Optimal Power Allocation for Amplify-And-Forward Relay Networks via ...

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

Optimal Power Allocation for Amplify-and-Forward Relay Networks via Conic Programming Tony Q.S. Quek and Moe Z. Win

Hyundong Shin

Marco Chiani

Laboratory for Information & Decision Systems Massachusetts Institute of Technology Cambridge, MA 02139, USA Email : {qsquek,moewin}@mit.edu

School of Electronics and Information Kyung Hee University Gyeonggi-do, Korea Email: [email protected]

IEIIT-BO/CNR University of Bologna Bologna, Italy Email: [email protected]

Abstract— Relay power allocation has been shown to provide substantial performance gain in wireless relay channels when perfect global channel state information (CSI) is available. In this paper, we show that by using a class of conic optimization theory, we can solve the relay power allocation problem for amplify-andforward (AF) relay networks in a straightforward manner. The problem formulation is such that the achievable rate with perfect global CSI is maximized under both the aggregate and individual relay power constraints for coherent and noncoherent AF relay networks. Numerical results quantify the performance gain using the optimal relay power allocation for both the coherent and noncoherent AF relay networks.

I. I NTRODUCTION The study of cooperation in wireless networks can be formulated as a relay channel, where one or more relays help a pair of terminals to communicate. Although the optimal protocol of relay channels is still unknown, one may consider several suboptimal relaying protocols. In particular, we focus on a network with multiple relay nodes with one pair of source and destination nodes, where each relay node employs the amplify-and-forward (AF) relaying protocol. This protocol is attractive due to its simplicity, less stringent synchronization, and power efficiency. Moreover, it has been shown to be the optimal relaying protocol is certain situations [1], [2]. When the relay node has access to its locally-bidirectional channel state information (CSI), it can perform distributed beamforming so that the relayed signals add up coherently at the destination node, i.e., coherent AF relaying protocol [1]–[3]. If distributed beamforming is not possible, the relay node can employ the noncoherent AF relaying protocol, where the relay simply sends a scaled version of its received observation to the destination node without performing any phase alignment [3], [4]. To further increase the throughput and power efficiency of the relay networks, it is important to investigate the fundamental performance gain promised by efficient resource allocation. In particular, we look at the relay power allocation problem as it applies to various applications. For example, consider a wireless sensor network, it is important to know the optimal distribution of the sensor batteries in order to prolong network lifetime [5]. As another example, consider a relay-enhanced cellular network, the power allocation problem can be used to manage inter-cell interference as well as to maximize

the throughput or extend coverage [6], [7]. Unlike previous results [8]–[10], we employ quasiconvex optimization and conic optimization theories to solve the relay power allocation problem in an efficient manner under perfect global CSI [11], [12]. In this paper, we consider the optimal relay power allocation problem for coherent and noncoherent AF relay networks under perfect global CSI. The problem formulation is such that the achievable rate is maximized under both the aggregate and individual relay power constraints. We show that the coherent AF relay power allocation problem can be formulated as a quasiconvex optimization problem, which can be solved efficiently through a sequence of convex feasibility problems using the bisection method [12]. Moreover, these convex feasibility problems can be cast as second-order cone programs (SOCPs) [11]. For the noncoherent AF relay power allocation problem, we show that the problem can also be approximated as a quasiconvex optimization problem by invoking linear approximation of modulus [13]. In this case, the main problem needs to be decomposed into 2L subproblems, where each subproblem is solved using bisection method via a sequence of conic feasibility problems. These convex formulations allow us to numerically compute the optimal solutions in an efficient manner [14]. The following notation is used. Boldface upper-case letters denote matrices, boldface lower-case letters denote column vectors, and plain lower-case letters denote scalars. The superscripts (·)T , (·)∗ , and (·)† denote the transpose, the complex conjugate, and the transpose conjugate respectively. tr(·), | · |, and
·
denote the trace operator, the absolute value, and the K standard Euclidean norm, respectively. RK + and R++ denote the nonnegative and positive orthants in Euclidean vector space of dimension K, respectively. B  0 and B  0 denote that the matrix is positive semi-definite and positive definite, respectively. II. P ROBLEM F ORMULATION We consider a wireless relay network consisting of K + 2 single-antenna nodes: a designated source-destination node pair and K relay nodes located randomly and independently in a domain of a fixed area (see Fig. 1). Throughout the paper, we assume that K is finite. Furthermore, we assume that there is

