average snr and ergodic capacity analysis for opportunistic df relaying with outage over rayleigh...

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 6, JUNE 2009 2807

Average SNR and Ergodic Capacity Analysis forOpportunistic DF Relaying with Outage over Rayleigh Fading Channels

Sungeun Lee, Student Member, IEEE, Myeongsu Han, Student Member, IEEE,and Daesik Hong Senior Member, IEEE

AbstractIn the paper, we deal with a single-selection oppor-tunistic relaying with the decode-and-forward (DF) protocol overRayleigh fading channels. The exact end-to-end average signal-to-noise ratios (SNR) and ergodic capacities of both proactiveand reactive opportunistic relaying are derived as a closed-formfor arbitrary link SNR. In addition, the effective ergodic capacitysatisfying the minimum required data rate without outage is alsoidentified for both relaying schemes. The analysis results are usedto demonstrate which relaying scheme outperforms the other forgiven system parameters.

Index TermsOpportunistic relaying, decode-and-forward, er-godic capacity.

I. INTRODUCTION

COOPERATIVE relaying techniques have recentlyachieved a great deal of popularity as an efficientway to mitigate fading in wireless networks [1]. Generally,cooperative relaying has focused on simultaneous transmissionfrom multiple relays [2]. However, recent research hasshown that carefully selected relay transmission incurs noperformance loss compared to multiple-relay transmission interms of the diversity-multiplexing gain tradeoff and outageprobability [3].

Cooperative opportunistic relaying describes a situationwhere a single relay among several relay nodes is selecteddepending on which relay provides the best end-to-end pathbetween the source and destination in a distributed wirelessnetwork [3]. In the opportunistic decode-and-forward (DF)relaying scheme, there are two variant techniques which areoutage-optimal: proactive (P-DF) and reactive (R-DF) [3]. InP-DF mode, a specified relay selected prior to the sourcetransmission participates in the cooperation, whereas in R-DFmode, relays that successfully decode the message take partin the cooperation.

Bletsas et al. studied an outage-based comparison of the twomethods where they showed that the outage probability for P-DF is exactly the same as that for R-DF [3]. Michalopoulos et

Manuscript received April 25, 2008; revised July 30, 2008 and November5, 2008; accepted December 19, 2008. The associate editor coordinating thereview of this letter and approving it for publication was K. B. Lee.

S. Lee and D. Hong are with the Department of Electrical and Elec-tronic Engineering, Yonsei University, Seoul, Korea (e-mail: {softmind,daesikh}@yonsei.ac.kr).

M. Han is with the Republic of Korea Marine Corps, Korea.This research was supported in part by the Ministry of Knowledge

Economy, Korea, under the ITRC (Information Technology Research Center)support program supervised by the IITA (Institute of Information TechnologyAssessment) (IITA-2008-C1090-0803-0002), and in part by Korea Scienceand Engineering Foundation through the NRL Program (Grant R0A-2007-000-20043-0).

Digital Object Identifier 10.1109/TWC.2009.080574

al. extended this result over the environment where the source-destination channel is taken into account [4], and showed thatthe R-DF scheme is slightly better than P-DF in terms ofoutage, since R-DF seems to take better advantage of thedirect source-destination channel. In addition, an approximateclosed-form of the bit-error probability (BEP) for both the P-DF and the R-DF modes is derived in [4]. However, evaluatingthe advantages and disadvantages of the two methods moreprecisely will require investigating the capacity in order toassess which opportunistic relaying method is superior to theother. Recently, the authors observe the end-to-end averagesignal-to-noise ratios (SNR) and ergodic capacities of twoopportunistic DF relay schemes based on the assumption of ahigh SNR [5]. From [5], it is shown that the ergodic capacityof P-DF scheme outperforms that of R-DF scheme under highSNR regime where channel qualities of all relays satisfy thepredetermined threshold.

