Unified Statistical Performance of FSO Link due to the Combined Effect of Weak Turbulence and Generalized Pointing Error with HD and IM/DD

1. Introduction

FREE -space optical communication (FSO) has been proposed as an attractive alternative to radio frequency communication in the sense that it provides wide bandwidth and high capacity without requirement of license. However, the scalability of FSO link is limited by pointing error, atmospheric turbulence, and loss [2]. Especially, when it comes to the FSO link between moving platforms, it is imperative works to analyze the statistical channel model considering accurate pointing errors and atmospheric turbulence at the same time [3].

The pointing error represents the displacement between the laser beam and the aperture of detector, and it can be analyzed by the horizontal and vertical displacements where both follow Gaussian distributions. Pointing error is composed of boresight and jitter. The boresight is the fixed displacement between each center while the jitter is random offset [4].

The performance analysis of pointing error combined with weak turbulence has been studied in [4]–[10]. Authors in [5] derived composite channel model considering pointing error and turbulence and optimized the beamwidth to maximize the capacity. A unified capacity expression that accounts for intensity modulation and direct detection (IM/DD) and heterodyne detection (HD) under the weak turbulence was derived in [6]. In [7]–[9], bit error rate (BER) and outage probability were investigated under weak turbulence with the effect of pointing error using M-ary amplitude shift keying modulation, subcarrier intensity modulation differential phase shift keying (SIMDPSK), and direct current biased optical orthogonal frequency division multiplexing (DCO-OFDM) respectively. However, the research groups in [6]–[9] assumed that the pointing error model follows Rayleigh distribution and utilized the composite probability density function (PDF) derived in [5]. Accordingly, the results represent only one specific pointing error model and the channel model is not suitable for describing various types of pointing errors described in [11]. In [4], authors derived composite PDF considering more general pointing error model with nonzero boresight and conducted performance analysis. Also, authors in [10] considered the nonzero boresight pointing error with Malaga modeled turbulence which is suitable for weak to strong turbulence conditions and derived the average BER expression for IM/DD FSO link. Nevertheless, the derivation of composite PDF for log-normal turbulence is ambiguous and the results in [4], [10] are limited to one pointing error model with IM/DD.

To the best of our knowledge, no studies have considered various types of pointing error model with weak turbulence. Accordingly, the BER analysis of both IM/DD and HD based on various pointing error models under weak turbulence has not been conducted. For the strong turbulence as Gamma Gamma fading, the statistical analysis with various pointing error models was dealt in [12]. In this paper, FSO links between ships or between shore and ship are considered and the effects of typical ship movements are analyzed. Then, by applying it to the pointing error models described in [11], we derive unified composite PDF results combining pointing error and weak atmospheric turbulence. Utilizing the unified composite PDF, we investigate the BER performance of FSO system based on both IM/DD and HD.

The rest of this paper is organized as follows. Section II presents a statistical model for weak atmospheric turbulence and introduces various pointing error models of moving platforms suggested in [11]. In Section III, we statistically obtain the composite PDF results for the pointing error models based on the weak turbulence model, and then with these results, we derive them as the unified composite PDF expression. Then, with the unified composite PDF result, we analyze the BER performance

in FSO communication based on IM/DD and HD in Section IV. In Section V, we describe the simulation setup and then crossverify these analytical results with the Monte-Carlo simulation results under various pointing error models and weak turbulence conditions. Finally, we conclude our paper in Section VI.

II. SYSTEM AND CHANNEL MODEL

Assuming that the transmitted signal x is distorted by channel gain h, and additive white Gaussian noise (AWGN) n with variance [TeX:] $$N_{0},$$ we can express the received signal y as

where [TeX:] $$\eta_{e}$$ is the effective photo-electric conversion ratio. In a typical FSO link, the channel gain h can be modeled as h = [TeX:] $$h_{l} h_{a} h_{p}$$ , where [TeX:] $$h_{l}$$ is the atmospheric loss factor, [TeX:] $$h_{a}$$ is the atmospheric fading factor, and[TeX:] $$h_{p}$$ is the pointing error factor. Note that [TeX:] $$h_{l}$$ is deterministic and both [TeX:] $$h_{a} \text { and } h_{p}$$ are random variables (RV). [TeX:] $$h_{l}$$ can be given as [TeX:] $$h_{l}=\exp (-\sigma z) \text { where } \sigma$$ is attenuation coefficient and z is link distance [13]. The statistical characteristic of random variables [TeX:] $$h_{a} \text { and } h_{p}$$ is presented below, respectively.