1-4244-0353-7/07/$25.00 ©2007 IEEE

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

no direct link between the source and destination nodes. All nodes are in half-duplex mode, so transmission occurs over two time slots via two hops. We consider that all the relay nodes operate in a common frequency band. In the first time slot, the relay nodes receive the signal transmitted by the source node. After processing the received signals, the relay nodes simultaneously transmit the processed data to the destination node during the second time slot while the source node remains silent. We assume perfect synchronization is attained at the destination node.1 The received signals at the relay and destination nodes can then be written y R = h B xS + z R yD = h TF x R + zD ,

(1)

Source

Relays

Fig. 1.

Wireless relay network

(2)

where xS is the transmitted signal from the source node to the relay nodes, x R is the K × 1 transmitted signal vector from the relay nodes to the destination node, y R is the K × 1 received signal vector at the relay nodes, yD is the ˜K (00, Σ R ) received signal at the destination node, z R ∼ N is the K × 1 circularly symmetric complex Gaussian noise 2 ) is a circularly symmetric vector, and zD ∼ CN (0, σD complex Gaussian random variable. Note that the different noise variances at the relay nodes are reflected in Σ R
2 2 , . . . , σR,K ). Moreover, z R and zD are independent diag(σR,1 and are mutually uncorrelated with xS and x R . With perfect global CSI at the destination node, h B and h F are K × 1 known deterministic channel vectors from the source node to the relay nodes and from the relay nodes to the destination node, respectively, where h B = [hB,1 , . . . , hB,K ]T ∈ CK and h F = [hF,1 , . . . , hF,K ]T ∈ CK . For convenience, we shall refer to h B as the backward channel and h F as the forward channel. At the source node, we impose an individual source power constraint PS , such that E{|xS |2 } ≤ PS . Similarly, at the relay nodes, we impose both an aggregate relay power constraint PR and an individual relay power constraint P , where Q R
xRx †R | h B }, tr (Q QR ) ≤ PR , and Pk
[Q QR ]k,k ≤ P for E{x k = 1, . . . , K. Note that Pk is the power allocated to the kth relay. The motivation for imposing these power constraints is twofold: (i) transmission power is a network resource which affects both the life-time and the scalability of the network; and (ii) regulatory agencies may limit total transmission power due to the fact that each transmission can cause interference to other users in the network. Our goal is to improve system performance by optimally allocating power to the relays. In our model, we will adopt the achievable rate as the quality-of-service metric and consider AF relaying protocol for each relay node. For the AF relaying protocol, the relay nodes simply transmit the exact signals they have received, scaled to meet the power constraints. In this case, x R in (2) is given by x R = Gy R ,

Destination

(3)

1 Exactly how to achieve this synchronization or the effect of small synchronization errors on performance is beyond the scope of this paper.

where G denotes the K × K diagonal relay gain matrix, and Q R becomes 
Q R = G PSh Bh †B + Σ R G † . (4) The diagonal structure of G implicitly assumes that each relay node only knows the signal it receives, and has no knowledge about the signals at the other relays. When each relay node has access to its own locally-bidirectionally CSI, it can perform distributed beamforming.2 As such, we denote this as coherent AF relaying and the kth diagonal element of G is given by [1]–[3] (k)

gcoh =



βk Pk

h∗B,k h∗F,k , |hB,k | |hF,k |

(5)

2 where βk
1/(PS |hB,k |2 + σR,k ) is defined for notational simplicity. On the other hand, in the absence of forward CSI at each relay node, the relay node simply forwards a scaled version of its received signal without additional processing. In this case, this AF protocol is denoted as noncoherent AF relaying and the kth diagonal element of G is given by [3], [4]  (k) (6) gnoncoh = βk Pk .

