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Abstract
The ultraviolet communication (UV) channel has been shown to have unique features that could be exploited for covert ground-to-ground communications in complex non-line-of-sight (NLOS) scenarios. A key challenge is the determination of optimal configuration of pointing directions of the UV nodes in unknown NLOS environments to maximize the link performance. In this paper, we proposed a novel steering optimization approach based on Finite Difference Stochastic Approximation (FDSA) to simultaneously optimize the transmitter (Tx) and receiver (Rx) pointing directions without any knowledge about the locations and relative orientations of the two nodes. We perform parametric analysis using Monte Carlo channel simulations to investigate and select appropriate key algorithmic parameters and analyze the performance of the proposed algorithm. We also carry out experimentation using our custom designed UV Tx and Rx gimbal systems and demonstrate the utility and efficiency of the proposed steering optimization approach and show that the received photon count can be increased significantly.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The ultraviolet communication (UVC) channel has been shown to have unique features, including robustness against jamming and enhanced low probability of detection (LPD) characteristics compared to radio frequency (RF) alternatives [1,2]. UVC also has low background solar radiation near the surface of the Earth [3], resulting in reduced background noise level experienced by nodes deployed on or near the ground. These unique features, coupled with significant advances in semiconductor emitter and detector technologies operating at these wavelengths, have led to the exploration of UVC as an alternative communications modality both for line-of-sight (LOS) and non-line-of-sight (NLOS) scenarios [4,5].
Several experimental test-beds have been developed to investigate the performance of UVC LOS and NLOS links and to verify channel models that have been developed [6–9]. Intuitively, the relative orientation of a UV source and detector has a significant impact on the channel gain, as is reported in the above experimental research efforts. In order to optimize communication performance (e.g., enhanced communication range or data rate) by maximizing the channel gain between the nodes, efficient and simultaneous optimization strategies for the source and detector steering angles are critical, especially in complex and dynamic NLOS scenarios.
There have been some research efforts focused on steering optimization of optical and RF based communication systems. Visible light communication networks have been proposed using LEDs as steerable sources, and, for example, Eroglu, et al. [10] study the steering of such LEDs, developing an extension of the $k$-means clustering algorithm to assign and steer beams over a LOS link to the users of the network. Khan, et al. [11] examine link discovery with directional antennas and employ adaptive algorithms for initial and subsequent link discovery in ad hoc networks. Kim, et al. [12] consider steering of directional antennas for mmWave indoor wireless networks. Most of these approaches can be framed as the optimization of an objective function. When gradients are available, gradient descent methods can be employed to achieve a local optimum utilizing the computed gradient. Stochastic gradient descent (SGD) reduces the computation at each step by randomly selecting a subset of the data to compute the gradient.
In this paper, we consider a gimbaled UVC system and optimize, in real time, the steering or pointing angles of both the transmitter (Tx) and the receiver (Rx) establishing a communication link. Since the closed-form analytical expression for the gradient is not available, we employ an algorithm to estimate the gradient based on noisy measurements. One technique that does not require direct access to the gradient is the finite difference stochastic approximation (FDSA) algorithm [13]. We describe and implement the gradient-free FDSA algorithm to efficiently and simultaneously optimize the steering angles of the UVC system by maximizing the channel gain. In the presence of dynamic conditions, such as mobile nodes or time-varying background noise (e.g., due to moving cloud cover, varying weather conditions, or other dynamics in the environment), the algorithm can be applied for adaptive tracking. Steering optimization of UV-based communication systems is needed for practical deployment of UVC networks and such an algorithm is not available in the literature. The novel algorithm presented in this paper should provide an invaluable tool for engineers and researchers implementing real-world systems. Specific contributions of the paper include the following:
- • A gradient-free steering optimization algorithm is developed with the goal of enhancing the signal strength (i.e., photon count) at a UVC Rx (potentially having only a NLOS link to the UVC Tx of interest) without the need for alignment or calibration. The proposed approach simultaneously optimizes the azimuth and inclination angles of both the transmitter and receiver gimbal systems.
- • Parametric analysis is performed to investigate key algorithmic parameters, namely the step size and perturbation parameters, using Monte Carlo based NLOS channel simulations. The selected parameters are then used for simulation based performance evaluation and demonstration of the steering optimization approach.
- • The performance of the proposed steering strategy for UVC in NLOS scenarios is investigated based on simulations and through experiments in outdoor NLOS scenarios using an implementation of the developed technique in an experimental test-bed.
