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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 4, MAY 2011
Outage Probability of Decode-and-Forward Opportunistic Relaying in a Multicell Environment Dongwoo Lee, Student Member, IEEE, and Jae Hong Lee, Fellow, IEEE Abstract—In this paper, a decode-and-forward (DF) opportunistic relaying is investigated in a multicell environment with intercell interference. The outage-optimal relay selection is investigated, and the closed-form expressions of the exact and the approximate outage probabilities are derived for the DF opportunistic relaying in a multicell environment over Rayleigh fading channels. Numerical results verify the validity of the theoretical analysis by comparison with Monte Carlo simulation.
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form expressions of the exact and the approximate outage probabilities for the DF opportunistic relaying in a multicell environment over Rayleigh fading channels. Numerical results verify the validity of our theoretical analysis by comparison with Monte Carlo simulation. This paper is organized as follows: In Section II, we describe the system model and the relay selection for the DF opportunistic relaying in a multicell environment. In Section III, we derive the closed-form expressions of the exact and the approximate outage probabilities for the DF opportunistic relaying in a multicell environment over Rayleigh fading channels. In Section IV, the numerical results verify the validity of our theoretical analysis by comparison between the analytical results and Monte Carlo simulation. Finally, conclusions are drawn in Section V.
Index Terms—Cooperative diversity, decode-and-forward (DF), intercell interference, opportunistic relaying, outage probability.
II. S YSTEM M ODEL I. I NTRODUCTION In wireless fading channel environments, cooperative diversity is well known as an efficient way to combat wireless impairments and achieve the spatial diversity gain [1]–[3]. In the cooperative diversity, multiple terminals with a single antenna share their resources and assist each other in data transmission to obtain the benefits of multipleinput–multiple-output (MIMO) systems. Since the concept of the cooperative diversity was introduced, various relaying protocols for the cooperative diversity have been investigated and analyzed in the literature [1]–[8]. Among these protocols, one of the most useful techniques for practical implementation is an opportunistic relaying [4], [5]. Since the best relay is selected by a certain policy among available relay candidates, the opportunistic relaying simplifies the system implementation and provides the same diversity gain as obtained by a distributed space-time code, which requires a quite difficult code design, global channel state information at the destination, and accurate synchronization among several transmitters distributed in different locations [6]–[8]. Most of previous works on the opportunistic relaying for the cooperative diversity have largely focused on single-cell environments with no interference. Although there are few prior works that investigate the opportunistic relaying with intercell interference [9]–[12], the statistical behaviors of the decode-and-forward (DF) opportunistic relaying in a multicell environment with intercell interference have not been sufficiently investigated. This paper investigates the DF opportunistic relaying in a multicell environment with intercell interference. We consider a synchronous and linear multicell environment where both the relay and the destination receive the interfering signals from multiple cochannel interferers in neighboring cells. The outage-optimal relay selection for the DF opportunistic relaying in a multicell environment with intercell interference is investigated. Then, we derive the closed-
Consider the DF opportunistic relaying in a synchronous and linear multicell environment consisting of M cells. Assume a single source sm communicating with a single destination dm with the help of a relay set Rm = {rm,1 , rm,2 , . . . , rm,K } of the K relays in the mth cell, each with a single antenna. Assume that the direct path between the source and the destination is blocked, whereas the relays can communicate with both the source and the destination [5]. Assume that a communication between the source and the destination is performed during two time slots under the half-duplex constraint. For the first time slot, at the kth relay in the mth cell, the received signal from the source and the interfering sources of the M − 1 neighboring cells is given by ysm ,rm,k = hsm ,rm,k xsm +
M
gsi ,rm,k xsi + nrm,k
(1)
i=1 i=m
for k = 1, 2, . . . , K, where hi,j is the fading channel coefficient from terminal i to terminal j, gi,j is the fading channel coefficient from the interfering terminal i to terminal j, xi is the transmitted data from terminal i, and nj is the additive noise at terminal j. Assume that the fading channel coefficient hi,j and gi,j , and the noise nj are independent zero-mean circularly symmetric complex Gaussian random variables with variance λ2i,j , μ2i,j , and N0 , respectively. Let Hi,j = |hi,j |2 and Gi,j = |gi,j |2 be the channel gain from terminal i to terminal j and that from the interfering terminal i to the terminal j, respectively. Then, the channel gain Hi,j and Gi,j are exponential random variables with hazard rates 1/λi,j and 1/μi,j , respectively. The signal-to-interference-plus-noise ratio (SINR) from the source to the kth relay is given by Psm Hsm ,rm,k
γsm ,rm,k =
N0 +
M
Psi Gsi ,rm,k
i=1 i=m
Manuscript received February 9, 2010; revised November 25, 2010 and January 13, 2011; accepted February 27, 2011. Date of publication March 10, 2011; date of current version May 16, 2011. This work was supported in part by the IT R&D program of the Ministry of Knowledge Economy/Korea Institute for Industrial Economics & Trade under Grant KI001809 (Intelligent Wireless Communication Systems in 3-Dimensional Environment) and in part by the National Research Foundation of Korea funded by the Korea government (Ministry of Education, Science and Technology) under Grant R01-2007-00011844-0. The review of this paper was coordinated by Dr. C.-C. Chong. The authors are with the Department of Electrical Engineering and INMC, Seoul National University, Seoul 151-742, Korea (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TVT.2011.2125807
=
ρsm ,rm,k Hsm ,rm,k 1+
M
(2)
ρsi ,rm,k Gsi ,rm,k
i=1 i=m
where Pi = |xi |2 is the transmit power from terminal i, ρsm ,rm,k = Psm /N0 is the signal-to-noise ratio (SNR) from the source to the kth relay, and ρsi ,rm,k = Psi /N0 is the interference-to-noise ratio (INR) from the interfering source si to the kth relay for i = 1, . . . , M and i = m.
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Suppose that the relay rm,b is selected from the set of relays that successfully decode the source data for relaying in the mth cell. Then, for the second time slot, at the destination, the received signal from the selected relay and the interfering selected relays of the M − 1 neighboring cells is given by yrm,b ,dm = hrm,b ,dm xrm,b +
M
gri,b ,dm xri,b + ndm .
Pri,b Gri,b ,dm
i=1 i=m
ρrm,b ,dm Hrm,b ,dm
=
1+
M
Pr[Dm ] =
rm,k ∈Dm
× 1+
A. Relay Selection Let Dm be the decoding set, which is the set of relays that successfully decode the source data in the mth cell, i.e., Dm ⊆ Rm . Then, the decoding set Dm is defined as Dm =
rm,k ∈ Rm
1 : log2 (1 + γsm ,rm,k ) ≥ R 2
1 log2 1 + min(γsm ,rm,b , γrm,b ,dm ) . 2
×
In terms of the outage probability, the optimal relay selection is to choose the relay that maximizes the mutual information between the source and the destination. Therefore, if the relay transmit power is the same for all relays, the optimal relay selection for the DF opportunistic relaying in a multicell environment can be expressed as [5, Th. 1]
i=1
γth
Ω sm
× 1+
χi,j (Ωs )
j=1
−j
Ωs[i]
rm,k ∈Dm
(5)
(6)
ρ(Ω (Ωs ) s ) τi
1−exp −
where R is a spectral efficiency from the source to the destination in bits per second per hertz. Suppose that relay rm,b is selected from the decoding set for the relay transmission. Then, the mutual information between the source and the destination is given by I=
γth exp − Ω sm
where ρrm,b ,dm = Prm,b /N0 is the SNR from the selected relay rm,b to the destination, and ρri,b ,dm = Pri,b /N0 is the INR from the interfering relays ri,b to the destination for i = 1, . . . , M and i = m.