However, the impact of predetermined threshold on ergodiccapacity is much more significant in low to medium SNR thanhigh SNR since only a part of relays can participate in the co-operation by the threshold in low to medium SNR. Therefore,it is very important to evaluate the exact performance in low tomedium SNR regime to compare these two relaying schemes.

In this paper, we analyze the exact end-to-end average SNRand the ergodic capacities of two opportunistic DF relayingschemes for arbitrary link SNRs over dual-hop Rayleighfading channels. In addition, we introduce a useful notion,effective ergodic capacity, which corresponds to the spectralefficiency of the successful transmissions excluding the outageevent. This concept is inferred from the threshold-based DFrelaying schemes [6][7]. This effective ergodic capacity isderived by discarding data packets that do not support arequired spectral efficiency. A potential advantage of thisapproach is that our analytic results can serve as a guide whenexamining the practical throughput of each relaying scheme,which reflecting in the outage events, and this can be usedto determine the superiority of one scheme over another forgiven system parameters.

This paper is organized as follows: Section II presents thecharacteristics of the two opportunistic DF relaying schemes.Section III derives the end-to-end average SNRs of the twomethods in order to explain the properties of the two relayingschemes. Section IV presents and analyzes the typical andeffective end-to-end ergodic capacities for both relaying types,and Section V provides the simulation and numerical resultsverifying the analysis as well as a discussion of the capacitytrends of the two schemes according to system parameters.Finally, our conclusions are presented in Section VI.

1536-1276/09$25.00 c 2009 IEEE

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2808 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 6, JUNE 2009

II. CHARACTERISTICS OF TWO OPPORTUNISTIC DFRELAY SYSTEMS

Let us consider a half-duplex dual-hop scenario where asingle source (S) communicates with a single destination (D)through a total of M DF relays (Rm for the mth relay withm V0 = {1, 2, . . . ,M}). We focus on the environmentthe direct path between the source and destination is blockedby an intermediate wall, while relays are located at theperiphery of the obstacle, i.e., the direct path from S to Dis not taken into account [3]. The instantaneous SNRs ofthe S-Rm and Rm-D channels are represented as 1,m and2,m, which yield to a statistically independent exponentialdistribution with a mean of 1/1 and 1/2, respectively [3].The cumulative distribution functions (CDF) of 1,m and 2,mcan be given by Fi,m(t) = 1eit for i = 1, 2, and t 0.As noted above, there are two kinds of opportunistic relayschemes, P-DF and R-DF. In P-DF, the best relay Rm ischosen to maximize the minimum of the channel strengthsbetween the links S-Rm and Rm-D for all M relays, i.e.,m = argmaxmV0 [min(1,m, 2,m)]. In R-DF, on the otherhand, all relays listen to the signal from S, and only those with1,m > T decode the received signal at the relay, where Tdenotes a predetermined SNR threshold indicating successfuldecoding at the receiver. The single relay with the best 2,mamong them then transmits to D.

Let us define VT as the index set of the relay for which theinstantaneous SNR of the S-Rm link satisfies the threshold,i.e., VT = {m | 1,m T for m V0}. In this case, theend-to-end instantaneous SNRs for P-DF and R-DF can thenbe represented as

P-DF = min(1,m, 2,m) (1)

where m = argmaxmV0 [min(1,m, 2,m)] as described,and

R-DF ={

0 , if VT = ,min(1,m , 2,m) , otherwise,

(2)

where m = argmaxmVT [2,m].Fig. 1 shows the schematic end-to-end SNR comparisons for

both modes when the required SNR (threshold) is T = 2. Fig.1(a) describes the low SNR case where the signal in the R-DFscheme is discarded at the relay since there is no decodablerelay during the first hop. The R-DF scheme drops the signalin advance, before the outage event happens at the destination.However, the P-DF scheme first transmits the signal regardlessof the outage, and then the outage occurs at the destinationafter the relay retransmits the signal. In Fig. 1(b), both P-DF and R-DF select the same relay to communicate, so theend-to-end SNRs for both schemes are identical to each other.For high SNR regions, on the other hand, the SNR of the P-DF outperforms that of the R-DF since the P-DF considersthe SNRs of both links when choosing the best relay. Thiswould seem to indicate that the behavior of the end-to-endaverage SNRs for two relaying schemes is different from thetrend of the outage probabilities [3]. From the perspectiveof outage probability, both P-DF and R-DF have exactly thesame results, regardless of the mean channel gain [3]. Onthe contrary, the end-to-end average SNR can be differentaccording to the mean channel gain for given threshold. At

