Effect of Relay Locations in Cooperative Networks†

Woong Cho, Hyun Seo Oh, and Dong Yong KwakElectronics and Telecommunications Research Institute (ETRI), Rep. of Korea, 305-700

Emails: {woongcho, hsoh5, dykwak}@etri.re.kr

Abstract—Cooperative networks have been appreciated fortheir spatial diversity benefits in wireless communications. Inthis paper, we will explore the error performance of cooperativenetworks depending on the relay locations. Both decode-and-forward (DF) and amplify-and-forward (AF) relaying protocolsare considered with an arbitrary number of relays. To providesimple transceiver structure, differential modulation scheme isadopted for both protocols. The effect of relay locations isinvestigated by considering the number of relays and transmitenergy including energy optimization. We show that there existsa common point where the error performance of both protocolscoincides. Based on these results, we can achieve the improvementin error performance by choosing the relaying protocol dependingon relay locations.

Index Terms—Cooperative systems, MIMO systems, resourcemanagement, error analysis

I. INTRODUCTION

In wireless communications, multi-input multi-output(MIMO) systems provide larger capacity by transmitting signalwith multiple antennas. However, these systems suffer fromthe antenna packing limitation such as small-sized deviceimplementation. To overcome this physical constraint and keepMIMO benefits, cooperative networks have been appreciatedfor their spatial diversity benefits, in which relay nodes be-tween the source and destination create virtual antennal arrays[6]. Cooperative networks have been adopted using coherentmodulations or nocoherent modulations, or space-time codingschemes, and their performance was analyzed in [2], [5], [7],[10], [13], [11], [14], [15].

Resource allocation has been considered as an importantresearch topic in cooperative networks to increase capacityand save system energy (power). The existing work on theresource allocation has been done in terms of optimal energy,location, and joint energy-location for the system resource [1],[3], [4], [12]. Among these optimizations, it has been shownthat the location of relay is critical to determine the systemperformance. Furthermore, the location information may beuseful for higher layer applications such as location basedrouting.

In this paper, we consider the effects of location in coopera-tive networks. We consider both decode-and-forward (DF) andamplify-and-forward (AF) relaying protocols with differentialmodulation. Based on the existing results in [3], [4], weanalyze the effect of relay locations on the error performance.An arbitrary number of relays and transmit energy on theerror performance are considered with relay locations. In

† This work is supported by the IT R&D program of MKE/IITA [2007-F-039-01, Vehicle Multi-hop Communication Technology Development]

addition, we show that there is a common point where theerror performance of both protocols is the same, and this pointcan be used for performance enhancement.

The rest of this paper is organized as follows. The signalrepresentation and demodulation rule are described in SectionII. The performance analysis depending on relay locations isdeveloped in Section III, and concluding remarks are given inSection IV.Notation:We use (·)∗ for conjugate, �{·} for the real part,CN (μ, σ2) for the complex Gaussian distribution with meanμ and variance σ2, and := for “is defined as”.

II. SYSTEM MODEL

Consider a network setup with one source node s, L relaynodes {rk}L

k=1, and one destination node d. We consider twoconventional relaying protocols, i.e., decode-and-forward (DF)and amplify-and-forward (AF) with differential modulation.For the DF protocol, the relay nodes demodulate the receivedsignal from the source node, then remodulate and forwardit to the destination node. Whereas for the AF protocol, therelay nodes simply amplify the received signal from the sourcenode and forward it to the destination node. We assume that adirect link between the source node and the destination nodeis blocked by obstacles.

A. Signal representation and channel model

With the nth phase-shift keying (PSK) symbol being assn = ej2πcn/M , cn ∈ {0, 1, ...,M − 1}; the correspondingtransmitted signal from the source is generated as xs

n =xs

n−1sn with the initial condition xs0 = 1. During the first

time slot of a transmission, the source node broadcasts thedifferentially encoded signal to all relays. The received signalat kth relay is given by

yrk,sn =

√Esh

rk,sn xs

n + zrk,sn , k = 1, 2, ..., L, (1)

Then, each relay node decodes (amplifies) the received signalfrom the source node, and these signal are transmitted tothe destination node during their distinct time slots. Letxrk

n denote the nth transmitted symbol from the kth relay,k = 1, 2, ..., L; which has a different form depending on therelaying protocol, then the received signal at the destinationnode can be represented as

yd,rkn =

√Erkhd,rk

n xrkn + zd,rk

n , k = 1, 2, . . . , L. (2)

In (1) and (2), Ei is the energy per symbol at node i, fadingcoefficient and the noise component of the channel i−j, ∀i, j ∈{s, rk, d} are hj,i

n ∼ CN (0, σ2j,i) and zi,j

n ∼ CN (0,N0),

978-1-4244-4067-2/09/$25.00 © 2009 IEEE Wireless VITAE’09737

respectively. Accordingly, the received instantaneous signal-to-noise ratio (SNR) of the channel i−j is γi,j = (|hi,j

n |2Ej)/N0,and the average received SNR is γi,j = (σ2

hi,jEj)/N0.

