Effects of haze particles and fog droplets on NLOS ultraviolet communication channels

Changming Xu; Hongming Zhang; Julian Cheng

doi:10.1364/OE.23.023259

1. Introduction

Ultraviolet (UV) light in the wavelength of 200nm to 290nm is solar-blind [1], which makes it feasible to transmit weak scattering light signals without the need to consider background noise [2]. Therefore, in free-space optical communication we can use the scattered solar-blind UV light to provide a non-line-of-sight (NLOS) link when the line-of-sight (LOS) channel between two transceivers is blocked [3]. UV scattering communication has been studied for decades [4–6 ], in which many studies focused on atmospheric transmission channels. In the transmission of light signals, ignoring the fluctuated turbulence, scattering and absorption are the two basic atmospheric effects on the channel path loss. For the NLOS UV scattering communication, scattering is a booster for the light receiving while absorption is an attenuation factor. Scattering and absorption are separately caused by small size air molecules and large size aerosols. The composition of air molecules are relatively constant in the low-altitude area, but the aerosols change frequently with weather conditions. The most common aerosol particles, also known as the Mie particles, are haze particles (the Mie particles in dry weather are collectively called haze particles) and fog droplets. With different Mie particles, the channel path loss can change differently.

In previous studies, some authors found that thick atmosphere or fog weather can help decrease the path loss [7–9 ], while others concluded that thick Mie particles can increase the path loss, which is harmful to the light transmission [10]. In [11], the authors obtained that, in a specific thick atmosphere, the Mie particles, depending on the communication range, can improve or weaken the received light signals. Thick atmosphere can reduce the path loss when the communication range is short, but this effect is reversed when the range increases. However, the relationship among the density of Mie particles, the communication range, and the path loss has yet been determined. Also, the fog weather was not considered. For the fog weather, we cannot analyze the channel simply through changing the scattering and absorption coefficients, as the fog droplet is larger than the common haze particle and has different asymptotic factor and scattering phase function. In [8], the authors described a calculation process for the single-scattering integral model in fog channel, but the influence of density and size of the fog on the path loss is still unclear. Besides, the calculation process in [8] is complex because it requires use a Mie-theory calculation program every time when a channel parameter changes.

In this paper, we improve a previous Monte-Carlo based multiple-scattering model to study the impact of different Mie particles (haze and fog) on UV scattering channels. We introduce the density and size of Mie particles as variables into the model and reveal how the path loss is influenced by these particles. We propose analytical functions for the Mie parameters, and these functions can eliminate the calculation of Mie theory in the Monte-Carlo process. By analyzing the multiple-scattering simulation results, we find out that the Mie particles can either enhance or degrade the path loss, depending on the communication range, the aerosol density, and size of the particles.

2. Channel model and Monte-Carlo process

To analyze the channel path loss, we first modify a Monte-Carlo based multiple-scattering model [12, 13]. Figure 1 shows the transmission and receiving scenario of UV scattering communication. This link is connected by multiple scatterings in practice. The range between the transmitter (Tx) and receiver (Rx) is denoted by r. Denote the transmitter elevation angle by θ ₁ and the receiver elevation angle by θ ₂. The angle ϕ ₁ is the Tx beam divergence and ϕ ₂ is the Rx field of view (FOV). The parameter θ_s is the rotation angle between the incident light and the scattered light. To perform the Monte-Carlo simulation, we need to obtain all atmospheric parameters and scattering phase functions.

Fig. 1 Structure of non-line-of-sight UV communication link.

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When a photon is emitted from the transmitter, it will propagate a distance Δs before it is scattered. The distance Δs between the transmitter and the first scattering point or between two scattering points is determined by

