Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels

Antonio García-Zambrana; Carmen Castillo-Vázquez; Beatriz Castillo-Vázquez

doi:10.1364/OE.19.013480

1. Introduction

Optical wireless communications using intensity modulation and direct detection (IM/DD) can provide high-speed links for a variety of applications [1], providing an unregulated spectral segment and high security. Recently, the use of atmospheric free-space optical (FSO) transmission is being specially interesting to solve the “last mile” problem, as well as a supplement to radio-frequency (RF) links [2, 3]. However, atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as atmospheric scintillation, severely degrading the link performance [4, 5]. Additionally, since FSO systems are usually installed on high buildings, building sway causes vibrations in the transmitted beam, leading to an unsuitable alignment between transmitter and receiver and, hence, a greater deterioration in performance. Error control coding as well as diversity techniques can be used over FSO links to mitigate turbulence-induced fading [6–11]. In [12, 13], selection transmit diversity is proposed for FSO links over strong turbulence channels, where the transmit diversity technique based on the selection of the optical path with a greater value of irradiance has shown to be able to extract full diversity as well as providing better performance compared to general FSO space-time codes (STCs) designs, such as conventional orthogonal space-time block codes (OSTBCs) and repetition codes (RCs). The combined effect of atmospheric and misalignment fading is analyzed in the case of single-input/single-output (SISO) FSO channels [14, 15]. In [16], the effects of atmospheric turbulence and misalignment considering aperture average effect were considered to study the outage capacity for SISO links. In [17, 18] the error rate performance for uncoded and coded FSO links over strong turbulence and misalignment fading channels is studied. The capacity calculation and the analysis of error rate performance for FSO links over log-normal and gamma-gamma turbulence and misalignment fading channels is presented in [19]. In [20] the error rate performance of PPM on terrestrial FSO Links with gamma-gamma turbulence and pointing errors is analyzed. However, to the best of our knowledge, the performance of MIMO FSO channels under the effects of atmospheric and misalignment fading has only been studied in [21], assuming repetition coding on the transmitter side and equal gain combining on the receiver side. In this work [21], the study of the outage probability and diversity gain has been considered for multiple-input/multiple-output (MIMO) FSO communication systems impaired by log-normal atmospheric turbulence and misalignment fading, showing that the diversity gain is independent of the number of transmitters and receivers, being conditioned only by misalignment parameters.

In this paper, the outage probability as a performance measure for MIMO FSO communication systems using IM/DD over strong atmospheric turbulence channels with pointing errors is analyzed, assuming different configurations as repetition coding and transmit laser selection on the transmitter side and equal gain combining and selection combining on the receiver side. Novel closed-form expressions for the outage probability as well as their corresponding asymptotic expressions are presented when the irradiance of the transmitted optical beam is susceptible to either strong turbulence conditions, following a negative exponential distribution, and pointing error effects, following a misalignment fading model, as in [16–18, 21], where the effect of beam width, detector size and jitter variance is considered. In this strong turbulence FSO scenario, obtained results show that the diversity order is independent of the pointing error when the equivalent beam radius at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver, showing the same slope of the outage performance versus average signal-to-noise ratio (SNR) as in a similar FSO scenario where misalignment fading is not considered. However, different coding gain, i.e. the horizontal shift in the outage performance in the limit of large SNR, is achieved as a consequence of the severity of the pointing errors effects and MIMO configuration assumed. Additionally, a significant improvement in performance is demonstrated when MIMO FSO links based on transmit laser selection with equal gain combining are adopted. Simulation results are further demonstrated to confirm the analytical results. Here, not only rectangular pulses are considered but also on-off keying (OOK) formats with any pulse shape, corroborating the advantage of using pulses with high peak-to-average optical power ratio (PAOPR). In contrast to [21], wherein it is concluded that the diversity gain for MIMO FSO communication systems impaired by log-normal atmospheric turbulence and misalignment fading is conditioned only by misalignment parameters, it is shown in this paper that the outage diversity of MIMO FSO links over strong turbulence and misalignment fading channels is independent of pointing errors when transmitters are adequately designed, providing an accurate control of their beamwidth in order to guarantee an equivalent beam radius at the receiver of at least twice the value of the pointing error displacement standard deviation.

2. System and channel model

We adopt a multiple-input/multiple-output array based on L laser sources, assumed to be intensity-modulated only and all pointed towards a distant array of M photodetectors, assumed to be ideal noncoherent (direct-detection) receivers. The transmit and receive apertures are physically situated so that all transmitters are simultaneously observed by each receiver. The use of infrared technologies based on IM/DD links is considered, where the instantaneous current y_lm(t) in the mth receiving photodetector corresponding to the information signal transmitted from the lth laser can be written as

y_{lm} (t) = η i_{lm} (t) x (t) + z_{m} (t)

where η is the detector responsivity, assumed hereinafter to be the unity, X ≜ x(t) represents the optical power supplied by the lth source and I_lm ≜ i_lm(t) the equivalent real-valued fading gain (irradiance) through the optical channel between the lth transmit and the mth receive aperture. Additionaly, the fading experienced between source-detector pairs I_lm is assumed to be statistically independent. Z_m ≜ z_m(t) is assumed to include any front-end receiver thermal noise as well as shot noise caused by ambient light much stronger than the desired signal at the mth detector. In this case, the noise can usually be modeled to high accuracy as AWGN with zero mean and variance

