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SECTION I

Optical wireless technologies are desirable for “last-mile” broadband access and all-optical networks. Optical wireless communication (OWC) systems are well suited for applications requiring high data rates, low deployment costs, enhanced security, and large bandwidths [1]. When an optical signal is transmitted through terrestrial free-space channels, it is subject to fading induced by atmospheric turbulence. Turbulence-induced fading can be a major source of system performance degradation in OWC links.

Direct detection based OWC systems have been extensively studied (see [2], [3], [4], [5], [6], [7], [8] and the references therein). Alternatively, optical coherent detection has been proposed and demonstrated with notable receiver sensitivity improvement for OWC links compared to direct detection [9]. Coherent OWC systems have the information bits encoded directly onto the electric field of the optical beam. A local oscillator is then used at the receiver to extract the information encoded on the optical carrier electric field. Recent advances in digital signal processing (DSP) [10] have made coherent OWC more applicable, which has drawn considerable attention due to its capabilities for excellent background noise/interference rejection, enhanced transmit power efficiency achieved through high receiver sensitivity, and improved spatial and frequency selectivities [1], [11], [12], [13], [14], [15], [16]. Successful coherent OWC systems have been built and tested in several recent investigations. For example, in [15] an OWC link based on coherent homodyne binary phase shift keying (BPSK) was experimentally demonstrated to transmit data at 5.625 Gb/s over a distance of 142 km. More recently, a polarization multiplexed quadrature phase shift keying based OWC system with coherent detection was demonstrated to have a transmission capability at 112 Gb/s with a testbed approximating 2 km propagation [16].

The benefits of using coherent OWC do not come without a cost. The accumulated phase noises introduced to the optical signal electric field necessitate alternative designs (compared to direct detection based systems) to achieve the full benefits of coherent OWC systems. Most investigations have focused on single-input multiple-output (SIMO) coherent OWC systems due to the inherent design complexities of adopting multiple transmitters in coherent OWC links.

Space-time coded systems have demonstrated their usefulness in RF applications [17]. It is found that the complexity of the space–time trellis coding increases exponentially as a function of the diversity order and transmission rate [18]. To reduce the decoding complexity, Alamouti proposed a space–time block coding (STBC) scheme for RF wireless transmission with two transmitters [17]. The Alamouti STBC can be readily adopted in direct detection based OWC systems with an added positive bias. Simon and Vilnrotter proposed a modified Alamouti code [19] for direct detection based OWC employing ON-OFF keying (OOK) and pulse position modulation (PPM). In a recent work, Park *et al.* [20] studied a subcarrier BPSK modulated OWC link using Alamouti-type STBC. Safari and Uysal [21] pointed out that a repetition coded system outperforms orthogonal STBC system with direct detection.

Unlike direct direction schemes, coherent OWC signals are detected based on the phase information carried on electric field. One can show that a repetition coded coherent system suffers severe system performance degradation due to losses in diversity order (from phase noise impacts). Furthermore, it was pointed out in [22, pp. 135-136] that no matter how perfect the phase compensation is performed in a repetition coded coherent receiver, the phase noise absolute difference between the different transmission branches will be present in the reconstructed signal. Efforts have therefore been made to pursue space-time coded coherent OWC systems. In [23], Haas *et al.* presented a space–time code design criteria for OWC links using heterodyne detection based on a large number of transmitters/receivers in lognormal turbulence. In [24], Bayaki and Schober presented simplified space–time code design criteria for coherent and differential OWC links in Gamma-Gamma turbulence. However, a detailed system architecture was not presented in both works. Ntogari *et al\.* recently studied an indoor 2 × 2 STBC OWC system using coherent detection and investigated its error rate performance numerically [25]. However, additional phase noises from different laser transmitters were not considered. Closed-form error rate expressions were not given either. Furthermore, since a flat indoor channel model was used, the authors did not consider turbulence-induced fading nor turbulence-induced phase noise. In [26], we introduced a coherent optical MIMO architecture and studied the error rates of such links employing quadrature phase-shift keying (QPSK) in [27]. More recently, Tang *et al.* introduced a binary polarization shift keying (BPOLSK) modulation scheme for a coherent optical MIMO link and studied the bit-error rate (BER) of BPOLSK using a square-law demodulator [28].

In this paper, we carry out a comprehensive investigation of coherent OWC systems using Alamouti-type STBC. Our system model and analysis take into account turbulence-induced fading as well as phase noises arising from the transmitter lasers, local oscillators, and turbulence channels. We propose two new 2 × 1 STBC OWC systems employing coherent detection. Using a series expansion approach, we study the error rate performance of the proposed systems over the Gamma-Gamma turbulence channels. Highly accurate error rate expressions are obtained in terms of infinite series. The diversity order and coding gain of the proposed coherent STBC systems are revealed with asymptotic analyses. It is shown that coherent detection and STBC can together offer substantial performance improvements for future OWC systems.

SECTION II

A typical OWC system employing coherent detection is shown in Fig. 1. The information bits are modulated on the electric field of an optical signal beam through an external modulator. For a single OWC link, the modulated electric field at a transmitter output is TeX Source $$e_{tx}(t) = E_{0} \exp\,(j\omega t + j\phi + j\phi_{t})\eqno{\hbox{(1)}}$$ where $E_{0}$ denotes the amplitude of the transmitted electric field, $\omega$ denotes the optical carrier frequency of the transmitters, $\phi$ denotes the encoded phase information, and $\phi_{t}$ is the phase noise from transmit laser. After transmission through a free-space channel and mixing with a local oscillator beam, we can express the mixed beam's electric field on the photodetector as TeX Source $$e_{\rm combine}(t) = E_{s}\exp(j\omega t + j\phi + j\phi_{tc}) + E_{LO}\exp(j\omega_{LO}t + j\phi_{LO})\eqno{\hbox{(2)}}$$ where $E_{s}$ is the received electric field amplitude (which is subject to optical scintillation), $E_{LO}$ denotes the amplitude of the local oscillator field, $\omega_{LO}$ is the frequency of the local oscillator beam, and $\phi_{tc} = \phi_{t} + \phi_{c}$ is the overall phase noise from the transmitter to the input of the coherent receiver where $\phi_{c}$ denotes the phase noise induced from a turbulent channel, and $\phi_{LO}$ denotes the local oscillator phase noise arising from the coherent receiver. In the coherent OWC model under construction, we assume that the small photodetector aperture size is on the order of $mm^{2}$ [29], [30]. The transverse phase coherence diameter of the incident beam, being on the order of centimeters [31], is much larger than this aperture size, and therefore the incident optical beam wavefronts can be assumed to be flat with negligible spatial phase noise. Thus, we focus on the temporal phase noise caused by atmospheric turbulence in our coherent system models.

