Subcarrier Intensity Modulated Optical Wireless ... - IEEE Xplore

Song et al.

VOL. 5, NO. 4/APRIL 2013/J. OPT. COMMUN. NETW. 349

Subcarrier Intensity Modulated Optical Wireless Communications in Atmospheric Turbulence With Pointing Errors Xuegui Song, Fan Yang, and Julian Cheng

Abstract—An optical wireless communication system using subcarrier intensity modulation is analyzed for gamma–gamma turbulence channels with pointing errors. We study the error rate performance of such a system employing M-ary phase-shift keying, differential phase-shift keying, and noncoherent frequency-shift keying. Highly accurate error rate approximations are derived using a series expansion approach. Furthermore, outage probability expressions are obtained for such a system. Asymptotic error rate and outage probability analyses are also presented. Our asymptotic analysis reveals some unique transmission characteristics of such a system. Our findings suggest that pointing error compensation is necessary as pointing errors can severely degrade the error rate and outage probability performance of an uncompensated system. Index Terms—Atmospheric turbulence; Optical wireless communications; Pointing errors; Subcarrier intensity modulation.

I. INTRODUCTION

O

wing to its simplicity and low cost, on–off keying (OOK) with irradiance modulation and direct detection (IM/DD) is commonly used in optical wireless communication (OWC1) applications [1,2]. However, because of random intensity fluctuations induced by atmospheric turbulence channels, an OOK IM/DD system requires adaptive thresholds in order to achieve its optimal performance. Such an adaptive OOK system can be costly to implement and is also subject to channel estimation errors. Pulse position modulation (PPM) has been proposed as an alternative to the OOK IM/DD modulation. PPM does not require adaptive thresholds, and it is mainly used in deep space communications [3,4]. However, the PPM modulation requires complex transceiver design due to tight synchronization requirements. In [5], Huang et al. proposed subcarrier intensity modulation (SIM) as an attractive Manuscript received September 14, 2012; revised February 7, 2013; accepted February 7, 2013; published March 28, 2013 (Doc. ID 176173). The authors are with School of Engineering, The University of British Columbia, Kelowna, BC, Canada (e-mail: [email protected]; e-mail: [email protected]; e-mail: [email protected]). 1 Outdoor OWC is also known as free space optical (FSO) communication. Digitial Object Identifier 10.1364/JOCN.5.000349

1943-0620/13/040349-10$15.00/0

alternative to the OOK IM/DD modulation for OWC applications and studied the error rate performance for differential phase-shift keying (DPSK) and M-ary phase-shift keying (MPSK) modulations over the lognormal turbulence fading. When the channel state information (CSI) is available, a SIM-binary phase-shift keying (BPSK) system, which uses a fixed detection threshold, has better error rate performance than an optimum OOK-based IM/DD system, which requires a time-varying adaptive detection threshold. When the CSI is not available, a SIM system can employ noncoherent modulations while an OOK-based IM/DD system is forced to use a predetermined fixed detection threshold, resulting in an undesirable error floor [6]. Therefore, SIM is an attractive technology for future FSO systems. In [6], Li et al. studied the error rate performance of uncoded and coded subcarrier intensity modulated systems employing MPSK modulation over the lognormal turbulence channels. In a later work, Popoola and Ghassemlooy studied the bit-error rate (BER) performance of subcarrier BPSK modulated systems over the gamma–gamma and the exponential turbulence channels [7]. More recently, Chatzidiamantis et al. studied adaptive subcarrier PSK modulated systems over the gamma– gamma turbulence channels [8]. In [9], Park et al. studied the BER performance of subcarrier BPSK with Alamouti coding using a moment generating function (MGF) approach and a series expansion of the modified Bessel function. Using a direct integration approach, Song et al. studied the error rate of subcarrier intensity modulated OWC systems employing BPSK, quadrature PSK (QPSK), DPSK, and noncoherent frequency-shift keying (NCFSK) over the gamma–gamma turbulence channels [10]. In [11,12], the same authors also studied the error rate performance of subcarrier MPSK modulated OWC systems over the K-distributed channels using an MGF approach and a series expansion of the modified Bessel function. Using a similar technique, Hassan et al. studied the error rate performance of subcarrier intensity modulated OWC systems employing general order rectangular quadrature amplitude modulation over the gamma–gamma, the Kdistributed, and the exponential turbulence channels [13,14]. To overcome the effect of atmospheric turbulence, we also studied the performance of multipleinput multiple-output (MIMO) OWC systems over the © 2013 Optical Society of America

350 J. OPT. COMMUN. NETW./VOL. 5, NO. 4/APRIL 2013

Song et al.

