Ergodic capacity comparison of optical wireless communications using adaptive transmissions

Md. Zoheb Hassan; Md. Jahangir Hossain; Julian Cheng

doi:10.1364/OE.21.020346

1. Introduction

Terrestrial optical wireless communication (OWC) systems are vulnerable to atmospheric turbulence and suffer random irradiance fluctuation, also known as scintillation, due to the random refractive index variation caused by inhomogeneities in both temperature and pressure. Scintillation is a major source of performance degradation, since it affects both the reliability and data rate of the OWC systems [1]. Several fading mitigation techniques have been proposed for OWC systems operating in atmospheric turbulence. These techniques include error control coding with interleaving, maximum-likelihood sequence detection, and spatial diversity using multiple apertures for transmission and/or reception. Adaptive transmission technique that varies basic transmission parameters according to the channel fading is also a promising fading mitigation solution. The main advantage of the adaptive transmission is the provision of high spectral efficiency without sacrificing bit-error rate (BER) requirements. Adaptive transmissions employing phase-shift keying (PSK) and quadrature amplitude modulation (QAM) were studied for the outdoor OWC systems [2, 3].

For an adaptive transmission, ergodic capacity, also known as average channel capacity, is an important information-theoretic performance metric which defines an upper bound for the achievable transmission rate with a negligible error over the fading channels [4, 5]. Recent works on ergodic capacity of OWC systems with coherent detection and intensity modulation with direct detection (IM/DD) are reviewed as follows. The authors in [6, 7] study the ergodic capacity and outage capacity of the coherent OWC systems with lognormal amplitude fluctuations and Gaussian phase fluctuations. The authors in [8] develop closed-form ergodic capacity expressions for a coherent OWC system with heterodyne detection by considering the combined impact of the Gamma-Gamma turbulence and pointing errors. Closed-form ergodic capacity expressions are derived for the IM/DD OWC systems over the Gamma-Gamma, K-distributed, and I-K distributed atmospheric turbulence channels in [9–12]. However, all the aforementioned closed-form ergodic capacity expressions are limited to the constant-power, variable-rate (CPVR) adaptive transmission policy where the channel state information (CSI) is available only at the receiver. Recently the authors in [13] present closed-form ergdoic capacity expressions for the variable-power, variable-rate (VPVR), CPVR, and truncated channel inversion with fixed rate (TCIFR) adaptive transmissions over the lognormal channels, which is commonly used to describe the irradiance fluctuation in a weak turbulence. Previously the authors in [14, 15] study ergodic capacity of the VPVR, CPVR, complete channel inversion with fixed rate (CCIFR), and TCIFR adaptive transmissions for subcarrier IM/DD OWC systems in Gamma-Gamma turbulence using a numerical integration based approach. In [16], the authors also derive ergodic capacity expressions of the VPVR, CPVR, CCIFR and TCIFR adaptive transmissions over the generalized-K fading channels (which is similar to the Gamma-Gamma fading channels) in terms of series solutions. However, their derived expressions for VPVR, CPVR and TCIFR adaptive transmissions are limited only to the integer values of m (a parameter of the generalized-K fading channel). In most of practical scenarios, the parameters of the Gamma-Gamma turbulence channels take non-integer values. As such the closed-form expressions developed in [16] for integer value of the generalized-K fading channel parameter can not be used to evaluate capacity performance of VPVR and TCIFR schemes over the Gamma-Gamma turbulence channels. In [17], the author also derives closed-form ergodic capacity expressions for the VPVR, CPVR, CCIFR and TCIFR adaptive transmission policies for subcarrier IM/DD over the Gamma-Gamma turbulence. However such closed-form results are presented in terms of the special Meijer’s G-function, which does not reveal any insight into the system performance. OWC systems employing polarization multiplexing (POLMUX) with coherent detection has also gained attention because of its ability to boost the data throughput. In [18], the authors analyze and experimentally verify a novel POLMUX system with coherent detection over a lognormal turbulence channel. In a related work, the authors in [19] analyze BER performance of a coherent OWC system with POLMUX in Gamma-Gamma turbulence and Gaussian phase noise. However, to the best of authors’ knowledge, no prior work has been reported on the comparison of ergodic capacities of subcarrier IM/DD, and coherent systems with and without POLMUX when adaptive transmissions are utilized.

The motivation of this work is to analyze and compare the ergodic capacity of the subcarrier IM/DD and coherent OWC systems with and without POLMUX using different adaptive transmission strategies when the CSI is available at both transmitter and receiver. In particular, we consider VPVR, CCIFR, and TCIFR adaptive transmission policies over the Gamma-Gamma turbulence channels. For all the adaptive transmission schemes, we assume the receiver can accurately estimate the channel, and an accurate channel estimation is available at the transmitter via a reliable feedback channel. Highly accurate ergodic capacity expressions are derived using a series expansion of the modified Bessel function and the Mellin transformation of the Gamma-Gamma random variable. Unlike [16], our series solutions for VPVR and TCIFR schemes work well when both parameters of the Gamma-Gamma turbulence channels take non-integer values. In addition, we carry out the asymptotic analysis of our ergodic capacity expressions and reveal some interesting insights into the capacity performance of the considered systems over the Gamma-Gamma turbulence channels. We also present a channel capacity comparison among the subcarrier IM/DD, and coherent OWC systems with and without POLMUX under an average transmitted optical power constraint. Although for non-adaptive transmission, the ergodic capacity is achievable with the assumption of independent turbulence/fading across time, capacity associated with adaptive transmission schemes namely, VPVR, CCIFR, and TCIFR is achievable without such independent assumption. While the detailed reason can be found in [34], for completeness of this paper, we discuss it here briefly. For the VPVR transmission, the time-varying channel can effectively be reduced to a set of time-invariant additive white Gaussian noise (AWGN) channels in parallel [34]. For each of these time-invariant channels, a capacity achieving encoder/decoder pair can be used in a time multiplexed fashion. As such there is no need to have the channel turbulence to be independent. The average capacity of the time-varying channel corresponds to the sum of capacities of the time-invariant channels weighted by the probabilities of having these time-invariant channels over time. For CCIFR transmission, since the transmitter adjusts the transmit signal power according to the channel turbulence in order to maintain a constant received SNR and transmits at a fixed rate, the channel appears to the encoder/decoder pair as a time-invariant AWGN channel. The capacity achieving encoder/decoder pair for the AWGN channel can be used and there is no need to have the channel turbulence to be independent. For TCIFR transmission, the transmitter adjusts the transmit signal power according to the channel turbulence in order to maintain a constant received power and transmits at a fixed rate as long as the received SNR is above a certain threshold, otherwise, the transmission is suspended. Similar argument as CCIFR can be made for TCIFR transmission. Therefore, our derived capacity expressions for the above mentioned three adaptive transmission schemes are valid without independent assumption of atmospheric turbulence over time. Since OWC systems transmit in the range of Gbits/sec, ergodicity assumption for such systems may not be valid. However, this is a standard assumption used in the literature [8–12, 20, 21]. This assumption corresponds to an ideal scenario and obtained results based on this assumption can be considered as theoretical upper bounds on the systems’ performance. Yet these results allow us to compare different systems under consideration and reveal insight into system’s performance improvement as well as design.