(8)
For the exact outage probability of the DF opportunistic relaying in a multicell environment, we derive the probability of a particular decoding set and the outage probability of the DF opportunistic relaying given a decoding set Dm as follows: Theorem 1 (Probability of Decoding Set): For the DF opportunistic relaying in a multicell environment, the probability of decoding set is given by
(4)
ρri,b ,dm Gri,b ,dm
i=1 i=m
Δ
Pr[Dm ] Pr[I < R|Dm ].
A. Exact Outage Probability
Prm,b Hrm,b ,dm M
(3)
The SINR from the selected relay rm,b to the destination is given by
N0 +
Pout (R) = Pr[I < R] =
Dm
i=1 i=m
γrm,b ,dm =
spectral efficiency R. By using the total probability law, the outage probability of the DF opportunistic relaying in a multicell environment is given by
γth Ω sm
ρ(Ω (Ωs ) s ) τi i=1
χi,j (Ωs )
j=1
−j
Ωs[i]
γth
Ω sm
(9)
where γth = 22R − 1 is the threshold SINR, Ωsm = ρsm ,rm,k λsm ,rm,k , Ωsi = ρsi ,rm,k μsi ,rm,k for i = 1, . . . , M and i = m, Ωs = diag(Ωs1 , Ωs2 , . . . , Ωsi , . . . , ΩsM ) for i = m, ρ(Ωs ) denotes the number of distinct diagonal elements of Ωs , Ωs[1] > Ωs[2] > · · · > Ωs[ρ(Ωs )] are the distinct diagonal elements in decreasing order, τi (Ωs ) is the multiplicity of Ωs[i] , and χi,j (Ωs ) is the (i, j)th characteristic coefficient of Ωs [13, eq. (129)]. Proof: See Appendix A. Theorem 2 (Outage Probability Conditioned on the Decoding Set): For the DF opportunistic relaying in a multicell environment, the outage probability conditioned on the decoding set is given by
ρ(Ωr ) τi (Ωr )
rm,b = arg ⇔ arg
max
γrm,k ,dm
max
Hrm,k ,dm
rm,k ∈Dm rm,k ∈Dm
Pr[I < R|Dm ] = (7)
where the equivalence follows from the fact that the relay selection M ρ G is independent of the interference term i=1,i=m ri,b ,dm ri,b ,dm in (4). It is worthy to note that, for implementing the DF opportunistic relaying in a multicell environment, the destination does not need to compute the SINR from the relay to the destination but only needs to estimate the channel gain from the relay to the destination. Therefore, the distributed method for implementing the DF opportunistic relaying in a multicell environment is equivalent to the distributed method proposed in [4]. III. P ERFORMANCE A NALYSIS The outage probability is defined as the probability that the mutual information between the source and the destination is below a given
i=1
χi,j (Ωr )
j=1
⎡ |Dm | ⎢ −j × ⎣1+Ωr[i],b
(−1)l
l=1 Am ⊆Dm |Am |=l
⎛ × exp⎝−
rm,k ∈Am
⎛ ×⎝
1 + Ωr[i],b
⎞ γth ⎠ Ωrm,k
rm,k ∈Am
γth Ωrm,k
⎞−j ⎤ ⎠ ⎥ ⎦
(10)
where Ωrm,k = ρrm,k ,dm λrm,k ,dm , Ωri,b = ρri,b ,dm μri,b ,dm for i = 1, . . . , M and i = m, Ωr = diag(Ωr1,b , Ωr2,b , . . . , Ωri,b , . . . ,
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ΩrM,b ) for i = m, and Ωr[1],b > Ωr[2],b > · · · > Ωr[ρ(Ωr )],b are the distinct diagonal elements in decreasing order, and |Dm | and |Am | are the cardinality of the set Dm and Am , respectively. Proof: See Appendix B. Then, by substituting (9) and (10) into (8), the closed-form expression of the exact outage probability of the DF opportunistic relaying in a multicell environment is given by (11), shown at the bottom of the page. Example 1 (Equal-Power Cochannel Interferers): When all of Ωsi and Ωri,b are identical, i.e., Ωsi = Ωs and Ωri,b = Ωr for i = 1, 2, . . . M and i = m, the characteristic coefficients of χi,j (Ωs ) and χi,j (Ωr ) become [13]
χ1,j (Ωs ) =
χ1,j (Ωr ) =
χi,1 (Ωs ) =
1−
j=1 j=i,m
χi,1 (Ωr ) =
M
1−
j=1 j=i,m
Ω sj Ω si
rm,k ∈Dm
0
−1 (15)
Ωri,b
for i = 1, 2, . . . , M and i = m, respectively.
×
i=1 ∞
(a)
j=1
rm,k ∈Dm
×
rm,k ∈Dm |Dm |
×
FW (γth )
rm,k ∈Dm
χi,j (Ωr )
p
Pout (R)
⎡ ⎣
z
Ω−j r[i],b Γ(j)
z
j−1
dz
Ωr[i],b
i=1
z
exp −
Ωr[i],b
dz
χi,j (Ωr )
j=1
(j + p − 1)! . (j − 1)!
Ωpr[i],b
rm,k ∈Dm |Dm |
×
γth Ωrm,k
|Dm |
p=0
rm,k ∈Dm
(17)
p
×
(16)
ρ(Ω (Ωr ) r ) τi
χi,j (Ωr )
i=1
Ωpr[i],b
γth exp − Ω sm
× 1+
rm,k ∈Dm
as ρrm,b ,dm → ∞.
where Γ(j) is the Gamma function defined as Γ(j) = (j − 1)! for a nonnegative integer j, (a) is obtained by using the approximation in (16), and (b) is obtained by replacing the integral term with the integral G(j, |Dm |; Ω−1 r[i],b ) in Appendix C. Note that (17) provides a reduced computational complexity compared with (10). By substituting (9) and (17) into (8), the closed-form expression of the approximate outage probability of the DF opportunistic relaying in a multicell environment is obtained as
×
,
Γ(j)
z j−1 exp −
ρ(Ω (Ωr ) r ) τi
γth Ωrm,k
|Dm | p=0
Ω−j r[i],b
γth (1 + z) Ωrm,k
j=1
(b)
For a large number of relays, the exact outage probability requires excessive computational time. To reduce this computational time, we derive the approximate outage probability of the DF opportunistic relaying in a multicell environment. By using the first-order approximation of Taylor series expansion, we can obtain the approximation ex 1 + x as x → 0. By using the aforementioned approximation for the exponential term, the cumulative distribution function (cdf) of W (in Appendix B) with respect to γth can be approximated as γth Ωrm,k
χi,j (Ωr )
ρ(Ωr ) τi (Ωr )
B. Approximate Outage Probability
γth (1 + z) 1 − exp − Ωrm,k
Dm
ρ(Ωr ) τi (Ωr )
=
(14)
=
(13)
−1
Ωrj,b
∞
i=1
respectively. Example 2 (Unequal-Power Cochannel Interferers): When all of Ωsi and Ωri,b are distinct, i.e., Ωsi = Ωsj and Ωri,b = Ωrj,b for i = j and i, j = m, the characteristic coefficients of χi,j (Ωs ) and χi,j (Ωr ) become [13]
M
Pr[I < R|Dm ]
(12)
j =M −1 otherwise
1, 0,
By using the integral G(a, b; z) in Appendix C, the cdf of γrm,b ,dm with respect to γth can be approximated as
0
j =M −1 otherwise
1, 0,
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Ω sm
× 1+
(j +p−1)! (j −1)!