A sufficiently large local oscillator power can be used to make shot noise be dominant, and the resulting noise variance becomes independent of the incident optical signal irradiance. The total optical power incident on the photodetector is
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$$P(t) = P_{s} + P_{LO} + 2\sqrt{P_{s}P_{LO}} \cos (\omega_{IF}t + \phi + \phi_{tc} + \phi_{LO})\eqno{\hbox{(3)}}$$ where $P_{s}$ is the received optical signal power, $P_{LO}$ is the local oscillator power $(P_{LO}\gg P_{s})$, and $\omega_{IF} \buildrel{\Delta}\over{=} \omega - \omega_{LO}$ is the intermediate frequency. Using the fact that the receiver photocurrent is the product of the responsivity $R$ and incident optical power, we can express the detected photocurrent as [13]
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$$i(t) = i_{dc} + i_{ac}(t) + n(t)\eqno{\hbox{(4)}}$$ where
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$$\eqalignno{i_{dc} = &\, R(P_{s} + P_{LO}) \approx RP_{LO}&\hbox{(5)}\cr\noalign{\vskip6pt} i_{ac}(t) = & \, 2R\rho \sqrt{P_{s}P_{LO}}\cos(\omega_{IF}t + \phi + \underbrace{\phi_{tc} + \phi_{LO}}_{\phi_{tr}})& \hbox{(6)}}$$ represent the DC and AC terms in the received photocurrent, respectively, and $n(t)$ is a zero-mean additive white Gaussian noise (AWGN) process due to shot noise^{1}. In (6), $\rho$ is a system constant that depends on the optoelectronic conversion efficiency and is assumed to be unity. We also define $\phi_{tr} = \phi_{tc} + \phi_{LO}$ as the total phase noise.

It is known that for coherent OWC transmit diversity will not achieve diversity order by simply sending the same signal from different transmitters [21], [23], i.e., employing repetition coding, due to the phase noises presented in (6). While receiver diversity schemes, such as maximum ratio combining, equal gain combining and selection combining, have been well studied for coherent OWC links (see [9], [11], [12], [13], [14] and references therein), only limited work can be found on transmit diversity using space–time coding technique [23], [24], [25]. The authors in [23], [24] considered space–time codes for coherent OWC links with optimal code design criteria. A single optical carrier frequency is considered in [23], [24], [25], while the work in [25] focuses on a turbulence-free OWC environment. Our work expands previous work in the following aspects. We propose two new architectures to implement STBC for coherent OWC systems in turbulence environments. The proposed architectures can handle the major impairments in coherent OWC links, and are of practical interest to explore transmit diversity. We analytically show that both of our proposed architectures can achieve the maximum diversity order in coherent OWC employing space–time codes (to be discussed in Sections 4.2 and 5). We provide series error rate expressions to evaluate the system performance for $M$-ary PSK (MPSK) in weak-to-strong turbulence conditions, and these analytical expressions allow us to perform asymptotic analysis in the large SNR regime. Though only 2 × 1 architectures are presented in this paper, our modeling approach can be generated to the multiple-receiver $(2 \times N)$ scenario for enhanced coherent link capabilities with increased diversity order.

We now propose an Alamouti-type STBC coherent MPSK OWC system with 2 transmitters and 1 receiver operating through atmospheric turbulence channels. One split laser beam followed by two external modulators are used as two transmitters. As long as the two external modulators are placed beyond the transverse coherence length, which is on the order of centimeters [3], the turbulence effects from two transmission paths can be considered to be independent. The system structure is shown in Fig. 2. A straightforward Alamouti decoding will not give the desired signals in coherent OWC links over atmospheric turbulence. Additional processing is required and is detailed as follows.

At the output of the transmitters, we have the modulated electric field from the $n$th modulator for the first and the second symbol durations as^{2}
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$$\cases{e_{1}^{(1)}(t) = E_{0}\exp(j\omega t + j\phi_{1} + j\phi_{t, 1}), & $n = 1$\cr\noalign{\vskip2pt} e_{1}^{(2)}(t) = -E_{0}\exp(j\omega t - j\phi_{2} + j\phi_{t, 1}), & $n = 1$\cr\noalign{\vskip2pt} e_{2}^{(1)}(t) = E_{0}\exp(j\omega t + j\phi_{2} + j\phi_{t, 2}), & $n = 2$\cr \noalign{\vskip2pt} e_{2}^{(2)}(t) = E_{0}\exp(j\omega t - j\phi_{1} + j\phi_{t, 2}), & $n = 2$}\eqno{\hbox{(7)}}$$ respectively. In (7), $\phi_{i}$, $i = 1, 2$ denotes the modulated phase information, and $\phi_{t, n}$ denotes the phase noise arising from the $n$th transmitter (laser phase noise).

Following the notational convention used in (7), we use subscript $n \in \{1, 2\}$ to denote elements corresponding to the first and the second transmitters, and we use superscripts with parentheses to denote the corresponding first and second symbol durations. For example, we let $i_{1}(t)$ and $i_{2}(t)$ denote the receiver photocurrent from the first and second transmitters, respectively, and let $i_{1}^{(1)}(t)$ and $i_{1}^{(2)}(t)$ denote the photocurrents received from the first transmitter during the first and second symbol durations, respectively.

The received optical signals at the input of the beam combiner can be written as TeX Source $$\eqalignno{e_{\rm mix}^{(1)}(t) = &\ E_{s, 1}\exp(j\omega t + j\phi_{1} + j\phi_{tc, 1}) + E_{s, 2}\exp(j\omega t + j\phi_{2} + j\phi_{tc, 2})&\hbox{(8)}\cr\noalign{\vskip6pt} e_{\rm mix}^{(2)}(t) = &\, -E_{s, 1}\exp(j\omega t - j\phi_{2} + j\phi_{tc, 1}) + E_{s, 2}\exp(j \omega t - j\phi_{1} + j\phi_{tc, 2})&\hbox{(9)}}$$ where $E_{s, n}$ is the received amplitude of the electric field from the $n$th transmitter, and $\phi_{tc, n}$ denotes the overall phase noise from the $n$th transmitter to the input of the coherent receiver. In obtaining (8) and (9), we have assumed the accumulated optical phase noise and turbulence-induced fading remain constant for at least two consecutive symbol durations. This is a valid assumption because typical turbulence channel statistics result in signal variations that are slow, being on the order of milliseconds, compared to the nanosecond symbol durations of the Gbit/s operational data rates in OWC applications.

