iet comm sep2011 duy

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Published in IET Communications

Received on 23rd June 2010

Revised on 29th March 2011

doi: 10.1049/iet-com.2010.0518

ISSN 1751-8628

Power allocation in wireless multiuser multi-relaynetworks with distributed beamformingD.H.N. Nguyen1 H.H. Nguyen2

1Department of Electrical and Computer Engineering, McGill University, 3480 University Street, Montreal, QC,

Canada H3A 2A72Department of Electrical and Computer Engineering, University of Saskatchewan, 57 Campus Drive, Saskatoon, SK,

Canada S7N 5A9

E-mail: [email protected]

Abstract:This article studies optimal power allocation schemes in a multi-relay cooperating network employing amplify-and-forward (AF) protocol with multiple sourcedestination pairs. It is assumed that full channel state information is available atthe relays. As such, distributed beamforming is employed in forwarding signals to the destinations. In this context, theauthors extend recent works on distributed beamforming for a single sourcedestination pair to the scenario where multiplesourcedestination pairs are competing for the power resource at the relays. Under orthogonal transmissions of each sourcedestination pair, considered are the two following power allocation problems: (i) minimise the sum relay power withguaranteed quality of service (QoS) in terms of signal-to-noise ratio (SNR) at the destinations, and (ii) jointly maximise theSNR margin at the destinations subject to individual power constraints at the relays. Although these optimisation problemscan be formulated as second-order conic programs (SOCP), the main contribution of this work are proposals of simple andfast converging numerical algorithms, based on the fixed point iteration framework, to efficiently solve these two problems.

1 Introduction

The last few years have witnessed a rapid expansion ofwireless communications with a growing number of

broadband wireless devices being deployed. In addition, theemergence of the so-called Internet of Things wouldconsiderably aggregate the number of wireless networkeddevices. This trend is expected to put a significant pressureto the current and future wireless network infrastructure tocope with demands for higher throughput, higher robustnessand better coverage. On the other hand, the resources thatare crucial for wireless communications, namely power and

bandwidth, are strictly limited. As a result, meeting thesedemands would certainly pose many technical challenges tonext generation wireless networks.

It is well known that communications over wirelesschannels usually suffer from poor coverage and lowrobustness because of the random nature of the wirelessmedium, which is rich of scattering and susceptible tofadings. Recently, the concept of cooperativecommunication [1] has been shown as a promisingtechnique that can significantly improve the capacity,reliability and coverage of wireless networks. By deployinga network of cooperating users (objects), one can allowsome users to cooperatively act as relays and assist a source

user in sending its information symbols to a destination.Since the relays cooperatively form a virtual array oftransmit antennas, and thereby providing diversitytransmission to the source signals, it widely known that

such cooperation can significantly improve the transmissionreliability [1]. When these relays know both the backward,that is, source-to-relay (S R) and forward, that is, relay-to-destination (R D) channels, they can beam theirretransmitted signals such that the received signals at thedestination are coherently constructed. This cooperativestrategy, referred to as distributed beamforming, wasinvestigated in [26]. In particular, Jing and Jafarkhani [2]considered the problem of controlling the power resource ateach relay in order to maximise the signal-to-noise ratio(SNR) at the destination. It shows that, depending on itsown bidirectional channels and other relays channels, eachrelay may not transmit at its maximum power to achieve themaximum SNR. Jing and Jafarkhani [2] also provide thecondition to determine the optimal transmit power at eachrelay. The same problem is also considered in [4] by thetechnique of conic programming. Khajehnouri and Sayed[3] studied a distributed relay strategy for wireless sensornetworks to obtain a certain target SNR at the destination,whereas Queket al. [5] investigated a similar problem withthe objective of minimising the sum of relay powers,referred to as sum relay power hereafter. More recently,distributed beamforming with second-order statistics of thechannel state information was examined in [6].

It is noted that most of the early works in distributed

beamforming consider the system with one source and onedestination, where all the power consumed at the relays isdevoted to only one sourcedestination pair (each sourcedestination pair is called a user hereafter). Multiuser

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multi-relay systems were first investigated in[7, 8], where therelay strategies were proposed to minimise the mean-squareerror between the source and received signals at thedestination. In addition, such systems allow the relays toshare their received signals from multiple sources, and thusrequire reliable links between the relays. In this work, thoseadditional links are not needed as the relays do not sharetheir received signals.

Cooperative multiuser beamforming in wireless ad hocnetworks was considered in [9], where a cluster of sourcescooperatively form beamformers towards multipledestinations. In particular, Li and Wang [9] studied theoptimal power allocation and beamforming weights at thesources, as well as provided an efficient iterative algorithmto jointly optimise them. On the other hand, distributed

beamforming in a multiuser system was studied in[10 12],where multiple pairs of source and destination weresimultaneously assisted by multiple relays on the samechannel, that is, over the same frequency band and at thesame time. The optimisation problem was formulated as anon-convex quadratically constrained quadratic program.Through convex relaxation technique, such a problem can

be efficiently solved by semi-definite programming [10] orby second-order conic programming (SOCP)[11]. Althoughthe approach in [10] is appealing in terms of spectralefficiency (which allow a higher network throughput thanthat with the orthogonal transmission), the received signalsat the destinations suffer from a very high level ofinterference. This is because the relays do not share theirreceived signals and cannot cooperatively suppress theinterference accumulated in the source relay transmissionstage, which then propagates to the destination. To obtaina high signal-to-noise-plus-interference target at thedestinations, numerical simulation shows that the systemusually requires many relays (order of tens) for a better

chance of selecting good relays, which can reduceinterference in the forwarding stage[12]. However, deployingmany relays is costly in practice and might not be suitable incertain applications. When there is only a small number ofrelays in the system, it is more preferable to employorthogonal transmissions for each sourcedestination pair tocompletely avoid the inter-user interference at thedestinations. At the other extreme, the multiuser multi-relaysystem model in[13]assumes orthogonality for each relaydestination transmission such that the maximal-ratiocombining is possible at the destination. Such an approach is,however, very spectrally inefficient.