ρ(Ω (Ωs ) s ) τi
Ωs[i]
1−exp −
j=1
χi,j (Ωs )
−j
i=1
γth
γth Ω sm
Ωs[i] Ω sm
j=1
ρ(Ω (Ωs ) s ) τi
γth
i=1 −j
χi,j (Ωs )
j=1
.
(18)
⎧ ⎡ ⎛ ⎞⎛ ⎞−j ⎫⎤ ⎪ ⎪ (Ωr ) |Dm | ⎨ ⎬ ⎢ρ(Ω r ) τi γth ⎠ ⎝ 1 γth ⎠ ⎥ −j l ⎝ Pout (R) = χ (Ω ) 1 + Ω (−1) exp − + ⎦ ⎣ i,j r r[i],b Ω Ω Ω ⎪ ⎪ r r r m,k m,k [i],b ⎩ ⎭ Dm i=1 j=1 l=1 Am ⊆Dm rm,k ∈Am rm,k ∈Am |Am |=l
ρ(Ω
−j s ) τi (Ωs ) Ωs[i] γth ×
exp −
rm,k ∈Dm
×
rm,k ∈Dm
Ω sm
χi,j (Ωs ) 1 +
i=1
γth 1 − exp − Ω sm
j=1
ρ(Ω (Ωs ) s ) τi i=1
j=1
Ω sm
χi,j (Ωs ) 1 +
γth
Ωs[i] Ω sm
−j γth
(11)
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Fig. 1. Outage probability versus SNR per hop for the DF opportunistic relaying system over Rayleigh fading channels with γth = 3 dB, λsm ,rm,k = λrm,k ,dm = 1 for k = 1, 2, . . . K, and μsi ,rm,k = μri,b ,dm = 0.005 for i = 1, . . . , M and i = m, M = 7, and K = 1, 2, 3, 4, 5.
Fig. 2. Outage probability versus number of relays for the DF opportunistic relaying system over Rayleigh fading channels with γth = 3 dB, λsm ,rm,k = λrm,k ,dm = 1 for k = 1, 2, . . . , K, and μsi ,rm,k = μri,b ,dm = 0.005 for i = 1, . . . , M and i = m, M = 7, and ρ = 15, 30 dB.
Compared to the exact outage probability, which requires excessive computational time, the approximate outage probability provides significantly reduced computational time to evaluate the outage probability of the DF opportunistic relaying in a multicell environment. IV. N UMERICAL R ESULTS Assume that the total transmit power is equally allocated to the source and the selected relay, i.e., ρ = ρsm ,rm,k = ρrm,b ,dm and ρ = ρsi ,rm,k = ρri,b ,dm for i = 1, . . . , M and i = m, where ρ is the SNR per hop. Suppose that the channel variance λsm ,rm,k = λrm,k ,dm = 1 for k = 1, . . . K and μsi ,rm,k = μri,b ,dm = 0.005 for i = 1, . . . , M and i = m, the number of cells M = 7, and the threshold SINR γth = 3 dB. Fig. 1 shows the outage probability versus SNR per hop for the DF opportunistic relaying over Rayleigh fading channels with K = 1, 2, . . . , 5. It is shown that the exact outage probability perfectly matches the simulation result. In addition, it is shown that the approximate outage probability is very close to the exact outage probability, particularly for small K. Compared to the exact outage probability, the approximate outage probability is a relatively simple and reasonably accurate way to evaluate the outage performance of the DF opportunistic relaying in a multicell environment. Due to the effects of the interference, the outage probability of the DF opportunistic relaying exhibits error floors at high-SNR region. Fig. 2 shows the outage probability versus the number of relays for the DF opportunistic relaying system over Rayleigh fading channels with ρ = 15, 30 dB. It is shown that the exact outage probability and the simulation result are in excellent agreement. In addition, the gap between the approximate and the exact outage probability is very small, particularly for small K, even though the gap is slightly widened when the number of relays is large at low SNR per hop. The DF opportunistic relaying system achieves the diversity gain, which increases with the number of relays. Fig. 3 shows the outage probability versus the number of interferers (which is equivalent to M − 1) for the DF opportunistic relaying system over Rayleigh fading channels with K = 3 and ρ = 15, 30 dB. It is shown that the exact outage probability agrees exactly with the simulation result. In addition, it is shown that the approximate outage
Fig. 3. Outage probability versus number of interferers for the DF opportunistic relaying system over Rayleigh fading channels with γth = 3 dB, λsm ,rm,k = λrm,k ,dm = 1 for k = 1, 2, . . . , K, and μsi ,rm,k = μri,b ,dm = 0.005 for i = 1, . . . , M and i = m, K = 3, and ρ = 15, 30 dB.
probability is very close to the exact outage probability. As expected, the outage probability of the DF opportunistic relaying becomes large as the number of interferers increases. V. C ONCLUSION In this paper, we have investigated the outage-optimal relay selection and derived the closed-form expressions of the exact and the approximate outage probabilities for the DF opportunistic relaying in a multicell environment over Rayleigh fading channels. The validity of our theoretical analysis has been verified by comparing the analytical results and Monte Carlo simulation. In addition, the approximate outage probability is a relatively simple and accurate way to evaluate the outage performance of the DF opportunistic relaying in a multicell environment.
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A PPENDIX A P ROBABILITY OF D ECODING S ET
A PPENDIX B O UTAGE P ROBABILITY C ONDITIONED ON THE D ECODING S ET
The SINR from the source to the kth relay in (2) can be reexpressed as
The SINR from the selected relay to the destination in (4) can be re-expressed as
ρsm ,rm,k Hsm ,rm,k
γsm ,rm,k =
M
1+
Uk 1 + Vk
=
ρsi ,rm,k Gsi ,rm,k
γrm,b ,dm = 1+
i=1 i=m
M
and the random variable Vk = i=1,i=m ρsi ,rm,k Gsi ,rm,k is the sum of M − 1 independent but not necessarily identically distributed (i.n.i.d.) exponential random variables. Then, since the mutual information between the source and the destination is independent of each relay, the probability that the kth relay is in the decoding set is given by Pr[rm,k ∈ Dm ] = Pr[γsm ,rm,k ≥ γth ] = 1−Fγsm ,rm,k (γth ) (20)
FUk (u) = 1 − exp −
ρ(Ωs ) τi (Ωs )
χi,j (Ωs )
j=1
u Ω sm
.
(21)
Ω−j s[i] Γ(j)
i=1 ∞
j=1
×
v 0
j−1
v j−1 exp −
v Ωs[i]
= 1−exp −
γ Ω sm
(22)
0
= 1−exp −
γ Ω sm
Γ(j)
i=1
γ 1 + Ωsm Ωs[i]
ρ(Ω (Ωs ) s ) τi i=1
Pr[Dm ] =
⎡ ⎢
×⎣1+
Γ(j)
χi,j (Ωr )
v dv
χi,j (Ωs ) 1+
Ωs[i] Ω sm
=
γ
rm,k ∈Dm
.
Ωrm,k
z
j−1
exp −
(27)
z
.
