Optimal Power Allocation for Decode-and-Forward Based Mixed MIMO-RF/FSO Cooperative Systems Neeraj Varshney†
Parul Puri†
Department of Electrical Engineering Indian Institute of Technology Kanpur Kanpur, India 208016 Email:
[email protected]
Department of Electronics and Communication Engineering Jaypee Institute of Information Technology Noida, India 201307 Email:
[email protected]
Abstract—This paper investigates a decode-and-forward (DF) based dual-hop mixed radio-frequency/free-space optical (RF/FSO) communication link, where the source and relay communicate with each other over a RF link and the relay and destination communicate over a FSO link. The RF channel experiences Rayleigh fading whereas the optical channel is affected by path loss, atmospheric turbulence, and pointing errors. Since, the FSO link has a higher data rate in comparison to the RF link, we employ multiple-input multiple-output (MIMO) with zero-forcing based linear receiver in the source-to-relay link, to enhance the RF link data rate. For the considered system and channel models, novel closed form expressions for the outage probability and average bit error rate are derived and verified using Monte Carlo simulations. Furthermore, a framework for optimal source-torelay power allocation is presented, which significantly improves the end-to-end performance of the system.
I. I NTRODUCTION Mixed radio-frequency/free-space optical (RF/FSO) relaying systems have generated a significant research interest due to their ability to connect a large number of RF users to last-mile access network through a single FSO link. This has been possible because of the features offered by the FSO technology, such as rapid deployment time, low cost, license-free bandwidths, and higher data-rates [1]. However, the performance of FSO systems is highly vulnerable to the unpredictable weather conditions. Additionally, the end-to-end data rate of RF/FSO systems is limited by the low speed RF link. In order to overcome this limitation, RF systems based on multiplexing, such as the multiuser systems and multipleinput multiple-output (MIMO) systems can be employed. Furthermore, the end-to-end performance of mixed RF/FSO cooperative systems critically hinges on the signal processing employed at the relay. It has been studied that the decode-andforward (DF) relaying outperforms the amplify-and-forward (AF) relaying protocol [2], [3]. Led by this, we investigate a three node mixed RF/FSO DF dual-hop cooperative system, where we consider multiple antennas at the source and relay nodes with zero-forcing based linear receiver [4], [5], and a point-to-point FSO link between the relay and the destination nodes. † both authors contributed equally to the work and authors name are arranged in alphabetical order.
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A. Literature The mixed RF/FSO systems were introduced in [6], where the authors presented the outage performance of the three node asymmetric dual-hop relayed transmission link. In the considered system, the RF link experienced Rayleigh fading and the atmospheric turbulent FSO link was modeled by the Gamma-Gamma distribution. The analysis was extended by considering an additional impairment that is characteristic to the FSO channel, i.e. the pointing errors (misalignment fading) in [7]. Recently, in [8], the authors analyzed the outage probability and bit-error-rate (BER) performance of a mixed dual-hop RF/FSO relaying system with outdated channel state information (CSI) at the relay. In [9], a similar system was analysed in terms of BER, outage performance, and capacity considering indirect modulation/direct detection (IM/DD) [10] and heterodyne detection schemes [11]. The analysis in [6], [7] was extended in [12], considering a newly proposed generalized fading model, the M -distribution [13]. Considering a similar