The rest of the paper is organized as follows. In Section 2, we describe the problem setup and formulation. In Section 3, a description of the proposed FDSA based steering optimization algorithm and the details of the implementation are presented. Parametric and performance analysis of the proposed algorithm using simulations and measurements in NLOS scenarios are discussed in Section 4, followed by concluding remarks in Section 5.
2. Problem formulation
This work considers the scenario where a UV link is established between a Tx and a Rx. We assume each transceiver is stationary but can control the pointing direction of its Tx beam or reception field of view. (Again, the approach is also appropriate for adaptive tracking applications in dynamic scenarios.) We assume no calibrated alignment between the devices, i.e., neither device has relative positional information of the other device. We are primarily concerned with the scenario in which line-of-sight between the devices is unavailable, a common case in practice as illustrated in Fig. 1, since the presence of line of sight may offer alternative alignment approaches. Regardless, the same concept can be applied to LOS links as well.
Fig. 1. A graphic of an example use case for NLOS UVC link in an urban environment is shown where the scattering from the atmosphere is being utilized to establish the NLOS link. The Tx and Rx are positioned on the ground. The Tx and Rx nodes (i.e., gimbal systems) that are used for experiments in this paper are also shown.
The steering optimization problem seeks to orient the Tx and Rx pointing directions to maximize the channel gain or, correspondingly, the photon count received within a given time interval (i.e., the incoming photon rate). This value depends on a number of device and environmental parameters. Device parameters include the pointing directions (inclination and azimuth angles) of both Tx and Rx, the Tx beam-width, the Rx field of view (FOV), the Rx filter efficiency and aperture area, and the Tx power (or number of photons transmitted). Environmental parameters include atmospheric scattering and extinction coefficients. Details of associated UVC models can be found in [9,14].
Tx and Rx gimbals are illustrated in Fig. 1, along with the steering angles. Since the steering problem is only concerned with the pointing directions, for convenience, we shall express the channel gain solely as a function of these parameters, i.e., $\Gamma (\theta _\text {t},\phi _\text {t},\theta _\text {r}, \phi _\text {r})$, where the pairs $(\theta _\text {t}, \phi _\text {t}) \in \mathbb {S}^{2}$ and $(\theta _\text {r}, \phi _\text {r}) \in \mathbb {S}^{2}$ are the pointing directions of the Tx and Rx nodes, respectively, in terms of the inclination and azimuth angles, and $\mathbb {S}^{2}$ corresponds to the spherical domain spanned by the angles $\theta$ and $\phi$. An exact expression of $\Gamma$ is not available (since the system geometry, device and environmental parameters are not known), and so values of $\Gamma$ must be measured. Due to shot noise and noise from the environment, this observation is the sum of two Poisson random variables, one with mean $P_\textrm {t}\Gamma G_\textrm {r}$ associated with the channel gain and the other with mean $\epsilon G_\textrm {r}$ associated with the background radiation noise, where $P_\textrm {t}$ is the transmitted power, $G_\textrm {r}$ is the Rx gain, and $P_\textrm {t}$, $G_\textrm {r}$, and $\epsilon$ are assumed to be constant regardless of the Tx and Rx orientations. That is, for particular pointing direction pairs, we observe the number of photons $f$ from $\text {Poisson}\{(P_\textrm {t}\,\Gamma +\epsilon )G_\textrm {r}\}$. Since $P_\textrm {t}$, $G_\textrm {r}$, and $\epsilon$ are constant, maximizing $\Gamma$ over the pointing directions is equivalent to maximizing $\mathbb {E}[f]$, the expected value of $f$. For convenience, Table 1 lists the parameters that are referred throughout the text.
Table 1. System Parameters
Explicitly solving for the optimal pointing directions requires full knowledge of all parameters, including positional information at both devices. Since this information is unavailable, we instead pursue incremental adjustments to the Tx and Rx that attempt to enhance the measured received signal power or photon rate. Note that, for simplicity in the presentation, we express this problem and our approach (in the next section) using pointing direction parameters with respect to a global coordinate reference frame of $\mathbb {S}^{2} \times \mathbb {S}^{2}$. Importantly, however, the algorithm never jointly employs the coordinate reference frames across the $\mathbb {S}^{2}$ domains, since it is assumed that the Tx and Rx have not coordinated or communicated details of the coordinate reference frames associated with the respective $\mathbb {S}^{2}$ domain. In the next section, we present a novel calibration-free steering optimization algorithm based on stochastic approximation that locally maximizes the objective function.
