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A unified statistical model for Málaga distributed optical scattering communications
Optics Communications 463 (2020) 125402
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A unified statistical model for Málaga distributed optical scattering communications Sudhanshu Arya, Yeon Ho Chung ∗ Department of Information and Communications Engineering, Pukyong National University, Busan, Republic of Korea
ARTICLE Keywords: Average bit error rate Málaga fading Optical scattering Outage probability
INFO
ABSTRACT In this paper, we present a novel analytical model for optical scattering communications over Málaga distributed irradiance fluctuation. The distribution of the received signal is quantified with each scattered optical path following Málaga distribution in the scattering communication links. The model efficiently includes the scattering by the off-axis eddies, in addition to the eddies on the propagation axis over a non-line-ofsight (NLOS) link, thus permitting accurate evaluation of the scattered ultraviolet (UV) communication. It is shown that as the effective number of large-scale cells reduces in the lower atmospheric channel, the probability density function (PDF) of the received signal-to-noise ratio (SNR) spreads widely with large variance and impacts the performance of optical scattering communications. Moreover, based on the derived model, closed-form expressions for the average bit error rate and the outage probability are also presented.
1. Introduction As a potential candidate for long-range non-line-of-sight (NLOS) connectivity in optical wireless communications (OWC), ultraviolet (UV) communication is gaining more and more interest from researchers. It comes with unique scattering properties and offers high security and low installation cost [1]. Although the optical scattering in the lower atmosphere enables NLOS links, UV links are highly susceptible to severe attenuation and fading under atmospheric turbulence channel, which limit the extensive utilization of the UV communication link. Recently, many turbulence distributions have been studied and proposed to model the optical scattering communication. One widely accepted model for ultraviolet communication over turbulence fading is the log-normal distribution. In [2], the authors derived an analytical model for the received signal distribution which is then applied to analyze the NLOS link performance. Considering the log-normal distribution for each scattered path, similar studies were also reported for NLOS ultraviolet communications [3,4]. However, the log-normal distribution is only valid for weak turbulence fading as reported in [5,6]. In addition, over a longer link range, the incident optical beam becomes increasingly incoherent, thereby making the log-normal distribution invalid [6]. In [7,8], the authors suggested an alternative distribution model, i.e. Gamma– Gamma distributed irradiance fluctuations, to analyze the performance of the UV communication systems. In this work, we develop a new and unified statistical model for an optical scattering communication link with each scattered path following Málaga distribution. The developed model is applicable to both plane and spherical waves under
all turbulence conditions. The model is developed for obtaining the probability density function (PDF) of the instantaneous received signalto-noise ratio (SNR). The derived model is then subsequently used to derive closed-form expressions for the average bit error rate and the outage probability. Results are also presented to yield an insight into the performance of the optical scattering communication over Málaga distributed channels. We briefly describe a comparative view of optical scattering distributions. 1.1. Gamma–Gamma, log-normal, and Málaga distributions The log-normal distribution model for the fading statistics is a widely reported model in UV communication systems [3,4]. It follows directly from the first Rytov approximation and fits well with the lowerorder moments of the intensity obtained from the experimental data under weak turbulence conditions. The log-normal model requires the magnitude of the scattered field wave to be small, compared with the unperturbed phase gradient as reported in [9]. Unfortunately, this assumption is only valid in a single scattering event characterized by the weak turbulence fading. Therefore, in the environment where multiple scatterings occur or when the propagation path is large, the incident light intensity becomes increasingly incoherent and the log-normal distribution becomes invalid [9,10]. In addition, it has been observed through experimental data that the log-normal PDF can underestimate the behavior of the peak and the tails. This leads to a significant
∗ Corresponding author. E-mail address: [email protected] (Y.H. Chung).
https://doi.org/10.1016/j.optcom.2020.125402 Received 29 December 2019; Received in revised form 23 January 2020; Accepted 26 January 2020 Available online 29 January 2020 0030-4018/© 2020 Elsevier B.V. All rights reserved.
S. Arya and Y.H. Chung
Optics Communications 463 (2020) 125402
performance degradation in radar and optical communications where detection and fade probabilities are calculated over the tails of the PDF [11,12]. The Gamma–Gamma distribution to model the turbulence fading over optical scattering communication link, i.e., the UV link, was suggested recently as a reasonable alternative to the log-normal distribution [7,13,14]. It is considered as a general distribution model for both weak and strong fading. This distribution model can be expressed as a modulation process which involves first-order and second-order log-amplitude perturbations. In this model, the irradiance is assumed to consist of scattering (small-scale fading) and refraction (large-scale fading) [15]. The small-scale fading is due to eddies cells smaller than the Fresnel zone or the coherence radius. The large-scale fading is attributed to the presence of eddies greater than the first Fresnel zone or the scattering disk [10]. In deriving Gamma–Gamma model, it is assumed that both large-scale and small-scale fluctuations follow Gamma distribution. The Gamma–Gamma distribution is experimentally proved to be valid under all turbulence conditions. In addition, the distribution parameters can completely be determined by the atmospheric condition. However, in the limit of strong turbulence fading, i.e., in a saturation regime and beyond, where the independent scatters become very large, the Gamma–Gamma model reduces to negative exponential. The Málaga distribution model is valid under all turbulence conditions ranging from weak to strong fluctuations. Unlike the log-normal distribution model, it remains valid in strong and saturation turbulence regimes. As reported in [16], the main advantage of Málaga distribution model is that it can lead to closed-form expressions for important figures of merit of the optical channels. In addition, it unifies most of existing statistical models for the turbulence fading and it shows an excellent match to the experimental data.
Fig. 1. NLOS UV communication link with Málaga distributed scattered path.
2. Statistical model for NLOS UV link The UV scattered beam that arrives at a photodetector depends on both the link geometry and the atmospheric characteristics. A typical NLOS UV configuration is illustrated in Fig. 1. 𝑑 represents the separation between the transmitter and the receiver. 𝜓𝑡 and 𝜓𝑟 are the transmitter and the receiver apex angles, respectively. 𝜙𝑡 is the transmit beam angle and 𝜙𝑟 denotes the receiver field-of-view (FOV). 𝑉𝑐 represents the common volume. Atmospheric turbulence causes intensity fluctuation in the received UV signal and is considered one of the main impairments in limiting the performance of UV communication systems. In addition, it also results in random phase variation in the optical wavefront at the receiver with limited detector size. This random phase variation tilts the received wavefront from normal at the receiver aperture by an angle termed the angle-of-arrival (AOA). AOA results in the jitter of diffraction pattern on the receiver plane [15]. Considering all these impairments, the instantaneous received UV signal can be modeled as
1.2. Contributions Unlike previous works on optical scattering communications, which mainly focus on the log-normal and Gamma–Gamma distributed turbulence fading, this work presents the first comprehensive statistical model for NLOS UV communication over Málaga distributed fading. The novel contributions of this work are presented below.
𝑦 = 𝑆ℎ𝑡 ℎ𝜀 ℎ𝑠 𝑃𝑡 𝐼 + 𝑛,
(1)
where 𝑆 is the receiver responsivity in A/W, ℎ𝑡 represents the fading √ due to atmospheric scintillation and can be expressed as ℎ𝑡 =
• To yield an insight into the performance analysis of optical scattering communication over Málaga distributed fading, we develop a new statistical model for the received irradiance over NLOS path. We quantify the distribution of the received irradiance, when the turbulence fading over each scattered path is Málaga distributed. • Subsequently, we derive the closed-form expression for the PDF of the received SNR for NLOS optical path. • Furthermore, we show that as the effective number of large-scale cells in the turbulence channel reduces, the PDF of the received SNR spreads over a wider range. This spreading has a varying impact on the performance of a NLOS UV link. • We further derive a closed-form expression of an important figure of merit, i.e., the average bit error rate. The outage performance is also evaluated. • We derive conditions for the proposed distribution to become the log-normal and Gamma–Gamma distributions. In addition, a comparative performance analysis is provided.
−
( ) 11∕6 11∕6 23.17𝑘7∕6 𝐶𝑛2 𝑑𝑡𝑣 +𝑑𝑣𝑟 5
10 [7], where 𝑘 = 2𝜋∕𝜆 represents the wave number and 𝜆 wavelength of the UV signal. 𝐶𝑛2 denotes the refractive index coefficient of atmospheric turbulence channel. ℎ𝜀 denotes the fading introduced by AOA fluctuations given by [17] ( √ ) ( √ ) ℎ𝜀 = 1 − 𝐽02 𝜋 𝐷 − 𝐽12 𝜋 𝐷 , (2) where 𝐷 is a constant greater than or equal to 1 and is determined by the ratio of the receiver FOV solid angle to the diffraction-limited solid angle. 𝐽0 (⋅) and 𝐽1 (⋅) represent the Bessel function of the first kink and order zero and one, respectively. ℎ𝑠 in (1) denotes the attenuation term due to atmospheric scattering. Considering single scattering phenomena, it can be modeled as [18]
ℎ𝑠 =
( )( ) 𝐴𝑟 𝛼𝑠 𝑞𝑠 𝜙2𝑡 𝜙𝑟 sin 𝜓𝑡 + 𝜓𝑟 12 sin2 𝜓𝑟 + 𝜙2𝑟 sin2 𝜓𝑡 ( ) ( ), 𝛼 𝑑 (sin 𝜓 +sin 𝜓 ) 𝜙 96𝑑 sin 𝜓𝑡 sin2 𝜓𝑟 1 − cos 2𝑡 exp 𝑡 sin 𝜓 𝑡+𝜓 𝑟 ( 𝑡 𝑟)
(3)
where 𝑃𝑡 is the transmit power. 𝑛 is the Gaussian noise. It should be noted that the selection of the noise model depends on both the detection process employed and the background noise level. A Gaussian model is preferable when the thermal noise is dominant. In addition, in a medium or high noise scenario, such as outdoor daytime operation or when transmitting using longer pulses, the Gaussian noise model is suitable [19,20]. 𝐼 denotes the optical scintillation.