2. Channel and system models

2.1. Atmospheric turbulence models

Several statistical models have been proposed to describe turbulence induced irradiance fluctuation. The Gamma-Gamma distribution recently emerges as a useful turbulence model since it has an excellent fit with measured data over a wide range of turbulence conditions [22]. In this paper, we model the optical irradiance I using a normalized Gamma-Gamma random variable (RV) with probability density function (PDF)

f_{I} (I) = \frac{2}{Γ (α) Γ (β)} {(α β)}^{\frac{α + β}{2}} I^{\frac{α + β}{2} - 1} K_{α - β} (2 \sqrt{α β I})

where Γ(·) is the Gamma function [23, Eq. 8.310] and K_α₋_β (·) is the modified Bessel function of the second kind of order α−β [23, Eq. 8.432(9)]. Here α is the effective number of large scale eddies and β is the effective number of small scale eddies. The shaping parameters α and β can not be arbitrary chosen in OWC applications, and they are related to the Rytov variance. Under an assumption of plane wave propagation with negligible inner scale (which corresponds to long propagation distance and small detector area), they can be determined by [24, Eqs. 2a, 2b]. Under this assumption, the inequality α > β holds for most scenarios [24]. Without loss of generality we assume α > β in this work. Scintillation index (

σ_{S I}^{2}

) is another important parameter to measure the turbulence level, and it is given by [25]

σ_{S I}^{2} = \frac{E [I^{2}]}{{(E [I])}^{2}} - 1 = \frac{1}{α} + \frac{1}{β} + \frac{1}{α β} .

2.2. Coherent OWC system

In a coherent OWC system, the information is modulated on the optical carrier amplitude and phase for transmission. After being transmitted through a free-space channel, the received optical signal is coherently mixed with the optical beam generated by a local oscillator (LO), and the combined optical beam is incident on the photodetector. We assume the phase noises from the turbulence and narrow-linewidth lasers change slowly, and they can be compensated using a phase locked loop (PLL) aided phase noise compensation mechanism [26]. This assumption is reasonable because atmospheric turbulence channel is a slowly varying fading channel with channel coherence time on the order of milliseconds [6], and the narrow-linewidth lasers have linewidths on the order of 10 kHz [27]. Consequently, the accumulated phase noises introduced by the atmospheric turbulence and/or lasers have millisecond variation on the timescales. In order to compensate such phase noises the PLL circuits should have a loop bandwidth (also known as the response rate of the PLL circuit) on the order of KHz [27]. Note that a practical PLL can work successfully up to 10 MHz [28]. Consequently, the accumulated phase noises due to the atmospheric turbulence and/or lasers can be tracked and corrected almost perfectly following the photodetection [18, 29]. Therefore, in the subsequent analysis we do not consider the impact of phase noises on the coherent detection. The optical power incident on the photodetector can be expressed as

P_{r} (t) = P_{s} + P_{L O} + 2 \sqrt{P_{s} P_{L O}} cos (ω_{I F} t + ϕ)

where P_s is the received optical signal power, P_LO is the LO power, ϕ is the phase information associated with the modulation order, and ω_IF = ω_c − ω_LO is the intermediate frequency, where ω_c and ω_LO denote the carrier frequency and the LO frequency, respectively. The photocurrent generated by the photodetector can be written as i_r_,_c(t) = i_DC + i_AC(t) + n_c(t) where i_DC = R(P_s + P_LO) and

i_{A C} (t) = 2 R \sqrt{P_{s} P_{L O}} cos (ω_{I F} t + ϕ)

are the DC and AC terms, respectively, R is the responsivity of phtotodetector, and n_c(t) is a shot noise limited AWGN process with variance

σ_{c}^{2}

. In a practical coherent OWC receiver, we have P_LO ≫ P_s, and hence, the DC term of the photocurrent is dominated by the term RP_LO. The shot noise generated by the LO is dominant compared to the background irradiance generated shot noise and/or receiver thermal noise. The shot noise variance can be written as

σ_{c}^{2} = 2 q R P_{L O} Δ f

, where q is the electronic charge, and Δf is the noise equivalent bandwidth of photodetector. The signal-to-noise ratio (SNR) at the input of an electrical demodulator can be written as the ratio of the time-averaged AC photocurrent power to the total noise variance. The received optical power P_s can be written as P_s = AI where A is the photodetector area, and I is the received optical irradiance. Assuming the mean of optical irradiance I is unity, the instantaneous SNR at the input of a electrical demodulator can be obtained as [33]

γ_{c} = \frac{〈 i_{A C}^{2} (t) 〉}{2 q R P_{L O} Δ f} = \frac{R A}{q Δ f} I = \bar{γ_{c}} I

where

\bar{γ_{c}}

is the average SNR per symbol. The average received power,

\bar{P_{s}}

, can be expressed as

\bar{P_{s}} = g \bar{P_{t}}

where g is a constant path-loss factor, and

\bar{P_{t}}

is the average transmitted optical power. The average SNR of the coherent OWC system can be written in terms of the average transmitted optical power as

\bar{γ_{c}} = \frac{R}{q Δ f} {\bar{P}}_{s} = \frac{R}{q Δ f} g {\bar{P}}_{t} .

2.3. POLMUX OWC system with coherent detection

In a coherent POLMUX OWC system, the optical signal from a continuous wave laser source is splitted into two orthogonal polarizations by a polarization beam splitter (PBS). The outputs of PBS are amplitude and/or phase modulated and then combined by a polarization beam combiner (PBC) before being transmitted. Since atmospheric turbulence channel is typically a weak depolarizing channel [30], the fluctuation of the state of polarization (SOP) of the transmitted optical beam is assumed to be in a controllable order [19]. The electric field of the received optical signal at the front end of optical receiver can be written as E_r(t) = [E_x_,_r (t), E_y_,_r(t)]^T where the subscripts x and y denote the two orthogonal channels, and where

\begin{array}{l} E_{x, r} (t) = \sqrt{\frac{P_{s}}{2}} e^{j (ω_{c} t + ϕ)} \\ E_{y, r} (t) = \sqrt{\frac{P_{s}}{2}} e^{j (ω_{c} t + ϕ)} . \end{array}

At the receiver, a polarization controller (PC) is employed to adjust the SOP of the received optical beam. The adjusted optical beam is splitted into two orthogonal channels using a second PBS. The output of the second PBS can be written as [19, Eq. 3]