In this work, we consider orthogonal transmissions for eachsourcedestination pair in addition to distributed

beamforming at the relays. This orthogonality considerationreadily applies in practical systems where several single-antenna source destination pairs operate in orthogonalchannels. Examples of such systems are time-divisionmultiple access (TDMA) and orthogonal frequency-divisionmultiple access (OFDMA) systems. In order to improve thecoverage and signal reception at the destinations, one coulddeploy multiple relays to concurrently assist thetransmission between each pair. Although the pairs areoperating in their own channels, they are actuallycompeting for the coupled power constraint at the relays.Consequently, efficient power utilisation for this network ofcooperating objects is critical for both maintaining the

source destination connections and prolonging the powerresources at the relays.

The focus of this paper is to study the optimal powerallocation schemes at the relays in a multiuser network

employing distributed beamforming with the amplify-and-forward (AF) protocol. By means of conic programming,we investigate two main power allocation schemes: (i)minimise the sum relay power with guaranteed quality ofservice (QoS) in terms of SNR at the destinations, and (ii)

jointly maximise the SNR margin at the destinations subjectto power constraints at the relays. These problems aresequentially investigated and shown to be closely related

with each other.Under scheme (i), considered are optimisation problems

with and without per-relay power constraints. As the twooptimisation problems are shown to be second-order conic

programs, they can be solved effectively by any conicsoftware package. However, as the required conic packageis not always readily available, the approach may not besuitable in real-time communications. To overcome thisdifficulty, this paper considers an alternative approach byapplying the fixed point iteration framework to the relaynetwork, and proposing two simple and fast numericalalgorithms to solve the two problems directly. In addition,the proposed algorithms can be implemented in adistributed manner, which allows a decentralised operationin practical networks.

Under scheme (ii), we study two optimal power allocationproblems corresponding to two different types of powerconstraints: sum relay power constraint and per-relay powerconstraints. Although the two problems can be effectivelysolved by the bisection method via SOCP feasibility

problem, we also propose two simple and fast convergingalgorithms to directly solve the two problems without theneed of a standard conic solution package.

Notations: Superscripts ()T, (.), and ()H stand fortranspose, complex conjugate and complex conjugatetranspose operations, respectively; diag(d1, d2, . . . , dM)

denotes an M M diagonal matrix with diagonal elementsd1, d2, . . . , dM; tr(.) denotes the trace of a square matrix; x

w

indicates the optimal value of the variable x; CN(0,s2)denotes a circularly symmetric Gaussian random variablewith variance s2.

2 System model

Consider an R-relay network with N pairs of sourcedestination users (SnDn, n 1, . . . , N), as illustrated inFig. 1. (This system model is also applicable to a one-source one-destination OFDM system or a multiple-sourcemulti-destination OFDMA system where N is interpreted asthe number of subcarriers.) All relays are assumed to workin a half-duplex mode, that is, they cannot receive and

Fig. 1 Block diagram of a distributed beamforming system with

R relays and N users

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transmit at the same time. Assume that there is no direct linkbetween any source and destination and the communicationbetween the two terminals of each user is assisted by all therelays, and implemented in two transmission stages. In thefirst stage, each users source broadcasts its signals to allthe relays, using TDMA or frequency-division multipleaccess FDMA. For the nth user, given sn as the sourcesignal, the received signals at the relays are given by

rn = fnsn +zrn [ CR1

(1)

where fn = [fn,1, . . . ,fn,R]T, and fn,i is the channel from the

nth source to the ith relay; zrn represents the additive whiteGaussian noise (AWGN) at the relays, whose componentsare i.i.d. CN(0,s2R) random variables.

At the ith relay, the received signal for the nth user isamplified by a complex beamforming weight wn,i, which is

to be designed. Letwn = [wn,1, . . . ,wn,R]T be the vector of

the beamforming weights for the nth user. Also defineWn diag(wn). Accordingly, by applying the AF protocol[1], the retransmitted signals from the relays scheduled for

the nth user are formed as

tn= Wnrn = Wn fnsn +Wnzrn (2)

In the second stage of transmission, all the relayssimultaneously transmit to the nth users destination.Similar to the first stage, the transmission to each usersdestination is carried out over orthogonal channels to avoidinter-user interference. Let gn= [g1,n, . . . ,gR,n]

T representthe channels from R relays to the nth destination. Thereceived signal at the nth destination is written as

yn= gTntn +zdn = g

Tn Wn fnsn +g

Tn Wnzrn +zdn (3)

where zdn CN(0,s2

D) is the AWGN at the destination.Define hn = [h

n,1, . . . ,h

n,R]

T = fngn= [fn,1g1,n, . . . ,fn,RgR,n]

T, where represents the component-wiseHadamard product. As a result, hn models the effectivechannel from source-n to destination-n through all therelays, excluding the beamforming factors. Then, one hasgTn Wnfn = h

Hn wn. Let s

2Sn= E[|sn|

2] be the average

transmit power of the nth source. Then, the SNR at the nthdestination is given by

SNRn =s2Sn |h

Hn wn|

2

s2

R

G1/2

n

wn

2 + s2

D

(4)

where Gn diag(|g1,n|2 , . . . , |gR,n|

2).Letpn be the total relay power allocated for the nth user,

then pnis calculated as

pn= E[tn2

] = wHn Dnwn (5)

where Dnis an R R diagonal matrix, with the ith diagonalelement [Dn]ii = s

2Sn|fn,i|

2 + s2R. On the other hand, thetransmit power at the ith relay is

Pi =N

n=1

E[|tn,i|2

] =N

n=1

wHn DnEiwn (6)

whereEiis anR Rmatrix whose elements are zero, exceptthe (i,i)-element, which is [Ei]ii 1. The total transmit power

of all the relays (and for all the users) is given by

Prelay=Ri=1

Pi =Nn=1

pn=Nn=1

wHn Dnwn (7)

In this proposed system model, it is assumed that the sourcesand the relays do not share a common power pool. This

assumption arises from practical implementation that thesources are separate entities and usually do not co-locatewith the relays. In addition, each source has its own powerlimit and also does not share its power to other sources.Our interests in this work are to investigate how relaysallocate the power between themselves to optimise differentsystem utilities, namely (i) power minimisation withguaranteed QoS or (ii) joint SNR maximisation with powerconstraints at the relays.

