BER analysis of DPSK–SIM over MIMO free space optical links with misalignment

Optik 125 (2014) 5176–5180

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BER analysis of DPSK–SIM over MIMO free space optical links with misalignment K. Prabu ∗ , D. Sriram Kumar Department of Electronics and Communication Engineering, National Institute of Technology (NIT), Tiruchirappalli 620015, India

a r t i c l e

i n f o

Article history: Received 6 September 2013 Accepted 1 May 2014 Keywords: Free space optics Differential phase shift keying subcarrier intensity modulation Strong atmospheric turbulence Pointing errors Single-input multiple-output Multiple-input multiple-output

a b s t r a c t Free-space optics (FSO) is a safe, license-free, high speed and high-bandwidth cordless technique for broadcasting signals at the same speed as fiber optics. But the main drawback of FSO is the dissolution of the signals due to atmospheric turbulence and thus degrading the link performance. So, to enhance the bit error rate (BER) performance, spatial diversity is employed over the FSO links for single-input multipleoutput FSO (SIMO-FSO) systems. In this paper, we analyze the BER performance of SIMO-FSO links with spatial diversity and multiple-input multiple-output FSO (MIMO-FSO) systems over gamma–gamma atmospheric turbulence fading channel with misalignment (pointing errors). The closed form expressions are derived for the same with various combining schemes in terms of the Meijer G function and also evaluate the results with 2D and 3D graphs. © 2014 Elsevier GmbH. All rights reserved.

1. Introduction Free space optical (FSO) communication is highly directional, secure, license-free, interference-free, high speed and wide bandwidth communication with low maintenance cost and less deployment time. FSO has outdoor wireless capabilities and thus gives greater mobility and flexibility to its users [1–3]. Presently, FSO communication systems have the speed of 1.25 Gbps (same speed as of fiber optics). Also, FSO has the potential capability of speeds as high as 10 Gbps possible as the data will be transmitted through air faster than the glass of the fiber optic cables [4,5]. This permits transmission and communication at the speed of light. Regardless of these major benefits, FSO faces several practical challenges for its commercialization. These include absorption, scattering and diffraction by the particles in the atmosphere, misalignment due to sway of buildings by wind loads, thermal expansion and weak earthquakes [6–10]. But, the blunder corruption of the signals is due to atmospheric turbulence. The main characteristic features of the optical domain are intensity, phase and polarization which are used to modulate and transmit signals. The signaling formats in FSO include on–off keying (OOK), pulse modulation (PM), subcarrier intensity modulation (SIM) and differential phase-shift keying (DPSK). OOK is used for its simplicity

∗ Corresponding author. Tel.: +91 9884888408. E-mail address: [email protected] (K. Prabu). http://dx.doi.org/10.1016/j.ijleo.2014.05.012 0030-4026/© 2014 Elsevier GmbH. All rights reserved.

[11], PM for power efficiency, SIM for low bandwidth requirement [12] while DPSK signaling can dissolve the scintillation effect to some extent. There are many methods to overcome the corruption effects of the signals due to turbulence-induced fading. Diversity techniques are employed to improve channel performance and dissolve in the FSO links. The various diversity techniques used in FSO communication are wavelength diversity, temporal diversity and space diversity [13–17]. Since the effect of atmospheric turbulence remains practically the same for all wavelengths. So, the wavelength diversity technique is less effective for FSO communication systems. Temporal diversity demands longer signal processing time [15]. Hence, we consider spatial diversity technique, where multiple receivers are used to receive the same bits of data. The spatial correlation between a pair of constituent transmitters in a multipleinput multiple-output (MIMO) system is investigated in [18]. The performance of the FSO communication system using binary phase shift keying subcarrier intensity modulation (BPSK-SIM) over a strong turbulence channel is investigated in [19]. In a recent work, the bit error rate (BER) performance of DPSK–SIM based MIMOFSO system over gamma–gamma distribution is evaluated [20]. The performance of multi-beam FSO system with diversity is analyzed in [21]. In this paper, the BER performance is analyzed for DPSK based SIM-FSO links over strong atmospheric turbulence with pointing errors and the closed-form mathematical expressions are derived for the average BER with various combining schemes and MIMO techniques.

