Subcarrier Intensity Modulated MIMO Optical ... - IEEE Xplore

Xuegui Song and Julian Cheng

VOL. 5, NO. 9/SEPTEMBER 2013/J. OPT. COMMUN. NETW. 1001

Subcarrier Intensity Modulated MIMO Optical Communications in Atmospheric Turbulence Xuegui Song and Julian Cheng

Abstract—We investigate spatial diversity techniques for subcarrier phase-shift keying (PSK)-modulated optical wireless communication links over the Gamma–Gamma channels. Both repetition code and the Alamouti-type orthogonal space–time block code (OSTBC) are considered. Highly accurate series error rate expressions are derived by using a moment generating function approach with a series expansion of the modified Bessel function. Truncation error analyses and asymptotic error rate analyses are also presented. Our asymptotic analyses show that the diversity order of the studied system depends only on the effective number of small-scale cells of the scattering process in the atmosphere. Our performance analyses confirm that the repetition code outperforms OSTBC for subcarrier PSK-based systems over the Gamma–Gamma channels. The asymptotic performance loss of the Alamouti-coded system with respect to the repetitioncoded system is also quantified analytically. Index Terms—Alamouti code; Atmospheric turbulence; Multiple-input multiple-output systems; Optical wireless communication.

I. INTRODUCTION

S

ubcarrier intensity modulation is an attractive technology for future optical wireless communication (OWC) systems. In [1], Huang et al. first proposed subcarrier intensity modulation for OWC applications and studied the error rate performance of differential phase-shift keying (DPSK) and M-ary phase-shift keying (MPSK) in lognormal channels. Thereafter, the error rate performance of subcarrier intensity-modulated single-input singleoutput (SISO) OWC systems over turbulence channels has been studied extensively [2–9]. The turbulenceinduced fading is a major source of system performance degradation for any outdoor OWC system. Similar to RF communications, a multiple-input multiple-output (MIMO) technique can be employed to mitigate the adverse effects of fading in an OWC system by using multiple lasers at the transmitter and multiple photodetectors at the receiver [10]. Manuscript received March 13, 2013; revised June 3, 2013; accepted July 12, 2013; published August 16, 2013 (Doc. ID 186969). X. Song and J. Cheng (e-mail: [email protected]) are with the School of Engineering, The University of British Columbia, Kelowna, British Columbia, Canada. http://dx.doi.org/10.1364/JOCN.5.001001

1943-0620/13/091001-09$15.00/0

Recently, performance analysis of MIMO OWC systems has gained increasing attention. In [11] and [12], Wilson et al. studied the error rate performance of MIMO OWC systems with pulse position modulation (PPM) in the lognormal and the Rayleigh channels. Later, Bayaki et al. studied the bit-error rate (BER) performance of a MIMO OWC system employing on–off keying (OOK) with adaptive detection threshold and PPM over the Gamma–Gamma channels [13]. In these studies, the authors adopted a repetition coding scheme across multiple lasers at the transmitter. In [14], Simon and Vilnrotter proposed a modified Alamouti code for an OWC system based on irradiance modulation and direct detection, employing OOK and PPM. In [15], García-Zambrana adopted a generalized approach to study space–time-coded OOK OWC systems based on the work in [14]. Later, Safari and Uysal extended the work presented in [14] for an arbitrary number of transmit lasers [16]. They discovered that the error rate performance of an OOK system employing orthogonal space–time block code (OSTBC) is inferior to its counterpart with repetition code, although both achieve full diversity order. In [17], Bayaki and Schober also studied the STBC design criterion and asymptotic pairwise error probability of an irradiance modulation and direct detection OWC system over the Gamma–Gamma channels. However, none of the above investigations studied the subcarrier intensity-modulated MIMO OWC system using OSTBC. In [18], using a moment generating function (MGF) approach and a series expansion of the modified Bessel function, Park et al. studied the BER performance of subcarrier binary phase-shift keying (BPSK)-modulated multiple-input single-output (MISO) OWC systems employing the Alamouti code. However, the authors did not study the truncation error of their series solutions, and they only considered subcarrier BPSK over the Gamma– Gamma channels. Recently, we investigated the error rate performance of subcarrier intensity-modulated MIMO OWC systems employing both Alamouti code and repetition code, and the preliminary results were presented in [19] and [20]. In this paper, we study the error rate performance of subcarrier intensity-modulated MIMO OWC systems employing both the Alamouti code and repetition code over the Gamma–Gamma channels. Highly accurate series error rate expressions are obtained. We also carry out a detailed study of the truncation error of our series solutions. Our asymptotic analysis shows that both schemes achieve © 2013 Optical Society of America

