Outage Performance of Opportunistic AF OFDM Relaying over Rician Fading Channel Sudhan Majhi,Piyush Kumar
Youssef Nasser
Indian Institute of Technology Patna, India Email:
[email protected],
[email protected],
American University of Beirut, Lebanon Email:
[email protected]
Abstract—This paper analyzes closed-form outage probability of opportunistic amplify-and-forward (AF) for orthogonal frequency division multiplexing (OFDM) relaying over Rician fading environment. The opportunistic relay node is selected by maximizing SNR over all the relay nodes. In this paper, we propose a method by which a closed form outage probability for opportunistic amplify-and-forward relaying for OFDM cooperative communication can be evaluated by utilizing the Marcum Q-function. The outage probabilities is derived over independent and identically distributed (i.i.d) fading channel and independent but not identically distributed (i.n.d) fading channel. To improve outage performance, more number of relay nodes has to be deployed irrespective of the number of subcarriers used by the transceivers. We investigate a closed-form outage performance over Rician fading channels at high SNR regime and the analytical results are validated through the Monte-Carlo simulation results. Index Terms—Opportunistic AF OFDM relaying, Amplifyand-forward relaying, Rician fading Channel, Outage performance.
I. I NTRODUCTION Cooperative communications, a promising technology for wireless transmissions, have taken a lot of attention in recent few years. It can benefit most of the leverages of multiple input multiple output (MIMO) systems without considering the conventional MIMO schemes [1]. Due to the higher degree of freedom, repetition-based relaying has its own importance in cooperative communications [2], [3]. Several relaying techniques such as amplify-and-forward (AF), decode-and-forward (DF), compress-and-forward (CF) and selective relaying have been presented in [1]. Recently, opportunistic relaying, in which only one relay node is used to forward source data to the destination, and its application over orthogonal frequency division multiplexing (OFDM) create a huge attention over cooperative communications due to its high spectral efficiency, power efficiency and simplicity for implementation [4], [5]. The OFDM scheme is underlying physical layer technology for IEEE802.11, Long Term Evolution (LTE), digital audio and video broadcast, etc. The single carrier outage performance of opportunistic DF and AF over asymmetric channels are provided in [6], [7] and [8] respectively. The outage analysis for multicarrier system is widely investigated in [9]. Design of cooperative OFDM (CO-OFDM) is provided in [10]. The selective OFDM
and OFDMA are introduced in [5], [11] and their outage performance over Rayleigh fading are provided in [12] and [13] respectively. The OFDMA relaying may provide better performance over OFDM relaying but it is very difficult to implement in practical scenarios [11]. Outage of unequal block based OFDM based AF relaying in frequency selective fading channel is investigated in [14]. The outage performance of opportunistic AF OFDM relaying over Rayleigh fading is introduced in [15]. In practice, cooperative communication often experiences Rician fading channel, however, none of the papers has provided performance for opportunistic AF OFDM relaying over Rician fading channel, which is the main objective of this paper. In this work, we provide a closed-form outage probability of opportunistic AF OFDM relaying over Rician fading channels. The analytical outage probability is derived at high signal-to-noise ratio (SNR) regime and validated through monte carlo simulation. A closed form derivation is obtained by using an approximation of Marcum Q-function. Two different scenarios has been analyzed in this paper; (1) opportunistic relaying with independent and identically distributed (i.i.d) fading over subcarriers and relays, (2) independent and identically distributed (i.i.d) fading over subcarriers and independent but not identically distributed (i.n.d) fading over relays. The rest of the article is organised as follows, section II describes about the opportunistic AF OFDM relaying system model. In section III, a closed-form expression for outage probability of opportunistic AF OFDM relaying over different scenarios has been considered. In section IV, analytical result is validated through Monte-Carlo simulation. Finally, section V draws the conclusion of the paper.
