# A Path Loss Model for Non-Line-of-Sight Ultraviolet Multiple Scattering Channels

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## Abstract

An ultraviolet (UV) signal transmission undergoes rich scattering and strong absorption by atmospheric particulates. We develop a path loss model for a Non-Line-of-Sight (NLOS) link. The model is built upon probability theory governing random migration of photons in free space, undergoing scattering, in terms of angular direction and distance. The model analytically captures the contributions of different scattering orders. Thus it relaxes the assumptions of single scattering theory and provides more realistic results. This allows us to assess the importance of high-order scattering, such as in a thick atmosphere environment, where short range NLOS UV communication is enhanced by hazy or foggy weather. By simulation, it is shown that the model coincides with a previously developed Monte Carlo model. Additional numerical examples are presented to demonstrate the effects of link geometry and atmospheric conditions. The results indicate the inherent tradeoffs in beamwidth, pointing angles, range, absorption, and scattering and so are valuable for NLOS communication system design.

### Keywords

Path Loss Elevation Angle Path Loss Model Scatter Phase Function Arrival Probability## 1. Introduction

In free space optical communication, the deep ultraviolet (UV) spectrum with wavelength 200 Open image in new window 280 nm is regarded as an appealing choice to overcome solar background radiation and relax pointing and tracking [3]. High altitude ozone absorbs most solar radiation in this band, yielding negligible background noise at sea level [4]. And, atmospheric scattering is very strong, enabling non-line-of-sight (NLOS) communications where the transmitter is not necessarily within the receiver field of view (FOV). However, in addition to scattering, strong atmospheric absorption leads to significant signal attenuation and so limits achievable rates.

These properties have motivated development of UV signal propagation models and communication systems for military and civilian applications, for example, long-range communication based on high-power UV lasers since the 1960s [5, 6, 7, 8]. Recent progress in deep UV light emitting diodes (LEDs) [9] and avalanche photodiodes (APDs) [10, 11] offers promise for deployment of low-cost/power and moderate bandwidth short-range UV communication systems [12, 13], including underwater communications [14, 15] and sensor networks [16, 17].

NLOS channel modeling is more complex than traditional LOS links. In addition to wavelength and device characteristics, NLOS path loss is a function of system geometry, including transmitter (Tx) beamwidth, communication range, receiver (Rx) FOV, the pointing elevation angles, as well as the optical properties of the atmosphere.

For simplicity and tractable analysis, single scattering models for NLOS communication links were developed [1, 18], with a corresponding single scattering assumption imposed; that is, each photon undergoes only a single interaction with the atmosphere before it reaches the detector. Recently the model has been further simplified for improved analytical tractability [19]. However, the single scattering assumption does not always lead to accurate link performance prediction, especially as the range increases, and with beam pointing at low elevation angles. Multiple scattering may occur when the particle density is large and/or the propagation distance is long. Alternatives to the single scattering model include an empirical path loss model [20] and a Monte Carlo statistical path loss model [2]. Those models are applicable for predicting path loss in a variety of scenarios and are generally more accurate than the single scattering model.

In this paper, following the same physical scattering law as the Monte Carlo statistical method [2, 21], we develop a stochastic analytical NLOS UV channel path loss model. We apply a stochastic analytical technique [22, 23] to theoretically derive the Open image in new window th scattered signal energy collected by the detector. The model assumes that the photons are stochastically scattered and/or absorbed by the atmospheric particles and involves probabilistic modeling of photon random moving direction, distance, energy loss, and receiver capture after a specified number of scatterings. In order to obtain the Open image in new window th-order scattered signal at the receiver, we trace the migration routes of a single photon through the medium. Its scattering distance and scattering angles follow certain probability distribution functions (PDFs) [24]. The propagation of a photon between two consecutive scatterers is modeled as a single scattering event. The probability of the photon arriving at the receiver is a function of the Open image in new window scattering events encountered. A similar technique has been applied to model multiply scattered lidar returns in cloudy media where significant scattering occurs [22]. To account for a divergent UV beam profile, the contributions of photons at all possible directions within the beam are integrated [25].

We assume a homogeneous atmosphere with constant scattering and absorption coefficients and ignore atmospheric turbulence. This assumes that different types of particles are well mixed and the environment is stationary. Photons are scattered elastically which conserves energy but incurs energy loss during propagation. The detector is assumed small in size compared with the propagation space and is regarded as a point detector with finite FOV.

Our model offers an analytical formulation for NLOS scattering channel path loss and provides a reference to easily check other models in a variety of system parameter settings. We consider the effects of different optical pointing geometries and find that path loss is relatively insensitive to the Tx beam angle but increases considerably with increased Tx/Rx elevation angles and decreased Rx FOV for medium communication range. Not surprisingly, numerical tests demonstrate good agreement with Monte Carlo simulation [2]. However, the current analytical approach can distinguish the contributions of different orders of multiple scattering, whereas Monte Carlo modeling can only capture the total scattering effect. A similar comparison of Monte Carlo simulation and analytical methods has also been described by Lavigne et al. [26].

The model reveals when high-order scattering plays a role, and thus offers more insight into the channel behavior. Extensive numerical tests show that high-order scattering may play a significant role for scenarios with large baseline ranges and elevation angles. Atmospheric conditions also affect the multiple scattering contributions. A tenuous atmosphere generally leads to higher path loss at a small range than a thick or extra thick atmosphere where rich multiple scattering enhances received signal strength. As a by-product, when contributions from only single scattering are retained, the corresponding single scattering path loss shows a good match with that predicted by Reilly's analytical single scattering model [1]. A disadvantage of the proposed model is that it cannot provide pulse spreading information, while the Monte Carlo model can [2].