The rest of the paper is organized as follows. In Section 2, the statistical model for NLOS UV link is presented. The marginal PDFs of the received irradiance and SNR are derived. The closed-form analytical expressions for the average bit error rate and the outage probability are also derived. Results and discussions are provided in Section 3. Section 4 draws the conclusions. 2
S. Arya and Y.H. Chung
Optics Communications 463 (2020) 125402
2.1. Marginal PDF of the received irradiance
2.2. Marginal PDF of the instantaneous SNR
In this subsection, we derive the marginal PDF of the intensity fluctuations at the receiver. Following a Málaga distribution, the PDF of the optical power arriving in the common volume is given by [16] ( √ ) ) ( 𝛽𝑣 𝛼𝑣 +𝑘 ∑ −1 ( ) 𝛼𝑣 𝛽𝑣 𝑘 2 𝐾𝛼𝑣 −𝑘 2 𝑓𝐼𝑣 𝐼𝑣 = 𝐴𝑣 𝑎𝑣𝑘 𝐼𝑣 , (4) 𝛾𝑣 𝛽𝑣 + 𝛺′ 𝑘=1
The instantaneous received SNR is given by ( )2 𝑆𝐻𝑃𝑡 𝐼 2 𝜒= , 𝜎𝑛2
where 𝐼 is normalized to unity and following this, the average received SNR is defined as [22] ( )2 𝑆𝐻𝑃𝑡 (𝐸 [𝐼])2 . (12) 𝜉= 𝜎𝑛2
where 𝛼𝑣 ⎧ ( )𝛽𝑣 + 𝛼𝑣 2𝛼𝑣 2 𝛾 𝑣 𝛽𝑣 2 ⎪ 𝐴𝑣 = 𝛼𝑣 ′ 𝛾 𝛽 +𝛺 1+ 𝑣 𝑣 ⎪ 𝛾𝑣 2 𝛤 (𝛼𝑣 ) . ⎨ ( ) 𝑘 𝑘 𝛽𝑣 − 1 (𝛾𝑣 𝛽𝑣 +𝛺′ )1− 2 ( 𝛺′ )𝑘−1 ( 𝛼𝑣 ) 2 ⎪ 𝑎 = ⎪ 𝑣𝑘 𝛾𝑣 𝛽𝑣 (𝑘−1)! 𝑘−1 ⎩
By applying the Jacobean transformation, the marginal PDF of the instantaneous SNR is given by
(5)
√ 𝜒 𝐼= 𝜉
𝑓𝜒 (𝜒) = |𝐽 |𝑓𝐼 (𝐼)|
where 𝛽𝑣 is a natural number. 𝐾𝛼𝑣 −𝑘 (⋅) denotes the modified Bessel function of the second kind and order 𝛼𝑣 − 𝑘. 𝛼𝑣 in (4) represents the effective number of large-scale cells of the scattering process and ′ determines the √ strength of the turbulence. 𝛾𝑣 = 2𝑏0 (1 − 𝜌𝑣 ) and 𝛺𝑣 = 𝛺𝑣 + 2𝜌𝑣 𝑏0 + 8𝑏0 𝛺𝑣 𝜌𝑐𝑜𝑠(𝜙𝐴 − 𝜙𝐵 ), where 𝛺𝑣 is the shaping parameter, and 𝜙𝐴 and 𝜙𝐵 are the deterministic phases of the scatter terms [16]. 𝜌𝑣 takes a positive number between 0 and 1 and denotes the fraction of the scattered power coupled to the unscattered component. 2𝑏0 represents the average total power of the scatter components and is given by [ ] | |2 | |2 2𝑏0 = 𝐸 |𝐼𝑆𝐶 | + |𝐼𝑆𝐺 | , (6) | | | | 𝐼𝑆𝐶
𝛽𝑣 ( ) 𝐴 𝐴 ∑ 𝑓𝜒 (𝜒) = 𝑣 𝑟 𝑎 𝛼 𝑘 4 𝑘=1 𝑣𝑘 𝑣
×
(
( ) 1 𝜉
𝛼𝑟 +𝑘 4
)
)−
(
𝛼𝑣 +𝑘 2
)
(14) √ √ ⎞ ⎛ 𝛼 𝛽 𝜒 𝑟 𝑟 ⎟. 𝐾𝛼𝑟 −𝑘 ⎜2 ⎜ 𝛾𝑟 𝛽𝑟 + 𝛺𝑟′ 𝜉⎟ ⎝ ⎠
Using exponential approximation for the Gaussian-𝑄 function [23] and using the identity [24, eq. (14)], the expression for ABER can be written as (16). (
( )− 𝛽𝑣 ( ) 𝛼𝑣 𝛽𝑣 𝐴 𝐴 ⎛∑ 𝐴𝐵𝐸𝑅 = 𝑣 𝑟 ⎜ 𝑎𝑣𝑘 𝛼𝑣 𝑘 4 ⎜𝑘=1 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′ ⎝
𝛼𝑣 +𝑘 2
)
⎞ ⎟ ⎟ ⎠
( ) ⎧ ( ) 𝛼𝑟 +𝑘 𝛽𝑟 ( 𝜒 ) ( 𝛼𝑟 +𝑘 −1) ∞ 4 ⎪ 1 ∑ 1 𝑎𝑟𝑘 exp − 𝜒 4 ×⎨ ∫0 𝜉 2 ⎪ 24 𝑘=1 ⎩ [( ] )√ | − 𝛼𝑟 𝛽𝑟 𝜒| 2,0 × 𝐺0,2 | 𝛼 −𝑘 𝛼 −𝑘 𝑑𝜒 𝛾𝑟 𝛽𝑟 + 𝛺𝑟′ 𝜉 || 𝑟2 , − 𝑟2
2 𝜎𝑅
where is the Rytov variance. With the UV photons arriving in the common volume forming a new source towards the receiver, the conditional distribution of the irradiance fluctuations 𝐼𝑟 at the receiver can then be written as ( √ ) ( ) 𝛽𝑟 𝛼𝑟 +𝑘 ∑ −1 ( ) 𝛼𝑟 𝛽𝑟 𝑘 2 𝑓𝐼𝑟 𝐼𝑟 ||𝐼𝑣 = 𝐴𝑟 𝐾𝛼𝑟 −𝑘 2 , (9) 𝑎𝑟𝑘 𝐼𝑟 𝛾𝑟 𝛽𝑟 + 𝛺𝑟′ 𝑘=1
(
(16)
)
( ) ( ) 𝛼𝑟 +𝑘 𝛽𝑟 ∞ 𝛼𝑟 +𝑘 4 −1 1∑ 1 + exp (−𝜒)𝜒 4 𝑎𝑟𝑘 ∫0 8 𝑘=1 𝜉 [( ] } )√ | − 𝛼𝑟 𝛽𝑟 𝜒| 2,0 ×𝐺0,2 | 𝛼 −𝑘 𝛼 −𝑘 𝑑𝜒 . 𝛾𝑟 𝛽𝑟 + 𝛺𝑟′ 𝜉 || 𝑟2 , − 𝑟2
By using the identity [25, eq. (07.34.21.0088.01)] in (16), the closedform expression for the ABER is derived as (17).
where 𝛼𝑟 , 𝐴𝑟 , and 𝑎𝑟𝑘 can similarly be obtained as 𝛼𝑣 , 𝐴𝑣 , and 𝑎𝑣𝑘 . The ( ) ( ) joint PDF of 𝐼𝑣 and 𝐼𝑟 is given by 𝑓𝐼𝑣 ,𝐼𝑟 (𝐼𝑣 , 𝐼𝑟 ) = 𝑓𝐼𝑟 𝐼𝑟 ||𝐼𝑣 𝑓𝐼𝑣 𝐼𝑣 . Replacing (4) and (9) therein and following the derivation steps provided in the Appendix, the closed-form expression of the marginal PDF of the irradiance scintillation at the receiver is derived as (10). (
𝑎𝑟𝑘 𝜒
)
𝛼𝑟 +𝑘 −1 4
𝛼𝑣 𝛽𝑣 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′
Assuming perfect channel state information, the average BER can be expressed by averaging the bit errors in the UV channel over the turbulence-induced fading distribution. Considering the non-return-tozero on–off keying (OOK) modulation scheme, the average BER is obtained ∞ (√ ) 𝜒 𝑓𝜒 (𝜒) 𝑑𝜒. (15) 𝐴𝐵𝐸𝑅 = 𝑄 ∫0
(8)
(
𝐴 𝐴 𝐴𝐵𝐸𝑅 = 𝑣 𝑟 4
( )− 𝛽𝑣 ⎛∑ ( ) 𝛼𝑣 𝛽𝑣 ⎜ 𝑎𝑣𝑘 𝛼𝑣 𝑘 ′ ⎜𝑘=1 𝛾𝑣 𝛽𝑣 + 𝛺 ⎝
𝛼𝑣 +𝑘 2
)
⎞ ⎟ ⎟ ⎠
( ) ( ) ⎧𝛽 ( ) 𝛼𝑟 +𝑘 ⎛ 𝛼𝑟 +𝑘 −1 𝑟 4 4 ⎪∑ 2 1 ⎜ × ⎨ 𝑎𝑟𝑘 ⎜ 24𝜋 𝜉 ⎪𝑘=1 ⎝ ⎩ [( ]) )2 4−𝛼𝑟 −𝑘 𝛼𝑟 𝛽𝑟 1 || 4,1 4 ×𝐺1,4 | 𝛾𝑟 𝛽𝑟 + 𝛺′ 8𝜉 || 𝛼𝑟 −𝑘 , 2−𝛼𝑟 −𝑘 , − 𝛼𝑟 −𝑘 , − 2−𝛼𝑟 −𝑘
)
)− 𝛼𝑣 +𝑘 2 𝛼𝑣 𝛽𝑣 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′ ( √ ) ( ) 𝛽𝑟 𝛼𝑟 +𝑘 ∑ −1 𝛼𝑟 𝛽𝑟 𝑘 2 × 𝑎𝑟𝑘 𝐼𝑟 𝐾𝛼𝑟 −𝑘 2 . 𝛾𝑟 𝛽𝑟 + 𝛺𝑟′ 𝑘=1
(
(
2.3. Average bit error rate
−1
(
𝛽𝑟 ∑ 𝑘=1
where and are statistically independent random processes and are related to the total received irradiance 𝐼𝑣 in the common volume as [16] ( ) 𝐶 𝐺 𝐼𝑣 = 𝐼𝐿𝑣 + 𝐼𝑆𝑣 + 𝐼𝑆𝑣 exp (𝛥 + 𝑗𝜍) , (7) √ √ 𝐶 = 𝐺𝜌2𝑏0 where 𝐼𝐿𝑣 = 𝐺𝛺𝑣 exp(𝑗𝜙𝐴 ) is the unscattered path, 𝐼𝑆𝑣 𝐺 exp(𝑗𝜙𝐵 ) is the quasi-forward scattered component, and 𝐼𝑆𝑣 = √ (1 − 𝜌𝑣 )𝐼𝑆 is the scattered component due to the off-axis eddies and is 𝐶 . 𝐺 is a Gamma distributed real statistically independent of 𝐼𝐿𝑣 and 𝐼𝑆𝑣 number with 𝐸[𝐺] = 1 and represents a slow fading of the unscattered component. 𝐼𝑆 is the circular Gaussian complex random variable. 𝛥 and 𝜍 are real random variables and denote the log-amplitude and phase perturbation induced due to the turbulence fading. Assuming the optical radiation to be a plane wave, 𝛼𝑣 is given by [21]
𝛽𝑣 ( ) 𝐴 𝐴 ∑ ( ) 𝑓𝐼𝑟 𝐼𝑟 = 𝑣 𝑟 𝑎 𝛼 𝑘 2 𝑘=1 𝑣𝑘 𝑣
(13)
,
where 𝐽 is the Jacobean matrix. After substituting the values in (11), the closed-form expression of 𝑓𝜒 (𝜒) is derived as (14).
𝐼𝑆𝐺
⎧ ⎛ ⎞ ⎫ 2 0.49𝜎𝑅 ⎪ ⎜ ⎟ ⎪ 𝛼𝑣 = ⎨exp ⎜ ( − 1 ⎬ , )7∕6 ⎟ 12∕5 ⎪ ⎜ 1 + 1.11𝜎 ⎟ ⎪ 𝑅 ⎩ ⎝ ⎠ ⎭
(11)
(10)
4
3
4
4
4
(17)
S. Arya and Y.H. Chung (
( ) 3 2
+
×
Table 1 Convergence of the proposed statistical model into existing models for NLOS UV links.
