Exact Capacity of Amplify-and-Forward Relayed Optical ... - IEEE Xplore

IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 27, NO. 8, APRIL 15, 2015

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Exact Capacity of Amplify-and-Forward 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 model a dual-hop subcarrier intensity modulated amplify-and-forward (AF) relayed optical wireless communication system with the assumption that direct communication between source and destination is not feasible. The optical channels are considered to be impaired with path loss, misalignment errors, and atmospheric turbulence, and are modeled assuming independent but not necessarily identically distributed Gamma–Gamma fading statistics. We derive exact analytical expressions for the average channel capacity of the system considering both type of AF relays, i.e., variable gain and fixed gain. Further to get valuable insight, an asymptotic analysis of channel capacity is presented at high signal-to-noise ratio to find the effect of various parameters on the performance of the system. Index Terms— Amplify and forward relay, channel capacity, Gamma-Gamma distribution, generalized bivariate Meijer’s G-function (BMGF).

I. I NTRODUCTION

O

PTICAL wireless communication (OWC) systems have gained significant research attention due to their ability to cater to high bandwidth demand. However, the optical links are vulnerable to adverse channel conditions caused by atmospheric turbulence and pointing errors [1]. The turbulence-induced scintillation and misalignment-fading is modeled using the Gamma-Gamma distribution [2] which is suitable for moderate to strong turbulence regimes. Multihop relaying is a popular technique used to improve the performance of OWC systems by mitigating the limiting effects of scintillation and misalignment-fading. Recently the amplify and forward (AF) relays have attracted a lot of research attention as they simply amplify and forward the incoming signal without performing any sort of decoding. The variable gain AF relays use the channel state information (CSI) of the preceding hop to control the relay gain. Whereas, a fixed-gain AF relay does not need instantaneous CSI of the first hop but employs amplifiers with a fixed gain which results in a signal with variable power at the relay output. The fixed-gain relays may not perform as well as variable-gain relays, but their low complexity and ease of deployment make them attractive from a practical viewpoint. The average channel capacity serves as one of the most important performance metrics to design OWC systems. Manuscript received October 14, 2014; revised January 29, 2015; accepted February 2, 2015. Date of publication February 4, 2015; date of current version March 24, 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.2015.2399953

The capacity performance of AF relayed OWC systems has been studied in [3] and [4] over moderate to strong turbulence regimes. However, in aforementioned papers, the equivalent end-to-end signal to noise ratio (SNR) of the system is upper bounded using the harmonic-geometric-mean (HGM) inequality and therefore could not contribute towards an exact capacity analysis. The aim of this letter is to go one step further by analyzing the exact capacity performance of relayed OWC systems employing both types of AF relays i.e. variable-gain and fixed-gain. The derived results are obtained in terms of special function known as bivariate Meijer’s G-function (BMGF). In this letter, we model a dual-hop subcarrier intensity modulation (SIM)-based relayed OWC system where the optical links are modeled by independent but not necessarily identically distributed (i.n.i.d) Gamma-Gamma fading statistics with misalignment fading. The analysis is carried out for both type of AF relays i.e. variable gain and fixed gain AF relays. Firstly, we find the moment generating function (MGF) of the reciprocal of instantaneous SNR and then utilize it to derive the exact analytical expression for average channel capacity of the system in terms of BMGF. Further to gain better insight of the system performance, an asymptotic analysis for variable gain relays is also presented at high SNR. II. S YSTEM M ODEL We consider a dual-hop, SIM-based relayed OWC system having three nodes source (s), destination (d) and relay (r ) with the assumption that direct communication between source and destination is not feasible due to practical reasons which may be either non-line-of-sight condition or the larger distance between the two. The system employs an AF relay and the transmitted electrical signal is modulated with binary phase shift keying (BPSK) modulation scheme. At the receiver, the incident optical signal at the photodetector is converted to an electrical signal through direct detection technique. In this case, the additive white Gaussian noise (AWGN) model can be used as an approximation to Poisson photon-counting detection model [1]. A. Signal to Noise Ratio The equivalent end to end SNR for variable-gain AF relayed vg OWC system, γeq , can be written as the normalized harmonic mean of instantaneous SNRs given as γsr γrd 1 −1 1 vg γeq = = + (1) γsr + γrd γsr γrd

