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Ergodic Capacity of SIM-Based DF Relayed Optical Wireless Communication Systems Mona Aggarwal, Student Member, IEEE, Parul Garg, Member, IEEE, and Parul Puri, Student Member, IEEE Abstract— In this letter, we consider a dual-hop, subcarrier intensity modulation-based relayed optical wireless communication system under the combined influence of atmospheric turbulence and pointing error impairments. The turbulence-induced fading is modeled by independent but not necessarily identically distributed Gamma–Gamma fading statistics with misalignment fading. The relaying protocol followed by the relay is decode and forward. We first, derive the statistics of instantaneous signal-to-noise ratio at the destination, followed by the novel and exact analytical expressions for ergodic capacity of the system in terms of mathematically tractable Meijer’s G-function and extended generalized bivariate Meijer’s G-function. Index Terms— Ergodic capacity, extended generalized bivariate Meijer’s G-function (EGBMGF), Gamma–Gamma distribution, pointing errors.
I. I NTRODUCTION
W
ITH the increasing demand for high data rate capacity, the optical wireless communication (OWC) systems are gaining considerable research interest in recent times. However, the performance of these systems is vulnerable to adverse channel conditions which occur mainly due to atmospheric turbulence-induced fading and pointing errors [1]. In case the source is not in line-of-sight (LOS) with the destination or the distance between the two is large enough to communicate, the relay-assisted communication can be employed to mitigate the adverse effects of atmospheric turbulence [2]. In literature, we study two type of relays i.e. amplify and forward (AF) and decode and forward (DF). The AF relay simply amplifies and forwards the incoming signal towards the destination whereas the DF relay decodes the incoming signal and then retransmits the decoded version towards the destination. The performance analysis of relay assisted OWC systems employing different system models (serial and/or parallel) has been done in [2]–[9]. The authors in [2] studied the outage probability of serial and parallel relaying transmission schemes employing both type of relays over weak turbulence regimes. The outage and error performance of AF relayed OWC systems have been studied in [3] over moderate to strong turbulence regimes. Another important issue of concern with OWC systems is pointing errors, which occur as a result of misalignment
between source and destination transceivers due to sway of high-rise buildings with dynamic wind loads, thermal expansions and weak earthquakes. The outage performance of DF relayed OWC systems have been analyzed in [4] over weak turbulence regimes with pointing errors. The error performance of DF relayed OWC systems in terms of average bit error rate (BER) and average symbol error rate (SER) has been studied in [5] and [6] using Gamma-Gamma distribution over strong turbulence channels. In [7], the ergodic capacity of AF relayed OWC system has been studied with the assumption that channel state information (CSI) is known only at the receiving terminals. Much of the earlier work is done considering traditional on-off keying (OOK) with intensity modulation and direct detection (IM/DD) systems which require adaptive thresholds for optimal performance and are subjected to channel estimation errors. Recently the subcarrier intensity modulation (SIM) has emerged as an attractive alternative to OOK IM/DD systems. In [8], the outage and SER performance of SIM based AF relayed OWC systems have been analyzed considering moderate to strong turbulence regimes with pointing error impairments whereas in [9], ergodic capacity of SIM-based AF relayed OWC system has been studied for various channel adaptive transmission schemes. In this letter, we investigate the ergodic capacity of SIM-based DF relayed OWC under the combined influence of turbulence and pointing error impairments which has not been analyzed yet. The turbulence channels are modeled by independent but not necessarily identically distributed (i.n.i.d) Gamma-Gamma fading distribution which models both smallscale and large-scale irradiance fluctuations and includes the well-known K-distributed model and the negative exponential model as the special cases. We derive the statistical characteristics of instantaneous signal to noise ratio (SNR) at the destination in terms of probability density function (pdf) and utilize it to obtain the exact analytical expression for ergodic capacity of the system. The derived expression is in terms of compact, mathematically tractable Meijer’s G-function and extended generalized bivariate Meijer’s G-function (EGBMGF). II. S TATISTICAL C HARACTERIZATION
Manuscript received April 9, 2014; revised November 13, 2014; accepted February 20, 2015. Date of publication March 2, 2015; date of current version April 29, 2015. (Corresponding author: Parul Garg.) The authors are with the Division of Electronics and Communication Engineering, Netaji Subhas Institute of Technology, New Delhi 110078, India (e-mail:
[email protected];
[email protected];
[email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LPT.2015.2407897
A. System Model We consider a SIM-based dual-hop OWC system with three nodes, a source (s), a destination (d) and a single relay (r) as in [9]. The transceiver mounted on relay consists of two directional apertures (one directed towards the source and other towards the destination) and the source and destination transceivers are equipped with single directional apertures
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AGGARWAL et al.: ERGODIC CAPACITY OF SIM-BASED DF RELAYED OPTICAL WIRELESS COMMUNICATION SYSTEMS
directed towards the relay. We assume that all the three transceivers are located on the high-rise buildings to enable LOS communication and the direct communication between source and destination is not feasible due to practical reasons described earlier. The relay follows a DF relaying protocol i.e. it receives the signal transmitted by source, decodes it and then re-transmits it towards the destination. At any time instant, the source transmits symbol x s to relay with average power normalized to unity and the relay transmits the previously decoded symbol xr to the destination where xr is given as, xr = xˆs . Here xˆs is the symbol decoded by the relay in the previous time instant. We further assume that CSI is present both at the relay and the destination and the irradiance of channel remains constant over a symbol duration. The received symbols at the relay, yr and the destination, yd can be written as, yr = h sr x s + n r yd = h rd xr + n d
(1)
where nr and n d are the additive white Gaussian noise (AWGN) of zero mean and variance N0 /2 at the relay and the destination respectively. Further h sr and h rd denote the channel coefficients for source to relay (s − r ) and relay to destination (r − d) links respectively.