II. S YSTEM M ODEL Consider a general 2-hop AF relying which consists of a source (S), M relays, Ri , i=1, 2, ..., M , and destination (D). In phase I, the received signal at D and Ri of the nth subcarrier can be modeled as √ εs,n Hsd (n)x(n) + Nsd (n) ysd (n) = √ ysri (n) = εs,n Hsri (n)x(n) + Nsri (n)
(1) (2)
where εri ,n is the transmitted power of each subcarrier by Ri , x0 (n) is the nth subcarrier transmitted signal by Ri , Nri d (n) is the zero mean complex Gaussian random variable with distribution N (0, Nri d ) and Hri ,d (n) the is channel gain of Ri -D link. III. O UTAGE P ROBABILITY A NALYSIS WITH O PPORTUNISTIC R ELAYING In this section, we analyze the outage probability of the opportunistic relaying in OFDM system. The selection of the relay is achieved according to the maximum instantaneous SNR available over the different relays. Moreover, we assume that the signal carried by all subcarriers will be forwarded by the same relay [16]. The equivalent instantaneous end-to-end SNR for opportunistic AF OFDM relaying can be written as [17] Fig. 1.
Illustration of Selection OFDM relaying
γd = γ0
N X n=1
where εs,n is the power of each subcarrier transmitted by S, x(n) is the data transmitted by the nth subcarrier from the source S, Nsd (n) and Nsri (n) are the zero mean complex Gaussian random variables with distribution N (0, Nsd ) and N (0, Nsri ) at D and Ri respectively, Hsd (n) and Hsri (n) are the fading coefficient of subcarrier n for the S-D and S-Ri (ith sublink of S-R link) links respectively. The channel coefficient Hsd (n) is defined as Hsd (n) =
L X
asd (l) exp(−j2πnτsd (l)/N ) + asd (n)
(3)
l=1
where N is the number of subcarrier, L is the number of non-line of sight (NLOS) components and asd (n) is the direct component of subcarrier n, τsd (l) is the delay of lth path, asd (l) (l 6= 0) is modeled as a circularly symmetric complex Gaussian random variable with zero mean and unit variance. asd (0) has the same distribution with non zero mean. This is a typical Rician fading whose instantaneous signal power between a node A and a node B is given by ξab = εa |Hab (n)|2 and it follows a non-central Chi-Squre distribution. The probability density function (PDF) of ξab is expressed as s ! Kab +1 − γ(Kξ¯ab +1) −Kab 4Kab (Kab +1)γ ab I0 fξab (γ)= ¯ e ξab ξ¯ab (4) where I0 (.) is the 0th order of the modified Bessel function of first kind. ξ¯ab = E[ξab ] , E(x) represents the expectation over K, Kab is the Rician factor and γ is the random variable of the Rician fading distribution. In phase II, the best or opportunistic relay node forwards the source’s data to the destination through AF relaying process. The received signal at the destination can be given as √ yri ,d (n) = εri ,n Hri d (n)x0 (n) + Nri d (n) (5)
γn,0 +
N X
γ0 αn,i γ0 βn,i (6) γ α + γ0 βn,i + 1 i={1,2,...,M } n=1 0 n,i max
where γn,0 = εs,n |Hsd (n)|2 , αn,i = εs,n |Hsri (n)|2 and βn,i = εri ,n |Hri d (n)|2 . Assuming all the noise variances are equal, i.e., Nsd = Nsri = Nri d = 1/γ0 , where γ0 is proportional the system SNR. To evaluate the outage probability of opportunistic AF OFDM relaying, the cumulative distribution function (CDF) of γd is to be evaluated. However, it is hard to compute the CDF of γd in a straight-forward, we use an upper bound of γd , which can be written as γ¯d
=
γ0 γsum,0 +
max
i={1,2,...,M }
(min (γ0 αsum,i , γ0 βsum,i ))
=
γ0 γsum,0 + γ0 γmax (7) PN PN where γsum,0 = γ , αsum,i = n=1 n=1 αn,i PNn,0 and βsum,i = β , and γ = n,i max n=1 max(ξmin,1 , ξmin,2 , ..., ξmin,M ) and ξmin,i = min (αsum,i , βsum,i ). The corresponding lower bound of outage probability of opportunistic AF OFDM relaying can be expressed as DM pOF = P r [γub < γ] out
(8) 2R
where γub = γsum,0 + γmax and γ = (2 − 1)/γ0 . The above outage probability is equivalent to the CDF of γub . To compute the CDF of γub , the problem turns out to derive the CDF of ξmin,i and CDF of γmax first. The CDF of the random variable ξmin,i is obtained by using Appendix A as Fξmin,i (γ) = 1 − 1 − Fαsum,i (γ) 1 − Fβsum,i (γ) s ! p 2(Ksri + 1)γ 2N Ksri , = 1 − QN ξ¯sri (9) s ! p 2(Kri d + 1)γ × QN 2N Kri d , ξ¯r d i
th
where QN (, ) is the N order Marcum Q-function, and Fαsum,i (γ) and Fβsum,i (γ) are the CDF of αsum,i and βsum,i ,
respectively. The corresponding PDF of ξmin,i is obtained by differentiating (9) as s ! p 2(Ksri + 1)γ 2N Ksri , fβsum,i (γ) fξmin,i (γ) = QN ξ¯sri s ! p 2(Kri d + 1)γ 2N Kri d , fαsum,i (γ) +QN ξ¯r d i
(10) where fαsum,i (γ) and fβsum,i (γ) are the PDF of αsum,i and βsum,i , respectively.