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where γx y is the instantaneous SNR for {x y} link for {x y} ∈ {sr, r d} defined as γx y = (ηh x y )2 /N0 where η is the optical to electrical conversion ratio, N0 is the variance of AWGN and h x y denotes the optical channel coefficient. Now considering the combined effects of channel impairments due to path loss, atmospheric turbulence-induced scintillation and misalignment-error, the probability density function (pdf) of instantaneous SNR γx y , modeled by Gamma-Gamma fading statistics is given in [5, eq. 10] as ξx y 2 f γx y (γx y ) = 2γx y (αx y )(βx y ) γx y 3,0 ×G 1,3 αx y βx y x y

2+1 ξ x y ξ 2, α , β xy xy xy

(2)

n,y

xy

2 for {y} ∈ {r, d} E denotes the expectation operator and σn,y is the variance of AWGN at the relay and at the destination η 2 h 2 A2 respectively. When ξx y 2 1, we obtain x y = σ 2l 0 . n,y Further, the end to end SNR for fixed-gain AF relayed OWC fg system γeq as in [7] is given as

γsr γrd = = γrd + C

1 C + γsr γsr γrd

−1 (3)

where C is gain constant. Proposition 1: Given a Gamma-Gamma distributed random variable γx y for {x y} ∈ {sr, r d}, the MGF of its reciprocal M 1 (s), is given by γx y

M

2αx y +βx y −1 ξx2y G 7,0 (s) = 4π(αx y )(βx y ) 2,7 ⎞ ⎛ ξx y 2 +1 ξx y 2 +2 , 2 αx2y βx2y s 2 ×⎝ × ξ 2 ξ 2 +1 α α +1 β β +1 ⎠ xy xy xy xy xy xy 16x y 2 , 2 , 2 , 2 , 2 , 2 ,0 1 γx y

(4) Proof: The MGF of reciprocal of γ , can be x y

∞ −s written as M 1 (s) = 0 exp γx y fγx y (γx y )dγx y . Now γx y

substituting f γx y (γx y ) from (2) and using the identity 1,0 e−x = G 0,1 x | −0 [8, eq. 11] and integrating with the help of [8, eq. 21], M 1 (s) may be obtained as in (4). γx y

M

C γsr γrd

(s) =

is given by

2 ξ2 2αsr +βsr +αrd +βrd −2 ξsr rd 16π 2 (αsr )(βsr )(αrd )(βrd ) 2 2 2 2 13,0 αsr βsr αrd βrd Cs × G 4,13 (16sr )(16rd ) ξsr 2 +1 ξsr 2 +2 ξrd 2 +1 ξrd 2 +2 , 2 , 2 , 2 × 2 (5) 1 , 2 , 0

where 1 = { ξrd2 , ξrd 2+1 , α2rd , αrd2+1 , β2rd , βrd2+1 } and 2 2 2 = { ξsr2 , ξsr 2+1 , α2sr , αsr2+1 , β2sr , βsr2+1 }. Proof: The MGF of γsrCγrd can be written as M C (s) = γsr γrd

∞∞ −sC f γsr (γsr ) f γrd (γrd )dγsr dγrd . Now follow0 0 exp γsr γrd 2

where (w) is the gamma function defined as ∞ (w)= 0 t w−1 exp (−t)dt, ξ = we /(2σs ) with we being the equivalent beam width at the receiver, σs being the standard deviation of the pointing√error displacement at the receiver, π/2 ra /w L , er f (.) denotes the A0 = er f 2 (ν), ν = error function, ra is the radius of the receiver aperture, − w L is the beam waist radius at distance L and G m,n p,q (. | − ) denotes the Meijer’s G-function [6, eq. 8.2.1]. The turbulence parameters α and β (α > 0, β > 0) are given in [5] and x y denotes the average electrical SNR given

