Sensing-Assisted Beam Tracking with Real-Time Beamwidth Adaptation for THz Communications

Among emerging applications for next-generation wireless systems, high mobility scenarios such as unmanned aerial vehicle (UAV) communications [1] and vehicle-to-everything (V2X) networks [2] require enhanced network capacity reaching up to Terabits per second (Tbps). To achieve the ultra-high data rate necessary for these applications, recent studies have considered terahertz (THz) communications as one of the key enablers [3] due to its substantial bandwidth which is typically around 10 gigahertz (GHz). However, obstacles exist in employing THz in high mobility scenarios. THz communications suffer from severe signal attenuation, primarily caused by free space path loss and molecular absorption [4]. To compensate for the attenuation, very large antenna arrays are employed to generate directional narrow beams with high beamforming gain. Nonetheless, the narrowness of these beams is likely to induce misalignment, as the target is prone to move out of the effective beamwidth [5, 6]. Such beam misalignment can cause significant performance degradation which presents major challenges in beam tracking and beamforming.

Beamwidth adaptation has been explored as an promising solution to counteract misalignment errors by balancing the trade-off between beam coverage and signal strength. Existing studies have employed precoders designed to generate beams with arbitrary predefined width [7, 8, 9]. Moreover, reference [7] has proposed a tracking algorithm triggered by outage events that automatically adapts the beamwidth to enhance reliability. However, given that the achievable data rate is subject to the target’s instantaneous velocity [10], the awareness of the target’s real-time mobility is crucial for effective beamforming and tracking in high mobility scenarios. Yet the aforementioned approaches solely operate under a static mobility assumption and cannot adapt beams in response to the real-time target motion.

Furthermore, recent advancements in integrated sensing and communications (ISAC) provide new opportunities in beamforming and tracking, benefiting from the Coordination Gain attained partially from sensing-assisted communications [11, 10]. In parallel, in reference [12], prior information obtained through conventional channel estimation has been utilised to track the fast-varying THz channels by predicting target motion. Although these studies do not directly address the scenarios in this paper, they inspire us to apply prior sensing assistance obtained by a specific ISAC design for motion prediction. Such an approach can help determine the desirable scale of beam coverage, and thus support effective beam tracking by incorporating real-time motion awareness.

Consequently, in this paper we propose a novel sensing-assisted beam tracking scheme with real-time beamwidth adaptation for THz communications that provides stable and continuous connectivity under various target mobility. Our approach commences by utilising the prior sensing information obtained periodically by the base station (BS) to predict the target’s motion path under a specific motion model. Based on the path prediction, the BS selects the optimal beam from a pre-calculated codebook. Furthermore, we optimise precoders with beamwidth adaptation in response to the target’s real-time mobility, which aims at maximising the average rate performance while adhering to constraints on minimum instantaneous rate and total transmit power. In particular, we transform the original optimisation problem into a single-variable unconstrained one by employing parameterised precoders and making necessary simplifications. Simulation results demonstrate that our proposed scheme outperforms conventional ones in terms of average achievable rate and outage probability by consciously aligning the beam with the range of target motion. The performance improvement is especially significant at high velocities.

The remainder of this paper is organised as follows. In Section II, we first introduce the THz communication model and then propose the sensing-assisted beam tracking scheme. In Section III, we optimise the mobility-aware precoder. Simulation results are presented in Section IV. Finally, conclusions are drawn in Section V.

Notation: Boldface letters represent vectors; $\lvert\cdot\rvert$ denotes the modulus of a complex number; $(\cdot)^{T}$ and $(\cdot)^{H}$ denote the transpose and conjugate transpose of a vector respectively; $\mathcal{CN}(\mu,\Sigma)$ denotes the Gaussian distribution with mean $\mu$ and covariance $\Sigma$; $\mathcal{U}(D)$ denotes the two-dimensional uniform distribution over area $D$.

We consider a typical THz multiple-input multiple-output (MIMO) system in this paper, where the BS employs a uniform linear array (ULA) with $N_{t}$ fully digital antennas, and each target employs a single isotropic antenna. Without loss of generality, we assume the targets to be independent of each other due to the sparsity of THz channels [13]. Therefore, we focus solely on the downlink transmission from the BS to a single target for the remainder of this paper.

In this section, we expect to design the optimal precoder to maximise the achievable data rate while providing extensive coverage of the predicted target path. To this end, we formulate the following optimisation problem

where the objective function is the time average of achievable data rate over a sensing period $\tau$, $C1$ is the minimum instantaneous rate constraint introduced to reduce outages caused by beam misalignment, and $C2$ is the transmit power constraint. Due to its non-convex nature and the fact that the objective function cannot be integrated in closed form, problem (6) is difficult to handle.

