872
IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 5, MAY 2013
Joint Power Allocation and Subcarrier Pairing for Cooperative OFDM AF Multi-Relay Networks Xueyi Li, Qi Zhang, Guangchi Zhang, and Jiayin Qin
Abstract—For conventional subcarrier pairing scheme in cooperative orthogonal frequency division multiplexing amplifyand-forward multi-relay networks, to avoid interference, each subcarrier pair (SP) is assigned to only one relay, over a specific subcarrier, the destination receives signals transmitted from only one relay. In this letter, we propose to assign each SP to all the relays. Thus, over a specific subcarrier, the destination receives signals transmitted from all the relays. We propose a joint power allocation and subcarrier pairing scheme which maximizes the transmission rate subject to total network power constraint. The problem is simplified and solved by using dual method. Index Terms—Orthogonal frequency division multiplexing (OFDM), multi-relay, subcarrier pairing, power allocation.
I. I NTRODUCTION
C
OOPERATIVE communications can improve the reliability of wireless transmission and extend its coverage [1]. For cooperative amplify-and-forward (AF) orthogonal frequency division multiplexing (OFDM) single relay networks, optimal power allocation and subcarrier pairing were studied to maximize the transmission rate in [2] and [3]. For cooperative AF multi-relay OFDM networks, some power allocation schemes have been proposed in [4]-[6]. In [6], joint optimization of subcarrier pairing, subcarrier-pair-torelay assignment, and power allocation was proposed under the individual power constraint on each node or the total network power constraint. For multi-relay networks, if multiple relays participate relaying simultaneously and different relays have different subcarrier pairs (SPs), interference may be introduced. The proposed scheme in [6] assigns each SP to only one relay to avoid interference. Over a specific subcarrier, the destination receives signals transmitted from only one relay. In this letter, we propose to assign each SP to all the relays. Thus, over a specific subcarrier, the destination receives signals transmitted from all the relays. We propose a joint power allocation and subcarrier pairing scheme which maximizes the transmission rate subject to total network power constraint.
Manuscript received December 3, 2012. The associate editor coordinating the review of this letter and approving it for publication was D. Popescu. This work was supported by the National Natural Science Foundation of China (61173148, 61102070, and 61202498), the Industry-UniversityResearch Project of Guangdong Province, and the Ministry of Education (2011B090400581), the Scientific and Technological Project of Guangzhou City (12C42051578 and 11A11060133). The authors are with the Department of Electronics and Communications Engineering, Sun Yat-Sen University, Guangzhou 510006, China. X. Li and G. Zhang are also with the School of Information Engineering, Guangdong University of Technology, Guangzhou 510006, China (email:
[email protected],
[email protected],
[email protected],
[email protected]). Digital Object Identifier 10.1109/LCOMM.2013.031913.122714
II. S YSTEM M ODEL We consider a two-hop cooperative OFDM AF multi-relay network, which consists of one source, one destination and K relays. In the relay network, the transmission bandwidth is uniformly divided into N subcarriers. The half duplex relaying scheme is employed. Thus, in the first time slot, the source transmits signals to all relays and the destination over all subcarriers. In the second time slot, all the relays amplify and forward the signals simultaneously to the destination, while the source keeps silent. The destination combines signals received in the two time slots and performs the optimal signal detection. The wireless channel between any two nodes on any subcarrier is assumed to be frequency-flat. Furthermore, we assume that the channel state information (CSI) of the whole network is perfectly known at every node. The joint power allocation and subcarrier pairing optimization is performed by a central processor which is located at the destination or the source. Since the subcarrier pairing is adopted, after receiving the signal from source on the ith , i ∈ {1, 2, · · · , N }, subcarrier in first time slot, the k th relay