2 η 2 h 2 A2 ξ 2 in [1] as x y = (ηE[h x y ])2 /N0 = σ 2l 0 ξ x2y+1 where

fg γeq

C γsr γrd

the MGF of

Proposition 2: Given γsr and γrd to be statistically independent Gamma-Gamma distributed random variables,

2

ing the procedure of [9], the MGF of as in (5).

C γsr γrd

may be obtained

III. AVERAGE C HANNEL C APACITY In this section, we find the analytical expression for the average channel capacity of the dual-hop relayed OWC system employing both types of AF relays i.e. variable gain and fixed gain. The channel capacity (in bits per second per hertz) of the system Cavg is given as [10] ∞ 1 1 Cavg = 1 − e−s M 1 (s) ds, (6) γeq log(2) 0 s where M 1 (s) denotes the MGF of inverse of γeq . γeq

A. Variable Gain Relays To find the exact capacity of dual-hop variable gain AF relayed OWC system, we first need to evaluate the MGF of vg inverse of γeq as given in (1). Further assuming γsr and γrd to be statistically independent which is generally true in most of the cases, we can write M

1 vg γeq

(s) = M

1 γsr

(s) × M

1 γrd

(s)

(7)

where M 1 (s) denotes the MGF of the reciprocal of γx y for γx y {x y} ∈ {sr, r d} as given in (4). Substituting (7) in (6), the vg expression of capacity for variable gain relays, Cavg can be written as ⎡ vg Cavg

∞ 1 ⎢ ⎢ = s −1 M 1 (s) M 1 (s) ds ⎢ γsr γrd log(2) ⎣ 0 I1

− 0

∞

e−s s −1 M

⎤ ⎥ ⎥ (s) M 1 (s) ds ⎥ γrd ⎦

1 γsr

(8)

I2

Further to simplify for Cavg , we evaluate the integrals I1 and I2 separately. Substituting the value of Mγx−1 (s) y using (4), we obtain the integral I1 given as κ2 κ5 κ7 κ8 ∞ 1 7, 0 7, 0 G κ G κ s s I1 = 1 4 2, 7 2, 7 κ ds (9) 16 π 2 0 s κ3 6

AGGARWAL et al.: EXACT CAPACITY OF AF RELAYED OWC SYSTEMS

κ1

where

α2 β 2

=

κ3 = κ5 =

βrd +1 2 , 0},

2 +1 ξ 2 +2 ξsr sr }, 2 , 2 2 β2 α αsr +1 βsr βsr +1 rd rd 2 , 2 , 2 , 0}, κ4 = { 16rd }, 2 +1 ξ 2 ξrd αrd αrd +1 βrd κ6 = { rd 2 , 2 , 2 , 2 , 2 ,

sr sr { 16 }, sr

ξ 2 ξ 2 +1 { 2sr , sr2 , α2sr , ξ 2 +1 ξ 2 +2 { rd2 , rd2 },

κ7

905

=

κ2

=

{

ξ2 ξ2

rd { (αsr )(βsrsr)(α }, rd )(βrd )

and

κ8 = {2αsr +βsr +αrd +βrd −2 }. Now integrating with the help of [6, eq. 2.24.1.1], the integral I1 evaluates as κ7 κ8 7 , 7 κ4 κ9 , κ5 I1 = (10) G 9,9 16 π 2 κ κ ,κ 1

where κ9

6

10

2−ξ 2 1−ξ 2 sr 1−αsr 2−βsr 1−βsr = { 2 sr , 2 sr , 2−α 2 , 2 , 2 , 2 1−ξ 2 −ξ 2 { 2 sr , 2sr }. Similarly Substituting