1,1 1.5 =

1,2 1 =

1,3 1.8 =

2,1 2 =

2,2 3 =

2,3 1.6 =

R1

R2

R3

S D

.

.P-DF path only

(a) low SNR, P-DF > R-DF = 0

1,1 1 =

1,2 2 =

1,3 2.5 =

2,1 2 =

2,2 2.2 =

2,3 1.8 =

R1

R2

R3

S D

Both P-DF & R-DF paths

.P-DF: R-DF:

(b) medium SNR, P-DF = R-DF

1,1 2 =

1,2 3 =

1,3 3 =

2,1 4 =

2,2 2 =

2,3 3 =

R1

R2

R3

S D

R-DF path

P-DF path

(c) high SNR, P-DF > R-DF

Fig. 1. Schematic comparisons of end-to-end SNR when the required SNR(threshold) is 2, i.e., T = 2. In case (a), the signal in the R-DF schemeis discarded whereas the signal in the P-DF is transmitted regardless of theoutage. Both schemes select the same path to communicate in case (b). Incase (c), in contrast, the end-to-end SNR for P-DF is better than that forR-DF.

this point, let us analyze the end-to-end average SNRs forthe P-DF and R-DF modes for a given threshold T , and theninvestigate how these SNRs are different by comparison.

III. ANALYSIS OF END-TO-END AVERAGE SNR

A. End-to-end Average SNRs for P-DF, P-DF

To calculate the CDF of the selected relay in the P-DFrelaying scheme, let min,m = min(1,m, 2,m) and P-DF =maxmV0

(min,m). By order statistics [8], the CDF of min,m is

1 e(1+2)t, and the CDF of the instantaneous SNR forthe selected relay, P-DF, can be described as

FP-DF(t) = P(P-DF t) = P (max

mV0(min,m) t

)

=[1 e(1+2)t

]M.

(3)

In order to calculate the end-to-end average SNR and ergodiccapacity, the following rule of integration by parts is used forthe random variable x:

x = E {x}ba = b

a

xfx(x) dx = [x Fx(x)]ba b

a

Fx(x) dx,

(4)where E{x}ba is the expectation of the x over the interval [a, b],and fx(x) and Fx(x) denote the probability density function(pdf) and cdf of the variable x, respectively.

Using the binomial theorem (x+y)n =n

k=0

(nk

)xnkyk for

(3) and applying the rule of integration by parts in (4), the



end-to-end average SNR for the P-DF relaying scheme canbe derived via some mathematical calculation from [9, Eq.0.155.4]:

P-DF = E {P-DF}0

=M

m=1

(M

m

)(1)m+1

m (1 + 2)

=1

1 + 2

Mm=1

1m

=HM

1 + 2,

(5)

where HM is the M -th harmonic number. From (5), it canbe easily inferred that the average SNR becomes greater asthe number of relays M increases. Actually, HM can beinterpreted as the SNR gain achieved by relay selection amongM relays since 11+2 is the end-to-end average SNR fora single relay. Note that this growth rate decelerates as thenumber of relays increases.

