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MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Optimal Power Allocation for Decode-and-Forward
Based Mixed MIMO-RF/FSO Cooperative Systems
Neeraj Varshney† and Parul Puri‡
†Department of Electrical Engineering,Indian Institute of Technology, Kanpur, India
neerajv@iitk.ac.in‡ Department of Electronics and Communication Engineering,
Jaypee Institute of Information Technology, Noida, Indiaparulpuri9@gmail.com
June 14, 2016
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Overview
1 Motivation
2 System Model
3 Performance AnalysisOutage AnalysisError Rate AnalysisDiversity Order and Optimal Power Allocation
4 Numerical Results
5 Conclusion
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Motivation
Mixed RF/FSO relaying systems have generated a significantresearch interest due to their ability to connect a large number ofRF users to last-mile access network.
because of the features offered by the FSO technology, such asrapid deployment time, low cost, license-free bandwidths, andhigher data-rates.However, the performance of FSO systems is highly vulnerableto the unpredictable weather conditions.In addition, the end-to-end data rate of RF/FSO systems islimited by the low speed RF link.
In order to overcome this limitation, multiple-input multiple-output(MIMO) technology can be employed.
On the other hand, the end-to-end performance can be significantlyenhanced using the optimal power allocation framework.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
System Model I
Figure: Schematic diagram of the DF based mixed RF/FSO cooperativesystem.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
System Model II
The transmission of vector x∈CNs×1 from S to R is given as
ySR =
√
P0
NsHSRx+ wSR , (1)
where
ySR∈CNr×1 denotes the corresponding received vector at the
relay node.
P0 is the available transmit power at the source.
HSR∈CNr×Ns is the SR MIMO channel matrix, where entries
are assumed as independent zero mean circularly symmetriccomplex Gaussian random variables with variance δ2SR .
wSR∈CNr×1 is the noise vector, having symmetric complex
Gaussian entries with variance η0/2 per dimension.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
System Model III
Now, employing the zero-forcing receiver, the instantaneousSNR for the i th symbol corresponding to the source-relay linkis given as1
γiSR =P0
NsN0[(HHSRHSR)−1]i ,i
, (2)
where i = 1, 2, · · · ,Ns and HHSRHSR is a Ns × Ns complex
Wishart distributed matrix with Nr degrees of freedom.
1B. K. Chalise and L. Vandendorpe, ”Performance analysis of linear receivers in a MIMO relaying system,”
IEEE Communications Letters, vol. 13, no. 5, pp. 330–332, 2009.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
System Model IV
The probability density function (PDF) and cumulativedistribution function (CDF) of γiSR can be written as2
fγiSR(x) =
(1
CSRδ2SR
)Nr−Ns+1 xNr−Ns exp(
− xCSRδ
2SR
)
Γ(Nr − Ns + 1), (3)
FγiSR(x) =
γ(
Nr − Ns + 1, xCSRδ
2SR
)
Γ(Nr − Ns + 1), (4)
where CSR = P0NsN0
, Γ(·) denotes the Gamma function, and
γ(·, ·) denotes the incomplete Gamma function3.2B. K. Chalise and L. Vandendorpe, ”Performance analysis of linear receivers in a MIMO relaying system,”
IEEE Communications Letters, vol. 13, no. 5, pp. 330–332, 2009.3A. Jeffrey and D. Zwillinger, Table of integrals, series, and products. Academic Press, 2007.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
System Model V
The relay decodes the received Ns bits using ZF receiver andforwards them to the destination over a FSO link.
The optical channel is modeled as4
HRD = Hl × Ha × Hp , (5)
where
Hl accounts for path loss,Ha represents the atmospheric turbulence-induced fading,modeled using the versatile Gamma-Gamma distribution, andHp represents the misalignment fading (pointing errors).
