## Abstract

Exact error rate performances are studied for coherent free-space optical communication systems under strong turbulence with diversity reception. Equal gain and selection diversity are considered as practical schemes to mitigate turbulence. The exact bit-error rate for binary phase-shift keying and outage probability are developed for equal gain diversity. Analytical expressions are obtained for the bit-error rate of differential phase-shift keying and asynchronous frequency-shift keying, as well as for outage probability using selection diversity. Furthermore, we provide the closed-form expressions of diversity order and coding gain with both diversity receptions. The analytical results are verified by computer simulations and are suitable for rapid error rates calculation.

©2010 Optical Society of America

## 1. Introduction

Free-space optical (FSO) communications has drawn considerable interest. Lower costs, larger bandwidths, better security, and greater flexibility are all FSO benefits [1]. FSO systems with diversity reception, in particular, achieve performance improvements by mitigating atmospheric turbulence. This has been shown for irradiance modulation and direct detection (IM/DD) system with OOK using optimal combining, equal gain combining (EGC), and selection combining [1, 2], and coherent detection has been successful in further improving the sensitivity and spectral efficiency [3].

With the benefits of coherent FSO in mind [3], exact BER expressions have been developed for differential phase-shift keying (DPSK) over *K*-distributed [4] and Gamma-Gamma distributed [5] turbulence. A heterodyne FSO system with pointing errors was studied by Sandalidis *et al.* [6]. Synchronous detection was then studied by Belmonte and Kahn [7, 8], and our recent work [9] demonstrated that maximum ratio combining (MRC) can provide excellent performance improvements at practical signal-to-noise ratios (SNRs).

In this work, both equal gain and selection diversity (SD) are analyzed for long-range and high-turbulence conditions with a *K* distribution [4,10]. Based on the new combiner SNR statistics, we derive an exact BER expression for binary phase-shift keying (BPSK) and an outage probability expression with EGC. For SD, we obtain outage probability and BER expressions with asynchronous DPSK and frequency-shift keying (FSK). The obtained error rates are highly accurate for all SNR regions. In particular, the asymptotic error rates are obtained based on the development of diversity order and coding gain.

## 2. The Coherent FSO System Model and Statistics

#### 2.1. Channel Model

For the *l*th branch of a coherent FSO system using BPSK, with received and local oscillator beams mixed in perfect spatial coherence over a sufficiently small photodetector area. Practical issues such as beam combining and polarization are discussed in [11]. The photodetector current is [9]

where *i _{dc,l}* =

*R*(

*P*+

_{s,l}*P*) and ${i}_{\mathit{ac},l}\left(t\right)=2R\sqrt{{P}_{s,l}{P}_{\mathit{LO}}}\mathrm{cos}({\omega}_{\mathit{IF}}t+\varphi +{\varphi}_{s,l})$ are the DC and AC terms, respectively,

_{LO}*R*is the detector responsivity, and

*n*(

_{l}*t*) is a shot-noise-limited additive white Gaussian noise (AWGN) process. The power PLO is sufficiently large so that shot noise is dominant and can be modeled as AWGN [12]. Here,

*P*is the received

_{s,l}*l*th optical signal power incident on the beamsplitter,

*P*is the local oscillator power assumed to be the same for all branches, and

_{LO}*ω*=

_{IF}*ω*

_{0}−

*ω*is the intermediate frequency, with

_{LO}*ω*

_{0}and

*ω*denoting the carrier frequency and local oscillator frequency, respectively. The phase information is

_{LO}*ϕ*∈ {0,

*π*}, and

*ϕ*is the

_{s,l}*l*th branch phase noise. It is assumed that each diversity branch turbulence is independent. For EGC cases, we assume the signal phase is well tracked at the coherent FSO detector and the phase noise can be fully compensated [3], as the optical phase varies slowly over 1 or 2 bits at a high data rate.