B. End-to-end Average SNR for R-DF, R-DF

In contrast to the P-DF method, the R-DF scheme firstchooses the relays {Rn} whose S-Rn link quality satisfiesthe threshold, i.e., n VT . Then, among these selected relays,only that one relay with the best Rn-D link quality transmitsthe signal. In the R-DF method, the probability PT,k that krelays will pass the threshold test (1,m T ) can be expressedby

PT,k =(M

k

)ek1T

(1 e1T )Mk . (6)

Let us define mk as the best relay index in R-DF schemeswhen there are k relay candidates that satisfy the thresholdcondition (1,m T ). In this case, the CDF of 1,m

kis

derived as(1 e1(tT ))u(t T ) since P (1,m

k< t) =

P (1,m < t|1,m T ), and the CDF of 2,mk

is [1e2t]k.Then, the CDF of the instantaneous SNR for k relay candi-dates, R-DFk , is given by

FR-DFk (t) = 1 P(min(1,m

k, 2,m

k) > t

)=(1 e2t)k

+(1 e1(tT )

)(1 (1 e2t)k)u(t T ).

(7)

Applying the rule in (4) for (7), the end-to-end average SNRfor R-DF can be given by

R-DFk = E{R-DFk

}0

=k

m=1

(k

m

)(1)m+1

{1 em2T

m2+

em2T

1 + m2

}.

(8)

Consequently, using (6) and (8), the total end-to-end averageSNR of the R-DF schemes is

R-DF =M

k=1

PT,k R-DFk . (9)

If the quality of the Rm-D link is improved continuously,i.e., 2 0, R-DFk converges to 1/1, meaning that R-DFperformance is bounded to the average SNR of S-Rm link ifthe average SNR of Rm-D is high enough [5].

IV. ANALYSIS OF END-TO-END ERGODIC CAPACITY

A. Typical and Effective Ergodic Capacities for P-DF, C P-DFand C P-DFeff

Let us define the end-to-end capacity as C = 12 log2(1+),where is the instantaneous end-to-end SNR. This can thenbe transformed into = 2 2C 1. If the CDF of C denotesFC(y), this can also be converted into F(22y 1). As aresult, using (3), the CDF of the capacity for the P-DF mode,FCP-DF(y), can be represented as

FCP-DF(y) = P(22C

P-DF 1 22y 1)

=[1 e(1+2)(22y1)

]M.

(10)

The integration of the exponential form in (10) can be rear-ranged as

0

ez(22y1)dy =

ez

2 ln 2E1(z), (11)

where E1(z) is the exponential integral function defined asE1(z) =

z

ett dt [9, ch.8.21]. Hence, exploiting (4) and

(10), the end-to-end ergodic capacity, C P-DF, is obtained in(12) at the bottom of the page.

In the P-DF scheme, the best relay, Rm, decodes thereceived message, and always re-transmits a re-encoded mes-sage to D regardless of the outage. Therefore, communicationthrough this relay can fail due to outage when either of thetwo hops (S-Rm or Rm-D link) fails; see Fig. 1(a). Althoughno reliable information is delivered to D for this case, theergodic capacity C P-DF still includes the case in the calculation.Consequently, C P-DF is not a good measure to evaluate therealistic system performance in terms of practical throughput.

Therefore, to measure the practical end-to-end performancewhile taking the outage into account, we use the effectiveergodic capacity C P-DFeff , which as defined here corresponds tothe average spectral efficiency of the successful transmissionsexcluding the outage event. C P-DFeff is derived in (13) at thebottom of the page where R = 12 log2(1 + T ) is the spec-tral efficiency required to decode the message successfully,and

R e

z(22y1)dy = ez

2 ln 2E1(z(T + 1)). This effective

C P-DF = E {CP-DF}0

=1

2 ln 2

Mm=1

(M

m

)(1)m+1em(1+2)E1 (m(1 + 2)) (12)

C P-DFeff = E{CP-DF}

R= R

[1

(1 e(1+2)T

)M]+

Mm=1

(M

m

)(1)m+1em(1+2)E1 (m (1 + 2) (T + 1))

2 ln 2(13)



ergodic capacity is evaluated within the interval [R,] toinclude only successful decoding cases. The first term in (13)represents the average capacity limited by the threshold, andthis term dominates C P-DFeff for low mean SNR regions belowthe threshold T . This term in the effective ergodic capacityenables us to observe the achievable data rate in the lowSNR region properly, instead of using the impractical ergodiccapacity. On the other hand, for high mean SNR values, i.e.,1/i T , C P-DFeff converges to C P-DF since the SNR quality ofthe selected relay link is good enough to exceed the threshold.Consequently, C P-DF can be an upper bound for C P-DFeff .