gamma–gamma turbulence channels [15,16]. However, none of the above work considered the effects of pointing errors owing to building sway caused by dynamic wind loads, thermal expansion, and weak earthquakes [17–20]. Recently, OWC systems with pointing errors have gained increasing attention. In [17–19], Arnon first analyzed the combined effect of atmospheric turbulence and misalignment on the BER performance of OOK-based IM/DD OWC systems assuming negligible detector size. In [20], Farid and Hranilovic proposed a composite OWC channel model by taking into consideration the atmospheric turbulence and misalignment. In their model, the authors explicitly considered the effect of beam width, detector size, and jitter variance. They derived a closed-form probability density function (PDF) of the composite channel with lognormal turbulence. However, the PDF of their composite channel model with gamma–gamma turbulence is given only as an integral. In [21], Sandalidis et al. investigated the BER performance of OWC systems over the Kdistributed turbulence channels with misalignment effects. They presented a closed-form expression for the PDF of the composite OWC channel in terms of Meijer’s G function. Later Sandalidis et al. extended this result to the gamma– gamma model for the atmospheric turbulence in [22]. Using this extended channel model and an approximation of the complementary error function, Gappmair et al. studied the error rate performance of a PPM-based system in [23]. In [24], Sandalidis also studied the error rate performance of coded OWC links over strong turbulence channels with misalignment. Furthermore, in obtaining the composite OWC channel model used in the above work, the authors assumed that the horizontal and vertical building sways are identically distributed with the assumption of independent Gaussian distributions. More recently, Gappmair et al. removed this assumption by considering different jitter in horizontal and vertical directions, and they modeled the composite OWC channel by a Hoyt distribution [25]. Based on this generalized composite channel model, they studied the BER performance of an OOK system by approximating the BER expression in terms of a finite series. Unfortunately, their series solution does not apply to all operation conditions. In [26], García-Zambrana et al. studied the outage performance of a MIMO OOK system over the exponential turbulence channels with pointing errors. Later, the same authors studied the asymptotic BER performance for OWC systems using transmit laser selection over the gamma–gamma atmospheric turbulence channels with pointing errors [27]. Their results are based on the Taylor series expansion of Meijer’s G function in the PDF of the composite channels. It is found that the diversity order of the system is independent of the pointing error only when the equivalent beam radius at the receiver is at least 2minfα; βg1∕2 times the value of the pointing error displacement standard deviation at the receiver. However, the presented asymptotic error rates in [27] may not be valid for any small to medium SNR value. In [28,29], Lee et al. investigated the effects of aperture averaging on OWC links in the presence of atmospheric turbulence and pointing errors. Their results suggest that the simple aperture averaging is useful in compensating pointing errors.

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

The contributions of this work are as follows. Using a series expansion approach, we obtain highly accurate series solutions to error rates and outage probability of subcarrier intensity modulated OWC systems in the gamma– gamma atmospheric turbulence with pointing errors. Unlike previous work in [25], our novel series solutions are general and can be applied to any operation condition for any finite SNR value. Our series approach can be applied to a variety of digital modulations. Furthermore, our series solutions are suitable for asymptotic analysis,2 and the obtained asymptotic expressions can reveal additional insights into these systems. The paper is organized as follows. In Section II, we briefly describe the system model of SIM. Section III reviews the statistical model for atmospheric turbulence with pointing errors. In Sections IV and V, we study the error rate performance of an OWC system susceptible to both atmospheric turbulence and pointing error effects under different operation conditions. In Section VI, we perform outage probability analysis for the system. Section VII concludes this paper and points out that severe performance degradation can be introduced by pointing errors.

II. SUBCARRIER INTENSITY MODULATION Figure 1 presents the block diagram of a subcarrier intensity modulated OWC system. At the transmitter, the data source is first used to modulate a radio frequency (RF) subcarrier signal. After being properly biased, the RF signal mt is used to modulate a continuous wave laser beam. The transmitted power, Pt t, of the modulated laser beam can be written as Pt t P1 ξmt;

(1)

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 power is converted into electrical signal through DD at the photodetector. In 2

For OOK- or PPM-based IM/DD systems, one can obtain similar asymptotic solutions by using Taylor series expansion of Meijer’s G function [26,27].

Song et al.


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) [1]. Therefore, for an atmospheric turbulence channel, the photocurrent at the receiver can be written as ir t PRIt1 ξmt nt;

(2)

where R is the photodetector responsivity, It is the instantaneous channel gain, and nt is a zero-mean AWGN attributable to thermal noise and has variance σ 2n . Here It is assumed to be a stationary random process caused by atmospheric turbulence and pointing errors. The sample I Itjtt0 at time instant t t0 gives the random variable (RV) I having PDF f I I. 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. To study the error rate performance of the considered system, we require the instantaneous SNR at the input of the electrical demodulator of an optical receiver. The instantaneous SNR is defined here as the ratio of the timeaveraged ac photocurrent power to the total noise variance [30], and it can be expressed as

relationship α > β always holds for OWC applications [33]. Without losing generality we will consider this assumption to be valid. By considering a circular detection aperture of radius r and a Gaussian beam, Farid and Hranilovic derived the PDF of Ip as [20] f Ip Ip