three-node system model, the authors in [14], derived expressions for the average capacity and BER, considering that the RF link experienced the Nakagami-m fading. Furthermore, to enhance the end-to-end data rate of the system, the dual-hop mixed RF/FSO scenario was extended to multiuser systems [15], [16]. In [15], the capacity of a multiuser AF relayed system based on selection diversity techniques was presented. The mixed RF/FSO systems discussed so far were based on the AF relaying protocol, in which the noise amplification due to the AF based relaying hampered the system performance. A DF based mixed multiuser RF/FSO system was introduced in [16]. B. Motivation Considering the aforementioned literature review, it is observed that limited work has been done to exploit the benefits of DF relayed mixed RF/FSO systems. Additionally, the literature lacks a MIMO based mixed RF/FSO system, in which the source and relay nodes are equipped with multiple antennas for parallel transmission and reception of RF modulated symbols. Such a system can increase the data rate of the RF link, thereby improving the end-to-end data rate of the system. Moreover,
the literature does not describe optimal source-relay power allocation techniques. C. Contribution Motivated by the above background, we present a performance analysis of a three node mixed MIMO-RF/FSO dual-hop system, in which the source and relay nodes communicate over a MIMO-RF link and relay and destination nodes communicate over a point-to-point FSO link. The main contributions of the paper are as follows (1) to increase the overall data-rate of dual-hop RF/FSO systems by employing MIMO transmission between source and relay nodes, (2) to derive closed form expressions for system outage and BER, (3) to derive asymptotic expressions for gaining physical insight of the derived expression of BER, (4) to carry out optimal-power allocation for the source-relay transmission, and (5) to characterize the system in terms of diversity order. Furthermore, the analysis is carried out considering an optical channel that is affected by path loss, atmospheric turbulence induced fading, and pointing errors. The mathematical analysis is accompanied with Monte-Carlo simulations and several numerical examples to illustrate the effect of the key system parameters. D. Structure The rest of the paper is organized as follows. Section II gives the system description including the system and channel models. The closed form expressions for the outage and error probabilities are derived in Section III and IV, respectively. Further, the diversity order analysis and optimal power allocation is carried out in section V. Section VI, provides the numerical results and finally, we conclude in section VII. II. S YSTEM
MODEL
We consider an asymmetric dual-hop mixed RF/FSO DF cooperative system having source, S, relay, R, and destination, D, as shown in Fig. 1. In the considered system, the source transmits Ns data symbols simultaneously to the relay over a Rayleigh faded RF channel. The relay decodes the received Ns symbols and forwards them sequentially to the destination over a FSO link. For this purpose, it is assumed that the source and relay nodes are equipped with Ns transmitting and Nr receiving antennas, respectively, for parallel transmission and reception of Ns RF modulated symbols. It is noted that the number of receive antennas, Nr , should be greater than or equal to the number of transmit antennas, Ns , in order to decode all the Ns transmitted symbols using zero-forcing receiver. Further, the relay is equipped with a single photoaperture transmitter and the destination is equipped with a single photo-detector, for the point-to-point FSO transmission. The transmission of vector x∈CNs ×1 from S to R is given as r P0 ySR = HSR x + wSR , (1) Ns where ySR ∈ CNr ×1 denotes the corresponding received vector at the relay node, P0 is the available transmit power