4,1 𝐺1,4
32𝜋 [(
Optics Communications 463 (2020) 125402
)
𝛼𝑟 +𝑘 −1 4
]⎫ )2 4−𝛼𝑟 −𝑘 𝛼𝑟 𝛽𝑟 ⎪ 3 || 4 . | 𝛾𝑟 𝛽𝑟 + 𝛺′ 32𝜉 || 𝛼𝑟 −𝑘 , 2−𝛼𝑟 −𝑘 , − 𝛼𝑟 −𝑘 , − 2−𝛼𝑟 −𝑘 ⎬ ⎪ 4 4 4 4 ⎭
Distribution model
Condition
Log-normal distribution [4]
𝜌𝑣 = 0 and 𝜌𝑟 = 0 Var[|𝐼𝐿𝑣 |] = 0 and Var[|𝐼𝐿𝑟 |] = 0 𝜌𝑣 = 1 and 𝜌𝑟 = 1 𝛾𝑣 = 0 and 𝛾𝑟 = 0 𝛺𝑣′ = 1 and 𝛺𝑟′ = 1
Gamma–Gamma distribution [7,13]
2.4. Outage probability In this subsection, we derive the outage probability for an NLOS UV link under Málaga distribution. It is defined as the probability that the achievable rate for a given UV link configuration is less than a predefined threshold 𝑅𝐵𝑇 ℎ , i.e., [ ] [ ] 𝑃𝑜𝑢𝑡 = Pr log2 (1 + 𝜒) < 𝑅𝐵𝑇 ℎ = Pr 𝜒 < 𝜒𝑇 ℎ , (18)
As can be seen, when
𝐺𝛺𝑣 𝛾𝑣
⟶ ∞ and
𝐺𝛺𝑟 𝛾𝑟
⟶ ∞, (21) and (22)
converge to the log-normal distribution. 3.2. PDF of Gamma–Gamma distributed scattered path
𝛥
where 𝜒𝑇 ℎ = 2𝑅𝐵𝑇 ℎ − 1 denotes the outage threshold. It is worth noting that the outage probability defined in (18) is independent of the transmission technique and coding strategy for a given outage threshold. The Málaga distributed turbulence results in SNR fluctuation, and depending on the turbulence strength, the receiver may experience an outage. With the derived PDF of 𝜒 in (14) and using the identity [24, eq. (26)] in (18), the closed-form expression for the outage probability is derived as (19). For proof of (19), please see Appendix. 𝑃𝑜𝑢𝑡
×
𝐴 𝐴 = 𝑣 𝑟 4
𝛽𝑟 ∑ 𝑘=1
[( ×
( 𝑎𝑟𝑘
( )− 𝛽𝑣 ⎛∑ ( ) 𝛼𝑣 𝛽𝑣 ⎜ 𝑎𝑣𝑘 𝛼𝑣 𝑘 ⎜𝑘=1 𝛾𝑣 𝛽𝑣 + 𝛺 ′ ⎝ (
𝛼𝑟 +𝑘
) 4 𝜒𝑇 ℎ 𝜉 )2 √
𝛼𝑟 𝛽𝑟 𝛾𝑟 𝛽𝑟 + 𝛺′
(
𝛼𝑣 +𝑘 2
)
)
As defined in [26], (7) can be written as 2 | 𝐶 𝐺 | 𝐼𝑣 = |𝐼𝐿𝑣 + 𝐼𝑆𝑣 + 𝐼𝑆𝑣 | exp (2𝛥) = 𝑌𝑣 𝑋𝑣 | | { 2 | 𝐶 + 𝐼 𝐺 | (small-scale fluctuations) , 𝑌𝑣 = |𝐼𝐿𝑣 + 𝐼𝑆𝑣 | 𝑆𝑣 | | 𝑋𝑣 = exp (2𝛥) (large-scale fluctuations)
where the PDFs of 𝑌𝑣 and 𝑋𝑣 can respectively be given as ( )𝛽𝑣 ) ( ( ) 𝛾𝑣 𝛽𝑣 𝑦𝑣 1 exp − 𝑓𝑌𝑣 𝑦𝑣 = 𝛾𝑣 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′ 𝛾𝑣 ( ) ′ 𝑦𝑣 𝛺𝑣 × 1 𝐹1 𝛽𝑣 ; 1; ( ) , 𝛾𝑣 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′
⎞ ⎟ ⎟ ⎠
2,1 𝐺1,3
(23)
(19)
(24)
and
] | 2−𝛼𝑟 −𝑘 𝜒𝑇 ℎ || 2 . 𝜉 || 𝛼𝑟 −𝑘 , − 𝛼𝑟 −𝑘 , − 𝛼𝑟 +𝑘 2 2 | 2
( ) 𝑓𝑋𝑣 𝑥𝑣 =
𝛼
( ) 𝛼𝑣 𝑣 ( ) 𝑥𝛼𝑣 −1 exp −𝛼𝑣 𝑥 . 𝛤 𝛼𝑣
(25)
where 1 𝐹1 (⋅; ⋅; ⋅) in (24) represents the Kummer confluent hypergeometric function of the first kind. It can be seen that the PDF of the large-scale fluctuations, 𝑋, follows the Gamma distribution. Following the steps in [16], the moment generating function (MGF) of 𝑌𝑣 and 𝑋𝑣 can readily be obtained as
3. Derivation of existing distribution models from the derived PDF In this section, we derive the conditions to generate the log-normal and Gamma–Gamma distribution from the proposed statistical model. Considering an NLOS UV link, we present the conditions for the PDFs derived in [4] and [13]. In [4], a log-normal distribution was assumed for each scattered path. Similarly, the marginal PDF of the received irradiance was derived in [13] when the fading statistics of each scattered path follow a Gamma–Gamma distribution.
[ ] ( ( ) 𝑀𝑌𝑣 (𝑠) = 𝓁 𝑓𝑌𝑣 𝑦𝑣 ; −𝑠 =
𝛾𝑣 𝛽𝑣 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′
)𝛽𝑣
( )𝛽 −1 1 − 𝛾𝑣 𝑠 𝑣
( 1−
𝛺𝑣′ 𝛾𝑣 𝛽𝑣 +𝛺𝑣′
− 𝛾𝑣 𝑠
) 𝛽𝑣 , (26)
3.1. PDF of log-normal distributed scattered path
and 𝛼 [ ] ( ) 𝛼𝑣 𝑣 𝑀𝑋𝑣 (𝑠) = 𝐿 𝑓𝑌𝑣 𝑦𝑣 ; −𝑠 = ( )𝛼 , 𝛼𝑣 − 𝑠 𝑣
Following (4) and (5) with 𝜌𝑣 set to zero, and assuming the variance of |𝐼𝐿,𝑣 |, that is, Var[|𝐼𝐿,𝑣 |]√ = 0, 𝐼𝐿,𝑣 will converge into a random 𝛺𝑣 . As can be seen, if 𝐸[𝛥] = 0, and variable with 𝐸[|𝐼𝐿,𝑣 |] = Var[𝛥] and Var[𝜍] are set to 0, then (7) can be written in terms of Born approximation as [15] ( ) |√ | 𝐼𝑣 = | 𝐺𝛺𝑣 exp 𝑗𝜙𝐴 + 𝐼𝑆 | . (20) | | As defined in Section 2, 𝐼𝑆 is a complex random variable with circular Gaussian distribution and owing to the fact that 𝜌𝑣 = 0, it follows that 𝐸[|𝐼𝑆 |2 ] approaches 2𝑏0 = 𝛾𝑣 . It yields the PDF of the optical power arriving in the common volume to follow the modified Rice–Nakagami distribution written as ) [ ] ( √ 2 𝐺𝛺𝑣 𝐼𝑣 ( ) 𝐺𝛺𝑣 + 𝐼𝑣 1 𝑓𝐼𝑣 𝐼𝑣 = exp − 𝐼0 , 𝐼𝑣 > 0, (21) 𝛾𝑣 𝛾𝑣 𝛾𝑣
(27)
where 𝓁[⋅] represents the Laplace transformation operation. Knowing the fact that 𝜌 = 1 yields 𝛾 = 0, it follows from (26) as ( )−𝛽𝑣 ( )−𝛽𝑣 𝛽𝑣 − 𝛺𝑣′ 𝑠 𝑀𝑌𝑣 (𝑠)lim 𝛾 →0 = 𝛺𝑣′ . (28) 𝑣 𝛺𝑣′ 𝛽𝑣 Applying the limit 𝛺𝑣′ ⟶ 1 into (28) yields ( 𝑀𝑌𝑣 (𝑠) =
1−
𝑠 𝛽𝑣
)−𝛽𝑣 .
(29)
(29) denotes the MGF of a Gamma function. Therefore, we can state that, under the limiting conditions of 𝜌𝑣 = 1, 𝛾𝑣 = 0, and 𝛺𝑣′ ⟶ 1, the distribution of small-scale fluctuations, 𝑌𝑣 , converges to a Gamma distribution. From (23), (25), and (29), we can state that the unconditional PDF of the irradiance, under the given conditions, converges to a Gamma–Gamma distribution. Similarly, the same conditions can also be derived for (9). Table 1 shows the conditions for the log-normal and Gamma–Gamma distributions.
where 𝐼0 (⋅) represents the modified Bessel function of the first kind and order zero. Similarly, the conditional received irradiance at the receiver can be obtained as ) [ ] ( √ 2 𝐺𝛺𝑟 𝐼𝑟 ( ) 𝐺𝛺𝑟 + 𝐼𝑟 1 | 𝑓𝐼𝑟 𝐼𝑟 |𝐼𝑣 = exp − 𝐼0 , 𝐼𝑟 > 0. (22) 𝛾𝑟 𝛾𝑟 𝛾𝑟 4
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Optics Communications 463 (2020) 125402
Table 2 Simulation parameters for different turbulence strength.
𝛼 𝛽 𝛺 𝑏0 𝜌
Weak
Moderate
Strong
50 14 1.0621 0.0216 0.86
2.55 22 0.4618 0.6525 0.988
2.2814 33 1.33 0.4231 0.84
Fig. 4. Average bit error rate relative to the average SNR.
Fig. 2. PDF of the received irradiance fluctuations.
Fig. 5. Outage probability relative to the average SNR (𝜒𝑇 ℎ = 5 dB).
