Transmit alternate laser selection with time diversity for FSO communications

Antonio García-Zambrana; Rubén Boluda-Ruiz; Carmen Castillo-Vázquez; Beatriz Castillo-Vázquez

doi:10.1364/OE.22.023861

1. Introduction

Atmospheric free-space optical (FSO) transmission using intensity modulation and direct detection (IM/DD) can be considered as an important alternative to consider for next generation broadband in order to support large bandwidth, unlicensed spectrum, excellent security, and quick and inexpensive setup [1]. Nonetheless, this technology is not without drawbacks, being the atmospheric turbulence one of the most impairments, producing fluctuations in the irradiance of the transmitted optical beam, which is known as atmospheric scintillation, severely reducing the link performance [2]. Additionally, an unsuitable alignment between transmitter and receiver due to vibrations in the transmitted beam can produce a greater performance degradation. In [3], the effects of atmospheric turbulence and misalignment considering aperture average effect were considered to study the outage capacity for single-input/single-output (SISO) links. In [4, 5], a wide range of turbulence conditions with gamma-gamma atmospheric turbulence and pointing errors is also considered on terrestrial FSO links, deriving closed-form expressions for the error-rate performance in terms of Meijer’s G-functions.

Error control coding as well as diversity schemes, including both spatial-domain and temporal-domain techniques, can be used over FSO links to mitigate turbulence-induced fading [6–13]. In case of temporal-domain techniques, interleaving is usually adopted to improve the channel coding performance [12]. In [14–16], 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 optimality of selection transmit diversity as an optimal power allocation strategy for shot noise limited FSO systems has been proved in [17], proposing an extension of this scheme to systems with limited feedback. In [18], a novel approximate closed-form bit error-rate (BER) expression is derived for FSO links with transmit laser selection over K-distributed atmospheric turbulence channels. In [19, 20], comparing different diversity techniques, a significant improvement in terms of outage and error-rate performance is demonstrated when multiple-input/multiple-output (MIMO) FSO links based on transmit laser selection are adopted over misalignment fading channels in the context of wide range of turbulence conditions, showing 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. Another remarkable conclusion is the fact that better performance is achieved when increasing the number of transmit apertures instead of the number of receive apertures in order to guarantee a same diversity order.

In this paper, a new transmit alternate laser selection (TALS) scheme with repetition coding for IM/DD FSO communications over atmospheric turbulence and misalignment fading channels is presented. Assuming channel state information (CSI) at the transmitter and receiver and a time diversity order (TDO) limited, we propose the transmit diversity technique based on the rotating selection of TDO out of the available L lasers corresponding to the optical paths with greater values of scintillation. Implementing repetition coding with blocks of TDO information bits, each information bit will be retransmitted TDO times using the TDO largest order statistics in an alternating way. Taking advantage of the temporal diversity limited to TDO, we can assume that the channel coefficients fade independently from one block to the next repetition of the same block. Closed-form asymptotic bit error-rate (BER) expressions are derived when the irradiance of the transmitted optical beam is susceptible to moderate-to-strong turbulence conditions, following a gamma-gamma (GG) distribution, and pointing error effects, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Fully exploiting the potential time-diversity TDO available in the turbulent channel, a significant diversity gain is achieved, obtaining a diversity order of (2L + 1 − TDO)TDO/2. In this way, diversity orders of 2L − 1 and 3L − 3 are achieved for time-diversity orders of 2 and 3, respectively. However, in spite of obtaining a higher diversity order as TDO increases, the adoption of a time diversity order superior to 2 translates to a remarkable coding gain disadvantage, not providing a significant better performance in the context of moderate turbulence when TDO ≥ 3 for FSO systems where BER targets as low as 10⁻⁹ are typically aimed to achieve. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results, showing that asymptotic expressions here obtained lead to simple bounds on the bit error probability in the range from low to high signal-to-noise ratio (SNR).

2. System and channel model

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

y_{m} (t) = η i_{m} (t) x (t) + z (t)

where η is the detector responsivity, assumed hereinafter to be the unity, X ≜ x(t) represents the optical power supplied by the mth source and I_m ≜ i_m(t) the equivalent real-valued fading gain (irradiance) through the optical channel between the mth laser and the receive aperture. Additionally, the fading experienced between source-detector pairs I_m is assumed to be statistically independent. Z ≜ z(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 detector. In this case, the noise can usually be modeled to high accuracy as AWGN with zero mean and variance σ² = N₀/2, i.e. Z ∼ N(0, N₀/2), independent of the on/off state of the received bit. 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_m ≜ y_m(t), however, can assume negative amplitude values. We use Y_m, X, I_m and Z to denote random variables and y_m(t), x(t), i_m(t) and z(t) their corresponding realizations.