B. Typical and Effective Ergodic Capacities for R-DF, C R-DFand C R-DFeff

As with the average SNR analysis in Section III-B, we firstevaluate the ergodic capacity of the R-DF scheme for eachk relay candidates to derive the total ergodic capacity. Then,using this capacity in conjunction with the probability that krelays will be selected, we obtain the total end-to-end ergodiccapacity for R-DF. The CDF of the capacity for the R-DFmode with k relay candidates, FCR-DFk (y), can be representedas

FCR-DFk

(y) =(1 e2(22y1)

)k+(1 e1(22y22R)

)(

1(1 e2(22y1)

)k)u(y R),

(14)

by substituting y = 12 log2(1 + t) into (7). After somemanipulation with (11), the end-to-end average capacity ofthe R-DF when k relays are selected, C R-DFk , is obtained by(15) at the bottom of the page. Using (15) in conjunction withthe relay selection probability (6), the total end-to-end averagecapacity of the R-DF relaying scheme can be obtained as

C R-DF =M

k=1

PT,k C R-DFk . (16)

As described above, the R-DF scheme investigates thesignal quality at the relay in advance to prevent outage events.Nevertheless, it is still possible to fail to communicate atthe destination due to outage since this scheme examinesthe S-Rm link quality only, not both S-Rm and Rm-D linkqualities: outage occurs if the quality of the Rm-D link is poor.Therefore, the R-DF scheme also needs an effective measurefor examining the practical ergodic capacity. The effectiveergodic capacity for R-DF schemes with k relay candidates,C R-DFk,eff , can be derived by (17) expressed at the bottom of thepage. This term is also evaluated in the interval [R,) to

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Mean SNR of the Hop [dB]

Endt

oE

nd E

rgod

ic C

apac

ity [b

its/s

/Hz]

EndtoEnd Typical and Effective Ergodic Capacity

P/RDF relaying [ Typical, analysis]P/RDF relaying [Effective, analysis]PDF relaying [ Typical, simulation]RDF relaying [ Typical, simulation]PDF relaying [Effective, simulation]RDF relaying [Effective, simulation]

R=1.5 & 3.0(T=7 & 63)

R=1.5(T=7)

R=3.0(T=63)

Fig. 2. Typical and effective ergodic capacity vs. mean channel gain forthe proactive and reactive DF relaying schemes. The number of relays andthe required spectral efficiency are M = 4 and R = 1.5, and 3 bps/Hz,respectively, with the assumption that 1 = 2. The lines represent theanalysis, and the markers denote the simulation results.

exclude decoding failure cases. The overall effective end-to-end ergodic capacity of R-DF scheme can then be describedas

C R-DFeff =M

k=1

PT,k C R-DFk,eff . (18)

V. SIMULATION AND NUMERICAL RESULTS

In this section, we provide simulation and numerical resultsto verify the analytic results in the previous section. Inaddition, we will use this data to examine the tendenciesof the typical and effective ergodic capacities for variousenvironments.

Fig. 2 evaluates the typical and effective end-to-end ergodiccapacities vs. the mean channel gain per hop for the P-DF andR-DF methods. The number of relays and the pre-determinedspectral efficiency are M = 4 and R = 1.5 and 3 [bps/Hz],respectively, with 1 = 2.

