Performance analysis of coherent wireless optical communications with atmospheric turbulence

Mingbo Niu; Xuegui Song; Julian Cheng; Jonathan F. Holzman

doi:10.1364/OE.20.006515

1. Introduction

Wireless optical communications (WOC) is well suited for applications requiring low costs, high data rates, enhanced security, and unlicensed broad bandwidths [1]. 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 [5] 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 [7]

i (t) = i_{dc} + i_{ac} (t) + n (t)

where i_dc = R(P_s + P_LO) and

i_{ac} (t) = 2 R \sqrt{P_{s} P_{L O}} cos (ω_{IF} t + ϕ)

represent the DC and AC terms, respectively, and n(t) is a zero-mean additive white Gaussian noise (AWGN) process due to shot noise. Since the local oscillator power, P_LO, is made to be sufficiently large, the shot noise can be modeled as AWGN with high accuracy [2,9]. In Eq. (1), R is the photodetector responsivity, P_s denotes the received optical signal power incident on the beamsplitter, the pre-modulated or differentially coded phase information is represented by ϕ, and ω_IF = ω₀ – ω_LO is the intermediate frequency, with ω₀ and ω_LO 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 [6].

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 [9]. 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 = RAI_s/(qΔf) where I_s 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 I_s as a Gamma-Gamma random variable with a probability density function (PDF) [10]

\begin{matrix} f_{I_{s}} (I_{s}) = \frac{2}{Γ (α) Γ (β)} {(α β)}^{\frac{α + β}{2}} I_{s}^{\frac{α + β}{2}} K_{α - β} (2 \sqrt{α β I_{s}}), & I_{s} > 0 \end{matrix}

where Γ(·) is the Gamma function, K_α−β (·) is the modified Bessel function of the second kind of order α − β. The positive parameters α and β are the effective number of large- and small-scale cells of atmospheric scattering processes, respectively. The parameters α and β are set by the Rytov variance [11], and the relationship α > β holds in WOC applications under an assumption of plane wave and negligible inner-scale [12].

3. Performance analysis of coherent WOC

3.1. Error rate analysis

The average error rate over an atmospheric turbulence channel can be expressed as

P_{e} = \int_{0}^{\infty} P_{e} (I_{s}) f_{I_{s}} (I_{s}) d I_{s} .

For BPSK, the conditional error rate P_e(I_s) is given by

P_{e} (I_{s}) = Q (\sqrt{γ}) = Q (\sqrt{I_{s} \bar{γ}})

where γ̄ denotes the average SNR and Q(·) is the Gaussian Q-function defined as

Q (x) = \int_{x}^{\infty} e^{- t^{2} / 2} / \sqrt{2 π} d t

. With a series expansion of the modified Bessel function [3, 13], the PDF of I_s can be expressed as [3]

f_{I_{s}} (I_{s}) = \frac{π}{sin [π (α - β)]} \sum_{p = 0}^{\infty} [a_{p} (α, β) I_{s}^{p + β - 1} - a_{p} (β, α) I_{s}^{p + α - 1}]

where we define

a_{p} (α, β) ≜ \frac{{(α β)}^{p + β}}{Γ (α) Γ (β) Γ (p - α + β + 1) p!} .

Using the alternative expression of the Gaussian Q-function, we can rewrite Eq. (3) as

P_{e} = \frac{1}{π} \int_{0}^{\frac{π}{2}} \int_{0}^{\infty} exp (- \frac{\bar{γ} I_{s}}{2 {sin}^{2} θ}) f_{I_{s}} (I_{s}) d I_{s} d θ .

Substituting 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

\begin{array}{r} P_{e}^{BPSK} = \frac{B (α - β, 1 - α + β)}{2 π} \sum_{p = 0}^{\infty} [a_{p} (α, β) Γ (p + β) B (\frac{1}{2}, p + β + \frac{1}{2}) {(\frac{\bar{γ}}{2})}^{- (p + β)} \\ - a_{p} (β, α) Γ (p + α) B (\frac{1}{2}, p + α + \frac{1}{2}) {(\frac{\bar{γ}}{2})}^{- (p + α)}] \end{array}

where we have used the identity B(x,y) = Γ(x)Γ(y)/Γ(x + y) as the Beta function [14, Eq. 8.384(1)] and the integral property [14, Eq. 3.621(1), Eq. 8.384(4)]

\int_{0}^{\frac{π}{2}} {(sin θ)}^{p + α} d θ = 2^{p + α - 1} B (\frac{p + α + 1}{2}, \frac{p + α + 1}{2}) = \frac{1}{2} B (\frac{1}{2}, \frac{p + α + 1}{2}) .

With 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

P_{e, asym}^{BPSK} = \frac{Γ (α - β) B (\frac{1}{2}, β + \frac{1}{2})}{2 π Γ (α)} {(\frac{\bar{γ}}{2 α β})}^{- β} .