B. Decision rule and demodulation

The received signals at the destination are independent,conditioned on the signal transmitted from the source node.With differential signal, the current received signal can berepresented by using the previous received signal. Hence, thelog likelihood functions (LLF) for given that xm

n is transmittedby the source is

ldm(yn) := �{(yd,rkn )∗yd,rk

n−1smn }, k = 1, 2, ..., L, (3)

where smn = ej2πm/M and m ∈ {0, 1, ..., M − 1}. Then, the

decision rule can be expressed as follow

m =arg maxm∈{0,1,...,M−1}

L∑k=1

ldm(yn), (4)

and sn = ej2πm/M . Notice that although the decision rule atthe destination node can be represented Eq. (4), the transmittedsignal at the relay node has a different form depending on therelaying protocol.

In the DF protocol, the received signal at each relay nodeis demodulated as:

m′=arg maxm�{(yrk,s

n )∗yrk,sn−1s

mn }, k = 1, 2, ..., L, (5)

where smn = ej2πm/M and srk

n = ej2πm′/M , which results in

xrkn = xrk

n−1srkn , k = 1, 2, ..., L. (6)

For the AF protocol, the received sinal at each relay node isamplified to generate

xrkn = Ark

yrk,sn , k = 1, 2, ..., L, (7)

where Arkis the amplification factor. To maintain a constant

average power at the relay output, the amplification factor is

Ark=

√1

σ2rk,sEs +N0

, k = 1, 2, ..., L. (8)

This factor is reasonable for both differential and noncoherentmodulations, since σ2

rk,s can be estimated by averaging thereceived signals without knowing the instantaneous channelstate information [3], [15]. For detailed formulation of thereceived signal, we refer the reader to [4] and [3] for DFprotocol and AF protocol, respectively.

III. EFFECT OF RELAY LOCATIONS

In this section, we will show the error rate performance ofthe cooperative networks using the system models in SectionII. Based on the error performance, we explore the effect ofrelay locations and find a common point where the DF andAF protocols have the same error performance.

A. Error Performance

With differential modulation, we have following results ofsymbol error rate (SER).

Proposition 1 With any given PDFe,rk

and PDFe,d , an upper

bound on PDFe can be found as :

PDFe ≤ PDF

e = 1−L∏

k=1

(1 − PDFe,rk

)(1− PDFe,d ), (9)

where, PDFe,rk

is the s-rk link SER and PDFe,d is the conditional

SER at the destination node provided that the symbol sn iscorrectly demodulated at all relays.

In Eq. (9), the SER of differential M -ary PSK (DMPSK)signaling in the s − rk link, PDF

e,rk, can be obtained as [8,

Ch.8]

Pe,rk=√

gPSK

2π

∫ 2/π

−2/π

Mγs(−[1−√1−gPSK cos θ])1−√1−gPSK cos θ

dθ, (10)

where gPSK � sin2 πM , Mγ(x) = 1/(1 − xγ), ∀x > 0,

and γ represents the average SNR. At the destination, thesignals from the L relays are combined to make a decision.Conditioned on that the symbol sn is correctly demodulatedand remodulated at all relay nodes, the conditional SER Pe,d

can be obtained by applying the results for L-diversity branchreception of M -phase signals in [9, Appendix C] as:

PDFe,d =

(−1)L−1(1−μ2)L

π(L−1)!

(∂L−1

∂bL−1

{1

b−μ2

[π

M(M−1)

− μ sin(π/M)√b−μ2cos2(π/M)

cot−1 −μ cos(π/M)√b−μ2cos2(π/M)

]})b=1

,(11)

where μ = γd,rk/(1 + γd,rk

). Notice that Eq. (9) is an upperbound since the cases where the rk − d link error correctsgiven the s− rk link error are ignored.

Proposition 2 At high SNR, the average SER of an L−relayAF system using DBPSK signaling can be approximated as:

PAFe ≈ C(L)

L∏k=1

[1

γrk,s+

1γd,rk

ln(γd,rk)]

, (12)

where C(L) = 122L−1

∑L−1n=0

(n+L−1

L−1

)∑L−1−nk=0

(2L−1

k

), which

is a constant depending on the number of the equivalentchannels.