σ_{m}^{2} = N_{0} / 2 M

, i.e. Z_m ∼ N(0, N ₀/2M), independent of the on/off state of the received bit. The factor M is consequence of the fact that the sum of the M receive aperture areas is the same as the aperture area of a system with no receive diversity, wherein the noise is usually modelled as AWGN with zero mean and variance σ ² = N ₀/2. This allows the systems to be compared fairly [6]. Since the transmitted signal is an intensity, X must satisfy ∀t x(t) ≥ 0. Due to eye and skin safety regulations, the average optical power is limited and, hence, the average amplitude of X is limited. The received electrical signal Y_lm ≜ y_lm(t), however, can assume negative amplitude values. We use Y_lm, X, I_lm and Z_m to denote random variables and y_lm(t), x(t), i_lm(t) and z_m(t) their corresponding realizations.

The irradiance is susceptible to either strong atmospheric turbulence conditions and pointing error effects. In this case, it is considered to be a product of two independent random variables, i.e. $I_{lm} = I_{lm}^{(a)} I_{lm}^{(p)}$ , representing $I_{lm}^{(a)}$ and $I_{lm}^{(p)}$ the attenuation due to atmospheric turbulence and the attenuation due to geometric spread and pointing errors, respectively, between lth transmitter and mth receiver. Although the effects of turbulence and pointing are not strictly independent, for smaller jitter values they can be approximated as independent [19]. Considering a limiting case of strong turbulence conditions [4, 22, 23], the turbulence-induced fading is modelled as a multiplicative random process which follows the negative exponential distribution, whose probability density function (PDF) is given by

f_{I_{lm}^{(a)}} (i) = e^{- i}, i \geq 0.

This PDF has also been adopted in different works [8, 9, 24] to describe turbulence-induced fading, leading to an easier mathematical treatment. Regarding to the impact of pointing errors, we use the general model of misalignment fading given in [16] by Farid and Hranilovic, wherein the effect of beam width, detector size and jitter variance is considered. Assuming a Gaussian spatial intensity profile of beam waist radius, ω_z, on the receiver plane at distance z from the transmitter and a circular receive aperture of radius r, the PDF of

I_{lm}^{(p)}

is given by

f_{I_{lm}^{(p)}} (i) = \frac{φ^{2}}{A_{0}^{φ^{2}}} i^{φ^{2} - 1}, 0 \leq i \leq A_{0}

where φ = ω_{z_eq}/2σ_s is the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter) at the receiver,

ω_{z_{eq}}^{2} = ω_{z}^{2} \sqrt{π} erf (v) / 2 v exp (- v^{2})

,

v = \sqrt{π} r / \sqrt{2} ω_{z}

, A ₀ = [erf(v)]² and erf(·) is the error function [25, eqn. (7.1.1)]. Here, independent identical Gaussian distributions for the elevation and the horizontal displacement (sway) are considered, being

σ_{s}^{2}

the jitter variance at the receiver. Using the previous PDFs for turbulence and misalignment fading, a closed-form expression of the combined PDF of I_lm was derived in [18] as

f_{I_{lm}} (i) = \frac{φ^{2}}{A_{0}^{φ^{2}}} i^{φ^{2} - 1} Γ (1 - φ^{2}, \frac{i}{A_{0}}), i \geq 0

where Γ(·,·) is the upper incomplete Gamma function [25, eqn. (6.5.3)]. Here, it must be commented that the pointing error model in Eq. (3) is not general enough to be directly applied to FSO systems with more than one photodetector (i.e. M > 1) since it depends on locations of the transmitter and receiver, requiring to take into account the relative position for each receive aperture regarding initial pointing in undisturbed position. This represents a fixed pointing error called boresight. As previously indicated, we adopt a MIMO array wherein the transmit and receive apertures are physically situated so that all transmitters are simultaneously observed by each receiver. For the sake of simplicity, it is here assumed that the receivers are placed very closely together and beam radius is large enough so that the relative displacement for each receiver regarding the initial pointing can be considered negligible, and hence the pointing error model in Eq. (3) can be approximately applied. This assumption can be reinforced by the fact that strong turbulence channels are more robust to boresight [19]. Additionally, as shown in the appendix, we consider OOK formats with any pulse shape and reduced duty cycle, allowing the increase of the PAOPR parameter [12, 23].

3. Outage performance analysis

In this section, the outage probability as a performance measure is evaluated for different MIMO FSO configurations over strong atmospheric turbulence channels with pointing errors. Together with the average error rate, outage probability, P _out, is another standard performance criterion of communications systems operating over fading channels [26]. It is defined as the probability that the instantaneous combined SNR, γ_T, falls below a certain specified threshold, γ_th, which represents a protection value of the SNR above which the quality of the channel is satisfactory, i.e.,

P_{out} = Prob (γ_{T} \leq γ_{th}) = \int_{0}^{γ_{th}} f_{γ T} (γ_{T}) d γ_{T} = F_{γ T} (γ_{th})

where f_γ_T (γ_T) and F_γ_T (γ_T) are the PDF and the cumulative distribution function (CDF) of γ_T, respectively. Next, the outage performance of a SISO FSO system is firstly evaluated in order to be considered as a benchmark in the analysis of the remaining MIMO configurations, corresponding to a multiple-input/single-output (MISO) system with a L × 1 array, a single-input/multiple-output (SIMO) system with a 1 × M array and, finally, a MIMO FSO system with a L × M array.

