Resource Allocation in Cooperative Relaying for Multicell OFDMA Systems_GAOQS

Resource Allocation in Cooperative Relaying

for Multicell OFDMA Systems

Abdelhalim Najjar1, Noureddine Hamdi2, and Ammar Bouallegue1

1 Communication Systems Lab, ENITManar University, Tunisia

2 Departement of Computer Science and Mathematics, INSATCarthage University, Tunisia

[email protected],[email protected],[email protected]

Abstract. In this paper, we propose an allocation algorithm that min-imizes the base station (BS) transmit power of broadcast link OFDMAcellular networks and satises the user rate requirements. By minimiz-ing the base station transmitted power, the co-channel cell interference(CCI) can be reduced eectively. To perform the mitigation of the CCI,a xed amplify and forward relay station (RS) that assists the BS tobroadcast data to mobiles in the cell edge region, has been adopted.Numerical results are used to show the improvement of the proposed op-timal cooperative scheme compared to the performance of similar schemewithout relays.

Keywords: Resource Allocation, Cooperative Relaying, OFDMA.

1 Introduction

Recently, intense interest has focused on OFDMA as an attractive multiple accesstechnique for broadband wireless applications. One of the biggest advantages ofOFDM systems is the ability to allocate power and rate optimally among subcar-riers. Several works have been proposed to perform ressource allocation such asin [1] where an ecient technique is based on a min-max formulation of the op-timization problem and is adopted to adhoc networks environment. The authorsof [2] and [3] investigate the issue of power control and subcarrier assignmentin a sectorized two-cell downlink OFDMA system impaired by co-channel cellinterference (CCI). Also, frequency hopping (FH) associated with OFDM mod-ulations oers the advantage of averaging interference as in CDMA system andenables us to adopt a frequency reuse factor (FRF) equal to 1 between co-channelcell in multicell context [4], [5]. Through FH and assuming that the channels arerandom frequency selective, the authors of [6] propose an allocation algorithmthat computes the optimum number of subcarrier per user that satises the raterequirement of all users and minimizes the total transmitted power.

In this paper, we perform the general problem for resource allocation pro-posed in [6] by introducing the cooperative relaying which was addressed in

C. Sacchi et al. (Eds.): MACOM 2011, LNCS 6886, pp. 107118, 2011.c Springer-Verlag Berlin Heidelberg 2011

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108 A. Najjar, N. Hamdi, and A. Bouallegue

our previous work [7]. For simplicity, we consider a sectored two-cell downlinkOFDMA system where a single RS by sector is placed at the edge region of eachcell. Downlink cooperation is triggered with the Non orthogonal Amplify andForward (NAF) cooperation protocol with one relay and two time slots [8]. Nu-merical results show that the proposed optimal cooperative scheme appreciablyoutperforms the non-cooperative scheme presented in [6].

2 System Model

In order to simplify our problem, the network is supposed to be one dimensionalcellular system that consists of a linear regular array of cells. We assume that agiven user is only subject to interference from the nearest interfering cell. Thuswe focus on two interfering sectors of two adjacent cells, say cell A and cell B asshown in Fig.1. We denote by D the radius of each cell . A single RS by sectoris placed at the edge region of each cell. We denote by N the total number ofusers in the cell, by W the channel bandwidth and by M the number of availablesubcarriers. We denote by n, the sharing factor associated with user n. Thus,the number of subcarriers modulated by user n is equal to nM .

Fig. 1. Two-cell system model

3 Resource Allocation

3.1 Single Cell Context

We denote by Rn the rate requirement for a given user n. For any modulate sub-carrier m of the OFDM symbol k, we denote by en = E(|sn(k,m)|2) the energytransmitted for a given user n. Similarly we dene pn = nen the average powertransmitted to user n. Denoting by PA =

Nn=1 pn the average power spent by

base station A during one OFDM block. Our aim is then the following: given arate vector R = [R1, ..., RN ]T , nd the power {pn}n{1,...,N} such that the totaltransmitted power PA is minimum. Furthermore, as we shall study a multicellenvironment, it is legitimate to limit the interference produced by base station A.Consequently, the power PA should not exceed a certain nuisance value whichis assumed to be a predened constant imposed by systems requirements. Tobe able to reach the capacity, we approximate the multi-cell interference as a

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Resource Allocation in Cooperative Relaying 109

Gaussian random variable as shown in several studies [2]. The ergodic capacityin the whole bandwidth is given by

