1. Introduction

Army tactical communications must contend with highly challenging environments that may include complex terrain, congestion, and clutter, such as may be found in urban areas. Ultraviolet (UV) communications has been proposed as a technology that may provide networking resilience [1–3], perhaps as a component of a highly heterogeneous network that leverages a number of distinct communication technologies with diverse characteristics. In particular, ultraviolet communications (UVC) can provide a short-range non-line-of-sight (NLOS) optical communication capability by leveraging increased atmospheric scattering at deep-UV wavelengths.

Of ever increasing concern for operating a tactical communications network is the possibility that a sophisticated adversary may detect the friendly transmissions. Once such a transmission is detected, an adversary may eavesdrop on, jam, localize, track, and/or target the transmission source or network. While many techniques have been proposed to decrease the detectability of conventional radio-frequency (RF) communications, the increased atmospheric absorption of deep-UV wavelengths implies that UVC has a natural low-probability-of-detection (LPD) characteristic [4,5]. In particular, the path loss associated with the UVC channel increases exponentially with range, suggesting that adversarial detection will be difficult except at very close ranges, even under ideal conditions for the adversary. Although a number of modulation techniques and quantitative analyses have been developed for RF LPD communications [6–10], much less attention has been paid to developing a rigorous understanding of the LPD properties of UVC. Such understanding is essential for optimizing the design and operation of UVC systems and networks and for predicting the quality of the LPD property in a given scenario. Without such a predictive capability, users would lack the guidance needed to know the extent and limit of their detectability, and this lack of awareness would substantially limit the usefulness of the LPD capability.

In this paper, we discuss a framework for rigorously characterizing the LPD properties of a UVC link by quantifying the standoff distance outside of which a UVC transmission is not detectable by an adversary. Specifically, we consider an adversary that measures photon counts for a given period of time and decides whether a transmission is present based on an optimal hypothesis test that assumes known noise statistics.

The paper is organized as follows. First, we present a number of representative operational scenarios that illustrate various line-of-sight (LOS) and NLOS configurations that our analysis can address. Next, we describe the LPD modeling framework, starting with the UVC LPD channel model geometry, followed by a description of the transmission requirements for a desired bit-error rate (BER), and concluding with the analysis of the resulting adversarial detection probability. Then, we discuss results of simulations studying the scenarios of interest, and finally we provide some concluding remarks.

2. Operational scenarios

In this section, we describe the operational scenarios that we are considering in our LPD analysis. While our analysis framework is meant to be general, the large number of parameters defining the combination of a friendly communication link and an associated adversarial detection link can make developing straightforward insights challenging. However, because we are ultimately most interested in worst-case scenarios in terms of the detectability of our communications, we can simplify the considered geometries to a great degree. For example, we will assume that the adversarial detector is optimally pointed to detect our transmitted signal, even though the adversary would most likely not be able to determine that direction prior to detection.

Now, the ultimate worst-case scenario occurs when the friendly communication link is operating via the NLOS scattering channel while the adversarial detector is in the LOS of the transmitter. While we are interested in examining that particular case, we note that this is not necessarily a likely scenario and that mitigation strategies can be pursued to further diminish the likelihood of such a situation. (For example, perhaps at some cost to the friendly communication performance or with the use of relays, transmitters can be constrained to point in directions where adversaries are deemed unlikely to be present.) Therefore, we will also consider other classes of configurations associated with the presence/absence of LOS for the friendly communication and adversarial detection links. Note, however, that within each of these classes, we are still guided by the goal of defending against worst-case scenarios. We describe these cases below.

2.1 Scenario 1: NLOS/NLOS

Our first scenario, depicted in Fig. 1, considers the case where neither the friendly receiver nor the adversarial detector can establish a LOS link. In particular, we assume that the transmitter, friendly receiver, and adversarial detector are positioned colinearly in that order with coplanar pointing directions, where the associated plane is oriented vertically. The friendly receiver and adversarial detector are pointed in the direction of the transmitter with inclination angles of $60^\circ$ and $75^\circ$, respectively, where the former inclination angle was chosen as representative of the pointing elevation that may be needed to get over small obstacles, and the latter inclination angle was chosen to approximately minimize the channel path loss associated with the adversarial detector link.

 

Fig. 1. Illustration of Scenario 1, which considers a NLOS friendly communications link and a NLOS adversarial detection link. Three subcases (Scenario 1a, 1b, and 1c) are considered in which different transmitter pointing directions are assumed. The specific geometry parameters are detailed in the text.