Fig. 1. Schematic representation of the DF based mixed RF/FSO
co-operative system
at the source, HSR ∈ CNr ×Ns is the source-relay MIMO channel matrix, and wSR ∈ CNr ×1 is the noise vector, having symmetric complex Gaussian entries with variance N0 /2 per dimension. Furthermore, the entries of HSR are assumed as independent zero mean circularly symmetric complex Gaus2 sian random variables with variance δSR . Now, employing the zero-forcing receiver, the instantaneous SNR for the ith symbol corresponding to the source-relay i link is given as, γSR = Ns N0 [(HHP0HSR )−1 ]i,i [17], where SR H i = 1, 2, · · · , Ns and HSR HSR is a Ns ×Ns complex Wishart distributed matrix with Nr degrees of freedom. The probability density function (PDF) and cumulative distribution function i (CDF) of γSR can be written as Nr −Ns +1 xNr −Ns exp − x 2 1 CSR δSR i fγSR (x) = , 2 CSR δSR Γ(Nr − Ns + 1) (2) γ Nr − Ns + 1, CSRxδ2 SR FγSR i (x) = , (3) Γ(Nr − Ns + 1) 0 where CSR = NPs N , Γ(·) denotes the Gamma function, and 0 γ(·, ·) denotes the incomplete Gamma function [18, (8.350.2)]. The relay decodes the received Ns bits using ZF receiver, i.e. multiply the pseudo-inverse of HSR with the received vector, ySR , and forward them to the destination over a FSO link. The optical channel is modeled as H = Hl × Ha × Hp , where Hl accounts for path loss, Ha represents the atmospheric turbulence-induced fading, modeled using the versatile Gamma-Gamma distribution, and Hp represents the misalignment fading (pointing errors). The corresponding PDF of the channel between relay and destination is given by [19, (12)] ! αβξ 2 αβh ξ2 3,0 fHRD (h) = G , A0 Hl Γ(α)Γ(β) 1,3 A0 Hl ξ 2 −1,α−1,β −1 (4) where Hl is the path loss, A0 is the fraction of the collected power at r = 0, r is the aperture radius, ξ = we /(2σs ), we is the equivalent beam-width radius, and σs is the standard deviation of the pointing error displacement at the receiver. a1 ,...,ap Further, Gm,n p,q (x| b1 ,...,bq ) is the Meijer’s G-function [20, (8.2.1.1)], and α and β are the large-scale and small-scale scintillation parameters, respectively, which depend on the 2 Rytov variance σR = 1.23Cn2 k 7/6 L11/6 . Here, k is the optical 2 wave number, Cn is the refractive index structure constant,
and L is the link distance. Now, employing a coherent optical receiver with heterodyne detection at the destination, the instantaneous SNR of the relay-destination link is given as, ℜA γRD ≈ q△f HRD [21, (6)]], where ℜ is the photodetector responsivity, A is photodetector area, q is the electronic charge, and △f denotes the noise equivalent bandwidth of the receiver. The PDF and CDF of γRD are given as ! ξ2 αβ ξ 2 + 1 3,0 G x , (5) fγRD (x) = xΓ(α)Γ(β) 1,3 γ¯ ξ 2 , α, β ! ξ2 αβ 1, ξ 2 + 1 3,1 FγRD (x) = G x , (6) Γ(α)Γ(β) 2,4 γ¯ ξ 2 , α, β, 0 2 ξ l A0 where γ¯ = ℜAH is the average SNR. q△f 1+ξ 2
′
where Pe (ǫ | x) is the first order derivative of conditional error probability (CEP) for the given SNR. For coherent and non-coherent binary modulation schemes, a unified expression for CEP is given as Pe (ǫ|γ) = Γ(p,qγ) 2Γ(p) , where Γ(·, ·) is the complementary incomplete gamma function defined in [23, (8.350.2)] and the values of parameters p and q corresponding to different modulation schemes are as summarized in Table I. TABLE I PARAMETERS p AND q
III. O UTAGE P ROBABILITY
The outage probability is one of the key performance criterion for evaluating a communication system. For a DF relayed system, the system is in outage when either or both of the links, (S − R) or (R − D) fail. For the ith transmitted symbol, where i ∈ {1, 2, ....Ns }, the outage probability is given by (i)
i i Pout = P r(min(γSR , γRD ) ≤ γth ) = Fγmin (γth ),
i i Fγmin (x) = 1 − P r(γSR > x)P r(γRD > x), i (x) + Fγ i = FγSR (8) RD (x) − FγSR (x)FγRD (x).