amplifies it and forwards it on the j th , j ∈ {1, 2, · · · , N }, subcarrier in second time slot. The subcarriers i and j form a SP (i, j). If different relays have different SPs, interference may be introduced. For example, if one relay has SP (i1 , j) and the other relay has SP (i2 , j), i1 = i2 , the received signal on the j th subcarrier at destination in the second time slot is the mixture of signals on subcarriers i1 and i2 in the first time slot. To avoid interference, we propose that all relays have same SPs in this letter. The wireless channel coefficient between the source and destination on the ith subcarrier is denoted as hi,0 . The channel coefficients between the source and k th relay and between the k th relay and destination on the ith subcarrier are denoted as hi,k,1 and hi,k,2 , respectively, i = 1, 2, · · · , N , k = 1, 2, · · · , K. At the k th relay and destination on the ith subcarrier, the received additive white Gaussian noises 2 2 (AWGNs) are with zero mean and variances of σi,k and σi,d , respectively. In the first time slot, the received signal at the destination on the ith subcarrier, denoted as yi,1 , is yi,1 = hi,0 xi + ni,d,1
(1)
where xi and ni,d,1 denote the signal transmitted from the source and the AWGN at the destination on the ith subcarrier, respectively. In the second time slot, for the SP (i, j), the received signal at the destination from all the relays on the j th subcarrier, denoted as yi,j,2 , is yi,j,2 =
K k=1
c 2013 IEEE 1089-7798/13$31.00
k
k hj,k,2 f(i,j) ejφ(i,j) (hi,k,1 xi + ni,k,1 ) + nj,d,2 (2)
LI et al.: JOINT POWER ALLOCATION AND SUBCARRIER PAIRING FOR COOPERATIVE OFDM AF MULTI-RELAY NETWORKS
where ni,k,1 denotes the AWGN at the k th relay on the ith subcarrier; nj,d,2 denotes the AWGN at the destination on the k k j th subcarrier; f(i,j) ejφ(i,j) denotes the complex amplification factor of the k th relay over SP (i, j). In (2), if let φk(i,j) = −∠hi,k,1 − ∠hj,k,2
(3)
where ∠• denotes the phase of •, the phase difference for signal xi on SP (i, j) between different relays at destination disappears. We can constructively combine the signals from different relays. The corresponding signal-to-noise ratio (SNR) for SP (i, j) in the second time slot is 2 + ζ) SNR2(i,j) = ξ 2 psi /(σj,d
(4)
where psi is the transmit power of source on the ith subcarrier, ξ = ζ =
K k=1 K
k |hj,k,2 |f(i,j) |hi,k,1 |
(5)
2 2 k σi,k |hj,k,2 |2 f(i,j) .
(6)
k=1
The received signal in the second time slot is constructively combined with that in the first time slot which has the SNR expressed as follows SNR1i
=
2 |hi,0 |2 psi /σi,d
.
(7)
Thus, the end-to-end transmission rate of multi-relay networks over the SP (i, j), denoted as R(i,j) , is expressed as 1 R(i,j) = log2 1 + SNR1i + SNR2(i,j) (8) 2 where the factor 12 is included because the signal is transmitted in two time slots. The transmitted power at the k th relay on the j th subcarrier is 2 k 2 pri,j,k = f(i,j) |hi,k,1 |2 psi + σi,k . (9) Let ρi,j ∈ {0, 1} denote the subcarrier pairing indicator as [6], where ρi,j = 1 and ρi,j = 0 mean that the ith subcarrier in the first hop is and is not paired with the j th subcarrier in the second hop, respectively. We have N
ρi,j = 1,
i=1
N
ρi,j = 1, ∀i, j.
(10)
j=1
The total network power constraint is N
psi +
i=1
N N K
ρi,j pi,j,k ≤ P.
(11)
i=1 j=1 k=1
III. J OINT O PTIMIZATION OF S UBCARRIER PAIRING AND P OWER A LLOCATION In this section, we propose the joint optimization approach of subcarrier pairing and power allocation to maximize the transmission rate. The optimization problem is formulated as max
N N i=1 j=1
ρi,j R(i,j)
s.t. (10) and (11).
(12)
873
k . The The unknown parameters in (12) are ρi,j , psi , and f(i,j) optimization problem (12) is difficult to solve directly. We propose a suboptimal approach. Assume that the variances of 2 2 = σi,d . AWGN at relays and destination are the same, i.e., σi,k k Since the noise at relays is amplified by the factor f(i,j) , at the destination, the accumulated noise introduced by the relays is much larger than the noise introduced by the destination, i.e., 2 k , especially when K or f(i,j) is large. Therefore, the ζ σj,d 2 SNR(i,j) in (4) is approximated as
SNR2(i,j) ≈ ξ 2 psi /ζ
(13)
k Taking partial derivative of (13) with respect to f(i,j) ,
∂SNR2(i,j) k ∂f(i,j)
=
2 k |hj,k,2 |2 f(i,j) 2ξ |hj,k,2 hi,k,1 | psi ζ −2ξ 2 psi σi,k
ζ2
.