However, the proposed sensing-assisted beam tracking scheme allows us to simplify the problem by considering a specific beam shape that features concentration within the predicted range, constant main-lobe beamforming gain, and adaptive beamwidth. More specifically, by constraining the beam shape, we can use limited parameters to control the physical direction and beamwidth, which significantly reduces the dimensionality of the decision variable. To this end, we formulate the precoder as the weighted linear combination of an infinite number of array response vectors within the angular range of predicted target path, which is defined as [7]

where $\beta$ is the power constraint coefficient, $\delta$ is introduced to indicate beam coverage, the offset parameter $\omega$ is introduced to optimise the beam shape, and

is the array response vector at the direction of $\theta_{p}=p+\theta_{m}$. Note that the integral in (7) can lead to unacceptable computational complexity when optimising the precoding vector. Therefore, several simplifications are employed. First, direction expressions are written as the sine of the target’s angular direction, transforming the domain from $[-\pi/2,\pi/2]$ to $[-1,1]$. Second, we define

and

where $\mathrm{sin}\phi_{0}$ and $\mathrm{sin}\phi_{\tau}$ are the target’s directions at time $0$ and $\tau$ respectively. Considering (9) and (10), we achieve symmetrical bounds of integration in (7), which simplify the calculation and also focus the beam coverage on the target’s motion path. In this way, the n-th component of $\boldsymbol{f}$ can be written as

where $\mathrm{Sa}(x)=\mathrm{sin}(x)/x$ is the sample function.

Because multi-path interference is considered negligible in this work, it is generally recognised that transmit power is positively related to the achievable data rate under the circumstances. As a result, $C2$ can be rewritten as

For simplicity, given a known $\delta$, let

Then according to (12), we have

By applying (14) in (11), we actually eliminate the need for $C2^{\prime}$ as a separate constraint. Now we are able to reformulate the original problem $Q1$ into a non-convex single-variable constrained optimisation problem $Q2$ as follows

The decision variable $\omega$ helps minimize the variation of beamforming gain within the range of target motion. To address $C1$ as well as further simplify the problem, we introduce a penalty function

and add it to the objective function. The penalty function is non-zero and inversely proportional to the degree of $C1$ violation when the constraint is violated, and is zero otherwise. The impact of penalty is controlled by the parameter $\alpha$. By transforming the constrained problem into an unconstrained one with this penalty, we finally obtain

which can be solved by applying particle swarm optimisation (PSO) [19]. Although the antiderivative of the integrand in (17) can not be written in closed form, we opt for numerical integration since the computations are manageable for pre-calculating the codebook.

Next, we need to establish the bounds for PSO. Based on (3), (11), (13), and (14), the beamforming gain can be presented as

where

and

are introduced to simplify the expression.

We present the following Theorem 1 to demonstrate the axial symmetry of $\lvert\boldsymbol{a}^{H}(\boldsymbol{S})\boldsymbol{f}(\delta,\omega)\rvert^{2}$, which consequently reduces the PSO search space by half.

By (5), (16), and (17), we can derive from Theorem 1 that the objective function in problem $Q3$ is also symmetric about $\omega=\frac{N_{t}-1}{2}\pi$. Additionally, it is evident that (18) can be considered zero for $\omega<0$, an interval where no optimal case exists. Therefore, we can implement a one-dimensional search using PSO on the interval $[0,\frac{N_{t}-1}{2}\pi]$ to find the optimal precoder.

In this section, we provide simulation results to evaluate the performance of the proposed sensing-assisted beamforming and tracking scheme in terms of achievable data rate and outage probability. The main simulation parameters are shown in Table I. We consider a THz MIMO communication system where the BS is equipped with a 128-antenna ULA and operates at the carrier frequency of 220 GHz with the bandwidth of 10 GHz. The noise power spectrum density $N_{0}$ is set to -174 dBm/Hz. To obtain the molecular absorption coefficient [4], we assume a relative humidity of 50%, a temperature of 14^∘C, and a pressure of 1012.6 hPa. We assume the target follows URM. Specifically, the target moves parallel to the ULA at a uniform velocity. The perpendicular distance between the target path and the line on which the ULA is located is set as 100 m. The start angle of target motion is at the broadside direction of ULA which is denoted as $\phi(\boldsymbol{S}|t=0)=0$ rad, and the end angle is set to 0.3 rad. This model represents the general case of linear motion, as both the target’s angle and distance from the BS vary with the motion.

Fig. 2 presents beam patterns generated by our proposed scheme at various target velocities within a single sensing interval. The sensing interval $\tau$ is set as 165 ms which will be elucidated later in this section. It can be observed that, aided by prior sensing information, the BS dynamically adjusts beamwidth in response to the target’s real-time motion. Narrower beams are utilised for low mobility targets, while wider beams are employed for high mobility targets, though this comes at the cost of reduced main-lobe beamforming gain.

Then we compare the proposed scheme with the event-based tracking scheme presented in [7] and the conventional scheme which we previously introduce in Section II. We set the transmit power to 40 dBm. For reproducing the event-based scheme, we set the time slot duration to 50 ms, the variance of random walk to 25, and the weight parameter to 0.1. We evaluate the performance of different schemes by calculating the time average of achievable date rate and outage probability over the target’s motion range $\{\phi(\boldsymbol{S})|0\leq\phi(\boldsymbol{S})\leq 0.3\}$. Note that the beam realignment in [7] depends on outages, which occur more frequently with higher target mobility. Consequently, a fair comparison in terms of sensing overhead (or pilot overhead in the conventional scheme) necessitates setting the sensing interval $\tau$ in our proposed scheme to 3.3 times of the time slot duration in [7]. Here, 3.3 represents the average number of elapsed time slots between two adjacent beam realignments, calculated across a range of target velocities from 10 m/s to 100 m/s in 10 m/s increments. Moreover, the interval between beam switches in the conventional scheme is set to the same value as $\tau$.