Download Full Size | PDF

Meanwhile, we consider three cases for the transmitter. For Scenario 1a, we assume the transmitter is pointed toward the receiver with an inclination angle of $60^\circ$. For Scenario 1b, we assume the transmitter is pointed vertically (i.e., with an inclination angle of $0^\circ$). Finally, for Scenario 1c, we assume that the transmitter is pointing away from the receiver with an inclination angle of $30^\circ$. Note that this last configuration actually violates the assumption that the adversary is placed at an advantaged position. Nevertheless, we consider this configuration since it can provide insight as to whether pointing away from the location of possible adversaries can mitigate the detection probability in the NLOS/NLOS configuration.

2.2 Scenario 2: NLOS/LOS

Continuing to focus on a NLOS friendly communication link, Scenario 2 (depicted in Fig. 2) considers the worst case positioning of an adversary, namely within the LOS of the transmission. This requires an elevated position for the adversary, such as might be obtained by placing the detector system on an unmanned aerial vehicle or drone. Otherwise, we consider analogous positions (colinear) and pointing directions (coplanar) as in Scenario 1. In particular, here we consider a transmitter pointing in the direction of the friendly receiver with an inclination angle of $60^\circ$, an inclination angle for the friendly receiver of $60^\circ$, and the adversarial detector directly in the LOS of and pointing directly at the transmitter.

 

Fig. 2. Illustration of Scenario 2, which considers a NLOS friendly communications link and a LOS adversarial detection link. The specific geometry parameters are detailed in the text.

Download Full Size | PDF

2.3 Scenario 3: LOS/NLOS

Finally, we consider the advantageous configuration for friendly communications, namely a LOS friendly communication link and a NLOS adversarial detection link, as depicted in Fig. 3. This configuration might be of particular interest when modeling, e.g., a network containing aerial relay nodes. Because the friendly communication link operates via LOS, the required transmit power will be signficantly reduced (relative to a NLOS link at a similar range). This reduced power implies that the transmitter will be able to tolerate a much closer adversary while still maintaining LPD operation. The particular configuration is the same as for Scenario 2 except that the positions of the friendly receiver and adversarial detector are reversed, and we again take an inclination angle of $75^\circ$ for the adversarial detector.

 

Fig. 3. Illustration of Scenario 3, which considers a LOS friendly communications link and a NLOS adversarial detection link. The specific geometry parameters are detailed in the text.

Download Full Size | PDF

3. UVC LPD analysis

In this section, we describe our model of an LPD UVC link and then employ that model to derive, as a function of a number of parameters, the standoff distance outside of which an adversary is unable to detect the transmission. We begin by defining a number of geometric parameters that can be used to parameterize the configuration of any UVC link, such as a friendly point-to-point communication link or the adversarial detection link. We note that, for any such geometric configuration, one can employ a well-known Monte Carlo channel model to obtain the associated path loss or channel gain as a function of, e.g., range. We then describe, for a given friendly communication link, the computation of BER and the implied minimum transmit power required to achieve a given BER rate, where we assume the system employs on-off keying (OOK) as a modulation format. For that transmit power, we then derive, for a given adversarial link, the probability of detection of that transmission. Finally, the resulting expression is used to determine the standoff distance that achieves a particular probability of detection.

3.1 Point-to-point channel model

We employ the parameterization of a point-to-point UV link geometry developed in [11,12], as depicted in Fig. 4. The coordinate system is defined with the receiver at the origin and the transmitter along the positive $y$-axis. The parameters of the geometry are then the inclination and azimuth angles ($\theta _{\textrm {R}}$ and $\phi _{\textrm {R}}$, respectively) of the receiver pointing direction, the field-of-view $\beta _{\textrm {R}}$ of the receiver, the inclination and azimuth angles ($\theta _{\textrm {T}}$ and $\phi _{\textrm {T}}$, respectively) of the transmitter pointing direction, the beam width $\beta _{\textrm {T}}$ of the transmitter, and the distance $r$ from the transmitter to the receiver.