1002 J. OPT. COMMUN. NETW./VOL. 5, NO. 9/SEPTEMBER 2013

full diversity, and the repetition coding scheme has better error rate performance than the Alamouti scheme. We also quantify the asymptotic performance loss of an Alamouticoded system with respect to a repetition-coded system analytically. Through our analyses, we discover that the previously made counterintuitive observation that a SISO system outperforms a 2 × 1 OSTBC-coded system in practical signal-to-noise ratio (SNR) regimes under weak turbulence conditions [16] will not occur under strong turbulence conditions. Thus, a MISO or MIMO system always outperforms a SISO system in terms of error rate under strong turbulence conditions. The paper is organized as follows. Section II briefly reviews the statistical models for atmospheric turbulence channels. In Section III, we describe the system model of subcarrier intensity-modulated MIMO OWC systems. In Sections IV and V, we study the error rate performance of MIMO OWC systems employing repetition code and the Alamouti code in atmospheric turbulence. Finally, Section VII concludes this paper.

II. ATMOSPHERIC TURBULENCE MODEL Several statistical models have been proposed to describe irradiance fluctuations in outdoor OWC systems. As discussed in [21] and [22], it has been well accepted that the lognormal model describes irradiance fluctuations under weak turbulence conditions. The K-distributed model is another useful turbulence model describing irradiance fluctuations under strong turbulence conditions [21,23,24]. The negative exponential turbulence model is suitable for describing the limiting case of saturated scintillation [3,22]. More recently, the Gamma–Gamma model has emerged as a useful turbulence model because it has excellent fit with simulation data over a wide range of turbulence conditions [22]. Specifically, the Gamma–Gamma model contains the K-distributed model (strong) and the negative exponential model (saturated) as special cases [21,22]. In this investigation, we model the atmospheric turbulence channel gain I as a Gamma–Gamma random variable (RV) whose probability density function (PDF) is given by αβ 2

f I I 2αβ ΓαΓβ I

αβ 2 −1

p K α−β 2 αβI ;

I > 0;


" β exp

0.51σ 2R

1 0.69σ 12∕5 R

! 5∕6

#−1 −1

;

(2b)

where σ 2R is the Rytov variance. Under this assumption, it has been shown that the inequality α > β always holds [25]. The relationships described in Eqs. (2a) and (2b) will change when a spherical wave or a finite inner scale is considered [21,22]. However, the ensuing performance analyses can be similarly applied to a scenario when a spherical wave or a finite inner scale is considered.

III. SYSTEM MODEL A. Subcarrier Intensity Modulated SISO OWC System Figure 1 presents the block diagram of a subcarrier intensity-modulated SISO OWC system. At the transmitter, the data source is first used to modulate an RF subcarrier signal. After being properly biased, the RF signal mt is used to modulate a continuous-wave optical beam. The transmitted power, Pt, of the modulated laser beam can be expressed as Pt P1 ξmt;

(3)

where P is the average transmitter power and ξ is the modulation index satisfying the condition −1 < ξmt < 1 to avoid overmodulation. For simplicity, the power of mt is normalized to unity. At the receiver, the received optical intensity is converted into electrical signal at the photodetector. In this work, we consider a PIN photodetector at the receiver. In most practical systems, the thermal noise and/or shot noise at the receiver can be modeled with high accuracy as additive white Gaussian noise (AWGN) [10]. The photocurrent at the receiver can be written as it PRIt1 ξmt nt;

(4)

where R is the photodetector responsivity, It is the instantaneous channel gain, and nt is a zero-mean AWGN having variance σ 2n . In Eq. (4), It is assumed to be a stationary

1

where Γ· denotes the Gamma function and K ν · is the modified Bessel function of the second kind of order v. The positive parameter α represents the effective number of large-scale cells of the scattering process in the atmosphere, and the positive parameter β represents the effective number of small-scale cells of the scattering process in the atmosphere. They are both determined by the Rytov variance [21]. Assuming plane wave propagation and negligible inner scale, one can obtain the parameters α and β of the Gamma–Gamma model as [21,22] " α exp

0.49σ 2R

1 1.11σ 12∕5 7∕6 R

!