At the beam combiner output, we denote the combined electric fields as
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$$\eqalignno{e_{\rm combine}^{(1)}(t) = &\, e_{\rm mix}^{(1)}(t) + E_{LO}\exp(j\omega_{LO}t + j\phi_{LO})&\hbox{(10)}\cr e_{\rm combine}^{(2)}(t) = &\, e_{\rm mix}^{(2)}(t) + E_{LO}\exp(j\omega_{LO}t + j\phi_{LO}).&\hbox{(11)}}$$ At the output of the 90 ° optical hybrid detector^{3}, we can express the converted photocurrent in the first and second symbol durations, respectively, as
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$$\eqalignno{i^{(1)}(t) = &\, RP_{s, 1} + RP_{s, 2} + RP_{LO}R\sqrt{P_{s, 1}P_{s, 2}}\exp \left(j(\phi_{1} - \phi_{2}) + j(\phi_{tc, 1} - \phi_{tc, 2})\right)\cr& + R\sqrt{P_{s, 2}P_{s, 1}}\exp\left(j(\phi_{2} - \phi_{1}) + j(\phi_{tc, 2} - \phi_{tc, 1})\right) + n^{(1)}(t) + R\sqrt{P_{LO}}\exp(j\omega_{IF} t)\cr& \times \left[\sqrt{P_{s, 1}}\exp(j\phi_{1} + j\phi_{tc, 1} - j\phi_{LO}) + \sqrt{P_{s, 2}}\exp (j\phi_{2} + j\phi_{tc, 2} - j\phi_{LO}) \right] + R\sqrt{P_{LO}}\exp(-j\omega_{IF} t) \cr& \times \left[\sqrt{P_{s, 1}}\exp(-j\phi_{1} - j \phi_{tc, 1} + j\phi_{LO}) + \sqrt{P_{s,\, 2}}\exp(-j\phi_{2} - j\phi_{tc, 2} + j\phi_{LO}) \right]&\hbox{(12)} \cr i^{(2)}(t) = &\, RP_{s, 1} + RP_{s, 2} + RP_{LO} - R\sqrt{P_{s, 1}P_{s, 2}}\exp\left(-j(\phi_{1} - \phi_{2}) + j(\phi_{tc, 2} - \phi_{tc, 1})\right) \cr& - R\sqrt{P_{s, 2}P_{s, 1}}\exp \left(-j(\phi_{2} - \phi_{1}) + j(\phi_{tc, 1} - \phi_{tc, 2})\right) + n^{(2)}(t) + R\sqrt{P_{LO}}\exp(j\omega_{IF} t) \cr& \times \left[ \sqrt{P_{s, 2}}\exp(-j\phi_{1} + j\phi_{tc, 2} - j\phi_{LO}) - \sqrt{P_{s, 1}}\exp(-j\phi_{2} + j\phi_{tc, 1} - j \phi_{LO}) \right] + R\sqrt{P_{LO}} \cr& \times \exp(-j\omega_{IF} t)\left[\sqrt{P_{s, 2}}\exp(j\phi_{1} - j \phi_{tc, 2} + j\phi_{LO}) - \sqrt{P_{s, 1}}\exp(j\phi_{2} - j\phi_{tc, 1} + j\phi_{LO}) \right]\qquad& \hbox{(13)}}$$ where $P_{s, n}$ denotes the power of the received signal sent from the $n$th transmitter, $n^{(1)}(t)$ and $n^{(2)}(t)$ are two AWGN processes due to shot noise with variance $\sigma_{n}^{2}$ and are independent and identically distributed (i.i.d.) to each other.

After filtering out the DC components in (12), (13), (9) and (11), we have the filtered photocurrent in the first and second symbol durations, respectively, as TeX Source $$\eqalignno{\mathtilde{i}^{(1)}(t) = &\, R\sqrt{P_{LO}} \exp(j\omega_{IF} t - j\phi_{LO})\left[\sqrt{P_{s, 1}}\exp(j\phi_{1} + j\phi_{tc, 1}) + \sqrt{P_{s,\, 2}}\exp(j \phi_{2} + j\phi_{tc, 2}) \right]\cr& + R\sqrt{P_{LO}}\times \exp(-j\omega_{IF} t + j\phi_{LO})\left[\sqrt{P_{s, 1}}\exp(-j\phi_{1} - j\phi_{tc, 1}) + \sqrt{P_{s, 2}}\exp(-j\phi_{2} - j\phi_{tc, 2}) \right]\cr& + \mathtilde{n}^{(1)}(t)&\hbox{(14)}\cr \mathtilde{i}^{(2)}(t) = &\, R\sqrt{P_{LO}}\exp(j\omega_{IF} t - j \phi_{LO})\left[\sqrt{P_{s, 2}}\exp(-j\phi_{1} + j\phi_{tc, 2}) - \sqrt{P_{s, 1}}\exp(-j\phi_{2} + j\phi_{tc, 1}) \right]\cr& + R\sqrt{P_{LO}}\exp(-j\omega_{IF} t + j\phi_{LO})\left[\sqrt{P_{s, 2}}\exp(j\phi_{1} - j\phi_{tc, 2}) - \sqrt{P_{s, 1}}\exp(j\phi_{2} - j\phi_{tc, 1}) \right]\cr& + \mathtilde{n}^{(2)}(t).&\hbox{(15)}}$$

Here, $\mathtilde{n}^{(1)}(t)$ and $\mathtilde{n}^{(2)}(t)$ are two filtered AWGN processes. According to the Alamouti decoding approach, the reconstructed signals are TeX Source $$\cases{\mathtilde{s}_{1} = h_{1}^{\ast} \mathtilde{i}^{(1)}(t) + h_{2} \left[\mathtilde{i}^{(2)}(t)\right]^{\ast}\cr \mathtilde{s}_{2} = h_{2}^{\ast} \mathtilde{i}^{(1)}(t) - h_{1} \left[\mathtilde{i}^{(2)}(t)\right]^{\ast}} \eqno{\hbox{(16)}}$$ where the superscript ${}^{\ast}$ denotes the complex conjugate.

Following [23] and [24], $h_{1} = \sqrt{P_{LO}P_{s, 1}}\exp(j\phi_{tc, 1} - j \phi_{LO})$ and $h_{2} = \sqrt{P_{LO}P_{s, 2}}\exp(j\phi_{tc, 2} - j\phi_{LO})$ are two complex quantities assumed to be known. This is a valid assumption because the turbulence induced fading as well as the accumulated phase noises from narrow-linewidth lasers and turbulence channels vary on the order of milliseconds. This is sufficiently slow for channel estimation (e.g., embedded DSP estimation circuits) compared to the targeted Gbit/s data rates^{4}.