2

Aφ0

2 −1

I pφ

0 ≤ I p ≤ A0 ;

;

(5)

where A0 erf v2 is the fraction of the collected power at radial distanceR0. The Gauss error function erf · is defined 2 as p erf p2π 0x e−t dt, and the parameter v is given by π x r v≜ 2 ωz with ωz denoting the beam waist (radius calculated ωz at e−2 ) at distance z. The parameter φ≜ 2σeqs in Eq. (5) is the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (σ s ) at the receiver. It will be shown that the parameter φ plays an important role when studying large SNR performance. The equivalent beam width ωzeq can be calculated as [20] h p i1 π erf v 2 6 : ωzeq ωz 2v exp−v 2 Using Eqs. (4) and (5), we obtain the composite PDF of I as

PRξ2 2 γ I γ¯ I2 ; σ 2n

(3) f I I

where γ¯ is the electrical SNR.3 We emphasize that the SNR γ defined in Eq. (3) is SNR per symbol.

III. ATMOSPHERIC TURBULENCE CHANNEL WITH POINTING ERRORS The irradiance of the transmitted laser is susceptible to both atmospheric turbulence and pointing error effects in an OWC system with misalignment. The channel gain I can be modeled as a product of two independent RVs, i.e., I Ia Ip , where Ia represents the attenuation attributable to atmospheric turbulence and Ip represents the attenuation caused by pointing errors [20,21]. In this work, we model the turbulence-induced fading I a as a gamma–gamma distributed RV whose PDF is given by [31] p αβ 2αβ 2 αβ −1 Ia2 K α−β 2 αβI a ; f Ia Ia ΓαΓβ

φ2

αβ 2

2φ2 αβ

2 Aφ0 ΓαΓβ

2 −1

Iφ

Z

∞

αβ

I a2

−φ2 −1

I∕A0

p K α−β 2 αβIa dI a : (7)

Substituting a series expansion of the modified Bessel function [10] " x∕22p−ν # ∞ X π Γp−ν1p! K ν x ; ν ∉ Z; (8) 2pν 2 sinπν p0 − x∕2 Γpν1p! into Eq. (7), we obtain the PDF of I as 2 f I I

Λα; β; φI

∞ 6 φ2 −1 X

2 Aφ0

p0

6 4

αβpβ

R∞ I∕A0

Ipβ−φ a

2 −1

Γp−αβ1p!

−

αβpα

R∞

I∕A0

I pα−φ a

3 dIa

2 −1

dI a

7 7; 5

(9)

Γpα−β1p!

where I a > 0;

(4)

where Γ· denotes the gamma function, and K α−β · is the modified Bessel function of the second kind of order α − β. The positive parameter α represents the effective number of large-scale cells of the scattering process, and the positive parameter β represents the effective number of smallscale cells of the scattering process in the atmosphere. It can be shown that the parameters α and β are determined by the Rytov variance [32]. In addition, under an assumption of plane wave and negligible inner scale the 3 The definition of electrical SNR follows [1]. It can be interpreted as the SNR without fading assuming normalized channel gain, i.e., EI 1.

Λα; β; φ ≜

πφ2 : ΓαΓβ sinπα − β

(10)

Denoting x

s −ω2zeq ln A0IIa 2

;

(11)

we can rewrite the PDF of I as f I I Λα; β; φ where

∞ X ap α; β; φ; A0 gp βIpβ−1 ; −ap β; α; φ; A0 gp αI pα−1 p0

(12)


Song et al.

pβ ap α; β; φ; A0 ≜

αβ A0

Γp − α β 1p!

and gp x is defined as Z∞ gp x ≜ expp x − φ2 tdt: 0

Z (13)

(14)

Letting Y I 2 denote a channel-dependent RV, we can obtain the MGF of Y as Λα; β; φ MY s Eexp−sY 2 " −pβ # ∞ X 2 ap α; β; φ; A0 gp βΓ pβ 2 s × pα −pα : p0 −ap β; α; φ; A0 gp αΓ 2 s 2

(15)

Both Eqs. (12) and (15) are important new analytical results for describing the statistics of the composite channel with gamma–gamma turbulence. We will use Eq. (15) to analyze the error rate performance for a subcarrier intensity modulated OWC system employing MPSK in Section IV.

IV. ERROR RATE PERFORMANCE WHEN α < φ2 The average error rate of a subcarrier modulated system over turbulence channels can be expressed as Z∞ Pe If I IdI; (16) Pe 0

where Pe I denotes the conditional error probability and f I I denotes the PDF of the channel gain. In this section, we will study the error rate performance of subcarrier intensity modulated OWC systems employing MPSK, DPSK, and NCFSK using a series expansion approach when α < φ2 .