#### 2.2. SNR Analysis

The SNR of an optical receiver is defined as the ratio of the time-averaged AC photocurrent power 〈*i*^{2}_{ac}(*t*)〉 to the total noise variance [12]. For EGC with *L* diversity branches, the received signals at different branches are co-phased and added with equal weight. In such a case, as shown in the Appendix, the instantaneous SNR at the output of the combiner becomes

where *q* denotes the electronic charge, Δ*f* is the noise equivalent bandwidth of the photodetector, *I _{N,l}* is the normalized

*K*turbulent coefficient (

*E*[

*I*] = 1) of the

_{N,l}*l*th branch,

*N*

_{0}/2 =

*qRP*is the noise power spectral density, and

_{LO}*E*=

_{b}*η*

^{2}

*AT*denotes the energy per bit. Here,

_{b}*E*[·] denotes the expectation operation,

*η*

^{2}is the mean irradiance,

*A*denotes the detector area, and

*T*is the bit duration. With the relationship

_{b}*P*=

_{s,l}*AI*where

_{s,l}*I*is the instantaneous received optical irradiance at the

_{s,l}*l*th branch, we can rewrite the instantaneous SNR as

where
$g\triangleq \sqrt{\frac{\mathit{RA}}{\left(\mathit{Lq}\mathrm{\Delta}f\right)}}$
is a scalar factor. For *L* = 1, the instantaneous SNR in Eq. (3) specializes to *γ _{EGC}* =

*RAI*

_{s,1}/(

*q*Δ

*f*), which agrees with the single-branch SNR obtained in [4,9]. Note that the SNR expression shown in Eq. (3) is fundamentally different from the SNR of EGC in the IM/DD applications, which is related to the sum of the irradiance [1,2]. Indeed, the EGC SNR for coherent FSO systems is related to the sum of the square root of the irradiance in each branch. This distinction can greatly impact the system performance.

Given the *l*th branch instantaneous SNR as *γ _{l}* =

*RP*/(

_{s,l}*q*Δ

*f*) [9], the instantaneous SNR for SD is

We can observe that the instantaneous combiner SNRs are independent of the local oscillator power for large *P _{LO}*. This is an important property of coherent FSO communication (in contrast to coherent radio frequency literature), as the local oscillator power does not affect coherent FSO system performance.

#### 2.3. Analysis for the Square Root of the Irradiance

To facilitate the performance analysis for the coherent EGC FSO systems, we can make use of the characteristic function (CF) for the square root of the irradiance *I _{s}*. In this work, we assume a signal irradiance

*I*with a

_{s}*K*-distributed model having a probability density function (PDF) given by [4]

where Γ(·) denotes the gamma function, *K _{α}*(·) is the modified Bessel function of the second kind of order

*α*, and

*α*is a channel parameter related to the scintillation index. According to the definition of the scintillation index [13]

one can show

Scintillation indices below unity correspond to weak turbulence conditions, which is commonly modeled with a log-normal distribution. For FSO applications, the scintillation index *σ*^{2}_{si} in the *K*-distributed model is (2,3) [4], and *α* is in (1,2). In the strong turbulence regime, the scintillation index can be calculated from the Rytov variance [13]

through the relation [13, Eq. (53)], where *C*^{2}_{n} is the index-of-refraction structure parameter, *k* is the optical wave number, and *d* is the propagation distance. For this coherent FSO system in the strong turbulence regime, one would expect *C*^{2}_{n} = 10^{−13}(*m*^{−2/3}) for a link length beyond 1*km* and aperture sizes below 1*mm*^{2}. This avoids aperture averaging (and a loss of coherence) and provides sufficient distance between individual photodetectors [10]. We denote the square root of the irradiance by
$z\triangleq \sqrt{{I}_{s}}$
. Then the PDF of *z* can be shown as [14]

which finds applications in modeling the displacement of a two-dimensional unbiased random walk. To obtain the CF of *z*, we let *z* = *xy*, where
$x\triangleq \sqrt{{I}_{x}}$
and
$y\triangleq \sqrt{{I}_{y}}$
are found to have Rayleigh and Nakagami (when *α* ≥ 1/2) PDFs, respectively, as

and

Averaging the CF of *z* conditioned on
$\sqrt{{I}_{x}}$
, one obtains the CF of *z* as

where the real and imaginary parts are respectively

and

Here, _{2}*F*_{1}(·, ·, ·; ·) is the Gaussian hypergeometric function, ℜ{·} and ℑ{·} denote the real and imaginary operators, respectively. The CF of *z* can also be obtained from the moment generating function (MGF) of a product of two independent Nakagami RVs [15,Eq. (3)].