with E denoting the expectation operator. The turbulence parameters α and β, both being greater than zero, are given by [6, eqs. (4) and (5)]. Further the cumulative γdistribution function (CDF) of x y is defined as Fx y (γ) = 0 f x y (γ)dγ as given in [6] as ξx2y Fx y (γ) = (αx y )(βx y ) 1, ξx2y + 1 γ 3,1 × G 2,4 αx y βx y . (3) x y ξx2y , αx y , βx y , 0 C. DF Relaying Protocol For DF relaying protocol, the equivalent instantaneous SNR at the destination γ D F , can be written as γ D F = min(sr , rd )
f x y (γx y ) =
F D F (γ) = Pr(sr < γ) + Pr(rd < γ) − Pr(sr < γ) Pr(rd < γ) = Fsr (γ) + Frd (γ) − Fsr (γ)Frd (γ)
ξx2y + 1 ξ 2, α , β xy xy xy
(5)
where Fsr (γ) and Frd (γ) are the CDF’s of sr and rd respectively. Further by taking the derivative of (5) with respect to γ, the pdf of γ D F , f D F (γ) can be obtained as follows f D F (γ) = f sr (γ) + frd (γ) − fsr (γ)Frd (γ) − f rd (γ)Fsr (γ).
ξx2y 2γx y (α x y )(βxy ) γx y 3,0 αx y β x y × G 1,3 x y
(4)
Therefore the CDF of γ D F , F D F (γ) can be derived as F D F (γ) = Pr(γ D F ≤ γ) = Pr(min(sr , rd ) ≤ γ) where sr and rd are instantaneous electrical SNRs of the s −r and r − d links respectively. Since sr and rd are independent, the CDF can be written as
B. Channel Model All the channels are assumed to be stationary, memoryless and ergodic with i.n.i.d fading statistics taking into account the combined effects of atmospheric turbulence-induced scintillation and misalignment-fading. The instantaneous electrical SNR, x y for {x y} ∈ {sr, r d} is given as x y = (ηh x y )2 /N0 where η is the optical to electrical conversion ratio, N0 is the variance of AWGN. The pdf of x y can be written, as in [9], as
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(6)
III. E RGODIC C APACITY (2)
where pointing error parameter ξ = we /(2σs ), with we being the equivalent beam radius at the receiver and σs being the standard deviation of the pointing error displacement at the receiver, (w) is the gamma ∞ function defined as (w) = 0 t w−1 exp (−t)dt, G m,n p,q is Meijer’s G-function [10, eq. (8.2.1.1)], x y denotes the average electrical SNR given as x y = (ηE[h x y ])2 /N0
The ergodic capacity of the system (in bits per second per hertz) Cerg , in terms of pdf of instantaneous SNR [11], is given as ∞ 1 ln(1 + γ) f D F (γ)dγ (7) Cerg = ln(2) 0 Substituting (6) in (7) and expressing logarithmic integrand in terms of Meijer’s G-function using ln(1 + x) = 1,2 1,1 x | 1,0 [10, eq. (8.4.6.5)], (7) may be re-written G 2,2 as (8), as shown at the bottom of this page.
