Optimal One-Dimensional Relay Placement inCognitive Radio Networks

Junhua Zhu and Jianwei HuangDepartment of Information Engineering

The Chinese University of Hong Kong, Hong KongEmail: {jhzhu, jwhuang}@ie.cuhk.edu.hk

Wei ZhangSchool of Electrical Engineering and Telecommunications

The University of New South Wales, Sydney, AustraliaEmail: wzhang@ee.unsw.edu.au

Abstract—In this work, we find the optimal relay location fora secondary user in a cognitive radio network subject to outageconstraints. With selection decode-and-forward relay protocol,we formulate this problem as a nonlinear optimization problem,and then derive an approximation formulation that gives aclose-optimal relay location, which means that we can use theapproximation instead to solve the relay placement problem incognitive radio networks. We find that the optimal relay locationis determined by the relative locations of the primary destination,the secondary source and the secondary destination. Moreover,we analyze the sensitivity of optimal relay location to other keyparameters such as path loss exponent, interference floor andoutage constraint of the primary user, and outage constraint ofthe secondary user.

I. INTRODUCTION

Cognitive radio technology [1] enables dynamic spectrumaccess [2] by letting unlicensed secondary users opportunisti-cally exploit under-utilized spectrum. In cognitive radio net-works, a secondary user can access licensed spectrum occupiedby primary users if it does not interfere with primary users’transmissions. There are two methods to achieve this: spectrumoverlay and spectrum underlay. In the first case, the secondaryuser cannot access the channel in presence of primary users’activities; while in the latter case, the secondary user canaccess the channel as long as the induced interference doesnot exceed the primary users’ interference floor. We study therelay placement problem via spectrum underlay in this paper.

Cooperative communication is a promising technique toimprove the performance of wireless networks, e.g., cellularnetworks (e.g., [3], [4]) and ad hoc networks (e.g., [5], [6]).When the source node cannot reach the destination node due toa poor channel condition, the relay node can help to relay thesignal for the source node and thus increase network capacity.This leads to the cooperative diversity that saves the source’stransmit power.

Recently, cooperative communication techniques have beenintroduced in cognitive radio networks [7]–[14]. While pre-vious work focused on resource allocation for cooperativecommunications in cognitive radio networks, we study therelay placement optimization in cognitive radio networks. Forsimplicity, we consider the one-dimensional case where thesource and the destination of secondary user and the relay areon the same line. The relay adopts the selection decode-and-forward protocol [6]. With a Rayleigh block fading channel

model, we calculate the maximum transmission rate of thesecondary user subject to outage constraints. We formulatethis problem as a nonlinear optimization problem, and derivean approximation formulation. We find that the optimal relaylocation is determined by the relative locations of the primaryuser and the secondary user, and the solution obtained from theapproximation is close to the optimal solution. Surprisingly,we find out that the optimal relay placement is indepen-dent of several system parameters, including the interferenceconstraint at the primary destination, the outage constraintof the primary user, and the background noise density. Wealso characterize the sensitivities of the optimal relay locationin terms of path loss exponent and outage constraint of thesecondary user.

The rest of this paper is organized as follows. Section II de-scribes the network model. Section III formulates and analyzesthe optimal relay placement problem. We obtain the optimalsolutions for both the original problem and the approximationformulation. Section IV illustrates the results through extensivenumerical results. Finally, Section V concludes this paper.

II. SYSTEM MODEL

We consider a network scenario illustrated in Fig. 1. Wehave a primary user PU and a secondary user SU , eachhas a source and a destination. A relay r is placed on thesame line determined by the secondary source SUs and thesecondary destination SUd. Let dsr be the distance betweenthe secondary source SUs and the relay r, dsp be the dis-tance between the secondary source SUs and the primarydestination PUd, dsd be the distance between the secondarysource SUs and the secondary destination SUd, and angleθ = ∠PUd − SUs − SUd. Then the distance between therelay r and the primary destination PUd is

drp =√d2sp + d2

sr − 2dspdsr cos θ. (1)