Δ s = - \frac{ln η^{(s)}}{k_{s}}

where η ^(s) is a uniform random variable between zero and one, k_s is the scattering coefficient, which is the sum of molecule (Rayleigh) scattering coefficient

k_{s}^{Ray}

and aerosol (Mie) scattering coefficient

k_{s}^{Mie}

, i.e.,

k_{s} = k_{s}^{Ray} + k_{s}^{Mie}

. We consider that the Mie particles are composed of haze particles and fog droplets, then we denote the Mie scattering coefficient

k_{s}^{Mie}

as the sum of haze scattering coefficient

k_{s}^{haz}

and fog scattering coefficient

k_{s}^{fog}

, i.e.,

k_{s}^{Mie} = k_{s}^{haz} + k_{s}^{fog}

. During the propagation, the photon also has a probability to be absorbed by the molecules or aerosols. The absorption probability is decided by the absorption coefficient k_a, which is the sum of molecule absorption coefficient

k_{a}^{Ray}

and aerosol absorption coefficient

k_{a}^{Mie}

, i.e.,

k_{a} = k_{a}^{Ray} + k_{s}^{Mie}

, and

k_{a}^{Mie}

is the sum of haze absorption coefficient

k_{a}^{haz}

and fog absorption coefficient

k_{a}^{fog}

, i.e.,

k_{a}^{Mie} = k_{a}^{haz} + k_{a}^{fog}

.

The Rayleigh scattering coefficient $k_{s}^{Ray}$ is given by [14]

k_{s}^{Ray} = \frac{8 π^{3}}{3} \frac{{(m - 1)}^{2}}{λ^{4} N} \frac{6 (1 + δ)}{6 - 7 δ} (3 + \frac{1 - δ}{1 + δ})

where m is the air refractive index, λ is wavelength in unit μm, N is the molecular density (2.54743×10²⁵ m⁻³) for standard atmosphere (atmosphere under standard pressure and temperature), and δ is the polarization defect factor (depolarization factor) which is the ratio of weak to strong polarization component and approximately constant (0.035) about wavelength. In the sea level area, m equals to m ₀ that can be obtained by [14, 15]

(m_{0} - 1) \times 10^{8} = 8060.51 + \frac{2480990}{132.274 - λ^{- 2}} + \frac{17455.7}{39.32957 - λ^{- 2}} .

The Rayleigh absorption coefficient $k_{a}^{Ray}$ is a combination of different gas absorptions such as H₂O, O₃, CO₂ and O₂. We obtain the Rayleigh absorption coefficient by the widely used software MODTRAN [16] and its value at sea level is given in Table 2. The MODTRAN inputs are as follows: the model of atmosphere is the 1976 US standard model; the type of atmospheric path is horizontal path; the visibility (VIS) is 23km (or 5km, does not affect the values of Rayleigh coefficients); the frequency is the reciprocal of wavelength; the observer height is 0km; and the other parameters are defaults. For other altitudes, the molecular density N and the air refractive index m can be modified by measured data or just simply referring to MODTRAN data [16].

In order to study how the path loss changes with the Mie particles’ density, we express the scattering and absorption coefficients for the Mie particles, respectively, as [17, 18]

k_{s}^{Mie} = N_{v} Q_{sca} (R) π R^{2}, and k_{a}^{Mie} = N_{v} Q_{abs} (R) π R^{2}

where R is the radius of the Mie particle and N_v is the density of Mie particles. The Mie scattering efficient factor Q_sca(R), absorption efficient factor Q_abs(R), and the extinction efficient factor Q_ext (R) can be obtained by [17, 19]

{\begin{array}{l} Q_{sca} (R) = \frac{2}{x^{2}} \sum_{n = 1}^{\infty} (2 n + 1) ({| a_{n} |}^{2} + {| b_{n} |}^{2}) \\ Q_{ext} (R) = \frac{2}{x^{2}} \sum_{n = 1}^{\infty} (2 n + 1) ℜ (a_{n} + b_{n}) \\ Q_{abs} (R) = Q_{ext} (R) - Q_{sca} (R) \end{array}

in which ℜ represents the real part, a_n and b_n are functions of size parameter x = 2πR/λ and complex refractive index of the Mie particle as [17]

{\begin{cases} a_{n} = \frac{m ψ_{n} (m x) {ψ_{n}}^{'} (x) - ψ_{n} (x) {ψ_{n}}^{'} (m x)}{m ψ_{n} (m x) {ξ_{n}}^{'} (x) - ξ_{n} (x) {ψ_{n}}^{'} (m x)} \\ b_{n} = \frac{ψ_{n} (m x) {ψ_{n}}^{'} (x) - m ψ_{n} (x) {ψ_{n}}^{'} (m x)}{ψ_{n} (m x) {ξ_{n}}^{'} (x) - m ξ_{n} (x) {ψ_{n}}^{'} (m x)} \end{cases}

where ψ_n(x) = xj_n(x) and

ξ_{n} (x) = x h_{n}^{(1)} (x)

, and j_n(x) and

h_{n}^{(1)} (x)

are respectively the spherical Bessel function and spherical Hankel function [17]. The limiting number to truncate the summation in Eq. (5) for a given parameter x is the closest integer to x + 4x ^1/3 + 2 [20].