4H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, ”Optical wireless communications with heterodyne
detection over turbulence channels with pointing errors,” IEEE/OSA Journal of Lightwave Technology, vol. 27, no.20, pp. 4440–4445, 2009.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
System Model VI
The corresponding PDF of the channel between relay anddestination is given by
fHRD(h)=
αβξ2
A0HlΓ(α)Γ(β)G 3,01,3
(
αβh
A0Hl
∣∣∣∣∣
ξ2
ξ2−1,α−1,β−1
)
, (6)
whereA0 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 atthe receiver.α and β are the large-scale and small-scale scintillationparameters, respectively, which depend on the Rytov varianceσ2R =1.23C 2
nk7/6L11/6. Here, k is the optical wave number, C 2
n
is the refractive index structure constant, and L is the linkdistance.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
System Model VII
Now, employing a coherent optical receiver with heterodynedetection at the destination, the instantaneous SNR of therelay-destination link is given as5,
γRD ≈ℜA
q △fHRD , (7)
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.
5P. Puri, P. Garg, and M. Aggarwal, ”Partial dual-relay selection protocols in two-way relayed FSO networks,”
IEEE/OSA Journal of Lightwave Technology, vol. 33, no. 21, pp. 4457–4463, 2015.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
System Model VIII
The PDF and CDF of γRD are given as
fγRD (x) =ξ2
xΓ(α)Γ(β)G 3,01,3
(
αβ
γ̄x
∣∣∣∣∣
ξ2 + 1
ξ2, α, β
)
, (8)
FγRD (x) =ξ2
Γ(α)Γ(β)G 3,12,4
(
αβ
γ̄x
∣∣∣∣∣
1, ξ2 + 1
ξ2, α, β, 0
)
, (9)
where γ̄ = ℜAHlA0q△f
(ξ2
1+ξ2
)
is the average SNR.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Outage AnalysisError Rate AnalysisDiversity Order and Optimal Power Allocation
Outage Analysis I
Theorem (Per-block average outage probability)
The Per-block average outage probability at the destination for theDF based mixed RF/FSO cooperative system is given by
Pout =1
Ns
Ns∑
i=1
[
1− exp
(
−γth
CSRδ2SR
)Nr−Ns∑
k=0
1
k!
(γth
CSRδ2SR
)k
×
{
1−ξ2
Γ(α)Γ(β)G 3,12,4
(
αβ
γ̄γth
∣∣∣∣∣
1, ξ2 + 1
ξ2, α, β, 0
)}]
. (10)
where γth is the threshold SNR.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Outage AnalysisError Rate AnalysisDiversity Order and Optimal Power Allocation
Outage Analysis II
For the i th transmitted symbol, where i ∈ {1, 2, ....Ns}, theoutage probability is given by
P(i)out = Pr(min(γiSR , γRD) ≤ γth) = Fγi
min(γth), (11)
Since, the links encounter fading independently, Fγimin
(x) canbe written as
Fγimin
(x) = 1− Pr(γiSR > x)Pr(γRD > x),
= FγiSR(x) + FγRD (x) −Fγi
SR(x)FγRD (x). (12)
Now, using (11) and (12), the per-block average outageprobability can be obtained as
Pout=1
Ns
Ns∑
i=1
[
FγiSR(γth)+FγRD (γth)−Fγi
SR(γth)FγRD (γth)
]
.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Outage AnalysisError Rate AnalysisDiversity Order and Optimal Power Allocation
Error Rate Analysis I
Theorem (Per-block average BER)
The per-block average error probability for the dual-hop mixed
RF/FSO DF cooperative system is given as Pe =1Ns
∑Ns
i=1 P(i)e ,
where the average BER P(i)e for i th bit is given as,
P(i)
e =1
2Γ(p)Γ(Nr−Ns+1)G 1,22,2
(
1
CSRδ2SRq
∣∣∣∣∣
1− p, 1
Nr−Ns+1, 0
)
+ξ2
2Γ(p)Γ(α)Γ(β)
Nr−Ns∑
k=0
qp
k! (CSRδ2SR)k(
q + 1CSRδ2SR
)p+k
× G 3,23,4
αβ
γ̄(
q + 1CSRδ2SR
)
∣∣∣∣∣
1− p − k , 1, ξ2 + 1
ξ2, α, β, 0
. (13)
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Outage AnalysisError Rate AnalysisDiversity Order and Optimal Power Allocation
Error Rate Analysis II
The average BER for the i th bit based on the CDF of theend-to-end SNR can be written as6
P(i)e = −
∫ ∞
0P
′
e(ǫ |x)Fγimin
(x)dx , (14)
where P′
e(ǫ |x) is the first order derivative of conditional errorprobability (CEP) for the given SNR.