0

The conditional BER of subcarrier BPSK modulation is given by [34] q P2 I Q (17) 2¯γ I2 ; R 2 where Qx 2π−1∕2 x∞ e−z ∕2 dz is the Gaussian Q function. Thus, using Eq. (12) and an alternative expression of the Gaussian Q function [35, Eq. (A.1)], we obtain the average BER as γ¯ I2 f I IdθdI exp − sin2 θ 0 0 2 pβ1 3 Γ 2 ap α;β;φ;A0 gp β −pβ ∞ 2 γ ¯ X Λα; β; φ pβ 6 7 p 4 5: (18) pα 2 π p0 Γpα1 a β;α;φ;A g α p 0 p − 2 2 − γ ¯ pα

1 P2 π

Z

∞

Z

π∕2

In deriving Eq. (18), we have used the integral identities [36, Eqs. 3.621(1) and 8.384(4)] and the relationship

1 1 px1 ; sin θpx dθ B ; 2 2 2

(19)

where B·; · is the beta function defined as Bx; y ΓxΓy∕Γx y [36, Eq. 8.384(1)]. We notice from the definition of gp x in Eq. (14) that this integral does not converge when φ2 < p x. We also observe from Eq. (18) that the terms in the infinite series converge to zero when p increases. Therefore, the average BER for subcarrier BPSK modulation in Eq. (18) can be approximated by a finite series as 2 pβ1 3 Γ 2 ap α;β;φ;A0 −pβ K γ¯ 2 2 −p−β Λα; β; φ X pβφ 4 5; p (20) P2 ≈ pα1 2 π p0 − Γ 2 ap β;α;φ;A0 γ¯ −pα 2 pαφ2 −p−α

where K ⌊φ2 − α⌋ > 0 and ⌊ · ⌋ is the floor function. With Eq. (20), we now examine the BER behavior in high SNR regimes. In OWC, typically we have α > β > 0 for pα gamma–gamma turbulence [33], so the term γ¯ − 2 decreases pβ faster than the term γ¯ − 2 in Eq. (20) for the same p value as γ¯ increases. Consequently, when γ¯ approaches ∞, the leading term of the series in Eq. (20) becomes the dominant term. Therefore, the BER for subcarrier BPSK modulation in high SNR regimes can be approximated by Λα;β;φAαβ β Γβ1 2 −2β ; 0 p P∞ 2 2 π Γβ1−αφ2 −ββ γ¯

21

when α < φ2 . Since α > β, the inequality φ2 > α also implies φ2 > β, which will ensure the error rate in Eq. (21) is positive. The above error rate analysis for BPSK can be generalized to the MPSK case using an MGF approach. From Eq. (3), we can express the SNR as γ γ¯ Y. The average symbol error rate (SER) of subcarrier MPSK modulation can be obtained as κ¯γ Λα;β;φ dθ 2 2π sin θ 0 # " pβ R ∞ X ap α;β;φ;A0 gp βΓ pβ γ − 2 0ηπ sin θpβ dθ 2 κ¯ ; × pα pα R γ − 2 0ηπ sin θpα dθ p0 −ap β;α;φ;A0 gp αΓ 2 κ¯

PM

A. Subcarrier MPSK Modulation

π∕2

1 π

Z

ηπ

MY

(22) where κ ≜ sin2 π∕M and η ≜ M − 1∕M. With the definition Z hp x; η ≜

ηπ 0

sin θpx dθ

(23)

we can obtain an integral identity as [15] p 1px πΓ 2 1 1−p−x 3 − cosηπF ; hp x; η ; ; cos2 ηπ ; 2 2 2 2Γ 1 px 2 (24) where F·; ·; ·; · is the hypergeometric function [36, Eq. (9.111)]. The average error rate for subcarrier MPSK modulation can then be obtained as

Song et al.

PM


Λα; β; φ 2π " pβ # ∞ X ap α; β; φ; A0 gp βhp β; ηΓ pβ γ − 2 2 κ¯ × : (25) pα pα γ − 2 p0 −ap β; α; φ; A0 gp αhp α; ηΓ 2 κ¯

Similarly, to ensure the convergence of gp β and gp α, the average error rate for subcarrier MPSK modulation can be approximated by a finite series as 2 3 pβ ap α;β;φ;A0 hp β;ηΓ pβ 2 − K 2 X κ¯γ Λα; β; φ 6 7 φ2 −p−β PM ≈ 4 5: (26) pα ap β;α;φ;A0 hp α;ηΓ pα 2π 2 − 2 p0 − κ¯ γ φ2 −p−α The asymptotic SER of subcarrier MPSK modulation can be similarly obtained as P∞ M

Λα; β; φh0 β; ηΓ

β αβ β 2

A0

2πΓβ 1 − αφ2 − β

β

κ¯γ −2 .

(27)

When M 2, it can be shown that Eq. (26) specializes to Eq. (20), and Eq. (27) specializes to Eq. (21).