Now, using (7) and (8), the per-block average outage probability can be obtained as Ns h i 1 X FγSR i (γth )+FγRD (γth )−Fγ i (γth )FγRD (γth ) . SR Ns i=1 (9) Substituting the CDF of the instantaneous SNRs of both the links in (9), using the identity γ(s, x) = (s − Ps−1 xk −x 1)! 1 − e k=0 k! , when s is a positive integer [18, (8.352.1)], and after some mathematical manipulations, the simplified expression for the per-block average outage probability is obtained as " NX k Ns r −Ns 1 X γth 1 γth Pout= 1− exp − 2 2 Ns i=1 CSR δSR k! CSR δSR k=0 ( !)# 1, ξ 2 + 1 ξ2 αβ G3,1 γ . (10) × 1− th 2 ξ , α, β, 0 Γ(α)Γ(β) 2,4 γ¯
Pout=
IV. E RROR P ERFORMANCE
The average BER for the ith bit based on the CDF of the end-to-end SNR can be written as [22, (21)] Z ∞ ′ (i) i Pe = − Pe (ǫ | x)Fγmin (x)dx, (11) 0
Modulation Scheme
p
q
Coherent binary frequency shift keying Coherent binary phase shift keying Non-Coherent binary frequency shift keying Differential binary phase shift keying
0.5 0.5 1 1
0.5 1 0.5 1 (i)
Differentiating Pe (ǫ | γ) and substituting in (11), the P e can be simplified as (i)
Pe =
(7)
where γth is the threshold SNR and Fγmin i (·) is the CDF of i the end-to-end SNR, γmin , for the ith symbol. Since, the links i encounter fading independently, Fγmin (x) can be written as
FOR BINARY MODULATION SCHEMES
qp 2Γ(p)
Z
0
∞ i e−qx xp−1 Fγmin (x)dx.
(12)
i Substituting the CDF, Fγmin (·), from (8) in (12), the expression
(i)
for P e can be written as (i) Pe
"Z
Z ∞ i (x)dx + e−qx xp−1 FγRD (x)dx e−qx xp−1 FγSR 0 0 {z } | {z } | I1 I2 # Z ∞ qp −qx p−1 i − e x FγSR (x)FγRD (x)dx . (13) 2Γ(p) 0 | {z } =
∞
I3
Now, one can simplify each integral in the above equation as i (x) from (3) in the integral I1 and follows. Substituting FγSR using the identity for γ(·, ·) function in terms of the Meijer’sG function [20, (8.4.16.1)] with [20, (2.24.3.1)], the integral I1 can be solved as ! 1 1 1 − p, 1 G1,2 . I1 = p 2 q N − N + 1, 0 q Γ(Nr − Ns + 1) 2,2 CSR δSR r s (14) Substituting CDF FγRD (x) from (6) in I2 , and integrating using [20, (2.24.3.1)], the integral I2 can be obtained as ξ2 I2 = p G3,2 q Γ(α)Γ(β) 3,4
αβ γ¯q
! 1 − p, 1, ξ 2 + 1 . ξ 2 , α, β, 0
(15)
i (x) and Fγ Substituting FγSR RD (x) from (3) and (6), respectively, in I3 and using the identity for γ(s, x) in terms of
exponential along with [20, (2.24.3.1)], the integral I3 can be evaluated as " ! ξ2 1 3,2 αβ 1 − p, 1, ξ 2 + 1 I3 = G Γ(α)Γ(β) q p 3,4 γ¯q ξ 2 , α, β, 0 NX r −Ns
1 p+k k 1 k=0 k! (CSR δ 2 ) q + 2 SR CSR δSR # 2 1 − p − k, 1, ξ + 1 αβ × G3,2 . 3,4 1 ξ 2 , α, β, 0 γ¯ q + 2 −
(16)
Now, substituting the expressions for I1 , I2 , and I3 from (14), (15), and (16), respectively, in (13), the average BER (i) P e for the ith bit can be written as ! 1 − p, 1 1 1 (i) 1,2 G Pe = 2 q N −N +1, 0 2Γ(p)Γ(Nr −Ns +1) 2,2 CSR δSR r s 2
