Error Rate Performance in OFDM-based Cooperative Networks

Error Rate Performance in OFDM-basedCooperative Networks

Victor K. Y. Wu and Ye (Geoffrey) LiSchool of Electrical and Computer Engineering

Georgia Institute of TechnologyAtlanta, Georgia

Email: [email protected], [email protected]

Marilynn Green, Tony Reid, and Peter WangNokia Siemens Networks

Dallas, TexasEmail: {marilynn.green, tony.reid, peter.wang}@nsn.com

Abstract— Cooperative relay networks have been shown toimprove performance in wireless communication systems asa form of spatial diversity. In this paper, we investigate theerror rate performance in a single path relay network and amultiple path relay network using orthogonal frequency divisionmultiplexing (OFDM) signals. Using the amplify-and-forwardand decode-and-forward relay algorithms, we derive input-outputrelations for the two networks. For the amplify-and-forward case,we consider two relay power allocation schemes. The first isconstant gain allocation, where the amplifying gain is constant forall subcarriers. The second is equal power allocation, where eachsubcarrier transmits the same power. The former scheme doesnot require channel state information (CSI), while the latter onedoes. For the decode-and-forward case, the transmitter and eachrelay are assumed to have uniform power allocations. We simulatethe word error rate (WER) performance for the two networks.For the single path relay network, amplify-and-forward givesvery poor performance, because as we increase the distancebetween the transmitter and receiver (and thus, add more relays),more noise and channel distortion enter the system. Decode-and-forward gives significantly better performance because noiseand channel distortion are eliminated at each relay. For themultiple path relay network, decode-and-forward again givesbetter performance than amplify-and-forward. However, theperformance gains are small compared to the single path relaynetwork case. Therefore, amplify-and-forward may be a moreattractive choice in this case due to its lower complexity.

I. INTRODUCTION

Recently, relays are being exploited to improve performancein wireless communications systems. The relays are a networkof transceiver nodes between the transmitter and receiver thatfacilitate the transfer of information. This type of scheme isknown as cooperation or cooperative communications in theliterature because the relay network is cooperating with thetransmitter and receiver to improve performance. It is alsoknown as meshed networking.

One current example of such technologies is the MIT-initiated One Laptop per Child (OLPC) project [1], whichaims to provide affordable laptops equipped with meshednetworking functionality to children in the developing world.Since cellular and Internet connectivity is sparse and sporadicin these regions, such laptops can cooperate to make the bestuse of available bandwidth.

In this paper, we restrict ourselves to a single path relaynetwork and a multiple path relay network in the context of

orthogonal frequency division multiplexing (OFDM) systems.The authors in [2], [3], [4] have provided several physical

layer relay algorithms. These include amplify-and-forward anddecode-and-forward. In amplify-and-forward, a node amplifiesits receive symbol, subject to a power constraint, before re-transmitting to the next node. This algorithm is obviously withlow complexity. In decode-and-forward, a node fully decodes asymbol, re-encodes it and then re-transmits it. In other words,this scheme attempts to eliminate channel distortion and noiseat each node.

The authors in [5], [6] have investigated cooperation for asingle path of relays connected in series. The motivation forthis network structure is that broader wireless coverage canbe achieved, while still maintaining a low power constraintat the transmitter. The authors consider analog relaying anddigital relaying as two possible relay algorithms. These areequivalent to the amplify-and-forward and decode-and-forwardalgorithms, respectively. A power budget is considered whereeach packet travelling through the network is only allowed toconsume a total fixed amount of power. As well, each nodehas a certain transmit power limit. The outage probability isthen minimized by allocating power among the relay networkunder these power constraints. This power allocation accountsfor the channel conditions in the network in order to achievethe optimal outage probability. Simulations indicate that 2 dBof total power can be saved for 5 relays by using optimal powerallocation instead of uniform power allocation. This is forthe decode-and-forward case. However, at high SNR values,the decode-and-forward case approximates the amplify-and-forward case.

The authors in [7] have investigated cooperation for multiplepaths of relays connected in parallel. In the conventionalscheme, all relays participate using amplify-and-forward. Thisis called all-participate amplify-and-forward (AP-AF). Theauthors also consider an algorithm where only one relay isselected in the transmission to maximize the mutual infor-mation. This is called selection amplify-and-forward (S-AF).S-AF selects the relay which results in the maximum mutualinformation between transmitter and receiver. Simulations ofoutage probability indicate that 5 dB of SNR can be saved for3 relays by using S-AF instead of AP-AF. The authors in [8]derive symbol error probabilities for multiple paths of relays.