#−1 −1

;

(2a)

Fig. 1. Block diagram of a subcarrier intensity-modulated SISO OWC system.



random process induced by atmospheric turbulence. The sample I Itjtt0 at time instant t t0 gives the RV I having PDF f I I given in Eq. (1). After removing the DC bias and demodulating through an electrical demodulator, we use the sampled electrical signal at the output of the optical receiver to recover the transmitted data. The instantaneous SNR is defined here as the ratio of the time-averaged AC photocurrent power to the total noise variance [26], and it can be expressed as PRξ2 2 γ I γ¯ I2 ; σ 2n

(5)

where γ¯ is the electrical SNR. We emphasize that the SNR γ defined in Eq. (5) is SNR per symbol. 1

Fig. 2. Block diagram of a subcarrier intensity-modulated MIMO OWC system.

where I R is the equivalent channel gain and zt is the combined AWGN having zero mean and variance Lσ 2n . It follows that the instantaneous SNR at the input of the electrical demodulator can be expressed as

B. Repetition-Coded MIMO OWC System

γR

We consider a MIMO OWC system with K 2 lasers and L ≥ 1 photodetectors. Figure 2 presents the block diagram of a subcarrier intensity-modulated MIMO OWC system. We assume that the two lasers illuminate the L photodetectors simultaneously. In this section, we consider a repetition-coded system where the two lasers are modulated with properly biased RF subcarrier signal mt. Each laser has average transmitter power P∕K in order to ensure that the total radiated power of a MIMO system is kept the same as that of a single transmitter laser system. We assume that the L photodetectors have the same responsivity R and photodetector area A. For a repetitioncoded MIMO OWC system over turbulence channels, the photocurrent at the lth photodetector is given by il t

K PR1 ξmt X I kl nl t; K k1

l 1; 2; …; L; (6)

K X L L X PR1 ξmt X I kl nl t K k1 l1 l1 |{z} |{z} IR

PRIR 1 ξmt zt; K 1

(8)

C. Alamouti-Coded MIMO OWC System In this section, we will first apply the Alamouti coding scheme to a subcarrier intensity-modulated MIMO OWC system with K 2 lasers and L 2 photodetectors. We will then generalize our result for a MIMO OWC system with K 2 lasers and arbitrary L ≥ 1 photodetectors. In the first symbol interval, transmitter laser 1 transmits signal m1 and transmitter laser 2 transmits signal m2 .2 Then, in the second symbol interval, transmitter laser 1 transmits −m2 and transmitter laser 2 transmits signal m1 . Assuming that the channels do not change during the two consecutive symbol intervals,3 we can express the photocurrents at the output of the two photodetectors during the two consecutive symbol intervals as

where Ikl is the instantaneous channel gain between the kth laser and the lth photodetector and nl t is the AWGN at the lth photodetector. We assume that the channel gains Ikl are independent and identically distributed (i.i.d.) RVs so that they have the same PDF f I I [13]. This assumption is well justified for link distances of the order of kilometers with aperture separation distances of the order of centimeters [12,13]. The AWGN nl t is also i.i.d., having zero mean and variance σ 2n . Assuming equal gain combining at the receiver, one can express the combined received photocurrent at the receiver as

ir t

PRξ2 2 γ¯ 2 I : I K 2 Lσ 2n R K 2 L R

i11

PRI11 1 ξm1 I21 1 ξm2 n11 ; K

(9a)

i12

PRI12 1 ξm1 I22 1 ξm2 n12 ; K

(9b)

i21

PRI11 1 − ξm2 I21 1 ξm1 n21 ; K

(9c)

i22

PRI 12 1 − ξm2 I22 1 ξm1 n22 ; K

(9d)

where iil denotes the received photocurrent at the lth photodetector in the ith symbol interval and nil represents

zt

(7)

The definition of electrical SNR follows [10]. It can be interpreted as the SNR without fading, assuming normalized channel gain, i.e., EI 1.

2

With T denoting the symbol interval, for simplicity, we drop the time index t and use m1 and m2 to denote mt and mt T, respectively. More generally, we use mi to denote mt i − 1T. 3 Such an assumption is well justified because the typical data rate of an OWC system is hundreds to thousands of Mbits/s, and the coherence time of the turbulence channel is of the order of milliseconds.


the AWGN terms with zero mean and the same variance σ 2n . After removing the DC bias, we can construct a photocurrent containing only m1 as K X L X ~i1 PRξm1 I 2 I 11 n11 I12 n12 I21 n21 I22 n22 K k1 l1 kl |{z} v1 |{z}


M I s ≜ Eexp−sI −pβ −pα ∞ X s s ap α; β ; (15) ap β; α αβ αβ p0 where E· denotes the expectation and

U

PRξm1 U v1 : K

ap α; β ≜

(10)

Similarly, we can construct a photocurrent containing only m2 as K X L X ~i2 PRξm2 I 2 I 21 n11 I22 n12 I11 n21 I12 n22 K k1 l1 kl |{z} v2

PRξm2 U v2 : K

(11)

Γα − βΓ1 − α βΓp β : ΓαΓβΓp − α β 1p!