[\begin{matrix} E_{x} (t) \\ E_{y} (t) \end{matrix}] = [\begin{array}{l} cos ε & - sin ε \\ sin ε & cos ε \end{array}] [\begin{matrix} E_{x, r} (t) \\ E_{y, r} (t) \end{matrix}]

where ε is the polarization control error which describes the SOP difference between the light after PBC at the transmitter and the light after PBS at the receiver. Following [18, Fig. 5], a post photodetection digital signal processing (DSP) aided coherent receiver is employed to recover the transmitted symbols from the two outputs of the PBS. For a fair comparison, we assume the LO power is equally distributed between two orthogonally polarized modes. We express the instantaneous SNR per orthogonal channel as [18, Eq. 20]

\begin{array}{l} γ_{x} = \frac{R^{2} P_{s} P_{L O} (cos ε + sin ε)}{2 q R P_{L O} Δ f} = λ_{1} \bar{γ_{p}} I \\ γ_{y} = \frac{R^{2} P_{s} P_{L O} (cos ε - sin ε)}{2 q R P_{L O} Δ f} = λ_{2} \bar{γ_{p}} I \end{array}

where

\bar{γ_{p}} = \frac{\bar{γ_{c}}}{2} = \frac{R A}{2 q Δ f}

is the normalized average SNR per channel, λ₁ = cosε + sinε, and λ₂ = cosε − sinε. For a system with fully equalized cross-polarization interference, we have ε = 0° or λ₁ = λ₂ = 1. The POLMUX systems with coherent detection are widely used for the fiber-optic applications since the need of significant high data rate predominates the implementation cost and complexity. The major impairments of the fiber-optics POLMUX systems are the polarization mode dispersion (PMD), polarization dependent loss (PDL), and depolarization [31]. However, such impairments are absent and/or negligible in atmospheric optical communications, and consequently, the atmospheric channels are more favourable to the coherent POLMUX systems. Atmospheric channel offers zero PMD (since atmosphere is not a birefringent propagation medium), negligible PDL, and depolarization [32]. Polarization rotation is the only effect that needs to be considered for the coherent POLMUX OWC systems. However, the polarization rotation is a slowly varying process, and the resultant cross-polarization interference can be equalized by a post-photodetection DSP. Consequently, coherent POLMUX OWC system is less complex compared to the fiber-optics POLMUX system with coherent detection. In addition, the recent development of integrated coherent receivers and post-photodetection high speed DSP have made this approach practically feasible [18]. However, a detailed study on the practical implementation of POLMUX system for OWC applications is beyond the scope of current paper and the interested readers are referred to [32] for further reading.

2.4. OWC system with subcarrier intensity modulation/direct detection

In a subcarrier IM/DD OWC link, an RF signal, s(t), pre-modulated with data source, is used to modulate the irradiance of a continuous wave optical beam at the laser transmitter after being properly biased. For an atmospheric turbulence channel, the received photocurrent after direct detection using photodetector can be expressed as

i_{r} (t) = R I (t) A [1 + ξ s (t)] + n (t)

where ξ is the modulation index satisfying the condition −1 ≤ ξs(t) ≤ 1 in order to avoid overmodulation, I(t) is assumed to be a stationary random process for the received irradiance fluctuation caused by atmospheric turbulence, and n(t) is the noise term caused by background radiation (i.e., ambient light) and/or thermal noise, and it is modeled as an AWGN process with variance

σ_{n}^{2}

. The sample I = I(t)|_t=t₀ at a time instant t = t₀ gives the RV I. Normalizing the power of s(t) to unity, the instantaneous SNR at the input of electrical demodulator can be written as [33]

γ_{s} = \frac{{(R A ξ)}^{2}}{2 Δ f (q R I_{b} + 2 k_{b} T_{k} / R_{L})} I^{2} = C_{s} I^{2}

where I_b is the background light irradiance, k_b is Boltzman constant, T_k is the temperature in Kelvin, R_L is the load resistance, and

C_{s} = \frac{{(R A ξ)}^{2}}{2 Δ f (q R I_{b} + 2 k_{b} T_{k} / R_{L})}

is a multiplicative constant. Assuming unit mean received irradiance, i.e., E[I] = 1, and using (2), we define the average SNR of the subcarrier IM/DD OWC system as

\bar{γ_{s}} = C_{s} E [I^{2}] = C_{s} \frac{(α + 1) (β + 1)}{α β}

, and relate

\bar{γ_{s}}

to the instantaneous SNR γ_s by

γ_{s} = \frac{α β \bar{γ_{s}}}{(α + 1) (β + 1)} I^{2} .

The average SNR can also be expressed as

\bar{γ_{s}} = \frac{{(R ξ)}^{2}}{2 Δ f (q R I_{b} + 2 k_{b} T_{k} / R_{L})} E [P_{s}^{2}]

. Using (2) and the relation

{\bar{P}}_{s} = g {\bar{P}}_{t}

, we write average SNR of the subcarrier IM/DD systems in terms of the average transmitted optical power as

\bar{γ_{s}} = \frac{{(R ξ)}^{2} (1 + \frac{1}{α}) (1 + \frac{1}{β})}{2 Δ f (q R I_{b} + 2 k_{b} T_{k} / R_{L})} g^{2} {\bar{P}}_{t}^{2} .

3. Ergodic capacity of coherent OWC system

3.1. Variable-power, variable-rate adaptive transmission

In a VPVR adaptive transmission scheme the transmitter simultaneously adapt the power and data rate in order to maintain a target BER at the receiver for all SNRs. With this adaptive transmission policy, more power and higher data rates are allocated when the channel condition is good, and the transmission is terminated when the received SNR falls below a cutoff level γ_o. For a VPVR adaptive scheme, the ergodic capacity (in bits/s/Hz) of a fading channel can be calculated by [34, Eq. 7]

{〈 C 〉}_{VPVR}^{coh} = \int_{γ_{o}}^{\infty} {log}_{2} (\frac{γ_{c}}{γ_{o}}) f_{γ_{c}} (γ_{c}) d γ_{c} = \frac{1}{ln 2} \int_{γ_{o}}^{\infty} ln (\frac{γ_{c}}{γ_{o}}) f_{γ_{c}} (γ_{c}) d γ_{c} .

Under an average transmit power constraint, the cutoff level γ_o must satisfy [34, Eq. 6]

\int_{γ_{o}}^{\infty} (\frac{1}{γ_{o}} - \frac{1}{γ_{c}}) f_{γ_{c}} (γ_{c}) d γ_{c} = 1

where in Eqs. (13) and (14) f_{γ_c} (γ_c) is the PDF of the SNR γ_c. Using a series expansion of the modified Bessel function of second kind [25, Eq. 15], we obtain the PDF and the cumulative distribution function (CDF) of γ_c, respectively, as

f_{γ_{c}} (γ_{c}) = Λ (α, β) \sum_{p = 0}^{\infty} [a_{p} (α, β) {(\frac{γ_{c}}{\bar{γ_{c}}})}^{p + β} γ_{c}^{- 1} - a_{p} (β, α) {(\frac{γ_{c}}{\bar{γ_{c}}})}^{p + α} γ_{c}^{- 1}]

and

F_{γ_{c}} (γ_{c}) = Λ (α, β) \sum_{p = 0}^{\infty} [\frac{a_{p} (α, β)}{p + β} {(\frac{γ_{c}}{\bar{γ_{c}}})}^{p + β} - \frac{a_{p} (β, α)}{p + α} {(\frac{γ_{c}}{\bar{γ_{c}}})}^{p + α}] .

where

Λ (α, β) ≜ \frac{Γ (α - β) Γ (1 - α + β)}{Γ (α) Γ (β)}

and

a_{p} (x, y) ≜ \frac{{(x y)}^{p + y}}{Γ (p - x + y + 1) p!}

.

3.1.1. Computation of Cutoff SNR γ_o

Equation (14) can be rewritten as

F_{γ_{c}} (γ_{o}) + γ_{o} (\underset{r_{1, c} (α, β, \bar{γ_{c}})}{\underset{︸}{\int_{0}^{\infty} γ_{c}^{- 1} f_{γ_{c}} (γ_{c}) d γ_{c}}} - \underset{r_{2, c} (α, β, γ_{o}, \bar{γ_{c}})}{\underset{︸}{\int_{0}^{γ_{o}} γ_{c}^{- 1} f_{γ_{c}} (γ_{c}) d γ_{c}}}) + γ_{o} = 1.