+
×
ξ 2Γ(p)Γ(α)Γ(β)
G3,2 3,4
k=0
αβ
γ¯ q +
1 2 CSR δSR
p
q p+k 2 ) k! (CSR δSR q + CSR1δ2 SR 1 − p − k, 1, ξ 2 + 1 . (17) ξ 2 , α, β, 0 k
Finally, the per-block average error probability for the dualhop mixed RF/FSO DF cooperative system is given as P e = PNs (i) (i) 1 th i=1 P e , where the average BER P e for i bit is given Ns in (17). V. D IVERSITY O RDER A NALYSIS AND O PTIMAL P OWER A LLOCATION
To carry out the diversity order analysis and optimal power allocation we first derive the high-SNR asymptotic approximation for BER. Let, the power factors for the source and relay be defined as a0 = P0 /P, a1 = P1 /P , respectively, where P0 is the transmitted power of source, P1 is the transmitted power of relay, and P is the total power. At high-SNRs, the term corresponding to k = 0 in (17), dominates the summation. Further, using the identity for the Meijer’s-G function [24, (18)], one can obtain the high-SNR asymptotic approximation of BER as Nr−Ns +1 ξ2 α β N0 N0 N0 N0 P e ≤ C1 + C2 +C3 +C4 , (18) P P P P where !Nr −Ns +1
2 Γ(Nr − Ns + p + 1) αβ ξ , C2 = 2Γ(p)Γ(Nr − Ns + 2) a1 q α 2 2 2 2 αβ ξ Γ(β − α)Γ(α + p) Γ(α−ξ )Γ(β −ξ )Γ(ξ + p) × ,C3= , 2Γ(p)Γ(α)Γ(β) a1 q 2(ξ 2 −α)Γ(p)Γ(β)Γ(1+α) αβ β ξ 2 Γ(α − β)Γ(β + p) and C4 = . 2 a1 q 2(ξ − β)Γ(p)Γ(α)Γ(1 + β)
C1 =
Ns 2 q a0 δSR
s.t.
P0 + P1 = P,
(19)
where C˜1 = C1 (a0 N0 )Nr −Ns +1 , C˜2 = C2 (a1 N0 )ξ , C˜3 = C3 (a1 N0 )α , and C˜4 = C4 (a1 N0 )β , respectively. Now, consider a scenario when β is less than α and ξ 2 . For this scenario, the first and last terms of (19) will dominate at high-SNRs. Considering the dominating terms, the above optimization problem can be solved by employing the Karush Kuhn Tucker (KKT) framework [25]. In the KKT framework, we differentiate the optimization objective with respect to P0 and set the value to zero and obtain a polynomial equation for P0 as 2
CSR δSR
NX r −Ns
power budget P , the convex optimization problem for optimal source-relay power allocation can be formulated as ( ξ2 α β) N −N +1 1 r s 1 1 1 ˜ ˜ ˜ ˜ min C1 + C2 + C3 + C4 , P0 ,P1 P0 P1 P1 P1
Using (18), one can readily observe that the net diversity of the considered mixed RF/FSO DF cooperative system is d = min{Nr − Ns + 1, ξ 2 , α, β}. Further, considering a total
P0Nr −Ns +2 − κ1 (P − P0 )β+1 = 0,
(20)
˜
s +1)C1 where the factor κ1 = (Nr −N . Similarly, one can obtain ˜4 βC the polynomial equations for the other scenarios as
P0Nr −Ns +2 − κ1 (P − P0 )κ2 +1 = 0, ˜ s +1)C1 where κ1 = (Nr −N , κ2 = ξ 2 for ˜2 ξ2 C ˜ s +1)C1 ξ 2 < α, β and κ1 = (Nr −N , κ2 = ˜3 αC 2
(21)
the scenario when α for the scenario
when α < ξ , β.