The PDF of the instantaneous SNR over different turbulence regimes is illustrated in Fig. 3. The identical NLOS UV links are assumed with 𝜉 set to 10 dB. It is interesting to note that decreasing 𝛼 causes a spreading of the PDF over a wider range of values. This spreading has a varying impact on the performance metrics derived from the PDF of the instantaneous received SNR for the NLOS UV link. The average bit error rate against the average received SNR over Málaga distributed turbulence from weak scintillations to strong fluctuations is illustrated in Fig. 4. As expected, the channel becomes more impaired as the effective number of large-scale cells reduces and results in a higher bit error rate. The outage probability against the average received SNR is illustrated in Fig. 5. As revealed from the results, under the stronger turbulence regime, a slow change in the slope of outage probability curve can be observed for lower SNR. It is interesting to compare the proposed statistical model with the log-normal and Gamma–Gamma distributions to validate the use of the proposed PDF model in practical NLOS UV applications under a wide variety of turbulence fading. Assuming a plane wave propagation, we consider the two existing PDF models, i.e., [4] and [13], for comparison. In [4], a log-normal distribution was assumed for each scattered path. Similarly, the marginal PDF of the received irradiance was derived in [13] when the fading statistics of each scattered path follow Gamma–Gamma distribution. In Fig. 6a, the normalized PDF of the received irradiance is plotted on a log-scale as a function of the received irradiance for the weak turbulence condition. For the performance comparison, we also plot the log-normal PDF obtained from [4] and Gamma–Gamma PDF derived
Fig. 3. PDF of the instantaneous SNR.
4. Results and discussions In this section, we present the received irradiance distribution and SNR distribution for an NLOS UV link, covering a range of turbulence strengths that extends from weak to strong irradiance fluctuations. We assume a plane wave propagation. For the sake of brevity, we set {𝛼𝑣 , 𝛽𝑣 , 𝜌𝑣 , 𝛾𝑣 , 𝛺𝑣 } = {𝛼𝑟 , 𝛽𝑟 , 𝜌𝑟 , 𝛾𝑟 , 𝛺𝑟 } = {𝛼, 𝛽, 𝜌, 𝛾, 𝛺} with values of 𝛼 representing different turbulence regimes. The simulation parameters are provided in Table 2 [16]. 𝜙𝑡 and 𝜙𝑟 are set to 𝜋∕180 and 𝜋∕3, respectively. 𝜓𝑡 and 𝜓𝑟 are set to 𝜋∕4 and 𝜋∕3, respectively. The baseline distance 𝑑 is set to 1000 m. All these angles are measured in radians. 𝐷 is set to 1.5. The outage threshold 𝜒𝑇 ℎ is set to 5 dB. Fig. 2 shows the PDF of the irradiance for an NLOS UV link over Málaga distribution, covering weak scintillations to strong fluctuations. It is observed that as the effective number of large-scale cells (𝛼) reduces, the turbulence effects become more significant. The results demonstrate that by decreasing 𝛼, the received irradiance variance increases, thus indicating a stronger turbulence effect. 5
S. Arya and Y.H. Chung
Optics Communications 463 (2020) 125402
condition, we find that the Gamma–Gamma model provides a better fit to the Málaga distribution than does the log-normal PDF. As shown in Fig. 6c, when we also consider the strong turbulence fading to make a comparison with the Gamma–Gamma distributed fading including inner scale effects, it is found to provide a better fit. The results shown in Fig. 6 demonstrate the high accuracy of the proposed statistical model. It can also be seen that the Gamma–Gamma PDF provides a better fit to the Málaga distribution than does the log-normal PDF derived in [4]. 5. Conclusion In this paper, a novel statistical model for an NLOS UV link over Málaga distributed irradiance fluctuation has been presented. We have developed a comprehensive and unified analytical model to evaluate the performance of the NLOS UV link. The performance impairment caused by the spreading of the PDF of the instantaneous received SNR for NLOS path has been investigated. It is shown that the performance degradation is significant when the effective number of large-scale cells of the scattering process reduces. CRediT authorship contribution statement Sudhanshu Arya: Conceptualization, Methodology, Writing- original draft. Yeon Ho Chung: Supervision, Validation, Writing - review & editing. Acknowledgment This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Republic of Korea (2018R1D1A3B07049858). Appendix
A.1. Proof of (10) ) ( ( ) The joint PDF of 𝐼𝑣 and 𝐼𝑟 is 𝑓𝐼𝑣 ,𝐼𝑟 (𝐼𝑣 , 𝐼𝑟 ) = 𝑓𝐼𝑟 𝐼𝑟 ||𝐼𝑣 𝑓𝐼𝑣 𝐼𝑣 . Therefore, the PDF of the received irradiance can readily be obtained as ∞ ( ) ( ) 𝑓𝐼𝑟 𝐼𝑟 = 𝑓𝐼𝑣, 𝐼𝑟 𝐼𝑣, 𝐼𝑟 𝑑𝐼𝑣 . (30) ∫0 Substituting (4) and (9) into (30) and using the identities [24, eq. (14)] with some mathematical manipulations, we obtained (31). ( √ ) ) ( 𝛽𝑟 𝛼𝑟 +𝑘 ∑ −1 ( ) 𝛼𝑟 𝛽𝑟 𝐼𝑟 2 𝐾𝛼𝑟 −𝑘 2 𝑓𝐼𝑟 𝐼𝑟 = 𝐴𝑣 𝐴𝑟 𝑎𝑟𝑘 𝐼𝑟 𝛾𝑟 𝛽𝑟 + 𝛺𝑟′ 𝑘=1 {𝛽 ( ( ) 𝛼𝑣 +𝑘 𝑣 ∞ ∑ −1 𝑎𝑣𝑘 2 2,0 (31) × 𝐼𝑣 𝐺0,2 ∫0 2 𝑘=1 [( ] )} )| − 𝛼𝑣 𝛽𝑣 | × 𝑑𝐼 . | 𝛼 −𝑘 𝛼 −𝑘 𝑣 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′ || 𝑣2 , − 𝑣2
Fig. 6. The normalized PDF of the received irradiance on the log-scale relative to the received irradiance over different turbulence fading conditions.
Since, the parameters 𝛼𝑣 , 𝛽𝑣 , and 𝛺𝑣′ for Málaga distributed fading must be real positive, the following conditions must hold true: )| | ( 𝛼 𝛽 | | 𝑣 𝑣 (32) |arg | < 𝜋, | 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′ || | and | 𝛼𝑣 − 𝑘 | 𝛼𝑣 + 𝑘 | | (33) | 2 |< 2 . | |
in [13]. As a typical weak turbulence fading, we use a Rytov variance 𝜎12 = 0.1. The inner scale value 𝑙0 is set to 0.5𝑅𝐹 , where 𝑅𝐹 represents the scale size of the Fresnel zone [27]. For the sake of brevity, we set the parameters 𝑎𝑣 = 𝑎𝑟 = 17.3 and 𝑏𝑣 = 𝑏𝑟 = 17.3, where the parameters 𝑎𝑣 , 𝑎𝑟 , 𝑏𝑣 , and 𝑏𝑟 describe the Gamma–Gamma model as described in [13]. For all the results, (𝜙𝐴 − 𝜙𝐵 ) is set to 𝜋∕2. As can be seen, under the weak turbulence condition, the proposed PDF generally fits the values predicted by the log-normal and Gamma–Gamma distribution model. From Fig. 6b, as a typical moderate turbulence
Finally, by knowing the conditions (32) and (33) and using the identity [28, eq. (7.811.4)] with some mathematical manipulations, we ( ) obtain the closed-form expression of 𝑓𝐼𝑟 𝐼𝑟 as shown in (10). 6
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Optics Communications 463 (2020) 125402
A.2. Proof of (19)
[11] R.J. Hill, R.G. Frehlich, W.D. Otto, The probability distribution of irradiance scintillation. [12] R.J. Hill, R.G. Frehlich, Probability distribution of irradiance for the onset of strong scintillation, J. Opt. Soc. Amer. A 14 (7) (1997) 1530–1540. [13] S. Arya, Y.H. Chung, A novel blind spectrum sensing technique for multi-user ultraviolet communications in atmospheric turbulence channel, IEEE Access 7 (2019) 58314–58323. [14] S. Arya, Y.H. Chung, Generic blind spectrum sensing scheme for all opticalwavelength multi-user free space optical communications, Opt. Commun. 450 (2019) 316–321. [15] L.C. Andrews, R.L. Phillips, Laser Beam Propagation Through Random Media, Vol. 152, SPIE press, Bellingham, WA, 2005. [16] A. Jurado-Navas, J.M. Garrido-Balsells, J.F. Paris, A. Puerta-Notario, J. Awrejcewicz, A unifying statistical model for atmospheric optical scintillation, in: Numerical Simulations of Physical and Engineering Processes, Vol. 181, InTech press, Croatia, 2011. [17] S. Huang, M. Safari, Free-space optical communication impaired by angular fluctuations, IEEE Trans. Wireless Commun. 16 (11) (2017) 7475–7487. [18] H. Ding, Modeling and Characterization of Ultraviolet Scattering Communication Channels Ph.D. thesis, UC Riverside, 2011. [19] A. Vavoulas, H.G. Sandalidis, D. Varoutas, Connectivity issues for ultraviolet uv-c networks, J. Opt. Commun. Netw. 3 (3) (2011) 199–205. [20] M.H. Ardakani, M. Uysal, Relay-assisted ofdm for nlos ultraviolet communication, in: 2015 17th International Conference on Transparent Optical Networks (ICTON), IEEE, 2015, pp. 1–4. [21] Z. Ghassemlooy, W.O. Popoola, E. Leitgeb, Free-space optical communication using subearrier modulation in gamma-gamma atmospheric turbulence, in: 2007 9th International Conference on Transparent Optical Networks, Vol. 3, IEEE, 2007, pp. 156–160. [22] H.E. Nistazakis, T.A. Tsiftsis, G.S. Tombras, Performance analysis of freespace optical communication systems over atmospheric turbulence channels, IET Commun. 3 (8) (2009) 1402–1409. [23] M. Chiani, D. Dardari, M.K. Simon, New exponential bounds and approximations for the computation of error probability in fading channels, IEEE Trans. Wireless Commun. 2 (4) (2003) 840–845. [24] V. Adamchik, O. Marichev, The algorithm for calculating integrals of hypergeometric type functions and its realization in reduce system, in: Proceedings of the International Symposium on Symbolic and Algebraic Computation, ACM, 1990, pp. 212–224. [25] The wolfram functions site wolfram research, inc. URL http://functions.wolfram. com. [26] A. Jurado-Navas, J. Garrido-Balsells, J.F. Paris, M. Castillo-Vazquez, A. PuertaNotario, Further insights on málaga distribution for atmospheric optical communications, in: 2012 International Workshop on Optical Wireless Communications (IWOW), IEEE, 2012, pp. 1–3. [27] A. Al-Habash, L.C. Andrews, R.L. Phillips, Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media, Opt. Eng. 40 (8) (2001) 1554–1563. [28] A. Jeffrey, D. Zwillinger, Table of Integrals, Series, and Products, Elsevier, 2007.