 

Fig. 4. UV link geometry [11,12]

Download Full Size | PDF

Given the geometric parameters of a given link, we then employ a standard Monte Carlo model [11,12] to obtain the channel path loss or, equivalently, the channel gain $\Gamma$. This model simulates the propagation of a large number of photons, probabilistically computing a trajectory of each photon based on models for the transmitter, receiver, and atmospheric interactions. Based on those trajectories, an estimate of the fraction of transmitted power that would impinge on the detector of aperture $S_{\textrm {R}}$ can be obtained, which is equivalent to $\Gamma$. Here, we assume that the transmitter has a Gaussian beam (with full-width half-max angle $\beta _{\textrm {T}}$), the receiver has a Gaussian-shaped angular sensitivity function (with full-width half-max angle $\beta _{\textrm {R}}$), and that the atmosphere is modeled using the functions and parameters in [11–14].

3.2 Analysis of friendly communication link

Given a particular friendly link geometry and associated channel gain $\Gamma$, we analyze the BER of an OOK communication link at a data rate of $R$. Let $P_{\textrm {T}}$ denote the transmit power. The number of transmitted photons when transmitting ON is given by $K_{\textrm {T}}\triangleq P_{\textrm {T}}\lambda /(Rhc)$, where $\lambda$ is the photon wavelength, $h$ is Planck’s constant, and $c$ is the speed of light. Let $N$ denote the average rate of all noise counts (e.g., combining optical noise and dark noise) measured by the receiver. Then $K_0\triangleq N/R$ is the expected number of received photons when OFF is transmitted. The expected number of received photons when ON is transmitted is given by $K_1 = K_0 + K_{\textrm {T}}\Gamma \eta$, where $\eta$ is the overall receiver efficiency (combining the receiver filter transmittance and the detector quantum efficiency). The number of received photons when OFF and ON is transmitted is Poisson distributed with parameter $K_0$ and $K_1$, respectively.

Assuming that transmissions of OFF and ON are equally likely, the maximum a posteriori (MAP) detector decides that OFF was transmitted if the number of detected photons for the corresponding symbol is no more than a threshold $\tau$ and otherwise that ON was transmitted, where

Note that the BER is a noninvertible function of the range $r$. Given a set of link (system, channel, and geometry) parameters, we are interested in determining the minimum transmit power $\check {P}_{\textrm {T}}$ required to achieve a target bit-error rate $\check {\textrm {BER}}$. Since (2) is noninvertible, we employ a (bisection) search to numerically compute $\check {P}_{\textrm {T}}$. Note that, depending on one’s particular modeling objective, it may be advantageous to precompute the channel gain $\Gamma$ as a function of $r$ (interpolating between evaluated values of $r$).

3.3 Analysis of adversarial detection

Next, for an adversary at some distance $r'$ from the transmitter, we compute the probability of detection $P_{\textrm {D}}$. This quantity is a function of a number of parameters associated with the geometry of the adversarial link and the detection algorithm. For the geometry, we assume that the adversary lies in a given direction relative to the transmitter pointing direction, so that the detector location is parameterized only by the range $r'$. We further assume that the pointing direction of the adversary is optimized to maximize the received signal. This clearly bounds the adversary’s performance, since prior to detection, the adversary would most likely not know the optimal pointing direction.

For a given adversarial pointing direction, we can therefore employ the same channel modeling approach as for the friendly communication link to obtain the adversary’s channel gain. For an arbitrary dummy range, we perform a search over adversary pointing directions to obtain the optimum. We then fix the pointing direction at the determined optimum when subsequently computing the channel gain $\Gamma '$ for any distance $r'$.

Now, given an adversarial detector that is measuring the number of received photons over some detection period $T\gg 1/R$, we perform a Neyman-Pearson hypothesis test for whether or not the source is transmitting. We further advantage the adversary by assuming the detector precisely knows the average rate $N'$ of the total noise counts (e.g., combining optical noise and dark noise) measured by the detector and that this rate does not vary over the detection period. These assumptions again imply that the detection performance we derive represents an upper bound on the adversary’s capability.

The probability density function (pdf) $f(k|H_0)$ of detector counts under the null hypothesis $H_0$ that there is not a transmitter emitting a signal is Poisson with parameter $K_0'=N'T$. On the other hand, the pdf $f(k|H_1)$ of detector counts under the alternative hypothesis $H_1$ that there is a transmitter emitting an OOK signal with total power $P_{\textrm {T}}/2$ is Poisson with parameter $K_1'=(P_{\textrm {T}}/2)T\lambda /(hc)\cdot \Gamma '\eta '$, where $\eta '$ is the overall efficiency of the adversary’s detector system (combining, e.g., the receiver filter transmittance and the detector quantum efficiency). It is therefore straightforward to use the log-likelihood ratio to show that the optimum decision rule takes the form of a threshold test

With $P_{\textrm {FA}}$ representing the probability that $n>\tau '$ when $H_0$ is true, we have

Finally, given a target $P_{\textrm {D}}$, we can perform a bisection search over $r'$ to determine the maximum range where an adversary can meet or exceed that target. This range represents the adversarial standoff distance outside of which the friendly communication link can be considered to be undetectable.