(16)

Since Ikl ’s are i.i.d. RVs having the same MGF as I, we PL P obtain the MGF of IR K k1 l1 I kl as M IR s M I sKL ∞ −2G KL X d s KL X bp KL − m; m ; m αβ m0 p0

(17)

where bp i; j is defined as

After demodulating through an electrical demodulator, the sampled electrical signal at the output of the receiver can be used to recover the transmitted data. Since nil ’s are i.i.d. zero-mean AWGN with variance σ 2n , we can obtain the variance of v1 and v2 as σ 2v Uσ 2n . Therefore, the instantaneous SNR at the input of the electrical demodulator can be expressed as γA

PRξ2 U 2 γ¯ 2 U: 2 2 K K Uσ n

(12)

The above analysis deals with an Alamouti-coded MIMO OWC system with K 2 lasers and L 2 photodetectors. It can be shown that the instantaneous SNR at the input of the electrical demodulator for an Alamouti-coded MIMO OWC system with K 2 lasers and L ≥ 1 photodetectors can be expressed as γA

K X L γ¯ X I2 : 2 K k1 l1 kl |{z}

(13)

j bp i; j ≜ ai p α; β ap β; α

(18)

and Gd ≜ p KL − mβ mα∕2. In Eq. (18), ai p α; β is calculated by convolving ap α; β with itself i − 1 times, e.g., a2 p α; β ap α; β ap α; β. We also note that 0 a1 p α; β ap α; β and ap α; β 1. In deriving Eq. (17) we have used two identities (Eqs. (0.314) and (0.316) of [27]). Using Eq. (17) and a property of Laplace transforms, we can obtain the PDF of IR as

f IR IR

∞ KL X KL X bp KL − m; mαβ2Gd m

m0

Γ2Gd

p0

2Gd −1 IR : (19)

To facilitate the ensuing analysis, we let Y R I 2R . The MGF of Y R can be obtained as M Y R s

YA

∞ KL X s −Gd KL X bp KL − m; mΓGd : m 2Γ2Gd α2 β 2 m0 p0 (20)

IV. ERROR RATE ANALYSES OF A REPETITION-CODED MIMO OWC SYSTEM A. Subcarrier MPSK Modulation

Starting with Eq. (20), we obtain the average error rate of a subcarrier MPSK-modulated MIMO OWC system employing repetition code as Pe;R

To facilitate the ensuing analysis, we make use of a series expansion of the modified Bessel function ∞ X π x∕22p−ν x∕22pν − ; K ν x 2 sinπν p0 Γp − ν 1p! Γp ν 1p! (14) where ν ∉ Z and jxj < ∞. Substituting Eq. (14) and π∕ sinπx ΓxΓ1 − x into Eq. (1), we obtain the MGF of I as

1 π

Z

M−1π M

0

MY R

π sin2 M ¯γ dθ K 2 L sin2 θ

∞ KL X KL X bp KL − m; mΓGd m

m0

Z

×

ηπ

0

2πΓ2Gd

p0

sin θpKL−mβmα dθ;

λR γ¯ −Gd (21)

where λR ≜ sin2 π∕M∕α2 β2 K 2 L and η ≜ M − 1∕M. Define Z gp x; η ≜

ηπ

0

sin θpx dθ;

(22)



we obtain an integral identity as [7]

B. Truncation Error Analysis

p 1px πΓ 2 1 1−p−x 3 − cosηπF ; gp x; η ; ; cos2 ηπ ; 2 2 2 2Γ 1 px 2

To evaluate the truncation error caused by eliminating the infinite terms after the first J 1 terms in Eq. (24), we first define the truncation error as

(23) where F·; ·; ·; · is the hypergeometric function (Eq. (9.111) of [27]). Then the average error rate of a subcarrier MPSKmodulated MIMO OWC system can be written as Pe;R

KL X KL m

m0

×

εR J ≜

2πΓ2Gd

p0

As demonstrated in the ensuing analysis, Eq. (24) can be simplified for BPSK and quadrature PSK (QPSK) modulations by obtaining alternative representations of gp x; η for specific η values.