According to the descriptions from MODTRAN [16] and IAMAP (International Association of Meteorology and Atmospheric Physics) [21], the haze particles consist of 70% water-soluble particles with a radius of 0.005μm and 30% dust particles with a radius of 0.5μm. The complex refractive index of the haze particle is 1.53+0.03i [21], where i ² = −1. The wavelength of UV light we use in this paper is 250nm. After calculation, we obtain the values of $Q_{sca}^{haz}$ and $Q_{abs}^{haz}$ of haze particles and present them in Table 2.

The size of fog droplets changes with weather, so we consider the radius of the fog droplets in our analysis. As shown from Eqs. (5) and (6), the Mie-theory summation functions are complicated and their calculation can be time-consuming. In the following, we provide an alternative method to perform the calculation directly.

The refractive index of the fog in light wavelength 250nm is approximately 1.362 [22] (actually it is a complex number and the ignoring imaginary part is about 1 × 10⁻⁹ [21]), and the size of the fog droplets is about 1μm to 10μm [2, 23]. As we ignore the imaginary part of the fog refractive index, the absorption efficient factor $Q_{abs}^{fog} (R_{f})$ (R_f represents the radius of fog droplet) is approximately zero. In this size region, we calculate the summation part of $Q_{sca}^{fog} (R_{f})$ as Fig. 2(a). The summation part can be fitted to a power function as $(5 \times 10^{14}) \cdot R_{f}^{1.977}$ . Thus, $Q_{sca}^{fog} (R_{f})$ can be expressed in terms of R_f as

\begin{array}{l} Q_{sca}^{fog} (R_{f}) & = \frac{2}{x^{2}} (5 \times 10^{14}) \cdot R_{f}^{1.977} \\ = \frac{5 \times 10^{14}}{2 π^{2}} λ^{2} R_{f}^{- 0.023} . \end{array}

Fig. 2 Mie-theory parameters fitting for fog droplets.

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When the propagating photon is scattered, the rotation angle θ_s is decided by its probability density function, which is also known as the phase function. Defining μ=cosθ_s, the total scattering phase function is a weighted sum of Rayleigh scattering phase function p_ray(μ), haze scattering phase function p_haz(μ), and fog scattering phase function p_fog(μ). We write the total scattering phase function as

p_{tot} (μ) = \frac{k_{s}^{Ray}}{k_{s}} p_{ray} (μ) + \frac{k_{s}^{haz}}{k_{s}} p_{fog} (μ) + \frac{k_{s}^{fog}}{k_{s}} p_{fog} (μ)

in which p_ray(μ) and p_haz(μ) are given, respectively, by [12]

p_{ray} (μ) = \frac{3 [1 + 3 γ + (1 - γ) μ^{2}]}{16 π (1 + 2 γ)}

where γ=0.0178, and

p_{haz} (μ) = \frac{1 - g_{h}^{2}}{4 π} [\frac{1}{{(1 + g_{h}^{2} - 2 g_{h} μ)}^{\frac{3}{2}}} + \frac{3 μ^{2} - 1}{4 {(1 + g_{h}^{2})}^{\frac{3}{2}}}]

where g_h is the asymmetric factor of haze particle. According to the Mie theory, the asymmetric factor g is given by [19]

g = \frac{4}{x^{2} Q_{sca}} \sum_{n = 1}^{\infty} [\frac{n (n + 2)}{n + 1} ℜ (a_{n} a_{n + 1}^{*} + b_{n} b_{n + 1}^{*}) + \frac{2 n + 1}{n (n + 1)} ℜ (a_{n} b_{n})] .