A. Atmospheric Turbulence

As we assume that a signal is transmitted by a plane wave in weak turbulence conditions, [TeX:] $$h_{a}$$ an be statistically modeled by lognormal fading. The PDF of [TeX:] $$h_{a}$$ is given by [14]

where [TeX:] $$\sigma_{X}^{2}$$ represents the log-amplitude variance and is approximately expressed by Rytov variance, [TeX:] $$\sigma_{R}^{2} \text { as } \sigma_{X}^{2} \approx \sigma_{R}^{2} / 4.$$ The Rytov variance is defined as [TeX:] $$\sigma_{R}^{2}=1.23 k^{7 / 6} C_{n}^{2} z^{11 / 6}$$ where k, z, and [TeX:] $$C_{n}^{2}$$ represent the optical wavenumber, the propagation distance, and the refractive index structure parameter, respectively [14]. The nth moment of [TeX:] $$h_{a}$$ is given by [11]

B. Pointing Error

Fig. 1(a) depicts main channel factors in FSO communication which are atmospheric turbulence and pointing error. Especially for the pointing error, there are two main movement factor as shown in Fig. 1(b), jitter and boresight, where the jitter is random offset of the beam center while the boresight is a fixed displacement between a detector aperture and an average beam footprint [4]. Movement of ships can be classified into rolling, yawing, and pitching as shown in Fig. 1(c) and the movements can lead to modeling the pointing error in FSO communication between ships or between shore and ship as follows. As yawing and pitching leads to displacement in only x or y axis while rolling occurs in both x and y axes, rolling can be modeled by the double sided pointing error model while yawing and pitching can be modeled by the single-sided pointing error. Further, if the ship is affected by two or more of the three main factors, the pointing error can be modeled as a double-sided error [1].

Assuming a Gaussian beam with beamwidth [TeX:] $$w_{z}$$ and detector with aperture radius of a, the fraction of the collected power can be approximated at the distance z as shown in [5]

In (4), as shown in Fig. 1(b), r is the radial displacement between the beam and the centers of detector, [TeX:] $$A_{0}=[\mathrm{erf}(v)]^{2}$$ is the fraction of collected power at r = 0 when the ratio between the beamwidth and the aperture radius is [TeX:] $$v=\sqrt{\left(a^{2} \pi / 2 w_{z}^{2}\right)},$$ and [TeX:] $$w_{z_{e q}}=\sqrt{w_{z}^{2} \frac{\sqrt{A_{0} \pi}}{2 v \exp \left(-v^{2}\right)}}$$ is the equivalent beamwidth, where [TeX:] $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}}$$ dt is the error function. The approximation in (4) is valid when [TeX:] $$w_{z}>6 a$$ which can be obtained by typical FSO communication systems [5]. Further, the radial displacement vector in Fig. 1(b) can be written as [TeX:] $$\mathbf{r}=\left[r_{x}, r_{y}\right]^{T}$$ where [TeX:] $$r_{x} \text { and } r_{y}$$ follow independent Gaussian distribution as [TeX:] $$r_{x} \sim \mathcal{N}\left(\mu_{x}, \sigma_{x}^{2}\right), \text { and } r_{y} \sim \mathcal{N}\left(\mu_{y}, \sigma_{y}^{2}\right),$$ , respectively. Then the radial displacement can be expressed as [TeX:] $$r=|\mathbf{r}|=\sqrt{r_{x}^{2}+r_{y}^{2}}.$$ According to its boresight and jitter, the pointing error models can be classified into 1) ‘nonzero boresight and identical jitter’ case, 2) ‘zero boresight and non-identical jitter’ case, and 3) ‘nonzero boresight and single jitter’ case as shown in [11]. In each case, the radial displacement r follows 1) Rician, 2) Hoyt, and 3) nonzero mean single sided Gaussian distribution, respectively and the corresponding PDF and nth momentum of [TeX:] $$h_{p}$$ are listed in Table 1 and Table 2, respectively. Utilizing them, we derive composite PDF of each case of the pointing error model with the weak turbulence and unified expression in the next section.

