Asymptotic BER analysis of FSO with multiple receive apertures over ℳ -distributed turbulence channels with pointing errors

Liang Yang; Mazen Omar Hasna; Xiqi Gao

doi:10.1364/OE.22.018238

1. Introduction

Recently, free-space optical (FSO) communication systems have received considerable attention due to their high capacity ability. Operating over the unlicensed optical spectrum, FSO techniques can provide high speed line-of-sight wireless transmission with low cost and good security [1–3].

To combat the effect of atmospheric turbulence on the system performance, the concept of conventional radio frequency (RF) multiple-input multiple-output (MIMO) has been applied to the FSO system [4–12]. The basic idea is to employ multiple apertures at both transmitter and receiver and create additional spatial degrees of freedom, which in turn enhances the quality of service (QoS) and increases the coverage. For instance, the information theoretic bounds for MIMO FSO links was first studied in [4] where they assume intensity-modulation/direct-detection (IM/DD) and log-normal turbulence channels. The authors in [5] investigated the outage probability performance of FSO systems with spatial diversity at both sides over log-normal fading channels. Later, bit-error rate (BER) performance expressions for MIMO FSO links for log-normal and K atmospheric turbulence channels were derived in [6] and [7], respectively. A comparison between the modified orthogonal space-time block coding (OSTBC) and repetition coding for FSO communication was studied in [8]. An iterative maximum-likelihood sequence detection method for MIMO optical wireless communications was proposed in [9]. The diversity-multiplexing trade-off in coherent FSO systems was analyzed in [10]. Circular MIMO FSO nodes with transmit selection and receive generalized selection diversity was investigated in [11]. More recently, the error rate performance comparison of coherent and sub-carrier intensity modulated optical wireless communications was analyzed in [12]. Here, they assume that only the receiver has multiple apertures and different combining schemes are used at the receiver.

Over the years, there have been many models used to describe the atmospheric turbulence for different degrees of strength, such as log-normal distribution, K distribution, gamma-gamma distribution and generalized ℳ -distribution [13].

Misalignment between transmitter and receiver due to building sway results in vibrations of the transmitted beam, and then the resulting pointing errors affect the performance of FSO links. The joint effects of turbulence and misalignment have been investigated in [14–18]. More specifically, in [14], the authors considered log-normal distributed and gamma-gamma distributed turbulence and the performance metrics are outage probability and capacity. Later, the average bit-error rate (BER) performance of FSO links over K-distributed turbulence was analyzed in [15]. Recently, performance analysis of optical wireless communications with pointing errors over ℳ -distributed fading was presented in [16,17].

For the MIMO or only receive diversity FSO, to the best of the authors’ knowledge, the references [4–12] assumed log-normal or K-distributed or gamma-gamma distributed fading channels. Also, they did not take the pointing errors into consideration. In realistic propagation environment, pointing errors caused by misalignment may result in the degradation of FSO links as well as atmospheric turbulence. In this letter, we extend the work in [12] to generalized ℳ distributed turbulence channels with pointing errors and investigate the joint effects on the system performance. As mentioned in [16,17,19,20], the ℳ -distributed model includes K, gamma-gamma, and negative exponential distributions. In particular, we consider two aperture combining schemes: optimal combining (OC) and selection combining (SC). In our work, we adopt the subcarrier intensity modulation (SIM) scheme [12,16,21] which is another attractive alternative to on-off keying (OOK) with IM/DD. To obtain some explicit insights into the effects of the related channel parameters on the system performance, we focus on the asymptotic performance analysis. Starting from the statistic analysis, we obtain the cumulative density function (CDF) of the signal-to-noise ratio (SNR) and then use the CDF-based method to analyze the asymptotic BER performance.