Using (1), (2) and (3), we can then compute the mutual information between xS and yD at the destination node conditioned on h B and h F as follows: hB , hF ) = H (yD |h hB , hF ) − H (¯ hF ) , I (xS ; yD |h zD |h

(7)

where H (·) denotes the differential entropy and z¯D represents the effective noise vector at the destination node and is given by z¯D = h TF Gz R + zD . With perfect global CSI of h B and h F at the destination node, the maximum entropy of yD is attained when yD is circularly symmetric complex Gaussian, which is the case when the input distribution of xS is circularly symmetric complex Gaussian.3 In this case, the achievable rate 2 Here, locally-bidirectional CSI refers to the knowledge of only h B,k and hF,k at the kth relay node. 3 Without loss of generality, we consider natural logarithm in the following analysis so that the capacity is in nats per complex dimension.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

of the AF relay network is given by   PSh TF Gh Bh †BG †h ∗F 1 CAF = log 1 + T . 2 2 h F GΣ RG †h ∗F + σD

(8)

Since logarithm is a monotonically increasing function, we can instead consider the equivalent maximization problem as follows:

Initialize tmin = fcoh (ζζ min ), tmax = fcoh (ζζ max ), and set tolerance ε ∈ R++ . (SOCP) in (12) by 1. Solve the feasibility program Pcoh fixing t = (tmax + tmin )/2. 2. If Scoh (t) = ∅, then set tmax = t else set tmin = t. 3. Stop if the gap (tmax −tmin ) is less than the tolerance ε. Go to Step 1 otherwise. (SOCP) in Step 1. 4. Output ζ ∗ obtained from solving Pcoh where fcoh (ζζ min ) and fcoh (ζζ max ) define a range of relevant values of fcoh (ζζ ), and the convex feasibility program in Step 1 can be formulated as an SOCP find ζ (SOCP) : (12) Pcoh subject to ζ ∈ Scoh (t), 0.

P h T Gh h † G † h ∗

maximizeP1 ,...,PK hST GFΣ GB†hB∗ +σ2F R F F D Q R ) ≤ PR , subject to tr (Q QR ]k,k ≤ P, 0 ≤ [Q

Although we can always initialize a very large interval, it takes exactly log2 ((tmax −tmin )/ε) iterations before the algorithm terminates [12]. We summarize these results in the following lemma. Lemma 1: The program Pcoh in Proposition 1 can be solved numerically by the bisection method:

∀k ∈ {1, . . . , K}. (9)

III. O PTIMAL R ELAY P OWER A LLOCATION A. Coherent AF Before we propose any algorithm in solving (9), we show that a more convenient and, yet, equivalent problem exists by reformulation. Specifically, we can redefine the coherent AF relay power allocation problem in (9) as a quasiconvex optimization problem with the upper-level set that satisfies an SOC constraint.4 In the following, we present some of our results. Proposition 1: The coherent AF relay power allocation problem in (9) is a quasiconvex optimization problem with a nonempty and compact set of maxima. It can be equivalently written as
T 2   (cc ζ ) P   maximizeζ fcoh (ζζ )
σD2S A Aζ 2 +1

K 2 Pcoh : subject to ≤ 1,  k=1 ζk √  0 ≤ ζk ≤ ηp , ∀k ∈ {1, . . . , K},  where ζk
PPRk is the decision variable of the optimization problem, ηp
P/PR denotes the ratio between the individual relay power constraint and the aggregate relay power constraint, and 0 < ηp ≤ 1. In addition, c = [c1 , . . . , cK ]T ∈ K×K are defined for RK + , and A = diag(a1 , . . . , aK ) ∈ R+ notational convenience, where the kth element of c and A are given by  (10) ck = βk PR |hB,k ||hF,k |, √ βk PR |hF,k |σR,k ak = . (11) σD Proof: See Appendix I.

Remark 1: Since there exists at least one feasible solution for Pcoh and this problem is a quasiconvex optimization program, it is well-known that we can solve it efficiently through a sequence of convex feasibility problems using the bisection method [12]. Furthermore, Proposition 1 allows us to solve a particular class of convex feasibility problems known as SOCP [11]. In order for the bisection method to work, it is important that we initialize an interval that contains ζ ∗ . 4 The upper-level set of a function f : Rn → R is defined as U (f, α) = x ∈ Rn : f (x x) ≥ α}. Similarly, the lower-level set of a function f : Rn → {x x ∈ Rn : f (x x) ≤ α}. R is defined as L(f, α) = {x