We assume that the primary user source PUs alwaystransmits to the primary user destination PUd. The SU trans-mits through the cooperative transmission without generatingexcessive interference to the PU. We consider the selectiondecode-and-forward relay protocol with time division (TD)orthogonal sub-channels similar as [6]. With this protocol,the cooperative transmission operates in two phases. First thesecondary source SUs transmits the signal to the relay r,

978-1-4244-7555-1/10/$26.00 ©2010 IEEE

Secondary User(Destination) SUd

dsp

dsr

dsd

!

drp

Primary User(Destination) PUd

Primary User(Source)PUs

SecondaryUser (Source)

SUsRelay r

Fig. 1. The coexisting primary user and secondary user scenario.

and then the relay r decodes and retransmits the receivedsignal to the secondary destination SUd. If the relay r cannotdecode the signal in the first phase, the secondary sourceSUs will retransmit the signal in the second phase. Forsimplicity, the time allocation for source’s transmission andrelay’s transmission are equal. These two phases togetheroccupy a time block.

A. Performance Metric: Maximum Rate Subject to OutageProbabilities

We consider a Rayleigh block fading channel model, wherethe fading coefficients are constant over a block and inde-pendent over different blocks. A good communication perfor-mance metric for this block fading environment is maximumtransmission rate of the secondary user subject to outage con-straints. Outage probability here is defined as the probabilityof the instantaneous achievable rate of a channel below itstarget transmission rate.

In a cognitive radio network, the transmission of a primaryuser should be protected from the interference caused byeither a secondary user or a relay. In spectrum underlay, theprimary user has a maximum tolerable interference level (e.g.,interference floor) P0 at the primary destination PUd [15].When the interference from the secondary user in the firstphase or the relay in the second phase exceeds P0, the primaryuser’s transmission will be corrupted. Thus, the secondaryuser and the relay have to control their transmit power levelssuch that the outage probability of the primary user, i.e.,the probability of the interference exceeding P0, is below anoutage constraint εp. The maximum transmission power levelsdepend both on εp and the locations of the secondary sourceSUs and relay r.

From the secondary user’s point of view, it wants to maxi-mize the data rate at the destination SUd subject to the outageprobability εs. The outage in this case is due to the fact thatthe channel conditions between the secondary source SUs, therelay r, and the secondary destination SUd can be poor dueto Rayleigh fading.

For simplicity, we assume that the maximum transmit powerof the transceivers of the secondary user and the relay are largeenough such that their maximum allowable transmit power are

only determined by the outage constraint of the primary user.We also assume that background noise, denoted by N0, ishomogeneous in the whole area.

III. OPTIMAL PLACEMENT OF COGNITIVE RELAY

We first formulate the optimal relay placement problem incognitive radio networks as a nonlinear optimization problem.Then we derive an approximation formulation that is easier tosolve and leads to more insights.

A. Transmission Powers of Secondary Source and Relay

First we consider the interference caused by the secondarysource and the relay at the primary destination. In the Rayleighblock fading channel, the received power at the destinationfollows exponential distribution with mean equal to Pd−α,where P is the transmit power of the source, d is thedistance between two nodes, and α is the path loss exponent1.Therefore, the outage probability of the primary destinationPUd during the source transmission phase is

pes,p , Pr[P rs,p > P0] = exp(− P0

Psd−αsp

), (2)

where P rs,p is the received interference at the primary destina-tion PUd during this phase, and Ps is the transmit power ofthe secondary source SUs. Similarly, the outage probability ofthe primary destination during the relay transmission phase is

per,p , Pr[P rr,p > P0] = exp(− P0

Prd−αrp

), (3)

where P rr,p is the received interference at the primary destina-tion PUd during this phase, and Pr is the transmit power ofthe relay r.