In this section, we assume that the channels are i.i.d fading over both subcarriers and relay nodes. It means that the mean of the channel gains of all the links are the same across subcarriers and relay nodes, however, different cooperative links (1st and 2nd hop) have different channel gain mean, i.e. ξ¯sri = ξ¯sr , ξ¯ri d = ξ¯rd , and fξmin,i (γ) = fξmin (γ) ∀ i. The Laplace transformation (LT) of the PDF of γub is obtained by using initial value theorem (IVT) given in Appendix B as γ→0
M! M fγ (γ) (fξmin (γ)) M s +1 sum,0
(11)
where fγsum,0 (γ) is the PDF of γsum,0 . The PDF of γub is obtained by applying the inverse LT (ILT) on the above, we can have fγub (γ) = lim γ M fγsum,0 (γ) (fξmin (γ))
M
γ→0
(12)
By integrating (12) and using the limiting value from Appendix B, the lower bound on outage probability or equivalent CDF of γub is obtained as 1 e−N Ksd (N M + N )((N − 1)!)M +1 N N Ksd + 1 −N Ksr Ksr + 1 × e ξ¯sd ξ¯sr N M Krd + 1 +e−N Krd γ N M +N ξ¯rd
DM pOF out,iid =
(13)
B. Fading is i.i.d over subcarriers but i.n.d over relays In opportunistic AF OFDM relaying, all subcarriers pass through the same relay nodes. Therefore, we assume that channels are i.i.d over the subcarriers, but i.n.d over relay nodes due to the different geographical locations of the relay nodes. Since the channels are i.i.d over the subcarriers, it provides the same PDF of ξmin,i given in (10). The PDF of γub over i.n.d with respect to relay nodes can be derived by using Appendix B as fγub (γ) = lim γ M fγsum,0 (γ) γ→0
M Y i=1
1 e−N Ksd (N M + N )((N − 1)!)M +1 N Y N M Ksd + 1 −N Ksri Ksri + 1 × e ξ¯sd ξ¯sri i=1 N K ri d + 1 γ N M +N +e−N Kri d ξ¯ri d
DM pOF out,ind =
(15)
IV. S IMULATION AND D ISCUSSION OF VARIOUS R ELAYING T ECHNIQUES
A. Fading is i.i.d over subcarriers and relays
L(fγub (γ)) = lim
By integrating (14) and substituting the limiting value, the lower bound on outage probability of AF OFDM relaying over i.n.d Rician fading channel is written as
fξmin,i (γ)
(14)
In this section, analytical and Monte-Carlo simulation results are presented. For i.i.d. channels, we set the same mean for all links and For i.n.d channels, we set different means for different S-Ri /Ri -D links. In the Rician fading channel, the Rician factor Kab is uniformly distributed [6], [7] and the mean γ¯ab of NLOS is uniformly distributed [1], [6]. The line of sight (LOS) components are derived for a given Kab and γ¯ab . We have also considered the Rayleigh fading channel for comparison purposes with Rician fading channel. The outage probability over Rician fading channel is obtained by substituting ξ sd = (Ksd + 1)/γsd in (22) and the outage probability over Rayleigh fading channel is obtained by substituting Ksri = 0 and Kri d = 0 in (22). The simulation results have been provided in Fig. 2 show that the analytical outage probability is tight bound with the simulation at high SNR regime for both Rician and Rayleigh fading channels. Fig. 3 shows the outage performance of the opportunistic AF OFDMA, the opportunistic AF OFDM and the opportunistic AF with single carrier cooperative communication systems. In