Substituting

γ_{c} = \bar{γ_{c}} I

into the first integral of Eq. (14), we can write r_1,_c(α, β,

\bar{γ_{c}}

) as

r_{1, c} (α, β, \bar{γ_{c}}) = \frac{1}{\bar{γ_{c}}} \int_{0}^{\infty} I^{- 1} f_{I} (I) d I = \frac{1}{\bar{γ_{c}}} E [I^{- 1}]

where E[I⁻¹] is the first negative integer moment of the Gamma-Gamma RV I. The Mellin transformation of a positive RV X provides all moments of X including positive integer, negative integer, and fractional moments [35]. Using the Mellin transformation of the Gamma-Gamma RV (see Appendix), we obtain r_1,_c(α, β,

\bar{γ_{c}}

) as

r_{1, c} (α, β, \bar{γ_{c}}) = \frac{1}{\bar{γ_{c}}} \frac{Γ (α - 1) Γ (β - 1) α β}{Γ (α) Γ (β)} .

Substituting Eq. (15) into the second integral of Eq. (14), after some algebraic manipulation, we can evaluate r_2,_c(α, β, γ_o,

\bar{γ_{c}}

) as

r_{2, c} (α, β, γ_{o}, \bar{γ_{c}}) = Λ (α, β) \sum_{p = 0}^{\infty} [\frac{a_{p} (α, β)}{(p + β - 1) γ_{o}} {(\frac{γ_{o}}{\bar{γ_{c}}})}^{p + β} - \frac{a_{p} (β, α)}{(p + α - 1) γ_{o}} {(\frac{γ_{o}}{\bar{γ_{c}}})}^{p + α}] .

Finally, applying Eqs. (16), (18), and (20) to Eq. (17), one obtains a series expression involving the cutoff SNR γ_o. For min{α, β} > 1, such expression can be numerically solved along with Eq. (14) in order to compute the cutoff SNR γ_o for a given average SNR. From Eqs. (16), (19) and (20), when

\bar{γ_{c}} \to \infty

we have F_{γ_c} (·) → 0, r_1,_c(α, β,

\bar{γ_{c}}

) → 0, and r_2,_c(α, β, γ_o,

\bar{γ_{c}}

) → 0. As a result, when the average SNR in Eq. (17) approaches infinity, γ_o approaches unity. Therefore, for the Gamma-Gamma turbulence channels, the value of the cutoff SNR, γ_o, is restricted to [0, 1].

3.1.2. Computation of ergodic capacity

In order to facilitate the computation of ergodic capacity of the VPVR adaptive scheme, we rewrite Eq. (13) as

{〈 C 〉}_{VPVR}^{coh} = \frac{1}{ln 2} [\underset{g_{1, c} (α, β, \bar{γ_{c}})}{\underset{︸}{\int_{0}^{\infty} ln (γ_{c}) f_{γ_{c}} (γ_{c}) d γ_{c}}} - \underset{g_{2, c} (α, β, γ_{o}, \bar{γ_{c}})}{\underset{︸}{\int_{0}^{γ_{o}} ln (γ_{c}) f_{γ_{c}} (γ_{c}) d γ_{c}}}] - {log}_{2} (γ_{o}) (1 - F_{γ_{c}} (γ_{o})) .

Here g_1,_c(α, β,

\bar{γ_{c}}

) in (21) is the expectation of lnγ_c, i.e., E[lnγ_c]. In order to estimate the expected value of lnγ_c, we first define Z = lnγ_c. The moment generating function of Z is

M_{Z} (s) = E [e^{s z}] = E [e^{s ln γ_{c}}] = E [γ_{c}^{s}] .

The expected value of lnγ_c can be obtained as

E [ln γ_{c}] = E [Z] = {\frac{d M_{Z} (s)}{d s} |}_{s = 0} = {\frac{d E [γ_{c}^{s}]}{d s} |}_{s = 0} .

It can be easily shown that if

γ_{c} = \bar{γ_{c}} I

, the k-th moment of γ_c is

E [γ_{c}^{k}] = {\bar{γ_{c}}}^{k} E [I^{k}]

, where E[I^k] is the k-th moment of RV I and it is given by [24, Eq. 3]

E [I^{k}] = \frac{Γ (α + k) Γ (β + k)}{Γ (α) Γ (β)} {(\frac{1}{α β})}^{k} .

Using [36, Eq. 06.05.20.0001.01] we can derive g_1,_c(α, β,

\bar{γ_{c}}

) as

g_{1, c} (α, β, \bar{γ_{c}}) = ψ (α) + ψ (β) + ln \bar{γ_{c}} - ln (α β)

where ψ(·) is the Euler’s digamma function [23, Eq. 8.360(1)]. In order to evaluate g_2,_c(α, β, γ_o,

\bar{γ_{c}}

) in Eq. (21), we first recall the fact that lim_z_→0z^a ln z = 0 for any real non-negative z and a. Substituting Eq. (15) into the second integral of Eq. (21) and using [23, Eq. 2.723(1)], we obtain g_2,_c(α, β, γ_o,

\bar{γ_{c}}

) as

\begin{array}{l} g_{2, c} (α, β, γ_{o}, \bar{γ_{c}}) = Λ (α, β) \sum_{p = 0}^{\infty} [a_{p} (α, β) w_{p} (α, β, γ_{o}) {(\frac{γ_{o}}{\bar{γ_{c}}})}^{(p + β)} \\ - a_{p} (β, α) w_{p} (β, α, γ_{o}) {(\frac{γ_{o}}{\bar{γ_{c}}})}^{(p + α)}] \end{array}

where

w_{p} (x, y, γ_{o}) ≜ \frac{ln γ_{o}}{p + y} - \frac{1}{{(p + y)}^{2}}

. To evaluate Eq. (26) numerically, we consider only a summation of finite K + 1 terms. Consequently, a truncation error is introduced due to elimination of infinite terms after the first K + 1 terms. Using a similar method described in [25], we can derive an upper bound for the truncation error, and show that the truncation error rapidly decreases with increasing K values and/or average SNR

\bar{γ_{c}}

. Finally, using Eqs. (25) and (26) one obtains the ergodic capacity for the coherent VPVR adaptive OWC system as

{〈 C 〉}_{VPVR}^{coh} = \frac{1}{ln 2} (g_{1, c} (α, β, \bar{γ_{c}}) - g_{2, c} (α, β, γ_{o}, \bar{γ_{c}})) - {log}_{2} (γ_{o}) (1 - F_{γ_{c}} (γ_{o})) .

