IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 2, FEBRUARY 2013 305

Robust Peer-to-Peer Relay Beamforming: A Probabilistic ApproachDhananjaya Ponukumati, Feifei Gao, and Chengwen Xing

Abstract—In this work, we design outage constrained collab-orative relay beamforming (CRBF) vectors for a peer-to-peeramplify-and-forward (AF) relay network with imperfect channelstate information (CSI) at the relays. Specifically, we modelchannel estimation error as a Gaussian random vector withknown statistical distribution. The objective is to minimize thetotal transmit power at relays subject to probabilistic quality ofservice (QoS) constraints at each receiver. To solve the originalnon-convex problem, we utilize Bernstein-type inequalities andrecast the original probabilistic constraints into conservativedeterministic linear matrix inequalities (LMI). An alternativemethod is to replace the probabilistic constraint with a conser-vative one by applying S-procedure. Employing rank relaxationtechnique, the two convex reformulations are numerically solvedwith semidefinite programming (SDP). Simulation results areprovided to corroborate our studies.

Index Terms—Distributed beamforming, peer-to-peer, outageprobability, semidefinite programming.

I. INTRODUCTION

TO enhance the utilization of the wireless network re-sources, energy efficient designs that enable communi-

cation of multiple source-destination pairs (peer-to-peer) overa shared channel are demanded. To realize this objective,collaborative relay beamforming (CRBF) algorithms for multi-node communication networks with perfect channel stateinformation (CSI) has been studied in [1]–[3] and in thereferences therein. For instance, a semidefinite programming(SDP) based method was proposed to compute distributedrelay beamforming weights in [2]. The problem of jointsource power control and general rank relay matrix designto minimize the total transmit power of the network subjectto quality-of-service (QoS) constraints at each destination wassolved in [3].

However, practically inevitable estimation errors and quan-tization errors in CSI should be considered in system design.One approach is to consider the worst-case based optimizationthat restricts the channel errors in a certain bounded region[4]–[6]. Since worst-case approach is too conservative, aprobabilistic approach is highly desired. In [7], the outage

Manuscript received September 26, 2012. The associate editor coordinatingthe review of this letter and approving it for publication was D. W. K. Ng.

This work was supported in part by the National Basic Research Programof China (973 Program) under Grant {2012CB316102, 2013CB336600}, bythe National Natural Science Foundation of China under Grant 61201187,and by the Tsinghua University Initiative Scientific Research Program underGrant 20121088074.

D. Ponukumati is with Qualcomm India Private Limited, Hyderabad,500001 India (e-mail: dhananjaysarma@gmail.com).

F. Gao is with Tsinghua National Laboratory for Information Sci-ence and Technology, Tsinghua University, Beijing 100084, China (e-mail:feifeigao@ieee.org).

C. Xing is with the School of Information and Electronics, Beijing Instituteof Technology, Beijing, 100081, China (e-mail: chengwenxing@ieee.org).

Digital Object Identifier 10.1109/LCOMM.2012.121912.122144

Fig. 1. A typical network with K source-destination pairs and L AF relays.

constrained CRBF method for a single source-destination pairrequires the presence of a large number of relays. To the bestof the authors’ knowledge, CRBF design for multiple peer-to-peer relay networks with stochastic knowledge of error inchannel gain has not been studied yet.

In this letter, channel estimation errors in the secondhop are modeled as Gaussian random variables with knowndistribution. Robust beamforming vectors are designed tominimize the average transmit power at relays subject to QoSconstraints, expressed as probabilistic constraints in terms ofsignal-to-interference-plus-noise ratio (SINR). With the aid ofBernstein-type inequality and the standard rank relaxation, thenon-convex robust design problem is transformed into convexSDP. From S-procedure [6], an alternative reformulation thatprovides an upper bound to the optimal objective is alsoproposed. Numerical simulations are performed to evaluatethe proposed studies.

Notation: Uppercase and lowercase bold letters denotematrices and vectors, respectively; | · | is the absolute value,‖·‖F is the Frobenius norm, and Tr(·) is the trace of a matrix;(·)ii is the ith diagonal element of the matrix while Diag(x)returns a diagonal matrix from the elements of x; �(·) denotesthe real part; � represents the positive semidefinite condition; ∼ means distributed as and Pr(·) denotes the probability.