Substituting (15) in (18) and using the identity [24, eq. (14)] with variable transformation with 𝜒 = 𝑍 2 , (18) can readily be written as ( )− 𝛽𝑣 ( ) 𝛼𝑣 𝛽𝑣 𝐴 𝐴 ⎛∑ 𝑃𝑜𝑢𝑡 = 𝑣 𝑟 ⎜ 𝑎𝑣𝑘 𝛼𝑣 𝑘 4 ⎜𝑘=1 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′ ⎝ ) ( √ ( ) 𝛼𝑟 +𝑘 𝛽𝑟 𝜒𝑇 ℎ ∑ 𝛼𝑟 +𝑘 4 1 2,0 × 𝑎𝑟𝑘 𝑍 4 −1 𝐺0,2 ∫0 𝜉 𝑘=1 [( ] ) − 𝛼𝑟 𝛽𝑟 𝑍 || × √ || 𝛼𝑟 −𝑘 𝑘−𝛼𝑟 𝑑𝑍. 𝛾𝑟 𝛽𝑟 + 𝛺𝑟′ 𝜉| 2 , 2
(
𝛼𝑣 +𝑘 2
)
⎞ ⎟ ⎟ ⎠ (34)
Finally, using the identity [24, eq. (26)] with some mathematical manipulations, we obtain (19). References [1] Z. Xu, B.M. Sadler, Ultraviolet communications: potential and state-of-the-art, IEEE Commun. Mag. 46 (5) (2008) 67–73. [2] H. Ding, G. Chen, A.K. Majumdar, B.M. Sadler, Z. Xu, Turbulence modeling for non-line-of-sight ultraviolet scattering channels, in: Atmospheric Propagation VIII, Vol. 8038, International Society for Optics and Photonics, 2011, p. 80380J. [3] Y. Zuo, H. Xiao, J. Wu, X. Hong, J. Lin, Effect of atmospheric turbulence on non-line-of-sight ultraviolet communications, in: 2012 IEEE 23rd International Symposium on Personal, Indoor and Mobile Radio Communications-(PIMRC), IEEE, 2012, pp. 1682–1686. [4] M.H. Ardakani, A.R. Heidarpour, M. Uysal, Performance analysis of relay-assisted nlos ultraviolet communications over turbulence channels, J. Opt. Commun. Netw. 9 (1) (2017) 109–118. [5] X. Zhu, J.M. Kahn, Performance bounds for coded free-space optical communications through atmospheric turbulence channels, IEEE Trans. Commun. 51 (8) (2003) 1233–1239. [6] G.R. Osche, Optical Detection Theory for Laser Applications, Vol. 2002, Wiley-VCH, ISBN: 0-471-22411-1, 2002, p. 424. [7] S. Arya, Y.H. Chung, Amplify-and-forward multihop non-line-of-sight ultraviolet communication in the gamma–gamma fading channel, J. Opt. Commun. Netw. 11 (8) (2019) 422–436. [8] S. Arya, Y.H. Chung, Non-line-of-sight ultraviolet communication with receiver diversity in atmospheric turbulence, IEEE Photonics Technol. Lett. 30 (10) (2018) 895–898. [9] S. Karp, R.M. Gagliardi, S.E. Moran, L.B. Stotts, Optical Channels: Fibers, Clouds, Water, and the Atmosphere, Springer Science & Business Media, 2013. [10] L.C. Andrews, R.L. Phillips, C.Y. Hopen, Laser Beam Scintillation with Applications, Vol. 99, SPIE press, 2001.
7
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
A unified statistical model for Málaga distributed optical scattering communications Sudhanshu Arya, Yeon Ho Chung ∗ Department of Information and Communications Engineering, Pukyong National University, Busan, Republic of Korea
ARTICLE Keywords: Average bit error rate Málaga fading Optical scattering Outage probability
INFO
ABSTRACT In this paper, we present a novel analytical model for optical scattering communications over Málaga distributed irradiance fluctuation. The distribution of the received signal is quantified with each scattered optical path following Málaga distribution in the scattering communication links. The model efficiently includes the scattering by the off-axis eddies, in addition to the eddies on the propagation axis over a non-line-ofsight (NLOS) link, thus permitting accurate evaluation of the scattered ultraviolet (UV) communication. It is shown that as the effective number of large-scale cells reduces in the lower atmospheric channel, the probability density function (PDF) of the received signal-to-noise ratio (SNR) spreads widely with large variance and impacts the performance of optical scattering communications. Moreover, based on the derived model, closed-form expressions for the average bit error rate and the outage probability are also presented.
1. Introduction As a potential candidate for long-range non-line-of-sight (NLOS) connectivity in optical wireless communications (OWC), ultraviolet (UV) communication is gaining more and more interest from researchers. It comes with unique scattering properties and offers high security and low installation cost [1]. Although the optical scattering in the lower atmosphere enables NLOS links, UV links are highly susceptible to severe attenuation and fading under atmospheric turbulence channel, which limit the extensive utilization of the UV communication link. Recently, many turbulence distributions have been studied and proposed to model the optical scattering communication. One widely accepted model for ultraviolet communication over turbulence fading is the log-normal distribution. In [2], the authors derived an analytical model for the received signal distribution which is then applied to analyze the NLOS link performance. Considering the log-normal distribution for each scattered path, similar studies were also reported for NLOS ultraviolet communications [3,4]. However, the log-normal distribution is only valid for weak turbulence fading as reported in [5,6]. In addition, over a longer link range, the incident optical beam becomes increasingly incoherent, thereby making the log-normal distribution invalid [6]. In [7,8], the authors suggested an alternative distribution model, i.e. Gamma– Gamma distributed irradiance fluctuations, to analyze the performance of the UV communication systems. In this work, we develop a new and unified statistical model for an optical scattering communication link with each scattered path following Málaga distribution. The developed model is applicable to both plane and spherical waves under
all turbulence conditions. The model is developed for obtaining the probability density function (PDF) of the instantaneous received signalto-noise ratio (SNR). The derived model is then subsequently used to derive closed-form expressions for the average bit error rate and the outage probability. Results are also presented to yield an insight into the performance of the optical scattering communication over Málaga distributed channels. We briefly describe a comparative view of optical scattering distributions. 1.1. Gamma–Gamma, log-normal, and Málaga distributions The log-normal distribution model for the fading statistics is a widely reported model in UV communication systems [3,4]. It follows directly from the first Rytov approximation and fits well with the lowerorder moments of the intensity obtained from the experimental data under weak turbulence conditions. The log-normal model requires the magnitude of the scattered field wave to be small, compared with the unperturbed phase gradient as reported in [9]. Unfortunately, this assumption is only valid in a single scattering event characterized by the weak turbulence fading. Therefore, in the environment where multiple scatterings occur or when the propagation path is large, the incident light intensity becomes increasingly incoherent and the log-normal distribution becomes invalid [9,10]. In addition, it has been observed through experimental data that the log-normal PDF can underestimate the behavior of the peak and the tails. This leads to a significant
∗ Corresponding author. E-mail address: [email protected] (Y.H. Chung).
https://doi.org/10.1016/j.optcom.2020.125402 Received 29 December 2019; Received in revised form 23 January 2020; Accepted 26 January 2020 Available online 29 January 2020 0030-4018/© 2020 Elsevier B.V. All rights reserved.
S. Arya and Y.H. Chung
Optics Communications 463 (2020) 125402
performance degradation in radar and optical communications where detection and fade probabilities are calculated over the tails of the PDF [11,12]. The Gamma–Gamma distribution to model the turbulence fading over optical scattering communication link, i.e., the UV link, was suggested recently as a reasonable alternative to the log-normal distribution [7,13,14]. It is considered as a general distribution model for both weak and strong fading. This distribution model can be expressed as a modulation process which involves first-order and second-order log-amplitude perturbations. In this model, the irradiance is assumed to consist of scattering (small-scale fading) and refraction (large-scale fading) [15]. The small-scale fading is due to eddies cells smaller than the Fresnel zone or the coherence radius. The large-scale fading is attributed to the presence of eddies greater than the first Fresnel zone or the scattering disk [10]. In deriving Gamma–Gamma model, it is assumed that both large-scale and small-scale fluctuations follow Gamma distribution. The Gamma–Gamma distribution is experimentally proved to be valid under all turbulence conditions. In addition, the distribution parameters can completely be determined by the atmospheric condition. However, in the limit of strong turbulence fading, i.e., in a saturation regime and beyond, where the independent scatters become very large, the Gamma–Gamma model reduces to negative exponential. The Málaga distribution model is valid under all turbulence conditions ranging from weak to strong fluctuations. Unlike the log-normal distribution model, it remains valid in strong and saturation turbulence regimes. As reported in [16], the main advantage of Málaga distribution model is that it can lead to closed-form expressions for important figures of merit of the optical channels. In addition, it unifies most of existing statistical models for the turbulence fading and it shows an excellent match to the experimental data.
Fig. 1. NLOS UV communication link with Málaga distributed scattered path.
2. Statistical model for NLOS UV link The UV scattered beam that arrives at a photodetector depends on both the link geometry and the atmospheric characteristics. A typical NLOS UV configuration is illustrated in Fig. 1. 𝑑 represents the separation between the transmitter and the receiver. 𝜓𝑡 and 𝜓𝑟 are the transmitter and the receiver apex angles, respectively. 𝜙𝑡 is the transmit beam angle and 𝜙𝑟 denotes the receiver field-of-view (FOV). 𝑉𝑐 represents the common volume. Atmospheric turbulence causes intensity fluctuation in the received UV signal and is considered one of the main impairments in limiting the performance of UV communication systems. In addition, it also results in random phase variation in the optical wavefront at the receiver with limited detector size. This random phase variation tilts the received wavefront from normal at the receiver aperture by an angle termed the angle-of-arrival (AOA). AOA results in the jitter of diffraction pattern on the receiver plane [15]. Considering all these impairments, the instantaneous received UV signal can be modeled as
1.2. Contributions Unlike previous works on optical scattering communications, which mainly focus on the log-normal and Gamma–Gamma distributed turbulence fading, this work presents the first comprehensive statistical model for NLOS UV communication over Málaga distributed fading. The novel contributions of this work are presented below.
𝑦 = 𝑆ℎ𝑡 ℎ𝜀 ℎ𝑠 𝑃𝑡 𝐼 + 𝑛,
(1)
where 𝑆 is the receiver responsivity in A/W, ℎ𝑡 represents the fading √ due to atmospheric scintillation and can be expressed as ℎ𝑡 =
• To yield an insight into the performance analysis of optical scattering communication over Málaga distributed fading, we develop a new statistical model for the received irradiance over NLOS path. We quantify the distribution of the received irradiance, when the turbulence fading over each scattered path is Málaga distributed. • Subsequently, we derive the closed-form expression for the PDF of the received SNR for NLOS optical path. • Furthermore, we show that as the effective number of large-scale cells in the turbulence channel reduces, the PDF of the received SNR spreads over a wider range. This spreading has a varying impact on the performance of a NLOS UV link. • We further derive a closed-form expression of an important figure of merit, i.e., the average bit error rate. The outage performance is also evaluated. • We derive conditions for the proposed distribution to become the log-normal and Gamma–Gamma distributions. In addition, a comparative performance analysis is provided.