Substituting (14) and (15), respectively into (16) yields TeX Source $$\eqalignno{\mathtilde{s}_{1}(t) = &\, RP_{LO}\exp(j\omega_{IF}t) \!\left[P_{s, 1}\exp (j\phi_{1}) \!+\! \sqrt{P_{s, 1}P_{s, 2}}\exp\!\left(j\phi_{2} \!+\! j(\phi_{tc, 2} \!-\! \phi_{tc, 1})\right)\right] \!+\! RP_{LO}\exp(-\!j\omega_{IF}t) \cr\noalign{\vskip4pt}&\times \left[P_{s, 1}\exp(-j\phi_{1} - 2j\phi_{ns, 1} + 2j\phi_{LO}) + \sqrt{P_{s, 1}P_{s, 2}}\exp\left(-j\phi_{2} - j(\phi_{tc, 1} + \phi_{tc, 2}) + 2j\phi_{LO}\right) \right]\cr\noalign{\vskip4pt}& +\! RP_{LO}\exp(j\omega_{IF}t) \!\left[P_{s, 2}\exp(j\phi_{1}) \!-\!\! \sqrt{P_{s, 1}P_{s, 2}}\exp\! \left(j\phi_{2} \!+\! j(\phi_{tc, 2} \!-\! \phi_{tc, 1})\right) \right] \!\!+\!\! RP_{LO}\exp(\!-\!j\omega_{IF}t) \cr\noalign{\vskip4pt}&\times \left[P_{s, 2}\exp(-j\phi_{1} + 2j\phi_{ns, 2} - 2j\phi_{LO}) - \sqrt{P_{s, 1}P_{s, 2}}\exp\left(-j\phi_{2} + j(\phi_{tc, 1} + \phi_{tc, 2}) - 2j\phi_{LO}\right) \right] \cr\noalign{\vskip3pt}& + h_{1}^{\ast}\mathtilde{n}^{(1)}(t) + h_{2}\left[\mathtilde{n}^{(2)}(t)\right]^{\ast}& \hbox{(17)}\cr \mathtilde{s}_{2}(t) = &\, RP_{LO}\exp(j\omega_{IF}t) \!\left[\sqrt{P_{s, 1}P_{s, 2}}\exp \left(j \phi_{1} \!+\! j(\phi_{tc,\, 1} \!-\! \phi_{tc,\, 2})\right) \!+\! P_{s,\, 2}\exp(j\phi_{2})\right] \!+\! RP_{LO} \exp(-j\omega_{IF}t) \cr\noalign{\vskip3pt}&\times \left[\sqrt{P_{s, 1}P_{s, 2}}\exp(-j\phi_{1} - j(\phi_{tc, 1} + \phi_{tc, 2}) + 2j\phi_{LO}) \right. \cr\noalign{\vskip3pt}& \qquad + P_{s, 2}\exp(-j\phi_{2} - 2j\phi_{ns, 2} + 2j\phi_{LO})\Big] - RP_{LO}\exp(j\omega_{IF}t) \cr\noalign{\vskip3pt}& \times \left[\sqrt{P_{s, 2}P_{s, 1}} \exp(j\phi_{1} + j(\phi_{tc, 1} - \phi_{tc, 2})) - P_{s, 1}\exp(j\phi_{2})\right] - RP_{LO}\exp(-j\omega_{IF}t) \cr \noalign{\vskip3pt}& \times \left[\sqrt{P_{s, 2}P_{s, 1}}\exp(-j\phi_{1} + j(\phi_{tc, 2} + \phi_{tc, 1}) - 2j \phi_{LO}) - P_{s, 2}\exp(-j\phi_{2} + 2j\phi_{ns, 1} - 2j\phi_{LO})\right] \cr\noalign{\vskip3pt}& + h_{2}^{\ast}\mathtilde{n}^{(1)}(t) - h_{1}\left[\mathtilde{n}^{(2)}(t)\right]^{\ast}.&\hbox{(18)}}$$ By observing (17) and (18), we note that those terms (signal terms and cross terms) containing $\exp(-j\omega_{IF}t)$ are corrupted by phase noises and so are of little use in signal reconstruction. We note that the cross terms containing $\exp(-j\omega_{IF}t)$ will be cancelled out leaving only the signal terms $\exp(j\phi_{1})$ or $\exp(j\phi_{2})$. To recover the desired signals, we multiply $\exp(-j\omega_{IF}t)$ to both (17) and (18) and filter out the high frequency components. As a result, we obtain the desired signals containing only $\exp(j\phi_{1})$ or $\exp(j \phi_{2})$ as TeX Source $$\eqalignno{\mathhat{s}_{1}(t) = &\, RP_{LO}\exp(j\phi_{1}) \underbrace{\left[P_{s, 1} + P_{s, 2}\right]}_{S} + \underbrace{h_{1}^{\ast}n^{(1)}(t)\exp(-j\omega_{IF}t) + h_{2}\left[n^{(2)}(t)\right]^{\ast}\exp(-j\omega_{IF}t)}_{z_{1}(t)}& \hbox{(19)}\cr\noalign{\vskip3pt} \mathhat{s}_{2}(t) = &\, RP_{LO}\exp(j\phi_{2}) \underbrace{\left[P_{s, 1} + P_{s, 2}\right]}_{S} + \underbrace{h_{2}^{\ast}n^{(1)}(t)\exp(-j\omega_{IF}t) - h_{1}\left[n^{(2)}(t)\right]^{\ast} \exp(-j\omega_{IF}t)}_{z_{2}(t)}.&\hbox{(20)}}$$ In (19) and (20), $z_{1}(t)$ and $z_{2}(t)$ denote two filtered AWGN processes. At the output of an electrical demodulator, after sampling, we find the first reconstructed signal as TeX Source $$\mathhat{s}_{1} = RP_{LO}\exp(j\phi_{1})S_{s} + z_{1}\eqno{\hbox{(21)}}$$ where $S_{s} = P_{s, 1} + P_{s, 2}$ is the summed receiving optical signal power, $z_{1}$ is a zero-mean Gaussian random variable (RV) with variance $\sigma_{z}^{2}$. Similarly, we obtain the second reconstructed signal as TeX Source $$\mathhat{s}_{2} = RP_{LO}\exp(j \phi_{2})S_{s} + z_{2}\eqno{\hbox{(22)}}$$ where $z_{2}$ is an AWGN term having the same variance $\sigma_{z}^{2}$ as $z_{1}$.

The signal-to-noise ratio (SNR) at the input of the demodulator of an optical receiver is defined as the ratio of the time-averaged AC photocurrent power to the total noise variance, and it can be calculated as TeX Source $$\gamma = {R^{2}P_{LO}^{2}S_{s}^{2} \over 2S_{s}qRP_{LO}^{2}\Delta f} = {R(P_{s, 1} + P_{s, 2}) \over 2q\Delta f} = \overline{\gamma} \underbrace{\sum_{n = 1}^{2} I_{s, n}}_{Y}\eqno{\hbox{(23)}}$$ where $A$ denotes the photodetector area and $I_{s, n}$ is the incident signal intensity from the $n$th laser transmitter. Here, we have used the relationship $P_{s, n} = AI_{s, n}$ and $\overline{\gamma} = (RA)/(2q\Delta f)$ is defined as the average SNR.