3.1.3. Asymptotic high SNR capacity

When the average SNR $\bar{γ_{c}}$ approaches infinity, the value of the cutoff SNR γ_o approaches unity, and consequently, log₂(γ_o) approaches zero. Also, it can be shown that in large SNR regimes $g_{2, c} (α, β, γ_{o}, \bar{γ_{c}}) ≪ g_{1, c} (α, β, \bar{γ_{c}})$ since in large SNR regimes ${(\frac{1}{\bar{γ_{c}}})}^{β} ≪ ln \bar{γ_{c}}$ . Hence, for asymptotically high average SNR we can evaluate the ergodic capacity of a coherent VPVR adaptive OWC system as

\begin{array}{l} {〈 C 〉}_{VPVR}^{coh, asym} & = \frac{ψ (α) + ψ (β)}{ln 2} - {log}_{2} (α β) + 3.3 {log}_{10} {\bar{γ}}_{c} \\ = \frac{ψ (α) + ψ (β)}{ln 2} - {log}_{2} (α β) + {log}_{2} (\frac{R g}{q Δ f}) + 0.33 (10 {log}_{10} {\bar{P}}_{t}) \end{array}

where we have used Eq. (5) to obtain the last equality. Equation (28) revels that the high SNR ergodic capacity of the coherent VPVR system gains 0.33 bits/s/Hz with 1 dB increase of average transmitted optical power.

3.2. Complete channel inversion with fixed rate

Under this adaptive transmission scheme, the transmitter adapts the transmit power (under an average transmit power constraint) according to the channel fading state in order to maintain a constant SNR at the recevier, i.e., inverts the channel fading while maintaining a constant transmission rate. The ergodic capacity (in bits/s/Hz) of a CCIFR adaptive transmission scheme is given by [34, Eq. 9]

{〈 C 〉}_{CCIFR}^{coh} = {log}_{2} (1 + \frac{1}{\int_{0}^{\infty} γ_{c}^{- 1} f_{γ_{c}} (γ_{c}) d γ_{c}}) .

The integral

\int_{0}^{\infty} γ_{c}^{- 1} f_{γ_{c}} (γ_{c}) d γ_{c}

has already been evaluated in (19). As a result, we can express the ergodic capacity of the CCIFR scheme as

{〈 C 〉}_{CCIFR}^{coh} = {log}_{2} (1 + \frac{(α - 1) (β - 1)}{α β} \bar{γ_{c}}) .

It can be shown that for a given channel capacity the CCIFR schemes over the Gamma-Gamma turbulence channels exhibit SNR penalty of

10 {log}_{10} (\frac{α β}{(α - 1) (β - 1)})

dB from the AWGN channel. Equation (30) is similar to the ergodic capacity of the CCIFR scheme over the generalized-K fading channel given by [16, Eq. 29]. However, our derivation approach is different from [16] and our capcity expression in Eq. (30) does not require that β is an integer. The asymptotic capacity in large SNR regimes can be obtained as

{〈 C 〉}_{CCIFR}^{coh, asym} = {log}_{2} (\frac{(α - 1) (β - 1)}{α β}) + {log}_{2} (\frac{R g}{q Δ f}) + 0.33 (10 {log}_{10} {\bar{P}}_{t})

. Hence, CCIFR scheme also gains 0.33 bits/s/Hz with 1 dB increase of average transmitted optical power.

3.3. Truncated channel inversion with fixed rate

Since the CCIFR scheme exhibits a large channel capacity penalty in severe fading, a modified channel inversion scheme is proposed where only the transmit power is adaptive according to the channel fading state provided that the received SNR is above a certain cutoff SNR γ_coh. The channel will not be used if the received SNR falls below γ_coh. For this adaptive transmission policy, the ergodic capacity (in bits/s/Hz) is given by [34, Eq. 12]

{〈 C 〉}_{TCIFR}^{coh} = {log}_{2} (1 + \frac{1}{\int_{γ_{coh}}^{\infty} γ_{c}^{- 1} f_{γ_{c}} (γ_{c}) d γ_{c}}) (1 - P_{o}^{coh} (γ_{coh}))

where

P_{o}^{coh} (γ_{coh})

is the outage probability given by

P_{o}^{coh} (γ_{coh}) = F_{γ_{c}} (γ_{coh})

. We can evaluate the integral

\int_{γ_{coh}}^{\infty} γ_{c}^{- 1} f_{γ_{c}} (γ_{c}) d γ_{c}

as

\int_{γ_{coh}}^{\infty} γ_{c}^{- 1} f_{γ_{c}} (γ_{c}) d γ_{c} = r_{1, c} (α, β, \bar{γ_{c}}) - r_{2, c} (α, β, γ_{coh}, \bar{γ_{c}})

where r_1,_c(·, ·, ·) and r_2,_c(·, ·, ·, ·) are obtained from Eqs. (18) and (20), respectively. Here, the cutoff level γ_coh is selected in order to maximize the channel capacity in Eq. (31) for a given average SNR. Our numerical result shows that the cutoff SNR that maximizes the channel capacity of a coherent TCIFR system increases with an increase of average SNR. Recall that the cutoff SNR of a coherent VPVR system is restricted to the range [0, 1]. Hence, due to increasing cutoff SNR values, coherent TCIFR systems exhibit higher outage probability compared to the coherent VPVR systems. Also note that, for coherent systems, Eqs. (30) and (32) are valid when min{α, β} > 1, which is typically satisfied for the Gamma-Gamma turbulence channels [26].

4. Ergodic capacity of coherent POLMUX OWC system

4.1. Variable-power, variable-rate adaptive transmission

In a coherent VPVR POLMUX scheme the transmit power on each channel is adapted subject to an average transmit power constraint per channel. For a fair comparison it is assumed that the total average transmit power of a coherent OWC system with and without polarization are same, and the total average transmit power is equally allocated between the two orthogonal channels in a POLMUX system. Under this assumption, the ergodic capacity of a coherent POLMUX VPVR scheme can be expressed as

{〈 C 〉}_{VPVR}^{pol} = \underset{{〈 C 〉}_{VPVR}^{pol, x}}{\underset{︸}{\int_{γ_{a}}^{\infty} {log}_{2} (\frac{γ_{x}}{γ_{a}}) f_{γ_{x}} (γ_{x}) d γ_{x}}} + \underset{{〈 C 〉}_{VPVR}^{pol, y}}{\underset{︸}{\int_{γ_{b}}^{\infty} {log}_{2} (\frac{γ_{y}}{γ_{b}}) f_{γ_{y}} (γ_{y}) d γ_{y}}}

where f_{γ_x} (γ_x) and f_{γ_x} (γ_x) are respectively PDFs of γ_x and γ_y, and γ_a and γ_b are the cutoff SNRs for the two orthogonal channels satisfying

\int_{γ_{a}}^{\infty} (\frac{1}{γ_{a}} - \frac{1}{γ_{x}}) f_{γ_{x}} (γ_{x}) d γ_{x} = 1

and

\int_{γ_{b}}^{\infty} (\frac{1}{γ_{b}} - \frac{1}{γ_{y}}) f_{γ_{y}} (γ_{y}) d γ_{y} = 1.