−
( ) 11∕6 11∕6 23.17𝑘7∕6 𝐶𝑛2 𝑑𝑡𝑣 +𝑑𝑣𝑟 5
10 [7], where 𝑘 = 2𝜋∕𝜆 represents the wave number and 𝜆 wavelength of the UV signal. 𝐶𝑛2 denotes the refractive index coefficient of atmospheric turbulence channel. ℎ𝜀 denotes the fading introduced by AOA fluctuations given by [17] ( √ ) ( √ ) ℎ𝜀 = 1 − 𝐽02 𝜋 𝐷 − 𝐽12 𝜋 𝐷 , (2) where 𝐷 is a constant greater than or equal to 1 and is determined by the ratio of the receiver FOV solid angle to the diffraction-limited solid angle. 𝐽0 (⋅) and 𝐽1 (⋅) represent the Bessel function of the first kink and order zero and one, respectively. ℎ𝑠 in (1) denotes the attenuation term due to atmospheric scattering. Considering single scattering phenomena, it can be modeled as [18]
ℎ𝑠 =
( )( ) 𝐴𝑟 𝛼𝑠 𝑞𝑠 𝜙2𝑡 𝜙𝑟 sin 𝜓𝑡 + 𝜓𝑟 12 sin2 𝜓𝑟 + 𝜙2𝑟 sin2 𝜓𝑡 ( ) ( ), 𝛼 𝑑 (sin 𝜓 +sin 𝜓 ) 𝜙 96𝑑 sin 𝜓𝑡 sin2 𝜓𝑟 1 − cos 2𝑡 exp 𝑡 sin 𝜓 𝑡+𝜓 𝑟 ( 𝑡 𝑟)
(3)
where 𝑃𝑡 is the transmit power. 𝑛 is the Gaussian noise. It should be noted that the selection of the noise model depends on both the detection process employed and the background noise level. A Gaussian model is preferable when the thermal noise is dominant. In addition, in a medium or high noise scenario, such as outdoor daytime operation or when transmitting using longer pulses, the Gaussian noise model is suitable [19,20]. 𝐼 denotes the optical scintillation.
The rest of the paper is organized as follows. In Section 2, the statistical model for NLOS UV link is presented. The marginal PDFs of the received irradiance and SNR are derived. The closed-form analytical expressions for the average bit error rate and the outage probability are also derived. Results and discussions are provided in Section 3. Section 4 draws the conclusions. 2
S. Arya and Y.H. Chung
Optics Communications 463 (2020) 125402
2.1. Marginal PDF of the received irradiance
2.2. Marginal PDF of the instantaneous SNR
In this subsection, we derive the marginal PDF of the intensity fluctuations at the receiver. Following a Málaga distribution, the PDF of the optical power arriving in the common volume is given by [16] ( √ ) ) ( 𝛽𝑣 𝛼𝑣 +𝑘 ∑ −1 ( ) 𝛼𝑣 𝛽𝑣 𝑘 2 𝐾𝛼𝑣 −𝑘 2 𝑓𝐼𝑣 𝐼𝑣 = 𝐴𝑣 𝑎𝑣𝑘 𝐼𝑣 , (4) 𝛾𝑣 𝛽𝑣 + 𝛺′ 𝑘=1
The instantaneous received SNR is given by ( )2 𝑆𝐻𝑃𝑡 𝐼 2 𝜒= , 𝜎𝑛2
where 𝐼 is normalized to unity and following this, the average received SNR is defined as [22] ( )2 𝑆𝐻𝑃𝑡 (𝐸 [𝐼])2 . (12) 𝜉= 𝜎𝑛2
where 𝛼𝑣 ⎧ ( )𝛽𝑣 + 𝛼𝑣 2𝛼𝑣 2 𝛾 𝑣 𝛽𝑣 2 ⎪ 𝐴𝑣 = 𝛼𝑣 ′ 𝛾 𝛽 +𝛺 1+ 𝑣 𝑣 ⎪ 𝛾𝑣 2 𝛤 (𝛼𝑣 ) . ⎨ ( ) 𝑘 𝑘 𝛽𝑣 − 1 (𝛾𝑣 𝛽𝑣 +𝛺′ )1− 2 ( 𝛺′ )𝑘−1 ( 𝛼𝑣 ) 2 ⎪ 𝑎 = ⎪ 𝑣𝑘 𝛾𝑣 𝛽𝑣 (𝑘−1)! 𝑘−1 ⎩
By applying the Jacobean transformation, the marginal PDF of the instantaneous SNR is given by
(5)
√ 𝜒 𝐼= 𝜉
𝑓𝜒 (𝜒) = |𝐽 |𝑓𝐼 (𝐼)|
where 𝛽𝑣 is a natural number. 𝐾𝛼𝑣 −𝑘 (⋅) denotes the modified Bessel function of the second kind and order 𝛼𝑣 − 𝑘. 𝛼𝑣 in (4) represents the effective number of large-scale cells of the scattering process and ′ determines the √ strength of the turbulence. 𝛾𝑣 = 2𝑏0 (1 − 𝜌𝑣 ) and 𝛺𝑣 = 𝛺𝑣 + 2𝜌𝑣 𝑏0 + 8𝑏0 𝛺𝑣 𝜌𝑐𝑜𝑠(𝜙𝐴 − 𝜙𝐵 ), where 𝛺𝑣 is the shaping parameter, and 𝜙𝐴 and 𝜙𝐵 are the deterministic phases of the scatter terms [16]. 𝜌𝑣 takes a positive number between 0 and 1 and denotes the fraction of the scattered power coupled to the unscattered component. 2𝑏0 represents the average total power of the scatter components and is given by [ ] | |2 | |2 2𝑏0 = 𝐸 |𝐼𝑆𝐶 | + |𝐼𝑆𝐺 | , (6) | | | | 𝐼𝑆𝐶
𝛽𝑣 ( ) 𝐴 𝐴 ∑ 𝑓𝜒 (𝜒) = 𝑣 𝑟 𝑎 𝛼 𝑘 4 𝑘=1 𝑣𝑘 𝑣
×
(
( ) 1 𝜉
𝛼𝑟 +𝑘 4
)
)−
(
𝛼𝑣 +𝑘 2
)
(14) √ √ ⎞ ⎛ 𝛼 𝛽 𝜒 𝑟 𝑟 ⎟. 𝐾𝛼𝑟 −𝑘 ⎜2 ⎜ 𝛾𝑟 𝛽𝑟 + 𝛺𝑟′ 𝜉⎟ ⎝ ⎠
Using exponential approximation for the Gaussian-𝑄 function [23] and using the identity [24, eq. (14)], the expression for ABER can be written as (16). (
( )− 𝛽𝑣 ( ) 𝛼𝑣 𝛽𝑣 𝐴 𝐴 ⎛∑ 𝐴𝐵𝐸𝑅 = 𝑣 𝑟 ⎜ 𝑎𝑣𝑘 𝛼𝑣 𝑘 4 ⎜𝑘=1 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′ ⎝
𝛼𝑣 +𝑘 2
)
⎞ ⎟ ⎟ ⎠
( ) ⎧ ( ) 𝛼𝑟 +𝑘 𝛽𝑟 ( 𝜒 ) ( 𝛼𝑟 +𝑘 −1) ∞ 4 ⎪ 1 ∑ 1 𝑎𝑟𝑘 exp − 𝜒 4 ×⎨ ∫0 𝜉 2 ⎪ 24 𝑘=1 ⎩ [( ] )√ | − 𝛼𝑟 𝛽𝑟 𝜒| 2,0 × 𝐺0,2 | 𝛼 −𝑘 𝛼 −𝑘 𝑑𝜒 𝛾𝑟 𝛽𝑟 + 𝛺𝑟′ 𝜉 || 𝑟2 , − 𝑟2
2 𝜎𝑅
where is the Rytov variance. With the UV photons arriving in the common volume forming a new source towards the receiver, the conditional distribution of the irradiance fluctuations 𝐼𝑟 at the receiver can then be written as ( √ ) ( ) 𝛽𝑟 𝛼𝑟 +𝑘 ∑ −1 ( ) 𝛼𝑟 𝛽𝑟 𝑘 2 𝑓𝐼𝑟 𝐼𝑟 ||𝐼𝑣 = 𝐴𝑟 𝐾𝛼𝑟 −𝑘 2 , (9) 𝑎𝑟𝑘 𝐼𝑟 𝛾𝑟 𝛽𝑟 + 𝛺𝑟′ 𝑘=1
(
(16)
)
( ) ( ) 𝛼𝑟 +𝑘 𝛽𝑟 ∞ 𝛼𝑟 +𝑘 4 −1 1∑ 1 + exp (−𝜒)𝜒 4 𝑎𝑟𝑘 ∫0 8 𝑘=1 𝜉 [( ] } )√ | − 𝛼𝑟 𝛽𝑟 𝜒| 2,0 ×𝐺0,2 | 𝛼 −𝑘 𝛼 −𝑘 𝑑𝜒 . 𝛾𝑟 𝛽𝑟 + 𝛺𝑟′ 𝜉 || 𝑟2 , − 𝑟2
By using the identity [25, eq. (07.34.21.0088.01)] in (16), the closedform expression for the ABER is derived as (17).