2. System and channel models

A. System model

Consider a MIMO FSO system with one transmit and M receive apertures as shown in Fig. 1. We assume that the signals are transmitted through the turbulence-induced fading channels and are corrupted at the receiver by the additive white Gaussian noise (AWGN). Assuming a SIM modulation, the transmitted optical power at the laser transmitter can be written as [21]

P (t) = P_{S} (1 + η_{S} x (t))

where P_S is the total transmit optical power at the transmitter, x(t) is the transmitted RF base-band signal with unit energy and which is used to modulate the irradiance of a continuous optical wave beam, and η_S is the electrical-to-optical conversion coefficient. Therefore, the received signal at the m receive aperture is given by [21]

y_{m} = {RA}^{'} (1 + η_{S} x (t)) I_{m} + n_{m}

where n_m is the AWGN at the mth receive aperture, R is the photodetector responsivity, A′ is the photodetector area, and I_m is the normalized channel fading coefficient between the mth receive aperture and the transmit aperture and represents the intensity fluctuations due to the joint effects of atmospheric turbulence and misalignment. As shown in [17], the channel state is assumed to be a product of two random components, i.e.,

I_{m} = I_{m}^{a} I_{m}^{p}

, where

I_{m}^{a}

is the attenuation due to atmospheric turbulence (modeled as a ℳ -distributed random variable) and

I_{m}^{p}

is the attenuation due to pointing errors.

Fig. 1 System model of a FSO system with multiple receive apertures.

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At the receiver, a photodetector converts its received optical signal into an electrical signal. Then, an electrical demodulator can be used to recover the transmitted data. If optimal combining is used, the resulting instantaneous SNR at the input of the electrical demodulator can be expressed as

γ_{OC} = \bar{γ} \sum_{m = 1}^{M} I_{m}^{2}

where γ̄ is defined as the average received SNR. If we apply the SC scheme, the selection policy is m^* = arg max I_m and the corresponding SNR is

γ_{SC} = \bar{γ} I_{m^{*}}^{2} .

B. Channel statistical model

When taking the combined effects of turbulence-induced scintillation and misalignment-induced fading into consideration, the combined unconditional probability density function (PDF) for generalized atmospheric optical channels has been derived in [17]. However, this PDF involves the Meijer G-function and makes the closed-form analysis untractable. Therefore, we intend to rewrite the PDF expression in terms of the generalized power series representation.

As derived in [17], the combined PDF of the irradiance I_m is given by

f_{I_{m}} (I) = \frac{g^{2} A}{2} I^{- 1} \sum_{k = 1}^{β} a_{k} {(\frac{α β}{μ β + Ω^{'}})}^{- \frac{α + k}{2}} G_{1, 3}^{3, 0} (\frac{α β}{μ β + Ω^{'}} {\frac{I}{A_{0}} |}_{g^{2}, α, k}^{g^{2} + 1})

where,

μ = E [{| U_{S}^{C} |}^{2}] = 2 b_{0} (1 - ρ)

, α is a positive parameter depending on the effective number of large-scale cells of the scattering process, β denotes the amount of fading parameter and is a natural number, while

Ω^{'} = Ω + 2 ρ b_{0} + 2 \sqrt{2 ρ b_{0} Ω} cos (φ_{A} - φ_{B})

represents the average power coming from the coherent contribution. Notice that the parameter ρ (0 ≤ ρ ≤ 1) denotes the amount of scattering power coupled to the line of sight (LOS) component, the parameters φ_A and φ_B are the deterministic phases of the LOS and the coupled-to-LOS component. In Eq. (4),

G_{p, q}^{m, n} (\cdot)

is the Meijer G-function, and the parameters A and a_k are defined as

A = \frac{2 α^{\frac{α}{2}}}{μ^{1 + \frac{α}{2}} Γ (α)} {(\frac{μ β}{μ β + Ω^{'}})}^{β + \frac{α}{2}}

a_{k} = (\begin{matrix} β - 1 \\ k - 1 \end{matrix}) \frac{{(μ β + Ω^{'})}^{1 - 0.5 k}}{(k - 1)!} {(\frac{Ω^{'}}{μ})}^{k - 1} {(\frac{α}{β})}^{0.5 k}

Furthermore, in Eq. (4), g = w_zeq/(2σ_s) is the ratio between the equivalent beam radius and the pointing error displacement standard deviation at the receiver σ_s, and A₀ is the fraction of the collected power at r = 0 (r radial distance) and the parameter w_zeq can be calculated by using the relations,

v = (\sqrt{π} a) / (\sqrt{2} w_{z})

and

w_{zeq} = w_{z}^{2} \sqrt{π} erf (v) / (2 v e^{- v^{2}}

, where w_z is the beam waist at distance z, a is the radius of a circular detection aperture, and erf() is the error function (See [16,17] for more details). As discussed in [13], the gamma-gamma and K distributions are special cases of the ℳ -distribution model, for example, (ρ = 1, Ω′ = 1), (ρ = 0, Ω = 0 or β = 1), and (ρ = 0, Ω = 0 or α → ∞) corresponding to gamma-gamma, K distribution, and negative exponential model, respectively.