II. PROBLEM FORMULATION

A. System Model

Consider a network wherein K source(S)–destination(D)pairs communicate through a set of L distributed relays {Ri},as shown in Fig. 1. We assume that each node in the networkhas a single antenna, and the direct link between each (Sl, Dl)pair is absent. Denote source-l and destination-l by Sl and Dl,respectively. The received signal in the first time slot at Ri is

yri =

K∑l=1

gilsl + ni, (1)

1089-7798/13$31.00 c© 2013 IEEE

306 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 2, FEBRUARY 2013

where sl is the data symbol transmitted from Sl, gil isthe Rayleigh flat fading channel from Sl to Ri, and ni ∼CN (0, σ2) is the zero mean circularly symmetric Gaussiannoise at each Ri with variance σ2. Without loss of generality,we assume that E{sl} = 0, E{|sl|2} = pl, and different sl’sare independent from each other. Due to the distributed nature,each Ri multiplies the received signal by a scalar weight wi

and then forwards it to the destination.The signal received at Dk in the second time slot is

ydk=

L∑i=1

hkiwi

(K∑l=1

gilsl + ni

)+ zk (2a)

=

L∑i=1

hkiwigiksk+

L∑i=1

hkiwi

K∑l=1,l �=k

gilsl

︸ ︷︷ ︸interference

+

L∑i=1

hkiwini+zk,︸ ︷︷ ︸noise

(2b)

where hki is the Rayleigh flat-fading channel from Ri to Dk,and zk ∼ CN (0, σ2) is the noise at Dk. Note that indepen-dence among channels {gil} and {hki} is also assumed. TheSINR at Dk can be expressed as

SINRk=

|L∑

i=1

hkiwigik|2pkK∑

l=1,l �=k

|L∑

i=1

hkiwigil|2pl+∑L

i=1 |hkiwi|2σ2+σ2

. (3)

Note that the noise at relays acts as self-interference ateach Dk, which is different from the downlink beamformingcase. The average total power transmitted at relays is

P =

L∑i=1

|wi|2(

K∑l=1

|gil|2pl + σ2

). (4)

Our objective is to minimize the transmit power at relayssubject to QoS constraints expressed in terms of SINR at eachdestination.

The optimization problem with perfect CSI can be mathe-matically formulated as

min{wi}

L∑i=1

|wi|2(

K∑l=1

|gil|2pl + σ2

)(5a)

s.t. SINRk ≥ γk, k = 1, . . . ,K. (5b)

where γk is the predefined threshold. The solution to (5)has been developed in [2] by applying semidefinite relaxationtechnique.

Since channels {gil} can be directly estimated at each Ri,and knowledge of channels {hki} at {Ri} depends upon thefeedback from {Dl}, the level of uncertainty is much higher inthe {Ri} → {Dl} links. To account for this imperfection, thechannel knowledge available at each relay is modeled as hk =hk + δk, where hk = [hk1, . . . , hkL]

H , hk is the estimate ofhk, and δk ∼ CN (0,Qk) is the corresponding error. Notethat Qk is the covariance of δk.

For convenience, let us introduce variables in vector nota-tion,

Gk =√pkDiag([g1k, . . . , gLk]

T ),

w = [w1, . . . , wL]T , W = wwH , vk = Gkw,

Z = σ2Diag[(wwH)11, . . . , (wwH)LL], ∀k = 1,. . . ,K. (6)

From (6), we know (3) is equivalent to

SINRk =|hH

k vk|2K∑

l=1,l �=k

|hHk vl|2 + hH

k Zhk + σ2

. (7)

B. Robust Optimization Problem Under Imperfect CSI

When the statistics of channel errors are known, QoS can beguaranteed in a probabilistic sense. The non-outage probabilityat Dk is mathematically defined as

Pr(

SINRk ≥ γk

)≥ 1− ρk, (8)

where γk is the SINR threshold and ρk is the outage proba-bility at Dk. Hence, the robust optimization problem can beformulated as

min{wi}

L∑i=1

|wi|2(

K∑l=1

|gil|2pl + σ2

)(9)

s.t. constraint (8), ∀ k = 1, . . . ,K.

III. ROBUST OPTIMIZATION

A. Conservative Reformulation Utilizing Bernstein-Type In-equality

Since it is difficult to compute a closed-form expression forleft hand side (LHS) of (8), a lower bound is applied here.From (7), we know (8) can be rewritten as

Pr(hHk Akhk ≥ σ2

)≥ 1− ρk, (10)

where Ak = Vk

γk− ∑

l=1,l �=k

Vl − Z, Vk = vkvHk .