3 Optimal power allocation with guaranteedQoS

3.1 Sum power minimisation without per-relaypower constraints

This section considers the optimal power allocation at therelays to minimise the sum relay power given a set of targetSNRs at the destinations. This design provides a relayingstrategy that can flexibly meet the QoS requirement at eachusers destination. The optimisation problem is formulatedas follows

minimisew1,...,wN

Nn=1

wHnDnwn

subject to SNRn

gn

, n

(8)

where gnis the target SNR at the nth destination. Obviously,this optimisation problem can be performed separatelythrough Nsmaller optimisation problems, each correspondsto one user. That is

minimisewn

wHn Dnwn

subject tos2Sn |h

Hn wn|

2

s2RG1/2n wn

2 + s2D gn

(9)

There are several approaches to solve the above optimisation

problem. The first approach, first proposed in [6], i s t oestablish wn in the direction ofD

1/2n xn, where xn is the the

principal eigenvector of D1/2

n (s2Snhnh

Hn gns

2RGn)D

1/2n .

Another approach is to cast the SNR constraint as a second-order conic constraint [5]

s2Sngn

h

Hn wn

sRG1/2n wnsD

(10)and then solve the optimisation problem as an SOCP. Thesolution to the problem can be obtained from any standardconic solution package, such as cvx [14]. We now consider

an alternative approach to solve problem (9) by optimisingthe relay power factor pn directly, instead of dealing withthe beamforming vector. The new approach, which does notrely on external conic solution software, also motivates a



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simple iterative fixed point algorithm. Besides, we note thatthe proposed algorithm is useful in solving the sum relay

power minimisation with per-relay power constraints inSection 3.2 and the inverse problems that maximise theSNR margin in Section 4.

First, problem (9) can also be recast as

minimisewn,pn

pn

subject tos2Sn |h

Hnwn|

2

s2RG1/2n wn

2 + s2D gn

wHn Dnwn = pn

(11)

Second, the following lemma establishes the relation betweenthe optimal beamforming vectorwn and the allocated relay

powerpn.

Lemma 1: Given pn as the relay power allocated for user-n,the optimal beamforming weights at the relays to maximisethe SNR of user-nare

wn,i =

dn

fwn,ig

w

i,n

pns2

R|gi,n|2 + s2D(s

2Sn|fn,i|

2 + s2R)(12)

where the normalisation factordnis

dn = pn

SRi=1(|fn,i|

2|gi,n|2(s2Sn |fn,i|

2 + s2R)/[pns2

R|gi,n|2

+ s2D(s2Sn|fn,i|

2 + s2R)]2)

(13)

The corresponding maximum SNR is

SNRn(pn) =Ri=1

pns2Sn|fn,i|

2|gi,n|2

pns2

R|gi,n|2 + s2D(s

2Sn|fn,i|

2 + s2R)(14)

Proof: The derivation of the optimal beamforming weightvectorwn is followed from the RayleighRitz theorem [15]

and Proposition 1 of [16]. Substituting s2D = (s2

DwHn

Dnwn/pn) into the SNR expression in (4) gives

SNRn=s2Snw

Hn hnh

Hn wn

wHn(s2

RGn + (s2

D/pn)Dn)wn

= pns2

Snw

H

n hnh

H

nwnwHn(pns

2RGn + s

2DDn)wn

pns2Snlmax

where lmax is the largest eigenvalue ofB1/2n hnh

Hn(B

Hn )

1/2,

with Bn= pns2

RGn + s2

DDn. The equality holds if

wn /B1n hn. We note that this result is consistent with the

work in [17] on the generalised Rayleigh Ritz theorem.More specifically, (13) of [17] states that wn is a scaledversion of the principal eigenvector ofB1n hnh

Hn , which is

indeedB1n hn.We proceed to provide the closed-form expression of each

beamforming weight in (12), and the normalisation factordnto ensure wHn Dnwn= pn. It is also noted that the largest

eigenvalue of B1/2n hnh

Hn(B

Hn )

1/2

is its only non-zeroeigenvalue, which is also its trace. Thus, the obtainedoptimal SNR value can be found in a closed-formexpression as stated in (14). A

By applying Lemma 1, one can optimise the powerallocation pn for user-n, then determine the optimal

beamforming vector accordingly. For notational simplicity, let

an,i =s2Sns2R

|fn,i|2, bn,i =

s2D(s2Sn|fn,i|

2 + s2R)

s2R|gi,n|2

Then, the achievable SNR can be written as

SNRn(pn) =Ri=1

an,i pn

bn,i +pn(15)

Interestingly, SNRn(pn) is a concave increasing function inpn.The optimisation problem is now restated as

minimisepn

pn

subject to

R

i=1

an,ipn

bn,i +pn gn

(16)

The above problem is convex, which then can be solvedefficiently by standard convex optimisation algorithms. Inaddition, the structure of the restated problem also revealsseveral interesting properties of the problem, including itsfeasibility and solution. Since pn/(bn,i+pn) , 1, one has

SNRn =Ri=1

an,i pn

bn,i +pn,

Ri=1

an,i

Thus, if the target SNRgn SRi=1an,i, the problem will be

infeasible.

Now, suppose that the target SNR is set such that theproblem is feasible. Since SRi=1(an,ipn/bn,i +pn) is a

monotonically increasing function, the constraintS

Ri=1(an,i pn/bn,i +pn) gn must be met with equality at

optimum. Thus, the unique solution of

Ri=1

an,i pn

bn,i +pn= gn (17)

is also the optimal solution to (16). It is then of interest tofind a simple and fast numerical algorithm to solve the

Rth polynomial in (17). The monotonicity of S

Ri=1(an,ipn/bn,i +pn) makes the bisection method

especially suitable to find the solution. Note that thestructure in (17) also motivates a simple iterative fixed pointalgorithm to find the optimal p

w

n . By rearranging (17), onehas the following simple iteration

p(t+1)n =

gn

SRi=1(an,i/bn,i +p

(t)n )

(18)

If (17) is feasible, then the above iteration will converge fromany initial point p(0)n 0. The convergence analysis of thefixed point iteration is based on the standard functionapproach introduced in [18]. Denote fn(p

(t)n ) =

gn/(SRi=1(an,i/bn,i +p

(t)n )), then the fixed point iteration

p(t+1)n = fn(p

(t)n ) will converge to a unique fixed point p

w

n ifthe function fn(pn) obeys the following properties[18]:

1. Positivity:fn(pn) . 0 for all pn . 0.



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2. Monotonicity: ifpn . pn , then fn(pn) . fn(pn).3. Scalability: ifa . 1, thenafn(pn) . fn(apn).