Fig. 3. Opportunistic AF OFDMA relaying provides 2.5 dB more diversity than opportunistic AF OFDM relaying and 8.5 dB more diversity than opportunistic AF with single carrier cooperative communication systems at outage of 10−4 . Although opportunistic AF OFDMA relaying outperforms the opportunistic AF OFDM relaying and opportunistic AF with single carrier, the opportunistic AF OFDM relaying is a viable candidate for cooperative communication due to its simplicity in the practical implementation. Fig. 4 and Fig. 5 show the outage performance of opportunistic AF OFDM relaying in Rician fading channel for different number of subcarriers and relays, respectively. It is clear that to improve outage performance, more number of relay nodes has to be deployed irrespective of the number of subcarriers used by the transceivers. V. C ONCLUSION In this paper, the outage performance of opportunistic AF OFDM has been investigated over Rician fading channel. A lower bound of the outage probability has been derived and validated through Monte-Carlo simulation results.
0
0
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Rician Simulation,M=3 Rician Analytical,M=3 Rayleigh Simulation,M=3 Rayleigh Analytical,M=3 Rician Simulation,M=6 Rician Analytical,M=6 Rayleigh Simulation,M=6 Rayleigh Analytical,M=6
Outage Probability
−2
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Outage Probability
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Rician Simulation,M=6 Rician Simulation,M=12
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−4
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−5
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−2
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−5
5
10
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0
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15
SNR [dB]
SNR [dB]
Fig. 2. The analytical and simulation result of outage probability of opportunistic OFDM relaying over Rician and Rayleigh fading channels for i.i.d channel and 16 subcarriers
Fig. 4. The outage probability of Opportunistic OFDM relaying over Rician fading channels as increasing number of relay nodes 0
10
0
N=16 N=32
10
Opportunistic OFDMA Opportunistic OFDM Opportunistic Single carrier
−1
10
−1
Outage Probability
Outage Probability
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−2
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−4
10 −4
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−5
10 −5
10
0
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SNR [dB]
SNR [dB]
Fig. 3. Simulation results of outage performance of Opportunistic OFDM, Opportunistic OFDM and Single Carrier
We show that the outage performance is better in Rician fading environment than Rayleigh fading environment. The increment in relay nodes improves the outage performance.
A PPENDIX A PN The PDF of γsum,0 = n=1 γn,0 is obtained by using the LT given in [18, eq.(29.3.81)]. The LT of the 0th order
Fig. 5. The outage probability of Opportunistic OFDM relaying over Rician fading channels as increasing number of subcarriers