(25)

The average BER of subcarrier BPSK can be obtained as

P2;R

∞ 1 KL X KL X bp KL − m; mΓ Gd 2 p λR γ¯ −Gd ; m p0 2 π Γ2Gd 1 m0

bp KL − m; mΓGd gp 2Gd − p; η 2πΓ2Gd KL−mβmα 1 : × p λR γ¯

(30)

Using the Taylor series expansion of xn ∕1 − x, we can simplify the summation term in Eq. (29) and obtain an upper bound of the truncation error as KL X 1 KL εR J ≤ p p J λR γ¯ − 1 λR γ¯ m0 m

For subcarrier BPSK, we have M 2 and η 1∕2. Substituting η 1∕2 into Eq. (23), we obtain p π Γ 1px 2 1 : gp x; 2 2Γ 1 px 2

(29)

where up;R m; η is defined as

λR γ¯ −Gd : (24)

m

m0

up;R m; η ≜

∞ X bp KL − m; mΓGd gp 2Gd − p; η

p 1 up;R m; η p ; λR γ¯ pJ1

X KL ∞ X KL

× maxp>J fup;R m; ηg.

(31)

After examining the first term in Eq. (30), we note that up;R m; η approaches zero when p approaches ∞; therefore the truncation error εR J diminishes with increasing J. Furthermore, we observe from Eq. (31) that the truncation error diminishes rapidly with increasing electrical SNR γ¯ . This observation suggests that our series solution is highly accurate in large-SNR regimes. We can therefore use it to perform asymptotic error rate analysis.

(26)

C. Asymptotic Error Rate Analysis where we have used the relationship Γ1 x xΓx. For subcarrier QPSK, we obtain the average symbolerror rate (SER) as P4;R

∞ KL X KL X bp KL − m; mΓGd m 2πΓ2Gd p0 1 1 − gp 2Gd − p; λR γ¯ −Gd ; × 2gp 2Gd − p; 2 4 (27)

m0

P∞ e;R

where gp x; 1∕4 can be shown to be [7] gp

We now study the asymptotic error rate performance of repetition-coded MIMO OWC systems over the Gamma– Gamma channels. In OWC, typically we have α > β > 0 for the Gamma–Gamma turbulence model [25], so the term λR γ¯ −α∕2 decreases faster than the term λR γ¯ −β∕2 as γ¯ increases. Consequently, when γ¯ increases, the error rate given in Eq. (24) in high-SNR regimes can be approximated by

g0 KLβ; ηΓ

KLβ 2

2πΓKLβ

F 1 ; px1 ; px3 ; 1 2 2 2 2 1 x; px1 4 2 2 p x 1

p Γα − β KL K Lαβ KLβ − KLβ γ¯ 2 : π Γα sin M (32)

(28)

with the help of the definition of the hypergeometric function F·; ·; ·; · (Eq. (9.111) of [27]). Note that Eqs. (24), (26), and (27) only involve the Gamma function and the hypergeometric function, which can be computed easily with standard scientific software.

Specifically, the asymptotic BER of subcarrier BPSK can be obtained as

P∞ 2;R

Γ

KLβ1 2

KβKLβ−1 αKLβ L p 2 π ΓKLβ

KLβ 2 −1

Γα − β KL − KLβ γ¯ 2 : Γα (33)



For a given β, the asymptotic error rates in Eqs. (32) and (33) approach the exact error rate faster with a greater α value.

V. ERROR RATE ANALYSES OF AN ALAMOUTI-CODED MIMO OWC SYSTEM A. Subcarrier MPSK Modulation Substituting Eq. (14) and π∕ sinπx ΓxΓ1 − x into Eq. (1), one can obtain the MGF of a RV X I2 as [18]

Pe;A

× gp 2Gd − p; ηλA γ¯ −Gd :

KL 1 X KL P2;A p 2 π m0 m

dp α; β ≜

2ΓαΓβΓp − α β 1p!