It is easy to verify that all these three properties are satisfiedby the functionfn(pn). Thus, the fixed point iteration (18) willsurely converge if (17) is feasible. Numerical results showthat the proposed algorithm converges in a few iterations.

3.2 Sum power minimisation with per-relay powerconstraints

In the previous section, sum relay power minimisation withguaranteed QoS at the destinations was considered, whereno restrictions on the individual power at each relay wereimposed. However, in practical relay communications, therelays are not normally co-located. In addition, each relay isequipped with its own amplifier and typically has its own

power limit. Under the per relay power constraints, thepower allocation scheme has to be modified accordinglywhile meeting the SNR requirement at each users receivingend. This section considers the approach to uniformly

minimise the margin Pi/P

max

i over all the relays, wherePmaxi denotes the maximum transmit power of the ith relay.

The approach of minimising the power consumption marginwas first investigated for the multiuser beamformingdownlink problem in point-to-point communications [19].When applied to relay networks, the idea is to serve all theusers, while maintaining the balance in power consumptionat the relays. The problem is stated as follows

minimisea,w1 ,...,wN

a

subject to SNRn gn, n

Pi aPmaxi , i

a 1

(19)

Here, the last two constraints are to ensure the powerconsumption at each relay not to exceed its maximumallowable amount. As presented before, the SNR constraintsare feasible ifgn S

Ri=1an,i, n. However, the feasibility of

the power constraints in (19) is much more difficult todetermine analytically. Thus, we drop the last constrainta 1 from problem (19) and consider the followingoptimisation

minimise

a,w1,...

,wN

aR

i=1

Pmaxi

subject tos2Sn |h

Hnwn|

2

s2RG1/2n wn

2 + s2D gn, n

Nn=1

wHn DnEiwn aP

maxi , i

(20)

which can be interpreted as a sum relay power minimisationproblem with per-relay power constraint awareness. Theresultant optimal aw from problem (20) then numericallydetermines the feasibility of problem (19). Note that thefeasibility of problem (20) only depends on the feasibility

of the SNR constraints, since a can be increased to meetthe per-relay power constraints. However, it might happenthataw . 1 at the optimal solution, that is, it is infeasibleto find the beamforming vectors that meet both the QoS

constraints and the strict per-relay power constraints. Tohandle this situation, an inverse problem, which tries tomaximise the SNR under strict per-relay power constraints

Pi Pmaxi , is desirable. We investigate such an inverse

problem in Section 4.It is also noted that the optimisation problem stated in (20)

is not readily convex. However, as the SNR constraints can berecast as an SOC constraint as in (10), the problem can be

transformed into a convex one, namely

minimisea,w1 ,...,wN

aRi=1

Pmaxn

subject to

s

2Sn

gn

h

Hnwn

sRG1/2n wn

sD

, n

Nn=1

wHn DnEiwn aP

maxi , i

(21)

Interestingly, the Lagrangians of the non-convex form in (20)and the convex form in (21) are the same. This property can

be easily verified by following the similar technique to that inProposition 1 of [19]. As strong duality holds for a convex

problem [20], strong duality also holds for problem (20).This means that the optimal value of problem (20) can befound by its dual problem. In the following, we investigatethe dual problem of (20), which then reveals both thestructure of the original problems solution and thealgorithm to solve it.

3.2.1 Beamforming duality:The Lagrangian of problem

(20) is established as

L(a,wn, l, m)

= aRi=1

Pmaxi +Ri=1

mi

Nn=1

wHn DnEiwn aP

maxi

Nn=1

lns2Sngn|hHn wn|

2 s2RG1/2n wn

2 s2D

(22)

Denote Q diag(m1, . . . ,mR)andP= diag(Pmax1 , . . . ,P

maxR ).

Rearrange the Lagrangian L(a, wn

,l, m) in (22), one has

L(a, wn, l, Q) =Nn=1

lns2

D +Nn=1

Ln(wn,ln, Q)

a[tr(QP) tr(P)] (23)

where Ln(wn,ln, Q) = wHn(DnQ (lns

2Sn

/gn)hnhHn +

lns2

RGn)wn only depends onwn,lnandQ. The dual functionof (23) is established as

g(Q, l) =Nn=1

lns2

D +Nn=1

minwn Ln(wn,ln,Q)

mina

{a[tr(QP) tr(P)]}



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It is clear that if

DnQlns

2Sn

gnhnh

Hn + lns

2RGn

is not a positive semidefinite matrix, there existswn to makeLnunbounded below. Similarly, if tr(QP)tr(P) . 0, it is

possible to find a . 0 to make a[tr(QP)2 tr(P)] 21.Thus, the dual problem is stated as

maximiseQ

maxl

Nn=1

lns2

D

subject to DnQ+ lns2

RGn Xlns

2Sn

gnhnh

Hn , n

tr(QP) tr(P), Q is diagonal, Q X 0

(24)

Since strong duality holds for problem (20), we have

awRi=1

Pmaxi =

Nn=1

lwns2

D (25)

In the next section, an interpretation via a virtual single-inputmultiple-output (SIMO) uplink channel shows that the dual

problem (24) is equivalent to the following minimax problem

maximiseQ

minl,wn

Nn=1

lns2

D

subject to lns2

Sn |hH

n wn|2

wHnDnQ wn + lns2

RwHn Gnwn

gn, n

tr(QP) tr(P), Q is diagonal, Q X 0

(26)

where wHn is interpreted as the receive beamforming vector of

the virtual uplink channel for user-n.

3.2.2 Interpretation via a virtual uplink channel: Inpoint-to-point multiuser communications, it is widelyknown that the optimal beamforming design for thedownlink multiple-input multiple-output (MIMO) channelcan be found via its equivalent uplink channel, which ismuch easier to handle. This property is known as uplinkdownlink duality [19, 2123]. Inspired by the uplinkdownlink duality property of the MIMO channel, here weintroduce the concept of a virtual uplink channel, and usesit to find the optimal power allocation scheme in a multi-relay network with per-relay power constraints.