An alternative STBC system using coherent detection can be achieved through the use of two distinct carrier frequencies at the transmitters. The resulting system block diagram is shown in Fig. 3. This system uses laser sources and local oscillators operating at two distinct frequencies. We express the received electric fields from the $n$th transmitter as TeX Source $$\cases{e_{1}^{(1)}(t) = E_{s, 1}\exp(j\omega_{1} t + j\phi_{1} + j \phi_{tc, 1}), & $n = 1$\cr e_{1}^{(2)}(t) = -E_{s, 1}\exp(j\omega_{1} t - j\phi_{2} + j\phi_{tc, 1}), & $n = 1$\cr e_{2}^{(1)}(t) = E_{s, 2}\exp(j\omega_{2} t + j\phi_{2} + j\phi_{tc, 2}), & $n = 2$\cr e_{2}^{(2)}(t) = E_{s, 2}\exp(j\omega_{2} t - j\phi_{1} + j\phi_{tc,\, 2}), & $n = 2$}\eqno{\hbox{(24)}}$$ where $\omega_{n}$ denotes the optical carrier frequency of the $n$th transmitter.

After filtering out the DC photocurrent and the downconversion processing, we can express the baseband-equivalent signal at the output of the receiver photodetectors during the two symbol intervals as
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$$\cases{i_{1}^{(1)}(t) = \sqrt{2}R \sqrt{P_{s, 1}P_{LO}}\exp(j \phi_{1})\exp(j\phi_{tr,\, 1}) + n_{1}^{(1)}(t)\cr i_{1}^{(2)}(t) = - \sqrt{2}R \sqrt{P_{s, 1}P_{LO}}\exp(-j \phi_{2})\exp(j\phi_{tr, 1}) + n_{1}^{(2)}(t)\cr i_{2}^{(1)}(t) = \sqrt{2}R \sqrt{P_{s, 2}P_{LO}}\exp(j\phi_{2}) \exp(j\phi_{tr,\, 2}) + n_{2}^{(1)}(t)\cr i_{2}^{(2)}(t) = \sqrt{2}R \sqrt{P_{s, 2}P_{LO}}\exp(-j\phi_{1})\exp(j \phi_{tr, 2}) + n_{2}^{(2)}(t)}\eqno{\hbox{(25)}}$$ where $P_{s, n}$ denotes the power of the received signal sent from the $n$th transmitter, $\phi_{tr, n}$ is the total phase noise within the $n$th OWC link, $n_{n}^{(1)}(t)$ and $n_{n}^{(2)}(t)$ are i.i.d. zero mean AWGN processes due to shot noise with equal variance $\sigma_{n}^{2}$. Then the photocurrent containing only $s_{1} = \exp(j\phi_{1})$ can be constructed by combining the received photocurrent as
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$$\eqalignno{\mathtilde{i}_{1}(t) = &\, \sqrt{2}R(P_{LO}P_{s, 1} + P_{LO}P_{s, 2})\exp(j\phi_{1}) + \sqrt{2}RP_{LO} \sqrt{P_{s, 1}P_{s, 2}}g^{2}(t)\exp(j\phi_{2})\exp\left(j(\phi_{tr, 2} - \phi_{tr, 1})\right)\cr & - \sqrt{2}RP_{LO}\sqrt{P_{s, 2}P_{s, 1}}g^{2}(t)\exp(j\phi_{2})\exp \left(-j(\phi_{tr, 1} - \phi_{tr, 2})\right) + \sqrt{P_{s, 1}P_{LO}}\exp(j\phi_{tr, 1}) \cr& \times \left[n_{1}^{(1)}(t) + n_{2}^{(1)}(t)\right] + \sqrt{P_{s, 2}P_{LO}}\exp(j\phi_{tr, 2}) \left[n_{1}^{(2)}(t) + n_{2}^{(2)}(t)\right].&\hbox{(26)}}$$ which can be further simplified to be
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$$\displaylines{\mathtilde{i}_{1}(t) = \sqrt{2}R(\underbrace{P_{LO}P_{s, 1} + P_{LO}P_{s, 2}}_{S})\exp(j\phi_{1})\hfill\cr\hfill + \underbrace{\sqrt{P_{s, 1}P_{LO}} \exp(j\phi_{tr, 1}) \left[n_{1}^{(1)}(t) + n_{2}^{(1)}(t)\right] + \sqrt{P_{s, 2}P_{LO}}\exp(j\phi_{tr, 2}) \left[n_{1}^{(2)}(t) + n_{2}^{(2)}(t)\right]}_{v_{1}(t)}.\quad\hbox{(27)}}$$ After the use of an electrical demodulator, the sampled signal can be found as
TeX Source
$$\mathtilde{i}_{1} = \sqrt{2}RS\exp(j\phi_{1}) + v_{1}\eqno{\hbox{(28)}}$$ where $S = P_{LO}(\sum_{n = 1}^{2}P_{s, n})$, and $v_{1}$ is an AWGN term having variance $\sigma_{v_{1}}^{2}$. Similarly, we can construct the photocurrent containing only $s_{2} = \exp(j\phi_{2})$ to be
TeX Source
$$\displaylines{\mathtilde{i}_{2}(t) = \sqrt{2}R(\underbrace{P_{LO}P_{s, 1} + P_{LO}P_{s, 2}}_{S})\exp(j\phi_{2})\hfill\cr\hfill + \underbrace{\sqrt{P_{s, 2}P_{LO}} \exp(-j\phi_{tr, 2}) \left[n_{1}^{(1)}(t) + n_{2}^{(1)}(t)\right] - \sqrt{P_{s, 1}P_{LO}}\exp(j\phi_{tr, 1}) \left[n_{1}^{(2)}(t) + n_{2}^{(2)}(t)\right]}_{v_{2}(t)}\quad\hbox{(29)}}$$ and its sampled signal is
TeX Source
$$\mathtilde{i}_{2} = \sqrt{2}RS\exp(j\phi_{2}) + v_{2}\eqno{\hbox{(30)}}$$ where $v_{2}$ is an AWGN term having variance $\sigma_{v_{2}}^{2}$. Because $n_{n}^{(1)}(t)$ and $n_{n}^{(2)}(t)$ are i.i.d. Gaussian noises with variance $\sigma_{n}^{2}$, the variances of $v_{1}$ and $v_{2}$ are the same as $\sigma_{v_{1}}^{2} = \sigma_{v_{2}}^{2} = 2S\sigma_{n}^{2}$. The SNR at the input of the demodulator in the optical receiver can be found as
TeX Source
$$\gamma = {2R^{2}S^{2} \over 4SqRP_{LO}\Delta f} = {R(P_{LO}P_{s, 1} + P_{LO}P_{s, 2}) \over 2qP_{LO}\Delta f} = \overline{\gamma} \underbrace{\sum_{n = 1}^{2} I_{s, n}}_{Y}.\eqno{\hbox{(31)}}$$ We note that the SNR in (31) has the same expression as that derived in (23). Therefore, the two proposed systems will offer the same error rate performance ^{5}. We will use this consistent SNR definition in our error rate analysis.