The irradiance is considered to be a product of three factors i.e., $I_{m} = ζ_{m} I_{m}^{(a)} I_{m}^{(p)}$ where ζ_m is the deterministic propagation loss, $I_{m}^{(a)}$ is the attenuation due to atmospheric turbulence and $I_{m}^{(p)}$ the attenuation due to geometric spread and pointing errors. ζ_m is determined by the exponential Beers-Lambert law as ζ_m = e^−Φd, where d is the link distance and Φ is the atmospheric attenuation coefficient. It is given by Φ = (3.91/V(km))(λ(nm)/550)^−q where V is the visibility in kilometers, λ is the wavelength in nanometers and q is the size distribution of the scattering particles, being q = 1.3 for average visibility (6 km < V < 50 km), and q = 0.16V + 0.34 for haze visibility (1 km < V < 6 km) [21]. To consider a wide range of turbulence conditions, the gamma-gamma turbulence model proposed in [2] is here assumed. Regarding to the impact of pointing errors, we use the general model of misalignment fading given in [3] by Farid and Hranilovic, wherein the effect of beam width, detector size and jitter variance is considered. A closed-form expression of the combined probability density function (PDF) of I_m was derived in [4] as

f_{I_{m}} (i) = \frac{α_{m} β_{m} φ_{m}^{2}}{A_{0} ζ_{m} Γ (α_{m}) Γ (β_{m})} G_{1, 3}^{3, 0} (\frac{α_{m} β_{m}}{A_{0} ζ_{m}} i | \begin{matrix} φ_{m}^{2} \\ φ_{m}^{2} - 1, α_{m} - 1, β_{m} - 1 \end{matrix}), i \geq 0

where

G_{p, q}^{m, n} [\cdot]

is the Meijer’s G-function [22, eqn. (9.301)] and Γ(·) is the well-known Gamma function. Assuming plane wave propagation, α and β can be directly linked to physical parameters through the following expresions [23]:

α = {[exp (0.49 σ_{R}^{2} / {(1 + 1.11 σ_{R}^{12 / 5})}^{7 / 6}) - 1]}^{- 1}

β = {[exp (0.51 σ_{R}^{2} / {(1 + 0.69 σ_{R}^{12 / 5})}^{5 / 6}) - 1]}^{- 1}

where

σ_{R}^{2} = 1.23 C_{n}^{2} κ^{7 / 6} d^{11 / 6}

is the Rytov variance, which is a measure of optical turbulence strength. Here, κ = 2π/λ is the optical wave number and d is the link distance in meters.

C_{n}^{2}

stands for the altitude-dependent index of the refractive structure parameter and varies from 10⁻¹³ m^−2/3 for strong turbulence to 10⁻¹⁷ m^−2/3 for weak turbulence [2]. It must be emphasized that parameters α and β cannot be arbitrarily chosen in FSO applications, being related through the Rytov variance. It can be shown that the relationship α > β always holds, and the parameter β is lower bounded above 1 as the Rytov variance approaches ∞ [24]. In relation to the impact of pointing errors [3], 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, 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}

, A₀ = [erf(v)]² and erf(·) is the error function [22, eqn. (8.250)]. Nonetheless, the PDF in Eq. (2) appears to be cumbersome to use in order to obtain simple closed-form expressions in the analysis of FSO communication systems. To overcome this inconvenience, the PDF is approximated by using the first two terms of the Taylor expansion at i = 0 as f_{I_m}(i) = a_mi^b_m−1 + c_mi^{b_m + O(ib_m+1}). As proposed in [25], we adopt the approximation f_{I_m}(i) ≈ a_mi^b_m−1 exp(ic_m/a_m). Different expressions for f_{I_m}(i), depending on the relation between the values of φ² and β, can be written as

f_{I_{m}} (i) \approx \frac{φ_{m}^{2} {(α_{m} β_{m})}^{β_{m}} Γ (α_{m} - β_{m})}{{(A_{0} ζ_{m})}^{β_{m}} Γ (α_{m}) Γ (β_{m}) (φ_{m}^{2} - β_{m})} i^{β_{m} - 1} e^{i \frac{α_{m} β_{m} (φ_{m}^{2} - β_{m})}{A_{0} ζ (α_{m} - β_{m} - 1) (β_{m} - φ_{m}^{2} + 1)}}, φ_{m}^{2} > β_{m}

f_{I_{m}} (i) \approx \frac{φ_{m}^{2} {(α_{m} β_{m})}^{φ_{m}^{2}} Γ (α_{m} - φ_{m}^{2}) Γ (β_{m} - φ_{m}^{2})}{{(A_{0} ζ_{m})}^{φ_{m}^{2}} Γ (α_{m}) Γ (β_{m})} i^{φ_{m}^{2} - 1}, φ_{m}^{2} < β_{m}