The organization of this paper is as follows. The stochastic path loss model based on a general NLOS UV channel configuration is developed in Section 2. It consists of modeling photon direction, distance traveled, probability of arrival at the receiver after scattering Open image in new window times, and total path loss. Numerical analysis for the multiple scattering NLOS channel is carried out and path loss performance is compared with both the single scattering model and Monte Carlo simulation in Section 3. Channel characteristics under different geometric parameters are analyzed in Section 4. Finally some conclusions are drawn.

## 2. Stochastic Path Loss Model

A NLOS UV channel involves rich scattering and absorption because of abundant suspended particulates in the atmosphere. For NLOS UV communication, scattering serves as the vehicle for information exchange between the Tx and Rx. The scattered photons that reach the receiver depend on the link geometry, the atmospheric optical properties, and their random migration, as described next.

### 2.1. Link Geometry

### 2.2. Atmospheric Optical Properties

From the communications viewpoint, scattering and absorption are two dominant types of photon interactions with the atmosphere. To describe a homogeneous atmospheric medium related to UV communication, we adopt the following coefficients [1, 26]: the Rayleigh (molecular) scattering coefficient Open image in new window , Mie (aerosol) scattering coefficient Open image in new window , absorption coefficient Open image in new window , and extinction coefficient Open image in new window . The total scattering coefficient is defined as the sum of the two scattering coefficients Open image in new window , and the extinction coefficient is given by the sum of the scattering and absorption coefficients as Open image in new window .

where Open image in new window , Open image in new window , and Open image in new window are relevant model parameters.

### 2.3. Elementary Events for Photon Random Migration

### 2.4. Modeling of Open image in new window Times Scattering

where Open image in new window is given by (9), Open image in new window by (10), and Open image in new window by (11). The integration limits for all variables cover their full ranges except the following: Open image in new window from Open image in new window to Open image in new window to ensure integration inside the source beam, and Open image in new window and Open image in new window within the solid angle of the receiver determined by the receiver area and distance Open image in new window . Note that no integration over Open image in new window is needed because it is a function of the other variables. Thus, there are a total of Open image in new window integration variables.

where for notational convenience Open image in new window represents the integrand product of functions over all Open image in new window , and Open image in new window for all differentials Open image in new window . During the process, we truncate the infinite integration limit for all Open image in new window (up to Open image in new window ) at a sufficiently large number to ensure that the integrand has decayed sufficiently close to zero. We have found generally that using a limit of several times Open image in new window works well.

It is important to clarify the difference between the *Monte Carlo integration* of a definite integral as used here and the *Monte Carlo simulation* of a stochastic process [2]. Even though both belong to the Monte Carlo family and involve random realizations, Monte Carlo integration is a numerical method based on the approximation of the averaged deterministic integrand function, whereas Monte Carlo simulation is a technique that relies on repeated samplings (numerous realizations) of a random process to compute a statistical expectation.

### 2.5. Energy Loss Model

Note that when Open image in new window , the energy represents the total received energy contributed by all scatterings and path loss represents the actual link performance. If we adopt decibels for path loss, then it becomes Open image in new window .

## 3. Modeling Performance

In this section we study modeling performance in terms of the received energy ratio and path loss results for different geometry scenarios. Through numerical simulation, multiple scattering effects are observed and analyzed. These results are also compared with Reilly's analytical single scattering model [1] as well as the Monte Carlo simulated multiple scattering model [2], respectively, under the same geometries and atmosphere parameters. We demonstrate good agreement with [1] when single scattering is of concern, and with [2] when multiple scatterings are considered.

Atmosphere model parameters.

Atmosphere | |||
---|---|---|---|

Tenuous | 0.266 | 0.284 | 0.972 |

Thick | 0.292 | 1.431 | 1.531 |

Extra thick | 1.912 | 7.648 | 1.684 |

## 4. NLOS UV Channel Characteristics

Since NLOS UV channel characteristics are crucial to communication system design, in this section, we apply the proposed multiple scattering model to further study the effects of different system parameters on the channel path loss, including link geometries and two typical atmosphere conditions. Referring to Figure 1, we study angle sensitivity by varying one angle and keeping the others fixed.

## 5. Conclusions

This paper proposed an analytical energy loss model for NLOS UV communication channels based on the probability theory of scattering and absorption. The model was developed by employing the PDFs of scattering distance and scattering angles. Multiple scattering was incorporated and contributions of different scattering orders were identified. The total energy loss was modeled by summing over the scattering order. The path loss predicted using only the contribution from the first-order agreed with Reilly's analytical single scattering model, as illustrated with different optical geometries. Multiple scattering becomes more dominant for some cases, especially longer propagation distance and larger elevation angles. Our model also provided results that are consistent with the Monte Carlo simulation model. Channel characteristics were investigated in detail, including the effects of varying system geometry and the effects of different atmosphere conditions.

Further study will be conducted to develop an analytical NLOS UV channel impulse response model from which our path loss model can be further validated and channel bandwidth can be predicted. An analytically more tractable path loss model is also a future topic of interest, which can enable more intuitive analysis for the effects of system geometry and parameters.

## Notes

### Acknowledgments

The authors gratefully acknowledge R. Drost for his help with the coding examples. This work was supported in part by the U.S. Army Research Office under Grant W911NF-09-1-0293 and the Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011.

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