III. STATISTICAL ANALYSIS OF COMPOSITE PDF

A. Special Cases

The statistical characteristics of channel gain [TeX:] $$h=h_{l} h_{a} h_{p}$$ can be assumed as [5]

A.1 Rician

We substitute (5) which is PDF of [TeX:] $$h_{p}$$ in case of Rician model and (2) into (11). After some mathematical manipulation, the composite PDF under weak turbulence can be expressed as

where [TeX:] $$I_{\nu}(\cdot)$$ denotes the th-order modified Bessel function of the first kind [15, eq. (8.431.1)]. We replace the [TeX:] $$I_{0}(\cdot)$$ in (12) with series representation of [15, eq. (8.445)], and change the order of the integral and infinite summation with mathematical manipulation as

By letting [TeX:] $$y=\ln \left(A_{0} h_{l} h_{a} / h\right),$$ (13) can be written as the function of y as

Then, with the help of an integral identity in [15, eq. (3.462.1)], the series representation of the composite PDF can be written as

where [TeX:] $$A(h)=\frac{\ln \left(\frac{h}{A_{0} h_{l}}\right)+2 \sigma_{X}^{2}+4 \sigma_{X}^{2} \frac{\varepsilon_{x}^{2}+\varepsilon_{y}^{2}}{2}}{2 \sigma_{X}} \text { and } D(\cdot)$$ is parabolic cylinder function [15, eq. (9.240)]. In this case, [TeX:] $$\varepsilon_{x}=\varepsilon_{y}=\varepsilon.$$

A.2 Hoyt

By substituting (6) and (2) into (11), the composite PDF of Hoyt pointing error model case can be expressed as

With similar procedure of replacing [TeX:] $$I_{0}(\cdot)$$ with series representation, swapping the integral and infinite summation, and change of variable, we can rewrite (16) as (17) where y = ln [TeX:] $$\left(A_{0} h_{l} h_{a} / h\right)$$ as Rician pointing error model case. After applying the integral identity of [15, eq. (3.462.1)], we can derive the composite PDF for Hoyt model as

A.3 Nonzero Mean Single-sided Gaussian

Along the same line, the composite PDF for nonzero mean single-sided case can be obtained by substituting (7) and (2) into (11) as

After the similar mathematical procedure and manipulation, (19) can be expressed as (20) which includes an integral form that the integral identity of [15, eq. (3.462.1)] can be applied to solve. Finally, we can obtain the composite PDF as

B. Unified Expression

Similarity of (15), (18), and (21) allows us to unify as

where

In Table 3, we specialize the unified PDF results given in (22) through functions of B(k) for each of the pointing error models where constants are listed in Table 4.

IV. PERFORMANCE ANALYSIS

BER for a given signal-to-noise ratio (SNR) is [17]

where p defines detection mechanism (i.e., p = 1 for IM/DD and [TeX:] $$p=\frac{1}{2}$$ for HD), while q denotes index for modulation type (i.e., q = 1 for PSK and [TeX:] $$q=\frac{1}{2}$$ for FSK).

A. IM/DD

For IM/DD technique, instantaneous electrical SNR is defined as [TeX:] $$\gamma=\left(\eta_{e}^{2} h^{2} / N_{0}\right)$$ [11]. Then, the average electrical SNR can be written as [TeX:] $$\mu_{I M / D D}=\left(\eta_{e}^{2} \mathbb{E}_{h}^{2}[h] / N_{0}\right)$$ . Since [TeX:] $$h_{a} \text { and } h_{p}$$ are statistically independent processes, and [TeX:] $$h_{l}$$ is deterministic, we can express the average electrical SNR as

Substituting (3) with n = 1 and first moment of [TeX:] $$h_{p} \text { with } \varepsilon^{2} \gg 1$$ for each pointing error model into (25), we obtain

where c can be given as, respectively

The SNR , can be written in respect to [TeX:] $$\mu_{I M / D D}$$ as

By utilizing (24) with (28), and PDF of [TeX:] $$\gamma, f_{\gamma}(\gamma),$$ the average BER in case of IM/DD can be obtained as

With [16, eq. (06.06.03.0008.01)], we can transform the gamma function in (29) into exponential function and by substituting (22), (29) can be written as

Using Taylor series expansion of exponential function and swapping the integration and summation with mathematical manipulation, we can express (30) as

where

Applying change of variable, we can rewrite (32) as

where

Therefore, we can express the average BER as

where

The special function [TeX:] $$\bar{P}(\alpha(k), \beta(n))$$ can be obtained as follows. By partitioning the integration interval in (34) into [TeX:] $$[-\infty, 0] \text { and }[0, \infty],$$ we can rewrite (34) as

where [TeX:] $$\bar{P}_{-}(\alpha(k), \beta(n))$$ represents the [TeX:] $$[-\infty, 0]$$ part and [TeX:] $$\bar{P}_{+}(\alpha(k), \beta(n))$$ represents the [TeX:] $$[0, \infty]$$ part. Using Taylor series, we transform [TeX:] $$\exp \left(2 \sigma_{X}\left(\frac{\varepsilon_{x}^{2}+\varepsilon_{y}^{2}}{2}+\beta(n)\right) A\right)$$ in [TeX:] $$\bar{P}_{+}(\alpha(k), \beta(n))$$ into series form as