3.1. SISO FSO system

In this subsection, the outage probability for the FSO system model previously presented with L = M = 1 is evaluated. According to Eq. (1) and the OOK signaling described in the appendix, a constellation of two equiprobable points in a one-dimensional space with an Euclidean distance of d, the statistical channel model corresponding to the SISO configuration can be written as

Y = XI + Z, X \in {0, d}, Z \sim N (0, N_{0} / 2) .

The received electrical SNR can be defined, as in [16], as

γ_{T_{SISO}} (i) = \frac{1}{2} \frac{d^{2}}{N_{0} / 2} i^{2} = \frac{4 P_{opt}^{2} T_{b} ξ}{N_{0}} i^{2} = γ ξ i^{2}

where γ represents the received electrical SNR in absence of turbulence when the classical rectangular pulse shape is adopted for OOK formats, T_b parameter is the bit period, P _opt is the average optical power transmitted and ξ represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, as explained in a greater detail in the appendix. Using Eq. (5), the outage probability for a SISO FSO system can be written as

P_{out} = Prob (γ ξ i^{2} \leq γ_{th}) = Prob (i^{2} \leq \frac{γ_{th}}{ξ γ}) = F_{I} (\sqrt{\frac{1}{ξ} \frac{γ_{th}}{γ}})

where the CDF of I, F_I (i), can be easily derived from Eq. (4) as

F_{I} (i) = 1 - {(\frac{i}{A_{0}})}^{φ^{2}} φ^{2} Γ (- φ^{2}, \frac{i}{A_{0}}), i \geq 0

by applying the differential relation in [25, eqn. (6.5.26)] and the fact that the series expansion corresponding to the upper incomplete Gamma function can be simplified by

Γ (a, z) \propto Γ (a) - (z^{a} / a) (1 - \frac{az}{1 + a} + O (z^{2}))

[27, eqn. (8.354.2)], where Γ(·) is the well-known Gamma function. The results corresponding to this FSO scenario are illustrated in Fig. 1, where rectangular pulse shapes with ξ = 1 are used, assuming different values of normalized beamwidth, ω_z/r, and normalized jitter, σ_s/r (i.e. ω_z/r = 5 with σ_s/r = {1,3,4,5} and ω_z/r = 10 with σ_s/r = {1,4,7,11}). Outage simulation results are furthermore included as a reference, demonstrating an excellent agreement with the analytical result in Eq. (8) for different misalignment fading conditions.

Fig. 1 Probability of outage versus normalized average SNR in FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors, assuming different values of normalized beamwidth, ω_z/r, and normalized jitter, σ_s/r.

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Additionally, we can take advantage of this series expansion in order to quantify the outage probability at high SNR, showing that the asymptotic performance of this metric as a function of the average SNR is characterized by two parameters: the diversity and coding gains. Therefore, for large enough SNR (γ → ∞), the asymptotic outage performance can be expressed after some algebraic manipulations as

P_{out} ≐ {(\frac{φ^{4}}{{(1 - φ^{2})}^{2} A_{0}^{2} ξ} \frac{γ_{th}}{γ})}^{1 / 2} + {(\frac{Γ {(1 - φ^{2})}^{- 2 φ^{2}}}{A_{0}^{2} ξ} \frac{γ_{th}}{γ})}^{φ^{2} / 2} .

Here, an even greater simplification can be deduced, leading to different asympotic expressions depending on the value of φ, as follows

P_{out} ≐ {(\frac{φ^{4}}{{(1 - φ^{2})}^{2} A_{0}^{2} ξ} \frac{γ_{th}}{γ})}^{1 / 2}, φ > 1

P_{out} ≐ {(\frac{Γ {(1 - φ^{2})}^{2 / φ^{2}}}{A_{0}^{2} ξ} \frac{γ_{th}}{γ})}^{φ^{2} / 2}, φ < 1

Results corresponding to this asymptotic analysis are also illustrated in the Fig. 1 for different cases in Eq. (11) . It is straightforward to show that the outage probability behaves asymptotically as (O_c(γ/γ_th))^–O_d, where O_d and O_c denote outage diversity and coding gain, respectively [26, 28]. At high SNR, if asymptotically the outage probability behaves as (O_c(γ/γ_th))^–O_d, the outage diversity O_d determines the slope of the outage performance versus average SNR curve in a log-log scale and the coding gain O_c (in decibels) determines the shift of the curve in SNR. Additionally, as previously reported by the authors [12, 13, 23], a relevant improvement in performance must be noted as a consequence of the pulse shape used, providing an increment in the average SNR of 10 log₁₀ ξ decibels.