An asymptotic expansion of Eqs. (34) and (35) reveals that both γ_a and γ_b are confined to the range [0, 1]. Using a similar method applied to the coherent OWC system, it can be shown

{〈 C 〉}_{VPVR}^{pol, x} = \frac{1}{ln 2} (g_{1, c} (α, β, λ_{1} \bar{γ_{p}}) - g_{2, c} (α, β, γ_{a}, λ_{1} \bar{γ_{p}})) - {log}_{2} (γ_{a}) (1 - F_{γ_{x}} (γ_{a})) .

and

{〈 C 〉}_{VPVR}^{pol, y} = \frac{1}{ln 2} (g_{1, c} (α, β, λ_{2} \bar{γ_{p}}) - g_{2, c} (α, β, γ_{b}, λ_{2} \bar{γ_{p}})) - {log}_{2} (γ_{b}) (1 - F_{γ_{y}} (γ_{b})) .

where F_{γ_x} (·) and F_{γ_y} (·) are respectively the CDFs of γ_x and γ_y, and they can be obtained by substituting

\bar{γ_{c}}

in Eq. (16) with

λ_{1} \bar{γ_{p}}

and

λ_{2} \bar{γ_{p}}

, respectively. Using a similar argument for obtaining Eq. (28), the asymptotic capacity at high SNR can be expressed as

\begin{matrix} {〈 C 〉}_{VPVR}^{pol, asym} = \frac{2 ψ (α) + 2 ψ (β)}{ln 2} - 2 {log}_{2} (α β) + {log}_{2} (λ_{1} λ_{2}) \\ + 2 {log}_{2} (\frac{R g}{2 q Δ f}) + 0.66 (10 {log}_{10} {\bar{P}}_{t}) . \end{matrix}

From Eq. (38), we observe that the high SNR ergodic capacity of coherent POLMUX VPVR adaptive scheme gains 0.66 bits/s/Hz with 1 dB increase of average transmitted optical power.

4.2. Complete channel inversion with fixed rate

The ergodic capacity of coherent POLMUX CCIFR scheme is obtained as

{〈 C 〉}_{CCIFR}^{pol} = {log}_{2} (1 + \frac{(α - 1) (β - 1)}{α β} λ_{1} \bar{γ_{p}}) + {log}_{2} (1 + \frac{(α - 1) (β - 1)}{α β} λ_{2} \bar{γ_{p}}) .

The asymptotic capacity at high SNR is expressed as

{〈 C 〉}_{CCIFR}^{coh, asym} = 2 {log}_{2} (\frac{(α - 1) (β - 1)}{α β}) + {log}_{2} (λ_{1} λ_{2}) + 2 {log}_{2} (\frac{R g}{2 q Δ f}) + 0.66 (10 {log}_{10} {\bar{P}}_{t}) .

4.3. Truncated channel inversion with fixed rate

The ergodic capacity of coherent POLMUX TCIFR scheme is obtained as

\begin{array}{l} {〈 C 〉}_{TCIFR}^{pol} & = {log}_{2} (1 + \frac{1}{r_{1, c} (α, β, λ_{1} \bar{γ_{p}}) - r_{2, c} (α, β, γ_{pol, x}, λ_{1} \bar{γ_{p}})}) (1 - F_{γ_{x}} (γ_{pol, x})) \\ + {log}_{2} (1 + \frac{1}{r_{1, c} (α, β, λ_{2} \bar{γ_{p}}) - r_{2, c} (α, β, γ_{pol, y}, λ_{2} \bar{γ_{p}})}) (1 - F_{γ_{y}} (γ_{pol, y})) . \end{array}

For a given average SNR or average transmitted power the cutoff SNRs γ_pol,x and γ_pol,y are selected in order to maximize

{log}_{2} (1 + \frac{1}{\int_{γ_{pol, i}}^{\infty} γ_{i}^{- 1} f_{γ_{i}} (γ_{i}) d γ_{i}})

(1 − F_{γ_i} (γ_pol,i)) where i = x, y. Similar to the coherent OWC, Eqs. (39) and (41) are valid for min{α, β} > 1.

5. Ergodic capacity of subcarrier IM/DD OWC system

5.1. Variable-power, variable-rate adaptive transmission

The ergodic capacity of a subcarrier IM/DD VPVR adaptive OWC system is given by

{〈 C 〉}_{VPVR}^{IM} = \frac{1}{ln 2} [\underset{g_{1, s}}{\underset{︸}{\int_{0}^{\infty} ln (γ_{s}) f_{γ_{s}} (γ_{s}) d γ_{s}}} - \underset{g_{2, s}}{\underset{︸}{\int_{0}^{γ_{e}} ln (γ_{s}) f_{γ_{s}} (γ_{s}) d γ_{s}}}] - {log}_{2} (γ_{e}) (1 - F_{γ_{s}} (γ_{e}))

where γ_e is the cutoff SNR satisfying the condition

\int_{γ_{e}}^{\infty} (\frac{1}{γ_{e}} - \frac{1}{γ_{s}}) f_{γ_{s}} (γ_{s}) d γ_{s} = 1.

In Eqs. (42) and (43), f_{γ_s} (γ_s) and F_{γ_s} (·) are respectively the PDF and CDF of the SNR of the subcarrier IM/DD OWC systems, and they are given by

f_{γ_{s}} (γ_{s}) = \frac{Λ (α, β)}{2} \sum_{p = 0}^{\infty} [b_{p} (α, β) {(\frac{γ_{s}}{\bar{γ_{s}}})}^{\frac{p + β}{2}} γ_{s}^{- 1} - b_{p} (β, α) {(\frac{γ_{s}}{\bar{γ_{s}}})}^{\frac{p + α}{2}} γ_{s}^{- 1}]

and

F_{γ_{s}} (γ_{s}) = Λ (α, β) \sum_{p = 0}^{\infty} [\frac{b_{p} (α, β)}{p + β} {(\frac{γ_{s}}{\bar{γ_{s}}})}^{\frac{p + β}{2}} - \frac{b_{p} (β, α)}{p + α} {(\frac{γ_{s}}{\bar{γ_{s}}})}^{\frac{p + α}{2}}] .

where in Eqs. (44) and (45)

b_{p} (x, y) ≜ a_{p} (x, y) {[\frac{(x + 1) (y + 1)}{x y}]}^{\frac{p + y}{2}}

. Similar to the coherent VPVR OWC systems, an asymptotic expansion of Eq. (43) reveals that for subcarrier IM/DD VPVR adaptive system the value of γ_e is also confined to the range [0, 1]. Using a similar approach in deriving Eq. (25), we can evaluate g_1,_s as

g_{1, s} = 2 ψ (α) + 2 ψ (β) + ln \bar{γ_{s}} - 2 ln (α β) .

Following Eq. (26), we obtain g_2,_s as

g_{2, s} = \frac{Λ (α, β)}{2} \sum_{p = 0}^{\infty} [b_{p} (α, β) v_{p} (α, β, γ_{e}) {(\frac{γ_{e}}{\bar{γ_{s}}})}^{\frac{p + β}{2}} - b_{p} (β, α) v_{p} (β, α, γ_{e}) {(\frac{γ_{e}}{\bar{γ_{s}}})}^{\frac{p + α}{2}}]

where

v_{p} (x, y, γ_{e}) ≜ \frac{2 ln γ_{e}}{p + y} - \frac{4}{{(p + y)}^{2}}

. Finally, using Eqs. (46) and (47) we can evaluate the ergodic capacity of the subcarrier IM/DD VPVR adaptive OWC system as

{〈 C 〉}_{VPVR}^{IM} = \frac{1}{ln 2} (g_{1, s} - g_{2, s}) - {log}_{2} (γ_{e}) (1 - F_{γ_{s}} (γ_{e})) .