:

(35)

Since I kl ’s are i.i.d. RVs P having PLthe 2same PDF f I I, we obtain the MGF of Y A K k1 l1 I kl as M Y A s M X sKL ∞ KL X s −Gd KL X cp KL − m; m 2 2 ; m p0 α β m0

(36)

To evaluate the truncation error caused by eliminating the infinite terms after the first J 1 terms in Eq. (40), we define the truncation error as

(37)

Pe;A

Z

M−1π M

0

MY A

! π sin2 M ¯γ dθ K 2 sin2 θ

∞ KL 1X KL X c KL − m; mλA γ¯ −Gd π m0 m p0 p Z ηπ × sin θpKL−mβmα dθ; 0

X p KL ∞ 1X 1 KL up;A m; η p ; π m0 m pJ1 λA γ¯

(42)

where up;A m; η is defined as up;A m; η ≜ cp KL − m; m

KL−mβmα 1 × gp 2Gd − p; η p : λA γ¯

Starting from Eq. (36), we obtain the average error rate of a subcarrier MPSK-modulated MIMO OWC system employing the Alamouti code as 1 π

(40)

B. Truncation Error Analysis

εA J ≜

and Gd p KL − mβ mα∕2 has been defined in Subsection IV.A. In deriving Eq. (36), we have used two identities from Eqs. (0.314) and (0.316) of [27].

λA γ¯ −Gd :

∞ KL 1X KL X c KL − m; m π m0 m p0 p

1 1 − gp 2Gd − p; λA γ¯ −Gd : × 2gp 2Gd − p; 2 4 (41)

where cp i; j is defined as j cp i; j ≜ di p α; β dp β; α

ΓGd 1

For QPSK modulation, we substitute M 4 and Eqs. (25) and (28) into Eq. (39) and obtain the average SER as

P4;A Γα − βΓ1 − α βΓ pβ 2

∞ cp KL − m; mΓ Gd 1 X 2 p0

(34) where

(39)

Using M 2 and Eq. (25), we simplify the average BER of subcarrier BPSK to

× M X s ≜ Eexp−sX pβ pα ∞ X s −2 s −2 dp α; β 2 2 ; dp β; α 2 2 α β α β p0

∞ KL 1X KL X c KL − m; m π m0 m p0 p

(43)

Using the Taylor series expansion of xn ∕1 − x, we can simplify the summation term in Eq. (42) and obtain an upper bound of the truncation error as

εA J ≤ (38)

where λA ≜ sin2 π∕M∕K 2 α2 β2 and η has been defined as η M − 1∕M in Subsection IV.A. Using the integral identity in Eq. (23), we obtain the average error rate of a subcarrier MPSK-modulated MIMO OWC system employing the Alamouti code as

KL X 1 KL maxfup;A m; ηg: p p J π λA γ¯ − 1 λA γ¯ m0 m p>J (44)

It can be shown that the truncation error εA J diminishes with increasing J. We also note that the truncation error diminishes rapidly with increasing electrical SNR γ¯ , which suggests that our series solution is suitable for asymptotic error rate analysis.


VOL. 5, NO. 9/SEPTEMBER 2013/J. OPT. COMMUN. NETW. 1007 0

C. Asymptotic Error Rate Analysis

10

Similar to the argument used in obtaining Eq. (32), the error rate of MPSK modulation in high-SNR regimes for a Alamouti-coded MIMO OWC system over the Gamma– Gamma channels can be approximated by

10

−2

P∞ e;A

BER

1KL 0 β g0 KLβ; η @Γα − βΓ 2 A Kαβ KLβ − KLβ γ¯ 2 : (45) ΓαΓβ sin Mπ 2KL π

−4

10

2x1 system −8

Γ

Kβ α p KLβ 2KL L π Γ 2

KLβ−1 KLβ

Γα − βΓ

−12

!KL β 2

ΓαΓβ

10

γ¯

− KLβ 2

From Eqs. (32) and (45), we can derive the asymptotic SNR loss in decibels for an Alamouti-coded system with respect to a repetition-coded system as ! " ! Γ 2β 2 10 ΓKLβ log 10 log Γβ β KL Γ KLβ 2 # 10 log 2KL − 1 − − 10 logL; KL

(47)

where log· is the logarithm function with base 10. It can be shown that this SNR loss factor is always greater than zero, indicating that a repetition-coded system has better error rate performance than an Alamouti-coded system. This observation agrees with the findings reported in [16] and [17] for OOK-based MIMO OWC systems. We observe from Eq. (47) that this SNR loss factor depends on the smaller channel parameter β only, and it decreases when L increases.