Consider a virtual SIMO uplink channel where a single-antenna transmitter with power pn wants to communicatewith an R-antenna receiver. The channel is modelled assSnhn [ C

1R. The effective additive Gaussian noise at the

receiver has the following covariance: s2DDnQ+ pns2

RGn.One can interprets2DDnQ as the added noise at the receiverand pns

2R

Gn as the noise induced by the transmitter, which

depends on the transmit power pn. Then, it is of interest tofind the optimal combiner at the receiver and the minimaltransmit power pn at the transmitter to obtain a certaintarget SNRgnat the virtual uplink channels receiving end.

LetwHn be the receive beamforming vector. The SNR at thereceiver can be expressed as

SNRn = pns2Sn |hHn wn|2s2DwHn DnQ wn + pns2RwHn Gn wn (27)Similar to the proof of Lemma 1, to maximise the above SNR,

using the RayleighRitz theorem [15], the optimal receivebeamformer is

wn= (s2

DDnQ+ pns2

RGn)1hn (28)

Given a specific value of the transmit power pn, the weight ofthe optimal combiner at the ith receive antenna only dependson the channel connected to itself, and is given by

wn,i = fwn,ig

w

i,n

(s2Sn|fn,i|

2 + s2R)s2

Dmn + pns2

R|gi,n|2

(29)

With the optimal combiner, the constraint on the SNR at the

receiver,SNRn gn, is now equivalent topns

2SnhHn (s

2DDnQ+ pns

2RGn)

1hn gn (30)

The next task is to determine the minimal uplink transmitpower pn. This problem is stated as

minimisepn

pn

subject to pns2SnhHn(s

2DDnQ+ pns

2RGn)

1hn gn

(31)

If the above inequality is reversed, the optimisation problem

can also be reversed into a maximisation problem as follows

maximisepn

pn

subject to pns2SnhHn(s

2DDnQ+ pns

2RGn)

1hn gn

(32)

It should be noted that the reversals of the constraint and theminimisation into a maximisation do not effect the solution of(31), as the two constraints in (31) and in (32) are met withequality at optimality.

Observe that the constraint in (32) is equivalent to

s2DDnQ+ pns2RGn Xpns

2S

n

gnhnhHn

(from Lemma 1 in [19]). Furthermore, identify pn = lns2

D,then the minimax problem in (26) must be equivalent to thedual problem (24). This allows us to solve the powerallocation problem with per-relay power constraints bysolving the minimax problem (26).

3.2.3 Numerical algorithm: The minimax problem (26)can be solved by iteratively solving N inner minimisation

problems on (wn,ln) and the outer maximisation problemon Q. For the outer problem, we compute the maximisation

maximiseQ

f(Q)

subject to tr(QP) tr(P), Q is diagonal, Q X 0

(33)



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where f(Q) is defined as

f(Q) = minimisel,wn

Nn=1

lns2

D

subject tolns

2Sn|hHn wn|

2

wHn DnQ wn + lns2

RwHn Gnwn

gn, n

For the inner problems, with a fixedQ the optimal combinerwn for the user-n is given by (28), and the optimal powerfactor ln is obtained by solving problem (31). Note that

problem (31) is equivalent to

minimisepn

pn

subject toRi=1

pnan,i

bn,imi + pn gn

(34)

Thus, the optimal value pwn can be obtained from the simplefixed point iteration, as presented in Section 3.1. Moreover,with ln= pn/s

2D, the fixed point iteration

l(t+1)n = gn

s2DR

i1(an,i/bn,imi + s2

Dl(t)n )

(35)

will surely converge to the optimal value lwn .Having known the optimal combiner of the virtual uplink

channel wn and its power factor lns2

D, the optimaldistributed beamformerwn for the user-n can be determined

by exploiting the relation between the two beamformers inthe next lemma.

Lemma 2: The optimal distributed beamforming vectorwninthe multiuser beamforming problem is a scaled version ofwn,that is, wn =

zn

wn.

Proof: From the KarushKuhnTucker conditions [20], thegradient of Lagrangian Ln(Q,ln,wn) vanishes at theoptimum ofwn, that is

Lnwn

= DnQlns

2Sn

gnhnh

Hn + lns

2RGn

wn = 0

Thus

wn = (DnQ+ lns2

RGn)1 lns2Sn

gnhnh

Hn wn

=lns

2Ds

2SnhHn wn

gnwn

which suggestszn

= (lns

2Ds

2Sn

/gn)hHnwn. However, this

expression still shows the dependence of zn on wn. Thenext step is to determine the value zn based on wn. As theSNR constraint in (20) is met with equality at optimum,that is, (s2Sn/gn)|h

Hn wn|

2 = s2RwHn Gnwn + s

2D. Substituting

wn =

zn

wn into the SNR constraint, one has

(zns2Sn

/gn)|hHn wn|

2 = zns2

RwHn Gn wn + s

2D Therefore

zn = gns

2D

s2Sn|hHn wn|

2 gns2

RwHn Gn wn

(36)

We now return to the maximisation problem off(Q) in (33).This problem can be computed by the subgradient projectionmethod, as presented next.

Lemma 3: The function f(Q) is concave in Q, and itssubgradient is given by diag(S

Nn=1wnw

Hn Dn), where wn is the

optimal distributed beamforming vector obtained fromLemma 2.