4. Results

In this section, we employ our modeling framework to investigate the LPD properties of the UVC scenarios described in Section 2. We assume the system parameters given in Table 1. Note that these are representative parameter values based on prior literature and experimental systems.

Table 1. Parameters used in our analyses.

View Table

First, Fig. 5 depicts the channel gain associated with certain links. Precomputing the channel gains as a function of range enables us to quickly perform analyses parameterized by the friendly communication range $r$, as well as to efficiently perform associated bisection searches on the adversarial detection range $r'$ when computing the corresponding standoff distances. In addition, we observe the well-documented short- and long-range behaviors of the channel gain function, namely that at short range, the channel gain falls off benignly (e.g., as $1/r$ or $1/r^2$ for a NLOS and LOS link, respectively), but the exponential atmospheric absorption at long range causes the channel gain to rapidly degrade beyond a few kilometers. This is a key characteristic that enables one mode of LPD operation. It implies that moderate increases in friendly communication range can be achieved with moderately more power, but this increased power is very quickly offset by the exponential absorption at long distances so that the adversarial standoff distance is minimally affected.

 

Fig. 5. Channel gain as a function of range for the two links that bound the performance of all the links considered in the paper. The lower bound occurs for the friendly communication link in Scenario 1c, and the upper bound occurs for the LOS links in Scenario 2 and Scenario  3.

Download Full Size | PDF

Now, having already assumed a number of parameters for the adversarial detection hardware, we note that there are even more parameters associated with the detection algorithm (e.g., the sensing time $T$, the permitted probability of false alarm $P_{\textrm {FA}}$, and the desired probability of detection $P_{\textrm {D}}$). One might be concerned that simply choosing a representative set of parameters is not bounding the worst-case detection probability. Fortunately, the LPD properties of UVC are relatively insensitive to the choice of parameters (e.g., because of the previously discussed exponential atmospheric absorption). For example, Figs. 6 and 7 illustrate the effect of different assumed values for $T$ and $P_{\textrm {FA}}$. (Note that we have chosen to focus on a representative scenario, namely the NLOS/NLOS Scenario 1a, which we find to be most interesting from an operational standpoint. A similar approach can be used to characterize the other scenarios.) For example, we see in Fig. 6 that despite $T$ ranging over three orders of magnitude, the range $r'$ at which $P_{\textrm {D}}=0.1$ varies by little over a factor of two. Similarly, Fig. 7 shows that even with $P_{\textrm {FA}}$ varying by four orders of magnitude, the range $r'$ at which $P_{\textrm {D}}=0.1$ varies by only about $50\%$.

 

Fig. 6. The probability of detection ($P_{\textrm {D}}$) in Scenario 1a for receiver times ($T$) ranging over three orders of magnitude. The probability of false alarm is set to $P_{\textrm {FA}}=0.01$, and the transmit power is set to $P_{\textrm {T}}=9.831$ mW, the minimum power needed to establish a 1 kb/s communication link at 50 m for the given scenario. Here, $P_{\textrm {D}}$ is computed by evaluating (5) for the given parameters and the channel gain associated with each range.

Download Full Size | PDF

 

Fig. 7. The probability of detection in Scenario 1a for probabilities of false alarm ($P_{\textrm {FA}}$) ranging over four orders of magnitude. The detection time is set to $T=1$ minute, and the transmit power is set to $P_{\textrm {T}}=9.831$ mW, the minimum power needed to establish a 1 kb/s communication link at 50 m for the given scenario. Here, $P_{\textrm {D}}$ is computed by evaluating (5) for the given parameters and the channel gain associated with each range.