Therefore, corresponding expressions for the coefficients a_m, b_m and c_m are given by

\begin{array}{l} a_{m} = {\begin{array}{l} \frac{φ_{m}^{2} {(α_{m} β_{m})}^{β_{m}} Γ (α_{m} - β_{m})}{{(A_{0} ζ_{m})}^{β_{m}} Γ (α_{m}) Γ (β_{m}) (φ_{m}^{2} - β_{m})}, & φ_{m}^{2} > β_{m} \\ \frac{φ_{m}^{2} {(α_{m} β_{m})}^{φ_{m}^{2}} Γ (α_{m} - φ_{m}^{2}) Γ (β_{m} - φ_{m}^{2})}{{(A_{0} ζ_{m})}^{φ_{m}^{2}} Γ (α_{m}) Γ (β_{m})}, & φ_{m}^{2} < β_{m} \end{array} \\ \begin{matrix} b_{m} = {\begin{array}{l} β_{m} - 1, & φ_{m}^{2} > β_{m} \\ φ_{m}^{2} - 1, & φ_{m}^{2} < β_{m} \end{array} & c_{m} = {\begin{array}{l} \frac{α_{m} β_{m} (φ_{m}^{2} - β_{m})}{A_{0} ζ (α_{m} - β_{m} - 1) (β_{m} - φ_{m}^{2} + 1)}, & φ_{m}^{2} > β_{m} \\ 0, & φ_{m}^{2} < β_{m} \end{array} \end{matrix} \end{array}

It can be noted that the second term of the Taylor expansion is equal to 0 when the diversity order is not independent of the pointing error effects, i.e. $φ_{m}^{2} < β_{m}$ . Assuming CSI at the transmitter and receiver and temporal diversity order equal to TDO, we propose the transmit diversity technique based on the rotating selection of TDO out of the available L lasers corresponding to the optical paths with greater values of scintillation to transmit a block of TDO information bits. Implementing repetition coding with these blocks of TDO information bits, each information bit will be retransmitted TDO times using the TDO largest order statistics in an alternating way. In this way, information bits in each block will be distributed among the TDO sources out of the available L lasers corresponding to greater values of scintillation, I_(L), I_(L−1) and I_(L−TDO+1), where I₍₁₎, I₍₂₎,..., I_(L) is a new sequence of L auxiliary random variables obtained by arranging the random sequence I₁, I₂,..., I_L in an increasing order of magnitude. Taking advantage of the temporal diversity limited to TDO, we can assume that the channel coefficients fade independently from one block to the next repetition of the same block. As a result of this, different TDO order statistics affecting to the same information bit can be considered independent since they are selected from the different groups of random variables and, hence, they do not influence each other. This fact is key to achieve a relevant improvement in performance.

Since atmospheric scintillation is a slow time varying process relative to typical symbol rates of an FSO system, wherein the coherence time ranges from a few milliseconds to tens of milliseconds [13], we consider the time variations according to the theoretical block-fading model due to the frozen-atmosphere characteristics of optical turbulence, where the channel fade remains constant during a block (corresponding to the channel coherence interval τ_c) and changes to a new independent value from one block to next. Hence, the channel may be assumed to be constant for hundreds of thousands of bits for gigabits per second (Gbps) signaling rates [2]. In other words, channel fades are assumed to be independent and identically distributed (i.i.d.). This temporal correlation can be overcome by means of long interleavers, being usually assumed both in the analysis from the point of view of information theory and error rate performance analysis of coded FSO links [8, 26, 27]. In [13] turbo product code (TPC) as the channel coding scheme is applied to FSO communications, investigating the efficiency of interleaving for different interleaving depths. However, as in [12, 28], we here assume that the interleaver depth can not be infinite and, hence, we can potentially benefit from a degree of time diversity limited equal to TDO. This consideration is justified from the fact that the latency introduced by the interleaver is not an inconvenience for the required application. For example, for a time diversity order available of TDO = 2, i.e. two channel fades i₁ and i₂ per frame, perfect interleaving can be done by simply sending the same information delayed at least the expected fade duration τ_c, as shown experimentally in [29]. In this case, a minimum required buffer size corresponding to (TDO − 1)R_bτ symbols must be assumed, being τ > τ_c and R_b the signaling rate. Based on the concept of temporal-domain diversity reception (TDDR), this idea has been applied for FSO links in [29–31], where two separate channels over the same transmit and receive path are implemented. Both channels carry the same data, but one of the channels is delayed by the expected fade duration. In [32] subcarrier time delay diversity (STDD) is proposed as an alternative means of mitigating the channel fading. In [33] the use of wavelength and time diversity in wireless optical communication systems that operate under different atmospheric turbulence conditions is analyzed, considering BER and outage probability as performance metrics. Decode-and-forward relay-assisted FSO communications using time-diversity are proposed in [34].