Ωr[i],b
j=1
|Dm |
i=1
z
j−1
0
γ(1+z) Ωrm,k
Ω−j r[i],b
exp −
(28)
z Ωr[i],b
z j−1 exp −
Γ(j)
⎛
(−1)l exp⎝−
z
|Dm |
χi,j (Ωr )⎣1+Ω−j r[i],b
j=1
⎛
⎞⎤ γ(1 + z) ⎠⎥ ⎦dz Ωrm,k
(−1)l
l=1 Am ⊆Dm |Am |=l
×exp⎝−
rm,k ∈Am
⎛ ×⎝
1 Ωr[i],b
+
Ωr[i],b
rm,k ∈Am
⎡ ⎢
dz
∞ 0
∞
(23)
1 − Fγsm ,rm,k (γth )
χi,j (Ωr )
ρ(Ωr ) τi (Ωr ) (b)
−j
Γ(j)
Γ(j)
1−exp −
|Am |=l
Ω−j r[i],b
Ω−j r[i],b
l=1 Am ⊆Dm
"
×
j=1
rm,k ∈Dm
j=1
i=1
χi,j (Ωs )
j=1
=
Ω−j s[i]
w
"
ρ(Ωr ) τi (Ωr ) (a)
dv
(26)
0 and (q) > 0 in [14, eq. (3.383.5)], with (x) denoting the real part of x. Finally, the probability of a decoding set is given by
!
(γ)
×
#∞
χi,j (Ωr )
j=1
! W
m,b ,dm
= Pr
(γth ).
From (27) and (28), the cdf of γrm,b ,dm is given by
Ω−j s[i]
ρ(Ω (Ωs ) s ) τi
v j−1 exp −
×
i=1
∞
fZk (z) =
=
γ(1+v) v exp − − Ω sm Ωs[i]
1 − exp −
ρ(Ωr ) τi (Ωr )
.
m,b ,dm
Similar to (22), the pdf of Zk is given by
1+Zk ρ(Ωr ) τi (Ωr )
FW (w) =
Fγr
χi,j (Ωs )
(25)
Then, by using the independence of Hrm,k ,dm for k = 1, · · · , K, the cdf of W is given by
i=1
Fγsm ,rm,k (γ) ! U " k = Pr 0 and (q) > 0 in [14, eq. (3.383.5)].
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A PPENDIX C I NTEGRAL I DENTITY To derive the approximate cdf of γrm,b ,dm , we define the integral G(a, b; z) as
∞ Δ
e−zy y a−1 (1+y)b dy,
G(a, b; z) =
(a) > 0, (z) > 0.
(30)
0
Then, the integral G(a, b; z) can be rewritten as G(a, b; z) = (a − 1)!Ψ(a, a + b + 1; z) (a)
= z −a (a − 1)!2 F0 (a, −b; −z −1 ) ∞
= z −a (a − 1)!
(a)n (−b)n
n=0
b (a + n − 1)! (−1)n b! (−1)n z −n
(a)
= z −a (a − 1)!
n=0
=z
−a
b b n=0
(−z −1 )n n!
n
(a − 1)!
(b − n)!
z −n (a + n − 1)!
n!
(31)
where Ψ(α, γ; z) is the confluent hypergeometric function of the second kind defined in [14, eq. (9.211.4)], 2 F0 (α, β; z) is the hypergeometric function defined in [14, eq. (9.14.1)], and (a)n is a Pochhammer symbol defined as (a)n = a(a + 1) . . . (a + n − 1) = Γ(a + n)/Γ(a) for nonnegative integer a and n. In (31), (a) follows from the identity 2 F0 (α, β; −x−1 ) = xα Ψ(α, α − β + 1; x) in [15, eq. (6.6.3)], and (b) follows from the fact that
$ (−b)n =
(−1)n b! , (b−n)!
0,
0≤n≤b n>b
[6] J. N. Laneman and G. W. Wornell, “Distributed space-time coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2415–2425, Oct. 2003. [7] G. Scutari and S. Barbarossa, “Distributed space-time coding for regenerative relay networks,” IEEE Trans. Wireless Commun., vol. 4, no. 5, pp. 2387–2399, Sep. 2005. [8] Y. Jing and B. Hassibi, “Distributed space-time coding in wireless relay networks,” IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3524–3536, Dec. 2006. [9] D. Lee and J. H. Lee, “Outage probability for opportunistic relaying on multicell environments,” in Proc. IEEE VTC 2009—Spring, Barcelona, Spain, Apr. 2009, pp. 1–5. [10] I. Krikidis, J. S. Thompson, S. McLaughlin, and N. Goertz, “Max-min relay selection for legacy amplify-and-forward systems with interference,” IEEE Trans. Wireless Commun., vol. 8, no. 6, pp. 3016–3027, Jun. 2009. [11] C. Zhong, S. Jin, and K.-K. Wong, “Dual-hop systems with noisy relay and interference-limited destination,” IEEE Trans. Commun., vol. 58, no. 3, pp. 764–768, Mar. 2010. [12] J. Si, Z. Li, and Z. Liu, “Outage probability of opportunistic relaying in Rayleigh fading channels with multiple interferers,” IEEE Signal Process. Lett., vol. 17, no. 5, pp. 445–448, May 2010. [13] H. Shin and M. Z. Win, “MIMO diversity in the presence of double scattering,” IEEE Trans. Inf. Theory, vol. 54, no. 7, pp. 2976–2996, Jul. 2008. [14] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. New York: Academic, 2007. [15] A. Erdelyi, Higher Transcendental Functions, vol. 1. New York: McGraw-Hill, 1953.
(32)
for a nonnegative integer b.
Low-Complexity Groupwise OSIC-ZF Detection for N × N Spatial Multiplexing Systems Yinman Lee, Member, IEEE, and Hong-Wei Shieh
Abstract—In this paper, we present a low-complexity groupwise ordered successive interference canceler (OSIC) with the zero-forcing (ZF) criterion for N × N spatial multiplexing systems. The proposed detection is composed of a number of processing stages, and at each stage, a fraction of the transmitted data streams are nulled to zero. In this way, the original matrix-inverse operations are replaced by a series of inversions with smaller sizes, and the computational complexity can then be largely reduced. We show that this groupwise OSIC-ZF approach can provide a good tradeoff between the computational complexity and the error-rate performance and, therefore, is very attractive as a new detection method standing between the linear ZF detection and the optimal OSIC-ZF detection. Index Terms—Groupwise ordered successive interference canceler (OSIC), multiple-input multiple-output (MIMO), spatial multiplexing, zero-forcing (ZF) criterion.
ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their careful reading and insightful comments. R EFERENCES [1] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity— Part I: System description,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1927–1938, Nov. 2003. [2] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [3] A. Nostratinia, T. E. Hunter, and A. Hedayat, “Cooperative communication in wireless networks,” IEEE Commun. Mag., vol. 42, no. 10, pp. 74– 80, Oct. 2004. [4] A. Bletsas, A. Khisti, D. P. Reed, and A. Lippman, “A simple cooperative diversity method based on network path selection,” IEEE J. Sel. Areas Commun., vol. 24, no. 3, pp. 659–672, Mar. 2006. [5] A. Bletsas, H. Shin, and M. Z. Win, “Cooperative communications with outage-optimal opportunistic relaying,” IEEE Trans. Wireless Commun., vol. 6, no. 9, pp. 3450–3460, Sep. 2007.
I. I NTRODUCTION In recent years, much attention has been paid to the development of multiple-input–multiple-output (MIMO) systems for wireless communications. With multiple antennas at both the transmitter and the Manuscript received August 9, 2010; revised November 23, 2010 and January 26, 2011; accepted February 9, 2011. Date of publication March 14, 2011; date of current version May 16, 2011. This work was supported by the National Science Council, Taiwan, under Grant NSC 99-2221-E-260-025. The review of this paper was coordinated by Dr. C. Cozzo. The authors are with the Graduate Institute of Communication Engineering, National Chi Nan University, Nantou 54561, Taiwan (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2011.2127499
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Outage Probability of Decode-and-Forward Opportunistic Relaying in a Multicell Environment Dongwoo Lee, Student Member, IEEE, and Jae Hong Lee, Fellow, IEEE Abstract—In this paper, a decode-and-forward (DF) opportunistic relaying is investigated in a multicell environment with intercell interference. The outage-optimal relay selection is investigated, and the closed-form expressions of the exact and the approximate outage probabilities are derived for the DF opportunistic relaying in a multicell environment over Rayleigh fading channels. Numerical results verify the validity of the theoretical analysis by comparison with Monte Carlo simulation.