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The rest of the paper is organized as follows: Section 2 discusses the system and channel model used. The average BER for SISO and MIMO-FSO systems are discussed in Sections 3 and 4 respectively. Section 5 describes the numerical results with graphical analysis. Finally, concluding remarks are highlighted in Section 6.

3. BER analysis for SISO-FSO system

2. System and channel model

1 1 Pec,SISO (h) = exp − SNR 2 2

The probability of conditional BER of the considered DPSK based SIM-FSO system depends upon the intensity fluctuation [23] and can be stated as





1 = exp 2



R2 A2 h2 − 4 2



(4)

2.1. System model The systems considered in this paper are single-input singleoutput (SISO) and MIMO-FSO in strong atmospheric turbulence with additive white Gaussian noise (AWGN). Here, M represents the transmission apertures, N represents the reception apertures over a discrete time ergodic channel. The data to be transmitted is modulated by using DPSK–SIM and transmitted into a strong atmospheric turbulence channel through telescope. The electrical signal acquired at the nth receive aperture is given by [14]



where R is the photo detector responsivity,  2 is the variance of the channel noise. The average BER Pe , for strong atmospheric turbulence channel can be expressed as



∞

Pe,SISO =

Pec,SISO (h)fh (h)dh

By using Eqs. (3) and (4) in (5), the average BER can be obtained as

M

rn = xR

hmn + vn ,

n = 1, 2, . . .N

(1) Pe,SISO =

m=1

where x is the transmitted signal, R is the detector responsivity, hmn is the irradiance from the mth transmitter and nth receiver, vn is the AWGN with zero mean and variance  v = N0 /2. The irradiance ‘hmn ’ models the optical intensity fluctuations resulting from atmospheric loss, turbulence and fading is given as [22] hmn = hlmn hsmn hpmn

(2)

where hlmn is the attenuation due to beam extinction and path loss from the mth transmitter and nth receiver, hsmn due to scintillation effects and hpmn due to the geometric spread and pointing errors.

˛ˇ 2 × A0 hl  (˛) (ˇ)

fhmn (hmn ) =

2 ˛mn ˇmn mn

A0mn hlmn  (˛mn ) (ˇmn )

3,0 G1,3



 2

˛mn ˇmn hmn  mn  2 A0mn hlmn −1+

mn

(3) ,˛mn −1,ˇmn −1

where ˛mn and ˇmn represent the effective number of large and small scale turbulent eddies,  (·) is the gamma function. The channel model for SISO is obtained by substituting m = n = 1.



∞

0.5 exp − 0

R2 A2 h2 4 2



 3,0 × G1,3

 2

˛ˇh   A0 hl −1+2 ,˛−1,ˇ−1

 dh

An exp(·) in the above integral can be expressed as the Meijer G function [24, Eq. (11)], the average BER of DPSK–SIM can be obtained in a closed form by using [24, Eq. (21)]

Pe,SISO =

2˛+ˇ−4  2 G1,6  (˛) (ˇ) 6,3



(1−2 /2),(2−2 /2),(1−˛/2),(2−˛/2),(1−ˇ/2),(2−ˇ/2)   2 2

4R2 A2 A20 h2l   2 ˛2 ˇ2





(6)

2.2. Channel model Here the strong atmospheric turbulence channel model is considered for MIMO-FSO systems using gamma-gamma distribution channel with misalignment. The probability density function (PDF) of the considered channel model in terms of Meijer G function is given as [19]

(5)

0

(7)

0,(− /2),(1− /2)