1930-529X/07/$25.00 © 2007 IEEEThis full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2007 proceedings.

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r0

r1

rm

rm+1n

(0)k

n(m−1)k

n(m)k

h(0)k

h(1)k

h(m−1)k

h(m)k

Transmitter

Receiver

Fig. 1. Single Path Relay Network

In this paper, we continue to investigate the series andparallel cooperative relay networks using OFDM signals. Weconsider a single path relay network and a multiple path relaynetwork. Using the amplify-and-forward relay algorithm, wederive the input-output relations for both networks. Usinga power constraint at each relay, we consider two relaypower allocation schemes. The first is constant gain allocation,where the amplifying gain is constant for all subcarriers.The second is equal power allocation, where each subcarriertransmits the same power. Using the decode-and-forward relayalgorithm, we derive input-output relations for both networks.We simulate word-error-rates (WERs) for the two networksusing both the amplify-and-forward and decode-and-forwardrelay algorithms.

The paper is organized as follows. In Section II, we considerthe single path relay network in [5], [6]. In Section III, weconsider a modified version of the multiple path relay networkin [7] where the transmitter-receiver direct link is removed.Notice that these two relay networks are series and parallelanalogs of each other. In Section IV, we present simulationsof the WERs for the two relay networks. Finally, Section Vconcludes the paper and provides future research directions.

II. SINGLE PATH RELAY NETWORK

A. Amplify-and-Forward

Figure 1 shows the single path relay network. In the figure,r0 is the transmitter, rm+1 is the receiver, and r1, . . . , rm

are m relay nodes connected in series forming a single pathlink between the transmitter and receiver. The relays performamplify-and-forward (AF) relaying. We assume that OFDMwith N subcarriers is used in the system.

h(0)k , . . . , h

(m)k are the complex subchannel gains at the kth

subcarrier in the link, for k = 1 to N . n(0)k , . . . , n

(m)k are

the corresponding noises, which are assumed to be mutuallyindependent and circular symmetric complex Gaussians allwith zero mean and variance N0B/N , where N0 is thepower spectral density of the underlying continuous time noiseprocess and B is the OFDM bandwidth of the system. Letp(0)k = Ptot/N be the transmit power on the kth subcarrier,

where Ptot is the net transmitter power. Let√

p(l)k be the

amplifying gain used in the amplify-and-forward algorithm atthe lth relay, for l = 1 to m. The kth receive symbol at rl is

amplified by√

p(l)k before it is forwarded to the next node.

Let x(0)k be the kth transmit symbol with zero mean and unit

variance. Let yk be the kth receive symbol at the receiver. Letx

(l)k be the kth receive symbol at the lth relay. Note that x

(l)k

is also the kth transmit symbol at the lth relay. Using Figure1, the input-output relation at the lth relay is

x(l)k = h

(l−1)k

√p(l−1)k x

(l−1)k + n

(l−1)k . (1)

The input-output relation at the receiver is

yk = h(m)k

√p(m)k x

(m)k + n

(m)k . (2)

(1) in closed form is

x(l)k =

(l−1∏i=0

h(i)k

√p(i)k

)x

(0)k +

l−1∑j=0

l−1∏

i=j+1

h(i)k

√p(i)k

n

(j)k . (3)

(2) in closed form is

yk =

(m∏

i=0

h(i)k

√p(i)k

)x

(0)k +

m∑j=0

m∏

i=j+1

h(i)k

√p(i)k

n

(j)k . (4)

We assume that the net transmit power at the transmitterand at each each relay is Ptot, that is,

N∑k=1

E

{∣∣∣∣√

p(l)k x

(l)k

∣∣∣∣2}

= Ptot. (5)

At the transmitter, we assume a uniform power distribution,that is, p

(0)k = Ptot/N . To derive the power constraint at each

relay, substitute (3) into (5) to arrive at

N∑k=1

p(l)k

N

b

(0)k

(l−1∏i=1

b(i)k p

(i)k

)+

1SNR

l−1∑j=0

l−1∏i=j+1

b(i)k p

(i)k

= 1, (6)

where b(i)k =

∣∣∣h(i)k

∣∣∣2, for i = 0 to m and SNR = Ptot/N0B.Note that (6) is defined recursively. The power constraint forp(l)k depends on p

(1)k , . . . , p

(l−1)k . p

(1)k is the base case in the

recursion, which follows from (6), when l = 1.One power allocation at the lth relay is to set p

(l)k constant

for all subcarriers. This results in moving p(l)k in (6) out of the

summation because it is no longer a function of k

p(l)k,ct = p

(l)ct =

NSNRN∑

k=1

SNRb

(0)k

(l−1∏i=1

b(i)k p

(i)ct

)+

l−1∑j=0

l−1∏i=j+1

b(i)k p

(i)ct

. (7)

We call this constant gain allocation (CT). Note that thispower allocation does not require each relay to have anychannel state information (CSI). That is, we do not actuallyneed to use (7). We can use (5) directly to solve for p

(l)ct .