with the feasible set Scoh (t) given by    cT ζ       2   tσD 1    : 0 , K 0 , Scoh (t) = ζ ∈ RK K +   PS  ζ  2  tσ  D PS Aζ  √ ζk ≤ ηp , ∀k ∈ {1, . . . , K} . (13) Proof: See Appendix II. B. Noncoherent AF First, we introduce the following lemma to allow us to transform the noncoherent AF relay power allocation problem into a quasiconvex one. Lemma 2 (Linear approximation of modulus [13]): The modulus of a complex number u ∈ C can be linearly approximated with the polyhedral norm given by       lπ lπ  u} cos u) = max Re {u u} sin , pL (u + Im {u l=1,...,L L L  u} and Im {u u} denote the real and imaginary parts where Re {u of u , and the polyhedral norm is bounded by
π  u| ≤ pL (u u) ≤ |u u| sec |u . 2L Remark 2: Geometrically, the main idea is to approximate the complex plane by a regular polygon with 2L vertices, and it u) approaches |u u| quadratically follows from Lemma 2 that pL (u as L → ∞. As a result, we can approximate the modulus of a complex number with arbitrary accuracy by increasing L. Using Lemma 2, we can reformulate the noncoherent AF relay power allocation problem in (9) with the following proposition.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

Lemma 3: The program Pnoncoh in Proposition 2 can be decomposed into 2L subproblems, where each subproblem can be solved numerically by bisection method via a sequence of convex feasibility problems in the form of SOCP. The optimal solution ζ ∗ is then given by the subproblem that gives the maximum fnoncoh (ζζ ∗ , L). Proof: See Appendix III. IV. N UMERICAL R ESULTS In this section, we illustrate the performance gain achieved by the relay power allocation for both the coherent and noncoherent AF relay networks using simple numerical results. Throughout the numerical results, we use the SeDuMi convex optimization MATLAB toolbox [15] to compute the optimal relay power allocations for ε = 0.001 using the bisection method in Section III. We assume that h B and h F are mutually independent circularly symmetric complex Gaussian noise vector, such that hB,k ∼ CN (0, 1) and hF,k ∼ CN (0, 1) for all k. Without loss of generality, we consider a symmetric network topology and we normalize the noise variances, such 2 2 = 1 and σD = 1. Since we are only interested in that σR,k 2 relay power allocation, we fix PS /σD = 10 log 30 dB. All plots are averaged over independent Monte Carlo simulation runs. Under perfect global CSI, we plot the ergodic capacities for both the coherent and noncoherent AF relay networks as a function of the number of relay nodes with ηP = 0.02 in Fig. 2. Clearly, we can observe that higher capacities can be achieved with optimal relay power allocation compared to uniform relay power allocation. However, this capacity gain is more substantial for the noncoherent AF relay network, thereby showing that relay power allocation is more beneficial in noncoherent AF relay network. Even with optimal relay power allocation, the capacity of the noncoherent AF relay

4.0

3.5 Ergodic Capacity (nats per sec/Hz)

Coherent AF

3.0

2.5

2.0

1.5

Noncoherent AF

1.0

0.5

0 10

20

30

40

50 60 Number of relays, K

70

80

90

100

Fig. 2. Effect of number of relays on the ergodic capacities of coherent and noncoherent AF relay networks, respectively. The solid and dashed lines indicate the uniform and optimal relay power allocations, respectively. 4.0

3.5 Ergodic Capacity (nats per sec/Hz)

Proposition 2: The noncoherent AF relay power allocation problem in (9) can be approximated as a quasiconvex optimization problem with a nonempty and compact set of maxima, and it can be equivalently written as    2 T  PS (pL (cc ζ ))   maximizeζ fnoncoh (ζζ , L)
σD2 Aζ 2 +1 A

K Pnoncoh : 2 ≤ 1, subject to  k=1 ζk √   ∀k ∈ {1, . . . , K}, 0 ≤ ζk ≤ ηp ,  where ζk
PPRk and pL (ccT ζ ) is the polyhedral approximation of |ccT ζ | from Lemma 2. In addition, we have defined c = K×K [c1 , . . . , cK ]T ∈ CK , and A = diag(a1 , . . . , aK ) ∈ R+ for notational convenience, where the kth element of c and A are given by  ck = βk PR hB,k hF,k , (14) √ βk PR |hF,k |σR,k ak = . (15) σD Proof: Proof is omitted as it follows straightforwardly the proof for Proposition 1 and Lemma 2.