At this point, it can be convenient to compare with the outage performance obtained in a similar context when misalignment fading is not present. In this case, knowing that the CDF corresponding to the negative exponential distribution in Eq. (2) is given by F _I^(a) (i) = 1 − exp(−i) and the fact that the series expansion for the exponential function can be simplified by exp(z) ∝ 1 + z + O(z ²), an asymptotic expression can be easily derived as $P_{out}^{exp} ≐ (γ_{th} / γ ξ))^{1 / 2}$ . From these results, it can be deduced that φ is a key parameter in order to optimize the outage diversity. So, it is shown that the diversity order is independent of the pointing errors when the equivalent beam radius at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver, i.e. φ > 1. Once this condition is satisfied, taking into account the coding gain in Eq. (11a), the impact of the pointing error effects translates into a coding gain disadvantage, D[dB], relative to strong atmospheric turbulence without misalignment fading given by

D [dB] ≜ 20 {log}_{10} (\frac{φ^{2}}{A_{0} (φ^{2} - 1)}) .

According to this expression, it can be observed in Fig. 1 that coding gain disadvantages of 23.7, 34.4 and 42.7 decibels are achieved for values of (ω_z/r,σ_s/r) = (5, 1), (ω_z/r,σ_s/r) = (10,1) and (ω_z/r,σ_s/r) = (10,4), respectively.

3.2. MISO FSO system with a L ×1 array

In this subsection, the outage probability for the FSO system model previously presented where the information signal is transmitted via L apertures and received by only one aperture (i.e. M = 1) is evaluated. Here, two different transmit diversity strategies are considered: repetition coding (RC) and transmit laser selection (TLS).

3.2.1. Transmit diversity using repetition coding

It is assumed that the same symbol is transmitted from the L transmitters at a given bit period [6, 10, 29]. According to Eq. (1), the statistical channel model corresponding to this MISO configuration can be written as

Y = X \frac{1}{L} \sum_{l = 1}^{L} I_{l} + Z, X \in {0, d}, Z \sim N (0, N_{0} / 2)

where I_l represents the equivalent irradiance through the optical channel between the lth transmit aperture and the photodetector. Here, the division by L is considered so as to maintain the average optical power in the air at a constant level of P _opt, being transmitted by each laser an average optical power of P _opt/L. In this way, the total transmit power is the same as in a FSO system with no transmit diversity. The resulting received electrical SNR can be defined as

γ_{T_{MISO-RC}} = γ ξ \frac{1}{L^{2}} {(\sum_{l = 1}^{L} i_{l})}^{2} = γ ξ \frac{1}{L^{2}} i_{T}^{2}

and, hence, the outage probability for a MISO FSO system with repetition coding can be written as

P_{out} = F_{I_{T}} (L \sqrt{\frac{1}{ξ} \frac{γ_{th}}{γ}})

Next, in order to address the cumbersome statistical problem related to the CDF of the sum of L variates, the PDF in Eq. (4) is approximated by a single polynomial term as f_{I_lm} (i) ≈ ai^b, based on the fact that the asymptotic behavior of the system performance is dominated by the behavior of the PDF near the origin, i.e. f_{I_lm} (i) at i → 0 determines high SNR performance [28]. Using the series expansion corresponding to the upper incomplete Gamma function [27, eqn. (8.354.2)], different asympotic expressions for Eq. (4), depending on the value of φ, can be written as

f_{I_{lm}} (i) \approx \frac{φ^{2}}{A_{0} (φ^{2} - 1)}, φ > 1

f_{I_{lm}} (i) \approx φ^{2} \frac{Γ (1 - φ^{2})}{A_{0}^{φ^{2}}} i^{φ^{2} - 1}, φ < 1

Since the variates are independent, knowing that the resulting PDF of their sum is the convolution of their corresponding PDFs, obtained via single-sided Laplace and its inverse transforms, approximate expressions for the CDF, F_{I_T} (i), of the combined variate can be easily derived as

F_{I_{T}} (i) \approx \frac{1}{Γ (L + 1)} {(\frac{φ^{2}}{A_{0} (φ^{2} - 1)})}^{L} i^{L}, φ > 1

F_{I_{T}} (i) \approx \frac{Γ {(1 + φ^{2})}^{L}}{Γ (1 + L φ^{2})} {(\frac{Γ (1 - φ^{2})}{A_{0}^{φ^{2}}})}^{L} i^{L φ^{2}}, φ < 1

and, then, the corresponding approximate expressions for the outage probability for a MISO FSO system with repetition coding can be obtained from Eq. (15) as follows

P_{out} ≐ {(\frac{L^{2}}{Γ {(L + 1)}^{2 / L}} \frac{φ^{4}}{A_{0}^{2} {(1 - φ^{2})}^{2} ξ} \frac{γ_{th}}{γ})}^{L / 2}, φ > 1

P_{out} ≐ {(\frac{L^{2} Γ {(1 + φ^{2})}^{2 / φ^{2}}}{Γ {(1 + L φ^{2})}^{2 / L φ^{2}}} \frac{Γ {(1 - φ^{2})}^{2 / φ^{2}}}{A_{0}^{2} ξ} \frac{γ_{th}}{γ})}^{L φ^{2} / 2}, φ < 1

Monte Carlo simulation results for Eq. (15) together with the asymptotic expressions in Eq. (18) are illustrated in Fig. 2, where rectangular pulse shapes with ξ = 1 are used, assuming a normalized beamwidth of ω_z/r = 5 and different values of normalized jitter of σ_s/r = 1 and σ_s/r = 4.