Proof: The proof of this lemma is similar to the proof ofProposition 3 in [19] for the point-to-point multiuserdownlink beamforming problem. Since f(Q) is the objectivefunction of the dual problem, it is a concave function bynature [20]. Now look back at the Lagrangian of thedistributed beamforming problem in (22). For a fixedQ,onehas

f(Q) = minw1 ,...,wn

minaL(a, wn, l,Q)

= minw1 ,...,wn

Nn=1

wHn DnQwn

subject tos2Sngn|hHn wn|

2 s2RG1/2n wn

2 + s2D

(37)

Following the same procedure in Proposition 3 of[19]one canrealise that diag(S

Nn=1wnw

Hn Dn) is the subgradient off(Q). In

particular, the subgradient of mi [Q]ii at the ith relay isS

Nn=1|wn,i|

2[Dn]ii. Compared to the result in Proposition 3 of[19] the difference here is the inclusion of Dn in thesubgradient, a direct consequence of the appearance ofDn inthe objective function of (37). A

Having derived the subgradient off(Q), Q is then updatedby applying the Euclidean projection PSQ of the subgradient

of f(Q) on the constraint set SQ = {Q: tr(QP) tr(P),Q X 0}, that is

Q(t+1) = PSQ Q

(t) + atdiagNn=1

wnwHn Dn

(38)

where at is an appropriate step size. This subgradientprojection method is guaranteed to converge to the globaloptimum of f(Q) [20]. We now summarise the iterativealgorithm to solve the distributed beamforming problemwith per-relay constraints with the property of distributedimplementation as follows:

1. Initialise Q(t). Sett 1.2. Repeat: fix Q

(t), then the relays transmit Q(t) to every

destination. Each destination then solves the fixed pointiteration in (35) to determine the required power lns

2D for

its corresponding virtual uplink channel. The optimalreceive beamformer wn and the scaling factor zn are thendetermined by the nth destination.3. Thenth destination broadcasts lnandznback to relays. Theith relay calculates the beamforming coefficients w1i,w2i, . . . ,wNiwith local information pertaining to the relay as

wn,i = zn

fw

n,igw

i,n

s2

D[(s2

Sn |fn,i|2

+ s2

R)mn + lns2

R|gi,n|2

]

4. The relays cooperates with each other to updateQ(t) as in (38).



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5. Sett t+ 1 and return to Step 2 until convergence.

It is worth nothing that the two proposed algorithms forpower minimisation (with and without per-relay powerconstraints) rely on the simple fixed-point iteration. Thisiteration is generally very fast in convergence, and also easyto implement in practical systems, instead of utilising anexternal optimisation software package.

4 Joint SNR-margin-maximised powerallocation

4.1 Sum relay power constraint

This section considers the power allocation scheme to jointlymaximise the minimal SNR margin subject to a sum relay

power constraint. Under such a constraint, the relays areallowed to share a common power pool, although they donot necessarily share their received signals from thesources. The problem is stated as

maximisew1,...,wN minn

SNRn

gn

subject to Prelay Pmaxrelay

(39)

Here, the sum relay power constraint is a firm systemrestriction. Even so, the problem is always feasible as it isalways possible to scale wn down to meet the sum relay

power constraint. The parameter gn is interpreted as theweight for user-ns SNR. By introducing the auxiliaryvariable t, denoted as the SNR margin, the problem can bereformulated as

maximiset,w1 ,...,wN

t

subject tos

2Sn|hHn wn|

2

s2RG1/2n wn

2 + s2D tgn, n

Nn=1

wHnDnwn P

maxrelay

(40)

Note that sincetis a variable, the SNR constraint is no longerconvex[4]. As a result, problem (40) is not a convex problem.However, for a fixed value of t, the problem can beformulated as a convex feasibility problem and is readilysolved by the bisection method[20]. This is presented next.

4.1.1 Bisection method: Define tw as the maximumattained value of t. For a specific target value of t, thefollowing SOCP feasibility problem is considered

find w1, . . . ,wN

subject to

s2Sntgn

h

Hn wn

sRG1/2n wn

sD

, n

Nn=1

wHnDnwn P

maxrelay

(41)

If the problem is feasible, which means thatt, tw

, then it ispossible to increase the target margint. Otherwise, ift, t

w,

the target margin should be reduced. The bisection method[20]is summarised as follows:

1. Initialise u andlas the upper and lower bounds oft.2. Repeat: t (u+ l)/2. Solve the feasibility problem (41).3. If (41) is feasible, then setl t, else, setu t.4. Return to Step 2 until u 2 l, 1,

where 1 is a small positive value.

4.1.2 Convex solution:In the previous section, an SOCP

approach using bisection method is presented to solve thejoint SNR margin maximisation problem. However, such anapproach is computationally expensive and unappealing,since it requires many iterations in the bisection method aswell as a standard conic solution package. Now, we makeuse of Lemma 1 to cast problem (40) as a convexoptimisation problem with t, p1, . . . , pNas the variables

S(Pmaxrelay) =

maximisep1 ,...,pN,t

t

subject to tgn Ri=1

an,i pn

bn,i +pn 0, n

N

n=1pn

P

max

relay

(42)

The above problem can be solved by any standard convexoptimisation algorithm. On the other hand, the connection

between the power minimisation with guaranteed QoSproblem presented in Section 3.1 and the joint SNR marginmaximisation S(Pmaxrelay) in (42) can be exploited to directlysolve (42). This is described next.

4.1.3 Modified fixed point iteration for findingpwn:We now establish the connection between powerminimisation problem Pn(gn) in (16) and the joint SNR

margin maximisation problem S(Pmaxrelay) in the following

lemma.

Lemma 4: The joint SNR margin maximisation problem (42)and the power minimisation problem in (16) are inverse

problems

t= SNn=1

Pn(tgn)

(43)

Pmaxrelay=Nn=1

Pn(gnS(Pmaxrelay)) (44)

Proof: This lemma is proved by contradiction and by themonotonicity of the function S

Ri=1(an,i pn/bn,i +pn).

Beginning with (43), suppose that pw

n is the optimal valueand also the optimal argument of Pn(tgn), then

SRi=1(an,i p

w

n/bn,i +pw

n) = tgn. Also, let tw

and pw

n, n bethe optimal value and arguments ofS(S

Nn=1p

w

n ). If tw, t,

there is a contradiction that pw

n s are also feasible solution

forS(SNn=1p

w

n ), and yet provide a higher objective value t.