B. Subcarrier DPSK/NCFSK Modulation The conditional BER for subcarriers DPSK and NCFSK is [34] 1 γ¯ 2 (28) Pe;k I exp − I ; 2 k where k 1 for DPSK and k 2 for NCFSK. The average BER can be obtained as Pe;k

1 2

Z

∞

0

γ¯ exp − I 2 f I IdI: k

(29)

Substituting Eq. (12) into Eq. (29) and using an integral identity [36, Eq. 3.326(2)], we obtain

Pe;k

" pβ # ∞ Λα; β; φ X ap α; β; φ; A0 gp β kγ¯ − 2 Γ pβ 2 pα γ¯ −pα : 4 2 p0 −Γ 2 ap β; α; φ; A0 gp α k (30)

Similar to the argument used in obtaining Eq. (20), the average BER for subcarrier DPSK/NCFSK modulations can be approximated by a finite series as

Pe;k

2 pβ 3 Γ 2 ap α;β;φ;A0 γ¯ −pβ K 2 X Λα; β; φ 6 φ 2 −p−β 7 k ≈ 4 5: Γ pα ap β;α;φ;A0 γ¯ −pα 4 2 p0 − 2 k φ2 −p−α

(31)

The asymptotic BER for subcarrier DPSK/NCFSK modulations can be obtained as

P∞ e;k

β Λα; β; φ Aαβ0 Γ 2β γ¯ −2β : 4Γβ 1 − αφ2 − β k

From Eqs. (21), (27), and (32), we observe that the diversity order of the system is 2β, which is independent of the pointing error, and it depends on the smaller atmospheric turbulence channel parameter β when α < φ2 . We also comment that with a given β the asymptotic error rates in Eqs. (21), (27), and (32) approach the exact error rate faster with a greater α value. To evaluate the performance loss of DPSK with respect to BPSK, we can derive an SNR penalty factor in dB from Eqs. (21) and (32) as 2p 32 π βΓ 2β 10 log 4 5 ; (33) SNRDPSK-BPSK β 2Γ β1 2

where log· is the log function with base 10. We observe from Eq. (33) that this SNR penalty factor depends on the smaller atmospheric turbulence channel parameter β and is the same as that derived for the atmospheric turbulence channel without pointing errors [10]. The above error rate analysis for OWC systems in the gamma–gamma turbulence channels with pointing errors requires that α < φ2 . This is a serious limitation of the series expansion approach using Eq. (12). To overcome this restriction, in Section V, we will use a modified series approach to study the error rate performance when α > φ2 .

C. Numerical Results In this section, we compare the approximate error rate with the exact error rate4 to verify our analytical results. We consider representative moderate (α 2.5, β 2.06) and strong (α 2.04, β 1.1) atmospheric turbulence conditions. We also set the jitter standard deviation σ s equal to the aperture radius r and the parameters ωz∕r 10 [25]. The modulation index is set to be ξ 0.9; the photodetector responsivity, R 0.5; and the noise standard deviation, σ n 10−7 A∕Hz [20,21]. We set the transmitted optical power P as the horizontal axis. In Fig. 2, SERs are presented for subcarriers QPSK and 8PSK over the gamma–gamma turbulence channels with pointing errors. The presented results show excellent agreement between the exact SERs and our series solutions. Figure 3 presents BERs with moderate and strong atmospheric turbulence conditions for subcarrier DPSK, NCFSK, and BPSK modulated OWC systems over the gamma–gamma turbulence channels with pointing errors. We again observe that our series solutions have excellent agreement with the exact BERs. As expected, the asymptotic BERs approach the exact BERs rapidly for α 2.04 and β 1.10 (strong turbulence condition). For α 2.50 and β 2.06 (moderate turbulence condition), the convergence of the asymptotic BERs is slow because in this case 4

(32)

The Rexact error rate Pe is calculated by numerical integration using Pe 0∞ Pe If I IdI, which has been confirmed by Monte Carlo simulations.


Song et al. 0

0

10

10

−1

10

−1

10

−2

10

−3

10

α=2.50 β=2.06 −4

10

−3

10

−4

NCFSK, Exact NCFSK, Series DPSK, Exact DPSK, Series BPSK, Exact BPSK, Series Asymptotic

10

QPSK, Exact QPSK, Series 8PSK, Exact 8PSK, Series

−5

10

α=2.04 β=1.10

−2

10 Average BER

Average SER

α=2.04 β=1.10

−5

10

−6

−10

−8

−6

−4 −2 0 2 4 Transmitted Optical Power, P (dBm)

6

8

10

Fig. 2. SERs of subcarrier QPSK and 8PSK modulated OWC systems over the gamma–gamma turbulence channels with pointing errors for ωz∕r 10 and σ s∕r 1. α

10 −10

−8

−6

α=2.50 β=2.06


6

8

10

Fig. 3. BERs of subcarrier intensity modulated OWC systems using NCFSK, DPSK, and BPSK over the gamma–gamma turbulence channels with pointing errors for ωz∕r 10, σ s∕r 1.