## 3. Performance Analysis of EGC FSO Systems

#### 3.1. Error Rate Analysis

The average BER of BPSK over a turbulence channel can be derived by averaging the conditional bit-error probability as

where *z _{s}* is the sum of

*L*square roots of the irradiance, i.e., ${z}_{s}={\mathrm{\Sigma}}_{l=1}^{L}\sqrt{{I}_{s,l}}$ . Here,

*f*(

*z*) is the PDF of

_{s}*z*and

_{s}*P*(

_{e}*z*) denotes the conditional bit-error probability given by

_{s}where *erfc*(·) is the complementary error function. In general, it is difficult to obtain a closed-form expression for the PDF of *z _{s}*. Hence, we use the Fourier inversion theorem to obtain the PDF of

*z*by

_{s}where *j*^{2} = −1,
${\varphi}_{{z}_{s}}\left(\omega \right)$
is the CF of *z _{s}* and
${\varphi}_{{z}_{l}}\left(\omega \right)$
is the CF of

*z*(with ${z}_{l}=\sqrt{{I}_{s,l}}$ ). With Eq. (15)–Eq. (17), one can obtain the average BER as

_{l}where ${\varphi}_{{z}_{s}}^{*}(\xb7)$ denotes the conjugate of ${\varphi}_{{z}_{s}}(\xb7)$ . Since the Fourier transform of the complementary error function is known as [16]

Equation (18) can be rewritten as

With a change of variable *ω* = tan*θ*, it is possible to rewrite Eq. (20) in terms of a definite integration for easier numerical evaluation

Though BPSK is only considered here, it is straightforward to extend our analysis to other coherent modulations such as quadrature phase-shift keying and *M*-ary quadrature amplitude modulation.

#### 3.2. Outage Probability Analysis

To find the outage probability, we make use of the Gil-Pelaez formula to express the CDF of *z _{s}* as [18]

Since the turbulence in each branch is independent,
${\varphi}_{{z}_{s}}\left(\omega \right)$
can be written directly as
${\varphi}_{{z}_{s}}\left(\omega \right)=\prod _{l=1}^{L}{\varphi}_{{z}_{l}}\left(\omega \right)$
. To continue, we express the CF of *z _{l}* in the polar form as

where the real and imaginary parts of the CF of *z _{l}* are obtained from Eq. (13) and Eq. (14). The numerator of the integrand in Eq. (22) can now be written as

With a change of variable, substitution of Eq. (24) into Eq. (22) gives the outage probability as

where *γ*^{*} is the specified outage threshold. Equation (25) can provide accurate outage calculation with a simple numerical integration algorithm.

## 4. Performance Analysis of SD FSO Systems

SD has the least complexity when compared to MRC and EGC, as it simply selects the branch with the strongest instantaneous received SNR. In the following, we present exact outage probability and exact error rates expressions for coherent FSO systems with SD reception.

#### 4.1. Outage Probability Analysis

After a change of variable and help from [17,Eq. 6.561(8)], we can obtain the CDF of the optical signal irradiance, *I _{s,l}*, at the

*l*th branch from Eq. (5) as

It is straightforward to show that the outage probability with SD becomes

#### 4.2. Error Rate Analysis for Differential and Asynchronous Detection

Unlike the outage probability analysis, the desired error rate analysis depends upon the modulation scheme and requires the PDF of *I _{s,SD}* ≜ max{

*I*,

_{s,l}*l*= 1, …,

*L*}. For identically distributed turbulence, we can show the PDF of

*I*as

_{s,SD}Differential and asynchronous detection are two alternatives for synchronous detection to minimize phase noise effects. In this work, coherent FSO with asynchronous detection refers to coherent photodetection at the receiver front-end, followed by envelope detection for signal recovery [12, P. 490]. The conditional BER with an average SNR of *$\overline{\gamma}$* is known to be

where *β* = 1/2 for DPSK (assuming moderate frequency offset) and *β* = 1/4 for asynchronous FSK [12]. Hence, the average BER for DPSK or FSK with SD is

## 5. Analysis of Diversity Order and Coding Gain

The analysis in this section allows us to gain valuable insight into BER behavior in the large SNR region for coherent FSO with both EGC and SD reception. The asymptotic BER can be expressed as
${P}_{e}\stackrel{.}{=}{\left({G}_{c}\overline{\gamma}\right)}^{-{G}_{d}}$
where *G _{d}* and

*G*respectively denote diversity order and coding gain [9]. Similarly we denote

_{c}*G*and

_{dl}*G*as the diversity order and coding gain of the

_{cl}*l*th branch.