Our goal in this paper is to maximize the transmission rate Rof the secondary user, thus the secondary source SUs and therelay r should use their maximum allowable transmit powersuch that the outage probabilities pes,p and per,p are equal toεp. From (2) and (3), we can derive

Ps =P0d

αsp

− ln εp, (4)

Pr =P0d

αrp

− ln εp. (5)

B. Problem Formulation

In the selection decode-and-forward scheme, the messageis first broadcasted by the source. If the relay can decodethe message, it will retransmit the message to the destinationby using the repetition coding. Otherwise, the relay remainssilent2, and the source will retransmit the message. Herewe assume that the achievable transmission rate is given bythe Shannon capacity of an additive white Gaussian noise(AWGN) channel. Using Shannon capacity allows us to fo-cus on the relay placement problem without worrying aboutthe choice of practical signaling and coding schemes. With

1For simplicity of presentation, we ignore shadow fading here.2We assume that there exist instant feedback from the relay to the source.

powerful coding schemes, transmission rates close to Shannoncapacity can be obtained in practice.

Let us denote the instantaneous signal-to-noise ratios(SNRs) of links (SUs, SUd), (SUs, r), and (r, SUd) as γsd,γsr, and γrd, respectively. We further let R be the transmissionrate of the secondary source SUs, and function

g(R) = 22R − 1.

Then the instantaneous achievable rate of the secondaryuser [6, Sec. IV.C] is

RDF =

12

log2(1 + 2γsd), γsr < g(R),

12

log2(1 + γsd + γrd), γsr ≥ g(R),(6)

The outage probability of the secondary user is

peDF , Pr[RDF < R]

= Pr[γsd <g(R)

2] · Pr[γsr < g(R)]

+ Pr[γsd + γrd < g(R)] · Pr[γsr ≥ g(R)].

(7)

In the Rayleigh block fading channel, the SNRs at thedestinations of links (SUs, SUd), (SUs, r), (r, SUd) all followexponential distribution with mean SNR γsd, γsr, and γrdrespectively. That is,

Pr[γsd < r] = 1− exp(− r

γsd

), (8)

Pr[γsr < r] = 1− exp(− r

γsr

), (9)

Pr[γrd < r] = 1− exp(− r

γrd

), (10)

where the average SNR values are

γsd =P s,dN0

=Psd−αsd

N0=

P0

−N0 ln εp

(dsddsp

)−α, (11)

γsr =P s,rN0

=Psd−αsr

N0=

P0

−N0 ln εp

(dsrdsp

)−α, (12)

γrd =P r,dN0

=Prd

−αrd

N0=

P0

−N0 ln εp

(drddrp

)−α. (13)

For simplicity, let

µ = P0/(−N0 ln εp), (14)

then we have

γsd = µ

(dsddsp

)−α, γsr = µ

(dsrdsp

)−α, γrd = µ

(drddrp

)−α.

(15)

Then, from Equation (7), the outage probability of the sec-ondary user is

peDF =(

1− exp(−g(R)

2γsd

))·(

1− exp(−g(R)γsr

))+(

1−(

γsdγsd − γrd

exp(−g(R)γsd

)+

γrdγrd − γsd

exp(−g(R)γrd

)))· exp

(−g(R)γsr

).

(16)

Note that Equation (16) does not hold when γsd = γrd. Inthat case, the locations of the secondary source and the relaycoincide with each other. However, this situation will not occuras the relay cannot offer any benefits to the secondary user.Thus, Equation (16) holds in real situations.

Our objective is to maximize the transmission rate R ofthe secondary user subject to the outage constraints of theprimary user and the secondary user. The primary user’soutage constraint has been taken care when calculating thetransmission powers in (4) and (5). The optimization problemcan be formulated as follows subject to outage constraint ofthe secondary user and location of the relay:

maxdsr,R

R

s.t. peDF ≤ εs,0 < dsr < dsd.

(17)

From the expressions of mean SNRs in (15), we find that alldistances dsd, dsr, and drd can be normalized with distancedsp. This means that the optimal relay location is determinedby the normalized distances and the angle θ. In other words,the optimal relay location is related to the relative locationsof the primary destination, the secondary source and thesecondary destination.