The value of g_h is calculated with

Q_{sca}^{haz}

and it is given in Table 2. Figure 3 depicts phase functions obtained by Mie theory computation for standard haze particle and two sizes of fog droplets. The size of fog droplets is much larger than the haze particles, and we can observe that the phase functions of fog droplets have much larger forward peak. For a greater control of forward-peaked and large-angle scattering [24], the phase function of fog needs to be fitted by a two-parameter function as [25]

p_{fog} (μ) = \frac{α g {(1 - g_{f}^{2} (R_{f}))}^{2 α}}{π {(1 + g_{f}^{2} (R_{f}) - 2 g_{f} (R_{f}) μ)}^{α + 1} [{(1 + g_{f} (R_{f}))}^{2 α} - {(1 - g_{f} (R_{f}))}^{2 α}]}

where g_f (R_f) is the asymmetric factor of fog droplet and can be calculated from Eq. (11). When α is 0.5, the two-parameter phase function is equal to the famous H-G phase function [25,26]. Similar to

Q_{sca}^{fog} (R_{f})

, we fit the summation part of Eq. (11) for g_f (R_f) as Fig. 2(a). The summation part of g_f (R_f) can also be fitted into a power function as

(2.594 \times 10^{14}) \cdot R_{f}^{1.993}

. Combined with Eq. (7), g_f (R_f) can be expressed as

\begin{array}{l} g_{f} (R_{f}) & = \frac{2}{x^{2} Q_{sca}^{fog} (R_{f})} (2.594 \times 10^{14}) \cdot R_{f}^{1.993} \\ = 1.0376 \cdot R_{f}^{0.016} . \end{array}

The values calculated for g_f (R_f) are shown in Fig. 2(b). Figure 2(b) also shows the values of g_f (R_f) that are calculated by Mie theory computation. One can find that direct fitting for the discrete values from Mie theory computation is difficult to be carried out. But our fitting method, which separately fits the summation part of g_f (R_f) and $Q_{sca}^{fog} (R_{f})$ , provides an effective way to obtain the fitting g_f (R_f), and the simulation results presented in the next section prove the accuracy of our fitting. The parameter α in Eq. (12) is given by [25]

α = \frac{ln (\frac{p_{fog} (1)}{p_{fog} (- 1)})}{4 arctanh (g_{f} (R_{f}))} - 1

where p_fog(1) and p_fog(−1) can be calculated from [19]

p_{fog} (μ) = \frac{λ^{2}}{8 π^{2}} ({| S_{1} (μ) |}^{2} + {| S_{2} (μ) |}^{2})

and where S ₁(μ) and S ₂(μ) are the complex scattering amplitudes in both orthogonal incident polarization directions, and are respectively given as [19]

{\begin{cases} S_{1} (μ) = \sum_{n = 1}^{\infty} \frac{2 n + 1}{n (n + 1)} [a_{n} π_{n} + b_{n} τ_{n}] \\ S_{2} (μ) = \sum_{n = 1}^{\infty} \frac{2 n + 1}{n (n + 1)} [a_{n} τ_{n} + b_{n} π_{n}] \end{cases}

where π_n and τ_n begin with π ₀ = 0 and π ₁ = 1, and they can be calculated by [17]

{\begin{cases} π_{n} = \frac{2 n - 1}{n - 1} μ π_{n - 1} - \frac{n}{n - 1} π_{n - 2} \\ τ_{n} = n μ π_{n} - (n + 1) π_{n - 1} . \end{cases}

Fig. 3 Phase functions for haze particle and fog droplets.

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Both p_fog(1) and p_fog(−1) can also be fitted to power functions, respectively, as

p_{fog} (1) = (1.91 \times 1029) \cdot R_{f}^{3.991}

and

p_{fog} (- 1) = (5.421 \times 10^{12}) \cdot R_{f}^{1.617} .

Thus α can be calculated by Eqs. (13), (18) and (19).

After obtaining all parameters and phase functions, we can perform the Monte-Carlo simulation to study the path loss for different densities or sizes of Mie particles without using the complex Mie-theory calculation. The detailed operation process of Monte-Carlo simulation can be referred to [13]. We program the Monte-Carlo based multiple-scattering simulation referring to the pseudo-code in [13], and the modifications we made for the pseudo-code are presented in Table 1. We next present and analyze the simulation results.