Cn(n, pn) = nE[log(1 + gnpnn

z)] (1)

where gn is the channel gain-to-noise ratio of user n. The single cell resourceallocation problem can be formulated as follow.Minimize PA with respect to {n, pn}n{1,...,N} under the following con-

straintsRn Cn, n = 1, ..., N. (2)

N

n=1

n = 1 (3)

N

n=1

pn (4)

Since the resource parameters of users in cell A are xed, it is straitforward toshow that the ergodic capacity Cn(n, pn) given by (1) is concave function ofn and pn (and hence Cn(n, pn) is convex). Thus the single cell resource allo-cation problem is convex in {n, pn}n{1,...,N} and can be solved using the La-grange Karush-kuhn-Tuker (KKT) conditions. At rst, we dene the LagrangianL associated with the proposed problem as

L(n, pn) = PA +N

n=1

n(Rn Cn) + (N

n=1

n 1) + (N

n=1

pn ) (5)

where n, n = 1, .., N are the Lagrange multipliers associated with constraints(2), the positives numbers and are the Lagrange multipliers associated withthe constraint (3) and (4) respectively. Considering the vector parameter X =[PTA ,

T ] where PA = [p1, ..., pN ]T and = [1, ..., N ]T The Lagrange(KKT)conditions are then written as

XL(n, pn) = 0 (6)

XPA N

n=1

nXCn + X(N

n=1

n) + X(N

n=1

pn) = 0 (7)

whereX denotes the gradient operator with respect to the vector x. The equa-tion(7) can be rewritten as the set of 2N equations given by L(n,pn)pn = 0 andL(n,pn)

n= 0 for n = 1, 2, ...N . The development of these equations leads to

E(z

1 + gn pnn z) =

1 + ngn

(8)

nE[log(1 + gnpnn

z) gnpnn

z

1 + gn pnn z] = (9)

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We dene the following function on R

F(x) = E[log(1 + xz)]E[ z1+xz ]

x (10)

using (8) and (10), (9) can be rewritten as

1 + gn

F(gnen) = (11)

thus en can be formulated as

en(, ) =1gnF1( gn

1 + ) (12)

where F1 dened in [0,+[ is the inverse of F with respect to composition.Dene for each x 0

L(x) = E[log(1 + F1(x)z)] (13)the sharing factors n, n = 1, ..., N , are given by

n(, ) =Rn

L( gn1+ )(14)

or from constraint (3) we have

N

n=1

n(, ) = 1 (15)

so and verify the following equation

N

n=1

Rn

L( gn1+ )= 1 (16)

The average power transmitted to user n is given by

pn(, ) = n(, )en(, ) (17)

=Rngn

F1( gn1+ )L( gn1+ )

Finally, the expression of PA can be formulated as

PA =N

n=1

Rngn

F1( gn1+ )L( gn1+ )

(18)

The solution to the single cell allocation problem is unique and can be solvedusing the following algorithm.

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Algorithm: 0.Repeat unique solution to (17).PA

Nn=1

Rngn

F1( gn1+ )L( gn1+ )

.

if PA > thenIncrease end ifuntilPA return ,

3.2 Asymptotic Regime

As shown in [6], the asymptotic regime is characterized by N, Wand NW where is a positive constant. The cell can be identied with acompact I = [,D] included in R+ since the cell is considered one dimensionalvariable. is a real number which can be chosen as small as needed. It is usefulto model gn as being directly related to the location xn by

gn =|xn|sN0

(19)

where |xn| is the distance between the mobile and base station and s is thepathloss component. We denote by t(x) = |x|

s

N0is a continuous function dened

on I to R+. In the following, we denote by rn = RnW the required rate for user nin nats/s. Without restriction, we assume that for each user n, rn [rmin, rmax]where rmax can be chosen as large as needed. The implementation of equation(18) needs two parameters of user n: Rn and gn which can be described by thecouple (rn, xn). Thus, the distribution of the set of couples {(rn, xn)}n{1,...,N}can be interpreted by the following measure (N) dened on the Borel sets ofR+ R+ as follows:

(N)(U, V ) =1N

N

n=1

rn,xn(U, V ) (20)

where U and V are any intervals of R+ and rn,xn is the Dirac measure at point(rn, xn). In order to have a clear idea on the meaning of this measure, (N)(U, V )is equal to /N where is the number of users located in V and requiring arate (in nats/s) in interval U . By replacing Rn by rnW in (18), we obtain:

P(N)A =

1W

N

n=1

rnF1( t(xn)(N)

1+(N))

t(xn)L(t(xn)(N)

1+(N))

(21)

=N

W

rmax

rmin

D

r

t(x)

F1( t(xn)(N)1+(N)

)

L( t(xn)(N)

1+(N))

d(N)(r, x)

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The new notations P (N)A and (N) are used to indicate the dependency of the

results on the number of users N . Intuitively, the asymptotic power per channeluse limNPNA can be obtained by replacing, N/W by ,

(N) by , (N) by and the distribution (N) by the asymptotic distribution of couples (rn, xn)as N . For more details on the convergence of measure, we can refer to [3].The convergence of P (N)A will come from the following assumption.