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form expressions of the exact and the approximate outage probabilities for the DF opportunistic relaying in a multicell environment over Rayleigh fading channels. Numerical results verify the validity of our theoretical analysis by comparison with Monte Carlo simulation. This paper is organized as follows: In Section II, we describe the system model and the relay selection for the DF opportunistic relaying in a multicell environment. In Section III, we derive the closed-form expressions of the exact and the approximate outage probabilities for the DF opportunistic relaying in a multicell environment over Rayleigh fading channels. In Section IV, the numerical results verify the validity of our theoretical analysis by comparison between the analytical results and Monte Carlo simulation. Finally, conclusions are drawn in Section V.
Index Terms—Cooperative diversity, decode-and-forward (DF), intercell interference, opportunistic relaying, outage probability.
II. S YSTEM M ODEL I. I NTRODUCTION In wireless fading channel environments, cooperative diversity is well known as an efficient way to combat wireless impairments and achieve the spatial diversity gain [1]–[3]. In the cooperative diversity, multiple terminals with a single antenna share their resources and assist each other in data transmission to obtain the benefits of multipleinput–multiple-output (MIMO) systems. Since the concept of the cooperative diversity was introduced, various relaying protocols for the cooperative diversity have been investigated and analyzed in the literature [1]–[8]. Among these protocols, one of the most useful techniques for practical implementation is an opportunistic relaying [4], [5]. Since the best relay is selected by a certain policy among available relay candidates, the opportunistic relaying simplifies the system implementation and provides the same diversity gain as obtained by a distributed space-time code, which requires a quite difficult code design, global channel state information at the destination, and accurate synchronization among several transmitters distributed in different locations [6]–[8]. Most of previous works on the opportunistic relaying for the cooperative diversity have largely focused on single-cell environments with no interference. Although there are few prior works that investigate the opportunistic relaying with intercell interference [9]–[12], the statistical behaviors of the decode-and-forward (DF) opportunistic relaying in a multicell environment with intercell interference have not been sufficiently investigated. This paper investigates the DF opportunistic relaying in a multicell environment with intercell interference. We consider a synchronous and linear multicell environment where both the relay and the destination receive the interfering signals from multiple cochannel interferers in neighboring cells. The outage-optimal relay selection for the DF opportunistic relaying in a multicell environment with intercell interference is investigated. Then, we derive the closed-
Consider the DF opportunistic relaying in a synchronous and linear multicell environment consisting of M cells. Assume a single source sm communicating with a single destination dm with the help of a relay set Rm = {rm,1 , rm,2 , . . . , rm,K } of the K relays in the mth cell, each with a single antenna. Assume that the direct path between the source and the destination is blocked, whereas the relays can communicate with both the source and the destination [5]. Assume that a communication between the source and the destination is performed during two time slots under the half-duplex constraint. For the first time slot, at the kth relay in the mth cell, the received signal from the source and the interfering sources of the M − 1 neighboring cells is given by ysm ,rm,k = hsm ,rm,k xsm +
M
gsi ,rm,k xsi + nrm,k
(1)
i=1 i=m
for k = 1, 2, . . . , K, where hi,j is the fading channel coefficient from terminal i to terminal j, gi,j is the fading channel coefficient from the interfering terminal i to terminal j, xi is the transmitted data from terminal i, and nj is the additive noise at terminal j. Assume that the fading channel coefficient hi,j and gi,j , and the noise nj are independent zero-mean circularly symmetric complex Gaussian random variables with variance λ2i,j , μ2i,j , and N0 , respectively. Let Hi,j = |hi,j |2 and Gi,j = |gi,j |2 be the channel gain from terminal i to terminal j and that from the interfering terminal i to the terminal j, respectively. Then, the channel gain Hi,j and Gi,j are exponential random variables with hazard rates 1/λi,j and 1/μi,j , respectively. The signal-to-interference-plus-noise ratio (SINR) from the source to the kth relay is given by Psm Hsm ,rm,k
γsm ,rm,k =
N0 +
M
Psi Gsi ,rm,k
i=1 i=m
Manuscript received February 9, 2010; revised November 25, 2010 and January 13, 2011; accepted February 27, 2011. Date of publication March 10, 2011; date of current version May 16, 2011. This work was supported in part by the IT R&D program of the Ministry of Knowledge Economy/Korea Institute for Industrial Economics & Trade under Grant KI001809 (Intelligent Wireless Communication Systems in 3-Dimensional Environment) and in part by the National Research Foundation of Korea funded by the Korea government (Ministry of Education, Science and Technology) under Grant R01-2007-00011844-0. The review of this paper was coordinated by Dr. C.-C. Chong. The authors are with the Department of Electrical Engineering and INMC, Seoul National University, Seoul 151-742, Korea (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TVT.2011.2125807
=
ρsm ,rm,k Hsm ,rm,k 1+
M
(2)
ρsi ,rm,k Gsi ,rm,k
i=1 i=m
where Pi = |xi |2 is the transmit power from terminal i, ρsm ,rm,k = Psm /N0 is the signal-to-noise ratio (SNR) from the source to the kth relay, and ρsi ,rm,k = Psi /N0 is the interference-to-noise ratio (INR) from the interfering source si to the kth relay for i = 1, . . . , M and i = m.
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Suppose that the relay rm,b is selected from the set of relays that successfully decode the source data for relaying in the mth cell. Then, for the second time slot, at the destination, the received signal from the selected relay and the interfering selected relays of the M − 1 neighboring cells is given by yrm,b ,dm = hrm,b ,dm xrm,b +
M
gri,b ,dm xri,b + ndm .
Pri,b Gri,b ,dm
i=1 i=m
ρrm,b ,dm Hrm,b ,dm
=
1+
M
Pr[Dm ] =
rm,k ∈Dm
× 1+
A. Relay Selection Let Dm be the decoding set, which is the set of relays that successfully decode the source data in the mth cell, i.e., Dm ⊆ Rm . Then, the decoding set Dm is defined as Dm =
rm,k ∈ Rm
1 : log2 (1 + γsm ,rm,k ) ≥ R 2
1 log2 1 + min(γsm ,rm,b , γrm,b ,dm ) . 2
×
In terms of the outage probability, the optimal relay selection is to choose the relay that maximizes the mutual information between the source and the destination. Therefore, if the relay transmit power is the same for all relays, the optimal relay selection for the DF opportunistic relaying in a multicell environment can be expressed as [5, Th. 1]
i=1
γth
Ω sm
× 1+
χi,j (Ωs )
j=1
−j
Ωs[i]
rm,k ∈Dm
(5)
(6)
ρ(Ω (Ωs ) s ) τi
1−exp −
where R is a spectral efficiency from the source to the destination in bits per second per hertz. Suppose that relay rm,b is selected from the decoding set for the relay transmission. Then, the mutual information between the source and the destination is given by I=
γth exp − Ω sm
where ρrm,b ,dm = Prm,b /N0 is the SNR from the selected relay rm,b to the destination, and ρri,b ,dm = Pri,b /N0 is the INR from the interfering relays ri,b to the destination for i = 1, . . . , M and i = m.