(14)
By letting λ(i,j) = ζ/ξ, (13) is maximized when k f(i,j) = λ(i,j)
|hi,k,1 | . 2 |h σi,k j,k,2 |
(15)
Substituting (15) into (4), we obtain 2 |hi,k,1 |2 K λ2(i,j) psi 2 k=1 σi,k . SNR2∗ (i,j) = |hi,k,1 |2 K 2 2 σj,d + λ(i,j) 2 k=1 σ
(16)
i,k
Then, the expression (16) is used in the following optimization. Substituting (15) into (9), the power at the k th relay over SP (i, j) is given by |hi,k,1 |2 psi |hi,k,1 |4 r∗ 2 + 2 . (17) pi,j,k = λ(i,j) 4 |h 2 σi,k σi,k |hj,k,2 |2 j,k,2 | Thus, the total power transmitted by all relays over SP (i, j), denoted as pri,j , is K K |hi,k,1 |2 psi |hi,k,1 |4 r r∗ 2 pi,j,k = λ(i,j) + 2 pi,j = 4 |h 2 σi,k σi,k |hj,k,2 |2 j,k,2 | k=1 k=1 (18) Then, we have λ2(i,j)
=
pri,j
·
psi
K k=1
|hi,k,1 |2 |hi,k,1 |4 + 4 2 |hj,k,2 |2 σi,k |hj,k,2 |2 σi,k K
−1 .
k=1
(19) Substituting (19) into (16), we obtain (20) shown on top of the next page. Let K K −1 |hi,k,1 |4 |hi,k,1 |2 (21) · ai,j = σ 4 |hj,k,2 |2 σ 2 |hj,k,2 |2 k=1 i,k k=1 i,k K −1 K |hi,k,1 |2 |hi,k,1 |2 2 bi,j = (22) · σj,d 2 σi,k σ 2 |hj,k,2 |2 k=1 k=1 i,k |hi,k,1 |2 |hi,k,1 |2 K K ci,j = di,j =
k=1
2 σi,k
K
2 |h 2 k=1 σi,k j,k,2 |
|hi,k,1 |4 4 |h 2 k=1 σi,k j,k,2 |
|hi,0 |2 . 2 σi,d
(23)
(24)
874
IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 5, MAY 2013
SNR2∗ (i,j)
=
psi pri,j
K |hi,k,1 |2 k=1
2 σi,k
2 2 σj,d
K k=1
|hi,k,1 |4 |hi,k,1 |2 |hi,k,1 |2 s 2 r + p σ + p i j,d i,j 2 4 2 σi,k |hj,k,2 |2 σi,k |hj,k,2 |2 σi,k
Using (21)-(24) and substituting (20) into (8), the transmission rate over SP (i, j) is expressed as ai,j bi,j ci,j psi pri,j 1 ∗ s R(i,j) = log2 1 + di,j pi + (25) 2 1 + ai,j psi + bi,j pri,j The equation (25) can be interpreted as follows. If all the relays are viewed as one virtual relay, for a given SP (i, j), di,j is the channel gain of the direct link; ai,j and bi,j are the channel gains of the first and second hops, respectively; ci,j can be viewed as a precoding gain at the source. At high SNR, the equation (25) is approximated as ai,j bi,j ci,j psi pri,j 1 ∗ s . (26) R(i,j) ≈ log2 1 + di,j pi + 2 ai,j psi + bi,j pri,j In this letter, we propose the joint power allocation and subcarrier pairing optimization scheme under the total network power constraint. The joint optimization problem is expressed as follows N N ai,j bi,j ci,j psi pri,j ρi,j s log2 1 + di,j pi + max 2 ai,j psi + bi,j pri,j i=1 j=1 s.t.
N
ρi,j = 1,
N
i=1
j=1
N
N N
i=1
psi +
ρi,j pri,j ≤ P.
s.t. psi + pri,j = Pi,j .