Assumption1: The measure (N) converges weakly to a measure as N.In order to further simplify the expression of the asymptotic power, it is realisticto assume that the limit joint distribution of rates distribution times a limitlocation distribution. This assumption come from the fact that in practice, therate requirement rn of a given user n is independent of its location xn.Assumption2: The measure satises d(r, x) = d(r)d(x) where is the

limit distribution of rates and is the limit distribution of the users locations.Here denotes the product of measures. Using these two assumptions, P (N)Aconverges to a constant PA dened by the following theoremTheorem1: Assume N in such a way that NW > 0 and that measure

N satises assumption 1 and assumption 2. Assume that t(x) is continuous andsatises t(x) > 0 on [,D]. The total power spent by the network P (N)A convergesto a constant PA given by the following form:

PA = r D

F1( t(x)1+ )t(x)L( t(x)1+ )

d(x) (22)

where r represents the average rate requirements per channel use in cell A givenby

r = rmax

rmin

rd(r) (23)

and (, ) is the unique solution of

r

D

d(x)

L( t(x)1+ )= 1 (24)

3.3 Multicell Context

For each c {A,B}, we denote by c the adjacent cell. We denote by P (i)c , thetotal required power per channel use transmitted by base station c at the ith

moment. The sequence{P

(i)c

}

{i0}is dened as follows:

P (0)c = 0 : For simplicity and without loss of generality, we initialize the totaltransmitted power to 0P (1)c = PA = PB : At the moment 1, we execute the allocation algorithm

and each BS will transmit a signal with the power PA .P (i)c = (P (i1)c , r) is the power delivered by BS c obtained by iterating

at moment i. (P (i1)c ), r) is the total power per channel use a cell c needs to

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transmit to reach the mean rate of r nats per channel use when the neighboringcell c transmits at power P (i1)c . It is given by the following expression:

(P (i1)c , r) = r D

F1(Tc(x)(P (i1)c ,r)1+(P

(i1)c ,r)

)

Tc(x)L(Tc(x)(P

(i1)c ,r)

1+(P(i1)c ,r)

)d(x) (25)

where ((P (i1)c , r), (P(i1)c , r)) is the unique solution to the equation

r

D

d(x)

L(Tc(x)(P(i1)c ,r)

1+(P(i1)c ,r)

)= 1 (26)

and

Tc(x) =|x|s

P(i1)c |2D x|s + N0

(27)

The convergence of the sequence

P (i)c = (P(i1)c , r) (28)

is treated by [6] and given by the following theorem

Theorem2: dene (r) in ]0,+[ as

(r) = r D

F1(T (x)(r)1+(r) )T (x)L(T (x)(r)1+(r) )

d(x) (29)

where T (x) is a continuous function that satises T (x) > 0 in I given by

T (x) =|x|s

|2D x|s + N0 (30)

and ((r), (r)) is the unique solution of:

r

D

d(x)

L(T (x)(r)1+(r) )= 1 (31)

For any value of P (0)c 0, there is a real rth > 0 unique solution of (r) = 1that satises

1.{P

(i)c

}

{i0}converge if r < rth.

2.{P

(i)c

}

{i0}diverge if r rth.

If we compare the expressions of the solutions given by equations (22) and (25),to the results of [6], we can see the presence of the term (1 + ) which doesnot exist in [6]. This is explained by the constraint (4) which is not taken intoaccount by [6].

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4 Cooperative Scheme with Fixed Relaying

4.1 Broadcasting Phase

The received signal at the mobile n directly from BS A and B during the broad-casting phase can be given by

r1,n(t) =

c{A,B}

L1

l=0

h(c)l sn(t v(c)l ) + Z1(t) (32)

h(c)l is the channel impulse of the l

th path within the cell c {A,B} and v(c)lis the corresponding time delay. Z1(t) is the noise component. The demodulatoroutput of the kth OFDM symbol at subcarrier m for user n is formulated by

R1,n(k,m) =

c{A,B}H

(c)1,n(k,m)s

(c)n,m + z1,m (33)

where H(c)(k,m)1,n , c {A,B} is the channel transfer function of the kth OFDMsymbol for mobile n at subcarrier m. In the following and without loss of gener-ality, we denote by H(c)(k,m),1,n as H

(c)1,n,m.