6J. M. Romero-Jerez and A. J. Goldsmith, ”Performance of multichannel reception with transmit antenna
selection in arbitrarily distributed Nakagami fading channels,” IEEE Transactions on Wireless Communications, vol.8, no. 4, pp. 2006-2013, 2009.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Outage AnalysisError Rate AnalysisDiversity Order and Optimal Power Allocation
Error Rate Analysis III
For coherent and non-coherent binary modulation schemes, aunified expression for CEP is given as7
Pe(ǫ|γ) =Γ(p, qγ)
2Γ(p)(15)
where
Γ(·, ·) is the complementary incomplete gamma function8.The values of parameters p and q corresponding to differentmodulation schemes are as summarized in Table 1.
7J. M. Romero-Jerez and A. J. Goldsmith, ”Performance of multichannel reception with transmit antenna
selection in arbitrarily distributed Nakagami fading channels,” IEEE Transactions on Wireless Communications, vol.8, no. 4, pp. 2006–2013, 2009.
8I. S. Gradshteyn, I. M. Ryzhik, and A. Jeffrey, Table of Integrals, series, and products, 7th ed. New York:
Academic Press, 2007.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Outage AnalysisError Rate AnalysisDiversity Order and Optimal Power Allocation
Error Rate Analysis IV
Table: Parameters p and q for binary modulation schemes
Modulation Scheme p q
Coherent binary frequency shift keying 0.5 0.5Coherent binary phase shift keying 0.5 1
Non-Coherent binary frequency shift keying 1 0.5Differential binary phase shift keying 1 1
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Outage AnalysisError Rate AnalysisDiversity Order and Optimal Power Allocation
Error Rate Analysis V
Differentiating Pe(ǫ |γ) and substituting in (14), the P(i)e can
be simplified as
P(i)e =
qp
2Γ(p)
∫ ∞
0e−qxxp−1Fγi
min(x)dx . (16)
Substituting the CDF, Fγimin(·), from (12) in (16), the
expression for P(i)e can be written as
P(i)
e =
[∫
∞
0
e−qxxp−1Fγ iSR(x)dx
︸ ︷︷ ︸
I1
+
∫∞
0
e−qxxp−1FγRD(x)dx
︸ ︷︷ ︸
I2
−
∫∞
0
e−qxxp−1Fγ iSR(x)FγRD
(x)dx
︸ ︷︷ ︸
I3
]
qp
2Γ(p). (17)
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Outage AnalysisError Rate AnalysisDiversity Order and Optimal Power Allocation
Error Rate Analysis VI
Using the identity for γ(·, ·) in terms of the Meijer’s-Gfunction [8.4.16.1]9 with [2.24.3.1]11 , each integral in (17) can
be solved to yield the final expression for P(i)e as,
P(i)
e =1
2Γ(p)Γ(Nr−Ns+1)G 1,22,2
(
1
CSRδ2SRq
∣∣∣∣∣
1− p, 1
Nr−Ns+1, 0
)
+ξ2
2Γ(p)Γ(α)Γ(β)
Nr−Ns∑
k=0
qp
k! (CSRδ2SR)k(
q + 1CSRδ2SR
)p+k
× G 3,23,4
αβ
γ̄(
q + 1CSRδ2SR
)
∣∣∣∣∣
1− p − k , 1, ξ2 + 1
ξ2, α, β, 0
. (18)
9A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series Volume 3: More Special
Functions, 1st ed. Gordon and Breach Science, 1986.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Outage AnalysisError Rate AnalysisDiversity Order and Optimal Power Allocation
Diversity Order Analysis I
At high-SNRs, the term corresponding to k = 0 in (18),dominates the summation. Further, using the identity for theMeijer’s-G function [18]10, one can obtain the high-SNRasymptotic approximation of BER as
Pe≤ C1
(N0
P
)Nr−Ns+1
+ C2
(N0
P
)ξ2
+C3
(N0
P
)α
+C4
(N0
P
)β
, (19)
where the terms C1,C2,C3 and C4 are given in the paper.