Increasing K

3.0

2.5

2.0

1.5

1.0 Coherent AF 0.5

0 −2 10

K=5 K = 50 K = 100 −1

10 η

0

10

p

Fig. 3. Effect of ηP on the ergodic capacity of coherent AF relay network. The solid and dashed lines indicate the uniform and optimal relay power allocations, respectively.

network fails to scale like log(K) as in the coherent case. This scaling behavior of noncoherent AF relay network is consistent with the results in [3]. This is attributed to the lack of locally-bidirectional CSIs at the relay nodes to enable coherent combining at the destination node. Therefore, optimal power allocation can only increase the effective SNR at the destination node for the noncoherent AF relay network, but not fully reap the gain obtained with coherent AF case due to the lack of distributed array gain. In Figs. 3 and 4, we show the effect of ηP on the ergodic capacities of the coherent and noncoherent AF relay networks with optimal relay power allocation, respectively. Under a fixed individual relay power constraint, a smaller ηP corresponds to a larger aggregate relay power. As a result, we can see that the capacities decrease in both figures as ηP increases since the aggregate relay power available in the network becomes smaller. As the number of relay nodes

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

4.0

3.5 Ergodic Capacity (nats per sec/Hz)

Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-331-D00336).

K=5 K = 50 K = 100 Noncoherent AF

A PPENDIX I P ROOF OF P ROPOSITION 1

3.0

2.5

2.0

Increasing K

1.5

1.0

0.5

0 −2 10

−1

10 η

0

10

p

Fig. 4. Effect of ηP on the ergodic capacity of noncoherent AF relay network. The solid and dashed lines indicate the uniform and optimal relay power allocations, respectively.

increases, we can observe that the capacity gap is wider for the coherent case than for the noncoherent case due to presence of distributed array gain for the coherent case. Hence, instead of performing relay power allocation to achieve better performance, it is preferable to increase number of relay nodes if possible for the coherent AF relay network. However, this is not true for the noncoherent case as we see that the gap closes in as K increases, and relay power allocation is more preferable. V. CONCLUSIONS In this paper, we solved the relay power allocation problem under perfect global CSI for coherent and noncoherent AF relay networks using quasiconvex optimization and conic optimization theories. We showed that these problems can be formulated as quasiconvex optimization problems, and can be solved using bisection method via a sequence of conic feasibility problems in the form of SOCPs. Numerical results showed that optimal relay power allocation can offer capacity gain for both the coherent and noncoherent AF relay networks. However, this capacity gain is more substantial for the noncoherent case. Although this approach only provides numerical optimal solutions instead of analytical solutions, it is attractive since it allows straightforward extension to take into account CSI uncertainty. Some of our work in this direction can be found in [16], [17]. ACKNOWLEDGMENT This research was supported, in part, by the Office of Naval Research Young Investigator Award N00014-03-1-0489, the National Science Foundation under Grants ANI-0335256 and ECS-0636519, DoCoMo USA Labs, the Charles Stark Draper Laboratory Reduced Complexity UWB Communication Techniques Program, the Institute of Advanced Study Natural Science & Technology Fellowship, the University of Bologna Grant “Internazionalizzazione”, and the Korea

First, to show that Pcoh is a quasiconvex program, we simply need to show that the objective function fcoh (ζζ ) is quasiconcave and the constraint set in Pcoh is convex. The constraint set in Pcoh is simply the intersection of an affine set with an SOC, which are both convex sets. Since the intersection of convex sets is convex, the constraint set in Pcoh is convex. Now, let us denote this convex domain of fcoh (ζζ ) in Pcoh as dom fcoh . Therefore, for any t ∈ R+ , the upper-level set of fcoh (ζζ ) is given by      PS (ccT ζ )2  U (fcoh , t) = ζ ∈ dom fcoh  2 ≥t Aζ
2 + 1 σD
A     cT ζ           2  tσD    = ζ ∈ dom fcoh    PS  K 0 .    2   tσD   Aζ PS

(16) It is clear that U (fcoh , t) is a convex set since it can be represented as an SOC. Since the upper-level set U (fcoh , t) is convex for every t ∈ R+ , fcoh (ζζ ) in Pcoh is, thus, quasiconcave. It is clear that a concave function is also quasiconcave.5 To further show that fcoh (ζζ ) is not concave, we consider two set of feasible solutions in dom fcoh , ζ 1 and ζ 2 , where ζ 1 = ζk ek and ζ 2 = δζk ek for 0 < δ < 1, and the index k is chosen arbitrarily. Note that there always exists such ζ 1 and ζ 2 in dom fcoh . For any λ ∈ [0, 1], we have fcoh (λζζ 1 + (1 − λ)ζζ 2 ) 2 PS /σD = a2 [λ2 +δ2 (1−λ)2 ] k + x2 [λc c2 [λ+δ(1−λ)]2 k

k

1 2 k +δck (1−λ)]