3. Proposed steering approach
In this section we provide the details of our recursive optimization algorithm, which is based on approximating the gradient components of the objective function $f$ from noisy samples. Optimization methods based on gradient approximation have been of interest for many decades and appeared in the literature as early as the mid-1950s [15]. They are particularly useful when the gradient information (or its direct measurement) is unavailable but where measurement of $f$ can yield noisy observations at a discrete set of points. Such approaches are generally referred to as stochastic approximation (SA) due to the randomness in the observation process [16]. One of the most popular gradient-free SA techniques is finite difference stochastic approximation (FDSA) [13]. In the rest of this section, we will briefly introduce FDSA and discuss the details of our FDSA-based algorithm followed by a description of implementation challenges.
3.1 Background on FDSA
FDSA is a gradient-free stochastic approximation technique that almost surely converges under certain conditions as discussed in [17,18]. In FDSA, each component of the gradient is approximated and used in the following recursive form:
where $\nu$ corresponds to an individual component of the $n-$dimensional vector $\Theta$ over which the objective function is maximized, $\nu _k$ is the value of the argument during the $k$th iterate, and $g_\nu$ is the estimate of the gradient component along $\nu$. The step-size parameter $\eta$ in Eq. (1) controls the rate of convergence of the algorithm and its significance is discussed in Section 4. The approximation for a single-sided FDSA is given by where $\Theta _k$ is the current position, $\Delta$ is a positive small perturbation and $\textbf {e}_{\nu }$ is an $n-$dimensional basis vector with the component associated with $\nu$ being 1 and all other components 0. In our approach, we implement the single sided FDSA, in which an iterate comprises a total of $n+1$ measurements to compute the gradient approximations for all components of $\Theta$.An alternative is the two-sided FDSA algorithm that uses additional measurements per iteration to improve the gradient estimation by replacing the term $f(\Theta _k)$ in Eq. (2) with $f(\Theta _k-\Delta \, \textbf {e}_{\nu })$ and $\Delta$ in the denominator with $2\Delta$.
3.2 Implementation of FDSA on $\mathbb {S}^{2}\times \mathbb {S}^{2}$
Recall that the domain of the objective function $f$ is $\mathbb {S}^{2}\times \mathbb {S}^{2}$, the product space of the spherical domains of the Tx and Rx pointing angles, respectively. Let the components of the vector $[\theta _{\textrm {t}}^{(k)}\;\; \phi _{\textrm {t}}^{(k)}\;\; \theta _{\textrm {r}}^{(k)}\;\; \phi _{\textrm {r}}^{(k)}]^{\textsf {T}} \in \mathbb {S}^{2}\times \mathbb {S}^{2}$ be the inclination and azimuth angles of the Tx and Rx determined at the $k$th iterate. These angles are referenced to the local coordinate frame of the associated gimbal, and these reference frames are not aligned or shared. The pointing directions are updated simultaneously at the end of each iteration using FDSA. To formulate the FDSA update, each spherical domain is embedded in $\mathbb {R}^{3}$ with the associated gimbal at the origin. The tangent planes in each $\mathbb {R}^{3}$ domain at the current iterate are then used to implement Eqs. (1) and (2), with derived directions projected back from these planes onto the $\mathbb {S}^{2}$ surfaces.
The FDSA algorithm involves first estimating the gradient and then computing the next iterate based on that gradient estimate. There is flexibility in the choice of perturbation directions; we next describe our choice. We naturally perturb the transmitter and receiver gimbals in sequence, holding the other at the current iterate. For the perturbations of a particular gimbal, we choose one direction in the associated plane to be that of maximal vertical component (which projects back to $\mathbb {S}^{2}$ as perturbing only the inclination angle), with the second then being perpendicular to that direction. It should be noted that one can certainly derive a gradient estimate based on an azimuth perturbation that coincides with our approach for infinitesimal perturbations. However, we observe that this approach potentially produces worse outcomes compared to our approach, especially if Tx-Rx orientations are close to the poles.