SECTION III

Turbulence-induced irradiance fluctuation phenomena have been extensively studied in the literature. There are several well-defined statistical models that can be used to describe irradiance fluctuation, including lognormal-Rician, lognormal, $K$, $I - K$, and Gamma-Gamma models [37]. It has been widely accepted that, among those proposed turbulence models, the lognormal turbulence model accurately describes irradiance fluctuations in weak turbulence conditions, and the $K$-distributed turbulence model accurately describes irradiance fluctuations in strong turbulence conditions. Recently, the Gamma-Gamma distribution has been demonstrated to be a useful turbulence model as it provides excellent fits with simulation data in a wide range of turbulence conditions [36], [37]. In this paper, we model the optical irradiance $I_{s, n}$ in (23) and (31) as a Gamma-Gamma RV having a probability density function (PDF) given by TeX Source $$f_{I}(I_{s, n}) = {2 \over \Gamma(\alpha)\Gamma(\beta)}(\alpha\beta)^{{\alpha + \beta \over 2}}I_{s, n}^{{\alpha + \beta \over 2} - 1} K_{\alpha - \beta} (2\sqrt{\alpha\beta I_{s, n}}), \quad I_{s, n}\ <\ 0\eqno{\hbox{(32)}}$$ where $\Gamma(\cdot)$ is the Gamma function, $K_{\alpha - \beta}(\cdot)$ denotes the modified Bessel function of the second kind of order $\alpha - \beta$. The positive parameter $\alpha$ represents the effective number of large-scale cells of the scattering process, and the positive parameter $\beta$ represents the effective number of small-scale cells of the scattering process in the atmosphere channel. We emphasize that the parameters $\alpha$ and $\beta$ cannot be arbitrarily chosen for OWC applications as they are determined by the Rytov variance $\sigma_{R}^{2}$ [37]. Assuming plane wave propagation and negligible inner-scale, the shape parameters of the Gamma-Gamma model have the relationship $\alpha\ > \ \beta$ and can be linked to the Rytov variance through [37] TeX Source $$\eqalignno{\alpha = &\, \left[\exp\left({0.49 \sigma_{R}^{2} \over \left(1 + 1.11\sigma_{R}^{12/5}\right)^{7/6}}\right) - 1 \right]^{-1}&\hbox{(33)}\cr \noalign{\vskip6pt} \beta = &\, \left[\exp\left({0.51\sigma_{R}^{2} \over \left(1 + 0.69\sigma_{R}^{12/5} \right)^{5/6}}\right) - 1 \right]^{-1}.&\hbox{(34)}}$$ The relationship $\alpha\ > \ \beta$ does not hold when the Rytov variance is small and when the finite inner-scale is non-negligible. However, our OWC system analysis through Gamma-Gamma turbulence can be similarly performed for $\alpha\ < \ \beta$ as the two shape parameters $\alpha$ and $\beta$ are symmetric in the Gamma-Gamma turbulence model.

We first look for the moment generating function (MGF) of $I_{s, n}$. To continue, we make use of a series expansion of the modified Bessel function of the second kind [6, Eq. (6)] TeX Source $$K_{\nu}(x) = {\pi\csc(\pi\nu) \over 2}\sum_{p = 0}^{\infty} \left[{(x/2)^{2p - \nu} \over \Gamma(p - \nu + 1)p!} - {(x/2)^{2p + \nu} \over \Gamma(p + \nu + 1)p!}\right]\eqno{\hbox{(35)}}$$ where $\nu \ \notin\ {\ssb Z}$ and $\vert x\vert\ <\ \infty$. In the Gamma-Gamma turbulence model, we have $\nu = \alpha - \beta$, and $\nu\ \notin\ {\ssb Z}$ can be satisfied with most practical scenarios. Substituting (35) and the relationship $\pi\csc(x\pi) = \Gamma(x)/\Gamma(1 - x)$ into (32) and using an integral identity [38], one can obtain the MGF of $I_{s, n}$ as [6] TeX Source $$\eqalignno{M_{I_{s, n}}(s) = &\, E\left[ \exp(sI_{s, n})\right] = \int\limits_{0}^{\infty} \exp(sI_{s, n})f_{I}(I_{s, n})dI_{s, n}\cr = &\, \left(\sum_{p = 0}^{\infty} a_{p}(\alpha, \beta)(-s)^{-(p + \beta)} + \sum_{p = 0}^{\infty} a_{p}(\beta, \alpha)(-s)^{-(p + \alpha)}\right)&\hbox{(36)}}$$ where we define TeX Source $$a_{p}(\alpha, \beta)\buildrel{\Delta}\over{=} {(\alpha\beta)^{p + \beta}\Gamma(p + \beta)B(\alpha - \beta, 1 - \alpha + \beta) \over \Gamma(\alpha)\Gamma(\beta)\Gamma(p - \alpha + \beta + 1)p!}.\eqno{\hbox{(37)}}$$ In (37), we have used the relationship $B(x, y) = \Gamma(x)\Gamma(y)/\Gamma(x + y)$ where $B(\cdot, \cdot)$ denotes the Beta function defined as $B(x, y)\buildrel{\Delta}\over{=}\int_{0}^{1} t^{x - 1}(1 - t)^{y - 1}dt$ [38].

We now derive the MGF of $Y = \sum_{n = 1}^{2} I_{s, n}$ defined in (23) or (31). As $I_{s, n}$'s are i.i.d. RVs, the MGF of $Y$ can be expressed as $M_{Y}(s) = [M_{I_{s, n}}(s)]^{2}$. With the help of a binomial expansion and [38, Eq. (0.316)], the MGF of the summed RV $Y$ can be obtained as TeX Source $$M_{Y}(s) = \sum_{m = 0}^{2} {2 \choose m} \sum_{p = 0}^{\infty} b(2 - m, m, \alpha, \beta) (-s)^{-p - 2\beta - m(\alpha - \beta)}\eqno{\hbox{(38)}}$$ where $b(x, y)$ is a computational coefficient defined as TeX Source $$b(x, y, \alpha, \beta) = a_{p}^{[x]}(\alpha, \beta) \ast a_{p}^{[y]}(\beta, \alpha).\eqno{\hbox{(39)}}$$ In (39), ∗ denotes the convolution operator, and $a_{p}^{[x]}(\alpha, \beta)$ denotes the convolution of $a_{p}(\alpha, \beta)$ with itself $x - 1$ times, e.g., $a_{p}^{[2]}(\alpha, \beta) = a_{p}(\alpha, \beta)\ast a_{p}(\alpha, \beta)$. Specifically, we have $a_{p}^{[0]}(\alpha, \beta) = 1$ and $a_{p}^{[1]}(\alpha, \beta) = a_{p}(\alpha, \beta)$. The obtained MGF expression in (38) can facilitate the ensuing performance evaluation and asymptotic analysis.