3. Error-rate performance analysis

In this section, an optical array based on L laser sources, all pointed towards a distant photodetector, is considered. We present simple bounds on the bit error rate using a pairwise error probability (PEP) analysis in the range from low to high SNR, taking advantage of the simpler expressions in Eq. (4). Here, it is assumed that the average optical power transmitted is P_opt, being adopted an on-off keying (OOK) signaling based on a constellation of two equiprobable points in a one-dimensional space with an Euclidean distance of $d_{E} = 2 P_{opt} \sqrt{T_{b} ξ}$ , where the parameter T_b is the bit period and ξ represents the square of the increment in Euclidean distance due to the use of a pulse shape of high peak-to-average optical power ratio (PAOPR), as explained in a greater detail in [19, appendix]. In the analysis, a perfect Gray code is also assumed and all symbols are transmitted with equal probability. The PEP represents the probability of choosing the space-time codeword X̂ when in fact the codeword X was transmitted. According to the novel transmit diversity transmit squeme here proposed, based on the rotating selection of TDO out of the available L lasers corresponding to the optical paths with greater values of scintillation to transmit a block of TDO information bits, the encoder takes a block of TDO bits x₁, x₂,...,x_TDO in each encoding operation and maps them to the transmit lasers according to a code matrix given by

\begin{matrix} X_{2} = [\begin{matrix} x_{1} & \bar{x_{2}} \\ x_{2} & x_{1} \end{matrix}], & X_{3} = [\begin{matrix} x_{1} & \bar{x_{3}} & \bar{x_{2}} \\ x_{2} & x_{1} & x_{3} \\ x_{3} & x_{2} & x_{1} \end{matrix}], & X_{4} = [\begin{matrix} x_{1} & \bar{x_{4}} & \bar{x_{3}} & \bar{x_{2}} \\ x_{2} & x_{1} & x_{4} & x_{3} \\ x_{3} & x_{2} & x_{1} & x_{4} \\ x_{4} & x_{3} & x_{2} & x_{1} \end{matrix}] \end{matrix}

for values of TDO of 2, 3 and 4, respectively. Each row is associated to the Lth, (L − 1)th, (L − 2)th,... order statistics corresponding to the scintillation and each column is related to the repetition of the block of TDO information bits using the TDO largest order statistics in an alternating way, and assuming that channel coefficients fade independently as a consequence of the temporal diversity. In order to suit space-time coded OOK formats using any pulse shape to optical wireless communications, the complement of a signal x_i can be redefined as

\bar{x_{i}} ≜ - x_{i} + d_{E}

, where by complement we mean that the signal waveform is obtained by reversing the roles of ”on” and ”off”, following the approach presented by Simon in [35]. For instance, according to Eq. (1) and for a time diversity order of 3, the received signals in the first, second and third bit intervals, respectively, are given by r = [r₁ r₂ r₃]^T so that

[\begin{matrix} r_{1} \\ r_{2} \\ r_{3} \end{matrix}] - [\begin{matrix} 0 \\ I_{(L), (2)} \\ I_{(L), (3)} \end{matrix}] \cdot d_{E} = [\begin{matrix} I_{(L), (1)} & I_{(L - 1), (1)} & I_{(L - 2), (1)} \\ I_{(L - 1), (2)} & I_{(L - 2), (2)} & - I_{(L), (2)} \\ I_{(L - 2), (3)} & - I_{(L), (3)} & I_{(L - 1), (3)} \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] + [\begin{matrix} z_{1} \\ z_{2} \\ z_{3} \end{matrix}]