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r0

r1

rm

rm+1

n(0|1)k

n(0|m)k

n(1|m+1)k

n(m|m+1)k

h(0|1)k

h(0|m)k

h(1|m+1)k

h(m|m+1)k

Transmitter Receiver

Fig. 2. Multiple Path Relay Network

A second power allocation is to choose p(l)k such that every

subcarrier transmits the same power at the lth relay. That is,

(5) becomes E

{∣∣∣∣√

p(l)k,eqx

(l)k

∣∣∣∣2}

= Ptot/N , for k = 1 to N .

This is equivalent to setting every summand on the left handside of (6) to 1/N . We then have

p(l)k,eq =

SNR

SNRb(0)k

(l−1∏i=1

b(i)k p

(i)k,eq

)+

l−1∑j=0

l−1∏i=j+1

b(i)k p

(i)k,eq

. (8)

We call this equal power allocation (EQ). Note that this powerallocation does require each relay to have the CSI of itsupstream channels.

B. Decode-and-Forward

In decode-and-forward (DF), each relay fully recovers theinformation bits (with possible errors) after receiving anOFDM symbol. It then converts the information bits back intoan OFDM symbol and then transmits it. The transmitter and allthe relays transmit with the same uniform power distribution.That is,

p(0)k = p

(l)k =

PtotN

, (9)

for k = 1 to N and for l = 1 to m.Let x

(0)k be the kth transmit symbol from the transmitter and

x(l)k be the kth transmit symbol from the lth relay, all with with

zero mean and unit variance. Let y(m+1)k be the kth receive

symbol at the receiver and y(l)k be the kth receive symbol at

the lth relay. Using Figure 1, the input-ouput relation at thelth relay is

y(l)k = h

(l−1)k

√PtotN

x(l−1)k + n

(l−1)k . (10)


y(m+1)k = h

(m)k

√PtotN

x(m)k + n

(m)k . (11)

III. MULTIPLE PATH RELAY NETWORK

A. Amplify-and-Forward

Figure 2 shows the multiple path relay network. In the fig-ure, r0 is the transmitter, rm+1 is the receiver, and r1, . . . , rm

are m relay nodes connected in parallel forming a multiplepath link between the transmitter and receiver. The relays

perform amplify-and-forward (AF) relaying. We assume thatOFDM with N subcarriers is used in the system.

h(0|1)k , . . . , h

(0|m)k , h

(1|m+1)k , . . . , h

(m|m+1)k are the complex

subchannel gains at the kth subcarrier in the link, for k =1 to N . n

(0|1)k , . . . , n

(0|m)k , n

(1|m+1)k , . . . , n

(m|m+1)k are the

corresponding noises, which are assumed to be mutually in-dependent, zero-mean, circular symmetric complex Gaussiansall with variance N0B/N , where N0 is the power spectraldensity of the underlying continuous time noise process andB is the OFDM bandwidth of the system. Let p

(0)k = Ptot/N

be the transmitter power on the kth subcarrier, where Ptot is

the net transmitter power. Let√

p(l)k be the amplifying gain

used in the amplify-and-forward algorithm at the lth relay, forl = 1 to m. The kth receive symbol at rl is amplified by√

p(l)k before it is forwarded to the next node.

Let xk be the kth transmit symbol with zero mean and unitvariance. Let y

(l)k be the kth receive symbol from the lth path

at the receiver. Using Figure 2, the input-output relation forthe lth path is

y(l)k =

(h

(0|l)k h

(l|m+1)k

√p(0)k

√p(l)k

)xk +

h(l|m+1)k

√p(l)k n

(0|l)k + n

(l|m+1)k . (12)

We assume that the net transmit power at the transmitterand at each each relay is Ptot, as in (5). At the transmitter, weassume a uniform power distribution, that is, p

(0)k = Ptot/N .