For detailed proofs and examples, we refer the reader to [4]and [3] for DF protocol and AF protocol, respectively.

B. SER depending on relay locations

To capture the effects of relay location, we make use of therelationship between the variance of channel fading coefficientσ2

ij and the inter-node distance Dij as follows:

σ2ij = C ·D−ν

ij , i, j ∈ {s, rk, d} , (13)

where ν is the path loss exponent of the wireless channel andC is a constant which we henceforth set it to 1 without loss of

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generality. Throughout this paper, we consider Dsrk+Drd =

Dsd = 1, i.e., line topology. To express the energy constraintby SNR, let us define the total SNR ρ := E/N0, the transmitSNR at the source node ρs := Es/N0, and the transmit SNRat the relay nodes ρrk

:= Erk/N0, where we use the total

energy constraint as ρ = ρs +∑L

k=1 ρrk.

In Figs. 1 and 2, we show that the average SER dependingon various relay locations for the DF protocol and AF protocol,respectively, with L = 2. We also plots the average SER forL = 3 with various locations in Fig. 3. We set ν = 4, ρj =10dB, j ∈ {s, rk}, k = 1, 2 and 3 . In Figs. 1 and 2, x axisrepresents the location of relay node 1 and y axis representsthe location of relay node 2. In both protocols, for the givenone relay’s location, the relay located at the same positionshows the minimum SER. This phenomenon can be observedfor L = 3. In Fig. 3, dotted lines and solid lines represent theSER of the DF and AF protocols, respectively, and (a, b, c)represents the location of relay node 1, 2, and 3. The caseswhen the relay nodes are located near the source node andmidpoint between the source node and destination node areconsidered. The Fig. 3 showed that the relay locations aremore critical for the DF protocol than AF protocol. It is worthmentioning that the effect of relay locations is more critical inthe DF protocol than in the AF protocol. For the DF protocol,the SER degrades dramatically when one relay moves from the(0.2, 0.2, 0.2) location, whereas the SER of the AF protocolis not severely fluctuating compared with the DF protocol.

In all cases, the co-located scenario shows better errorperformance than the scenarios of various relay locations.Intuitively, this is due to the fact that the degradation oferror performance in one bad link results in the overall errorperformance degradation. Therefore, it is good choice to selectco-located relay nodes to achieve better performance. How-ever, the trend of SER is different depending on the relayingprotocol. For the DF protocol, we can achieve minimum SERby locating relay nodes near the source node. The minimumSER of the AF protocol can be obtained by locating relaynodes at the midpoint between the source node and destinationnode.

From Figs. 1, 2, and 3, we may find a common point wherethe SER of the DF protocol and AF protocol has the same.By considering co-located scenario, we find a location wheretwo relaying protocols have the same SER. This location canbe found by treating SER Eqs. (9) and (12) as a function ofthe distance and finding common point where two SER’s areequal. We will consider DBPSK signaling. Then, PDF

e,rkand

PDFe,d in Eq. (9) can be simplified to

PDFe,rk

= 1/ [2(1 + γrk,s)] , (14)

and

PDFe,d =

12

[1− μ

L−1∑k=0

(2k

k

)(1− μ2

4

)k]

, (15)

respectively, where μ = γd,rk/(1 + γd,rk

).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

0.5

110

−4

10−3

10−2

10−1

Dsr

1

Dsr

2

Ave

rage

SE

R

Fig. 1. The average SER depending on various relay locations (L =2, DF, ρj=10dB, where j ∈ {s, r1, r2}).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

0.2

0.4

0.6

0.8

110

−3

10−2

10−1

Dsr

1

Dsr

2

Ave

rage

SE

R

Fig. 2. The average SER depending on various relay locations (L =2, AF, ρj=10dB, where j ∈ {s, r1, r2}).

0 5 10 15 20 2510

−6

10−5

10−4

10−3

10−2

10−1

SNR (dB)

Ave

rage

SE

R

AFDF(0.5,0.5,0.5)(0.2,0.2,0.2)(0.5,0.5,0.2)(0.2,0.2,0.5)

Fig. 3. The average SER depending on various relay locations (L =3, DF and AF protocols, SNR= ρj , where j ∈ {s, r1, r2, r3}).

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910

−6

10−5

10−4

10−3

10−2

10−1

100

Dsr

Ave

rage

SE

R

AF protocolDF protocol

ρs=ρ

r=10, 20, 30, and 40 dB

Fig. 4. The average SER depending on various relay locations andtransmit energies (L = 1, DF and AF protocols).