which can be rewritten in a matrix form as r − r_o= H · x + z. Here, I_(1),(n), I_(2),(n),..., I_(L),(n) represent the order statistics corresponding to the nth bit interval. Taking advantage of the temporal diversity, it must be noted that the different order statistics I_(L), I_(L−1) and I_(L−2), affecting to the same information bit, can be considered independent since they are selected from the different groups of random variables and, hence, they do not influence each other. It must be noted that this approach cannot only be viewed as a two-stage transmit laser selection scheme where a space-time block coding scheme is applied to the system using the selected transmit lasers as proposed in antenna-subset selection, technique well known for RF systems [36, 37]. Here, according to the code matrix in (6), the available temporal diversity is fully exploited in order to achieve different channel gains corresponding to each column. In this way, the space-time coding takes into account the repetition of the block of TDO information bits using different laser-subset selections and, hence, different order statistics affecting to the same information bit will not mutually influence each other, since they are selected from different subsets of random variables. For the sake of clarity, it must be emphasized that the space-time codeword in (6) is here defined by the repetition of the block of TDO information bits using the TDO largest order statistics from different TDO subsets in an alternating way. Assuming perfect CSI and a receiver that implements a maximum-likelihood metric, i.e., one that minimizes the metric

m (r, x) = {‖ r - r_{o} - H \cdot x ‖}^{2}

where the squared norm of a matrix is defined by the sum of the magnitude squared of all its elements, the conditional PEP with respect to scintillation coefficients of greater value, I_(L), I_(L−1),... and I_(L−TDO+1), for the shortest error event is given as [38]

P (X \to \hat{X} | {I_{(r)}}_{L - TDO + 1 \leq r \leq L}) = Q (\sqrt{\frac{{(\frac{d_{E}}{TDO})}^{2}}{2 N_{0}} \sum_{k = 1}^{TDO} i_{k}^{2}})

where Q(·) is the Gaussian-Q function. Here, the division of d_E by TDO 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/TDO. Substituting the value of d_E gives

P (X \to \hat{X} | {I_{(r)}}_{L - TDO + 1 \leq r \leq L}) = Q (\sqrt{\frac{2 γ ξ}{{TDO}^{2}} \sum_{k = 1}^{TDO} i_{k}^{2}})

where

γ = P_{opt}^{2} T_{b} / N_{0}

is the average receiver electrical SNR in the presence of the turbulence, knowing that PDF in (2) is normalized. Under the assumption of time diversity, we can exploit independency among fading coefficients to obtain the average PEP, P(X → X̂), by averaging (10) as follows

P (X \to \hat{X}) = \underset{TDO-fold}{\underset{︸}{\int_{0}^{\infty} \int_{0}^{\infty} \dots \int_{0}^{\infty}}} Q (\sqrt{\frac{2 γ ξ}{{TDO}^{2}} \sum_{k = 1}^{TDO} i_{k}^{2}}) \prod_{r = L - TDO + 1}^{L} f_{I_{(r)}} (i_{r - L + TDO}) d i_{r - L + TDO}

According to order statistics [39], the PDF corresponding to I_(r) can be written as

f_{I_{(r)}} (i) = \frac{Γ (L + 1)}{Γ (r) Γ (L - r + 1)} f_{I} (i) {(1 - F_{I} (i))}^{L - r} {(F_{I} (i))}^{r - 1}

being F_I(i) the cumulative density function (CDF) corresponding to the turbulence model. Using the binomial expansion to approximate the term (1 − F_I(i))^L−r and taking advantage of the simpler expressions in Eq. (4) wherein f_I(i) ≈ ai^b−1 exp(ic/a), the PDF in Eq. (12) can be simplified as

f_{I_{(r)}} (i) ≃ \frac{Γ (L + 1)}{Γ (r) Γ (L - r + 1)} f_{I} (i) {(F_{I} (i))}^{r - 1} = \frac{Γ (L + 1) a^{r} b^{1 - r}}{Γ (r) Γ (L - r + 1)} exp (\frac{c (b r + 1)}{a (b + 1)} i) i^{b r - 1}

An union bound on the average BER can be found as [38]

P_{b} (E) \leq \frac{1}{n_{c}} \sum_{X} P (X) \sum_{X \neq \hat{X}} n (X, \hat{X}) P (X \to \hat{X})

where P(X) is the probability that the codeword X is transmitted, n(X, X̂) is the number of information bit errors in choosing another codeword X̂ instead of X and n_c is the number of information bits per transmission. Next, if we were to choose to approximate the average BER by considering only codeword errors of minimum distance and knowing that Gray coding is adopted, we can use (11) to obtain

P_{b} (E) ≃ \frac{TDO + 1}{2} P (X \to \hat{X}) .