, 1},

κ10 = for Mγx−1 (s) using (4) and integrating with the help of y [11, eq. 07.34.21.0081.01], the integral I2 evaluates to κ7 κ8 1,0 : 7,0 : 7,0 1 κ2 κ5 κ1 , κ4 G I2 = (11) 16 π 2 1,0 : 2,7 : 2,7 − κ κ 3

6

,n 1 : m 2 ,n 2 : m 3 ,n 3 G mp11,q 1 : p2 ,q2 : p3 ,q3

where (.) is the BMGF as given in [11]. vg Now substituting (10) and (11) in (8), we get Cavg as κ7 κ8 vg 7 , 7 κ4 κ9 , κ5 Cavg = G 9,9 16 π 2 log(2) κ1 κ6 , κ10 1 κ2 κ5 1,0 : 7,0 : 7,0 κ − G 1,0 , κ 1 4 : 2,7 : 2,7 − κ3 κ6 (12) In (12), we obtain the final closed form expression channel capacity of variable gain AF relayed OWC system in terms of BMGF. It is worth noting that the BMGF is not a built-in function available in standard mathematical software packages. However, using the two-fold Mellin Barnes representation of BMGF [12, eq. 1.1], an efficient MATLAB software implementation of this function is obtained with the help of [13]. B. Fixed Gain Relays To obtain the exact capacity of dual-hop fixed gain AF relayed OWC system, we again need to evaluate fg the MGF of inverse of γeq as given in (3). Assuming γsr and γrd to be statistically independent, we can write M 1 (s) = M 1 (s) × M C (s). Substituting the values γsr

fg γeq

γsr γrd

of M 1 (s) and M C (s) from (4) and (5), and following γsr γsr γrd the procedure same as variable gain relays, the exact capacity for fixed gain OWC systems may be obtained as κ7 κ8 κ13 fg Cavg = 64 π 3 log(2) κ9 , κ11 13 , 7 : 7,0 : 13,0 − G 1,0 × G 11 , 15 κ4 C 1,0 : 2,7 : 4,13 κ12 , κ10 1 κ2 κ11 × (13) (κ1 ), (κ4 κ1 C) − κ3 κ12 2 +1 ξ 2 +2 ξ 2 +1 ξ 2 +2 ξsr sr rd rd 2 , 2 , 2 , 2 }, κ12 2 2 βrd βrd +1 ξsr ξsr +1 αsr αsr +1 βsr 2 , 2 , 2 , 2 , 2 , 2 , 2

2 ξ 2 +1 ξrd rd 2 , 2 , βsr +1 2 , 0} and

where κ11 = {

= {

αrd 2

,

, αrd2+1 ,

κ13 = {

2 ξsr

2αsr +βsr −1

(αsr )(βsr )

}.

C. Asymptotic Capacity Analysis for Variable Gain Relays at High SNR To get better insight of the system performance, we present asymptotic capacity analysis for dual hop variable gain AF relayed OWC system. At high SNR values, the equivalent end-to-end-SNR as defined in (1), can be approximated as 1 min(γsr , γrd ) (14) 2 As a special case of asymptotic capacity analysis at high SNR, we assume independent and identically distributed (i.i.d) fading statistics with γsr = γrd and therefore using (1) and (2), the pdf of equivalent end-to-end SNR γeq , can be written as 2γ ξ 2 + 1 ξ2 3,0 αβ G (15) f γeq (γ ) = 2γ (α)(β) 1,3 ξ 2 , α, β γeq =

where αsr = αrd = α, βsr = βrd = β, sr = rd = and ξsr = ξrd = ξ . Now using the definition of average channel ∞ 1 capacity given as Cavg = ln(2) 0 ln(1 + γ ) f γeq (γ )dγ [4], substituting f γeq (γ ) using (15) and expressing logarithmic integrand in terms of Meijer’s G-function using 1,2 1,1 x | 1,0 , and then integrating using ln(1 + x) = G 2,2 i.i.d. is [6, eq. 2.24.1.1], the closed form expression of Cavr obtained in terms of Meijer’s G-function. Further using the Meijer’s G-function expansion as given in [8, eq. 18] with lim x→0 a Fb [c; d; x] = 1, the average channel capacity asymptotically, at high SNR, can be written as 8 2 2 ψ2,k ξ 2 2α+β−1 α β i.i.d. ∼ Cavr = 4π ln(2) (α)(β) 8 k=1