While the pointing perturbations and iterate update could be derived in the given gimbal reference frames, a change of basis simplifies the development. In particular, we consider reference frames in which the positions of the Tx and Rx at the current iterate are at the poles of their respective domains; we refer to these as “polar” bases, in contrast with “gimbal” bases. Let
To compute a finite difference approximation of the gradient, we require five measurements; i.e., each iteration comprises five measurements of the objective function, each at a different steering configuration. We first measure $f_0$, the objective function at the current iterate. Next, two measurements are taken under orthogonal perturbations of the Tx pointing direction. The first measurement pointing direction for the Tx (in the polar bases) is given by $(\Delta,\phi _\text {t}^{(k)})$, and the second is given by $(\Delta,\phi _\text {t}^{(k)}+90^{\circ })$. (Note that for simplicity, we choose to perturb an arc length $\Delta$, which approximates moving a distance $\Delta$ in the tangent plane and projecting to $\mathbb {S}^{2}$.) Applying $m$ to these pointing directions yields gimbal Tx directions for measurements $f_1$ and $f_2$. Analogous treatment for the Rx yields the Rx directions for measurements $f_3$ and $f_4$.
Still considering the polar bases for both nodes, we can now estimate the gradients restricted to the tangent planes with axes corresponding to the perturbed directions (prior to projection back onto $\mathbb {S}^{2}$). See Fig. 2 for an illustration of these directions. The gradient estimate for all components is given by
Fig. 2. A diagram depicting the geometry of the FDSA based approach on $\mathbb {S}^{2}$. The current position is perturbed along two orthogonal directions. The magnitude and direction (i.e., $\omega$) of the gradient are estimated from $f_0$, $f_1$, and $f_2$ based on Eq. (7).
Based on this gradient estimate, we compute the pointing direction of the nodes at the start of the next iterate. Focusing on the Tx side (still in the polar basis) and referencing Fig. 2, the direction of the gradient relative to the gradient measurement axes is defined by the angle $\omega _\text {t} = \text {atan2}(g_2,g_1)$. The magnitude of the gradient projected onto the Tx domain is $\rho _\text {t}=\sqrt {g_1^{2}+g_2^{2}}$, and so the next iterate moves the Tx direction a distance $\eta \rho _\text {t}$, where $\eta$ is the step size parameter. We again interpret this distance as an arc length in $\mathbb {S}^{2}$, as opposed to interpreting it as a distance in the extended domain $\mathbb {R}^{3}$, which would require a subsequent projection. Therefore the next iterate (still in the polar basis) is given by $(\eta \rho _\text {t},\,\phi _\text {t}^{(k)}+ \text {atan2}(g_2,g_1))$. Applying $m$ to $(\eta \rho _\text {t},\,\phi _\text {t}^{(k)}+ \text {atan2}(g_2,g_1))$ yields the next iterate in the gimbal bases. Algorithm 1 summarizes the FDSA based optimization method for the described above to simultaneously optimize the pointing directions of two gimbal systems defined in $\mathbb {S}^{2}\times \mathbb {S}^{2}$ to maximize the received photon counts. All angles in Algorithm 1 are in radians.
3.3 Implementation considerations
Next, we describe implementation challenges that need to be taken into account to improve the utility of the proposed approach in practice. Note that each measurement of $f$ includes the background noise in addition to the signal counts that we seek to maximize. To compensate for the contribution of background noise, one approach is to take, for each measurement of $f$, a separate noise measurement while the UV source is off, and then subtract the measured noise counts from the measurement of $f$. However, this approach doubles the number of measurements and time requirements. Another approach would be sampling the background noise count when the gimbals are updating their orientations between measurements. In our implementation, we assume the noise distributions at $f(\Theta _k)$ and $f(\Theta _k+\Delta \, \textbf {e}_{\nu })$ are the same provided that $\Delta$ and the time difference between the measurements are small enough. So, the expected value of the background noise contribution in the difference $f(\Theta _k+\Delta \, \textbf {e}_{\nu }) - f(\Theta _k)$ given in Eq. (2) is assumed to be $0$.
Algorithm 1. Iterative optimization of UVC steering (Angles are in radians)
Another source of noise that impacts the estimates is measurement noise or shot noise, which correspond to the fluctuations of the photon rates measured. Considering the photon arrival statistics to be Poisson distributed, the variance of the fluctuations around the mean has a linear relationship with the number of counts detected. Therefore, although background noise contributions are assumed to be compensated as mentioned, at high levels they cause fluctuations to become larger, which increases the uncertainty in $f(\Theta _k+\Delta \, \textbf {e}_{\nu }) - f(\Theta _k)$ and degrades the accuracy of the estimates.