To simplify the expression in (11), we use the approximation for the Q-function presented in [40, eq. (14)] (i.e., Q(x) ≃ (1/12)exp(−x²/2) + (1/4)exp(−2x²/3)), finally obtaining

\begin{array}{l} P_{b} (E) & ≃ \frac{TDO + 1}{24} \prod_{r - L - TDO + 1}^{L} \int_{0}^{\infty} exp (- \frac{γ ξ}{{TDO}^{2}} i_{r - L + TDO}^{2}) f_{I_{(r)}} (i_{r - L + TDO}) d i_{r - L + TDO} \\ + \frac{TDO + 1}{8} \prod_{r = L - TDO + 1}^{L} \int_{0}^{\infty} exp (- \frac{4 γ ξ}{3 {TDO}^{2}} i_{r - L + TDO}^{2}) f_{I_{(r)}} (i_{r - L + TDO}) d i_{r - L + TDO} \end{array}

This expression can be simplified as

P_{b} (E) ≃ \frac{TDO + 1}{8} (\frac{1}{3} \prod_{r = L - TDO + 1}^{L} T_{r} (\frac{γ ξ}{{TDO}^{2}}) + \prod_{r = L - TDO + 1}^{L} T_{r} (\frac{4 γ ξ}{3 {TDO}^{2}}))

where T_r(ρ) is defined as follows

T_{r} (ρ) = \int_{0}^{\infty} exp (- ρ i^{2}) f_{I_{(r)}} (i) d i

Taking advantage of the simpler expression in Eq. (13) for the PDF corresponding to the rth order statistic, this integral can be solved in terms of Hermite polynomials H_n(x) [22, eqn. (8.950.1)]. By using the integral representation of the parabolic cylinder function

D_{p} (z) = (1 / Γ (- p)) exp (- z^{2} / 4) \int_{0}^{\infty} x^{- p - 1} exp (- x^{2} / 2 - x z) d x

[22, eqn. (9.241.2)], and the fact that the parabolic cylinder function is connected to the Hermite polynomials as

D_{n} (z) = 2^{- n / 2} exp (- z^{2} / 4) H_{n} (z / \sqrt{2})

[22, eqn. (9.253)], the function T_r(ρ) can be written as

T_{r} (ρ) = \frac{Γ (L + 1) a^{r} b^{1 - r}}{Γ (r) Γ (L - r + 1)} Γ (b r) ρ^{- b r / 2} H_{- b r} (- \frac{c (b r + 1)}{2 a (b + 1) \sqrt{ρ}})

Substituting this expression in Eq. (17) and after some algebraic manipulations, the approximation of average BER is given by

\begin{array}{l} P_{b} (E) & ≃ \frac{TDO + 1}{8} {(b Γ (L + 1))}^{TDO} {(\frac{a {TDO}^{b}}{b {(γ ξ)}^{b / 2}})}^{\frac{TDO}{2} (L + L^{'})} (\prod_{k = L^{'}}^{L} \frac{Γ (b k)}{Γ (k) Γ (L - k + 1)}) \\ \times (\frac{1}{3} \prod_{k = L^{'}}^{L} H_{- b k} (A (b k + 1)) + {(\frac{3^{b / 2}}{2^{b}})}^{\frac{TDO}{2} (L + L^{'})} \prod_{k = L^{'}}^{L} H_{- b k} (\frac{\sqrt{3}}{2} A (b k + 1))) \end{array}

where L′ = L − TDO + 1 and

A = - \frac{c TDO}{2 a (b + 1) {(γ ξ)}^{1 / 2}}

. Considering now that the PDF in Eq. (2) is approximated by using the first term of the Taylor expansion, i.e. assuming in Eq. (20) a value of c = 0, it is straightforward to show that the average BER behaves asymptotically as (Λ_cγξ)^−Λ_d, where Λ_d and Λ_c denote diversity order and coding gain, respectively. At high SNR, if asymptotically the error probability behaves as (Λ_cγξ)^−Λ_d, the diversity order Λ_d determines the slope of the BER versus average SNR curve in a log-log scale and the coding gain Λ_c (in decibels) determines the shift of the curve in SNR. With the purpose of analyzing the diversity order achieved for the TALS scheme here proposed when L transmit lasers are available and since c = 0 implies that A = 0 and, hence, the argument of the Hermite polynomials is 0, we can use in Eq. (20) that