Using (19), one can readily observe that the net diversity ofthe considered mixed RF/FSO DF cooperative system is
d = min{Nr − Ns + 1, ξ2, α, β}. (20)
10V. S. Adamchik and O. I. Marichev, ”The algorithm for calculating integrals of hypergeometric type function
and its realization in reduce system,” in Proceedings of International Conference Symbolic Algebraic Comput,Tokyo, Japan, 1990.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Outage AnalysisError Rate AnalysisDiversity Order and Optimal Power Allocation
Optimal Power Allocation I
Further, considering a total power budget P , the convexoptimization problem for optimal source-relay power allocationcan be formulated as
minP0,P1
{
C̃1
(1
P0
)Nr−Ns+1
+ C̃2
(1
P1
)ξ2
+ C̃3
(1
P1
)α
+ C̃4
(1
P1
)β}
,
s.t. P0 + P1 = P , (21)
where
C̃1 = C1(a0N0)Nr−Ns+1, C̃2 = C2(a1N0)
ξ2 , C̃3 = C3(a1N0)α,
and C̃4 = C4(a1N0)β , respectively.
a0 = P0/P , a1 = P1/P are the power factors at the source andrelay respectively.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Outage AnalysisError Rate AnalysisDiversity Order and Optimal Power Allocation
Optimal Power Allocation II
Theorem (Optimal Power Allocation)
For a scenario when β is less than α and ξ2, one non-negative zeroof the polynomial equation below
PNr−Ns+20 − κ1(P − P0)
β+1 = 0, (22)
determines the optimal power P∗0 at the source and hence, the
optimal power at the relay is P∗1 = P − P∗
0 for a given total
cooperative power budget P, where κ1 =(Nr−Ns+1)C̃1
βC̃4.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Outage AnalysisError Rate AnalysisDiversity Order and Optimal Power Allocation
Optimal Power Allocation III
Similarly, one can obtain the polynomial equations for theother scenarios as
PNr−Ns+20 − κ1(P − P0)
κ2+1 = 0, (23)
where
κ1 =(Nr−Ns+1)C̃1
ξ2C̃2, κ2 = ξ2 for the scenario when ξ2 < α, β.
κ1 =(Nr−Ns+1)C̃1
αC̃3, κ2 = α for the scenario when α < ξ2, β.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Simulation Result I: Outage Performance
0 5 10 15 20 25 30 35 40 45 5010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Average SNR (dB)
Ou
tag
e P
rob
ab
ility
Analytical (equal power)Simulated (equal power)Analytical (optimal power)Simulated (optimal power)
ξ=1.6758; α=5.4181; β=3.7916; δ
SR2 =10; N
s=2;N
r=3
ξ=1.6785; α=3.9974; β=1.6524; δ
SR2 =1; N
s=2;N
r=3
ξ=1.6785; α=3.9974; β=1.6524; δ
SR2 =1; N
s=N
r=2
ξ=1.6758; α=5.4181; β=3.7916; δ
SR2 =10; N
s=N
r=2
Figure: Outage performance of the DF based mixed RF/FSO cooperativesystem under various channel conditions.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Simulation Result II: Error rate Performance
0 5 10 15 20 25 30 35 40 45 5010
−6
10−5
10−4
10−3
10−2
10−1
100
Average SNR (db)
BE
R
Analytical (equal power)Simulated (equal power)Analytical (optimal power)Simulated (optimal power)Asymptotic bound
(a): ξ=1.6768; α=3.9928; β=1.7056; δ
SR2 =0.1; N
s=N
r=2
(b): ξ=1.6885; α=5.0711; β=1.1547; δ
SR2 =0.5;N
s=N
r=2
(c): ξ=1.6768; α=3.9928; β=1.7056; δ
SR2 =0.1; N
s=2;N
r=3
(d): ξ=1.6885; α=5.0711; β=1.1547; δ
SR2 =0.5; N
s=2,N
r=3
Figure: BER performance of the DF based mixed RF/FSO co-operativesystem under various channel conditions.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
MotivationSystem Model
Performance AnalysisNumerical Results
Conclusion
Conclusion
Closed form analytical expressions have been derived for theend-to-end SNR outage probability, BER, diversity order andoptimal power allocation for a DF based mixedMIMO-RF/FSO cooperative system.
It has been observed that the optimal power factors derivedusing the proposed framework, significantly improve theend-to-end performance of the MIMO-RF/FSO cooperativesystem.
Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems
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