⎡ Cerg
1 ⎢ ⎢ = ⎢ ln(2) ⎣
∞
0
1,2 G 2,2
γ
∞ 1, 1 1,2 G 2,2 γ 1, 0 f sr (γ)dγ + 0 I1
∞
−
0
1, 1 1, 0 f rd (γ)dγ I2
∞ 1, 1 1,2 1,2 G 2,2 G 2,2 γ γ fsr (γ)Frd (γ)dγ − 1, 0 0 I3
⎤
⎥ 1, 1 ⎥ (γ)F (γ)dγ f ⎥ sr 1, 0 rd ⎦ I4
(8)
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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 27, NO. 10, MAY 15, 2015
Fig. 1. Ergodic Capacity (in b/s/Hz) Vs average electrical SNR per hop (dB) under moderate and strong turbulence regimes for different values of normalized beam width we /ra and normalized jitter σs /ra = 2 with link length L=2 km.
Fig. 2. Ergodic Capacity (in b/s/Hz) Vs average electrical SNR hop (dB) for different values of ξ under moderate turbulence strength for link length L=2 km.
Now we will evaluate the integrals given in (8) separately. To evaluate the integral I1 , we substitute f sr (γ) from (2) and integrate with the help of [10, eq. (2.24.1.1)] and [12, eq. (07.34.03.0002.01)], to obtain I1 as I1 =
2 2αsr +βsr −1 ξsr 4π ln(2)(αsr )(βsr ) ⎞ ⎛ 2 +2 ξsr 0, 1, ⎟ 2 7,1 ⎜ ⎟ (9) ⎜κ1 2 × G 3,7 ⎠ ⎝ ξsr αsr αsr + 1 βsr βsr + 1 , , , , , 0, 0 2 2 2 2 2 α2 β 2
sr sr where κ1 = { 16 }. In a similar fashion, a closed form sr expression of I2 can be obtained by replacing sr with r d in (9). Further substituting (2) and (3) in (8) and re-writing with the help of [10, eq. (8.2.2)], the integral I3 can be re-written as
∞ 1, 1 1 1,2 κ7 κ8 G I3 = γ 8π 2 ln(2) 0 γ 2,2 1, 0
κ5 κ 2 6,0 6,2 G 4,8 κ4 γ dγ (10) × G 2,6 κ1 γ κ3 κ6
where κ2 2 ξsr
{
2 +1 ξ 2 +2 ξsr sr 2 , 2 },
κ4
=
α2 β 2
rd rd { 16 }, κ3 rd
, α2sr , αsr2+1 , β2sr , βsr2+1 } , κ5 = { 12 , 1,
2 +1 ξrd
2
2
=
2 +2 ξrd
2 , 2 2 2 +1 ξrd ξrd κ6 = { 2 , 2 , α2rd , αrd2+1 , β2rd , βrd2+1 , 0, 12 }, κ7 2 ξ 2 ξrd }, and κ8 = {2αsr +βsr +αrd +βrd −3 }. { (αsr )(βsrsr)(α rd )(βrd )
{
,
2 +1 ξsr
=
},
=
Now integrating with the help of [12, eq (07.34.21.0081.01)], the integral I3 is evaluated as
κ7 κ8 0, 1 κ2 κ5 2,1:6,0:6,2 (κ I3 = G ), (κ ) (11) 1 4 8π 2 ln(2) 2,2:2,6:4,8 0, 0 κ3 κ6 ,n 1 :m 2 ,n 2 :m 3 ,n 3 where G mp11,q is the EGBMGF as given in 1 : p2 ,q2 : p3 ,q3 (.) [12, eq. (07.34.21.0081.01)]. Similarly a closed form expression of I4 can also be obtained by interchanging sr and r d in (11). Now substituting (9) and (11) in (8), we obtain Cerg as given in (12), as shown at the top of the next page,
Fig. 3. Ergodic Capacity (in b/s/Hz) Vs normalized jitter σs /ra under different turbulence regimes for total link length L=4 with normalized beam width being fixed to we /ra = 10. 2 +1 ξ 2 +2 2 +1 ξrd ξ 2 ξrd αrd rd = { rd 2 , 2 }, κ10 2 , 2 , 2 , 2 2 ξsr +1 ξsr +2 αrd +1 βrd βrd +1 1 2 , 2 , 2 }, κ11 = { 2 , 1, 2 , 2 } and κ12 = ξ 2 ξ 2 +1 { 2sr , sr2 , α2sr , αsr2+1 , β2sr , βsr2+1 , 0, 12 } . Therefore (12) gives
where κ9
=
{
an exact analytical expression for ergodic capacity of the DF relayed SIM OWC system in terms of Meijer’s G-function and EGBMGF. To the best of authors’ knowledge the Meijer’s G function is available in standard mathematical softwares but EGBMGF can be evaluated by using the Mathematica implementation as given in [13, table (II)]. Further it can be easily shown that for the non-pointing error case i.e. (ξi → ∞), the expression of ergodic capacity reduces to (13), as shown at the top of the next page, where ψ1 = { α2sr , αsr2+1 , β2sr , βsr2+1 }, ψ2 = { α2rd , αrd2+1 , βrd βrd +1 1 2 , 2 }, ψ3 = { (αsr )(βsr )(αrd )(βrd ) } and ψ4 = {2αsr +βsr +αrd +βrd }. IV. N UMERICAL R ESULTS In this section we investigate the ergodic capacity of the system under the combined influence of atmospheric turbulence with pointing error impairments and illustrate it through numerical plots. The analysis is done for moderate