Using a similar argument in obtaining Eq. (28), we obtain the ergodic capacity of the subcarrier IM/DD VPVR system at high SNR as

\begin{array}{l} {〈 C 〉}_{VPVR}^{IM, asym} & = \frac{2 ψ (α) + 2 ψ (β)}{ln 2} - 2 {log}_{2} (α β) \\ + {log}_{2} (\frac{{(R ξ g)}^{2} (1 + \frac{1}{α}) (1 + \frac{1}{β})}{2 Δ f (q R I_{b} + 2 k_{b} T_{k} / R_{L})}) \end{array} + 0.66 (10 {log}_{10} {\bar{P}}_{t}) .

We observe from Eq. (49) that the ergodic capacity of subcarrier IM/DD VPVR system gains 0.66 bits/s/Hz with 1 dB increase of average transmitted optical power. In high SNR region with 1 dB increase of average transmitted optical power, the subcarrier IM/DD systems provide higher spectral efficiency improvement than the coherent OWC systems without POLMUX. This can be explained by the following analytical arguments. The average SNR of the coherent OWC systems is proportional to the average transmitted optical power whereas the average SNR of the subcarrier IM/DD systems is proportional to square of the the average transmitted optical power, as observed from Eqs. (5) and (12) respectively. Therefore, the high SNR ergodic capacity of the subcarrier IM/DD VPVR scheme has a steeper slope (2 times) with increasing average transmitted optical power compared to the high SNR ergodic capacity of the coherent VPVR scheme. Due to this steeper slope, in high SNR region subcarrier IM/DD systems provide higher spectral efficiency improvement with 1 dB increase in average transmitted optical power than the coherent detection systems. As a result, although the coherent systems significantly outperform the subcarrier IM/DD systems, the capacity performance gap between the coherent and subcarrier IM/DD systems is reduced with the increase of average transmitted optical power. Such a observation is also depicted by our numerical results.

5.2. Channel inversion with fixed rate

The ergodic capacity of a subcarrier IM/DD CCIFR scheme is given by

{〈 C 〉}_{CCIFR}^{IM} = {log}_{2} (1 + \frac{1}{\int_{0}^{\infty} γ_{s}^{- 1} f_{γ_{s}} (γ_{s}) d γ_{s}}) .

Substituting

γ_{s} = \frac{α β \bar{γ_{s}}}{(α + 1) (β + 1)} I^{2}

into Eq. (50) and using Mellin transformation of Gamma-Gamma RV, the ergodic capacity of the subcarrier IM/DD CCIFR scheme can be expressed as

{〈 C 〉}_{CCIFR}^{IM} = {log}_{2} (1 + \frac{Γ (α) Γ (β)}{α β (α + 1) (β + 1) Γ (α - 2) Γ (β - 2)} \bar{γ_{s}}) .

An asympttoic analysis of Eq. (51) reveals that ergodic capacity of subcarrier IM/DD CCIFR scheme gains 0.66 bits/s/Hz with 1 dB increase of average transmitted optical power. The ergodic capacity of a subcarrier IM/DD TCIFR scheme is given by

{〈 C 〉}_{TCIFR}^{IM} = max_{γ_{th} > 0} {log}_{2} (1 + \frac{1}{\int_{γ_{th}}^{\infty} γ_{s}^{- 1} f_{γ_{s}} (γ_{s}) d γ_{s}}) (1 - P_{o}^{IM} (γ_{th}))

where γ_th is the cutoff SNR below which no power adaption is accomplished, and

P_{o}^{IM} (γ_{th})

is the outage probability of the IM/DD OWC systems, and it is given by

P_{o}^{IM} (γ_{th}) = F_{γ_{s}} (γ_{th})

. The integral

\int_{γ_{th}}^{\infty} γ_{s}^{- 1} f_{γ_{s}} (γ_{s}) d γ_{s}

can be expressed as

\int_{γ_{th}}^{\infty} γ_{s}^{- 1} f_{γ_{s}} (γ_{s}) d γ_{s} = \underset{r_{1, s}}{\underset{︸}{\int_{0}^{\infty} γ_{s}^{- 1} f_{γ_{s}} (γ_{s}) d γ_{s}}} - \underset{r_{2, s} (γ_{t h})}{\underset{︸}{\int_{0}^{γ_{t h}} γ_{s}^{- 1} f_{γ_{s}} (γ_{s}) d γ_{s}}}

where

r_{1, s} = \frac{(α + 1) (β + 1)}{\bar{γ_{s}}} \frac{Γ (α - 2) Γ (β - 2) α β}{Γ (α) Γ (β)}

and

r_{2, s} (γ_{t h}) = Λ (α, β) \sum_{p = 0}^{\infty} [\frac{b_{p} (α, β)}{(p + β - 2) γ_{e}} {(\frac{γ_{t h}}{\bar{γ_{s}}})}^{\frac{p + β}{2}} - \frac{b_{p} (β, α)}{(p + α - 2) γ_{e}} {(\frac{γ_{t h}}{\bar{γ_{s}}})}^{\frac{p + α}{2}}] .

For the subcarrier IM/DD systems, Eqs. (51), (54), and (55) are valid when min{α, β} > 2.

6. Numerical results

In this section, we compare the ergodic capacity of the subcarrier IM/DD, and coherent OWC systems with and without POLMUX using our series solutions with the exact ergodic capacity obtained using numerical integration. All series solutions are calculated using the first 31 terms. We consider the following turbulent OWC scenarios where the path loss factor is empirically expressed in terms of visibility, and the turbulence strength is assumed to increase with propagation distance [26] : 1) a 2 Km haze optical channel (in strong turbulence with α = 2.04 and β = 1.10) with 4.35 dB/Km path loss; 2) a 900 m light smoke optical channel (in moderate turbulence with α = 2.50 and β = 2.06) with 9.56 dB/Km path loss; 3) a 700 m light fog optical channel (in weak turbulence α = 4.43 and β = 4.39) with 11.5 dB/Km path loss. In order to generate the numerical plots of capacity versus the average transmitted optical power, we also make the following assumptions [26]: the modulation index ε = 0.85, the photodetector responsivity R = 0.75 A/W, load resistance R_L = 50 Ω, the bit duration T = 1ns with approximate transmission bandwidth 1 GHz, thermal noise variance 3.3 × 10⁻¹³ Amp², and the background noise variance 10⁻¹⁵ Amp². Assuming a typical LO power 10⁻² W [37], the local oscillator-induced shot noise variance is 5 ×10⁻¹² Amp². Also, in our numerical results we have selected the range of the average transmitted optical power from −12 dBm to 6 dBm so that the assumption P_LO ≫ P_s remains valid.

Figures 1(a) and 1(b) respectively present the ergodic capacity of the coherent and coherent POLMUX OWC systems over a strong Gamma-Gamma turbulence channel. In both figures, excellent agreement is observed between the exact channel capacity and capacity obtained by our series approximations. From these figures, it is obvious that the VPVR scheme achieves the largest channel capacity for both the coherent and coherent POLMUX OWC systems. From Fig. 1(b) it is depicted that due to the polarization control error the ergodic capacity of a coherent POLMUX system gets reduced. However, a polarization control error with ε = 10° has no noticeable influence on the ergodic capacity performance. Asymptotic channel capacities for the VPVR and CCIFR schemes are also shown in both figures, and they converge to the exact capacity at large SNR values, as expected.