The solution for above optimization problem is
bi,j ci,j di,j +bi,j ci,j ξi,j Pi,j ; bi,j ci,j > di,j 2 +b s∗ ξi,j i,j ci,j ξi,j pi = Pi,j ; bi,j ci,j ≤ di,j
ai,j bi,j c2i,j −ai,j ci,j di,j Pi,j ; bi,j ci,j > di,j 2 +b r∗ ξi,j i,j ci,j ξi,j pi,j = 0; bi,j ci,j ≤ di,j ξi,j =
ai,j bi,j c2i,j + bi,j ci,j di,j − ai,j ci,j di,j .
k=1
1 = log2 (1 + αi,j Pi,j ) 2 where αi,j is the equivalent end-to-end channel gain,
bi,j ci,j (ξi,j +di,j )2 bi,j ci,j > di,j (ξi,j +bi,j ci,j )2 αi,j = . di,j bi,j ci,j ≤ di,j
(20)
This end-to-end equivalent channel gain simplifies the optimization of power allocation. For all possible SPs, we only need to optimize the variables {Pi,j }, {ρi,j }. Thus, the optimization problem (27) is simplified as N N ρi,j
max
{Pi,j },{ρi,j }
i=1 j=1 N
s.t.
2
log2 (1 + αi,j Pi,j )
ρi,j = 1,
i=1
N
ρi,j = 1, ∀i, j
j=1
N N
ρi,j Pi,j ≤ P.
(34)
i=1 j=1
Since the optimization problem (34) satisfies the time-sharing condition and the duality gap can be negligible as number of subcarriers becomes sufficiently large [6]-[7], we can solve it by using dual method. The solution obtained by dual method is asymptotically optimal [7]. The Lagrange dual function of primal problem (34) is given by g(μ) =
N N ρi,j
max
{Pi,j },{ρi,j }
i=1 j=1
2
log2 (1 + αi,j Pi,j ) (35)
i=1 j=1
(27)
(28)
(29)
(30)
where μ ≥ 0 denotes the dual variable. The dual optimization problem is expressed as min g(μ) μ
s.t. μ ≥ 0.
(36)
For a given SP (i, j), let 1 (37) log2 (1 + αi,j Pi,j ) − μPi,j . 2 Obviously, Li,j is a concave function of Pi,j . Thus, for the problem that maximizes Li,j subject to Pi,j ≥ 0, applying Karush-Kuhn-Tucher (KKT) conditions, we obtain
+ 1 1 ∗ − (38) Pi,j (μ) = 2μ αi,j Li,j =
where x+ = max(0, x). Substituting (38) into (35), we obtain an alternative expression of the dual function (31)
Substituting (29)-(30) into (26), the maximum transmission rate over SP (i, j) subject to the power constraint Pi,j is ∗ R(i,j)
k=1
−1
N N − μ( ρi,j Pi,j − P )
Similar to [2], given an SP (i, j), the optimal psi and pri,j that maximize the end-to-end transmission rate over SP (i, j) under the constraint that psi + pri,j = Pi,j , 0 ≤ Pi,j ≤ P are obtained by solving
where
K
ρi,j = 1, ∀i, j
i=1 j=1
∗ , max Ri,j
K
(32)
(33)
g(μ) = max
{ρi,j }
where L∗i,j (μ)
1 log2 = 2
N N
ρi,j L∗i,j (μ) + μP
(39)
i=1 j=1
αi,j 2μ
+ −μ
1 1 − 2μ αi,j
+ .
(40)
Define an N × N matrix L = L∗i,j (μ) . In order to maximize g(μ) in (39), we should select one element in each row and each column in L such that the sum of them is the largest.
LI et al.: JOINT POWER ALLOCATION AND SUBCARRIER PAIRING FOR COOPERATIVE OFDM AF MULTI-RELAY NETWORKS
50 PASP, N=30, K=4 PASPRA, N=30, K=4 SPOA, N=30, K=4 CCE−S, N=30, K=4 ES, N=2, K=2 PASP, N=2, K=2
Average transmission rate (b/s/Hz)
45 40 35 30 25
4.5
20 4
15
19
20
21
10 5 0
Fig. 1.
5
15 P (dB)
20
25
Average transmission rate versus P when N = 30, K = 4.