From [22, Eq. (07.34.06.0002.01)], the Meijer G-function can be expressed in terms of the generalized power series representation as

\begin{array}{l} G_{p, q}^{m, n} (z |_{b_{1}, \dots, b_{m}, \dots b_{q}}^{a_{1}, \dots, a_{n}, \dots, a_{p}}) = \sum_{t = 1}^{m} \frac{Π_{j = 1, j \neq t}^{m} Γ (b_{j} - b_{t})}{Π_{j = n + 1}^{p} Γ (a_{j} - b_{t})} z^{b_{t}} \\ \cdot \sum_{i = 0}^{\infty} \frac{Π_{j = n + 1}^{p} {(1 - a_{j} + b_{t})}_{i} Π_{j = 1}^{n} Γ (1 - a_{j} + b_{t} + i)}{Π_{j = 1}^{m} {(1 - b_{j} + b_{t})}_{i} Π_{j = m + 1}^{q} Γ (1 - b_{j} + b_{j} + i)} {({(- 1)}^{p - m - n} z)}^{i} \end{array}

where q > p, and (a)_n is the Pochhamer number defined as (a)_n = a(a + 1)· · · (a + n−1), with (a)₀ = 1. Using Eq. (7), we can rewrite Eq. (4) as

f_{I_{m}} (I) = \frac{g^{2} A}{2} \sum_{k = 1}^{β} a_{k} \sum_{t = 1}^{3} \frac{Π_{j = 1, j \neq t}^{3} Γ (b_{j} - b_{t})}{Γ (g^{2} + 1 - b_{t})} \sum_{i = 0}^{\infty} \frac{{(b_{t} - g^{2})}_{i} δ^{b_{t} + i - \frac{α + k}{2}}}{Π_{j = 1}^{3} {(1 - b_{j} + b_{t})}_{i} A_{0}^{b_{t} + i}} I^{b_{t} + i - 1}

where δ = αβ/(μβ + Ω′), b₁ = g², b₂ = α, and b₃ = k. From Eq. (8), the PDF of

I_{m}^{2}

can be obtained as

f_{I_{m}^{2}} (I) = \frac{1}{2} I^{- \frac{1}{2}} f_{I_{m}} (\sqrt{I})

3. Performance analysis

In this section, we analyze the BER performance of the above two FSO systems over generalized turbulence fading channels with pointing errors. More specifically, assuming BPSK modulation, we use the CDF-based method to evaluate the BER performance [23], namely,

P_{e} = E_{X} [F (\frac{X^{2}}{2})]

where X is a standard normal-distributed random variable and F(·) is the CDF expression.

A. BER analysis for OC

From Eq. (10), to compute the BER, we need to find the CDF of γ_OC. Combining Eq. (8) and Eq. (9), we express the moment generating function (MGF) of $I_{m}^{2}$ as

\begin{array}{l} M_{I_{m}^{2}} (s) = \frac{g^{2} A}{4} \sum_{k = 1}^{β} a_{k} \sum_{t = 1}^{3} \frac{Π_{j = 1, j \neq t}^{3} Γ (b_{j} - b_{t})}{Γ (g^{2} + 1 - b_{t})} \\ \cdot \sum_{i = 0}^{\infty} \frac{{(b_{t} - g^{2})}_{i} δ^{b_{t} + i - \frac{α + k}{2}}}{Π_{j = 1}^{3} {(1 - b_{j} + b_{t})}_{i} A_{0}^{b_{t} + i}} Γ (\frac{b_{t} + i}{2}) s^{- \frac{b_{t} + i}{2}} \end{array}

Let

Y = Σ_{m = 1}^{M} I_{m}^{2}

. Since Y is the sum of M independent random variables, the MGF of Y is readily given by