(8)
For the exact outage probability of the DF opportunistic relaying in a multicell environment, we derive the probability of a particular decoding set and the outage probability of the DF opportunistic relaying given a decoding set Dm as follows: Theorem 1 (Probability of Decoding Set): For the DF opportunistic relaying in a multicell environment, the probability of decoding set is given by
(4)
ρri,b ,dm Gri,b ,dm
i=1 i=m
Δ
Pr[Dm ] Pr[I < R|Dm ].
A. Exact Outage Probability
Prm,b Hrm,b ,dm M
(3)
The SINR from the selected relay rm,b to the destination is given by
N0 +
Pout (R) = Pr[I < R] =
Dm
i=1 i=m
γrm,b ,dm =
spectral efficiency R. By using the total probability law, the outage probability of the DF opportunistic relaying in a multicell environment is given by
γth Ω sm
ρ(Ω (Ωs ) s ) τi i=1
χi,j (Ωs )
j=1
−j
Ωs[i]
γth
Ω sm
(9)
where γth = 22R − 1 is the threshold SINR, Ωsm = ρsm ,rm,k λsm ,rm,k , Ωsi = ρsi ,rm,k μsi ,rm,k for i = 1, . . . , M and i = m, Ωs = diag(Ωs1 , Ωs2 , . . . , Ωsi , . . . , ΩsM ) for i = m, ρ(Ωs ) denotes the number of distinct diagonal elements of Ωs , Ωs[1] > Ωs[2] > · · · > Ωs[ρ(Ωs )] are the distinct diagonal elements in decreasing order, τi (Ωs ) is the multiplicity of Ωs[i] , and χi,j (Ωs ) is the (i, j)th characteristic coefficient of Ωs [13, eq. (129)]. Proof: See Appendix A. Theorem 2 (Outage Probability Conditioned on the Decoding Set): For the DF opportunistic relaying in a multicell environment, the outage probability conditioned on the decoding set is given by
ρ(Ωr ) τi (Ωr )
rm,b = arg ⇔ arg
max
γrm,k ,dm
max
Hrm,k ,dm
rm,k ∈Dm rm,k ∈Dm
Pr[I < R|Dm ] = (7)
where the equivalence follows from the fact that the relay selection M ρ G is independent of the interference term i=1,i=m ri,b ,dm ri,b ,dm in (4). It is worthy to note that, for implementing the DF opportunistic relaying in a multicell environment, the destination does not need to compute the SINR from the relay to the destination but only needs to estimate the channel gain from the relay to the destination. Therefore, the distributed method for implementing the DF opportunistic relaying in a multicell environment is equivalent to the distributed method proposed in [4]. III. P ERFORMANCE A NALYSIS The outage probability is defined as the probability that the mutual information between the source and the destination is below a given
i=1
χi,j (Ωr )
j=1
⎡ |Dm | ⎢ −j × ⎣1+Ωr[i],b
(−1)l
l=1 Am ⊆Dm |Am |=l
⎛ × exp⎝−
rm,k ∈Am
⎛ ×⎝
1 + Ωr[i],b
⎞ γth ⎠ Ωrm,k
rm,k ∈Am
γth Ωrm,k
⎞−j ⎤ ⎠ ⎥ ⎦
(10)
where Ωrm,k = ρrm,k ,dm λrm,k ,dm , Ωri,b = ρri,b ,dm μri,b ,dm for i = 1, . . . , M and i = m, Ωr = diag(Ωr1,b , Ωr2,b , . . . , Ωri,b , . . . ,
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 4, MAY 2011
ΩrM,b ) for i = m, and Ωr[1],b > Ωr[2],b > · · · > Ωr[ρ(Ωr )],b are the distinct diagonal elements in decreasing order, and |Dm | and |Am | are the cardinality of the set Dm and Am , respectively. Proof: See Appendix B. Then, by substituting (9) and (10) into (8), the closed-form expression of the exact outage probability of the DF opportunistic relaying in a multicell environment is given by (11), shown at the bottom of the page. Example 1 (Equal-Power Cochannel Interferers): When all of Ωsi and Ωri,b are identical, i.e., Ωsi = Ωs and Ωri,b = Ωr for i = 1, 2, . . . M and i = m, the characteristic coefficients of χi,j (Ωs ) and χi,j (Ωr ) become [13]
χ1,j (Ωs ) =
χ1,j (Ωr ) =
χi,1 (Ωs ) =
1−
j=1 j=i,m
χi,1 (Ωr ) =
M
1−
j=1 j=i,m
Ω sj Ω si
rm,k ∈Dm
0
−1 (15)
Ωri,b
for i = 1, 2, . . . , M and i = m, respectively.
×
i=1 ∞
(a)
j=1
rm,k ∈Dm
×
rm,k ∈Dm |Dm |
×
FW (γth )
rm,k ∈Dm
χi,j (Ωr )
p
Pout (R)
⎡ ⎣
z
Ω−j r[i],b Γ(j)
z
j−1
dz
Ωr[i],b
i=1
z
exp −
Ωr[i],b
dz
χi,j (Ωr )
j=1
(j + p − 1)! . (j − 1)!
Ωpr[i],b
rm,k ∈Dm |Dm |
×
γth Ωrm,k
|Dm |
p=0
rm,k ∈Dm
(17)
p
×
(16)
ρ(Ω (Ωr ) r ) τi
χi,j (Ωr )
i=1
Ωpr[i],b
γth exp − Ω sm
× 1+
rm,k ∈Dm
as ρrm,b ,dm → ∞.
where Γ(j) is the Gamma function defined as Γ(j) = (j − 1)! for a nonnegative integer j, (a) is obtained by using the approximation in (16), and (b) is obtained by replacing the integral term with the integral G(j, |Dm |; Ω−1 r[i],b ) in Appendix C. Note that (17) provides a reduced computational complexity compared with (10). By substituting (9) and (17) into (8), the closed-form expression of the approximate outage probability of the DF opportunistic relaying in a multicell environment is obtained as
×
,
Γ(j)
z j−1 exp −
ρ(Ω (Ωr ) r ) τi
γth Ωrm,k
|Dm | p=0
Ω−j r[i],b
γth (1 + z) Ωrm,k
j=1
(b)
For a large number of relays, the exact outage probability requires excessive computational time. To reduce this computational time, we derive the approximate outage probability of the DF opportunistic relaying in a multicell environment. By using the first-order approximation of Taylor series expansion, we can obtain the approximation ex 1 + x as x → 0. By using the aforementioned approximation for the exponential term, the cumulative distribution function (cdf) of W (in Appendix B) with respect to γth can be approximated as γth Ωrm,k
χi,j (Ωr )
ρ(Ωr ) τi (Ωr )
B. Approximate Outage Probability
γth (1 + z) 1 − exp − Ωrm,k
Dm
ρ(Ωr ) τi (Ωr )
=
(14)
=
(13)
−1
Ωrj,b
∞
i=1
respectively. Example 2 (Unequal-Power Cochannel Interferers): When all of Ωsi and Ωri,b are distinct, i.e., Ωsi = Ωsj and Ωri,b = Ωrj,b for i = j and i, j = m, the characteristic coefficients of χi,j (Ωs ) and χi,j (Ωr ) become [13]
M
Pr[I < R|Dm ]
(12)
j =M −1 otherwise
1, 0,
By using the integral G(a, b; z) in Appendix C, the cdf of γrm,b ,dm with respect to γth can be approximated as
0
j =M −1 otherwise
1, 0,
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Ω sm
× 1+
(j +p−1)! (j −1)!