Moreover, the CSI error can also be expressed as

δk = Q1/2k ek, (11)

where ek ∼ CN (0, IL×L) is the standard circularly symmetricGaussian random vector. Utilizing (11), (10) can be written as

Pr(eHk Akek + 2�(eHk pk) + ck ≥ σ2

)≥ 1− ρk, (12)

where

Ak=Q1/2k AkQ

1/2k , pk=Q

1/2k Akhk, ck = hH

k Akhk. (13)

Note that we have expressed the non-outage probability at Dk

in terms of the standard Gaussian vector in (12).Lemma 1: [8]: Let G = vHPv + 2�(vHu) where P ∈

SN is a Hermitian matrix, u ∈ CN , and v ∼ CN (0, I). Thenfor any ζ ≥ 0, we have

Pr

(G≥Tr(P)−

√2ζ√|P‖2F+2‖u‖2−ζs−(P)

)≥1−e−ζ, (14)

where s−(P) � max{αmax(−P), 0} in which αmax denotesthe maximum eigenvalue, and ‖ · ‖F is the Frobenius norm.

PONUKUMATI et al.: ROBUST PEER-TO-PEER RELAY BEAMFORMING: A PROBABILISTIC APPROACH 307

Introducing auxiliary variable ζk, and applying (14) in (12),if ζk ≥ 0 and if

σ2−ck≤Tr(Ak)−√2ζk(‖Ak‖2F+2‖pk‖2)−ζks

−(Ak),

(15)

we obtain 1−e−ζk ≥ 1−ρk. By introducing auxiliary variablesxk and yk, the constraint (15) is equivalently written as

xk + ζkyk ≤ Tr(Ak) + ck − σ2, (16a)√2ζk

√‖Ak‖2F+2‖pk‖2 ≤ xk, (16b)

ykIL×L +Ak � 0, (16c)

where IL×L is an L × L identity matrix. Note that wehave replaced the probabilistic constraint (8) by conservativedeterministic constraints in (16).

From (4), the average power transmitted from all relays canalso be expressed as

P = Tr

(W(

K∑k=1

GkGHk + σ2IL×L)

). (17)

The conservative formulation of the robust optimizationproblem (9) is then

min{W},ζk

Tr

(W(

K∑k=1

GkGHk + σ2IL×L)

)(18a)

s.t. 1− e−ζk ≥ 1− ρk, (18b)

constraints (16a) − (16c), (18c)

Rank(W) = 1, ∀k = 1, . . . ,K.

From (18b), the smallest ζk that minimizes the transmit powerin (18a) is ζk = −ln(ρk). Although the objective function andconstraints of (18) are linear in W, the rank constraint is stillnon-convex.

By removing the rank constraint, we can convert (18) intoa convex SDP problem that can be numerically solved withinterior point algorithms. This widely adopted technique ispopularly known as rank relaxation [9]. Subsequently, theconservative reformulation is

min{W}

Tr

(W(

K∑k=1

GkGHk + σ2IL×L)

)(19)

s.t. constraints (16a) − (16c), ∀k = 1, . . . ,K.

Formulation (19) has a linear objective and constraints thatare in the form of linear matrix inequalities (LMI). In casethe obtained solution W may have a general rank, a standardway [9] is to utilize randomization to approximate W by arank-1 matrix. In some cases, when W is directly rank-1, thenits principal eigenvector is the optimal solution to (19).

B. Conservative Reformulation Utilizing S-Procedure

In this section, following the approach in [10], we obtain alower bound to the LHS of (12) in closed-form and numeri-cally solve the conservative reformulation. Let

eHk Akek + 2�(eHk pk) + hHk Akhk ≥ σ2, (20)

−4 −3 −2 −1 0 1 2 3 4 5−5

0

5

10

15

20

25

Target SINR in dB

Po

we

r re

qu

ire

d in

dB

Proposed method−I, ρ = 0.1

Proposed method−II, ρ = 0.1

Proposed method−I, ρ = 0.05

Proposed method−II, ρ = 0.05Perfect CSI

Fig. 2. Performance of the proposed method with ε = 0.01, L = 8.

if ek belongs to the set

S = {ek|Pr(eHk ek ≤ r2k

)≥ 1− ρk} (21)

where rk ≥ 0 is the radius of an L dimensional hyper-sphere. Note that ‖ek‖2 is a Chi-square random variablewith 2L degrees of freedom, zero mean and variance 1/2.From (21), the lower bound of rk for which (21) is always

true is rk =

√CDF−1(1 − ρk)

2, where CDF−1(x) is the

inverse cumulative distribution function of Chi-square randomvariable at x. From (20) and (21), the inequalities

r2k − eHk ek ≥ 0 (22)

eHk Akek + 2�(eHk pk) + hHk Akhk − σ2 ≥ 0 (23)

hold true with probability greater than 1 − ρk. Note that wehave replaced the probabilistic constraint of SINRk with twodeterministic constraints (22) and (23).