On the other hand, if tw , t, there is a contradiction thatpwn . p

w

n to make twgn . tgn, then the constraint

SNn=1 p

w

n SNn=1p

w

n in S(SNn=1p

w

n ) cannot be true. The prooffor (44) follows the same line. A

Using the results in Lemma 4, problem S(Pmaxrelay) in (42) canbe solved by iteratively solving N problems Pn(tgn) fordifferent values of t until S

Nn1Pn(tgn) = P

maxrelay. Then the



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optimal arguments pwn of Pn(twgn) are also optimal to

S(Pmaxrelay). Therefore pw

n must also satisfy the fixed pointiteration (18) with gn replaced by t

wgn. Unfortunately, tw

needs to be determined as well. However, the condition onoptimality S

Nn=1p

w

n = Pmaxrelay allows a modified fixed point

iteration to overcome this difficulty as follows. Let

pn =

gn

SRi=1(an,i/bn,i +p(t)n ) (45)

Then normalise the result such that the sum relay power isequal to the maximum allowable power

p(t+1)n =

Pmaxrelay

SNl=1 pl

(46)

The iteration gets back to (45) until convergence. Theconvergence analysis of the proposed algorithm is similar tothose in Section 3.1, based on the standard functionapproach proposed in [18]. Numerous simulations show a

rapid convergence rate of the modified fixed point iteration.

4.2 Per-relay power constraints

With the same arguments as in Section 3.2, it might bedesirable to have the power constraint at each relay. In thissection, with such strict constraints imposed, we examinethe optimal power allocation scheme to jointly maximisethe joint SNR margin at the destinations. This problem isstated as

maximisew1,...,wN

minn

SNRngn

subject to Pi Pmaxi , i

(47)

By introducing the auxiliary variable t, this problem isrestated as

maximiset,w1 ,...,wN

t

subject tos2Sn |h

Hn wn|

2

s2RG1/2n wn

2 + s2D tgn, n

N

n=1

wHnDnEiwn P

maxi , i

(48)

4.2.1 Bisection method: Similar to the optimisationproblem (40), problem (47) is not convex. However, with afixed value oft, the problem can be formulated as a convexfeasibility problem. Thus, this problem can be solved by the

bisection method [20]. Define tw as the maximum attainedvalue of t. For a specific value of t, the following convexSOCP feasibility problem is considered:

find w1, . . . , wN

subject to

s2Sntgn

h

Hn wn

sRG1/2n wn

sD

, n

Nn=1

wHnDnEiwn P

maxi , i

(49)

If the problem is feasible, then t, tw

. Otherwise t. tw

.The bisection method is the same as the one in Section4.1.1. It should be pointed out that instead of solving thefeasibility problem in (49), one can solve the total powerminimisation problem under per-relay power constraints(20) withtg1, . . . ,tgNas the target SNR at the destinations:

minimisea,w1 ,...,wN aR

i=1P

maxi

subject to

s2Sntgn

h

Hn wn

sRG1/2n wn

sD

, n

Nn=1

wHn DnEiwn aP

maxi , i

(50)

The resultant optimal aw could be used to determine thefeasibility of the distributed beamforming problem with thetarget SNR margin t. More specifically, if the optimalaw . 1, it means that at least one of the per-relay power

constraints is violated, that is, it is infeasible to meet the SNRtargets tg1, . . . , tgN without compromising the per-relay

power constraints. Thus, the target margin t needs to beadjusted to a smaller value. Conversely, ifa

w, 1, one can

scale up the beamforming vectorwn to improve the targetSNR margin t without violating the per-relay powerconstraints. This suggests that at the optimal SNR margintarget tw, aw 1. Thus, the aforementioned bisectionmethod can be modified to solve problem (50) with differenttarget SNR margin until a

w 1. The algorithm proposed in

Section 3.2 can be readily used to quickly solve (50).

4.2.2 Iterative algorithm for finding wnw: In this

section, by adapting the algorithm outlined in Section 3.2, anovel iterative algorithm is proposed to directly solve thejoint SNR margin maximisation problem (47). It should beemphasised that unlike the iterative algorithm in finding theoptimal distributed beamforming design with a fixed targetSNR at each destination, the SNR at the nth destinationtgnis now a variable. More specifically, t is an optimisationvariable, not a parameter, and has to be determined as well.At first, we revisit the optimisation problem (50). Note thatstrong duality holds for the optimisation problem (50), thatis, awS

Ri=1P

maxi = S

Nn=1l

w

n s2

D. Furthermore, the bisectionmethod in Section 4.2.1 states that at the optimal value tw,aw 1, which also means that S

Nn=1l

w

ns2

D = SNn=1P

maxi at

optimum. This can be met by adjusting the fixed point

iteration in (35). In the algorithm presented here to jointlymaximise the SNR margin with per-relay power constraints,the modified fixed point iteration is taken at Step 2. Thealgorithm is as follows:

1. Initialise Q(t). Sett 1.2. Repeat: fix Q

(t), solve the fixed point ln by iterativefunction

ln = gn

s2DSRi=1(an,i/bn,imi + s

2Dln)

(51)

then normalise the result

ln=lnS

Ri=1P

maxi

s2DSNl=1

ll(52)



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so that SNn=1lns

2D = S

Ri=1P

maxi , then return to (51) until

convergence.3. Find the optimal receive beamformers of the virtualchannels as

wn = (s2

DDnQ(t) + lns

2Ds

2RGn)

1hn (53)

4. Determine the achievable SNR of the virtual uplinkchannel for each user

gn =lns

2Sn|wHn hn|

2

wHn DnQ(t)wn + lns

2Rw

HnGnwn

(54)

5. Update the distributed downlink beamformerswn =

zn

wn, where

zn = gns

2D

s2Sn|hHn wn|

2 gns2

RwHn Gn wn

(55)

6. UpdateQ(t) using subgradient projection method with step

size a t:

Q(t+1) = PSQ Q

(t) + atdiagNn=1

wnwHn Dn

(56)

7. Sett t+ 1 and return to Step 2 until convergence.

Although the above algorithm takes several steps to find to

the optimal solution, the calculation at each step is quitesimple and devised specifically for this problem. As aresult, the complexity of this approach is fairly low withshort running time. In fact, numerous simulations withrandom channel realisations show a much fasterconvergence time of the proposed algorithm as compared tothe bisection method in Section 4.2.1.

5 Numerical results

This section presents the numerical results on the relay powerconsumptions to maintain guaranteed QoS at the destinationsand on the achievable SNR margin when strict power

constraints are imposed at the relays. Also presented are theconvergence plots of the proposed iterative algorithms. Thenetwork being considered is equipped with four relays.The number of users to be served by the network is 3. Thesource power is set at 10 for all the users sources in all the

simulations and the noise variances s2R ands2

D are set tounity. Flat Rayleigh fading is assumed in all the channels,where each S R and R D channel coefficients isassumed to be i.i.d. CN(0, 1). When the per-relay powerconstraints are imposed, the maximum per-relay power isset at 10. The target SNRgnis set at 5 for all the destinations.

