at the lower rate. On the contrary, the ML needs to reduce the pilot period too much. This can be noticed from Table I, where the overhead reduction is increased even for the small array sizes. Finally, the overhead reduction for beam coherence and the outage probability with the rate are almost the same, but both have different pilot periods.
Overhead Reduction  
Array Size  4  9  16  36  64  144  256  512  1024  
Beam Coherence  19%  29%  36%  46%  53%  62%  67%  72%  76%  
19%  29%  35%  44%  52%  60%  65%  71%  75%  
How to choose T  Outage Probability  60%  60%  61%  73%  76%  80%  83%  85%  87% 
Now, we consider the CDF to compare between the two proposed methods of choosing the pilot period . In Fig. 15, we present the CDF of the received SNR for array size for three different cases: (1) outage with (2) outage with (3) beam coherence with . For case 1, with pilot period , and 0.498 ms for ML, and ML respectively. It is clear ML outperforms the ML, where it can maintain SNR up to 600 for the same outage probability. Case 2, with pilot period , and 0.206 ms still shows superiority in terms of SNR but not significant as in case 1. However, the pilot overhead reduction is significantly increased in case 2 to 83% instead of 65% as in case 1. Case 3 shows that both ML and ML nearly have the same average SNR as designed but the ML has lower probability for the smaller SNR values. Albeit, choosing the pilot period by the beam coherence method aimed to have the same average drop in the SNR, but the ML is more likely to have lower SNR values in that case due to additional errors caused by estimating the slope variations rather than the AoA estimation errors.
Fig. 16, involve the three cases in Fig. 15 but in terms of the achievable rate. We can see that for case 1, is satisfied for both ML, and ML at the 50% of the maximum rate. Moreover, the ML surpass the ML roughly by 4 bps/Hz for the same outage without reducing the pilot period. For case 2, both ML, and ML are nearly the same with minor advantage to the ML at the higher rate. Similar to our discussion for Fig. 15, ML obligated to reduce its pilot period by almost 59% to support instead of while ML only reduce its period by 17%. Case 3, shows both algorithms can have but only for a very small rate. Still we can notice an advantage to ML in terms of outage probability since we force both algorithms to drop to the same level which worsen the situation of the ML due to additional slope variations errors. From our discussions about Fig. 15, and 16, we can say that its better to choose the pilot period based on the outage probability definition due to:(1) the beam coherence method is not directly related to the operational rate and the outage probability (2) the beam coherence method is not a fair choosing method especially if an additional estimation errors exists in the ML.
Up to this point, we have argued the overhead reduction. Now, we measure the performance of the ML in terms of the effective rate given by equation 38 when the achievable rate is either the outage rate or the average rate . First, for the effective outage rate as shown in Fig. 17, the ML outperforms the ML for all pilot training intervals values. Moreover, we can notice that ML is more resilient to discontinuities such that it reduced only by 0.5 bps/Hz for 1024 array element. On contrary, the ML is more sensitive to pilots disruptions causing a rate drop by 2 bps/Hz for .
Increasing the operational rate to increases the pilot overhead reduction which gives the advantage to the ML algorithm. This makes ML to surpass the ML by 0.25 bps/Hz for the smallest pilot training interval . Moreover, ML has a rate drop by nearly 1 bps/Hz for while ML suffers a rate drop up to 8 bps/Hz. Finally, the higher pilot disruptions makes the efficiency slope decreases too fast, and in that case the array gain is not able to compensate. This noticed for the ML with , where the rate begins to drastically drop instead of increasing with the array size.
Now, we consider the effective average achievable rate as shown in Fig. 19, and Fig. 20. as a verification way of the ContinuousDiscrete advantage. Not only the ML gain from the overhead reduction or higher efficiency, but also it is more likely to have the higher received SNR than ML. The average rate is more significant in case of than since in the later case both algorithm are expected to operate near the edge of the maximum rate. This makes the average of both algorithms to be almost the same.
Partially, we have assumed fixed for both approaches. Two more scenarios can be considered when both approaches are assigned the same pilot period and that period is designed for; (1) the Discrete tracking algorithms and (2) the ContinuousDiscrete tracking algorithms. In these two scenarios, the ContinuousDiscrete tracking algorithm will have longer than the Discrete algorithm.
Viii Conclusion
Beam tracking is crucial for maintaining the quality of established links and avoiding the highcost initial link establishment process. Traditional solutions for the beam tracking problem rely on discrete measurement updates which occur at the instances of pilot signal arrival. By nature, such an approach ignores the information on the first order variation of the beams in between channel measurements. On the contrary, by considering the continuous nature of channel variation over time, a smarter beam tracking solution should exploit the gradual beam variation, and thus attempt to continuously and actively adjust the beam directions even when no measurements are available. This can be achieved by deriving a continuous state transition model for the channel.
In this paper, we propose a ”ContinuousDiscrete” beam tracking solution, which exploits system information like the first derivative of the beam angles, to allow continuous beam updates while still relying on discrete measurements. Our solution requires less frequent pilot symbols (less overhead) while maintaining similar tracking performance to discrete tracking, and it can achieve much better performance if the pilot frequency is kept the same. The performance is studied under different SNR levels, array sizes, pilot periods, and different array configurations, i.e., ULA and UPA. We show that our ContinuousDiscrete solution outperforms Discrete tracking algorithms in terms of overhead reduction by up to 87%. We also show that increasing the array size for fixed SNR and pilot period increases the tracking accuracy as long as the pilot period is comparable to the beam coherence time.
Another interesting result we show is that larger MIMO arrays do not necessarily lead to improved beam tracking performance. This is due to (1) shorter beam coherence time, and (2) higher link disruption (more frequent pilot symbols), especially for pilot training time in tens of microseconds. Furthermore, we demonstrate that in certain situations, uniform planar arrays may provide improved beam tracking performance over ULAs of similar size, in terms of average SNR, due to longer beam coherence time.
Appendix A Proof of Theorem 1 Part 1
Proof.
From equations 31 and 30, and given the assumption of and replacing , and by and respectively, then:
(40) 
Using the approximation given in (23), then is given by:
(41) 
The discrete representation of the AoA model in (4) using pilot period is given by:
(42) 
Now, we simplify the cosine difference in (41) using the discrete representation in (42) and as follows:
(43)  