,

(17)

where fcoh (λζζ 1 +(1−λ)ζζ 2 ) is clearly convex in ζk . Hence, we have showed that there exists ζ 1 , ζ 2 ∈ dom fcoh and λ ∈ [0, 1], such that fcoh (λζζ 1 +(1−λ)ζζ 2 ) ≤ λfcoh (ζζ 1 )+(1−λ)fcoh (ζζ 2 ). As a result, fcoh (ζζ ) is not concave on dom fcoh . Since the second constraint in Pcoh is bounded, it follows that dom fcoh is bounded. Moreover, the upper-level set U (fcoh , t) is an SOC, which is closed, so fcoh (ζζ ) is also closed. Therefore, by Weierstrass Theorem, the set of maxima of fcoh over dom fcoh is nonempty and compact. A PPENDIX II P ROOF OF L EMMA 1 We first show that for each given t, the convex feasibility program is an SOCP. From (16), it follows that the first constraint in (13) is an SOC constraint. Clearly, the aggregate relay power constraint in Pcoh can be cast as an SOC constraint. Lastly, linear constraints are equivalently SOC 5 If X ⊆ Rn is a convex set, then f : X → R is concave on X if for every x +(1−λ)y y ) ≥ λf (x x)+(1−λ)f (y y ). x , y ∈ X , and 0 ≤ λ ≤ 1, we have f (λx

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

(SOCP)

constraints [11]. In summary, Pcoh is an SOCP since S(coh) (t) is equivalent to the intersection of K + 2 SOC constraints and the objective function is linear. A PPENDIX III P ROOF OF L EMMA 3 Using Proposition 2, we can still use the bisection method to solve for Pnoncoh . However, unlike Pcoh , we now have 2L level sets instead of only one due to the approximation. Since the polyhedral norm in Lemma 2 requires the calculation of the maximum of 2L vertices, we end up with 2L convex sets to represent these vertices. The convex set that maximizes this norm not only yields the best approximation, but also gives the optimal solution of Pnoncoh . For completeness, we present the modified bisection method that solves for Pnoncoh , as follows: 0. Initialize tmin = fnoncoh (ζζ min ), tmax = ), and set tolerance ε ∈ R . fnoncoh(ζζ max ++        + Im c T ζ sin lπ ≥ 0, 1. If Re c T ζ cos lπ L L (SOCP) solve Pnoncoh,1 (l) in (18) using bisection method as follows: 1-1. Solve for t = (tmax + tmin )/2. 1-2. If Snoncoh,1 (t, l) = ∅, then set tmax = t else set tmin = t. 1-3. Stop if the gap (tmax − tmin ) is less than the tolerance ε. Go to Step 1-1 otherwise.         + Im c T ζ sin lπ < 0, 2. If Re c T ζ cos lπ L L (SOCP) solve Pnoncoh,2 (l) in (19) using the bisection method as follows: 2-1. Solve for t = (tmax + tmin )/2. 2-2. If Snoncoh,2 (t) = ∅, then set tmax = t else set tmin = t. 2-3. Stop if the gap (tmax − tmin ) is less than the tolerance ε. Go to Step 2-1 otherwise. 3. Output ζ ∗ that maximizes fnoncoh (ζζ ∗ , L) obtained from solving the 2L subproblems in Step 1 and 2. where fnoncoh (ζζ min ) and fnoncoh (ζζ max ) define a range of relevant values of fnoncoh (ζζ ), and each lth feasibility program in Step 1 can be formulated as a SOCP, as shown below: find ζ (SOCP) (18) Pnoncoh,1 (l) : subject to ζ ∈ Snoncoh,1 (t, l), with the feasible set Snoncoh,1 (t, l) given by Snoncoh,1 (t, l)          T  + Im c T ζ sin lπ Re c ζ cos lπ  L L    2   tσD K  K 0, = ζ ∈ R+ : PS    2  tσD  A ζ PS     Re c T ζ cos(lπ/L) + Im c T ζ sin(lπ/L) ≥ 0,    √ 1 K 0, ζk ≤ ηp , ∀k ∈ {1, . . . , K} . ζ and each lth feasibility program in Step 2 can also be formulated as a SOCP, as shown below: ζ find (SOCP) Pnoncoh,2 (l) : (19) subject to ζ ∈ Snoncoh,2 (t, l),