To derive the power constraint at each relay and thus, possiblepower allocations, we use a derivation similar to the one inSection II-A to arrive at

N∑k=1

p(l)k

N

(b(0|l)k +

1SNR

)= 1, (13)

where b(i|j)k =

∣∣∣h(i|j)k

∣∣∣2, for i = 0 to m, for j = 1 to m + 1,

i �= j, and SNR = Ptot/N0B.Constant gain allocation (CT) in this case is

p(l)k,ct = p

(l)ct =

NSNRN∑

k=1

(SNRb

(0)k + 1

) . (14)

Again, this power allocation does not require each relay tohave any channel state information (CSI). The lth relay onlyhas to multiply its entire OFDM receive symbol by a constant,√

p(l)ct , such that the total transmit power is Ptot, similar to

constant gain allocation in Section II-A.Equal power allocation (EQ) in this case is

p(l)k,eq =

SNR

SNRb(0|l)k + 1

. (15)

Note that this power allocation does require each relay to havethe CSI of its upstream channel.


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input

output 1

output 2

output 3

Fig. 3. Convolutional encoder.

B. Decode-and-Forward

In decode-and-forward (DF), each relay fully recovers theinformation bits (with possible errors) after receiving anOFDM symbol. It then converts the information bits back intoan OFDM symbol and then transmits it. The transmitter and allthe relays transmit with the same uniform power distribution.That is,

p(0)k = p

(l)k =

PtotN

, (16)

for k = 1 to N and for l = 1 to m.Let x

(0)k be the kth transmit symbol from the transmitter and

x(l)k be the kth transmit symbol from the lth relay, all with with

zero mean and unit variance. Let y(m+1)k be the kth receive

symbol at the receiver and y(l)k be the kth receive symbol at

the lth relay. Using Figure 2, the input-ouput relation at thelth relay is

y(l)k = h

(0|l)k

√PtotN

x(0)k + n

(0|l)k . (17)


y(m+1)k =

m∑i=1

(h

(i|m+1)k

√PtotN

x(i)k + n

(i|m+1)k

). (18)

Note that in this situation, there are m (modified in general)copies of the original kth transmit symbol x

(0)k , namely,

x(1)k , . . . , x

(m)k . Therefore, the receiver has to assume (incor-

rectly in general) that all m relays perform perfect recoveryof the information bits so that x

(l)k = x

(0)k . That is, the receiver

assumes the kth receive symbol is

y(m+1)k =

(m∑

i=1

h(i|m+1)k

√PtotN

)x

(0)k +

m∑i=1

n(i|m+1)k . (19)

This allows the receiver to use a filter that is matched tom∑

i=1

h(i|m+1)k .

IV. SIMULATION RESULTS

We simulate WERs versus SNR for both the amplify-and-forward and decode-and-forward cases and for both relaynetworks. At the transmitter (and at the transmitter structureof a relay using decode-and-forward), each information wordcontains 83 bits. Using the convolutional encoder shown inFigure 3, the information word is encoded into a 255 bit

codeword. A zero bit is padded at the end to make 256 bits.The bits are then interleaved and modulated onto N = 128QPSK (quadrature phase shift keying) subcarriers to form oneOFDM symbol. At the receiver (and at the receiver structure ofa relay using decode-and-forward), the codeword is recovered(with possible errors) using a matched filter and deinterleaving.A Viterbi decoder is used to decode the codeword. Both harddecisions and soft decisions are used.

We consider the m = 2 and 4 relays cases. We assume thatall distances between any two adjacent transceiver nodes arethe same. Therefore, all path loss effects are normalized to0 dB. Shadowing is assumed to be log-normally distributed.That is, the received power gain due to shadowing in dB is azero-mean Gaussian with variance of 8 dB, which is typicalfor cellular land mobile applications [9]. We model frequencyselective fading as Typical Urban (TU) channels [9]. We usean OFDM bandwidth of 800 kHz divided into N = 128 equalblocks. Maintaining OFDM orthogonality, this translates intoan OFDM symbol period of Ts = 160 µs. The simulationresults are shown in Figure 4.

As shown in the plots, there are significant error rateperformance gains when using decode-and-forward instead ofamplify-and-forward for the single path relay network case.The gains are even larger when we increase the distancebetween the transmitter and receiver (and thus, add morerelays). The amplify-and-forward error rates suffer becausemore channel distortion and noise enter the system. Thedecode-and-forward error rates suffer only slightly becausenoise and channel distortion are eliminated at each relay. Thisresults in the large performance gains for m = 4. In terms ofpower allocation when using amplify-and-forward, CT is thepreferable choice since EQ requires CSI and only results insmall performance gains over CT.