SECTION IV

In a coherent OWC turbulence link, the average symbol error rate (SER) of MPSK can be expressed as [13, Eq. (12)] TeX Source $$P_{e} = {1 \over \pi} \int\limits_{0}^{{(M - 1)\pi \over M}} M_{Y}\left(- {\sin^{2}(\pi/M)\overline{\gamma} \over 2\sin^{2} \theta}\right)d\theta.\eqno{\hbox{(40)}}$$ Substituting (38) into (40), we can find the SER for an Alamouti-coded OWC system with coherent detection as TeX Source $$\eqalignno{P_{e, M} = &\, {1 \over \pi} \sum_{m = 0}^{2} {2 \choose m} \sum_{p = 0}^{\infty} (\mu_{M}\overline{\gamma})^{-p - 2\beta - m(\alpha - \beta)} b(2 - m, m, \alpha, \beta) \int\limits_{0}^{{(M - 1)\pi \over M}} (\sin^{2}\theta)^{p + 2\beta + m(\alpha - \beta)}d \theta\cr\noalign{\vskip3pt} = &\, {1 \over \pi} \sum_{m = 0}^{2} {2 \choose m} \sum_{p = 0}^{\infty} (\mu_{M} \overline{\gamma})^{-p - 2\beta - m(\alpha - \beta)} b(2 - m, m, \alpha, \beta) \varphi\left(p + 2\beta + m(\alpha - \beta), {M - 1 \over M}\right)\qquad&\hbox{(41)}}$$ where we define $\mu_{M} = (\sin^{2}(\pi/M))/2$. In deriving (41), we have used the following integral identity [39]: TeX Source $$\varphi(x, \eta) = \int\limits_{0}^{\eta\pi}\sin^{2x}\theta d \theta = {\pi^{3/2}\sec(\pi x) \over 2\Gamma(x + 1)\Gamma\left({1 \over 2} - x\right)} - \cos(\eta\pi) \, {}_{2}F_{1}\left({1 \over 2}, {1 \over 2} - x;{3 \over 2};\cos^{2}\eta\pi\right)\eqno{\hbox{(42)}}$$ where ${}_{2}F_{1}(\cdot, \cdot;\cdot;\cdot)$ denotes the Gaussian hypergeometric function defined in [38, Eq. 9.100].

Note that when $M = 2$, i.e., binary phase-shift keying (BPSK) modulation, $\varphi(x, (1/2))$ reduces to TeX Source $$\varphi\left(x, {1 \over 2}\right) = {\sqrt{\pi}\Gamma\left(x + {1 \over 2}\right) \over 2\Gamma(x + 1)} = {1 \over 2}B\left({1 \over 2}, x + {1 \over 2}\right).\eqno{\hbox{(43)}}$$ Therefore, we can find the average BER for Alamouti-coded coherent systems using BPSK as TeX Source $$P_{e, 2} = {1 \over 2\pi} \sum_{m = 0}^{2} {2 \choose m} \sum_{p = 0}^{\infty} b(2 - m, m, \alpha, \beta) B\left({1 \over 2}, p + 2\beta + m(\alpha - \beta) + {1 \over 2}\right) \left({\overline{\gamma} \over 2}\right)^{-p - 2\beta - m(\alpha - \beta)}.\eqno{\hbox{(44)}}$$ When QPSK modulation is considered, one can derive the average SER for Alamouti-coded coherent systems using QPSK as TeX Source $$\displaylines{P_{e, 4} = {1 \over 2\pi} \sum_{m = 0}^{2} {2 \choose m} \sum_{p = 0}^{\infty} b(2 - m, m, \alpha, \beta)\hfill\cr\noalign{\vskip4pt}\hfill \times \left[B\left({1 \over 2}, p + 2\beta + m(\alpha - \beta) + {1 \over 2}\right) - \varphi\left(p + 2\beta + m(\alpha - \beta), {3 \over 4}\right) \right] \left({\overline{\gamma} \over 4}\right)^{-p - 2\beta - m(\alpha - \beta)}\quad \hbox{(45)}}$$ where $\varphi(x, (3/4))$ is obtained by making use of the Gaussian Hypergeometric function definition as TeX Source $$\varphi\left(x, {3 \over 4}\right) = \int\limits_{0}^{{\pi \over 4}}\sin^{2x}\theta d\theta = 2^{-x - {3 \over 2}} \int\limits_{0}^{1} y^{x - {1 \over 2}}\left(1 - {y \over 2} \right)^{- {1 \over 2}}dy = {\sqrt{2}{}_{2}F_{1}\left({1 \over 2}, x + {1 \over 2};x + {3 \over 2};{1 \over 2} \right) \over 2^{x + 1}(2x + 1)}.\eqno{\hbox{(46)}}$$ The error rate solutions derived in (41), (43) and (45) contain only the Gamma function, and the Gaussian Hypergeometric function can be rapidly calculated with standard MATLAB or Maple software.

The asymptotic error rate of a digital communication system in fading channels can be obtained with diversity order $G_{d}$ and coding gain $G_{c}$ as TeX Source $$P_{e}^{\infty}\approx (G_{c}\overline{\gamma})^{-G_{d}}.\eqno{\hbox{(47)}}$$ For the proposed Alamouti-type STBC OWC systems using coherent detection, we observe that the infinite summation terms appearing in (41), (43) and (45) diminish rapidly with increasing $\overline{\gamma}$. This implies that the derived series solutions with finite terms can be highly accurate in large SNR regions.

Since the terms in (41) also decrease with the increasing index $p \ (p \in {\ssb N})$, the dominant term in the SER solution would only contain $(\mu_{M}\overline{\gamma})^{-2\min\{\alpha, \beta\}}$. As a result, when $\overline{\gamma}$ approaches $\infty$, the leading term when $p = 0$ in (41) becomes the dominant term. Therefore, one obtains the asymptotic SER approximation in large SNR regimes for MPSK modulated Alamouti-type STBC systems over the Gamma-Gamma turbulence channels as TeX Source $$P_{e, M}^{\infty}\approx {\left[a_{0}(\alpha, \beta)\right]^{2}\varphi\left(2\min\{\alpha, \beta\}, {M - 1 \over M}\right) \over \pi} \, (\mu_{M}\overline{\gamma})^{-2\min\{\alpha, \beta\}}.\eqno{\hbox{(48)}}$$

The closed-form expression of asymptotic error rate shows that the coding gain is $G_{c} = \mu_{M}\pi^{(1/(2\min\{\alpha, \beta\}))}[a_{0}^{2}(\alpha, \beta)\phi(2 \min\{\alpha, \beta\}, ((M-1)/M))]^{- (1/(2\min\{\alpha, \beta\}))}$, and diversity order is $G_{d} = 2\min\{\alpha, \beta\}$ which only depends on the smaller channel parameter. However, the convergence of the asymptotic error rate depends on the absolute difference between $\alpha$ and $\beta$ in the Gamma-Gamma turbulence channels. When the absolute difference between $\alpha$ and $\beta$ is small, typically if $\vert\alpha - \beta\vert\ <\ 0.5$, the leading term containing $(\mu_{M}\overline{\gamma})^{-2\min\{\alpha, \beta\}}$ (when $p = 0$) decreases at a similar rate to that of the second leading term containing $(\mu_{M}\overline{\gamma})^{-(\alpha + \beta)}$ (when $p = 0$) as $\overline{\gamma}$ increases. Thus, the asymptotic error rates obtained in (48) will slowly converge to the exact error rates. The asymptotic error rates approach the exact error rates rapidly when the difference of $\vert\alpha - \beta\vert$ is large, typically if $\vert \alpha - \beta\vert\ 1$. Such asymptotic error rate behavior will be observed from our numerical examples.