Next we rewrite the Problem (17) into an equivalent butsimpler form. Given a fixed parameter µ in (14), we find thatthe relay placement dsr that maximizes transmission rate Rin Problem (17) also maximizes g(R)/µ with the same setof constraints. This is because the g(R)/µ is an increasingfunction of R. We also observe that all mean SNRs in (15)have a common coefficient µ. Let ρ(R) = g(R)/µ. WithEquation (16), the optimal relay location can be obtained bysolving the following problem:

maxdsr,R

ρ(R)

s.t. peDF (ρ(R)) ≤ εs,0 < dsr < dsd,

(18)

where

peDF (ρ(R)) =(

1− exp(−ρ(R)

(dsddsp

)α))·(

1− exp(−ρ(R)

(dsrdsp

)α))

+

1−

(dsddsp

)−α(dsddsp

)−α−(drddrp

)−α exp(−ρ(R)

(dsddsp

)α)

+

(drddrp

)−α(drddrp

)−α−(dsddsp

)−α exp(−ρ(R)

(drddrp

)α)

· exp(−ρ(R)

(dsrdsp

)α).

(19)

Equation (19) suggests that the only variables in Problem (18)are ρ(R) and dsr. The optimal solution d∗sr of Problem (18)only depends on the normalized distances and path losscoefficient α (in (19)) as well as the outage constraint εs ofthe secondary user (in the first constraint of Problem (18)).

As Problem (17) and (18) have the same optimal relaylocation, then such location is independent of interference floorP0, the background noise N0, and the outage constraint εp ofthe primary user. We note, however, the maximum achievablerate R depends on these parameters.

C. Approximation

Ideally, we want to obtain the closed form optimal solutionsof Problem (18). However, this turns out to be very difficult (ifnot impossible) due to the complex nature of Equation (16).To obtain more insights, we will look at an approximationformulation that leads to very close to optimal relay location.The effectiveness of the approximation will be demonstratedthrough extensive numerical results in Section IV.

The starting point of approximation is the assumption thatthe outage probability of secondary user εs should be small.Suppose εs → 0, we have g(R)/µ→ 0 [16]. Then we have

limg(R)µ →0

peDF =

limg(R)µ →0

Pr[γsd <g(R)

2]Pr[γsr < g(R)]+

limg(R)µ →0

Pr[γsd + γrd < g(R)]Pr[γsr ≥ g(R)]

=g(R)2

2γsdγsr+

g(R)2

2γsdγrd.

(20)

Combined with the fact that peDF = εs at the optimal solution,we have

εs =(

12γsdγsr

+1

2γsdγrd

)· g(R)2. (21)

That is,

R =12

log2

1 + µ√εs

√√√√√√(dsddsp

)−α(drddrp

)−α(dsrdsp

)−α(drddrp

)−α+(dsrdsp

)−α .

(22)So our rate maximization problem (18) can be approximated

as follows:

maxdsr

12

log2

1+ µ√εs

√√√√√√(dsddsp

)−α(drddrp

)−α(dsrdsp

)−α(drddrp

)−α+(dsrdsp

)−α

s.t. 0 < dsr < dsd.(23)

Finding the optimal solution of Problem (23) is equivalentof solving the following problem:

maxdsr

(dsddsp

)−α(drddrp

)−α(dsrdsp

)−α(drddrp

)−α+(dsrdsp

)−αs.t. 0 < dsr < dsd.

(24)

Compared with Problem (18), the approximation formula-tion (24) is independent of the outage constraint εs of thesecondary user. This is because the rate function under lowoutage probability in (22) is a monotonic increasing functionof the product of the outage constraint εs and the polynomialexpression of normalized distances. Thus, the optimal relaylocation under a value of εs is also optimal under any othervalue of εs. However, the rate function with an arbitrary outageprobability does not have such a property, and hence theoptimal relay placement of Problems (17) and (18) depends onthe value of εs. In the next section, we will demonstrate theeffectiveness of the approximation formulation (23) throughextensive simulation results.

IV. RESULTS AND DISCUSSION

In this section, we provide some results about the optimalrelay location in cognitive radio networks. Problem (17) and(23) are both nonlinear optimization problems which are diffi-cult to solve in closed form. However, they both have only onevariable dsr3. Thus, we can use an efficient one-dimensionalsearch algorithm to find the exact optimal solutions for bothproblems.

