Optimal Power Allocation for Decode-and-Forward …home.iitk.ac.in/~neerajv/docs/ppt_SPCOM16_1.pdfbecause of the features oﬀered by the FSO technology, such as rapid deployment time,

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

[email protected]‡ Department of Electronics and Communication Engineering,

Jaypee Institute of Information Technology, Noida, [email protected]

June 14, 2016

Neeraj Varshney† and Parul Puri‡ Optimal Power Allocation for MIMO-RF/FSO Cooperative Systems



Conclusion

Overview

1 Motivation

2 System Model

3 Performance AnalysisOutage AnalysisError Rate AnalysisDiversity Order and Optimal Power Allocation

4 Numerical Results

5 Conclusion




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.




Conclusion

System Model I

Figure: Schematic diagram of the DF based mixed RF/FSO cooperativesystem.




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.




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.




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.




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.




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.




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.




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.




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.




Conclusion


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)

]

.




Conclusion


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)




Conclusion


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.




Conclusion


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.




Conclusion


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




Conclusion


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)




Conclusion


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.




Conclusion


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.




Conclusion


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.




Conclusion


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.




Conclusion


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.




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.




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.




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.




Conclusion

Thank You


Documents

Optimal Power Allocation for Decode-and-Forward …home.iitk.ac.in/~neerajv/docs/ppt_SPCOM16_1.pdfbecause of the features oﬀered by the FSO technology, such as rapid deployment time,