SECTION V

In this section, we present error rate curves for the Alamouti-type STBC optical systems using coherent detection over the Gamma-Gamma turbulence channels. The series error rates are computed by eliminating the infinite terms after calculating the first $J + 1$ terms. The exact error rates are calculated from numerical integration through $P_{e} = \int_{0}^{\infty} P_{e}(\gamma)f_{\gamma}(\gamma)d\gamma$ where $P_{e}(\gamma)$ is the conditional error probability and $f_{\gamma}(\gamma)$ denotes the PDF of the instantaneous SNR. In the following examples, error rates are shown against the average SNR. For a fair comparison, we assume the power of the single transmitter in the SISO system equals the total power of the two transmitters in the 2 × 1 STBC systems.

SER curves for 8PSK-modulated signals with Alamouti-type STBC using coherent detection are presented in Fig. 4 for weak $(\alpha = 3.6, \beta = 3.3)$, moderate $(\alpha = 2.5, \beta = 2.1)$, strong $(\alpha = 2.0, \beta = 1.1)$ turbulence conditions. The series SERs are compared with the exact SERs, and excellent agreement is seen between these two types of SER curves. This validates our series solutions. Asymptotic error rates are also shown to agree with our analytical error rate results when the average SNR is asymptotically large. The results show that our proposed Alamouti-type STBC systems using coherent detection can effectively mitigate the turbulence impacts. A substantial SER improvement can be found in moving from the SISO system to the 2 × 1 Alamouti-type STBC systems. For example, when $\alpha = 3.6$ and $\beta = 3.3$ and at 35 dB SNR, a SISO coherent system has an error rate of $10^{-6}$, while an Alamouti 2 × 1 system achieves an error rate of $10^{-10}$.

Fig. 5 compares the series SER and exact SER for QPSK-modulated OWC systems using coherent detection. Again, excellent agreements are observed from the SER plots in Fig. 5 between our series solutions and exact SERs. Agreement between asymptotic and exact SERs also becomes clear as the average SNR increases. As expected, the asymptotic SER approaches the exact SER much faster in the case of $\alpha = 2.0$ and $\beta = 1.1$ (parameters $\alpha$ and $\beta$ having a large difference) than the case of $\alpha = 3.6$ and $\beta = 3.3$ (parameters $\alpha$ and $\beta$ having a small difference). A substantial SER improvement can also be seen for the SISO system compared to the 2 × 1 Alamouti systems. At an average SNR of 35 dB in a turbulent channel with parameters $\alpha = 2.5$ and $\beta = 2.1$, a SISO coherent system has a SER of $3 \times 10^{-5}$, while an Alamouti 2 × 1 system achieves a SER of $3 \times 10^{-10}$.

A space–time codes design criterion is provided in [24] for coherent and differential OWC systems. In the Gamma-Gamma turbulence, optimal space–time codes need to maximize the following parameter [24], Eq. (27) TeX Source $$J_{\rm OWC} = {(d_{11}d_{22})^{\min\{\alpha, \beta\}} \over {}_{2}F_{1} \left(\min\{\alpha, \beta\}, \min\{\alpha, \beta\};1;\xi^{2}\right)}\eqno{\hbox{(49)}}$$ where $d_{11}$ and $d_{22}$ are the diagonal elements of the Gram matrix obtained from an error matrix ${\bf E} = {\bf C}_{\bf 1} - {\bf C}_{\bf 2}$ with ${\bf C}_{\bf 1}$ and ${\bf C}_{\bf 2}$ denoting two possible transmitted space–time codewords, and $0\ \leq\ \xi\ <\ 1$ is given in [24, Eq. (20)]. Alamouti's STBC is found to maximize $J_{\rm OWC}$. From Figs. 4 and 5, it is seen that our proposed systems can obtain a diversity order with $G_{d} = 2\min\{\alpha, \beta\}$ in the Gamma-Gamma turbulence. The obtained diversity order attains the maximum diversity order can be achieved by space–time codes with the design criterion described in [24]. Such a fact suggests that the Alamouti-type STBC considered in this paper is one optimal space–time code for coherent OWC systems. It also demonstrates the usefulness of the proposed architectures for implementing STBC in coherent OWC links.

It should be pointed out that the authors in [24] use different performance metrics from this work. In [24], the numerical results are presented in terms of BER versus received laser power per bit, and show the performance improvement of coherent detection based STBC PSK systems over IM/DD based repetition coded OOK/PPM systems in a strong turbulence condition $(\alpha = 2.5, \beta = 1)$. In this section, average SERs are shown versus the average SNR per branch. From the presented numerical examples, we conclude that the two proposed Alamouti-type STBC OWC systems using coherent detection can indeed improve the error rate performance, beyond that of SISO OWC systems, for a wide range (weak-to-strong) of turbulence conditions. Furthermore, the derived simple series error rate solutions are found to be efficient and easy to apply.

SECTION VI

Transmit diversity was explored for coherent OWC applications through the use of space–time codes. We proposed two novel STBC systems for coherent optical wireless links over atmospheric turbulence channels. The presented coherent STBC systems can effectively overcome the phase noise distortions arising from applying direct detection based STBC architectures. The error rate performance of OWC links employing these structures was studied with PSK modulations. The derived error rate solutions allow a highly accurate and efficient estimation of the system performance for coherent STBC implementations. The presented structures and techniques can be useful for emerging OWC systems using coherent detection.

Corresponding author: J. Cheng (e-mail: julian.cheng@ubc.ca).

^{1}A practical OWC system is employed with $P_{LO} \gg P_{s}$, and the DC photocurrent is approximated by the dominant term $RP_{LO}$ in (5), where the photocurrent due to thermal noise and the dark current have been neglected due to the large value of $RP_{LO}$.

^{2}Using a narrow linewidth laser, we can assume that the transmitter laser phase noises $\phi_{t, i}$ for $i = 1, 2$ are constant over two consecutive symbol periods.

^{3}Implementations and the detailed structure of an optical hybrid receiver can be found in [32]. In practice, two circuits can be used to process the real and imaginary parts separately.

^{5}Optical filtering may induce additional power loss, which is ignored in this paper. The proposed system in Fig. 2 using a single carrier frequency can have superior performance over the proposed system in Fig. 3 using multiple carrier frequencies when such a power loss is taken into consideration.

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