Figure 2 shows the optimal relay location dsr and thecorresponding optimal transmission rate R of the secondaryuser obtained by the original formulation (17) and the ap-proximation formulation (23) respectively. For each curve, wefix the angle θ and change the distance between the secondarysource and the primary destination dsp.

From Fig. 2a, we find that solving the approximationformulation (23) leads to a very close-optimal relay location

3The variable rate R in Problem (17) can be eliminated by setting the firstinequality constraint to equality.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

dsp (normalized by dsd)

Opt

imal

rela

y lo

catio

n (n

orm

alize

d by

dsd

)

OptimalApprox.

! = 0

! = "/4

! = "/2

! = "! = 3"/4

(a) Optimal relay location.

1000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

dsp (normalized by dsd)

Sour

ce tr

ansm

issio

n ra

te R

10−0.4 1000

0.5

1

OptimalApprox.Fixed

! = 0

! = "

(b) Optimal rate.

Fig. 2. The optimal relay location dsr and the corresponding optimal transmission rate R with α = 4, εp = 0.05, P0/N0 = 10dB, and εs = 0.1.

0 45 90 135 1800.4

0.45

0.5

0.55

!

Opt

imal

rela

y lo

catio

n (n

orm

aliz

ed b

y d sd

)

" = 2, Original" = 2, Approx." = 3, Original" = 3, Approx." = 4, Original" = 4, Approx.

(a) Path loss exponent α.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.46

0.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

Outage probablity !s of secondary user

Opt

imal

rela

y lo

catio

n (n

orm

alize

d by

dsd

)

OriginalApprox.

" = 0

" = #/4

" = #/2

" = 3#/4

" = #

(b) Outage constraint εs of secondary user.

Fig. 3. Sensitivity analysis of other parameters.

for the secondary user. When the distance between the sec-ondary source and the primary destination dsp is large (e.g.,dsp ≥ 10), the optimal location of relay r is very close themiddle point of the secondary source SUs and the secondarydestination SUd. This is the consistent with the result inthe literature without interference constraint. By comparingcurves with different values of θ ∈ [0, π], we observe thatwhen θ increases, the optimal location of the relay r movestowards the secondary source SUs.

From Fig. 2b, we observe that when dsp/dsd ≥ 1, the im-pact of θ on the determination of the optimal relay location islimited. This implies that when the primary user is sufficiently

far away from the secondary user, we can ignore its locationand just consider its interference floor P0 and outage constraintεp.

Next we study the sensitivity of the optimal relay locationto other key system parameters. Figure 3a shows the influenceof path loss exponent α on the optimal relay location. Fromthis figure, we see that α has a very small impact on theoptimal location selection, and the approximation formulationcan find the near-optimal location. Figure 3b shows the impactof outage constraint εs of secondary user on the optimalrelay location. From the analysis in the previous section, weknow that it has no impact on the optimal solution obtained

by the approximation formulation in the low outage proba-bility regime, and we validate our analysis with this figure.However, as implied by the curves obtained with the originalformulation, εs has non-negligible influence on the optimallocation of a relay when it is relatively large. In practice, theoutage probability is typically much less than 0.1, and thusthe approximation formulation leads to very close to optimalrelay position.

V. CONCLUSION

In this paper, we study the optimal relay placement problemin cognitive spectrum underlay networks. For simplicity, Wefocus on the one-dimensional case. We consider the selectiondecode-and-forward relay protocol with Rayleigh blockingfading channel model. In such an environment, a naturalmetric to evaluate the performance of the secondary user is itsmaximum transmission rate subject to outage constraints. Weformulate this problem as a nonlinear optimization problem,and further derive an equivalent formulation that illustrateseveral interesting insights of the optimal relay position. Wefurther derive an approximation formulation in the low outageprobability regime.