5.1 Relay power consumptions with guaranteedQoS at destinations

Fig. 2illustrates the power consumptions at the relays for 50different channel realisations. At each channel realisation, thesum relay power, and the highest relay power level of the fourrelays are plotted and compared between the two relayingschemes: with and without per-relay power constraints. Ascan be seen from the figure, imposing the per-relay powerconstraints does increase the sum relay power, comparedwith the optimal scheme that does not impose theconstraints. However, the chief advantage of applying the

per-relay power constraints is that it balances the powerconsumption at the relays and does not overuse any of

them. Consequently, the highest relay power level of thefour relays with the per-relay power constraints is alwayssmaller than that without the constraints.

The convergence of the proposed algorithms is illustratedin Figs. 3 and 4. We utilise the following randomisedchannel realisation to generate both figures (see (57))

Fig. 3plots the evolution of the sum relay power allocatedfor user-1,p1and the corresponding SNR1after each iteration

Fig. 2 Power consumptions at the relays over 50 channel

realisations with different power constraints: with per-relay power

constraints (solid lines), without per-relay power constraints

(dashdot lines)

[f1,f2, f3] =

1.384+ 0.060i 0.805 0.227i 0.449 0.870i

0.356 1.417i 0.149+ 0.874i 0.425+ 0.746i

1.318 0.348i 0.841 0.446i 0.389 0.080i

0.240+ 0.326i 0.789 1.644i 0.777+ 0.268i

[g1, g2, g3] =

0.667 + 0.415i 0.719 0.055i 0.867 0.008i

1.499 0.177i 0.128 + 0.628i 0.492+ 0.645i0.455 + 0.339i 1.075 + 1.632i 0.005+ 0.039i

0.498 + 0.472i 0.027 + 0.371i 0.553 0.782i

(57)



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by the iterative fixed point algorithm (18). It can be seen thatthe algorithm converges very quickly after only a fewiterations to the optimal p

w

1 from various arbitrary startingpoints, whereas the corresponding SNR also converges toits target value g1 5. Fig. 4 displays the convergence ofthe proposed iterative algorithm in Section 3.2.3 in findingthe optimal distributed beamformers w

w

n with per-relaypower constraints. The step-size at 1/t is used for thesubgradient update of the iterative algorithm. Thesummation S

Nn=1wn w

w

n , which is the norm residue ofthe beamformers, plotted after each iteration clearly showsthe convergence of the proposed algorithm. Numeroussimulations also show that the proposed algorithmconverges in a small fraction of the running time required

by the cvx package[14].

5.2 Achievable SNR margin

Fig. 5 illustrates the achievable SNR margin at thedestination, that is, minn SNRn/gn for 50 different channelrealisations. At each channel realisation, lowest SNR

margin, the highest relay power levels of the four relays areplotted and compared between two different constraints:sum relay power constraint and per-relay power constraints.As can be observed in the figure, imposing the per-relay

power constraints does not decrease achievable SNR marginmuch in most of the simulations, compared with theoptimal scheme that imposes the sum relay powerconstraint. To loosely explain this observation, it is noticedthat the objective function (the minimum SNR margin) israther flat near its optimality, such that a change in the

power allocation at each relay does not change the objectivefunction much. However, at the optimal solution, theallocated power may be highly inclined to one of the relays,which can be seen from the figure. Although the highesttransmit power at each relay is strictly under or equal to 10with the per-relay power constraints, it may reach to 25with the sum power constraint. This is a clear benefit ofimposing the per-relay power constraints in terms of powerconsumption at each relay.

Fig. 6 illustrates the convergence of the modified fixedpoint algorithm in Section 4.1.3. Plotted are the evolution

of the allocated relay power pn and the correspondingSNRn, n 1, 2, 3 for the three users after each iteration. Itcan be seen that the algorithm converges very quickly afteronly a few iterations, as the allocated relay power for each

Fig. 4 Convergence of the proposed algorithm in finding the

optimal distributed beamformers with per-relay power constraints

Fig. 3 Convergence of the iterative fixed point algorithm (18) with

different starting points and the achievable SNR at user-1s

destination after each iteration

Fig. 5 Achievable SNR margintand the power consumptions at

the relays over 50 channel realisations with different power

constraints: with per-relay power constraints (solid lines), and

without per-relay power constraints (dash dot lines)

Fig. 6 Convergence of the relay power for each user and the

corresponding achievable SNR at each users destination by the

modified iterative fixed point algorithm



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user converges to its optimal value. The corresponding SNRalso converges to the same optimal value, as all users SNRsare set at the same weight.

Finally, Fig. 7 displays the convergence of the proposediterative algorithm in Section 4.2.2 in finding the optimaldistributed beamformers w

w

n with per-relay powerconstraints to maximise the SNR margin. Again, thesummation S

Nn=1wn w

w

n plotted after each iterationclearly shows the convergence of the proposed algorithm. Itis noted that the same randomised channel realisation in(57) is also applied to obtain Figs. 6and7.

6 Conclusions

In this paper, we have studied the optimal power allocationschemes in a multiuser multi-relay network to eitherminimise the total relay power with guaranteed QoS at thedestinations or maximise the SNR margin subject to powerconstraints at the relays. By means of convex optimisationtechniques, it was shown that these problems can beformulated and solved via SOCP. Optimal solutions to thetwo problems can be obtained by any conic solution

package. In addition, by applying the fixed-point iterationframework to the relay network, we also proposed simpleand fast iterative algorithms to directly solve the twooptimisation problems. As the proposed algorithms workwithout the need of external software package, they can beeasily implemented and are more suitable in real-timecommunications.

7 Acknowledgments

The authors would like to thank the anonymous reviewer formany insightful and constructive comments, which helped toimprove the presentation and clarity of our paper. This workwas supported by an NSERC Discovery Grant.

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Fig. 7 Convergence of the proposed algorithm in finding the

optimal distributed beamformers with per-relay power constraints

to jointly maximise the SNR margin



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