For L = 1, by equating Eqs. (9) and (12), we have

1−[1− 1

2(1 + ρsD−νsr )

] [1− 1

2(1 + ρr(1 −Dsr)−ν)

]

=12

[1

ρsD−νsr

1ρr(1−Dsr)−ν)

ln(D−νrd )

], (16)

where left hand side (LHS) and right hand side (RHS) arethe SER of DF protocol and AF protocol, respectively. In theabove equation, the log term and path loss exponent rendera closed-form solution intractable. Although an analyticalsolution is not readily available, one could resort to thenumerical search using Eq. (16). Since the Eq. (16) cannotbe solved algebraically, we represent a solution graphically.We plot SER of the both protocols in Fig. 4 for various relaylocations Dsr and transmit energy ρj , j ∈ {s, r}. The dottedline and solid line represent SER of the DF protocol andthe AF protocol, respectively. The figure shows that only onecrossing point exists when transmit energy is very high, andthat point gives us a common point where the SER of the bothDF and AF protocols is the same.

For L ≥ 2, we can also resort numerical search to find acommon point using the Eqs. (9) and (12). However, noticethat the SER in Eq. (9) is an upper bound, and the gapbetween the real SER and an upper bound increases as thenumber of relays increases. Furthermore, the SER of the AFprotocol in Eq. (12) is an approximated value under high SNRassumption. Therefore, it is reasonable to find a common pointby simulations because of the difference between Eqs. (9) and(12). We will find a common point by considering the numberof relays and transmit energy.

C. Simulations

In this subsection we consider the effect of relay locationswith various scenarios. We assume that all relay nodes locatedat the same position, and the transmit energy of each node isthe same, i.e., Dsrk

= Dsr, ∀k and ρr = ρrj , ∀j with ρs = ρr.Fig. 5 depicts the average SER depending on the number ofrelays for both DF and AF protocols. We use ρs = ρr = 10dB

0 0.2 0.4 0.6 0.8 110

−6

10−5

10−4

10−3

10−2

10−1

Dsr

Ave

rage

SE

R

AF protocolDF protocol

L=1, 2, 3, and 4

Fig. 5. The average SER depending on the relay locations withvarious number of relay nodes (ρs = ρr = 10dB, ν = 4).

with ν = 4. The figure reveals that there exist a commonpoint between the DF protocol and AF protocol except forL = 1, which agrees with our analytical result. Notice that thecommon point moves towards the source node as L increases.These results suggest that lower SER can be obtained byappropriately choosing the relaying protocol with respect toa certain location. This method can be regarded as a simpleadaptive relaying protocol which uses the information of relaylocations. We can use DF protocol when the relay nodes arelocated near the source node. When the relay nodes movetowards the destination node, lower SER can be achieved byswitching the relaying protocol to the AF protocol at a certainpoint. This protocol selection depending on the relay locationsprovides us lower SER compared with the case where weuse one relaying protocol. Fig. 6 represents the average SERdepending on the transmit energy with L = 2. The figureshows that as the transmit energy increases the crossing pointmoves towards the source node. The Figs. 5 and 6 indicate thatdeciding a common point depends on not only the number ofrelays but also the transmit energy where this common pointmoves towards the source node as the number of relays andtransmit energy increase.

Now, let us consider the effect of relay locations combinedwith energy optimization. We will see how the common pointwill be changed by adopting energy optimization. As in [4],[3], we consider following energy optimization scheme.

In a relay system with path loss exponent ν and source-destination distance Ds,d, for any given relay locations (Ds,r

and Dr,d), and the total energy per symbol ρ, we will deter-mine the optimum energy allocation ρs and ρr, which:

minimizes PDFe or PAF

e

subject to ρs + Lρr = ρ. (17)

For the results of optimization, we refer the reader to [4],[3]. In this paper, we compare the effect of relay locationswith and without energy optimization. For the system with-out optimization we assign uniform energy allocation, i.e.,

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0 0.2 0.4 0.6 0.8 110

−5

10−4

10−3

10−2

10−1

Dsr

Ave

rage

SE

R

AF protocolDF protocol

ρs=ρ

r=5, 10, 15, and 20 dB

Fig. 6. The average SER depending on the relay locations withvarious transmit energies (L = 2, ν = 4).

0 0.2 0.4 0.6 0.8 110

−5

10−4

10−3

10−2

10−1

Dsr

Ave

rage

SE

R

L=1L=2L=3

DFAF

Fig. 7. The average SER depending on the relay locations, the systemwith uniform energy allocation (ρs = ρr = ρ/(L + 1), ρ = 15dB,ν = 4).