Using the MGF of a *K*-distributed RV and following [9], one can show that the diversity order and coding gain offered by the *l*th branch are *G _{dl}* = 1 and

*G*= 2

_{cl}*βη*

^{2}(

*α*−1)/

*α*respectively. Consequently, the diversity order of SD becomes

*G*= ∑

_{d,SD}

^{L}_{l=1}

*G*=

_{dl}*L*and the coding gain of SD can be derived as

For BPSK EGC, the diversity order and coding gain offered by the *l*th branch can be found to be *G _{dl}* = 1 and

*G*= 2

_{cl}*η*

^{2}(

*α*− 1)/

*α*respectively. The overall EGC diversity order is

*G*=

_{d,EGC}*L*, and the expression for the overall EGC coding gain,

*G*can be found in [19,Eq. (12)]. The diversity order here is similar to the diversity order observed in an RF system (with the exception of the Nakagami-

_{c,EGC}*m*fading model).

## 6. Numerical Results

Coherent FSO numerical case studies are presented. The average SNR per branch is *$\overline{\gamma}$* = *RAη*^{2}/(*q*Δ*f*). BER curves for BPSK-modulated signals with EGC diversity reception (*α* = 1.8) are presented in Fig. 1 for average SNRs from 0 to 30 dB. Asymptotic BERs are also shown and agree with our analytical results when the SNR is large. When the number of diversity branches increases, the BER performance improves. At an average SNR=30 dB, a three-branch EGC reception achieves an error rate of 2.5 × 10^{−8}. Though it is not shown in Fig. 1, all analytical results have been verified with computer simulations and they have excellent agreement.

Figure 2 compares the BER performance of MRC and EGC for BPSK-modulation and shows that coherent FSO systems using EGC offer a close error rate performance to that of MRC. At a BER of 10^{−7}, 1 dB SNR (or less) is required for EGC to achieve the MRC performance. Hence, EGC offers a good balance between system performance and complexity.

To further reduce the system complexity, SD becomes a viable alternative. The received optical signals need to be co-phased for EGC, but this procedure is not required for SD reception. Therefore, SD can reduce the receiver complexity. The SD outage probability is shown in Fig. 3 along with EGC and MRC. It is apparent that the outage probability can be improved by using systems with increased complexity (e.g. MRC) and higher diversity orders. With little knowledge of the turbulence phase noise, asynchronous DPSK and FSK with SD become the preferred choices. BERs for these two modulation schemes are presented in Fig. 4 and Fig. 5. It is apparent that SD reception mitigates turbulence effects for differential and asynchronous modulations. Moreover, the performance improves as the effective aperture size increases. The asymptotic solutions are also shown in both Fig. 4 and Fig. 5. It is found that they agree with the analytical results when the SNR becomes large. Turbulence effects are shown in Fig. 6 for *σ*^{2}_{si} ranging from 2.1 to 2.9 for both MRC and EGC. As expected, BER performance improves for coherent FSO communication as the value of *σ*^{2}_{si} decreases.

It is interesting to observe from Fig. 6 that EGC with *σ*^{2}_{si} = 2.1 outperforms MRC with *σ*^{2}_{si} = 2.5 for the same number of diversity branches. For practical applications, a modest parametrization error in estimating the scintillation indices or channel parameters will overshadow the benefits provided by MRC. In such a case, EGC may be a better choice due to its reduced complexity.

## 7. Conclusion

In this work, we have studied the performance of diversity reception for coherent FSO communication systems in strong atmospheric turbulence. The exact BER expressions and outage probability with EGC and SD reception have been obtained. Our numerical results show that coherent FSO transmission with EGC outperforms that with SD and gives comparable performance to that of MRC with reduced complexity, while the numerical results have demonstrated that the new analytical expressions are useful in providing highly accurate error rate estimation. In addition, we studied the diversity order and coding gain for coherent FSO combining showing the performance impact of optical systems and channel parameters. Our analytical study can be easily extended to coherent FSO communications under different turbulence conditions.

## Appendix

At the output of the *l*th photodetector, the received photocurrent is given in Eq. (1). Let
${c}_{l}=2R\sqrt{{P}_{\mathit{LO}}{P}_{s,l}}$
, and the AC current can be expressed as follows

Then, two real filters are used to implement the complex filtering in the downconverter process. The real and imaginary parts of the baseband signal are, respectively, obtained as

and

After a lowpass filter, the inphase and quadrature components become

and

Now we combine (35) and (36) to obtain the equivalent baseband signal as

The last step is to compensate the phase noise, giving the branch signal before combining as

Consequently, with the assumption of equal noise variance in each diversity branch, the instantaneous SNR at the output of the equal gain combiner becomes

where *σ*^{2}_{n,l} = 2*qRP _{LO}*Δ

*f,l*= 1, …,

*L*, is the noise variance in the

*l*th diversity branch at the receiver side, and it is assumed to be the same for all diversity branches.

## Acknowledgments

The research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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