Download Full Size | PDF

Figure 8 provides an example of the principle output of our analysis framework, namely the adversarial standoff distance as a function of the friendly communication range, which we refer to as the LPD-characteristic curve. That is, for the scenario described in the caption of Fig. 8, the plot shows how close an adversary would need to be to detect a transmission that is strong enough to establish a friendly communication link at a given range (on the $x$-axis of the plot). This is computed for each friendly link range $r$ by first employing a bisection search of (2), using thresholds $\tau$ computed with (1), for the transmitted power needed to achieve a desired friendly communication performance. This transmitted power, along with the other scenario parameters, are then used in a bisection search of (5), using the threshold $\tau '$ computed from (4), for the adversarial range $r'$ that achieves the target probability of detection $P_{\textrm {D}}$.

 

Fig. 8. The LPD-characteristic curve (i.e., the adversarial standoff distance as a function of the friendly communication range) for Scenario 1a for three choices of the probability of detection ($P_{\textrm {D}}$). The probability of false alarm ($P_{\textrm {FA}}$) and the detection time ($T$) are set to $10^{-4}$ and 1 minute, respectively.

Download Full Size | PDF

We first note that, similar to the previous figures, the analysis is not sensitive to the specifically chosen values for $P_{\textrm {D}}$. Second, we highlight an interesting feature of the curve, namely that there are three operational regimes, as apparent from a plateau in the center region of the curve. This demonstrates how the exponential atmospheric absorption translates to unique LPD characteristics distinct from conventional RF systems.

In particular, at short range, both the friendly receiver and adversarial detector experience the short-range channel effects, namely the benign $1/r$ range dependence of the NLOS channel gain or the $1/r^2$ range dependence of the LOS channel. As such, the increased power needed to extend the friendly communication range similarly improves the adversary’s ability to detect the transmission. Analogously, at long range, the channel gain for the friendly receiver and adversarial detector are both dominated by the exponential absorption effects, and, again, increased transmission power has similar effects on both the communication range and standoff distance. However, at moderate communication ranges, the friendly communication receiver continues to experience the benign channel degradation with range, while the adversarial detector experiences the exponential attenuation. The resulting plateau in the LPD-characteristic curve indicates that the atmospheric absorption is providing additional protection against adversarial detection, and that friendly communications can be established to address a range of operations with a relatively subdued impact on its detectability by an adversary.

Finally, Fig. 9 depicts the LPD-characteristic curves for each of the scenarios described in Section 2. We note that the curves for Scenarios 1a–1c are all similar to each other, providing a key insight that, given a particular friendly communication performance target, steering of the transmitter does not appear to be an effective strategy for improving the LPD characteristics of the link. Second, we see that, not surprisingly, Scenario 2 (NLOS friendly communications and LOS adversarial detection) results in a communication link with the worst LPD features. However, even in this disadvantaged case, the degradation in the LPD properties is mild, with the standoff distance being about double that of Scenario 1. Again, the exponential absorption that the adversary must contend with helps to limit the impact of the scenario modification. Finally, not surprisingly, Scenario 3 (LOS friendly communications and NLOS adversarial detection) achieves the best LPD characteristic. In this case, the adversarial standoff distance is approximately equal to the friendly communication range, providing a rule of thumb that, when communicating via LOS UVC, an adversary would need to be in the midst of the communication systems to be able to detect the transmission NLOS. Also, we note that, because the friendly communication range and adversarial standoff distance are comparable in this case, we do not observe the previously discussed plateau feature. However, the lack of the plateau is the result of enhanced LPD properties and is therefore of no concern.

 

Fig. 9. The LPD-characteristic curve (i.e., the adversarial standoff distance as a function of the friendly communication range) for the five scenarios described in Section 2. The probability of false alarm ($P_{\textrm {FA}}$), the probability of detection ($P_{\textrm {D}}$), and the detection time ($T$) are set to $10^{-4}$, $10^{-3}$, and 1 minute, respectively.

Download Full Size | PDF

5. Conclusions

In this paper, we have outlined a framework for analyzing the LPD properties of UVC. Although it had been previously hypothesized that the atmospheric absorption of the UVC channel would provide LPD features, the lack of a rigorous analysis has resulted in limited insight for the design and operation of UV systems with respect to LPD communications. The example results presented here demonstrate some of the types of insights that can be obtained using the proposed framework. For example, we observe that LOS detection of a NLOS communication link is not as significant of a concern as one might fear, that steering of a UVC transmitter does not seem to be an effective detection mitigation strategy, and that a LOS link provides LPD standoff distances that are commensurate with the communication range. In future work, we will extend this framework and its application to consider the effects of a wider number of friendly and adversarial system parameters, as well as broader operations scenarios, including UVC networks.