4. BER analysis for MIMO-FSO system The BER performance in a strong atmospheric turbulence channel can be improved by employing spatial diversity technique. It can be implemented either at the transmitter (MISO) or at the receiver (SIMO) or at both the sides (MIMO). The average BER for MIMO-FSO links can be calculated from [14]



⎛

∞

Pe,MIMO =

fh (h) · 0

R2 A2

1 exp ⎝− 2 4 2

M N   n=1

2 ⎞ ⎠ dh hMN

(8)

m=1

where fh (h) is the joint PDF of vector h = (h11 , h12 , . . . hMN ) of length MN. By using Eq. (3) in (8), the average BER can be obtained as

Pe,MIMO

2 ˛mn ˇmn mn × = A0mn hlmn  (˛mn ) (ˇmn )

0

∞

⎛

R2 A2  0.5 exp ⎝− 4 2 N

n=1

M  m=1

2 ⎞   2 ˛mn ˇmn hmn  mn 3,0 ⎠ hMN × G1,3 A0mn hlmn 

 dhMN

(9)

2 ,˛ −1+mn mn −1,ˇmn −1

By expressing the exp(·) as Meijer G function [24, Eq. (11)], the average BER of DPSK–SIM can be expressed in a closed-form by using [24, Eq. (21)]

⎡ Pe,MIMO =

M N m=1

2 2˛mn +ˇmn −4 mn 1,6 ⎢ × G6,3 ⎣ n=1  (˛mn ) (ˇmn )

 1−   

2

2− 2

mn , mn , 2 2 4R2 A2 A20 h2l mn mn 2 M2N2 − 2 1− 2  2 ˛2mn ˇmn 0, 2mn , 2mn

1−˛mn 2

,

2−˛mn 2

,

1−ˇmn 2

,

2−ˇmn 2

⎤ ⎥ ⎦

(10)

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Cost-effective key in strong atmospheric turbulence is to implement multiple receive apertures that dissolve the turbulence induced irradiance variation. In SIMO, diversity techniques used in the receiver include optimal combining (OC), equal gain combining (EGC) and selection combining (SC).

The average BER of SIMO-FSO system with EGC diversity can be expressed from Eq. (5) as

Pe,EGC = 0.5 ×

SNROC =

2N 2

n=1

h2n

N Pe,EGC = ˘n=1

⎡

(11)

⎢ ⎣

The conditional BER of the system can be computed as Pec,OC (h) = Q



 SNROC

 =Q

R 2 A2  N h2n 2N 2 n=1





∞

Pe,OC =

Q

2N 2

0

n=1

h2n

Pe,OC ≈

1 12



fh (h) dh

(13)

+

4 n=1

∞

h2 n=1 n

− (R2 A2 /4N )

fhn (hn )e



∞

− (R2 A2 /3N 2 )

 1−   

n=1

fhn (hn )e

h2 n

2

2− 2

dhn

Pe,SC = 0.5 ×

dhn



RA 2N

Pe,OC ≈

12

⎢

n=1 (˛n )(ˇn )

G1,6 6,3 ⎣

+

4

2˛n +ˇn −3 n2

 1−   

n=1 (˛n )(ˇn )

⎢

G1,6 6,3 ⎣

 1−   

1−ˇn 2

,

2−ˇn 2

⎤ ⎥ ⎦

(19)

erfc

n=1

h2n

fh (hSC )dhSC

(21)

where the selective channel model PDF can be expressed as

2

2− 2

n, n, 2 2 4R2 A2 A20 h2l N 2 ˛2n ˇn2 0, −n2 , 1−n2 2 2

⎡ 1 N

,



N

 d F (hSC ) = dhSC hSC N

⎡

2−˛n 2

(14)

By applying Eq. (3) in (14) and express the exp(·) as Meijer G function [24, Eq. (11)], the average BER of DPSK–SIM can be evaluated using [24, Eq. (21)] as

2˛n +ˇn −3 n2

,

(20)

∞

0

0

1 N

1−˛n 2

The average BER of the receiver with SC diversity can be expressed by using the conditional BER probability and selective channel model as