β

the terms γ¯ −2 and γ¯ −2 are similar for small to medium transmitted power P, which corresponds to small to medium β SNRs. Therefore, the term γ¯ −2 in the series solutions becomes dominant only at large transmitted power P.

V. ERROR RATE PERFORMANCE WHEN α > φ2 To study the error rate performance when α > φ2 , we will adopt a modified series expansion approach.

where Γ·; · is the incomplete gamma function. Since I∕A0 is positive, the truncation error RB decreases with increasing values of B (e.g., RB < 10−12 when B 5). Thus, we can always choose a proper value for B to achieve the desired accuracy. In obtaining the last approximation in Eq. (34), we have also used the series expansion of the modified Bessel function in Eq. (8). To obtain the BER expression, we can omit the RB term. Recall Newton’s generalized binomial theorem,

A. Subcarrier BPSK Modulation By introducing an auxiliary parameter B and partitioning the integration interval in Eq. (7) into I∕A0 ; I∕A0 B and I∕A0 B; ∞, the PDF of the gamma–gamma fading with pointing errors can be represented as

2

x yμ

∞ X μ i0

i

xμ−i yi ;

μ ∈ C;

i ∈ Z;

(37)

where

3 p Z αβ dI 2 αβI K p α−β a a 2 6 I∕A0 7 2φ2 αβ 2 φ2 −1 I∕A0 B αβ 2 2 −φ −1 7 dI 2 αβI I φ −1 6 I I K R f I I φ2 a p α−β a a B 2 αβ 4 R 5 −φ2 −1 I∕A0 ∞ A0 ΓαΓβ Aφ0 ΓαΓβ I∕A I a2 K α−β 2 αβIa dI a 0 B 8 h pβ−φ2 pβ−φ2 i > αβpβ I B − AI0 ∞ > < 2 X A Γp−α−β1p!pβ−φ Λα; β; φ φ2 −1 0 (34) ≈ I h pα−φ2 pα−φ2 i 2 > αβpα I I Aφ0 p0> : − Γpα−β1p!pα−φ ; B − 2 A0 A0 αβ 2φ2 αβ 2

R I∕A0 B

αβ

Ia2

−φ2 −1

where RB is the truncation error given by Z RB

∞ I∕A0 B

αβ

Ia2

−φ2 −1

p K α−β 2 αβI a dIa :

(35)

It can be shown that RB is upper bounded by RB <

! r 2 2 22φ −αβ1 αβφ −αβ∕2 K α−β I∕A0 BI∕A0 Bα−β 2 ; 2 αβI∕A B Γ α β − 2φ ; p α−β p 0 α − β α − β2 I∕A0 B2 exp − α − β2 I∕A0 B2

(36)

Song et al.


μ i

μμ − 1 μ − i 1 μi i! i!

(38)

N Λα;β;φ X p φ2 2 π A0 i0 9 8 2 pβ −φ2 pα−φ2 > iφ2 > Γ iφ2 1 PJ > > − > βp − αp γ¯ 2 > > > = < A0 Bi iφ2 p0 i i × : > > > > βi Γ iβ1 αi Γ iα1 > > 2 2 −iβ −iα > > 2 2 − γ ¯ − γ ¯ ; : iβ−φ2 iα−φ2

P2 ≈

is the generalized binomial coefficient. In Eq. (38), ·i is the Pochhammer symbol standing for a falling factorial. When μ is a nonnegative integer n, the generalized binomial coefficients for i > n become zeros and Eq. (37) specializes to the well-known binomial formula x yn

n X n i0

i

i ∈ Z:

xn−i yi ;

μ X ∞ I μ μ−i I i B B ; i A0 A0 i0

i ∈ Z;

iβ

A0 B

iα

(45) (39)

Applying the generalized binomial theorem to AI0 Bμ (μ ∈ R), i.e.,

A0 B

(40)

we can rewrite the PDF in Eq. (34) as

From Eq. (45), we identify that the diversity order is minfφ2∕2; α∕2; β∕2g. The asymptotic BER in Eq. (21) is derived when α < φ2 . When α > φ2 > β, it can be shown from Eq. (45) that the asymptotic BER is the same as Eq. (21). When α > β > φ2 , the PDF of Y near the origin can be expressed as αβ φ2 αβ 2 2 lim f Y y φ2 yφ ∕2−1 y→0 A0 ΓαΓβ Z ∞ αβ p −φ2 −1 × I a2 K α−β 2 αβIa dI a ; (46) 0

f I ≈

Λα; β; φ

∞ X

2

Aφ0

p0

8 9 > > > > 2 > > P∞ p β − φ > 1 1 iφ2 −1 − pβ−1 > > > > > βp iI 2 I > > i0 pβ−φ B A < = 0 A0 B i × ; 2 > > P∞ p α − φ > > 2 −1 > > 1 1 iφ pα−1 > −αp > I − > > 2 I i0 A0 Bi > > A0 Bpα−φ > > i : ; (41) where

where the integral in Eq. (46) equals a constant. Using Proposition 1 in [37], we obtain diversity order Gd φ2∕2 and coding gain 1 aΓt 3∕2 −t1 ; (47) Gc p 2 π t 1 where a is a