M_{Y} (s) = {[M_{I_{m}^{2}} (s)]}^{M}

If we use Eq. (11), we note that the expansion in Eq. (12) will be very complicated. Therefore, in this paper, we focus on the asymptotic analysis. Let i = 0, Eq. (11) simplifies to

M_{I_{m}^{2}} (s) \to \frac{g^{2} A}{4} [A (1, g^{2}) s^{- \frac{g^{2}}{2}} + A (2, α) s^{- \frac{α}{2}} + A (3, k) s^{- \frac{k}{2}}]

where the constants A(1, g²), A(2, α), and A(3, k) are given by

A (t, x) = \sum_{k = 1}^{β} a_{k} \frac{Π_{j = 1, j \neq t}^{3} Γ (b_{j} - x)}{Γ (g^{2} + 1 - x)} \frac{δ^{x - \frac{α + k}{2}}}{A_{0}^{x}} Γ (\frac{x}{2})

By substituting Eq. (13) into Eq. (12), the asymptotic MGF of Y can be expressed as

M_{Y} (s) \to \frac{{(g^{2} A)}^{M}}{4^{M}} \sum_{r = 0}^{M} (\begin{matrix} M \\ r \end{matrix}) \sum_{t = 0}^{M - r} {(A (1, g^{2}))}^{M - r - t} {(A (2, α))}^{t} \sum_{v = r}^{r β} λ_{v} s^{- \frac{(M - r - t) g^{2} + t α + v}{2}}

where λ_v is the coefficient of

s^{- \frac{v}{2}}

in the expansion of

{(A (3, k) s^{- \frac{k}{2}})}^{r}

. Using the inverse Laplace transform, we can obtain the CDF of γ_OC as

\begin{matrix} F_{γ_{OC}} (γ) \to \frac{{(g^{2} A)}^{M}}{4^{M}} \sum_{r = 0}^{M} (\begin{matrix} M \\ r \end{matrix}) \sum_{t = 0}^{M - r} {(A (1, g^{2}))}^{M - r - t} {(A (2, α))}^{t} \\ \cdot \sum_{v = r}^{r β} \frac{λ_{v}}{Γ (\frac{M - r - t) g^{2} + t α + v}{2} + 1)} {(\frac{γ}{\bar{γ}})}^{\frac{(M - r - t) g^{2} + t α + v}{2}} \end{matrix}

Substituting Eq. (15) into Eq. (10) yields

\begin{array}{l} P_{e}^{OC} & \to \frac{{(g^{2} A)}^{M}}{2^{2 M + 1} \sqrt{π}} \sum_{r = 0}^{M} (\begin{matrix} M \\ r \end{matrix}) \sum_{t = 0}^{M - r} {(A (1, g^{2}))}^{M - r - t} {(A (2, α))}^{t} \\ \cdot \sum_{v = r}^{r β} \frac{λ_{v} Γ (\frac{(M - r - t) g^{2} + t α + v + 1}{2})}{Γ (\frac{(M - r - t) g^{2} + t α + v}{2} + 1)} {\bar{γ}}^{- \frac{(M - r - t) g^{2} + t α + v}{2}} \end{array}

For large SNR, the diversity order is determined by the smallest exponent of γ̄. Therefore, we can conclude that the diversity order of a FSO system with multiple receive apertures over generalized fading channels with pointing errors and OC is

M (min {\frac{g^{2}}{2}, \frac{α}{2}, \frac{β}{2}})

.

B. BER analysis for SC

From Eq. (13), the CDF of $γ_{m} = \bar{γ} I_{m}^{2}$ can be obtained as

F_{γ_{m}} (γ) \to \frac{g^{2} A}{4} [\frac{A (1, g^{2})}{Γ (\frac{g^{2}}{2} + 1)} {(\frac{γ}{\bar{γ}})}^{\frac{g^{2}}{2}} + \frac{A (2, α)}{Γ (\frac{α}{2} + 1)} {(\frac{γ}{\bar{γ}})}^{\frac{α}{2}} + \frac{A (3, k)}{Γ (\frac{k}{2} + 1)} {(\frac{γ}{\bar{γ}})}^{\frac{k}{2}}]