PASP PASPRA SPOA CCE−S
30
Average transmission rate (b/s/Hz)
10
25
20
assignment scheme (denoted as “PASPRA”) proposed in [6], power allocation scheme with subchannel selection (denoted as “CCE-S”) proposed in [4] and single parameter optimization approach to optimal power allocation scheme (denoted as “SPOA”) proposed in [5]. For CCE-S and SPOA, it is assumed that source power is equivalent to total relays power in the simulations. For simplicity, we assume that hi,k,1 , hi,k,2 ∼ CN (0, 1) and hi,0 ∼ CN (0, 1/8). The variances of AWGN at 2 2 = σi,d = 1. the relays and destination are σi,k In Fig. 1, we compare the average transmission rates of our proposed PASP scheme with those of the PASPRA, CCE-S, SPOA schemes for different values of P when N = 30, K = 4. It is shown that our proposed PASP scheme outperforms the other schemes at the high SNR regime. In Fig. 1, we also compare our proposed scheme with the optimal solution of (12) obtained by exhaustive search (denoted as “ES” in the legend) when N = 2, K = 2 since the exhaustive search for the case when N = 30, K = 4 is too complex. It is found from Fig. 1, the average transmission rate of our proposed suboptimal solution is very close to that of optimal solution. In Fig. 2, the average transmission rates achieved by different schemes are compared for different numbers of subcarriers N when P = 20 dB, K = 4. From Fig. 2, it is observed that the improvement of our proposed PASP scheme over the other schemes increases with the increase of the number of subcarriers.
15
V. C ONCLUSIONS 10
5
0
Fig. 2.
875
0
10
20
30 N
40
50
60
Average transmission rate versus N when P = 20 dB, K = 4.
In this letter, by assigning a specific SP to all the relays, we proposed a joint power allocation and subcarrier pairing scheme for two-hop cooperative OFDM AF multi-relay networks. It is shown from simulation results that our proposed scheme performs better than the other schemes in [4]-[6] at the high SNR regime and the performance improvement increases with the increase of the number of subcarriers. R EFERENCES
This is a standard linear assignment problem which is solved by Hungarian method [8]. In summary, assuming that the initial value of μ is given, the power allocation for every possible SP is obtained using (38) where ρi,j is ignored. Then, the matrix L is computed. By applying Hungarian method, the optimal SP indicators {ρ∗i,j } are obtained to maximize the dual function (35). Since the dual function is convex, the subgradient method [7] is employed to compute the optimal dual solution μ∗ which minimizes g(μ) where the convergence is guaranteed. After obtaining μ∗ , the optimal solution of (34) can be obtained, using (29), (30) and (25) the transmission rate can be computed. IV. S IMULATION R ESULTS In this section we compare the Monte-Carlo simulation results of our proposed joint power allocation and subcarrier pairing scheme (denoted as “PASP” in the legend) with that of joint power allocation with subcarrier pairing and relay
[1] 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. [2] Y. Li, W. Wang, J. Kong, W. Hong, X. Zhang, and M. Peng, “Power allocation and subcarrier pairing in OFDM-based relaying networks,” in Proc. 2008 IEEE ICC, pp. 2602–2606. [3] Y. Li, W. Wang, J. Kong, and M. Peng, “Subcarrier pairing for amplifyand-forward and decode-and-forward OFDM relay links,” IEEE Commun. Lett., vol. 13, no. 4, pp. 209–211, Apr. 2009. [4] M. Ibrahimi and B. Liang, “Efficient power allocation in cooperative OFDM system with channel variation,” in Proc. 2008 IEEE ICC, pp. 3022–3028. [5] Y. Zhang and Q. Wang, “Single parameter optimization approach to the optimal power allocation of OFDM relaying system,” in Proc. 2009 IEEE ICASSP, pp. 2653–2656. [6] W. Dang, M. Tao, H. Mu, and J. Huang, “Subcarrier-pair based resource allocation for cooperative multi-relay OFDM systems,” IEEE Trans. Wireless Commun., vol. 9, no. 5, pp. 1640–1649, May 2010. [7] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimization of multicarrier systems,” IEEE Trans. Commun., vol. 54, no. 7, pp. 1310– 1322, July 2006. [8] H. Khun, “The Hungarian method for the assignment problems,” Naval Research Logistics Quarterly 2, pp. 83–97, 1995.