H_{ν} (0) = \sqrt{π} 2^{ν} / Γ (\frac{1 - ν}{2})

[41, eqn. (07.01.03.0001.01)]. It is easy to deduce that the average BER behaves asymptotically as 1/γ^{bTDO(2L−TDO−1)/4}, corroborating a diversity gain of (2L − TDO − 1)TDO/2 in relation to the absence of TALS scheme, wherein the average BER varies as 1/γ^b/2 [20]. In this way, diversity orders of 2L − 1 and 3L − 3 are achieved for time-diversity orders of 2 and 3, respectively. The results corresponding to this FSO scenario with rectangular pulse shapes and ξ = 1 are illustrated in Fig. 1, corroborating previous conclusions. Different weather conditions are considered: haze visibility of 4 km with

C_{n}^{2} = 1.7 \times 10^{- 14} m^{- 2 / 3}

and average visibility of 16 km with

C_{n}^{2} = 8 \times 10^{- 14} m^{- 2 / 3}

, corresponding to moderate and strong turbulence, respectively. Together with λ = 1550 nm, a link distance of d = 3 km and values of normalized beamwidth and jitter of (ω_z/r, σ_s/r) = (5, 1), α and β are calculated from Eq. (3). Monte Carlo simulation results are furthermore included as a reference, confirming the accuracy and usefulness of the analytical expressions here obtained. Due to the long simulation time involved, simulation results only up to BER=10⁻⁸ are included. Simulation results corroborate that asymptotic expressions lead to simple bounds on the bit error probability in the range from low to high SNR. Here, it must be emphasized that a value of TDO = 1 corresponds to the case of transmit laser selection (TLS), obtaining expressions with better accuracy than previously reported in the literature in this general FSO scenario wherein GG fading model with pointing errors has been considered. At this point, it must be noted that the TLS scheme based on the selection of the optical path with a greater value of irradiance has shown to be able to provide better performance compared to general FSO STCs designs, such as conventional OSTBCs and repetition codes [14]. In this way, the TLS scheme, i.e. TALS with TDO = 1, can serve us as a benchmark to show the improvement in performance with the TALS scheme here proposed. In spite of obtaining a higher diversity order as TDO increases, the adoption of a time diversity order superior to 2 translates to a remarkable coding gain disadvantage, not providing a significant better performance in the context of moderate turbulence when TDO ≥ 3 for FSO systems where BER targets as low as 10⁻⁹ are typically aimed to achieve. This can be justified from Eq. (20) wherein the value of TDO is related to the SNR as (γ/TDO²)^{−bTDO(2L−TDO−1)/4}, as a consequence of maintaining 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/TDO, as previously considered in Eq. (9).

Fig. 1 BER performance using TALS over atmospheric turbulence and misalignment fading channels, when different weather conditions (a) $C_{n}^{2} = 1.7 \times 10^{- 14} m^{- 2 / 3}$ and (b) $C_{n}^{2} = 8 \times 10^{- 14} m^{- 2 / 3}$ are assumed for a link distance of d = 3 km and values of normalized beamwidth and jitter of (ω_z/r, σ_s/r) = (5, 1).

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Taking a more in-depth look at the impact of pointing errors and the relation between the values of φ² and β, BER performance in FSO links using TALS with/without pointing errors in the context of moderate turbulence is displayed in Fig. 2, assuming L = {1, 3}, a link distance of d = 5 km and values of normalized beamwidth of ω_z/r = 5 in Fig. 2(a) and ω_z/r = 10 in Fig. 2(b) and normalized jitter of σ_s/r = {1, 3, 6}. Additionally, the numerical simulation of Eq. (15) has been also included for L = 3 with TDO = {1, 2} and different pointing errors in order to contrast the accuracy of the average BER obtained through accounting for error event paths of minimum distance. To analyze the BER performance obtained in a similar context when mis-alignment fading is not present and knowing that the impact of pointing errors in our analysis can be suppressed by assuming A₀ → 1 and φ² → ∞ [3], the corresponding approximate BER expression can be easily derived from (20) as follows

\begin{array}{l} P_{b} (E) & ≃ \frac{TDO + 1}{8} {(\frac{{(β Γ (L + 1))}^{\frac{2}{L + L^{'}}} Γ (α - β) {(α TDO)}^{β}}{β^{- β + 1} Γ (α) Γ (β) {(γ ξ)}^{β / 2}})}^{\frac{TDO}{2} (L + L^{'})} (\prod_{k = L^{'}}^{L} \frac{Γ (β k)}{Γ (k) Γ (L - k + 1)}) \\ \times (\frac{1}{3} \prod_{k = L^{'}}^{L} H_{- β k} (A (β k + 1)) + {(\frac{3^{β / 2}}{2^{β}})}^{\frac{TDO}{2} (L + L^{'})} \prod_{k = L^{'}}^{L} H_{- β k} (\frac{\sqrt{3}}{2} A (β k + 1))) \end{array}

where L′ = L − TDO + 1 and

A = \frac{{(α β)}^{- β + 1} TDO}{2 (β + 1) (α - β - 1) {(γ ξ)}^{1 / 2}}

.