×

(1 + ψ2,k − ψ1,1 )8j =1, j =k (ψ2, j − ψ2,k ) 4j =2 (ψ1, j − ψ2,k )

(16)

where ψm,n denotes the n t h term of ψm with 2 2 2 2 ψ1 = {0, 1, ξ 2+1 , ξ 2+2 } and ψ2 = { ξ2 , ξ 2+1 , α2 , α+1 2 , β β+1 2 , 2 , 0, 0}.

The asymptotic expression for the average channel capacity of the system is dominated by the minimum of {ξ, α, β, 1}. More specifically, when the difference between the above mentioned parameters is less than one, asymptotic capacity expression is dominated by min{ξ, β, 1} where α > β. Further when the difference between the parameters exceeds one, asymptotic capacity is dominated by the sum of two least valued terms out of {ξ, α, β, 1}. IV. N UMERICAL R ESULTS Now we demonstrate the effect of various channel impairments on the average channel capacity of dual-hop AF relayed OWC system. The analysis is done under moderate (Cn2 = 3 × 10−14 ) and strong (Cn2 = 1 × 10−13 ) turbulence regimes with operating wavelength λ = 1550 nm and the radius of receiver aperture ra = 10 cm [14, Table III]-[16]. For i.i.d. statistics case, Fig.1 plots average channel capacity of the system against average electrical SNR per hop for different values of normalized jitter σs /ra with normalized beam width being fixed to we /ra = 10. The moderate turbulence (Cn2 = 3 × 10−14 ) is considered with link lengths

906


Another important observation is that turbulence has adverse effect on the capacity performance of the system. Further, the effect of turbulence is more prominent at high SNR values. V. C ONCLUSION In this letter, exact analytical expressions have been derived for the average channel capacity of dual-hop relayed OWC system for both type of AF relays. The numerical examples demonstrated the effect of turbulence strength, average electrical SNR per hop, pointing error impairments (normalized jitter and normalized beam width) on the system performance. It is observed that the results are useful for designing transceivers for relayed OWC systems. Fig. 1. Average channel capacity vs average electrical SNR per hop (dB) for dual-hop variable gain AF relayed OWC system under moderate turbulence (Cn2 = 3 × 10−14 ) with total link length L=2000 m.

Fig. 2. Average channel capacity vs average electrical SNR per hop (dB) for the dual-hop fixed gain AF relayed OWC system under moderate 2 (Cn = 3 × 10−14 ) and strong (Cn2 = 1 × 10−13 ) turbulence with total link length L=2000 m. The gain constant is fixed to C = 2.

L sr = L rd = 1000 m. It has been observed that with increase in the value of σs /ra , capacity performance of the system deteriorates. For comparison purposes, the exact analytical results obtained using (12) have been compared with the asymptotic results obtained by (16) and it has been observed that at high SNR values, asymptotic results follow the exact analytical results. Further simulations results have been plotted to verify the correctness of our derived analytical results. In Fig. 2, we plot the average channel capacity of the system as a function of average electrical SNR per hop (dB) of the system under different turbulence regimes with pointing error parameter being fixed to ξ = 2. The simulations have been done with 105 number of samples and it has been observed that the obtained results exactly follow the derived analytical results. Further, for comparison purposes, the upper bounds obtained using HGM inequality [3] are also plotted. As expected, with increase in average electrical SNR per hop, the average channel capacity the system increases.

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