For each c {A,B}, the received signal at the relay within cell c is given by

y(c)1,r(t) =

L1

l=0

c(c)l s

(c)n (t (c)l,r ) + c

(c)l s

(c)n (t

(c)l,r ) + z

(c)r (t) (34)

where c(c)l is the channel impulse between BS c and the corresponding relay. c(c)l

is the channel impulse between BS c and the relay within the cell c. (c)l,r and

(c)l,r are respectively the times delay of the channel impulse c

(c)l and c

(c)l . In our

study, we consider a free space loss (FSL) model characterized by a path-lossexponent s = 2. The basic equation for path-loss in decibels are

FSL : PLdB(dkm) = 20log10(dkm) + 97.5 (35)

where dkm is the distance in kilometers between the serving BS and the mo-bile. The average channel gain between the serving BS A and the mobile n onsubcarrier m can be written as:

G(A)1,n,m = 10

PLdB(d(A)n /10)E(|H(A)1,n,m|2) (36)

The received SINR for mobile n on subcarrier m during the rst time slot canbe expressed as:

1,n,m =G

(A)1,n,mp

(A)n,m

G(B)1,n,mP

B + N0

(37)

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4.2 Relaying Phase

Each relay amplies and forwards its received signal while all the BS are silent.Consequently, the received signal at the mobile n from the relays during therelaying phase can be developed as

r2,n(t) =

c{A,B}

L1

l=0

cq(c)l y1,r(t (c)l ) + Z2(t) (38)

=

c{A,B}

L1

l=0

cq(c)l [c

(c)l s

(c)n (t (c)l,r (c)l )

+ c(c)l s

(c)n (t

(c)l,r (c)l ) + z(c)r (t (c)l )] + Z2(t)

(39)

where q(c)l is the channel impulse of the lth between BS c and the mobile n. (c)l

is the corresponding time delay. c is the amplication factor used at the RSwithin cell c and given by the following expression

(c) =1

L1l=0 |c(c)l |2 + N0EsTs

, (40)

Es denotes the average energy per transmitted symbol and Ts is the symbolduration. The demodulated signal sample of the kth OFDM symbol at subcarrierm for user n can formulated by:

R2,n(k,m) = H(A)2,n,msA,n,m + H

(B)2,n,ms2,n,m + z2,m (41)

whereH

(A)2,n,m =

(A)Q(A)n,mC(A)r,m (42)

H(B)2,n,m =

(A)Q(A)n,mC(B)r,m +

(B)Q(B)n,mC(B)r,m (43)

For c {A,B}, C(c)r,m denotes the channel transfer function between BS c andits RS on subcarrier m. Q(c)n,m is the channel transfer function between the relayof BS c and mobile n on subcarrier m. The channel gain between the relay ofcell (c) and mobile n on subcarrier m can be expressed as

G(c)2,n,m = 10

PLdB(d(c)r,n/10)E(|H(c)2,n,m|2) (44)d(c)r,n denotes the distance between mobile n and the relay of cell (c). During the

second time slot, the SINR for mobile n on subcarrier m is given by the followingformula

2,n,m =G

(A)2,n,mp

(A)n,m

G(B)2,n,mP

B + N0

(45)

The received SINR at the mobile n on subcarrier m in broadcasting phase andrelaying phase are maximum ratio combined (MRC) as follows

n,m = 1,n,m + 2,n,m (46)

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5 Scheduling

To enhance the performance of the proposed scheme, the proportional fair schedul-ing (PFS) algorithm is considered in this paper. In mathematical terms, the PFscheduling decision at the time slot t may be expressed as .

ns = argmaxn

(an,m(t)an(t)

) (47)

where an,m(t) is given by

an,m(t) = log2(1 + n,m(t)) (48)

an is an estimate average rate for user n in a past window of tc slots, and it isupdated each time slot according to:

an(t + 1) = (1 1tc

)an(t), n = ns (49)

ans(t + 1) = (11tc

)ans(t) +ans,m(t)

tc(50)

6 Numerical Results

In Fig.2, we compute the total power PA W required by base station A to reach amean rate r = 2Bits/Sec/Hz and r = 1Bits/Sec/Hz per channel use, versus thecell radius. Asymptotic approximations provided by (22), (23), (25) and (26) areconsidered. We can deduce from theses curves that the required power increaseswith the cell radius and the average rate. Also, the required power in multicellenvironment is greater than the single cell context. This increase is caused bymulticell interference that disturbs each base station. Fig.3 illustrates the re-ceived SINR versus the distance between the serving BS (A) and the consideredmobile n for the proposed scheme and the scheme proposed in [6]. As shown inthis gure, the proposed cooperative scheme gives better performance than thenon-cooperative scheme [6]. The obtained gain compared to [6] is about 10dB