VI. NUMERICAL RESULTS

AND

5

10

15

20

In Fig. 3, we present BERs of subcarrier BPSKmodulated SISO and MIMO OWC systems over a Gamma–Gamma channel with α 5.05 and β 1.16, a representative strong turbulence condition with the Rytov variance σ 2R 7. We observe that our series solutions (with J 20) have excellent agreement with the simulated BERs. As expected, the asymptotic BERs approach the simulated BERs in high-SNR regimes. From Fig. 3, we also observe that both the Alamouti-coded system and the repetition-coded system can achieve the same diversity order of KLβ∕2. However, a repetition-coded system has better BER performance than an Alamouti-coded system. When

30

35

40

45

50

Fig. 3. BERs of subcarrier BPSK-modulated SISO/MIMO OWC systems over a Gamma–Gamma channel with α 5.05, β 1.16, and J 20.

the BER level is at 10−5 , the SNR loss of a 2 × 1 Alamouti-coded system with respect to a 2 × 1 repetitioncoded system is found to be SNRA–R 2.0 dB, while that of 2 × 2 systems is found to be SNRA–R 1.5 dB at BER 10−10 , agreeing with the theoretical predictions of 2.06 and 1.51 dB, respectively, from our asymptotic analysis. Under the assumed strong turbulence condition, the counterintuitive observation that a SISO system outperforms a 2 × 1 OSTBC-coded system in practical SNR regimes [16] cannot be made, indicating that the MISO or MIMO technique is an effective way to reduce the adverse effects of strong turbulence fading in OWCs. Figure 4 presents BERs of subcarrier BPSK-modulated SISO and MIMO OWC systems over a Gamma–Gamma channel with α 11.65 and β 10.12, a representative weak turbulence condition with Rytov variance σ 2R 0.2. 0

10

DISCUSSION

In this section, we use our analytical results to study the performance of subcarrier intensity-modulated MIMO OWC systems over the Gamma–Gamma channels. Simulation results are used to corroborate the analytical results.

25

Electrical SNR (dB)

:

(46)

SNRA–R

SISO Series Asymptotic Alamouti code, simulation Repetition code, simulation

−10

10

SISO Alamouti code Repetition code

−1

10

−2

10

BER

P∞ 2;A

KLβ1 2

2x2 system

10

Specifically, the asymptotic BER of BPSK modulation can be obtained as

−6

10

−3

10

2x1 system

−4

10

2x2 system

−5

10

−6

10

0

2

4

6

8

10

12

14

16

18

20

Electrical SNR (dB)

Fig. 4. BERs of subcarrier BPSK-modulated SISO and MIMO OWC systems over a Gamma–Gamma channel with α 11.65 and β 10.12.

1008 J. OPT. COMMUN. NETW./VOL. 5, NO. 9/SEPTEMBER 2013 2.6

MIMO OWC system with respect to a repetition-coded system depend only on the smaller channel parameter β. Furthermore, the performance gap of an Alamouti-coded MIMO OWC system can be narrowed by increasing the number of receiving photodetectors.

α=4.39, β = 2.56 α=4.12, β = 1.44 α=13.82, β = 1.00

2.4 2.2 2

SNRA−R (dB)


1.8

REFERENCES

1.6 1.4 1.2 1 0.8 1

2

3

4

5

6

7

8

9

10

Number of photodetectors L

Fig. 5. SNR loss in decibels for an Alamouti-coded system with respect to a repetition-coded system as a function of L for different α and β values.

From Fig. 4, we observe that a SISO system outperforms a 2 × 1 Alamouti-coded system when the SNR value is less than 12 dB, which agrees with the counterintuitive observation made in [16] for the lognormal weak turbulence channels. However, a 2 × 2 Alamouti-coded system has better BER performance than a SISO system over the entire SNR range considered. Under the weak turbulence condition, the MIMO technique has noticeable advantages when the SNR value is large. In Fig. 5, we plot the asymptotic SNR loss in decibels for an Alamouti-coded system with respect to a repetitioncoded system as a function of the number of photodetectors L under different turbulence conditions. We observe that this SNR loss factor is a decreasing function of L. This indicates that the performance gap between an Alamouticoded system and a repetition-coded system becomes smaller as the number of photodetectors increases. We also observe that SNRA–R decreases with decreased values of the channel parameter β. In other words, the performance gap between an Alamouti-coded system and a repetitioncoded system will become smaller under stronger turbulence conditions. In the limiting case of the K-distributed channel where β 1, this performance gap can achieve its minimum value.