ρs = ρr = ρ/(L + 1). We consider ρ = 15dB for L = 1, 2,and 3 with ν = 4. Figs. 7 and 8 depict the SER of thesystem with uniform energy allocation and optimum energyallocation, respectively. For both cases, there exist a commonpoint where both protocols’ SER is the same except L = 1.Two figures show that the crossing point is almost the same inboth energy allocation schemes although the crossing pointsare little bit different. This implies that energy optimization isnot critical for finding a common point, i.e., finding a commonpoint is mainly handled by the property of relaying protocolnot by the energy optimization of each relaying protocol.

IV. CONCLUSIONS

In this paper, we investigated the error performance of coop-erative network depending on the relay locations. To achievebetter SER, we show that the co-located relay nodes are goodchoice. We also show that the trend of error performance isdifferent depending on the relaying protocol. With an arbitrarynumber of relays and transmit energy, the effect of relaylocations is observed for both DF and AF relaying protocols.Based on these results, we reveal that there is a common point

0 0.2 0.4 0.6 0.8 110

−5

10−4

10−3

10−2

10−1

Dsr

Ave

rage

SE

R

L=1L=2L=3

AF

DF

Fig. 8. The average SER depending on the relay locations, the systemwith optimum energy allocation (ρs = ρo

s, ρr = ρor, ρ = 15dB [4],

[3], ν = 4).

between the DF protocol and AF protocol where the SERcoincides. Finally, we show that it is possible to enhance theerror performance by properly choosing the relaying protocoldepending on the relay locations.

REFERENCES

[1] R. Cao and L. Yang, “Practical issues in resource optimization of relaynetworks,” in Proc. of CISS, 2008.

[2] D. Chen and J. N. Laneman, “Modulation and demodulation forcooperative diversity in wireless systems,” IEEE Trans. on WirelessCommunications, vol. 5, no. 7, pp. 1785–1794, Jul. 2006.

[3] W. Cho, R. Cao, and L. Yang, “Optimum resource allocation for amplify-and-forward relay networks with differential modulation ,” IEEE Trans.on Signal Processing, vol. 56, no. 11, pp. 5680–5691, Nov. 2008.

[4] W. Cho and L. Yang, “Optimum resource allocation for relay networkswith differential modulation,” IEEE Trans. on Communications, vol. 56,no. 4, pp. 531–534, Apr. 2008.

[5] Y. Jing and H. Jakarkhani, “Distributed differential space-time codingfor wireless relay networks,” IEEE Trans. on Communications, vol. 56,no. 7, pp. 1092–1100, Jul. 2008.

[6] J. N. Laneman and G. W. Wornell, “Energy-efficient antenna sharingand relaying for wireless networks,” in Proc. of WCNC, vol. 1, 2000,pp. 7–12

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

[8] M. K. Simon and M. S. Alouini, Digital Communication over FadingChannels, 2nd ed. Wiley, 2004.

[9] J. Proakis, Digital Communications, 4th ed. McGraw-Hill, New York,2001.

[10] A. Ribeiro, X. Cai, and G. B. Giannakis, “Symbol error probabilities forgeneral cooperative links,” IEEE Trans. on Wireless Communications,vol. 4, no. 3, pp. 1264–1273, May 2005.

[11] K. T and B. S. .Rajan, “Partially-coherent distributed space-time codeswith differential encoder and decoder, ” IEEE Journal on Selected Areasin Communications, vol. 25, no. 2, pp. 426–433, Feb. 2007.

[12] M. Yu, J. Li and H. Sadjadpour, “Amplify-forward and decode-forward:The impact of location and capacity contour,” in Proc. of MILCOM,vol. 3, 2005, pp. 1609–1615.

[13] G. Wang, Y. Zhang, and M. Amin, “Differential distributed space-time modulation for cooperative networks,” IEEE Trans. on WirelessCommunications, vol. 5, no. 11, pp. 3097–3180, Nov. 2006.

[14] T. Wang, Y. Yao, and G. B. Giannakis, “Non-coherent distributed space-time processing for multiuser cooperative transmissions,” IEEE Trans.on Wireless Communications, vol. 5, no. 12, pp. 3339–3343, Dec. 2006.

[15] Q. Zhao and H. Li, “Differential modulation for cooperative wirelesssystems,” IEEE Trans. on Signal Processing, , vol. 55, no. 5, pp. 2273–2283, May 2007.

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