αβpβ Bpβ−φ Γp − α − β 1p!p β − φ2

2

αp

αβpα Bpα−φ : Γp α − β 1p!p α − φ2

(43)

Then the BER of subcarrier BPSK modulation can be expressed as Z ∞ q Λα;β;φ Q 2¯γ I 2 f IdI ≈ p φ2 P2 0 2 π A0 9 8 > > iφ2 1 pβ−φ2 pα−φ2 > > > > 2 P∞ Γ > > 2 > > −iφ 2 > > β α γ ¯ − i iφ2 p p > > p0 A B ∞ = < 0 X i i × : > > > iα1 i0 > > > iβ βi Γ iβ1 α Γ iα i > > 2 2 > > − γ¯ − 2 − γ¯ − 2 > > 2 2 > > A0 Biα−φ iα ; : A0 Biβ−φ iβ (44) Keeping i ≤ N and p ≤ J in Eq. (44), we obtain a finite series approximation of Eq. (44) as

Z

∞

2 Aφ0 ΓαΓβ 0

αβ

I a2

−φ2 −1

p K α−β 2 αβIa dIa

(48)

2

aΓt 3∕2 −t1 P∞ γ¯ : 2 p 2 π t 1

42

and

αβ 2

and t φ2 − 1. The asymptotic BER with BPSK modulation can be calculated as [37]

2

βp

φ2 αβ

(49)

Following the approach used for the α < φ2 case, we can straightforwardly derive error rate expressions for subcarrier MPSK, DPSK, and NCFSK modulations when α > φ2 . We omit the derivations here because of space limitations. Some numerical results are presented in Fig. 4. From Eqs. (21), (45), and (49), we conclude that the diversity order of the subcarrier OWC system in the gamma–gamma turbulence channels with pointing errors depends on the smaller value of φ2 and β. This observation agrees with the conclusion made in [27] for an OOK-based IM/DD system. More specifically, the diversity order is Gd 12 minfβ; φ2 g, or more generally Gd 12 minfα; β; φ2 g. Therefore, to avoid the adverse impact of the pointing error on the diversity order of the system, we require φ2 > minfα; βg. It follows that the minimump required equivalent beam width should be ωzeq > 2σ s minfα;βg. Under this condition, an optimal value of the equivalent beam width that minimizes the error rate can be obtained by further studying our asymptotic solutions. Numerical techniques can be adopted to find the optimum beam width.5 5

A detailed discussion on the beam width optimization problem can be found in [20,21,27].


Song et al. 0

10

B. Numerical Results

−1

10

We follow the numerical settings in Subsection IV.C and consider a weak turbulence condition (α 4.03, β 3.45) and a normalized jitter standard deviation σ s∕r 3. Under this operation condition, we have α > φ2 .

−2

10

Average BER

−3

In Fig. 4, we set B 5, N J 30 in Eq. (45) to approximate the exact BER of BPSK in the gamma– gamma fading with pointing errors. BER curves of DPSK and NCFSK are also presented. It can be seen from Fig. 4 that the approximate BERs can achieve remarkable accuracy over a wide range of transmitted power. Asymptotic BERs are also presented to reveal the BER behavior in large transmitted optical power regions.

10

−4

10

−5

10

NCFSK, Exact NCFSK, Series DPSK, Exact DPSK, Series BPSK, Exact BPSK, Series Asymptotic

−6

10

−7

10

−8

10 −10

Figure 5 presents a BER comparison of subcarrier BPSK modulated OWC systems with and without pointing errors. We observe that the pointing error degrades the error rate performance of an uncompensated system severely. The pointing error becomes the dominant factor that affects the system performance, and it can make the system performance unacceptable. Therefore, pointing error compensation is necessary in a practical OWC system.