N

(18)

fh (h) dh

hSC = max(h1 , h2 , . . .hn )



N 2

h2n

2˛n +ˇn −4 n2 1,6 G (˛n )(ˇn ) 6,3

n, n, 2 2 4R2 A2 A20 h2l N 2  2 ˛2n ˇn2 0, −n2 , 1−n2 2 2

0

n=1 N

1



n=1

The simplest combining scheme which is used in SIMO is selection combining. In which the received signal having high SNR is selected and other received signals are discarded. The selection is made according to [14]

The integral in Eq. (13) is simplified by using the approximation of Q-function [25, Eq. (14)] to obtain the average BER as N



N

4.3. Selection combining



R A N 2 2



(12)

The OC diversity is applied at the receiver, then the average BER of SIMO-FSO is written as



RA 2N

The above integral can be evaluated by expressing the complementary error function as Meijer G function [26, Eq. (8.4.14.2)] and using [24, Eq. (21)]

The electrical signal to noise ratio (SNR) of the FSO system with OC can be described as R A N

erfc 0

4.1. Optimal combining

2 2

∞

2

2− 2

n, n, 2 2 16R2 A2 A20 h2l 3N 2 ˛2n ˇn2 0, −n2 , 1−n2 2 2

1−˛n 2

1−˛n 2

,

,

2−˛n 2

2−˛n 2

,

fhSC (hSC ) =

N 

fhi (hSC )Fhj (hSC )

(22)

i=1 j=1,j = / i

,

1−ˇn 2

1−ˇn 2

,

,

2−ˇn 2

2−ˇn 2

⎤ ⎥ ⎦

⎤ ⎥ ⎦

(15)

By applying Eq. (22) in (21), we get the average BER 4.2. Equal gain combining In EGC diversity technique, N number of photo detectors in the receiver is realized. The signals from N photo detector are unified and then demodulated. The electrical SNR of the FSO system with EGC can be described as SNREGC

R 2 A2  N = h2n 2N 2 n=1

(16)



 SNREGC

= 0.5 × erfc

RA 2N



N n=1

 h2n

Pe,SC = 0.5×

i=1

/ i j=1,j =

(17)

∞

RA 2N

erfc 0

 N

n=1

× Fhj (hSC )dhSC

h2n

 fhi (hSC ) (23)

where the cumulative distribution function (CDF), Fh (hSC ) can be expressed as [12] n2 G3,1 FhSC (hSC ) =  (˛n ) (ˇn ) 2,4

The probability of conditional BER can be defined as Pec,EGC (h) = Q



N N



1,n2 +1

 ˛n ˇn hSC  A0 hl



(24) n2 ,˛n ,ˇn ,0

The integral in Eq. (23) can be evaluated by the Gaussian quadrature rule (GQR).

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Fig. 1. BER against SNR for SISO and SIMO-FSO systems with ˛ = 4, ˇ = 1 and ˛ = 1, ˇ = 1.

5. Numerical results and discussions Here, the conclusive mathematical results of the detailed expressions derived in Sections 3 and 4 are presented. For the numerical valuations, the FSO system parameters are noise standard deviation  n = 10−7 A/Hz, photo detector responsivity  = 0.5 A/W, beam radius wL ∼ = 2.5 m at 1 km distance and jitter standard deviation  s ∼ = 30 cm. Fig. 1 demonstrates that the use of the SIMO system in terms of SNR, improves the average BER performance. That is, the average BER performance gets enhanced by boosting the number of receiving apertures and employing the combining schemes. The BER performance of SISO and SIMO-FSO systems with OC, EGC and SC are compared for the various values of ˛ = 4, ˇ = 1, ˛ = ˇ = 1 and N = 3, 4, 6 as shown in Fig. 2. From Fig. 2, it is observed that the best error rate performance is accomplished with SC diversity (N = 6, ˛ = 4, ˇ = 1) with BER of 10−4 for SNR = 50 dB. Fig. 3, illustrates that the BER comparison of SISO, SIMO, MISO and MIMO-FSO systems for ˛ = 4, ˇ = 1 and ˛ = ˇ = 1. Among the considered schemes, MIMO with four transmitting and four receiving aperture (MIMO, M = N = 4) gives the best BER performance of 10−17 and 10−14 for ˛ = 4, ˇ = 1, ˛ = ˇ = 1, respectively. The 3D plots shown in Fig. 4 depicts that the negotiation between the average BER against the effective number of large scale (˛) and small scale (ˇ) turbulent eddies for SISO, SIMO with OC, EGC and SC combining schemes respectively. The 3D graphs are plotted only for the values of ˛ = ˇ = 4. From Fig. 4, it is inferred that the best error rate performances are achieved for large values of ˛ = ˇ = 4.