Other than maximizing photon counts, we are also interested in the amount of time that lapses between iterations and total time at the end of the optimization process. The elapsed time for an iteration cycle is the accumulation of time required for taking measurements, movements of the pointing directions, and processing data. One should select the sampling duration for $f_i, i \in \{0,1,2,3,4\}$ by taking the incoming photon rate into account. Longer sampling time results in collecting more photons and getting a better estimate of the distribution, i.e., the true mean value and variance. Hence, there is a tradeoff between time per iteration and accuracy of gradient estimates.
Another practical challenge is that the measured photon counts are only available at the Rx node. Therefore, a coordination mechanism is required between the two nodes that enables the Rx to share the measurements with the Tx. In an ideal system, both nodes can be set as a transceiver and data can be transferred via a full duplex UV channel. In our test-bed implementation, we use a designated RF link to send the information from Rx to Tx.
4. Simulation and experimental results
In this section, we investigate the performance of the proposed steering optimization algorithm via simulation and experimentation. First, we obtain values of the parameters $\eta$ and $\Delta$ through a parametric analysis that shows their impact on the amount of time that the algorithm needs to reach the optimal solution. We then present performance evaluation of the proposed steering optimization approach via simulations, where the $\eta$ and $\Delta$ values selected based on the parametric analysis discussed above. We utilize channel data acquired from Monte-Carlo simulations for both the parametric analysis and the optimization simulations. We also show the results of an experimental investigation by implementing the algorithm in LabVIEW using a UVC test-bed system.
4.1 Parametric analysis
We find the optimal values for $\eta$ and $\Delta$ through simulation analysis. The term $\eta \rho$ described in Section 3.2 is the step taken towards the solution from the current position at the end of each iterate. The larger the magnitude of $\eta \rho$, the fewer iterations needed to reach a solution, but at the same time there is an increased risk of overshooting. The value of $\eta$ has a significant impact on this trade-off. The second parameter that needs to be investigated is $\Delta$, which is the amount of perturbation along one dimension in degrees. A good approximation of the partial derivative ${\partial f}/{\partial \nu }$ via computation of the slope suggests $\Delta$ being infinitesimal. However, due to the shot noise present in the system, this will cause photon counts at $\Theta _k + \Delta \textbf {e}_{\nu }$ and ${\Theta _k}$ to be similar, which exacerbates the uncertainty of the gradient estimation. Therefore, we consider values of $\Delta$ that are higher than $1^{\circ }$.
Here, we provide the details of the sensitivity analysis of $\eta$ and $\Delta$ by considering a representative scenario for a NLOS UVC link. This scenario consists of a two-node network, where Tx and Rx nodes are separated by 100 m. The background noise level and the transmitted power are set to 1000 counts/s and 10 mW, respectively. The LOS path is blocked and the Tx inclination angle $\theta _\text {t}$ is restricted to be less than $60^{\circ }$. We use the results of a Monte-Carlo based approach described in [19] to estimate the channel gain as a function of the pointing directions of the nodes. The initial configuration of the nodes are set to $(\theta _\text {t},\phi _\text {t}) =(40.0^{\circ },-70.0^{\circ }), (\theta _\text {r},\phi _\text {r})=(50.0^{\circ },75.0^{\circ })$, at which the (true) photon rate is 2624. The global solution is achieved at $(60.0^{\circ },-90.0^{\circ }), (75.0^{\circ },90.0^{\circ })$ when the (true) rate becomes 9011, which is $243\%$ higher than the starting photon rate. Note that $\pm 90^{\circ }$ azimuth angles suggests that the nodes are facing towards each other. There is also a local maxima at $(60.0^{\circ },-90.0^{\circ }), (40.5^{\circ },90.0^{\circ })$, where the photon rate becomes 4765. It should be noted while we are presenting the results based on a specific scenario here, we have investigated several other settings for the maximum inclination angle and the initial conditions and the variation in the optimal $\eta$ and $\Delta$ values is not significant.