modified Bessel function of first kind is obtained as ( s ) 1 Ksd (Ksd + 1)γ ¯ = eKsd (Ksd +1)/sξsd (16) L Io 2 ¯ s ξsd The LT of the PDF of γn,0 is obtained by using the shifting property as ! Ksd (Ksd +1) (Ksd + 1) −Ksd 1 (ξ¯sd s+Ksd +1) L fξsd (γ) = e e K +1 ξ¯sd s + ξsd ¯sd (17)
Therefore, the LT of the PDF of γsum,0 is equal to the N multiplication of L fξsd (s) which can be obtained as
L(fγub (γ)) = L(fγsum,0 (γ))L(fγmax (γ))
L fγsum,0 (γ) ×
Since γub is the sum of γsum,0 and γmax , the LT of the PDF of the random variable γub can be written as
N −N Ksd
= (Ksd + 1) e
1 ξ¯sd s + Ksd + 1
N
By using LT of differentiation, we can write (18)
N Ksd (K+1)
−1) L(fγ(M (γ)) = sM −1 L(fγmax (γ)) − sM −2 fγmax (0) max
e ξ¯sd s+Ksd +1
To obtain inverse LT (ILT) of (18), we use [18, eq.(29.3.81)] (N −1)/2 1 N Ksdξ¯(Kssd +1) γ sd L−1 e = sN N Ksd (Ksd + 1)/ξ¯sd s ! N Ksd (Ksd + 1)γ ×IN −1 2 ξ¯sd (19) Now, by using (19) and the shifting property of ILT, the ILT of (18) provides the PDF of γsum,0 as (N +1)/2 (K +1)γ Ksd + 1 −N Ksd − sd ¯ ξ sd fγsum,0 (γ) = e ξ¯sd s ! (20) (N −1)/2 γ N Ksd (Ksd + 1)γ × IN −1 2 N Ksd ξ¯sd The cumulative distribution function (CDF) of γsum,0 is obtained by integrating (20) and by substituting N Ksd = b2 /2 +1)ξ = x2 /2 yields and (Ksd ξ¯
−2) −... − fγ(M (0) max
(26)
(M −2)
(1)
Since fγmax (0) = fγmax (0) = ... = fγmax (0) = 0, by using the initial value theorem of LT, we can write −1) fγ(M (γ) = lim sM L(fγmax (γ)) max lim γ→0
(27)
s→∞
By using (27) and (24), we can write L(fγmax (γ)) = lim
γ→0
lim s→∞
M! (fξ (γ))M sM min,i
(28)
Similarly, by using the above procedures the LT of fγsum,0 (γ) can be written as L(fγsum,0 (γ)) = lim
γ→0
1 fγ (γ) s sum,0
(29)
By substituting (28) and (29) in (25), we can have L(fγub (γ)) = lim
M!
fγsum,0 (γ)(fξmin,i (γ)) γ→0 sM +1
sd
q
(25)
M
(30)
By using the series form of Marcum Q-function, we can have s b 0 p 2(Ksr +1)γ Z ∞ 2N Ksr + x N −1 x2 +b2 ¯sr ξ 2(K + 1)γ sr 2 2N Ksr , = e− e− 2 IN −1 (bx)dx QN = 1− r x 2(Ksd +1)γ ξ¯sr b ¯ ξ s sd ! n/2 ∞ s ! X N Ksr ξ¯sr N Ksr (Ksr + 1)γ p 2(Ksd + 1)γ × In 2 2N Ksd , = 1 − QN (21) (Ksr + 1)γ ξ¯sr n=1−N ξ¯sd (31) where QN (, ) is the N th order Marcum Q-function Now, we will find the limiting value by using the above series Z
Fsum,0 (γ) =
2(Ksd +1)γ ¯ ξ
x
x N −1
e−
x2 +b2 2
IN −1 (bx)dx
The CDF of γmax is derived by using as M Fγmax (γ) = Fξmin,i (γ)
(22)
and the corresponding PDF of γmax is obtained by differentiating above as M −1 (23) fγmax (γ) = M fξmin,i (γ) Fξmin,i (γ) Since Fξi (0) = 0, all the derivatives of (23) bellow (M − 1) order is zero when γ → 0 and the derivative of (M − 1)th order for γ → 0 as γ0 → ∞, can be written as ∂ (M −1) fγmax (γ) γ=0 = lim M !(fξmin,i (γ))M (M −1) γ→0 ∂γ
lim fγsum,0 (γ)(fξmin,i (γ))M = lim fγsum,0 (γ) γ→0 s p 2(Ksri + 1)γ 2N Ksri , fβsum,i (γ) QN ξ¯sri s M p 2(Kri d + 1)γ +QN 2N Kri d , fαsum,i (γ) ξ¯ri d γ→0
A PPENDIX B
(24)
(32)
By substituting (31) and (20) in (32), we can write as 1 e−N Ksd ((N − 1)!)M +1 N N Ksd + 1 −N Ksr Ksr + 1 × e ξ¯sd ξ¯sr N M Krd + 1 +e−N Krd γ N M +N −M −1 ξ¯rd (33)
lim fγsum,0 (γ)(fξmin,i (γ))M =
γ→0
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