AGGARWAL et al.: ERGODIC CAPACITY OF SIM-BASED DF RELAYED OPTICAL WIRELESS COMMUNICATION SYSTEMS
⎞ 2 +2 ξsr 0, 1, 2 2αsr +βsr −1 ⎟ ξsr 2 7,1 ⎜ ⎟ ⎜ κ1 2 = G 3,7 ⎠ ⎝ α α β β ξ + 1 + 1 4π ln(2)(αsr )(βsr ) sr sr sr sr sr , , , , , 0, 0 2 2 2 ⎛ 2 2 ⎞ 2 +2 ξrd 0, 1, ⎜ ⎟ 2 2αrd +βrd −1 ξrd 2 ⎟ 7,1 ⎜ + G 3,7 ⎜κ4 2 ⎟ ⎝ ξrd αrd αrd + 1 βrd βrd + 1 ⎠ 4π ln(2)(αrd )(βrd ) , , , , , 0, 0 2 2 2 2 2
κ7 κ8 0, 1 κ2 κ5 0, 1 κ9 κ11 2,1:6,0:6,2 2,1:6,0:6,2 (κ1 ), (κ4 ) + G 2,2:2,6:4,8 (κ4 ), (κ1 ) − 2 G 2,2:2,6:4,8 8π ln(2) 0, 0 κ3 κ6 0, 0 κ10 κ12
0, 1 0, 1 2αsr +βsr 2αrd +βrd 6,1 6,1 G G = κ1 κ4 + ψ1 , 0, 0 ψ2 , 0, 0 4π ln(2)(αsr )(βsr ) 2,6 4π ln(2)(αrd )(βrd ) 2,6 1 1 2,1 , 1 − − ψ3 ψ4 0, 1 0, 1 2,1:4,0:4,2 2,1:4,0:4,2 2 κ 1 , κ 4 + G κ 4 , κ 1 − G 2,2:0,4:2,6 2,2:0,4:2,6 16π 2 ln(2) 0, 0 ψ1 ψ2 , 0, 12 0, 0 ψ2 ψ1 , 0, 12
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⎛
Cerg
Cerg
(Cn2 = 3 × 10−14 ) and strong (Cn2 = 1 × 10−13 ) turbulence [6] with the wavelength λ = 1550 nm and radius of receiver aperture ra = 10 cm which is half of receiver aperture diameter [9]. Without loss of generality, we take turbulence parameters for s − r and r − d links as αsr = αrd = α, βsr = βrd = β and ξsr = ξrd = ξ with average SNR per hop sr = rd = . Fig. 1 shows a plot of ergodic capacity (in bits per second per hertz) with respect to average electrical SNR per hop and demonstrates the effect of both atmospheric turbulence and pointing error impairments on the capacity performance of the system. It can be seen that for fixed value of normalized jitter, the capacity performance of the system deteriorates with increase in turbulence strength and/or decrease in normalized beam width. In Fig. 2 we plot ergodic capacity (in bits per second per hertz) as a function of average electrical SNR per hop showing the effect of pointing errors on the system performance. We observe that for given turbulence strength, the ergodic capacity of the system increases with decrease in pointing errors (i.e. ξ increases) and shows the best result for no pointing errors case when (ξ → ∞). Fig. 3 demonstrates the variations of the ergodic capacity of relayed OWC system with respect to normalized jitter standard deviation σs /ra under different turbulence regimes. In can be observed that with normalized beam width being fixed to we /ra = 10, the ergodic capacity of the system decreases with increase in σs /ra and saturates at a certain level. Further it is observed that the effect of turbulence is less prominent at high values of σs /ra . V. C ONCLUSION In this letter, we evaluated the capacity performance of dual-hop SIM-based OWC system with DF relay operating over Gamma-Gamma turbulence channels with pointing error impairments. We derived the pdf of instantaneous SNR at the destination followed by the novel closed form expression for ergodic capacity of the system. Finally the various numerical examples are included to demonstrate the effect of turbulence and pointing error on the system performance.
(12)
(13)
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