Table 1. Modifications for the steps of pseudo-code in [13]

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3. Numerical results and analyses

The parameters in the UV scattering link geometry are described as follows. The transmitter divergence angle is given a small value (1°) for long-range scattering transmission. The receiver field-of-view (FOV) and receiving area are chosen according to practical devices [27,28], and are shown in Table 2. The elevation angles of transmitter and receiver are chosen to be (10°, 10°), (20°, 20°) and (30°, 30°). The largest scattering order in the multiple-scattering process of a single photon is 3.

Table 2. Simulation Parameters

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Figure 4 shows the changing path losses with density of haze particles without considering the fog droplets (scattering and absorption coefficients are assumed to be zero). The range of haze density is chosen to be 10⁹ m⁻³ to 10¹⁰ m⁻³, which corresponds to the visibility range from 23km to 4km [29]. We depict the path loss with three communication ranges (200m, 500m and 1000m). When the communication range is shorter than 500m, as shown from Fig. 4, an increase of haze density (decreasing of visibility) can reduce the path loss and, as a result, the communication rate can be increased. However, when the communication range is increased to, say, 1000m, thick haze does not always give a lower path loss. The path loss will fall first and then rise with the haze density. This is because when the communication range is extended, the absorption effect will play a dominant role in the scattering process. As the increase of haze density strengthens the absorption effect, this makes thick haze degrade the performance of long-range UV scattering communications.

Fig. 4 Path loss changing with haze density. No fog droplets exist.

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Fog is an attenuation factor in LOS FSO channels [30], but the strong scattering effect of fog can sometimes improve the transmission of UV NLOS communication. We now analyze the path loss in fog weather with different densities and sizes of fog droplets. The density of haze particles is chosen to be 10⁹ m⁻³, and the density of fog droplets in common fog weather ranges from 10⁷ m⁻³ to 10⁹ m⁻³ [23].

Figure 5 depicts the path losses of fog channels with different fog droplets densities. The size of the fog droplets is chosen to be 1μm. For a certain short communication range, thicker fog density can decrease the path loss. But this improvement will be restricted to thinner density region with increasing communication range, or increasing elevation angles. These results are similar to the case of the haze particles.

Fig. 5 Path loss changing with fog density. Haze density is 10⁹ m⁻³ and fog radius is 1μm.

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Figure 6 shows the path losses change with fog size. The density of fog droplets is chosen to be a fixed value of 10⁸ m⁻³. As we can see from Fig. 6, path losses always fall first and then rise with the drop radius, and the inflection point moves to smaller radius when communication range increases. These results indicate that an increase of fog radius has similar path loss effects to the increase of haze or fog density. For comparison, we present the results for different fog drop sizes obtained by full Mie theory computation in Fig. 6. Due to the calculation precision of Mie theory, the path loss results of full computation fluctuate tightly in the lines obtained by fitting equations. This proves that our fitting is effective and applicable in the simulation of UV scattering path loss for different size of fog droplets.

Fig. 6 Path loss changing with fog radius. Haze density is 10⁹ m⁻³ and fog density is 10⁸ m⁻³.

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We also simulate path losses of different haze particles and fog droplets for a larger transmitter beam divergence angle of 15°. We can see from Figs. 4 –6 that, 15° Tx divergence angle makes no noticeable difference to the path loss for larger elevation angles such as (20°, 20°) and (30°, 30°) compared with 1° divergence angle. But for small elevation angles such as (10°, 10°), the path losses of 15° divergence angle have a distinct decrease from those of 1° divergence angle. This phenomenon tells that a larger Tx divergence angle will help reduce the channel path loss for UV scattering communications in haze and fog weather when the transceivers elevation angles are small.