Fig. 3 compares the rate and outage performance under various target velocities among different schemes. The range of velocity is set to [10 m/s, 20 m/s, …, 100 m/s]. We can observe from Fig. 3LABEL:sub@cap_vel that our proposed sensing-assisted scheme achieves a higher average rate than the other two schemes while there is a downtrend in performance as velocity increases for all schemes. Additionally, Fig. 3LABEL:sub@out_vel shows that our proposed scheme maintains a lower outage probability, staying below 10% even at the velocity of 100 m/s. Note that the fluctuation in the event-based scheme’s line is due to the non-monotonic decrease in the frequency of beam realignment with increasing velocity.

Fig. 4 presents the instantaneous achievable rate for all three schemes over the time range from the initialisation to the first beam realignment in the event-based scheme. Simulations are run with a target velocity of 20 m/s. We can observe that our sensing-assisted scheme adjusts its beams to fit within the range of target motion, which accounts for its improved rate performance and reduced occurrence of outages. In contrast, the other schemes direct the center of their beams solely towards the target’s initial position, leading to almost half of the transmit power being wasted due to the absence of motion awareness. Besides, it illustrates the definition of $\tau$ and the time slot duration. Further, we can observe that the rate line for our scheme shows approximate axial symmetry within each sensing interval. It is because that, under given conditions, the target’s distance from the BS has only trivial effect on achievable rate, which justifies the uniformity of our beam shaping.

Finally, Fig. 5 compares the average rate over a range of transmit power for all three schemes at the velocity of 100 m/s, demonstrating the effectiveness of our proposed scheme in high mobility scenarios.

Effective beam tracking is crucial in mitigating the performance degradation caused by beam misalignment in THz communications within high-mobility scenarios. In this paper, a sensing-assisted beam tracking scheme with real-time beamwidth adaptation has been proposed, which ensures stable and continuous connectivity for targets with various mobility. Specifically, by periodically collecting the prior sensing information, the BS can predict the path of target motion and then adapt the beamwidth and transmit direction accordingly. To enable such adaptation, we have optimised parameterised precoders that maximise the average achievable rate within each sensing period. Simulation results demonstrate that our scheme significantly outperforms existing methods by achieving improved rate performance and reduced outage occurrence across various mobility conditions, laying the groundwork for future high-mobility applications.

[Proof of Theorem 1] First, we provide Lemma 1 to prove the axial symmetry of the first summation term on the right-hand side of (18).

Subsequently, we provide Lemma 2 and Lemma 3 to prove the axial symmetry of the second summation item on the right-hand side of (18). Let the domain of $(m,n)$ be

Let

where $q_{m,n}(\omega)$ is introduced for simplification.

Lemma 2 implies that for the same value of $m-n$, the shape of $q_{m,n}(\omega)$ remains unchanged. Similar to Lemma 1, we can prove that $q_{m,n}(\omega)$ is symmetric about $\omega=\frac{m+n-2}{2}\pi$, i.e., the position of $q_{m,n}(\omega)$ is determined by the value of $m+n$. For random variable $(m,n)\sim\mathcal{U}(D)$, it is trivial to prove that the probability mass function of $m+n$ is symmetric about $N_{t}+1$. Then we prove Lemma 3 to further present the symmetry of $\sum_{(m,n)\in D}q_{m,n}(\omega)$.

Recall that the position of $q_{m,n}(\omega)$ is determined $m+n$ and that the shape of $q_{m,n}(\omega)$ is determined $m-n$. Lemma 3 implies that for every $(m,n)\in D$ that satisfies $m+n=\rho$, where $\rho\in\mathbb{Z}$, there exists a corresponding $(m^{\prime},n^{\prime})\in D$ that satisfies $m^{\prime}+n^{\prime}=2(N_{t}+1)-\rho$ and $m^{\prime}-n^{\prime}=m-n$. Consequently, for every $q_{m,n}(\omega)$, $(m,n)\in D$, there exists a symmetric counterpart $q_{m^{\prime},n^{\prime}}(\omega)$ with $(m^{\prime},n^{\prime})\in D$ that is symmetric to $q_{m,n}(\omega)$ about $\omega=\frac{N_{t}+1-2}{2}\pi$ and has the same shape as $q_{m,n}(\omega)$. Hence, we can conclude that $\sum_{(m,n)\in D}q_{m,n}(\omega)$ is symmetric about $\omega=\frac{N_{t}-1}{2}\pi$.

In summary, the expression on the right-hand side of (18) is symmetric about $\omega=\frac{N_{t}-1}{2}\pi$.