In the simulations, we include the following sets of values for $\eta$ and $\Delta$: $\eta \in \{10^{-3}, 10^{-4}, 5\times 10^{-5}, 10^{-5}, 5\times 10^{-6}, 10^{-6}\}$ and $\Delta \in \{1.35^{\circ }, 1.80^{\circ }\, 2.25^{\circ }, 2.7^{\circ }\}$. For each pair $(\eta, \Delta )$, the simulation is repeated 1000 times. At the end of each run, we record the final configurations of the nodes and the total time needed to reach that configuration in terms of estimated time taking into account the movement of the actual gimbal system, not the simulation time. For a particular configuration (transmitted power, range, constraints on the orientations, etc.) the global and local solutions are known, determined through the results of Monte-Carlo simulations. The algorithm stops when one of the following conditions occurs: i) reaching a near-optimal global solution (Gbl) configuration, at which the (true) photon rate is within $3\%$ of the global solution and the arc lengths between optimal and final configurations are less than 0.1; ii) reaching a local solution (Lcl); or iii) not being able to converge within a specified time limit, e.g., 1 hour (NC). We perform simulations taking into account realistic time needed for gimbal steering and data collection as well as for other signal processing and computation to estimate the total time needed to reach a solution based on the selected $\eta$ and $\Delta$ parameters. Based on these constraints the convergence rates for different pairs of $(\eta,\Delta )$ is given in Table 2. When $\Delta \geq 1.8^{\circ }$, both $\eta =10^{-5}$ and $\eta =5\times 10^{-6}$ give a near-optimal solution with a rate of greater than $99.8\%$.
Table 2. Convergence results of the sensitivity analysis considering the three stopping criteria 1) global (Gbl), 2) local (Lcl), and 3) no convergence (NC)
In addition to the convergence rates, there is also another key factor in the selection of $\eta$ and $\Delta$: elapsed time, or number of iterations that is needed to reach a solution. Figure 3 shows results of the same analysis from a perspective of average elapsed time. Out of 1000 runs, we consider only the ones that reach a near-optimal solution. The least amount of time is obtained when $\eta =10^{-5}$ and $\Delta =2.25^{\circ }$. It is also worth mentioning that variations in $\Delta$ have significantly less impact on the performance compared to that of $\eta$.
Fig. 3. Sensitivity analysis for $\eta$ and $\Delta$. Out of 1000 runs, only those that converged to the global solutions are averaged.
4.2 Performance validation via simulations
Once the optimal values of $\eta$ and $\Delta$ are determined, we investigate the performance of the proposed algorithm based on channel data using Monte Carlo simulations. Here, we consider a different scenario to demonstrate the validity of the selected $\eta$ and $\Delta$ values acquired in the previous section when applied to other scenarios. Specifically, we choose the noise level to be 2000 counts/s and a range of 75 m. Moreover, the initial pointing directions are configured to further away from the optimal orientations as $(\theta _\text {t},\phi _\text {t}) =(35.0^{\circ },-65.0^{\circ }), (\theta _\text {r},\phi _\text {r})=(40.0^{\circ },60.0^{\circ })$ at which the (true) photon rate is 3131. The global solution occurs at $(60.0^{\circ },-90.0^{\circ }), (75.0^{\circ },90.0^{\circ })$ with a (true) rate of 11931, which is $281\%$ higher than the initial rate. Recall that Tx inclination is restricted to $60^{\circ }$ maximum. Figure 4 illustrates an instance of the change in photon rates for this particular configuration using the global stopping criteria. The algorithm stops when the pointing directions reach $(\theta _\text {t},\phi _\text {t}) =(60.0^{\circ },-91.2^{\circ }), (\theta _\text {r},\phi _\text {r})=(75.2^{\circ },93.5^{\circ })$, where the (true) rate is 11639 counts/s, which corresponds to an improvement of $271\%$ within 20 minutes.
Fig. 4. Simulation of the FDSA algorithm, where $\eta =10^{-5}$ and $\Delta =2.25^{\circ }$. After $15$ minutes, the photon rate becomes more than three times higher than the initial value.
Another result that provides insights into the behavior of the steering iterates at both the Rx and Tx nodes during the optimization process is given in Fig. 5. This result shows the convergence paths at the Rx and Tx for the same simulation scenario considered in Fig. 4. It takes 40 iterations to simultaneously optimize Rx and Tx. The starting orientation and the optimal solution on both sides are shown as red and magenta markers, respectively. The small markers on the lines correspond to starting point at the beginning of the iterations. Note that the Rx does not show any improvement for the first 12-15 iterations and its convergence is slower compared to Tx in this case. The Rx steering initially tends to a local maxima before correcting toward the global maximum. This behavior demonstrates that the convergence time will depend on the starting point chosen.
Fig. 5. The convergence paths of the Rx (on the left) and Tx nodes on $\mathbb {S}^{2}$ associated with the simulation result given in Fig. 4. The two nodes adjust their pointing directions to maximize the received photon rate iteratively and simultaneously. The blue markers on the lines show the configurations at the beginning of the iterations.