Fig. 3. BER against SNR for SISO and MIMO-FSO systems with (a) ˛ = 4, ˇ = 1 and (b) ˛ = 1, ˇ = 1.

Fig. 4. 3D plot for BER variation with respect to ˛ and ˇ for (a) SISO (b) OC, (c) EGC and (d) SC.

Fig. 2. BER against SNR for SIMO-FSO system with OC, EGC, SC for (a) ˛ = 4, ˇ = 1 and (b) ˛ = 1, ˇ = 1.

Fig. 5. 3D plot for BER variation with respect to ˛ and ˇ for MIMO-FSO system, M = N = 2.

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The minimum obtained BER values are 0.25, 3 × 10−2 , 5 × 10−2 and 1.5 × 10−2 for SISO, SIMO with OC, EGC and SC diversity using three receive apertures respectively (N = 3). Fig. 5, depicts the tradeoff between the average BER against the ˛ and ˇ for MIMO-FSO systems. The better average BER performance 10−4 is achieved through 2 × 2 MIMO systems. 6. Conclusions In this paper, the average BER performance of DPSK–SIM based SISO, SIMO and MIMO-FSO systems over gamma–gamma atmospheric turbulence channel model with misalignment is analyzed. Also the closed-form expressions for SIMO with various combining schemes in terms of the Meijer G function are derived. It is noted that SC presents the best BER performance amongst all the combining schemes considered. It is also demonstrated that the BER performance develops with the boosting of receiving apertures. This work presents the simulation of BER performances of SIMO and MIMO with a maximum of six receive apertures. The best BER performance is achieved by using MIMO 4 × 4 is 10−17 at SNR = 50 dB. References [1] H.A. Willebrand, B.S. Ghuman, Fiber optics without fiber, IEEE Spectr. 38 (2001) 40–45. [2] T. Ohtsuki, Multiple-subcarrier modulation in optical wireless communications, IEEE Commun. Mag. 41 (3) (2003) 74–79. [3] D. Kedar, S. Arnon, Urban optical wireless communication networks: the main challenges and possible solutions, IEEE Commun. Mag. 42 (2004) S2–S7. [4] H. Le-Minh, D. O’Brien, G. Faulkner, et al., A 1.25-Gb/s indoor cellular optical wireless communications demonstrator, IEEE Photonics Technol. Lett. 22 (2010) 1598–1600. [5] C.J. Henderson, D.G. Leyva, T.D. Wilkinson, Free space adaptive optical interconnect at 1.25 Gb/s with beam steering using a ferroelectric liquid-crystal, J. Lightwave Technol. 24 (5) (2006) 1989–1997. [6] S. Arnon, Optimization of urban optical wireless communication systems, IEEE Trans. Wirel. Commun. 2 (2003) 626–629. [7] H.G. Sandalidis, Coded free-space optical links over strong turbulence and misalignment fading channels, IEEE Trans. Commun. 59 (3) (2011) 669–674. [8] S. Arnon, Effects of atmospheric turbulence and building sway on optical wireless-communication systems, Opt. Lett. 28 (2003) 129–131.

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