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 is

\begin{array}{l} P_{e}^{MPSK} & = \frac{B (α - β, 1 - α + β)}{π} \\ \times \sum_{p = 0}^{\infty} [\frac{a_{p} (α, β) Γ (p + β) g_{p} (β)}{{sin}^{2 p + 2 β} (\frac{π}{M})} {(\frac{\bar{γ}}{2})}^{- (p + β)} - \frac{a_{p} (β, α) Γ (p + α) g_{p} (α)}{{sin}^{2 p + 2 α} (\frac{π}{M})} {(\frac{\bar{γ}}{2})}^{- (p + α)}] \end{array}

where g_p(x) denotes an integral identity from Mathematica^® defined as

\begin{array}{r} g_{p} (x) ≜ \frac{M - 1}{M} \int_{0}^{π} {sin}^{2 p + 2 x} (\frac{M - 1}{M} t) d t = [\frac{π^{\frac{3}{2}} sec (p π + π x)}{2 Γ (p + x + 1) Γ (\frac{1}{2} - p - x)} \\ - cos (\frac{M - 1}{M} π) {}_{2}F_{1} (\frac{1}{2}, \frac{1}{2} - p - x; \frac{3}{2}; {cos}^{2} \frac{(M - 1) π}{M})] \end{array}

and ₂F₁(·,·;·;·) is the Gaussian hypergeometric function [14, Eq. 9.100].

DPSK has been shown to be an effective alternative to coherent PSK in coherent WOC systems [15] 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 as

\begin{array}{l} P_{e}^{DPSK} & = & \frac{B (α - β, 1 - α + β)}{2} \\ \times \sum_{p = 0}^{\infty} [a_{p} (α, β) Γ (p + β) {(\frac{\bar{γ}}{2})}^{- (p + β)} - a_{p} (β, α) Γ (p + α) {(\frac{\bar{γ}}{2})}^{- (p + α)}] . \end{array}

The BER for high SNR regimes can be similarly obtained as

P_{e, asym}^{DPSK} = \frac{Γ (α - β)}{2 Γ (α)} {(\frac{\bar{γ}}{2 α β})}^{- β} .

Using Eq. (9) and Eq. (13), we obtain the SNR penalty factor for DPSK with respect to BPSK as

{SNR}_{DPSK-BPSK} = 10 {log}_{10} [\sqrt{π} Γ (β + 1) / Γ (β + 1 / 2)] / β

. 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 I_s in terms of a series as

F_{I_{s}} (I_{0}) = \int_{0}^{I_{0}} f_{I_{s}} (I_{s}) d I_{s} = B (α - β, 1 - α + β) \sum_{p = 0}^{\infty} [\frac{a_{p} (α, β)}{p + β} I_{0}^{p + β} - \frac{a_{p} (β, α)}{p + α} I_{0}^{p + α}] .

With γ = γ̄I_s, one obtains the outage probability as

P_{o} (γ_{th}) = B (α - β, 1 - α + β) \sum_{p = 0}^{\infty} [\frac{a_{p} (α, β)}{(p + β) {\bar{γ}}^{p + β}} γ_{t h}^{p + β} - \frac{a_{p} (β, α)}{(p + α) {\bar{γ}}^{p + α}} γ_{th}^{p + α}]

where γ_th denotes the specific outage probability threshold.

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 as

ε_{J} = | B (α - β, 1 - α + β) | \sum_{p = J + 1}^{\infty} | \frac{a_{p} (α, β)}{(p + β) {\bar{γ}}^{p + β}} γ_{th}^{p + β} - \frac{a_{p} (β, α)}{(p + α) {\bar{γ}}^{p + α}} γ_{th}^{p + α} | .

The truncated error ε_J can then be rewritten as

ε_{J} = \frac{| B (α - β, 1 - α + β) |}{Γ (α) Γ (β)} \sum_{p = J + 1}^{\infty} {(\frac{α β γ_{th}}{\bar{γ}})}^{p} \frac{1}{p!} | b_{p} (α, β) - b_{p} (β, α) |

where we have defined

b_{p} (α, β) ≜ \frac{{(α β γ_{th})}^{β}}{(p + α) Γ (p - α + β + 1) {\bar{γ}}^{β}} .

The summation term in Eq. (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

ε_{J} < \frac{| B (α - β, 1 - α + β) |}{Γ (α) Γ (β)} max_{p > J} {c_{p} (α, β)} exp (\frac{α β γ_{th}}{\bar{γ}})

where c_p(α,β) ≜ |b_p(α,β)−b_p(β,α)|. It is straightforward to show that b_p(α,β) or c_p(α,β) decreases to zero when p approaches ∞, and the truncation error diminishes with increasing index p. From Eqs. (18) and (19), it is also clear that the truncation error diminishes rapidly with increasing average SNR. It can be shown that ε_J 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.

Fig. 1 (a) Exact and approximated BERs for BPSK and DPSK over turbulent Gamma-Gamma channels with J = 35; (b) Exact and approximated SERs for QPSK and 8PSK over turbulent Gamma-Gamma channels with J = 35.

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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 γ_th = 3 dB. Again, our series solutions match the exact error rate and outage curves.

Fig. 2 (a) Exact and approximated outage probability over turbulent Gamma-Gamma channels with J = 30; (b) Outage truncation error over moderately turbulent Gamma-Gamma channels when α = 2.711, β = 2.319.

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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).

5. Conclusion

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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