4. Conclusion

In this paper, we studied the channel properties with different density and size of Mie particles for the NLOS UV scattering communications. We improved an existing Monte-Carlo model and analyzed the influence of different Mie particles to channel path loss. For calculation convenience, we developed Mie-theory fitting functions to replace the complicated Mie-theory computations in the Monte-Carlo process. The results show that the influence of Mie particles on the path loss depends on both the density or size of the particles, as well as the communication range. For haze particles, when the communication range is short (<500m), an increase of density of haze particles can reduce the channel path loss, which can in turn enhance the performance of UV communications. When the communication range is large (>500m), the path loss will decrease first and then rise with the haze density. For fog droplets with a specific drop size (such as 1μm), the change of fog density has similar effect to the path loss as the haze particles. This phenomenon also exists in the case of different drop sizes.

Acknowledgments

This work was supported by National Key Basic Research of China (Grant No. 2013CB329203), National High Technology Research and Development Program of China (Grant No. 2013AA013601), and Science and Technology Program of State Grid Corporation of China (Grant No. SGHAZZ00FCJS1500238).

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Steps in [13]	Modifications in this article
k_a in Step 1, Fig. 3	$k_{a} = k_{a}^{Ray} + k_{s}^{Mie}$ . $k_{a}^{Ray}$ is calculated by Modtran and $k_{a}^{Mie}$ by Eq. (4).
k_r in Step 1, Fig. 3	$k_{s}^{Ray}$ , calculated by Eq. (2)
k_m in Step 1, Fig. 3	$k_{s}^{Mie}$ , calculated by Eq. (4)
g in Step 1, Fig. 3	g, calculated by Eq. (11). For fog droplet, g can be simply calculated by Eq. (13).
Input in Step 1, Fig. 3	Item α is added, calculated by Eq. (14)
phase function of Eq. (5) in [13], corresponding to Step 46 in Fig. 3 and Step 13 in Fig. 4	For fog droplet, the phase function is replaced with Eq. (8), where p_fog (μ) is calculated by (12).

Wavelength λ (nm)	250	Divergence angle ϕ ₁(°)	1
FOV angle ϕ ₂(°)	30	Receiving area (cm²)	1.77
Rayleigh scattering coefficient $k_{s}^{Ray} (m^{- 1})$	3.2117 × 10⁻ ⁴	Rayleigh absorption coefficient $k_{a}^{Ray} (m^{- 1})$	1.0926 × 10⁻ ³
Haze scattering efficiency factor $Q_{sca}^{Haz}$	3.3742	Haze efficiency factor $Q_{abs}^{Haz}$	0.4803
Haze particle radius (μm)	0.1535	Haze asymmetric factor g_h	0.7667

Steps in [13]	Modifications in this article
k_a in Step 1, Fig. 3	$k_{a} = k_{a}^{Ray} + k_{s}^{Mie}$ . $k_{a}^{Ray}$ is calculated by Modtran and $k_{a}^{Mie}$ by Eq. (4).
k_r in Step 1, Fig. 3	$k_{s}^{Ray}$ , calculated by Eq. (2)
k_m in Step 1, Fig. 3	$k_{s}^{Mie}$ , calculated by Eq. (4)
g in Step 1, Fig. 3	g, calculated by Eq. (11). For fog droplet, g can be simply calculated by Eq. (13).
Input in Step 1, Fig. 3	Item α is added, calculated by Eq. (14)
phase function of Eq. (5) in [13], corresponding to Step 46 in Fig. 3 and Step 13 in Fig. 4	For fog droplet, the phase function is replaced with Eq. (8), where p_fog (μ) is calculated by (12).

Wavelength λ (nm)	250	Divergence angle ϕ ₁(°)	1
FOV angle ϕ ₂(°)	30	Receiving area (cm²)	1.77
Rayleigh scattering coefficient $k_{s}^{Ray} (m^{- 1})$	3.2117 × 10⁻ ⁴	Rayleigh absorption coefficient $k_{a}^{Ray} (m^{- 1})$	1.0926 × 10⁻ ³
Haze scattering efficiency factor $Q_{sca}^{Haz}$	3.3742	Haze efficiency factor $Q_{abs}^{Haz}$	0.4803
Haze particle radius (μm)	0.1535	Haze asymmetric factor g_h	0.7667

Effects of haze particles and fog droplets on NLOS ultraviolet communication channels

Abstract

1. Introduction

2. Channel model and Monte-Carlo process

3. Numerical results and analyses

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Tables (2)

Equations (19)

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