For independent and identically distributed γ_m, the CDF of γ_SC is given by

\begin{array}{l} F_{γ_{SC}} (γ) & = {[F_{γ_{m}} (γ)]}^{M} \\ \to \frac{{(g^{2} A)}^{M}}{4^{M}} \sum_{r = 0}^{M} (\begin{matrix} M \\ r \end{matrix}) \sum_{t = 0}^{M - r} {(\frac{A (1, g^{2})}{Γ (\frac{g^{2}}{2} + 1)})}^{M - r - t} \\ \cdot {(\frac{A (2, α)}{Γ (\frac{α}{2} + 1)})}^{t} \sum_{v = r}^{r β} χ_{v} {(\frac{γ}{\bar{γ}})}^{\frac{(M - r - t) g^{2} + t α + v}{2}} \end{array}

where χ_v is the coefficient of

{(\frac{γ}{\bar{γ}})}^{\frac{v}{2}}

in the expansion of

{(\frac{A (k)}{Γ (\frac{k}{2} + 1)} {(\frac{γ}{\bar{γ}})}^{\frac{k}{2}})}^{r}

. Then, combining Eq. (10) and Eq. (18), we have

\begin{matrix} P_{e}^{SC} (γ) \to \frac{{(g^{2} A)}^{M}}{2^{2 M + 1} \sqrt{π}} \sum_{r = 0}^{M} (\begin{matrix} M \\ r \end{matrix}) \sum_{t = 0}^{M - r} {(\frac{A (1, g^{2})}{Γ (\frac{g^{2}}{2} + 1)})}^{M - r - t} \\ \cdot {(\frac{A (2, α)}{Γ (\frac{α}{2} + 1)})}^{t} \sum_{v = r}^{r β} χ_{v} \frac{Γ (\frac{(M - r - t) g^{2} + t α + v + 1}{2})}{{\bar{γ}}^{\frac{(M - r - t) g^{2} + t α + v}{2}}} \end{matrix}

We can see that the diversity order for SC scheme is also

M (min {\frac{g^{2}}{2}, \frac{α}{2}, \frac{β}{2}})

.

4. Numerical results

In this section, we illustrate our analysis with some numerical results for the channel model under consideration. For the FSO networks, we note that the current references seldom present the simulation results. In this work, we provide the simulation results to verify our analysis. For the BER performance, we assume that BPSK is used.

The BER performance for different values of g, α, and β is plotted in Fig. 2, where we set M = 3. It should be noted that α = 3.6, β = 1, and ρ = 0, and α = 4.2, β = 2, and ρ = 1 correspond to the K and gamma-gamma distributions, respectively. It is clearly shown that the asymptotic results approach the exact values at high SNR regions. As expected, we can observe that increasing the channel parameters α, β, and g can improve the system performance. Furthermore, we can see that the optimal combining has a better performance than the selection scheme. However, for given α, β, and g, OC and SC have the same diversity order since the slopes of BER curves for SC and OC are the same. In Fig. 3, we plot the curves for different pairs of M and g for α = 4.2, β = 2, and ρ = 1. It is clearly shown that increasing M can improve the system performance. Also, the increasing of g can improve the BER performance significantly because g → ∞ means no pointing errors.

Fig. 2 BER of a FSO network with multiple receive apertures over ℳ -distributed channels with pointing errors. M = 3

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Fig. 3 Simulation of BER of a FSO network with multiple receive apertures over ℳ -distributed channels with pointing errors. α = 4.2, β = 2, and ρ = 1.

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5. Conclusions

In this letter, we analyzed the asymptotic BER performance of a FSO system with multiple receive apertures over ℳ -distributed channels with pointing errors. Results show that the diversity order for both OC and SC are $M min {\frac{g^{2}}{2}, \frac{α}{2}, \frac{β}{2}}$

Acknowledgments

The authors would like to acknowledge the support of Ooredoo under the project QUEX-Qtel-09/10-10, the National Natural Science Foundation of China (NSFC) under Grants 61372096 and 61320106003, and the Open Research Fund of the State Key Laboratory of Integrated Services Networks ( ISN15-07).

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Asymptotic BER analysis of FSO with multiple receive apertures over ℳ -distributed turbulence channels with pointing errors

Abstract

1. Introduction

2. System and channel models

A. System model

B. Channel statistical model

3. Performance analysis

A. BER analysis for OC

B. BER analysis for SC

4. Numerical results

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (3)

Equations (20)

Optics Express