Coherent wireless optical communication systems with heterodyne detection are analyzed for binary phase-shift keying (BPSK), differential PSK (DPSK), and M-ary PSK over Gamma-Gamma turbulence channels. Closed-form error rate expressions are derived using a series expansion approach. It is shown that, in the special case of K-distributed turbulence channel, the DPSK incurs a 3 dB signal-to-noise ratio (SNR) penalty compared to BPSK in the large SNR regime. The outage probability is also obtained, and a detailed outage truncation error analysis is presented and used to assess the accuracy in system performance estimation. It is shown that our series error rate expressions are simple to use and highly accurate for practical system performance estimation.
© 2012 Optical Society of America
Wireless optical communications (WOC) is well suited for applications requiring low costs, high data rates, enhanced security, and unlicensed broad bandwidths . Simple irradiance modulation with direct detection (IM/DD) and on-off keying (OOK) are commonly used in such optical wireless systems [2, 3], although, significant challenges can arise for WOC from increased scintillation in atmospheric turbulence channels. Optical irradiance amplitude and phase suffer from random fluctuations when the channel refractive index is distorted by time-varying temperatures and pressures. Turbulent distortion forms a critical limitation for IM/DD OOK systems as an adaptive time-varying threshold is necessary for optimal performance, and simpler fixed threshold systems have been shown to have irreducible error floors [4, 5].
To overcome the threshold challenges of OOK, subcarrier  and coherent WOC systems with phase-shift keying (PSK) have been introduced to achieve optimal performance without an adaptive threshold. WOC with coherent detection can further enhance the system performance by improving the noise rejection capability, achieving higher sensitivity and increasing spectral efficiency, although such benefits come at a cost of higher system complexity [6–8].
In this paper, we analyze coherent WOC systems in Gamma-Gamma turbulence channels. We present infinite series solutions to the outage probability and average error rates for WOC systems with binary PSK (BPSK), M-ary PSK (MPSK) and differential PSK (DPSK). Additional insights into coherent WOC systems are provided through asymptotic error performance analyses, and a detailed study of the truncation error is presented for the outage series solution.
2. Coherent WOC system
2.1. Receiver model
A PSK-modulated coherent WOC system with heterodyne detection is considered within an atmospheric optical transmission link. Given that the received optical beam and local oscillator beam are mixed in perfect spatial coherence over a sufficiently small photodetector area, the photocurrent at the output of the photodetector can be expressed as 2,9]. In Eq. (1), R is the photodetector responsivity, Ps denotes the received optical signal power incident on the beamsplitter, the pre-modulated or differentially coded phase information is represented by ϕ, and ωIF = ω0 – ωLO is the intermediate frequency, with ω0 and ωLO denoting the carrier frequency and local oscillator frequency, respectively. In the following analysis, we assume the phase noise can be fully compensated at the receiver, as the phase noise is sufficiently slow compared to the extremely high data rates of WOC systems .
The signal-to-noise ratio (SNR) of an optical receiver is defined as the ratio of time-averaged AC photocurrent power to total noise variance . Given that optical power is the product of the photodetector area A and irradiance, the SNR for coherent WOC systems at the input of the demodulator is γ = RAIs/(qΔf) where Is is the turbulence-dependent optical irradiance, Δf is the noise equivalent bandwidth of the photodetector, and q is the electronic charge.
2.2. Turbulence model
Common turbulence models are the log-normal model (for weak turbulence regimes), K-distributed model (for strong turbulence regimes), and Gamma-Gamma model (from weak to strong turbulence regimes). In this work, we model the optical irradiance Is as a Gamma-Gamma random variable with a probability density function (PDF) 11], and the relationship α > β holds in WOC applications under an assumption of plane wave and negligible inner-scale .