VI. N UMERICAL A NALYSIS
AND
D ISCUSSION
The analysis and simulations are done considering the transmission of coherent binary phase shift keying (BSPK) modulated symbols, with Ns = 2, Nr ∈ {2, 3}, and average 2 source-relay RF channel gain, δSR ∈ {0.1, 0.5, 1, 10}. Further, the FSO channel undergoes moderate (Cn2 = 3 × 10−14 m−2/3 ) and strong (Cn2 = 1 × 10−13 m−2/3 ) atmospheric turbulence with link length, L ∈ {1 km, 2 km}, aperture radius, r = 0.1 m, and laser wavelength, λ = 1.55 × 10−6 m. For these values the FSO parameters are obtained as (ξ, Hl , α, β) ∈ {(1.6758, 0.8159, 3.9974, 1.6524), (1.6758, 0.9033, 5.4181, 3.7916), (1.6768, 0.9033, 3.99, 1.71), (1.6885, 0.8159, 5.0711, 1.1547)}. Figs. 2-3 demonstrate the outage probability and BER performance of the system with equal and optimal power allocation. It is observed that the analytical curves match the simulation curves, thereby validating the derived analytical results. Furthermore, it can be seen that the optimal power factors derived using (20) and (21), significantly improve the end-to-end performance of the system. One can also notice that the end-to-end performance can be enhanced by increasing the number of receiving antennas at the relay, as the decoding error at the relay decreases with more number of antennas. In Fig. 3, the asymptotic bound derived in (18) is plotted to demonstrate the achieved diversity order of the system under various channel conditions. It can be seen in Fig. 3 that the achieved diversity order for the scenarios (a), (b), (c), and (d) are 1, 1, 1.7056, 1.1547, respectively.
0
10
ξ=1.6785; α=3.9974; 2 β=1.6524; δSR=1; Ns=Nr=2
−1
Outage Probability
10
ξ=1.6758; α=5.4181; 2 β=3.7916; δSR=10; Ns=Nr=2
−2
10
−3
10
−4
10
ξ=1.6785; α=3.9974; 2 β=1.6524; δSR=1; Ns=2;Nr=3 ξ=1.6758; α=5.4181; β=3.7916; δ2SR=10; Ns=2;Nr=3
−5
10
−6
10
−7
10
0
Analytical (equal power) Simulated (equal power) Analytical (optimal power) Simulated (optimal power) 5
10
15
20
25
30
35
40
45
50
Average SNR (dB) Fig. 2. Outage performance of the DF based mixed RF/FSO co-
operative system under various channel conditions 0
10
(a): ξ=1.6768; α=3.9928; 2 β=1.7056; δSR=0.1; Ns=Nr=2
−1
10
(b): ξ=1.6885; α=5.0711; β=1.1547; δ2 =0.5;N =N =2
−2
BER
10
SR
s
40
45
r
−3
10
(c): ξ=1.6768; α=3.9928; 2 β=1.7056; δSR=0.1; Ns=2;Nr=3
−4
10
−5
10
−6
10
0
Analytical (equal power) Simulated (equal power) Analytical (optimal power) Simulated (optimal power) Asymptotic bound 5
10
15
(d): ξ=1.6885; α=5.0711; 2 β=1.1547; δSR=0.5; Ns=2,Nr=3 20
25
30
35
50
Average SNR (db) Fig. 3. BER performance of the DF based mixed RF/FSO co-operative
system under various channel conditions
VII. C ONCLUSION In this paper, we presented novel closed-form expressions for BER, outage probability, and diversity order for a dualhop mixed MIMO-RF/FSO DF cooperative system, where FSO link experienced Gamma-Gamma fading distribution with path loss and pointing errors. In order to increase the datarate of the source-relay link, we considered the multiple antenna arrangement at the source and relay for parallel transmission and reception, respectively. Further, the DF based relay employed zero-forcing linear receiver to decode the data symbols transmitted by the source over Rayleigh fading channel. This paper also presented an optimal source-relay power allocation scheme which significantly improved the end-to-end performance of the system. R EFERENCES [1] L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. Bellingham, WA, USA: SPIE, 2005. [2] A. Nosratinia, T. E. Hunter, and A. Hedayat, “Cooperative communication in wireless networks,” IEEE Communications Magazine, vol. 42, no. 10, pp. 74–80, 2004.
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