−8

−6


2

pβ pβ ∞ Λα; β; φ X ap α; β; φ; A0 gp β¯γ − 2 γ 2 −1 : (50) f γ γ pα pα 2 γ − 2 γ 2 −1 p0 −ap β; α; φ; A0 gp α¯

P 6 Po γ th ≈ Λα; β; φ K p0 4

0

ap α;β;φ;A0 pβφ2 −p−β a β;α;φ;A

p 0 − pαφ 2 −p−α

f γ γdγ Λα; β; φ

2 ∞ X 4 × p0

β

pβ 3 ap α;β;φ;A0 gp β γ 2

Λα;β;φαβ P∞ o γ th βAβ φ2 −βΓβ−α1

th

−

Po γ th ≍

ap β;α;φ;A0 gp α pα

Λα; β; φ

0

γ¯

pβ

pα 5; γ th γ¯

8 > > ∞ > < X

2

Aφ0

pβ 3 γ th γ¯

γ th γ¯

β γ th γ¯

2

:

> :

2

52

53

(51)

2

Using Eq. (41), we obtain the outage probability as

9 i iφ2 > P∞ h p β − φ2 p α − φ2 γ th 2 > > βp − αp γ¯ = p0 i i ; iβ iα > βi γ th 2 αi γ th 2 > > − − ; iβ−φ2 iα−φ2 γ¯ γ¯

1 A0 Bi iφ2

i0 > >

2

7 pα 5 ;

when we have assumed α < φ2 . The asymptotic outage probability can be obtained as

Thus the outage probability can be obtained as γ th

10

where γ th is the outage threshold. Similar to the argument used in obtaining Eq. (20), the outage probability can be approximated by a finite series as

Using Eq. (12), we obtain the PDF of γ as

Z

8

Fig. 4. BERs of subcarrier intensity modulated OWC systems using NCFSK, DPSK, and BPSK over the gamma–gamma turbulence channels with pointing errors for α 4.03, β 3.45, ωz∕r 10, and σ s∕r 3 (α > φ2 ).

VI. OUTAGE PROBABILITY PERFORMANCE

Po γ th ≜

6

A0 B

iβ

A0 B

(54)

iα

when α > φ2 . Keeping i ≤ N and p ≤ J in Eq. (54), we obtain a finite series approximation of Eq. (54) as

Po γ th ≈

Λα; β; φ 2

Aφ0

8 > > N > < X > :

i0 > >

1 A0 Bi iφ2

−

9 i iφ2 > p β − φ2 p α − φ2 γ th 2 > > βp − αp γ¯ = p0 i i : iβ iα > βi γ th 2 αi γ th 2 > > − ; iβ−φ2 iα−φ2 γ¯ γ¯

PJ

A0 B

h

iβ

A0 B

Using Proposition 5 in [37], we obtain the asymptotic outage probability as

iα

(55)

Song et al.


0

10

outage probability performance of an uncompensated system severely.

−1

10

VII. CONCLUSIONS

Average BER

With Pointing Error −2

10

−3

10

Without Pointing Error −4

10

Strong Turbulence without Pointing Errors Strong Turbulence with Pointing Errors Moderate Turbulence without Pointing Errors Moderate Turbulence with Pointing Errors

−5

10

0

5

10

15

20 25 30 Electrical SNR (dB)

35

40

45

50

Fig. 5. BER comparison of subcarrier BPSK modulated OWC systems over the gamma–gamma turbulence channels with or without pointing errors for ωz∕r 10, σ s∕r 1.

2a P∞ o γ th φ2

φ 2 γ th γ¯

2

56

;

where a is given in Eq. (48). In Fig. 6, we present outage probabilities of subcarrier intensity modulated systems over the gamma–gamma turbulence fading with pointing errors. Two scenarios are considered. In scenario one, we consider a strong (α 2.04, β 1.1) atmospheric turbulence condition with ωz∕r 10, σ s∕r 1, where α < φ2 . In scenario two, we consider a weak (α 4.03, β 3.45) atmospheric turbulence condition with ωz∕r 10, σ s∕r 3, where α > φ2 . From Fig. 6, we again observe that the approximate outage probabilities can achieve remarkable accuracy over a wide range of SNR values. Asymptotic outage probabilities are also presented to reveal the outage behavior in large SNR regimes. In Fig. 6, we also conclude that the pointing error can degrade the

−2

10

α=2.04 β=1.10

Outage Probability

−4

−6

10

−8

10

Exact, Scenario 1 Series, Scenario 1 Exact, Scenario 2 Series, Scenario 2 Asymptotic Exact, No Pointing Error

−10

10

−12

10

25

30

35

40

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In this work, using a series expansion approach, we have developed highly accurate series error rate and outage probability expressions for SIMs over the gamma–gamma turbulence channels with pointing errors. Our novel series solutions are general and agree with the exact error rate and outage probability results over a wide range of operation conditions. Therefore, the series method used in this work is a powerful analytical tool in studying the performance of a subcarrier intensity modulated OWC system. Asymptotic analysis based on our series solutions shows that the diversity order of the studied system is determined by 12 minfα; β; φ2 g. When φ2 > minfα; βg, the impact of the pointing error on the diversity order of the system can be mitigated, and such a condition can be satisfied by p setting the equivalent beam width greater than 2σ s minfα; βg. An optimal value of the equivalent beam width that minimizes the error rate or outage probability can be further obtained by studying our asymptotic solutions. Our results also indicate that pointing error compensation is necessary in a practical OWC system; otherwise the pointing error can degrade the error rate performance and outage probability performance of the system severely.

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