For the multiple path relay network case, the performancegains resulting from using decode-and-forward instead ofamplify-and-forward diminish as we add more paths betweenthe transmitter and receiver (and thus, add more relays). Thisis because for m = 4 relays and thus, for 4 paths, the systemis already resistant to noise and channel distortion. There-fore, decode-and-forward cannot provide any more significantimprovement over amplify-and-forward. In such a situation,amplify-and-forward might be a more attractive choice due toits lower complexity. In terms of power allocation when usingamplify-and-forward, CT is the preferable choice because itgives better performance than EQ and does not require CSI.

As expected, soft decisions give better performance thanhard decisions in Viterbi decoding.

V. CONLCUSIONS

In this paper, we investigate cooperation by applying OFDMsignals to a single path relay network and a multiple pathrelay network. In particular, we look at the WER performanceresulting from such systems using the amplify-and-forwardand decode-and-forward relay algorithms. Simulations indicatethat decode-and-forward is a good choice for the singlepath relay network, because of its ability to eliminate noise


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0 3 6 9 12 15 18 21 2410

−2

10−1

100

WE

R

SNR (dB)

hard, AF, CT

hard, AF, EQhard, DFsoft, AF, CT

soft, AF, EQsoft, DF

(a) Single Path Relay Network, m=2

0 3 6 9 12 15 18 21 2410

−2

10−1

100

WE

R

SNR (dB)

hard, AF, CT



(b) Single Path Relay Network, m=4

0 3 6 9 12 15 18 21 2410

−2

10−1

100

WE

R

SNR (dB)

hard, AF, CT



(c) Multiple Path Relay Network, m=2

0 3 6 9 12 15 18 21 2410

−2

10−1

100

WE

R

SNR (dB)

hard, AF, CT



(d) Multiple Path Relay Network, m=4

Fig. 4. WERs in single path and multiple path relay networks with TU channels using AF and DF. N = 128, m = 2 and 4.

and channel distortion. For the multiple path relay network,amplify-and-forward may be the better choice for its lowcomplexity.

Future research includes investigating relay algorithms otherthan amplify-and-forward and decode-and-forward in OFDM-based cooperative networks. For example, hybrid algorithmscombining the two can be considered. That is, depending onthe channel conditions, a relay can choose different relayalgorithms for different subcarrier symbols in an OFDMsignal. It can amplify-and-forward, decode-and-forward, oreven just discard subcarrier symbols. This in turn leads tomore possibilities for relay power allocation. In this paper, weonly investigate the single path relay network and the multiplepath relay network. Other general relay networks need to beconsidered in the context of OFDM as well. This will lendmore insight into developing a general theoretical frameworkfor OFDM in cooperative relay networks.

REFERENCES

[1] (2005) One laptop per child. [Online]. Available:http://laptop.media.mit.edu/

[2] J. N. Laneman, “Cooperative diversity in wireless networks: algorithmsand architectures,” Ph.D. dissertation, MIT, Cambridge, MA, Sept. 2002.

[3] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity inwireless networks: Efficient protocols and outage behavior,” IEEE Trans.Inform. Theory, vol. 50, pp. 3062–3080, Dec. 2004.

[4] J. N. Laneman, G. Wornell, and D. N. C. Tse, “An efficient protocol forrealizing cooperative diversity in wireless networks,” in Proc. IEEE Int.Symp. Information Theory (ISIT), Washington D.C., June 2001, p. 294.

[5] M. O. Hasna and M. Alouini, “Performance analysis of two-hop relayedtransmission over rayleigh-fading channels,” in Proc. IEEE VehicularTechnology Conf., Vancouver, BC, Canada, Sept. 2002, pp. 1992–1996.

[6] ——, “Optimal power allocation for relayed transmissions over rayleigh-fading channels,” IEEE Trans. Wireless Commun., vol. 3, pp. 1999–2004,Nov. 2004.

[7] Y. Zhao, R. Adve, and T. J. Lim, “Improving amplify-and-forward relaynetworks: Optimal power allocation versus selection.”

[8] A. Ribeiro, X. Cai, and G. B. Giannakis, “Symbol error probabilities forgeneral cooperative links,” IEEE Trans. Wireless Commun., vol. 4, pp.1264–1273, May 2005.

[9] G. L. Stuber, Principles of Mobile Communications, 2nd ed. Norwell,MA: Kluwer Academic Publishers, 2001.


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Error Rate Performance in OFDM-based Cooperative Networks

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