Fig. 2 BER performance using TALS over atmospheric turbulence and misalignment fading channels when a link distance of d = 5 km is assumed in the context of moderate turbulence together with values of normalized beamwidth of (a) ω_z/r = 5 and (b) ω_z/r = 10 and normalized jitter of σ_s/r = {1, 3, 6}.

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As was also concluded in [20], it can be seen that pointing errors can in fact impact the diversity order under assumption of φ² < β, as corroborated for the cases (ω_z/r, σ_s/r) = (5, 3) in Fig. 2(a) and (ω_z/r, σ_s/r) = (10, 6) in Fig. 2(b). Therefore, the main aspect to consider in order to optimize the error-rate performance is the relation between the values of φ² and β, being the adoption of transmitters with accurate control of their beamwidth especially important to satisfy the condition φ² > β in order to maximize the diversity order gain. Once this condition is satisfied, the impact of the pointing error effects translates into a coding gain disadvantage, D[dB], relative to GG channel model without misalignment fading given by

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

As in [19, 20], it is also here corroborated that the impact of pointing errors is not related to the diversity order, obtaining the same coding gain disadvantage regardless the number of transmit lasers. As a result, an identical coding gain of 10.6 decibels between values of normalized beamwidth and jitter of (ω_z/r, σ_s/r) = (5, 1) and (ω_z/r, σ_s/r) = (10, 1) can be contrasted comparing Fig. 2(a) and Fig. 2(b) for L = 1 as well as for L = 3, in excellent agreement with the result obtained from Eq. (22). Nonetheless, when the adoption of transmitters with precise control of their beamwidth is not possible the adoption of the TALS scheme here proposed is an attractive alternative as can be seen in Fig. 2 for values of normalized beamwidth and jitter of (ω_z/r, σ_s/r) = (5, 3) and (ω_z/r, σ_s/r) = (10, 6) when L = 3 and a time diversity order of 2 is available. For instance, an improvement in average SNR above 20 decibels can be achieved for values of normalized beamwidth and jitter of (ω_z/r, σ_s/r) = (5, 3) when considering BER=10⁻⁹ as a practical performance target.

4. Conclusions

In this paper, a new transmit alternate laser selection scheme with repetition coding for IM/DD FSO communications over atmospheric turbulence and misalignment fading channels is analyzed. Assuming CSI at the transmitter and receiver and a time diversity order limited, we propose the transmit diversity technique based on the rotating selection of TDO out of the available L lasers corresponding to the optical paths with greater values of scintillation. Implementing repetition coding with blocks of TDO information bits, each information bit will be retransmitted TDO times using the TDO largest order statistics in an alternating way. Closed-form asymptotic BER expressions using a pairwise error probability analysis in the range from low to high SNR are derived when the irradiance of the transmitted optical beam is susceptible to moderate-to-strong turbulence conditions, following a gamma-gamma distribution, and pointing error effects, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Fully exploiting the potential time-diversity TDO available in the turbulent channel, a significant diversity gain is achieved, obtaining a diversity order of (2L + 1 − TDO)TDO/2. In this way, diversity orders of 2L − 1 and 3L − 3 are achieved for time-diversity orders of 2 and 3, respectively. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results. In relation to the impact of pointing errors, TALS scheme can be adopted as an alternative to achieve a relevant improvement in performance when the optimization of the transmit laser beamwidth as presented in [20] is not possible. Additionally, it must be emphasized that expressions with better accuracy than previously reported in the literature have been here presented for the transmit laser selection scheme (i.e., for the case TDO = 1) in this general FSO scenario wherein GG fading model with pointing errors has been considered.

Acknowledgments

The authors wish to acknowledge the financial support given by Spanish MINECO Project TEC2012-32606.

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Transmit alternate laser selection with time diversity for FSO communications

Abstract

1. Introduction

2. System and channel model

3. Error-rate performance analysis

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (2)

Equations (24)

Optics Express