3. Performance analysis of coherent WOC
3.1. Error rate analysis
The average error rate over an atmospheric turbulence channel can be expressed as3, 13], the PDF of Is can be expressed as  Eq. (3) as Eq. (4) into Eq. (6) and using integral identities [14, Eq. 3.478(1), Eq 3.621(1), Eq. 8.384(4)], one obtains the average bit-error rate (BER) for BPSK coherent WOC systems as 14, Eq. 8.384(1)] and the integral property [14, Eq. 3.621(1), Eq. 8.384(4)] Eq. (7), one can examine error rate behavior in large SNR regimes with insight into coherent WOC systems. Since α > β, the term (γ̄/2)−(p+α) decreases faster than the term (γ̄/2)−(p+β) in Eq. (7) for the same p values as γ̄ increases. Consequently, when γ̄ approaches ∞, the leading term of the series in Eq. (7) becomes dominant, and the error rate in large SNR regimes can be obtained as
The above analysis can be generalized to the MPSK case. With [7, Eq. (12)] and a change of variables, one can obtain a closed-form expression of the average symbol-error rate (SER) for MPSK over Gamma-Gamma atmospheric turbulence channels. The result is14, Eq. 9.100].
DPSK has been shown to be an effective alternative to coherent PSK in coherent WOC systems  as it does not require optical carrier phase estimation. Thus, we now present a closed-form BER expression for this important modulation format. The conditional BER for DPSK is given by 0.5 exp(−γ/2). Integrating over γ gives the average BER of DPSK asEq. (9) and Eq. (13), we obtain the SNR penalty factor for DPSK with respect to BPSK as . Note that this penalty factor only depends on the smaller channel parameter β, and the SNR penalty factor for DPSK is exactly 10log10(2) ≈ 3 dB over the K-distributed turbulence channels when β = 1.
3.2. Outage probability analysis
Outage probability is an important criterion for communication system performance evaluation. To facilitate the outage probability analysis, we use Eq. (4) to express the cumulative distribution function (CDF) of Is in terms of a series as
3.3. Truncation error analysis
To evaluate the accuracy of the closed-form expressions we obtained for the coherent WOC system, we now examine the outage truncation error caused by an elimination of the infinite terms after the first J + 1 terms in Eq. (15). We first define the outage truncation error asEq. (17) can be simplified through a Taylor series expansion of the exponential function. The result gives an upper bound of the truncation error in Eq. (16) through Eqs. (18) and (19), it is also clear that the truncation error diminishes rapidly with increasing average SNR. It can be shown that εJ is upper bounded by a term on the order of γ̄−β when p and γ̄ are large. Although only the outage truncation error analysis is presented here, similar truncation error analyses can be performed for differential and coherent PSK.
4. Numerical examples
In Fig. 1(a), BERs for weak (α = 3.553, β = 3.339), moderate (α = 2.711, β = 2.319) and strong (α = 1.967, β = 1.236) Gamma-Gamma turbulence channels are plotted for coherent WOC with BPSK and DPSK. Exact error rate is found by numerical integration based on Eq. (3), which has been confirmed by Monte Carlo simulations. Excellent agreement is seen between our series solutions and the exact BERs. As predicted in Section 3.1, the SNR penalty factor depends on channel turbulence levels (via the parameter β). For weak turbulence with α = 3.553 and β = 3.339 at an average BER of 10−10, the SNR penalty factor is 1.6 dB, which agrees with the 1.58 dB theoretical prediction. For moderate turbulence with α = 2.711 and β = 2.319 at an average BER of 10−8, the SNR penalty factor is 2 dB, which agrees with the 1.96 dB theoretical prediction. The SNR penalty factor increases as β decreases.
Exact and approximated SER curves with QPSK and 8PSK are plotted in Fig. 1(b) for moderate (α = 2.711, β = 2.319) and strong (α = 1.967, β = 1.236) turbulence. The SER degrades as M increases for MPSK, as expected, due to average symbol power constraints. In Fig. 2(a), outage probability is presented for the same weak to strong turbulence conditions as Fig. 1(a) with γth = 3 dB. Again, our series solutions match the exact error rate and outage curves.
To quantify the series solution accuracy for finite terms, outage truncation error is shown in Fig. 2(b). Our series solution is seen to be highly accurate in practical SNR regimes. For example, the truncation error is on the order of 10−18 at an average SNR of 30 dB. Also shown in Fig. 2(b), the outage truncation error diminishes with increasing average SNR in a linear fashion on a log-log plot when p and average SNR is large (as discussed in Section 3.3).
We have studied the performance of coherent WOC systems and developed closed-form error rate and outage probability expressions that give highly accurate performance estimations of differential and coherent PSK optical wireless systems over the Gamma-Gamma atmospheric turbulence. Our truncation error analysis has also demonstrated that our series solution converges to exact results rapidly with a finite number of terms for all turbulence conditions.
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