4.3 Experimental validation
4.3.1 System and measurement setup
We conduct outdoor experiments to investigate the performance of the proposed algorithm using a custom designed UV-band communication system. The system includes a Tx and a Rx node (a simplex channel), where each node comprises a gimbal structure with two stepper motors as shown in Fig. 1. The Tx and Rx gimbals are equipped with a UV source and a UV detector, respectively. The pointing direction of gimbals can be precisely controlled by the motors rotating the azimuth and inclination angles. The minimum step on both directions is $0.1125^{\circ }$. A data acquisition device (DAQ) is used for communicating with the gimbals via a PC interface. It is also used for counting the number of photons detected, adjusting the motors, and synchronization of Tx and Rx. We implement the FDSA algorithm in LabVIEW with the Rx sharing the information of received photon counts with the Tx at the end of each iteration. We establish a designated RF link between the nodes for this purpose. It should be noted that a duplex UV channel could be used for the coordination instead of a separate RF link.
4.3.2 Measurement results
The first outdoor experiment is conducted in an open space when the nodes are separated by 22.5 m. A UV block hardboard is placed in between the nodes to remove any LOS link. The nodes are first aligned to each other, and then the system is brought to the initial configuration before the optimization process started. Both $\theta _\text {t}$ and $\theta _\text {r}$ are restricted to be less than $60^{\circ }$. The starting orientations are $(\theta _\text {t},\phi _\text {t})=(45^{\circ },170^{\circ })$ and $(\theta _\text {r},\phi _\text {r})=(45^{\circ },10^{\circ })$, at which the background noise level is recorded as 3000 counts/s, and the signal rate (after subtracting noise) is estimated as 3300 counts/s. We halted the optimization process after 20 minutes. The change in the received photons per second is given in Fig. 6. Note that counts correspond to accumulation of both noise and signal parts. At the end of 20 minutes both nodes reach the maximum inclination ($60^{\circ }$). The final orientations become $(\theta _\text {t},\phi _\text {t})=(60^{\circ },177.4^{\circ })$, $(\theta _\text {r},\phi _\text {r})=(60^{\circ },1.9^{\circ })$. The noise level and signal rates are obtained as 3000 counts/s and 8800 counts/s after optimization, which yields an increase of the data rate about 167%.
Fig. 6. Outdoor experiment to validate the proposed steering algorithm for the first scenario. The range between the Tx and Rx was 22.5 m and the LOS link was blocked.
For the second experiment we place the nodes near adjacent sides of a low profile building. In this case the separation is 44 m and alignment of the nodes is not made since the LOS link is blocked with the building in between. The noise counts at the start and end of the process are recorded as 2200 counts/s and 1600 counts/s, while total counts (signal plus noise) are recorded as 3500 counts/s and 5000 counts/s, respectively. An improvement of 161% in data rate is achieved at the end of the optimization. The overall progress in the received counts is shown in Fig. 7. Similar to the previous scenario, both $\theta _\text {t}$ and $\theta _\text {r}$ start from $45^{\circ }$ and reach $60^{\circ }$, this time after 40 minutes. Note that these angles are referenced to the local coordinate system as the nodes are not aligned with each other.
Fig. 7. Outdoor experiment to validate the proposed steering algorithm for the second scenario. The range between the Tx and Rx was 44 m and the LOS link was blocked by a large building.
5. Conclusions and future work
In this paper, we presented a novel approach to simultaneously optimize the Tx and Rx pointing directions of UVC nodes in NLOS scenarios by maximizing the channel gain and consequently the photon rate at the receiver. We have developed an FDSA based approach to approximate the gradient of the channel gain from noisy measurements. We carried out a parametric analysis and investigated the performance of the proposed approach through simulations. Through simulation and experimentation using our custom designed UV communication system, we have demonstrated the utility of the proposed steering optimization approach.