Fig. 2 Probability of outage versus normalized average SNR in MISO and SIMO FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors, assuming a normalized beamwidth of ω_z/r = 5 and values of normalized jitter of (a) σ_s/r = 1 and (b) σ_s/r = 4.

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3.2.2. Transmit diversity using laser selection

When channel side information is not only available at the receiver but also on the transmitter, an alternative transmit diversity scheme can be adopted. Following the transmit laser selection (TLS) scheme based on the selection of the optical path with a greater value of fading gain (irradiance) [12], our MISO system model can be considered as an equivalent SISO system model where the channel irradiance corresponding to the TLS scheme, I_max, can be written as I_max = max_l _=1,2,… _L I_l. In this way, having a SISO system as a reference, the statistical channel model corresponding to this MISO configuration can be written as

Y = X I_{max} + Z, X \in {0, d}, Z \sim N (0, N_{0} / 2)

where, in order to maintain the average optical power transmitted at the same constant level P _out, the division by L is not included in Eq. (19) because of not more than one laser is simultaneously used. According to order statistics [30], the CDF, F_{I_max} (i), of the resulting channel irradiance, I_max, is given by F_{I_max} (i) = [F_I (i)]^L and, hence, the outage probability can be expressed as

P_{out} = F_{I_{max}} (\sqrt{\frac{1}{ξ} \frac{γ_{th}}{γ}}) = {(F_{I} (\sqrt{\frac{1}{ξ} \frac{γ_{th}}{γ}}))}^{L}

In the same way, its corresponding asympotic expressions can be easily derived from Eq. (11) as

P_{out} ≐ {(\frac{φ^{4}}{{(1 - φ^{2})}^{2} A_{0}^{2} ξ} \frac{γ_{th}}{γ})}^{L / 2}, φ > 1

P_{out} ≐ {(\frac{Γ {(1 - φ^{2})}^{2 / φ^{2}}}{A_{0}^{2} ξ} \frac{γ_{th}}{γ})}^{L φ^{2} / 2}, φ < 1

Results for the exact outage probability in Eq. (20) together with the corresponding asymptotic expressions in Eq. (21) are also illustrated in Fig. 2. Monte Carlo simulation results are furthermore included as a reference, demonstrating an excellent agreement with the analytical result in Eq. (20). Comparing both transmit diversity techniques and their corresponding asymptotic expressions in Eq. (18) and Eq. (21) , respectively, it can be stated that the superiority of the TLS scheme translates into a coding gain advantage, A_TLS[dB], relative to the RC scheme given by

A_{TLS} [dB] ≜ 10 {log}_{10} (\frac{L^{2}}{Γ {(L + 1)}^{2 / L}}), φ > 1

A_{TLS} [dB] ≜ 10 {log}_{10} (\frac{L^{2} Γ {(1 + φ^{2})}^{2 / φ^{2}}}{Γ {(1 + L φ^{2})}^{2 / L φ^{2}}}), φ < 1

As observed in Fig. 2, it can be deduced from expressions in Eq. (22) that the greater the number of transmit apertures, the better performance in terms of coding gain advantage. Additionally, this superiority is even more significant when φ < 1 and the value of φ is decreasing.

3.3. SIMO FSO system with a 1 × M array

In this subsection, the outage probability for the FSO system model previously presented is evaluated when the information signal is transmitted via only one aperture (i.e. L = 1) and received by M apertures. Here, two different receive diversity strategies are considered: equal gain combining (EGC) and selection combining (SC).

3.3.1. Receive diversity using equal gain combining

Following this combining technique, the receiver adds the information corresponding to M photodetectors and, hence, according to Eq. (1), the statistical channel model for this SIMO configuration can be written as

Y = X \frac{1}{M} \sum_{m = 1}^{M} I_{m} + Z_{EGC}, X \in {0, d}, Z_{EGC} \sim N (0, N_{0} / 2)

where I_m represents the equivalent irradiance through the optical channel between the transmit aperture and the mth photodetector. Here, the division by M is considered to ensure that the area of the receive aperture in SISO links has the same size as in the sum of M receive aperture areas of SIMO links [6]. It can be noted that the resulting expression is equivalent to the expression in Eq. (13), obtained for MISO FSO links assuming RC at the transmitter side [10]. Therefore, we can interchange L by M in Eq. (15) and Eq. (18) to obtain the outage probability and its corresponding asympotic expressions for SIMO FSO links using equal gain combining.