ρ(Ω (Ωs ) s ) τi
Ωs[i]
1−exp −
j=1
χi,j (Ωs )
−j
i=1
γth
γth Ω sm
Ωs[i] Ω sm
j=1
ρ(Ω (Ωs ) s ) τi
γth
i=1 −j
χi,j (Ωs )
j=1
.
(18)
⎧ ⎡ ⎛ ⎞⎛ ⎞−j ⎫⎤ ⎪ ⎪ (Ωr ) |Dm | ⎨ ⎬ ⎢ρ(Ω r ) τi γth ⎠ ⎝ 1 γth ⎠ ⎥ −j l ⎝ Pout (R) = χ (Ω ) 1 + Ω (−1) exp − + ⎦ ⎣ i,j r r[i],b Ω Ω Ω ⎪ ⎪ r r r m,k m,k [i],b ⎩ ⎭ Dm i=1 j=1 l=1 Am ⊆Dm rm,k ∈Am rm,k ∈Am |Am |=l
ρ(Ω
−j s ) τi (Ωs ) Ωs[i] γth ×
exp −
rm,k ∈Dm
×
rm,k ∈Dm
Ω sm
χi,j (Ωs ) 1 +
i=1
γth 1 − exp − Ω sm
j=1
ρ(Ω (Ωs ) s ) τi i=1
j=1
Ω sm
χi,j (Ωs ) 1 +
γth
Ωs[i] Ω sm
−j γth
(11)
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Fig. 1. Outage probability versus SNR per hop for the DF opportunistic relaying system over Rayleigh fading channels with γth = 3 dB, λsm ,rm,k = λrm,k ,dm = 1 for k = 1, 2, . . . K, and μsi ,rm,k = μri,b ,dm = 0.005 for i = 1, . . . , M and i = m, M = 7, and K = 1, 2, 3, 4, 5.
Fig. 2. Outage probability versus number of relays for the DF opportunistic relaying system over Rayleigh fading channels with γth = 3 dB, λsm ,rm,k = λrm,k ,dm = 1 for k = 1, 2, . . . , K, and μsi ,rm,k = μri,b ,dm = 0.005 for i = 1, . . . , M and i = m, M = 7, and ρ = 15, 30 dB.
Compared to the exact outage probability, which requires excessive computational time, the approximate outage probability provides significantly reduced computational time to evaluate the outage probability of the DF opportunistic relaying in a multicell environment. IV. N UMERICAL R ESULTS Assume that the total transmit power is equally allocated to the source and the selected relay, i.e., ρ = ρsm ,rm,k = ρrm,b ,dm and ρ = ρsi ,rm,k = ρri,b ,dm for i = 1, . . . , M and i = m, where ρ is the SNR per hop. Suppose that the channel variance λsm ,rm,k = λrm,k ,dm = 1 for k = 1, . . . K and μsi ,rm,k = μri,b ,dm = 0.005 for i = 1, . . . , M and i = m, the number of cells M = 7, and the threshold SINR γth = 3 dB. Fig. 1 shows the outage probability versus SNR per hop for the DF opportunistic relaying over Rayleigh fading channels with K = 1, 2, . . . , 5. It is shown that the exact outage probability perfectly matches the simulation result. In addition, it is shown that the approximate outage probability is very close to the exact outage probability, particularly for small K. Compared to the exact outage probability, the approximate outage probability is a relatively simple and reasonably accurate way to evaluate the outage performance of the DF opportunistic relaying in a multicell environment. Due to the effects of the interference, the outage probability of the DF opportunistic relaying exhibits error floors at high-SNR region. Fig. 2 shows the outage probability versus the number of relays for the DF opportunistic relaying system over Rayleigh fading channels with ρ = 15, 30 dB. It is shown that the exact outage probability and the simulation result are in excellent agreement. In addition, the gap between the approximate and the exact outage probability is very small, particularly for small K, even though the gap is slightly widened when the number of relays is large at low SNR per hop. The DF opportunistic relaying system achieves the diversity gain, which increases with the number of relays. Fig. 3 shows the outage probability versus the number of interferers (which is equivalent to M − 1) for the DF opportunistic relaying system over Rayleigh fading channels with K = 3 and ρ = 15, 30 dB. It is shown that the exact outage probability agrees exactly with the simulation result. In addition, it is shown that the approximate outage
Fig. 3. Outage probability versus number of interferers for the DF opportunistic relaying system over Rayleigh fading channels with γth = 3 dB, λsm ,rm,k = λrm,k ,dm = 1 for k = 1, 2, . . . , K, and μsi ,rm,k = μri,b ,dm = 0.005 for i = 1, . . . , M and i = m, K = 3, and ρ = 15, 30 dB.
probability is very close to the exact outage probability. As expected, the outage probability of the DF opportunistic relaying becomes large as the number of interferers increases. V. C ONCLUSION In this paper, we have investigated the outage-optimal relay selection and derived the closed-form expressions of the exact and the approximate outage probabilities for the DF opportunistic relaying in a multicell environment over Rayleigh fading channels. The validity of our theoretical analysis has been verified by comparing the analytical results and Monte Carlo simulation. In addition, the approximate outage probability is a relatively simple and accurate way to evaluate the outage performance of the DF opportunistic relaying in a multicell environment.
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A PPENDIX A P ROBABILITY OF D ECODING S ET
A PPENDIX B O UTAGE P ROBABILITY C ONDITIONED ON THE D ECODING S ET
The SINR from the source to the kth relay in (2) can be reexpressed as
The SINR from the selected relay to the destination in (4) can be re-expressed as
ρsm ,rm,k Hsm ,rm,k
γsm ,rm,k =
M
1+
Uk 1 + Vk
=
ρsi ,rm,k Gsi ,rm,k
γrm,b ,dm = 1+
i=1 i=m
M
and the random variable Vk = i=1,i=m ρsi ,rm,k Gsi ,rm,k is the sum of M − 1 independent but not necessarily identically distributed (i.n.i.d.) exponential random variables. Then, since the mutual information between the source and the destination is independent of each relay, the probability that the kth relay is in the decoding set is given by Pr[rm,k ∈ Dm ] = Pr[γsm ,rm,k ≥ γth ] = 1−Fγsm ,rm,k (γth ) (20)
FUk (u) = 1 − exp −
ρ(Ωs ) τi (Ωs )
χi,j (Ωs )
j=1
u Ω sm
.
(21)
Ω−j s[i] Γ(j)
i=1 ∞
j=1
×
v 0
j−1
v j−1 exp −
v Ωs[i]
= 1−exp −
γ Ω sm
(22)
0
= 1−exp −
γ Ω sm
Γ(j)
i=1
γ 1 + Ωsm Ωs[i]
ρ(Ω (Ωs ) s ) τi i=1
Pr[Dm ] =
⎡ ⎢
×⎣1+
Γ(j)
χi,j (Ωr )
v dv
χi,j (Ωs ) 1+
Ωs[i] Ω sm
=
γ
rm,k ∈Dm
.
Ωrm,k
z
j−1
exp −
(27)
z
.