We find that the optimal relay location is determined by therelative locations of the primary destination, the secondarysource, and the secondary destination. It is independent of theinterference constraint at the primary destination, the outageconstraint of the primary user, and the background noisedensity. We also find that the approximation formulation givesa close to optimal relay location. This implies that we can usethe approximation formulation when determining the optimalrelay location in cognitive radio networks. The sensitivities ofoptimal relay location to other key system parameters are alsoinvestigated. The path loss exponent has only small impacton the optimal relay location, for both original formulationand approximation formulation. The outage constraint of sec-ondary user can influence the optimal relay location, but theimpact is small under a low outage probability as in practice.

ACKNOWLEDGEMENT

This work is supported by the General Research Funds(Project Number 412308 and 412509) established under theUniversity Grant Committee of the Hong Kong Special Ad-ministrative Region, China. W. Zhang’s work is supported inpart by the Australian Research Council Discovery ProjectDP0987944.

REFERENCES

[1] J. Mitola III and G. Maguire Jr., “Cognitive radio: making softwareradios more personal,” IEEE In Personal Communications, vol. 6,no. 4, pp. 1–18, Aug. 1999.

[2] I. F. Akyildiz, W.-Y. Lee, M. C. Vuran, and S. Mohanty, “NeXtgeneration/dynamic spectrum access/cognitive radio wireless networks:A survey,” Computer Networks, vol. 50, no. 13, pp. 2127–2159,September 2006.

[3] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity- part i: System description,” IEEE Transactions on Communications,vol. 51, no. 11, pp. 1927–1938, Nov. 2003.

[4] ——, “User cooperation diversity - part ii: Implementation aspects andperformance analysis,” IEEE Transactions on Communications, vol. 51,no. 11, pp. 1939–1948, Nov. 2003.

[5] J. N. Laneman and G. W. Wornell, “Distributed space-time-codedprotocols for exploiting cooperative diversity in wireless networks,”IEEE Transactions on Information Theory, vol. 49, no. 10, pp.2415–2425, October 2003.

[6] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversityin wireless networks: Efficient protocols and outage behavior,” IEEETransactions on Information Theory, vol. 50, no. 12, pp. 3062–3080,Dec. 2004.

[7] T. Fujii and Y. Suzuki, “Ad-hoc cognitive radio - development tofrequency sharing system by using multi-hop network,” in Proc. IEEEDySPAN, 2005.

[8] J. Mietzner, L. Lampe, and R. Schober, “Distributed transmit powerallocation for relay-assisted cognitive-radio systems,” in Proc. IEEEACSSC, 2007.

[9] K. Hamdi and K. B. Letaief, “Cooperative communications forcognitive radio networks,” in Proc. PGNet, 2007.

[10] K. B. Letaief and W. Zhang, “Cooperative communications forcognitive radios,” Proc. of the IEEE, vol. 97, no. 5, pp. 878–893, May2009.

[11] M. Xie, W. Zhang, and K.-K. Wong, “A geometric approach to improvespectrum efficiency for cognitive relay networks,” IEEE Transactionson Wireless Communications, vol. 9, no. 1, pp. 268–281, Jan. 2010.

[12] Q. Zhang, J. Jia, and J. Zhang, “Cooperative relay to improve diversityin cognitive radio networks,” IEEE Communications Magazine, vol. 47,no. 2, pp. 111–117, Feb. 2009.

[13] K. Lee and A. Yener, “Outage performance of cognitive wireless relaynetworks,” in Proc. IEEE GLOBECOM, 2006.

[14] H. A. Suraweera, P. J. Smith, and N. A. Surobhi, “Exact outageprobability of cooperative diversity with opportunistic spectrum access,”in Proc. IEEE ICC Workshops, 2008.

[15] Federal Communication Commission, “Establishment of interferencetemperature metric to quantify and manage interference and to expandavailable unlicensed operation in certain fixed mobile and satellitefrequency bands,” ET Docket 03-289, Notice of Inquiry and ProposedRulemaking, 2003.

[16] A. S. Avestimehr and D. N. C. Tse, “Outage capacity of the fading relaychannel in the low-SNR regime,” IEEE Transactions on InformationTheory, vol. 53, no. 4, pp. 1401–1415, April 2007.