where 𝛼𝑟 , 𝐴𝑟 , and 𝑎𝑟𝑘 can similarly be obtained as 𝛼𝑣 , 𝐴𝑣 , and 𝑎𝑣𝑘 . The ( ) ( ) joint PDF of 𝐼𝑣 and 𝐼𝑟 is given by 𝑓𝐼𝑣 ,𝐼𝑟 (𝐼𝑣 , 𝐼𝑟 ) = 𝑓𝐼𝑟 𝐼𝑟 ||𝐼𝑣 𝑓𝐼𝑣 𝐼𝑣 . Replacing (4) and (9) therein and following the derivation steps provided in the Appendix, the closed-form expression of the marginal PDF of the irradiance scintillation at the receiver is derived as (10). (
𝑎𝑟𝑘 𝜒
)
𝛼𝑟 +𝑘 −1 4
𝛼𝑣 𝛽𝑣 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′
Assuming perfect channel state information, the average BER can be expressed by averaging the bit errors in the UV channel over the turbulence-induced fading distribution. Considering the non-return-tozero on–off keying (OOK) modulation scheme, the average BER is obtained ∞ (√ ) 𝜒 𝑓𝜒 (𝜒) 𝑑𝜒. (15) 𝐴𝐵𝐸𝑅 = 𝑄 ∫0
(8)
(
𝐴 𝐴 𝐴𝐵𝐸𝑅 = 𝑣 𝑟 4
( )− 𝛽𝑣 ⎛∑ ( ) 𝛼𝑣 𝛽𝑣 ⎜ 𝑎𝑣𝑘 𝛼𝑣 𝑘 ′ ⎜𝑘=1 𝛾𝑣 𝛽𝑣 + 𝛺 ⎝
𝛼𝑣 +𝑘 2
)
⎞ ⎟ ⎟ ⎠
( ) ( ) ⎧𝛽 ( ) 𝛼𝑟 +𝑘 ⎛ 𝛼𝑟 +𝑘 −1 𝑟 4 4 ⎪∑ 2 1 ⎜ × ⎨ 𝑎𝑟𝑘 ⎜ 24𝜋 𝜉 ⎪𝑘=1 ⎝ ⎩ [( ]) )2 4−𝛼𝑟 −𝑘 𝛼𝑟 𝛽𝑟 1 || 4,1 4 ×𝐺1,4 | 𝛾𝑟 𝛽𝑟 + 𝛺′ 8𝜉 || 𝛼𝑟 −𝑘 , 2−𝛼𝑟 −𝑘 , − 𝛼𝑟 −𝑘 , − 2−𝛼𝑟 −𝑘
)
)− 𝛼𝑣 +𝑘 2 𝛼𝑣 𝛽𝑣 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′ ( √ ) ( ) 𝛽𝑟 𝛼𝑟 +𝑘 ∑ −1 𝛼𝑟 𝛽𝑟 𝑘 2 × 𝑎𝑟𝑘 𝐼𝑟 𝐾𝛼𝑟 −𝑘 2 . 𝛾𝑟 𝛽𝑟 + 𝛺𝑟′ 𝑘=1
(
(
2.3. Average bit error rate
−1
(
𝛽𝑟 ∑ 𝑘=1
where and are statistically independent random processes and are related to the total received irradiance 𝐼𝑣 in the common volume as [16] ( ) 𝐶 𝐺 𝐼𝑣 = 𝐼𝐿𝑣 + 𝐼𝑆𝑣 + 𝐼𝑆𝑣 exp (𝛥 + 𝑗𝜍) , (7) √ √ 𝐶 = 𝐺𝜌2𝑏0 where 𝐼𝐿𝑣 = 𝐺𝛺𝑣 exp(𝑗𝜙𝐴 ) is the unscattered path, 𝐼𝑆𝑣 𝐺 exp(𝑗𝜙𝐵 ) is the quasi-forward scattered component, and 𝐼𝑆𝑣 = √ (1 − 𝜌𝑣 )𝐼𝑆 is the scattered component due to the off-axis eddies and is 𝐶 . 𝐺 is a Gamma distributed real statistically independent of 𝐼𝐿𝑣 and 𝐼𝑆𝑣 number with 𝐸[𝐺] = 1 and represents a slow fading of the unscattered component. 𝐼𝑆 is the circular Gaussian complex random variable. 𝛥 and 𝜍 are real random variables and denote the log-amplitude and phase perturbation induced due to the turbulence fading. Assuming the optical radiation to be a plane wave, 𝛼𝑣 is given by [21]
𝛽𝑣 ( ) 𝐴 𝐴 ∑ ( ) 𝑓𝐼𝑟 𝐼𝑟 = 𝑣 𝑟 𝑎 𝛼 𝑘 2 𝑘=1 𝑣𝑘 𝑣
(13)
,
where 𝐽 is the Jacobean matrix. After substituting the values in (11), the closed-form expression of 𝑓𝜒 (𝜒) is derived as (14).
𝐼𝑆𝐺
⎧ ⎛ ⎞ ⎫ 2 0.49𝜎𝑅 ⎪ ⎜ ⎟ ⎪ 𝛼𝑣 = ⎨exp ⎜ ( − 1 ⎬ , )7∕6 ⎟ 12∕5 ⎪ ⎜ 1 + 1.11𝜎 ⎟ ⎪ 𝑅 ⎩ ⎝ ⎠ ⎭
(11)
(10)
4
3
4
4
4
(17)
S. Arya and Y.H. Chung (
( ) 3 2
+
×
Table 1 Convergence of the proposed statistical model into existing models for NLOS UV links.
4,1 𝐺1,4
32𝜋 [(
Optics Communications 463 (2020) 125402
)
𝛼𝑟 +𝑘 −1 4
]⎫ )2 4−𝛼𝑟 −𝑘 𝛼𝑟 𝛽𝑟 ⎪ 3 || 4 . | 𝛾𝑟 𝛽𝑟 + 𝛺′ 32𝜉 || 𝛼𝑟 −𝑘 , 2−𝛼𝑟 −𝑘 , − 𝛼𝑟 −𝑘 , − 2−𝛼𝑟 −𝑘 ⎬ ⎪ 4 4 4 4 ⎭
Distribution model
Condition
Log-normal distribution [4]
𝜌𝑣 = 0 and 𝜌𝑟 = 0 Var[|𝐼𝐿𝑣 |] = 0 and Var[|𝐼𝐿𝑟 |] = 0 𝜌𝑣 = 1 and 𝜌𝑟 = 1 𝛾𝑣 = 0 and 𝛾𝑟 = 0 𝛺𝑣′ = 1 and 𝛺𝑟′ = 1
Gamma–Gamma distribution [7,13]
2.4. Outage probability In this subsection, we derive the outage probability for an NLOS UV link under Málaga distribution. It is defined as the probability that the achievable rate for a given UV link configuration is less than a predefined threshold 𝑅𝐵𝑇 ℎ , i.e., [ ] [ ] 𝑃𝑜𝑢𝑡 = Pr log2 (1 + 𝜒) < 𝑅𝐵𝑇 ℎ = Pr 𝜒 < 𝜒𝑇 ℎ , (18)
As can be seen, when
𝐺𝛺𝑣 𝛾𝑣
⟶ ∞ and
𝐺𝛺𝑟 𝛾𝑟
⟶ ∞, (21) and (22)
converge to the log-normal distribution. 3.2. PDF of Gamma–Gamma distributed scattered path
𝛥
where 𝜒𝑇 ℎ = 2𝑅𝐵𝑇 ℎ − 1 denotes the outage threshold. It is worth noting that the outage probability defined in (18) is independent of the transmission technique and coding strategy for a given outage threshold. The Málaga distributed turbulence results in SNR fluctuation, and depending on the turbulence strength, the receiver may experience an outage. With the derived PDF of 𝜒 in (14) and using the identity [24, eq. (26)] in (18), the closed-form expression for the outage probability is derived as (19). For proof of (19), please see Appendix. 𝑃𝑜𝑢𝑡
×
𝐴 𝐴 = 𝑣 𝑟 4
𝛽𝑟 ∑ 𝑘=1
[( ×
( 𝑎𝑟𝑘
( )− 𝛽𝑣 ⎛∑ ( ) 𝛼𝑣 𝛽𝑣 ⎜ 𝑎𝑣𝑘 𝛼𝑣 𝑘 ⎜𝑘=1 𝛾𝑣 𝛽𝑣 + 𝛺 ′ ⎝ (
𝛼𝑟 +𝑘
) 4 𝜒𝑇 ℎ 𝜉 )2 √
𝛼𝑟 𝛽𝑟 𝛾𝑟 𝛽𝑟 + 𝛺′
(
𝛼𝑣 +𝑘 2
)
)
As defined in [26], (7) can be written as 2 | 𝐶 𝐺 | 𝐼𝑣 = |𝐼𝐿𝑣 + 𝐼𝑆𝑣 + 𝐼𝑆𝑣 | exp (2𝛥) = 𝑌𝑣 𝑋𝑣 | | { 2 | 𝐶 + 𝐼 𝐺 | (small-scale fluctuations) , 𝑌𝑣 = |𝐼𝐿𝑣 + 𝐼𝑆𝑣 | 𝑆𝑣 | | 𝑋𝑣 = exp (2𝛥) (large-scale fluctuations)
where the PDFs of 𝑌𝑣 and 𝑋𝑣 can respectively be given as ( )𝛽𝑣 ) ( ( ) 𝛾𝑣 𝛽𝑣 𝑦𝑣 1 exp − 𝑓𝑌𝑣 𝑦𝑣 = 𝛾𝑣 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′ 𝛾𝑣 ( ) ′ 𝑦𝑣 𝛺𝑣 × 1 𝐹1 𝛽𝑣 ; 1; ( ) , 𝛾𝑣 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′
⎞ ⎟ ⎟ ⎠
2,1 𝐺1,3
(23)
(19)
(24)
and
] | 2−𝛼𝑟 −𝑘 𝜒𝑇 ℎ || 2 . 𝜉 || 𝛼𝑟 −𝑘 , − 𝛼𝑟 −𝑘 , − 𝛼𝑟 +𝑘 2 2 | 2
( ) 𝑓𝑋𝑣 𝑥𝑣 =
𝛼
( ) 𝛼𝑣 𝑣 ( ) 𝑥𝛼𝑣 −1 exp −𝛼𝑣 𝑥 . 𝛤 𝛼𝑣
(25)
where 1 𝐹1 (⋅; ⋅; ⋅) in (24) represents the Kummer confluent hypergeometric function of the first kind. It can be seen that the PDF of the large-scale fluctuations, 𝑋, follows the Gamma distribution. Following the steps in [16], the moment generating function (MGF) of 𝑌𝑣 and 𝑋𝑣 can readily be obtained as
3. Derivation of existing distribution models from the derived PDF In this section, we derive the conditions to generate the log-normal and Gamma–Gamma distribution from the proposed statistical model. Considering an NLOS UV link, we present the conditions for the PDFs derived in [4] and [13]. In [4], a log-normal distribution was assumed for each scattered path. Similarly, the marginal PDF of the received irradiance was derived in [13] when the fading statistics of each scattered path follow a Gamma–Gamma distribution.
[ ] ( ( ) 𝑀𝑌𝑣 (𝑠) = 𝓁 𝑓𝑌𝑣 𝑦𝑣 ; −𝑠 =
𝛾𝑣 𝛽𝑣 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′
)𝛽𝑣
( )𝛽 −1 1 − 𝛾𝑣 𝑠 𝑣
( 1−
𝛺𝑣′ 𝛾𝑣 𝛽𝑣 +𝛺𝑣′
− 𝛾𝑣 𝑠
) 𝛽𝑣 , (26)
3.1. PDF of log-normal distributed scattered path
and 𝛼 [ ] ( ) 𝛼𝑣 𝑣 𝑀𝑋𝑣 (𝑠) = 𝐿 𝑓𝑌𝑣 𝑦𝑣 ; −𝑠 = ( )𝛼 , 𝛼𝑣 − 𝑠 𝑣
Following (4) and (5) with 𝜌𝑣 set to zero, and assuming the variance of |𝐼𝐿,𝑣 |, that is, Var[|𝐼𝐿,𝑣 |]√ = 0, 𝐼𝐿,𝑣 will converge into a random 𝛺𝑣 . As can be seen, if 𝐸[𝛥] = 0, and variable with 𝐸[|𝐼𝐿,𝑣 |] = Var[𝛥] and Var[𝜍] are set to 0, then (7) can be written in terms of Born approximation as [15] ( ) |√ | 𝐼𝑣 = | 𝐺𝛺𝑣 exp 𝑗𝜙𝐴 + 𝐼𝑆 | . (20) | | As defined in Section 2, 𝐼𝑆 is a complex random variable with circular Gaussian distribution and owing to the fact that 𝜌𝑣 = 0, it follows that 𝐸[|𝐼𝑆 |2 ] approaches 2𝑏0 = 𝛾𝑣 . It yields the PDF of the optical power arriving in the common volume to follow the modified Rice–Nakagami distribution written as ) [ ] ( √ 2 𝐺𝛺𝑣 𝐼𝑣 ( ) 𝐺𝛺𝑣 + 𝐼𝑣 1 𝑓𝐼𝑣 𝐼𝑣 = exp − 𝐼0 , 𝐼𝑣 > 0, (21) 𝛾𝑣 𝛾𝑣 𝛾𝑣
(27)
where 𝓁[⋅] represents the Laplace transformation operation. Knowing the fact that 𝜌 = 1 yields 𝛾 = 0, it follows from (26) as ( )−𝛽𝑣 ( )−𝛽𝑣 𝛽𝑣 − 𝛺𝑣′ 𝑠 𝑀𝑌𝑣 (𝑠)lim 𝛾 →0 = 𝛺𝑣′ . (28) 𝑣 𝛺𝑣′ 𝛽𝑣 Applying the limit 𝛺𝑣′ ⟶ 1 into (28) yields ( 𝑀𝑌𝑣 (𝑠) =
1−
𝑠 𝛽𝑣
)−𝛽𝑣 .