Ωr[i],b
j=1
|Dm |
i=1
z
j−1
0
γ(1+z) Ωrm,k
Ω−j r[i],b
exp −
(28)
z Ωr[i],b
z j−1 exp −
Γ(j)
⎛
(−1)l exp⎝−
z
|Dm |
χi,j (Ωr )⎣1+Ω−j r[i],b
j=1
⎛
⎞⎤ γ(1 + z) ⎠⎥ ⎦dz Ωrm,k
(−1)l
l=1 Am ⊆Dm |Am |=l
×exp⎝−
rm,k ∈Am
⎛ ×⎝
1 Ωr[i],b
+
Ωr[i],b
rm,k ∈Am
⎡ ⎢
dz
∞ 0
∞
(23)
1 − Fγsm ,rm,k (γth )
χi,j (Ωr )
ρ(Ωr ) τi (Ωr ) (b)
−j
Γ(j)
Γ(j)
1−exp −
|Am |=l
Ω−j r[i],b
Ω−j r[i],b
l=1 Am ⊆Dm
"
×
j=1
rm,k ∈Dm
j=1
i=1
χi,j (Ωs )
j=1
=
Ω−j s[i]
w
"
ρ(Ωr ) τi (Ωr ) (a)
dv
(26)
0 and (q) > 0 in [14, eq. (3.383.5)], with (x) denoting the real part of x. Finally, the probability of a decoding set is given by
!
(γ)
×
#∞
χi,j (Ωr )
j=1
! W
m,b ,dm
= Pr
(γth ).
From (27) and (28), the cdf of γrm,b ,dm is given by
Ω−j s[i]
ρ(Ω (Ωs ) s ) τi
v j−1 exp −
×
i=1
∞
fZk (z) =
=
γ(1+v) v exp − − Ω sm Ωs[i]
1 − exp −
ρ(Ωr ) τi (Ωr )
.
m,b ,dm
Similar to (22), the pdf of Zk is given by
1+Zk ρ(Ωr ) τi (Ωr )
FW (w) =
Fγr
χi,j (Ωs )
(25)
Then, by using the independence of Hrm,k ,dm for k = 1, · · · , K, the cdf of W is given by
i=1
Fγsm ,rm,k (γ) ! U " k = Pr 0 and (q) > 0 in [14, eq. (3.383.5)].
1930
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 4, MAY 2011
A PPENDIX C I NTEGRAL I DENTITY To derive the approximate cdf of γrm,b ,dm , we define the integral G(a, b; z) as
∞ Δ
e−zy y a−1 (1+y)b dy,
G(a, b; z) =
(a) > 0, (z) > 0.
(30)
0
Then, the integral G(a, b; z) can be rewritten as G(a, b; z) = (a − 1)!Ψ(a, a + b + 1; z) (a)
= z −a (a − 1)!2 F0 (a, −b; −z −1 ) ∞
= z −a (a − 1)!
(a)n (−b)n
n=0
b (a + n − 1)! (−1)n b! (−1)n z −n
(a)
= z −a (a − 1)!
n=0
=z
−a
b b n=0
(−z −1 )n n!
n
(a − 1)!
(b − n)!
z −n (a + n − 1)!
n!
(31)
where Ψ(α, γ; z) is the confluent hypergeometric function of the second kind defined in [14, eq. (9.211.4)], 2 F0 (α, β; z) is the hypergeometric function defined in [14, eq. (9.14.1)], and (a)n is a Pochhammer symbol defined as (a)n = a(a + 1) . . . (a + n − 1) = Γ(a + n)/Γ(a) for nonnegative integer a and n. In (31), (a) follows from the identity 2 F0 (α, β; −x−1 ) = xα Ψ(α, α − β + 1; x) in [15, eq. (6.6.3)], and (b) follows from the fact that
$ (−b)n =
(−1)n b! , (b−n)!
0,
0≤n≤b n>b
[6] J. N. Laneman and G. W. Wornell, “Distributed space-time coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2415–2425, Oct. 2003. [7] G. Scutari and S. Barbarossa, “Distributed space-time coding for regenerative relay networks,” IEEE Trans. Wireless Commun., vol. 4, no. 5, pp. 2387–2399, Sep. 2005. [8] Y. Jing and B. Hassibi, “Distributed space-time coding in wireless relay networks,” IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3524–3536, Dec. 2006. [9] D. Lee and J. H. Lee, “Outage probability for opportunistic relaying on multicell environments,” in Proc. IEEE VTC 2009—Spring, Barcelona, Spain, Apr. 2009, pp. 1–5. [10] I. Krikidis, J. S. Thompson, S. McLaughlin, and N. Goertz, “Max-min relay selection for legacy amplify-and-forward systems with interference,” IEEE Trans. Wireless Commun., vol. 8, no. 6, pp. 3016–3027, Jun. 2009. [11] C. Zhong, S. Jin, and K.-K. Wong, “Dual-hop systems with noisy relay and interference-limited destination,” IEEE Trans. Commun., vol. 58, no. 3, pp. 764–768, Mar. 2010. [12] J. Si, Z. Li, and Z. Liu, “Outage probability of opportunistic relaying in Rayleigh fading channels with multiple interferers,” IEEE Signal Process. Lett., vol. 17, no. 5, pp. 445–448, May 2010. [13] H. Shin and M. Z. Win, “MIMO diversity in the presence of double scattering,” IEEE Trans. Inf. Theory, vol. 54, no. 7, pp. 2976–2996, Jul. 2008. [14] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. New York: Academic, 2007. [15] A. Erdelyi, Higher Transcendental Functions, vol. 1. New York: McGraw-Hill, 1953.
(32)
for a nonnegative integer b.
Low-Complexity Groupwise OSIC-ZF Detection for N × N Spatial Multiplexing Systems Yinman Lee, Member, IEEE, and Hong-Wei Shieh
Abstract—In this paper, we present a low-complexity groupwise ordered successive interference canceler (OSIC) with the zero-forcing (ZF) criterion for N × N spatial multiplexing systems. The proposed detection is composed of a number of processing stages, and at each stage, a fraction of the transmitted data streams are nulled to zero. In this way, the original matrix-inverse operations are replaced by a series of inversions with smaller sizes, and the computational complexity can then be largely reduced. We show that this groupwise OSIC-ZF approach can provide a good tradeoff between the computational complexity and the error-rate performance and, therefore, is very attractive as a new detection method standing between the linear ZF detection and the optimal OSIC-ZF detection. Index Terms—Groupwise ordered successive interference canceler (OSIC), multiple-input multiple-output (MIMO), spatial multiplexing, zero-forcing (ZF) criterion.
ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their careful reading and insightful comments. R EFERENCES [1] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity— Part I: System description,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1927–1938, Nov. 2003. [2] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [3] A. Nostratinia, T. E. Hunter, and A. Hedayat, “Cooperative communication in wireless networks,” IEEE Commun. Mag., vol. 42, no. 10, pp. 74– 80, Oct. 2004. [4] A. Bletsas, A. Khisti, D. P. Reed, and A. Lippman, “A simple cooperative diversity method based on network path selection,” IEEE J. Sel. Areas Commun., vol. 24, no. 3, pp. 659–672, Mar. 2006. [5] A. Bletsas, H. Shin, and M. Z. Win, “Cooperative communications with outage-optimal opportunistic relaying,” IEEE Trans. Wireless Commun., vol. 6, no. 9, pp. 3450–3460, Sep. 2007.
I. I NTRODUCTION In recent years, much attention has been paid to the development of multiple-input–multiple-output (MIMO) systems for wireless communications. With multiple antennas at both the transmitter and the Manuscript received August 9, 2010; revised November 23, 2010 and January 26, 2011; accepted February 9, 2011. Date of publication March 14, 2011; date of current version May 16, 2011. This work was supported by the National Science Council, Taiwan, under Grant NSC 99-2221-E-260-025. The review of this paper was coordinated by Dr. C. Cozzo. The authors are with the Graduate Institute of Communication Engineering, National Chi Nan University, Nantou 54561, Taiwan (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2011.2127499
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