3.3.2. Receive diversity using selection combining

This type of combining is based on the selection of the receive aperture corresponding to the optical path with a greater value of fading gain (irradiance) [6, 10]. In this way, in a similar way as previously presented for MISO FSO links with transmit laser selection, our SIMO system model can be considered as an equivalent SISO system model where the channel irradiance corresponding to the SC scheme, I_max, can be written as I_max = max_m _=1,2,··· _M I_m and, hence, the statistical channel model can be written as

Y = X \frac{1}{M} I_{max} + Z_{SC}, X \in {0, d}, Z_{SC} \sim N (0, N_{0} / 2 M)

where M represents the same role as previously explained when EGC is adopted. It can be observed that the noise variance is given by N ₀/2M since only one of the M receive apertures is used. Taking into account Eq. (7), the resulting received electrical SNR can be defined as

γ_{T_{SIMO-SC}} = γ ξ \frac{1}{M} I_{max}^{2}

and, hence, the outage probability for a SIMO FSO system with selection combining can be written as

P_{out} = F_{I_{max}} (\sqrt{\frac{M}{ξ} \frac{γ_{th}}{γ}}) = {(F_{I} (\sqrt{\frac{M}{ξ} \frac{γ_{th}}{γ}}))}^{M} .

In the same way, its corresponding asympotic expressions, depending on the value of φ, can be easily derived from Eq. (11) as

P_{out} ≐ {(M \frac{φ^{4}}{{(1 - φ^{2})}^{2} A_{0}^{2} ξ} \frac{γ_{th}}{γ})}^{M / 2}, φ > 1

P_{out} ≐ {(M \frac{Γ {(1 - φ^{2})}^{2 / φ^{2}}}{A_{0}^{2} ξ} \frac{γ_{th}}{γ})}^{M φ^{2} / 2}, φ < 1

Results for the exact outage probability in Eq. (26) together with the corresponding asymptotic expressions in Eq. (27) are also illustrated in Fig. 2. Monte Carlo simulation results are furthermore included as a reference, demonstrating an excellent agreement with the analytical result in Eq. (26). Comparing both receive diversity techniques and their corresponding asymptotic expressions, a different relative performance is deduced depending on the value of φ, corroborating the superiority of EGC scheme when φ > 1 and the superiority of both SC scheme or EGC scheme when φ < 1, depending on the values of M and φ. This improvement in performance of the EGC scheme when φ > 1 translates into a coding gain advantage, A_EGC[dB], relative to the SC scheme as follows

A_{EGC} [dB] ≜ 10 {log}_{10} (\frac{Γ {(M + 1)}^{2 / M}}{M}), φ > 1

corroborating the fact that the greater the number of apertures, the higher coding gain advantage. Nonetheless, it can be noted that there is no difference between both diversity techniques for a value of M = 2. In a similar way, the superiority of SC scheme when φ < 1 translates into a coding gain advantage, A_SC[dB], relative to the EGC scheme as follows

A_{SC} [dB] ≜ 10 {log}_{10} (\frac{M Γ {(1 + φ^{2})}^{2 / φ^{2}}}{Γ {(1 + M φ^{2})}^{2 / M φ^{2}}}), φ < 1

Nonetheless, it can be concluded from this expression that the superiority of the SC scheme is only valid for low values of φ and when a small number of receive apertures is considered, being only justified the adoption of this receive diversity technique in the context of very severe pointing errors. As depicted in Fig. 3 for a normalized beamwidth of ω_z/r = 10, better performance for the EGC scheme is again achieved as the number of receive apertures increases and the value of the normalized jitter, σ_s/r, is lower and lower. So, the value of φ becomes higher and higher, and hence it is closer to the unity.

Fig. 3 Coding gain advantage for the SC scheme relative to the EGC scheme, assuming a normalized beamwidth of ω_z/r = 10 and values of normalized jitter of 6, 7, 8 and 9, corresponding to values of φ of 0.83, 0.71, 0.62 and 0.55, respectively.

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3.4. MIMO FSO system with a L ×M array

According to results in previous subsections, the outage probability for MIMO FSO IM/DD links is here evaluated when transmit laser selection and equal gain combining are the diversity strategies adopted in the transmit and receive side, respectively. The statistical channel model for this MIMO configuration can be written as

Y = X \frac{1}{M} \sum_{m = 1}^{M} I_{m} + Z_{EGC}, X \in {0, d}, Z_{EGC} \sim N (0, N_{0} / 2)

where I_m represents the equivalent irradiance through the optical channels between the L transmit apertures and the mth photodetector, based on the selection of the optical path with a greater value of irradiance, i.e. I_m = max_l _=1,2,··· _L I_lm. The resulting received electrical SNR can be defined as

γ_{T_{MIMO}} = γ ξ \frac{1}{M^{2}} {(\sum_{m = 1}^{M} i_{m})}^{2} = γ ξ \frac{1}{M^{2}} i_{T}^{2}

and, hence, the outage probability is given by

P_{out} = F_{I_{T}} (M \sqrt{\frac{1}{ξ} \frac{γ_{th}}{γ}})

where I_T represents the sum of M variates following the Lth-order statistics distribution for L observations from the distribution in Eq. (4). As in previous subsections, in order to address the cumbersome statistical problem related to the CDF of the sum of these M variates and making use of Eq. (16) , we use an approximated expression for the PDF of I_m, which is given by

f_{I_{m}} (i) \approx L {(\frac{φ^{2}}{(φ^{2} - 1) A_{0}})}^{L} i^{L - 1}, φ > 1

f_{I_{m}} (i) \approx L φ^{2} {(\frac{Γ (1 - φ^{2})}{A_{0}^{φ^{2}}})}^{L} i^{L φ^{2} - 1}, φ < 1

according to order statistics [30]. Since the variates are independent, knowing that the resulting PDF of their sum is the convolution of their corresponding PDFs, obtained via single-sided Laplace and its inverse transforms, approximate expressions for the CDF, F_{I_T} (i), of the combined variate can be easily derived as