(29)
(29) denotes the MGF of a Gamma function. Therefore, we can state that, under the limiting conditions of 𝜌𝑣 = 1, 𝛾𝑣 = 0, and 𝛺𝑣′ ⟶ 1, the distribution of small-scale fluctuations, 𝑌𝑣 , converges to a Gamma distribution. From (23), (25), and (29), we can state that the unconditional PDF of the irradiance, under the given conditions, converges to a Gamma–Gamma distribution. Similarly, the same conditions can also be derived for (9). Table 1 shows the conditions for the log-normal and Gamma–Gamma distributions.
where 𝐼0 (⋅) represents the modified Bessel function of the first kind and order zero. Similarly, the conditional received irradiance at the receiver can be obtained as ) [ ] ( √ 2 𝐺𝛺𝑟 𝐼𝑟 ( ) 𝐺𝛺𝑟 + 𝐼𝑟 1 | 𝑓𝐼𝑟 𝐼𝑟 |𝐼𝑣 = exp − 𝐼0 , 𝐼𝑟 > 0. (22) 𝛾𝑟 𝛾𝑟 𝛾𝑟 4
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Optics Communications 463 (2020) 125402
Table 2 Simulation parameters for different turbulence strength.
𝛼 𝛽 𝛺 𝑏0 𝜌
Weak
Moderate
Strong
50 14 1.0621 0.0216 0.86
2.55 22 0.4618 0.6525 0.988
2.2814 33 1.33 0.4231 0.84
Fig. 4. Average bit error rate relative to the average SNR.
Fig. 2. PDF of the received irradiance fluctuations.
Fig. 5. Outage probability relative to the average SNR (𝜒𝑇 ℎ = 5 dB).
The PDF of the instantaneous SNR over different turbulence regimes is illustrated in Fig. 3. The identical NLOS UV links are assumed with 𝜉 set to 10 dB. It is interesting to note that decreasing 𝛼 causes a spreading of the PDF over a wider range of values. This spreading has a varying impact on the performance metrics derived from the PDF of the instantaneous received SNR for the NLOS UV link. The average bit error rate against the average received SNR over Málaga distributed turbulence from weak scintillations to strong fluctuations is illustrated in Fig. 4. As expected, the channel becomes more impaired as the effective number of large-scale cells reduces and results in a higher bit error rate. The outage probability against the average received SNR is illustrated in Fig. 5. As revealed from the results, under the stronger turbulence regime, a slow change in the slope of outage probability curve can be observed for lower SNR. It is interesting to compare the proposed statistical model with the log-normal and Gamma–Gamma distributions to validate the use of the proposed PDF model in practical NLOS UV applications under a wide variety of turbulence fading. Assuming a plane wave propagation, we consider the two existing PDF models, i.e., [4] and [13], for comparison. In [4], a log-normal distribution was assumed for each scattered path. Similarly, the marginal PDF of the received irradiance was derived in [13] when the fading statistics of each scattered path follow Gamma–Gamma distribution. In Fig. 6a, the normalized PDF of the received irradiance is plotted on a log-scale as a function of the received irradiance for the weak turbulence condition. For the performance comparison, we also plot the log-normal PDF obtained from [4] and Gamma–Gamma PDF derived
Fig. 3. PDF of the instantaneous SNR.
4. Results and discussions In this section, we present the received irradiance distribution and SNR distribution for an NLOS UV link, covering a range of turbulence strengths that extends from weak to strong irradiance fluctuations. We assume a plane wave propagation. For the sake of brevity, we set {𝛼𝑣 , 𝛽𝑣 , 𝜌𝑣 , 𝛾𝑣 , 𝛺𝑣 } = {𝛼𝑟 , 𝛽𝑟 , 𝜌𝑟 , 𝛾𝑟 , 𝛺𝑟 } = {𝛼, 𝛽, 𝜌, 𝛾, 𝛺} with values of 𝛼 representing different turbulence regimes. The simulation parameters are provided in Table 2 [16]. 𝜙𝑡 and 𝜙𝑟 are set to 𝜋∕180 and 𝜋∕3, respectively. 𝜓𝑡 and 𝜓𝑟 are set to 𝜋∕4 and 𝜋∕3, respectively. The baseline distance 𝑑 is set to 1000 m. All these angles are measured in radians. 𝐷 is set to 1.5. The outage threshold 𝜒𝑇 ℎ is set to 5 dB. Fig. 2 shows the PDF of the irradiance for an NLOS UV link over Málaga distribution, covering weak scintillations to strong fluctuations. It is observed that as the effective number of large-scale cells (𝛼) reduces, the turbulence effects become more significant. The results demonstrate that by decreasing 𝛼, the received irradiance variance increases, thus indicating a stronger turbulence effect. 5
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Optics Communications 463 (2020) 125402
condition, we find that the Gamma–Gamma model provides a better fit to the Málaga distribution than does the log-normal PDF. As shown in Fig. 6c, when we also consider the strong turbulence fading to make a comparison with the Gamma–Gamma distributed fading including inner scale effects, it is found to provide a better fit. The results shown in Fig. 6 demonstrate the high accuracy of the proposed statistical model. It can also be seen that the Gamma–Gamma PDF provides a better fit to the Málaga distribution than does the log-normal PDF derived in [4]. 5. Conclusion In this paper, a novel statistical model for an NLOS UV link over Málaga distributed irradiance fluctuation has been presented. We have developed a comprehensive and unified analytical model to evaluate the performance of the NLOS UV link. The performance impairment caused by the spreading of the PDF of the instantaneous received SNR for NLOS path has been investigated. It is shown that the performance degradation is significant when the effective number of large-scale cells of the scattering process reduces. CRediT authorship contribution statement Sudhanshu Arya: Conceptualization, Methodology, Writing- original draft. Yeon Ho Chung: Supervision, Validation, Writing - review & editing. Acknowledgment This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Republic of Korea (2018R1D1A3B07049858). Appendix
A.1. Proof of (10) ) ( ( ) The joint PDF of 𝐼𝑣 and 𝐼𝑟 is 𝑓𝐼𝑣 ,𝐼𝑟 (𝐼𝑣 , 𝐼𝑟 ) = 𝑓𝐼𝑟 𝐼𝑟 ||𝐼𝑣 𝑓𝐼𝑣 𝐼𝑣 . Therefore, the PDF of the received irradiance can readily be obtained as ∞ ( ) ( ) 𝑓𝐼𝑟 𝐼𝑟 = 𝑓𝐼𝑣, 𝐼𝑟 𝐼𝑣, 𝐼𝑟 𝑑𝐼𝑣 . (30) ∫0 Substituting (4) and (9) into (30) and using the identities [24, eq. (14)] with some mathematical manipulations, we obtained (31). ( √ ) ) ( 𝛽𝑟 𝛼𝑟 +𝑘 ∑ −1 ( ) 𝛼𝑟 𝛽𝑟 𝐼𝑟 2 𝐾𝛼𝑟 −𝑘 2 𝑓𝐼𝑟 𝐼𝑟 = 𝐴𝑣 𝐴𝑟 𝑎𝑟𝑘 𝐼𝑟 𝛾𝑟 𝛽𝑟 + 𝛺𝑟′ 𝑘=1 {𝛽 ( ( ) 𝛼𝑣 +𝑘 𝑣 ∞ ∑ −1 𝑎𝑣𝑘 2 2,0 (31) × 𝐼𝑣 𝐺0,2 ∫0 2 𝑘=1 [( ] )} )| − 𝛼𝑣 𝛽𝑣 | × 𝑑𝐼 . | 𝛼 −𝑘 𝛼 −𝑘 𝑣 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′ || 𝑣2 , − 𝑣2
Fig. 6. The normalized PDF of the received irradiance on the log-scale relative to the received irradiance over different turbulence fading conditions.
Since, the parameters 𝛼𝑣 , 𝛽𝑣 , and 𝛺𝑣′ for Málaga distributed fading must be real positive, the following conditions must hold true: )| | ( 𝛼 𝛽 | | 𝑣 𝑣 (32) |arg | < 𝜋, | 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′ || | and | 𝛼𝑣 − 𝑘 | 𝛼𝑣 + 𝑘 | | (33) | 2 |< 2 . | |
in [13]. As a typical weak turbulence fading, we use a Rytov variance 𝜎12 = 0.1. The inner scale value 𝑙0 is set to 0.5𝑅𝐹 , where 𝑅𝐹 represents the scale size of the Fresnel zone [27]. For the sake of brevity, we set the parameters 𝑎𝑣 = 𝑎𝑟 = 17.3 and 𝑏𝑣 = 𝑏𝑟 = 17.3, where the parameters 𝑎𝑣 , 𝑎𝑟 , 𝑏𝑣 , and 𝑏𝑟 describe the Gamma–Gamma model as described in [13]. For all the results, (𝜙𝐴 − 𝜙𝐵 ) is set to 𝜋∕2. As can be seen, under the weak turbulence condition, the proposed PDF generally fits the values predicted by the log-normal and Gamma–Gamma distribution model. From Fig. 6b, as a typical moderate turbulence
Finally, by knowing the conditions (32) and (33) and using the identity [28, eq. (7.811.4)] with some mathematical manipulations, we ( ) obtain the closed-form expression of 𝑓𝐼𝑟 𝐼𝑟 as shown in (10). 6
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A.2. Proof of (19)
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Substituting (15) in (18) and using the identity [24, eq. (14)] with variable transformation with 𝜒 = 𝑍 2 , (18) can readily be written as ( )− 𝛽𝑣 ( ) 𝛼𝑣 𝛽𝑣 𝐴 𝐴 ⎛∑ 𝑃𝑜𝑢𝑡 = 𝑣 𝑟 ⎜ 𝑎𝑣𝑘 𝛼𝑣 𝑘 4 ⎜𝑘=1 𝛾𝑣 𝛽𝑣 + 𝛺𝑣′ ⎝ ) ( √ ( ) 𝛼𝑟 +𝑘 𝛽𝑟 𝜒𝑇 ℎ ∑ 𝛼𝑟 +𝑘 4 1 2,0 × 𝑎𝑟𝑘 𝑍 4 −1 𝐺0,2 ∫0 𝜉 𝑘=1 [( ] ) − 𝛼𝑟 𝛽𝑟 𝑍 || × √ || 𝛼𝑟 −𝑘 𝑘−𝛼𝑟 𝑑𝑍. 𝛾𝑟 𝛽𝑟 + 𝛺𝑟′ 𝜉| 2 , 2
(
𝛼𝑣 +𝑘 2
)
⎞ ⎟ ⎟ ⎠ (34)
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