Fig. 1 Ergodic capacity of coherent and coherent POLMUX OWC systems in Gamma-Gamma turbulence.

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Figures 1(c) and 1(d) plot the ergodic capacities of the coherent and coherent POLMUX OWC systems over a moderate Gamma-Gamma turbulence channel. Excellent agreement is also observed between the exact channel capacity and the capacity obtained by our series approximations. In both figures we observe that the performance gap between different adaptive transmission schemes gets decreased with a decrease of turbulence strength. From Figs. 1(c) and 1(d), the SNR gap between the VPVR and CCIFR schemes is approximately 3 dB for both the coherent and coherent POLMUX OWC systems when α = 2.50 and β = 2.06. This result agrees with the predicted SNR gap of 3.06 dB between the VPVR and CCIFR schemes from our asymptotic ergodic capacity analysis. Asymptotic capacities are also shown in this figure, and they converge to the exact ergodic capacity when $\bar{γ_{c}} \geq 20 dB$ or $\bar{γ_{p}} \geq 20 dB$ .

Figure 2(a) compares the ergodic capacities of the subcarrier IM/DD and coherent POLMUX VPVR schemes over a strong Gamma-Gamma turbulence channel, while Fig. 2(b) compares the ergodic capacities of the subcarrier IM/DD and coherent POLMUX CCIFR schemes over a weak turbulence channel with the same average transmitted power constraint. Figs. 2(a) and 2(b) clearly demonstrate that the coherent POLMUX schemes significantly outperform the coherent and subcarrier IM/DD schemes for the same average transmitted optical power. For example, at an average transmitted optical power of −12 dBm, the coherent POLMUX VPVR scheme achieves 26.28 bits/s/Hz and 25.37 bits/s/Hz channel capacity with polarization control error ε = 0° and ε = 30°, respectively. On the other hand, for the same average transmitted optical power, the coherent and subcarrier IM/DD VPVR schemes attain channel capacity of 14.19 bits/s/Hz and 5.03 bits/s/Hz, respectively. From Fig. 2(b), at an average transmitted optical power of −4 dBm coherent POLMUX CCIFR scheme achieves 32.74 bits/s/Hz and 31.83 bits/s/Hz channel capacity with polarization control error ε = 0° and ε = 30°, respectively. On the other hand, for the same average transmitted optical power, the coherent and subcarrier IM/DD CCIFR schemes obtain channel capacity of 17.42 bits/s/Hz and 9.738 bits/s/Hz, respectively. Both Figs. 2(a) and 2(b) illustrate that the performance gap between the coherent and the subcarrier IM/DD systems gets narrowed with an increase of transmitted optical power. However, the performance gap between coherent POLMUX and subcarrier IM/DD systems remains constant with an increase of transmitted optical power. This is because both coherent POLMUX and subcarrier IM/DD systems have the same slope of 0.66 bits/s/Hz whereas the coherent systems have a slope of 0.33 bits/s/Hz with respect to the dB values of average transmitted optical power.

Fig. 2 Ergodic capacity comparison among the subcarrier IM/DD, and coherent OWC systems with and without POLMUX subject to the same average transmitted optical power.

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7. Conclusion

We have developed highly accurate series ergodic capacity expressions for the subcarrier IM/DD and coherent OWC systems with and without POLMUX employing different adaptive transmission schemes. Our ergodic capacity analysis reveals that the coherent POLMUX OWC systems significantly outperform the coherent and subcarrier IM/DD OWC systems even in the presence of polarization control error.

Appendix

Mellin transformation of a RV I is defined as $M_{I} (z) = E [I^{z - 1}] = \int_{0}^{\infty} I^{z - 1} f_{I} (I) d I$ . Assuming I is a Gamma-Gamma RV, one can show I = XY, where X and Y are two independent Gamma RVs with PDFs respectively given by $f_{X} (x) = \frac{α^{α} x^{α - 1}}{Γ (α)} exp (- α x)$ and $f_{Y} (y) = \frac{β^{β} y^{β - 1}}{Γ (β)} exp (- β y)$ . Since the Mellin transformation of a product of two independent RVs is a product of Mellin transformations of the two RVs [38], we obtain M_I (z) = M_X (z)M_Y (z). Using an integral identity [23, Eq. 3.326(2)], we obtain the Mellin transformations of RVs X and Y, respectively, as $M_{X} (z) = \frac{Γ (α + z - 1)}{Γ (α)} \frac{1}{α^{z - 1}}$ and $M_{Y} (z) = \frac{Γ (β + z - 1)}{Γ (β)} \frac{1}{β^{z - 1}}$ . Finally, the Mellin transformation of I becomes

M_{I} (z) = \frac{Γ (α + z - 1) Γ (β + z - 1)}{Γ (α) Γ (β)} {(\frac{1}{α β})}^{z - 1} .

It follows that E[I⁻¹] and E[I⁻²], which are required by Eqs. (18) and (54), can be obtained by letting z = 0 and z = −1 in Eq. (56) [35].

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Ergodic capacity comparison of optical wireless communications using adaptive transmissions

Abstract

1. Introduction

2. Channel and system models

2.1. Atmospheric turbulence models

2.2. Coherent OWC system

2.3. POLMUX OWC system with coherent detection

2.4. OWC system with subcarrier intensity modulation/direct detection

3. Ergodic capacity of coherent OWC system

3.1. Variable-power, variable-rate adaptive transmission

3.1.1. Computation of Cutoff SNR γ_o

3.1.2. Computation of ergodic capacity

3.1.3. Asymptotic high SNR capacity

3.2. Complete channel inversion with fixed rate

3.3. Truncated channel inversion with fixed rate

4. Ergodic capacity of coherent POLMUX OWC system

4.1. Variable-power, variable-rate adaptive transmission

4.2. Complete channel inversion with fixed rate

4.3. Truncated channel inversion with fixed rate

5. Ergodic capacity of subcarrier IM/DD OWC system

5.1. Variable-power, variable-rate adaptive transmission

5.2. Channel inversion with fixed rate

6. Numerical results

7. Conclusion

Appendix

References and links

Cited By

Figures (2)

Equations (56)

Optics Express

Abstract

1. Introduction

2. Channel and system models

2.1. Atmospheric turbulence models

2.2. Coherent OWC system

2.3. POLMUX OWC system with coherent detection

2.4. OWC system with subcarrier intensity modulation/direct detection

3. Ergodic capacity of coherent OWC system

3.1. Variable-power, variable-rate adaptive transmission

3.1.1. Computation of Cutoff SNR γo

3.1.2. Computation of ergodic capacity

3.1.3. Asymptotic high SNR capacity

3.2. Complete channel inversion with fixed rate

3.3. Truncated channel inversion with fixed rate

4. Ergodic capacity of coherent POLMUX OWC system

4.1. Variable-power, variable-rate adaptive transmission

4.2. Complete channel inversion with fixed rate

4.3. Truncated channel inversion with fixed rate

5. Ergodic capacity of subcarrier IM/DD OWC system

5.1. Variable-power, variable-rate adaptive transmission

5.2. Channel inversion with fixed rate

6. Numerical results

7. Conclusion

Appendix

References and links

Cited By

Figures (2)

Equations (56)

Optics Express

3.1.1. Computation of Cutoff SNR γ_o