VII. CONCLUSIONS In this investigation, we developed highly accurate series error rate expressions for subcarrier intensitymodulated MIMO OWC systems employing both the repetition code and the Alamouti code. Our truncation analysis demonstrated that the presented series solutions can converge to the exact error rate results rapidly, especially in high-SNR regimes. Our performance analysis confirms that a repetition code outperforms an OSTBC, although both schemes achieve full diversity. Our asymptotic analysis shows that the diversity order of the system and the asymptotic SNR loss factor for an Alamouti-coded

[1] W. Huang, J. Takayanagi, T. Sakanaka, and M. Nakagawa, “Atmospheric optical communication system using subcarrier PSK modulation,” IEICE Trans. Commun., vol. E76-B, pp. 1169–1177, Sept. 1993. [2] J. Li, J. Q. Liu, and D. P. Tayler, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 55, pp. 1598–1606, Aug. 2007. [3] W. O. Popoola and Z. Ghassemlooy, “BPSK subcarrier intensity modulated free-space optical communications in atmospheric turbulence,” J. Lightwave Technol., vol. 27, pp. 967–973, Apr. 2009. [4] N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, and S. Arnon, “Adaptive subcarrier PSK intensity modulation in free space optical systems,” IEEE Trans. Commun., vol. 59, pp. 1368–1377, May 2011. [5] X. Song, M. Niu, and J. Cheng, “Error rate of subcarrier intensity modulations for wireless optical communications,” IEEE Commun. Lett., vol. 16, pp. 540–543, Apr. 2012. [6] Md. Z. Hassan, X. Song, and J. Cheng, “Subcarrier intensity modulated wireless optical communications with rectangular QAM,” J. Opt. Commun. Netw., vol. 4, pp. 522–532, June 2012. [7] X. Song and J. Cheng, “Optical communication using subcarrier intensity modulation in strong atmospheric turbulence,” J. Lightwave Technol., vol. 30, pp. 3484–3493, Nov. 2012. [8] X. Song, Md. Z. Hassan, and J. Cheng, “Subcarrier DQPSK modulated optical wireless communications in atmospheric turbulence,” Electron. Lett., vol. 48, pp. 1224–1225, Sept. 2012. [9] X. Song, F. Yang, and J. Cheng, “Subcarrier intensity modulated optical wireless communications in atmospheric turbulence with pointing errors,” J. Opt. Commun. Netw., vol. 5, pp. 349–358, Apr. 2013. [10] X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 50, pp. 1293–1300, Aug. 2002. [11] S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun., vol. 53, pp. 1402–1412, Aug. 2005. [12] S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE J. Sel. Areas Commun., vol. 23, pp. 1901–1910, Sept. 2005. [13] E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in Gamma–Gamma fading,” IEEE Trans. Commun., vol. 57, pp. 3415–3424, Nov. 2009. [14] M. K. Simon and V. Vilnrotter, “Alamouti-type space–time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun., vol. 4, pp. 35–39, Jan. 2005.



[15] A. García-Zambrana, “Error rate performance for STBC in freespace optical communications through strong atmospheric turbulence,” IEEE Commun. Lett., vol. 11, pp. 390–392, May 2007. [16] M. Safari and M. Uysal, “Do we really need OSTBCs for freespace optical communication with direct detection?” IEEE Trans. Wireless Commun., vol. 7, pp. 4445–4448, Nov. 2008. [17] E. Bayaki and R. Schober, “On space–time coding for freespace optical systems,” IEEE Trans. Commun., vol. 58, pp. 58–62, Jan. 2010. [18] J. Park, E. Lee, and G. Yoon, “Average bit-error rate of the Alamouti scheme in Gamma–Gamma fading channels,” IEEE Photon. Technol. Lett., vol. 23, pp. 269–271, Feb. 2011. [19] X. Song and J. Cheng, “Performance of subcarrier intensity modulated MIMO wireless optical communications,” in 26th Queen’s Biennial Symp. on Communications, Kingston, Ontario, Canada, May 28–29, 2012. [20] X. Song and J. Cheng, “Alamouti-type STBC for subcarrier intensity modulated wireless optical communications,” in IEEE Global Communications Conf. (GLOBECOM), Anaheim, CA, Dec. 3–7, 2012.

[21] L. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation With Applications. Philadelphia, PA: SPIE, 2001. [22] A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng., vol. 40, pp. 1554–1562, Aug. 2001. [23] E. Jakeman and P. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett., vol. 40, pp. 546–550, Feb. 1978. [24] E. Jakeman, “On the statistics of K-distributed noise,” J. Phys. A, vol. 13, pp. 31–48, Jan. 1980. [25] N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the Gamma–Gamma atmospheric turbulence model,” Opt. Express, vol. 18, pp. 12824–12831, June 2010. [26] G. P. Agrawal, Fiber-Optical Communication Systems, 3rd ed. New York: Wiley, 2002. [27] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. San Diego: Academic, 2000.