with the feasible set Snoncoh,2 (t, l) given by Snoncoh,2 (t, l)           − Im c T ζ sin lπ −Re c T ζ cos lπ  L L    2   tσD K  K 0, = ζ ∈ R+ : PS    2  tσD  A ζ PS     Re c T ζ cos(lπ/L) + Im c T ζ sin(lπ/L) ≤ 0,    √ 1 K 0, ζk ≤ ηp , ∀k ∈ {1, . . . , K} . ζ Clearly from above, we can see that for a given t and l, each feasibility program in Step 1 and 2 are SOC constraints. As a (SOCP) (SOCP) result, Pnoncoh,1 (l) and Pnoncoh,2 (l) are simply SOCPs and ζ ∗ which maximizes fnoncoh (ζζ ∗ , L) lies in the intersection of the set of feasible solutions obtained from these 2L subproblems. R EFERENCES [1] M. Gastpar and M. Vetterli, “On the capacity of large gaussian relay networks,” IEEE Trans. Inform. Theory, vol. 51, no. 3, pp. 765–779, Mar. 2005. [2] A. F. Dana and B. Hassibi, “On the power efficiency of sensory and ad hoc wireless networks,” IEEE Trans. Inform. Theory, vol. 52, no. 7, pp. 2890–2914, July 2006. [3] H. B¨olcskei, R. Nabar, O. Oyman, and A. Paulraj, “Capacity scaling laws in MIMO relay networks,” IEEE Trans. Wireless Commun., vol. 5, no. 6, pp. 1433–1444, June 2006. [4] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inform. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [5] I. Maric and R. D. Yates, “Cooperative multicast for maximum network lifetime,” IEEE J. Select. Areas Commun., vol. 23, no. 1, pp. 127–135, Jan. 2005. [6] H. Wu, C. Qiao, S. De, and O. Tonguz, “Integrated cellular and ad hoc relaying systems: iCAR,” IEEE J. Select. Areas Commun., vol. 19, no. 10, pp. 2105–2115, Oct. 2001. [7] R. Pabst, et al, “Relay-based deployment concepts for wireless and mobile broadband radio,” IEEE Commun. Mag., vol. 42, no. 9, pp. 80– 89, Sept. 2004. [8] I. Maric and R. D. Yates, “Bandwidth and power allocation for cooperative strategies in Gaussian relay networks,” in Proc. Asilomar Conf. on Signals, Systems and Computers, vol. 2, Pacific Grove, CA, Nov. 2004, pp. 1907 – 1911. [9] I. Hammerstrom and A. Wittneben, “On the optimal power allocation for nonregenerative OFDM relay links,” in Proc. IEEE Int. Conf. on Commun., Istanbul, TURKEY, June 2006. [10] Y. Zhao, R. Adve, and T. J. Lim, “Improving amplify-and-forward relay networks: Optimal power allocation versus selection,” in Proc. IEEE Int. Symp. on Inform. Theory, Seattle, WA, July 2006, pp. 1234–1238. [11] M. S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret, “Applications of second-order cone programming,” Linear Algebra and Its Appl., vol. 284, pp. 193–228, Nov. 1998. [12] S. Boyd and L. Vandenberghe, Convex Optimization. New York, NY: Cambridge University Press, 2004. [13] K. Glashoff and K. Roleff, “A new method for chebyshev approximation of complex-valued functions,” Mathematics of Computation, vol. 36, no. 153, pp. 233–239, Jan. 1981. [14] Y. Nesterov and A. Nemirovsky, Interior point polynomial algorithms in convex programming. Philadelphia, PA: SIAM, 1994. [15] J. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optim. Meth. Softw., vol. 11-12, pp. 625–653, Aug. 1999. [16] T. Q. S. Quek, H. Shin, M. Z. Win, and M. Chiani, “Robust power allocation for amplify-and-forward relay networks,” in Proc. IEEE Int. Conf. on Commun., Glasgow, SCOTLAND, 2007. [17] ——, “Robust power allocation in wireless relay networks,” in preparation.