F_{I_{T}} (i) \approx \frac{{(L Γ (L))}^{M}}{Γ (LM + 1)} {(\frac{φ^{2}}{(φ^{2} - 1) A_{0}})}^{LM} i^{LM}, φ > 1

F_{I_{T}} (i) \approx \frac{Γ {(L φ^{2} + 1)}^{M}}{Γ (LM φ^{2} + 1)} {(\frac{Γ (1 - φ^{2})}{A_{0}^{φ^{2}}})}^{LM} i^{LM φ^{2}}, φ < 1

and, then, the corresponding approximate expressions for the outage probability for a MIMO FSO system with transmit laser selection and equal gain combining can be obtained from Eq. (32) as follows

P_{out} ≐ {(\frac{(M^{2} Γ {(L + 1)}^{2 / L}}{Γ {(LM + 1)}^{2 / LM}} \frac{φ^{4}}{{(φ^{2} - 1)}^{2} A_{0}^{2} ξ} \frac{γ_{th}}{γ})}^{\frac{LM}{2}}, φ > 1

P_{out} ≐ {(\frac{M^{2} Γ {(L φ^{2} + 1)}^{2 / L φ^{2}}}{Γ {(LM φ^{2} + 1)}^{2 / LM φ^{2}}} \frac{Γ {(1 - φ^{2})}^{2 / φ^{2}}}{A_{0}^{2} ξ} \frac{γ_{th}}{γ})}^{LM φ^{2} / 2}, φ < 1

As expected, these expressions can be reduced to those corresponding to MISO and SIMO FSO scenarios in the special case of M = 1 and L = 1, respectively. In this way, it can be noted that expressions in Eq. (35) are equivalent to the expressions in Eq. (21) when a MISO FSO system with transmit laser selection is assumed. In the same way, expressions in Eq. (35) can also be reduced to those corresponding to SIMO FSO system with equal gain combining, being equivalent to the expressions in Eq. (18) when L is interchanged by M. As previously commented in section 3.3.1, expressions corresponding to a SIMO FSO system using equal gain combining are equivalent to those corresponding to a MISO FSO system assuming RC at the transmitter side. Monte Carlo simulation results for Eq. (32) together with the asymptotic expressions in Eq. (35) are illustrated in Fig. 4, where rectangular pulse shapes with ξ = 1 are used, assuming a normalized beamwidth of ω_z/r = 5 with a normalized jitter of σ_s/r = 1 and a normalized beamwidth of ω_z/r = 10 with a normalized jitter of σ_s/r = 7 and values of L = {2, 4} and M = {2, 4}.

Fig. 4 Probability of outage versus normalized average SNR in MIMO FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors, assuming a normalized beamwidth of ω_z/r = 5 with a normalized jitter of σ_s/r = 1 and a normalized beamwidth of ω_z/r = 10 with a normalized jitter of σ_s/r = 7, corresponding to values of φ of 2.55 and 0.71, respectively. Additionally, results assuming the optimum beamwidth corresponding to values of jitter of σ_s/r = 1 and σ_s/r = 7 are also included for L = 4 and M = 2.

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4. Outage optimization

From obtained results, it can be concluded that the outage diversity is independent of the pointing error effects when the equivalent beam radius value at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver, i.e. φ > 1, being φ a main parameter to consider in order to optimize the outage performance. Once this condition is satisfied, comparing different diversity techniques and their corresponding asymptotic expressions, it can be noted that better performance is achieved when increasing the number of transmit apertures, i.e. L, instead of the number of receive apertures, i.e. M, in order to guarantee a same diversity order, as shown in Fig. 4 when assuming a normalized beamwidth of ω_z/r = 5 with a normalized jitter of σ_s/r = 1 and a diversity order of 8 with values of {L,M} = {4,2} and {L,M} = {2,4}. Taking into account expressions in Eq. (35a) and Eq. (21a), this can be quantified as a coding gain disadvantage, D_MIMO[dB], for the MIMO FSO system relative to the MISO FSO system using laser selection as follows

D_{MIMO} [dB] ≜ 20 {log}_{10} (\frac{M Γ {(O_{d} / M + 1)}^{M / O_{d}}}{Γ {(O_{d} + 1)}^{1 / O_{d}}}) .

where O_d and M denote outage diversity and the number of receive apertures, respectively. In agreement with this expression, it can be seen in Fig. 4 that a difference of 2.13 decibels is achieved for a diversity order of 8 and values of M = 2 and M = 4 when a normalized beamwidth of ω_z/r = 5 with a normalized jitter of σ_s/r = 1 is assumed. Lastly, since proper FSO transmission requires transmitters with accurate control of their beamwidth, the optimization procedure is finished by finding the optimum beamwidth, ω_z/r, that gives the minimum outage [16, 17, 31]. From Eq. (35a), it can be observed that this is equivalent to minimize the expression in Eq. (12) corresponding to the SISO configuration, being independent of the MIMO configuration adopted. In this way, the optimum beamwidth can be achieved using numerical optimization methods for different values of normalized jitter, σ_s/r [32]. Numerical results for the optimum beamwidth are illustrated in Fig. 5 for a range of values in the normalized jitter from σ_s/r = 1 to σ_s/r = 10 in discrete steps of 0.5 when the stochastic function minimizer simulated annealing is used. From this figure, it can be deduced that the outage optimization provides numerical results following a linear performance. This leads to easily obtain a first-degree polynomial as follows