Applying S-procedure [6], the uncertainty variable ek canbe eliminated from (22) and (23), resulting in an equivalentLMI [

Ak + λkIL×L pk

pHk ck

]≥ 0, (24)

where ck = hHk Akhk−λk r

2k−σ2, and λk ≥ 0 is an auxiliary

random variable.After rank relaxation, the alternative conservative reformu-

lation of (9) is

min{W}

Tr

(W(

K∑k=1

GkGHk + σ2IL×L)

)(25)

s.t. constraints (24), ∀k = 1, . . . ,K.

IV. NUMERICAL RESULTS

In this section, we numerically examine the performance ofour proposed robust design. In all examples, the parametersare set as K = 3 and σ2 = 0.1. All channel estimates

308 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 2, FEBRUARY 2013

−4 −3 −2 −1 0 1 2 3 4 5−5

0

5

10

15

20

25

Target SINR in dB

Po

we

r re

qu

ire

d in

dB

Proposed method−I, ρ = 0.1

Proposed method−II, ρ = 0.1

Proposed method−I, ρ = 0.05

Proposed method−II, ρ = 0.05 Perfect CSI

Fig. 3. Performance of the proposed method with ε = 0.04, L = 8.

are assumed unit norm, i.e., ‖hk‖2 = 1, and pk = 1,∀ k = 1, . . . ,K . The channel covariance matrix is modeled asQ

k= ε2I, ε is the variance of error. The outage probabilities

and target SINR of each MU are set to be equal, i.e. ρi = ρ,γi = γ, ∀i = 1, . . . ,K . In our simulations, we compare theperformance of the proposed methods with the perfect CSIcase. The reformulation utilizing Bernstein-type inequality isreferred to as Method-I and the one with S-procedure isreferred to as Method-II. Interestingly, the solutions of Vk

obtained in all numerical simulations are rank-1 matricesthemselves. Hence, the obtained solutions offer tight boundsto the optimal objective of (9).

In Fig. 2 and Fig. 3, we plot the minimum power requiredversus the target SINR with L = 8 for different values of ρ.It is observed that for each ρ, there exists a critical SINRbeyond which the optimization problem becomes infeasible.For instance in Fig. 2, 4 dB is the critical SINR for ε =0.01, and ρ = 0.1. Since the minimum power required withε = 0.01, and ρ = 0.1 is around 2 dB more than the perfectCSI case for a majority of feasible SINR values, the proposedMethod-I provides tight bounds for relatively small values ofε, that are reasonable in a practical scenario. Moreover, theminimum power required is sensitive to the outage probabilityas the target SINR approaches the critical SINR. In Fig. 3, it isnoticed that the critical SINR shifts to the left with an increasein ε for all values of ρ.

In Fig. 4, the minimum power required versus the targetSINR was plotted for L = 6, and ε = 0.04. As expected, fora given γ and ρ, the power required increases with a decreasein L. Since the minimum power required with Method-I isless than that of Method-II for all values of target SINR, itis numerically demonstrated that the former offers a tighterbound. Moreover, Method-I could satisfy higher target SINRthan Method-II due to the fact that replacing probabilisticconstraint in SINR by a hyper-spherical bound on the CSIerror is a very coarse approximation.

−4 −3 −2 −1 0 1 2−5

0

5

10

15

20

25

Target SINR in dB

Pow

er

required in d

B

Proposed method−I, ρ = 0.1

Proposed method−II, ρ = 0.1

Proposed method−I, ρ = 0.05

Proposed method−II, ρ = 0.05Perfect CSI

Fig. 4. Performance of the proposed method with ε = 0.04, L = 6.

V. CONCLUSIONS

In this work, we designed robust distributed relay beam-formers with imperfect CSI at relays for a multiuser peer-to-peer network. The channel error was modeled as a Gaussianrandom variable and QoS constraints were expressed in termsof outage probability of SINR. The optimization criterion wasto minimize the transmit power at relays subject to proba-bilistic SINR constraints at each destination. Two conservativereformulations to the original non-convex problem were de-veloped and numerically solved using SDP. Simulation resultsdemonstrate that the method with Bernstein-type inequalityobtains tighter bounds.

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