With integral identity [16, eq. (07.41.21.0014.01)], [TeX:] $$P_{+}(\alpha(k), \beta(n))$$ can be derived as

Utilizing [16, eq. (07.41.26.0043.01)], [TeX:] $$P_{-}(\alpha(k), \beta(n))$$ can be expressed through Meijer G function [15, eq. (9.301)] as

and with integral identity [15, eq. (7.813.2)], we can derive [TeX:] $$\bar{P}_{-}(\alpha(k), \beta(n))$$ as

B. Heterodyne Detection

The instantaneous electrical SNR is given as [TeX:] $$\gamma= \ \eta_{e} h / N_{0}$$ [11]. Similarly, we can express the average electrical SNR as [TeX:] $$\mu_{H D}=\left(\eta_{e} A_{0} h_{l} c / N_{0}\right),$$ where c is given as (27), and the instantaneous electrical SNR can be expressed in terms of [TeX:] $$\mu_{H D}$$ as

With (24) and (42), we can derive average BER for HD as

Applying series expansion of incomplete gamma function [16, eq. (06.06.06.0002.01)], (43) leads to

In a similar way as IM/DD case, (44) can also be expressed through [TeX:] $$\bar{P}(\alpha(k), \beta(n)) \text { and } \bar{\nu}\left(\beta(n) \mid \varepsilon_{x}, \varepsilon_{y}, s, \rho_{x}, \rho_{y}\right)$$ as

V. NUMERICAL RESULTS

In this section, with exact Monte-Carlo simulations, our derived results of average BER (i.e., (35) and (45)) are crossverified with simulation results. Using the parameter setting listed in Table 5, average BER curves are plotted against the average electrical SNR [TeX:] $$\mu$$ for IM/DD and HD in Figs. 2 and 3, respectively. Both Figs. 2and 3 considered the three pointing errors models (i.e., Rician, Hoyt and Nonzeromean) with parameter values in Table 6 and weak turbulence models with different Rytov variance value (i.e., [TeX:] $$\sigma_{R}^{2}=0.05 \text { and } \left.\sigma_{R}^{2}=0.2\right).$$

First, as the analytical results and simulation results match each other, we confirm that expressions (35) and (45) provide a precise evaluation. Since greater value of Rytov variance leads to stronger turbulence, it can be easily found from Figs. 2 and 3 that the BER performance of [TeX:] $$\sigma_{R}^{2}=0.2$$ is worse than that of [TeX:] $$\sigma_{R}^{2}=0.05.$$ From the parameter values in Table 6, the Rician model represents the worst pointing error model with boresight and jitter in two axes. Also, the Hoyt model describes the error caused by only jitter which gives constant movement with larger value in the y axis than x axis. Lastly for the non-zero mean model, the error occurs by only x axis with dominant value of boresight.

From Fig. 2, we find that the Rician model shows the worst error rate performance and the curve of Rician model with [TeX:] $$\sigma_{R}^{2}=0.05$$ shows worse performance than that of other pointing error models with [TeX:] $$\sigma_{R}^{2}=0.2.$$ It can be confirmed that with the parameter values listed in Table 6, the effect of Rician pointing error is greater than that of turbulence with [TeX:] $$\sigma_{R}^{2}=0.2.$$ Also, we note that as the SNR value increases up to a certain point, the BER curves of Hoyt model and the curves of non-zero mean model are reversed at that point. Therefore, we can infer that the effect of boresight is dominant compared to jitter in low SNR region, and as the SNR value increases, the effect of the jitter becomes dominant compared to that of boresight. In other words, the fixed displacement between transmitter and receiver has greater effect on BER performance in low SNR region, and the random offset has greater effect in high SNR region. Comparing Figs. 2 and 3, we can see the performance of IM/DD and that of HD are similarly affected by the pointing error and turbulence. However, it is obvious that HD shows greater error performance.

VI. CONCLUSION

In this paper, we pointed out the pointing error models that can occur in FSO communication between shore and ship or

ships. Based on the modeled results, we statistically derived the unified composite PDF containing all possible pointing error models based on weak turbulence model. In addition, we analyzed BER performance in FSO communication with IM/DD and HD technology based on the derived unified composite PDF results. Then, with exact Monte-Carlo simulations, our derived analytical results of average BER (i.e., (35) and (45)) were cross-verified with simulation results under various pointing error models and weak turbulence conditions. From some selected results, the effects of Rytov dispersion, aiming, and jitter on the BER performance were identified.