Table 1. Simulation parameters

Parameters Values

channel bandwidth 5 MHz

Carrier frequency 1.8 GHz

Number of subcarriers 300

White noise power density -174 dBm/Hz

Relay transmit power 33dBm (2W)

Minimum mobile to BS distance 100m

Distance between BS and Relay 800m

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0 500 1000 1500

104

103

102

101

100

101

D (m)

P W

Multicell context, ( 2 bits/Sec/Hz )Single cell context, ( 2 bits/Sec/Hz )Multicell context, ( 1 bits/Sec/Hz )Single cell context, ( 1 bits/Sec/Hz )

Fig. 2. Required power versus cell radius D

0 500 1000 150020

10

0

10

20

30

40

dn(m)

SINR

(dB)

OptNon Coop [6]OptCoop

Fig. 3. The received SINR versus the distance between user and the BS

0 10 20 30 40 501.2

1.4

1.6

1.8

2

2.2

2.4

User index

Aver

age

Spec

tral E

fficie

ncy

(Bit/S

ec/H

z)

OptCoopOptNon coop [6]

Fig. 4. The average spectral eciency versus the number of users with tc = 100 slots

in the cell edge. In Fig.4, we evaluate the performance of the proposed optimalcooperative scheme in terms of average spectral eciency versus the number ofusers in the cell A. As seen in this gure, the proposed optimal scheme with xedrelaying performs much better than the optimal scheme without cooperation. Wenotice a performance improvement of about O.7 bits/Sec/Hz in comparison withscheme presented in [6].

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7 Conclusion

In this paper, we have investigated the resource allocation problem in cooperativerelaying for sectorized downlink OFDMA system. In multicell context, asymp-totic algorithm can provides a tractable expression of the minimal power whichdepends only on the global rate requirement. Minimizing the total transmittedpower assisted by xed relays is a powerful solution for interference avoidancein the edge of the cell.

References

1. Wiczanowski, M., Stanczak, S., Boche, H.: Providing quadratic convergence of de-centralized power control in wireless networks-The method of min-max functions.IEEE Trans. Signal Process. 56(8), 40534068 (2008)

2. Ksairi, N., Bianchi, P., Ciblat, P., Hachem, W.: Resource allocation for downlinksectorized cellular OFDMA systems: Part I-Optimal allocation. IEEE Trans. SignalProcess. 58(2), 720734 (2010)

3. Ksairi, N., Bianchi, P., Ciblat, P., Hachem, W.: Resource Allocation for DownlinkCellular OFDMA Systems Part II: Practical Algorithms and Optimal Reuse Factor.IEEE Trans. Signal Process. 58(2), 720734 (2010)

4. Stamatiou, K., Proakis, J.: A performance analysis of coded frequency-HoppedOFDMA. In: Proc. IEEE Wireless Commun. Networking Conf., pp. 11321137(2005)

5. Laroia, R., Uppala, S., Li, J.: Designing a mobile broadband wireless access network.IEEE Signal Process Mag., 2028 (September 2004)

6. Gault, S., Hachem, W., Ciblat, P.: Performance Analysis of an OFDMA Trans-mission System in a Multicell Environment. IEEE Transaction on Communica-tions 55(4) (April 2007)

7. Najjar, A., Hamdi, N., Bouallegue, A.: Fractional Frequency Reuse Scheme in Coop-erative Relaying For Multi-cell OFDMA Systems. In: Vinel, A., Bellalta, B., Sacchi,C., Lyakhov, A., Telek, M., Oliver, M. (eds.) MACOM 2010. LNCS, vol. 6235,pp. 199210. Springer, Heidelberg (2010)

8. Nabar, R.U., Bolcskei, H., Kneubuhler, F.W.: Fading relay channels: performancelimites and space-time signal design. IEEE Journals on Sel. Areas in Telecom-mun. 22(6), 10991109 (2004)

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Resource Allocation in Cooperative Relaying for Multicell OFDMA SystemsIntroductionSystem ModelResource AllocationSingle Cell ContextAsymptotic RegimeMulticell Context

Cooperative Scheme with Fixed RelayingBroadcasting PhaseRelaying Phase

SchedulingNumerical ResultsConclusionReferences

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Resource Allocation in Cooperative Relaying for Multicell OFDMA Systems_GAOQS