ω_{z} / r_{optimum} \approx 2.85 (σ_{s} / r - 1) + 2.6

considering the values of optimum beamwidth corresponding to normalized jitters of σ_s/r = 1 and σ_s/r = 10, respectively. As can be seen in this figure, it is clearly depicted that the approximative analytical expression remains very accurate to numerical results. The use of this expression is shown in Fig. 4, where results assuming the optimum beamwidth corresponding to values of jitter of σ_s/r = 1 and σ_s/r = 7 are also included for L = 4 and M = 2. As expected, a greater improvement in performance is achieved when σ_s/r = 7 compared to the improvement in performance corresponding to the normalized jitter of σ_s/r = 1 since not only coding gain but also diversity gain is obtained, making outage diversity performance independent of the impact of misalignment fading.

Fig. 5 Optimum normalized beamwidth versus normalized jitter, σ_s/r in FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors.

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5. Conclusions

In this paper, the outage probability as a performance measure for MIMO FSO communication systems using IM/DD over strong atmospheric turbulence channels with pointing errors is analyzed, assuming different configurations as repetition coding and transmit laser selection on the transmitter side and equal gain combining and selection combining on the receiver side. Novel closed-form expressions for the outage probability as well as their corresponding asymptotic expressions are presented when the irradiance of the transmitted optical beam is susceptible to either strong turbulence conditions, following a negative exponential distribution, and pointing error effects, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. In this strong turbulence FSO scenario, obtained results show that the diversity order is independent of the pointing error effects when the equivalent beam radius at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver, showing the same slope of the outage performance versus average SNR as in a similar FSO scenario where misalignment fading is not considered. However, different coding gain, i.e. the horizontal shift in the outage performance in the limit of large SNR, is achieved as a consequence of the severity of the pointing errors effects and MIMO configuration assumed. Additionally, a significant improvement in performance is demonstrated when MIMO FSO links based on transmit laser selection with equal gain combining are adopted. Simulation results are further demonstrated to confirm the analytical results. The asympotic expressions here obtained are especially useful for performance analysis of MIMO FSO systems, providing an easier understanding of performance limiting factors when operating under different misalignment fading conditions and various diversity techniques are adopted. From the relevant results here obtained, investigating in this FSO scenario the impact on the diversity order of different receive configurations, e.g., rectangular and circular arrays of receive apertures, wherein the displacement for each photodetector regarding the initial pointing is considered as well as the possible correlation among their corresponding shifting, is an interesting topic for future research.

Appendix

We consider OOK formats with any pulse shape and reduced duty cycle, allowing the increase of the PAOPR parameter [12, 23]. A new basis function ϕ (t) is defined as $ϕ (t) = g (t) / \sqrt{E_{g}}$ where g(t) represents any normalized pulse shape satisfying the non-negativity constraint, with 0 ≤ g(t) ≤ 1 in the bit period and 0 otherwise, and $E_{g} = \int_{- \infty}^{\infty} g^{2} (t) dt$ is the electrical energy. In this way, an expression for the optical intensity can be written as

x (t) = \sum_{k = - \infty}^{\infty} a_{k} \frac{2 T_{b} P_{opt}}{G (f = 0)} g (t - k T_{b})

where G(f = 0) represents the Fourier transform of g(t) evaluated at frequency f = 0, i.e. the area of the employed pulse shape, and T_b parameter is the bit period. The random variable (RV) a_k follows a Bernoulli distribution with parameter 1/2, taking the values of 0 for the bit “0” (off pulse) and 1 for the bit “1” (on pulse). From this expression, it is easy to deduce that the average optical power transmitted is P _opt, defining a constellation of two equiprobable points (x ₀ = 0 and x ₁ = d) in a one-dimensional space with an Euclidean distance of

d = 2 P_{opt} \sqrt{T_{b} ξ}

where ξ = T_b E_g/G ²(f = 0) represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, alternative to the classical rectangular pulse.

Acknowledgments

The authors would like to thank the anonymous reviewers for their useful comments that helped to improve the presentation of the paper. The authors are grateful for financial support from the Junta de Andalucía (research group “ Communications Engineering ( TIC-0102)”).

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Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels

Abstract

1. Introduction

2. System and channel model

3. Outage performance analysis

3.1. SISO FSO system

3.2. MISO FSO system with a L ×1 array

3.2.1. Transmit diversity using repetition coding

3.2.2. Transmit diversity using laser selection

3.3. SIMO FSO system with a 1 × M array

3.3.1. Receive diversity using equal gain combining

3.3.2. Receive diversity using selection combining

3.4. MIMO FSO system with a L ×M array

4. Outage optimization

5. Conclusions

Appendix

Acknowledgments

References and links

Cited By

Figures (5)

Equations (48)

Optics Express