The results provide insights into how the proposed approach performs in practice although there remain some challenges that require further investigation. One observation is that, for the five to ten minutes, the pointing directions often wander close to the initial configuration before breaking out on a path towards a satisfactory local (or global) optimum. There are certain aspects of the proposed algorithm that require further investigation including asymptotic rate of convergence associated with a choice of the FDSA parameters. Another important aspect is the selection of the initial pointing directions of the Tx and Rx. The ultimate convergence of the algorithm depends on the choice of initial conditions, and one potential future work is a systematic investigation and mitigation to address the issue of the algorithm not converging due to poor initial conditions. Another challenge that is left for future work is devising an appropriate stopping criteria. The received number of photons can change significantly with changes in the environment. Common occurrences, such as changing cloud cover or the movement of the sun can significantly effect the photon count. A stopping criteria will need to be adaptive and not based only on photon count. Yet another potential direction for future research is the implementation of alternative stochastic approaches, such as simultaneous perturbation stochastic approximation, which may offer potential savings in computation and measurement time.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. Z. Xu and B. M. Sadler, “Ultraviolet communications: Potential and state-of-the-art,” IEEE Commun. Mag. 46(5), 67–73 (2008). [CrossRef]
2. M. J. Weisman, F. T. Dagefu, T. J. Moore, C. H. Arslan, and R. J. Drost, “Analysis of the low-probability-of-detection characteristics of ultraviolet communications,” Opt. Express 28(16), 23640–23651 (2020). [CrossRef]
3. L. R. Koller, Ultraviolet Radiation (John Wiley & Sons, 1965).
4. R. J. Drost and B. M. Sadler, “Survey of ultraviolet non-line-of-sight communications,” Semicond. Sci. Technol. 29(8), 084006 (2014). [CrossRef]
5. A. Vavoulas, H. G. Sandalidis, N. D. Chatzidiamantis, Z. Xu, and G. K. Karagiannidis, “A survey on ultraviolet c-band (UV-C) communications,” IEEE Commun. Surv. Tutorials 21(3), 2111–2133 (2019). [CrossRef]
6. G. A. Shaw, A. M. Siegel, and J. Model, “Extending the range and performance of non-line-of-sight ultraviolet communication links,” in Unattended Ground, Sea, and Air Sensor Technologies and Applications VIII, vol. 6231E. M. Carapezza, ed., International Society for Optics and Photonics (SPIE, 2006), pp. 93–104.
7. G. Chen, F. Abou-Galala, Z. Xu, and B. M. Sadler, “Experimental evaluation of LED-based solar blind NLOS communication links,” Opt. Express 16(19), 15059–15068 (2008). [CrossRef]
8. N. Raptis, E. Pikasis, and D. Syvridis, “Power losses in diffuse ultraviolet optical communications channels,” Opt. Lett. 41(18), 4421–4424 (2016). [CrossRef]
9. C. H. Arslan, F. T. Dagefu, M. J. Weisman, and R. J. Drost, “Channel model validation for and extensions of an ultraviolet networking optimization framework,” in 2019 IEEE 90th Vehicular Technology Conference (VTC2019-Fall), (IEEE, 2019), pp. 1–7.
10. Y. S. Eroglu, C. K. Anjinappa, I. Guvenc, and N. Pala, “Slow beam steering and NOMA for indoor multi-user visible light communications,” IEEE Trans. on Mobile Comput. 20(4), 1627–1641 (2021). [CrossRef]
11. Z. Khan, J. J. Lehtomäki, V. Selis, H. Ahmadi, and A. Marshall, “Intelligent autonomous user discovery and link maintenance for mmWave and teraHertz devices with directional antennas,” IEEE Trans. Cogn. Commun. Netw. 7(4), 1200–1215 (2021). [CrossRef]
12. M. Kim, Y. S. Kim, and W. Lee, “Analysis of directional neighbour discovery process in millimetre wave wireless personal area networks,” IET Networks 2(2), 92–101 (2013). [CrossRef]
13. J. C. Spall, Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control (John Wiley & Sons, Inc., 2003).
14. R. J. Drost, M. J. Weisman, F. T. Dagefu, and C. H. Arslan, “A low-complexity approach to UV communication channel model evaluation,” in 2019 International Conference on Military Communications and Information Systems (ICMCIS) (IEEE, 2019), pp. 1–7.
15. J. Kiefer and J. Wolfowitz, “Stochastic estimation of the maximum of a regression function,” Ann. Math. Stat. 23(3), 462–466 (1952). [CrossRef]
16. D. Chin, “Comparative study of stochastic algorithms for system optimization based on gradient approximations,” IEEE Trans. Syst., Man, Cybern. B 27(2), 244–249 (1997). [CrossRef]
17. J. R. Blum, “Approximation methods which converge with probability one,” Ann. Math. Stat. 25(2), 382–386 (1954). [CrossRef]
18. V. Fabian, “Stochastic approximation methods,” Czech. Math. J. 10(1), 123–159 (1960). [CrossRef]
19. R. J. Drost, T. J. Moore, and B. M. Sadler, “UV communications channel modeling incorporating multiple scattering interactions,” J. Opt. Soc. Am. A 28(4), 686–695 (2011). [CrossRef]