Recall that R corresponds to a fixed value associatedwith the pre-determined threshold T , and may vary fromapplication to application; in fact, it determines the number ofrelays that belong to VT in the R-DF scenario, and whetheror not the information is delivered to D from the selectedrelay for measuring the effective capacity. For a given spectral

C R-DFk = E{CR-DFk }0

=k

m=1

(k

m

)(1)m+1 e

2m

2 ln 2

[E1 (2m) E1 (2m (T + 1)) + e1(T+1) E1 ((1 + 2m) (T + 1))

] (15)

C R-DFk,eff = E{CR-DFk }R = R [1 (1 e2T )k]+

km=1

(k

m

)(1)m+1 e

2m

2 ln 2

[e1(T+1) E1 ((1 + 2m) (T + 1))

](17)



efficiency R, we can show that both the typical and effectiveergodic capacity analysis in (12), (13), (16), and (18) are inexact agreement with the simulation results. Therefore, we canconfirm that both ergodic capacities of the P-DF and R-DF canbe evaluated by these results for variable circumstances.

First, let us focus on the typical ergodic capacities C P-DFand C R-DF which do not take outage events at D into con-sideration. In the figure, solid lines denote the typical ergodiccapacities for both schemes, P-DF and R-DF. Note that, inthe low SNR regions below the threshold, C P-DF has onlyone mono-increasing curve, whereas C R-DF has two curves,corresponding to each required rate R, both of which arealmost zero restricted by the threshold. The P-DF scheme doesnot concern itself with any successful decoding constraint andalways retransmits the signal to D irrespective of the outagecondition. Consequently, the typical capacity curve for P-DFis independent of the threshold. In contrast, the R-DF schememay suffer a pause in transmission at the relay if outage isexpected at D due to poor S-Rm link quality. Therefore, thetypical capacity for R-DF has a tiny value for mean SNRvalues below the threshold. As a result, there is a big gapbetween the typical ergodic capacity of P-DF and R-DF inthe low SNR regions.

In addition, for high mean SNR regions in which all Mrelays are supposed to be chosen as candidates, C P-DF out-performs C R-DF. This is because the P-DF scheme considersboth links when selecting the best relay to transmit, while onlythe Rm-D link is considered in the R-DF method. Actually,it is an interesting result that the typical ergodic capacitiesof both schemes differ according to the mean SNR since theoutage probabilities of both schemes are exactly the same forthe same mean SNR [3][5].

To combine these different results simultaneously, we in-troduce and evaluate the effective ergodic capacity concerningthe decoding failure at D. A remarkable fact of these capacityresults is that the effective ergodic capacities of both schemes,P-DF and R-DF, are almost the same for the low-to-mediumSNR regions. This means that the practical system throughputfor both schemes is very similar over low-to-medium SNRregions, even though two schemes use different methodologiesto select the best relay. In addition, the outage tendencyof the P-DF scheme can be inferred from the effective er-godic capacity curve, compared to the typical capacity curve.Accordingly, a fairer comparison of the capacities betweenthe P-DF and R-DF relaying schemes can be done, especiallyfor the low-to-medium SNR regime, by applying the effectiveconcept, which takes the outage into consideration.

Moreover, in a high mean SNR regime, the effective ergodiccapacities C P-DFeff and C R-DFeff approach the general ergodiccapacities C P-DF and C R-DF. Therefore, we can state that theeffective ergodic capacity is bounded to the typical capacityfor high SNRs. Since C P-DF > C R-DF for high SNR, , C P-DFeffis also better than C R-DFeff for this regime.

Fig. 3 describes how the effective ergodic capacity can bevaried as the number of relays M changes. The thresholdsare fixed to T = 7 in the analysis. As shown in the figure,the effective ergodic capacity of R-DF becomes saturated veryquickly compared to that of the P-DF scheme, and this trendis more prominent for high mean SNR values. For example,

2 4 6 8 10 12 14 16 18 20 22 240

0.5

1

1.5

2

2.5

3

3.5

4

Number of Relays

Endt

oE

nd E

rgod

ic C

apac

ity [b

its/s

/Hz]

EndtoEnd Effective Ergodic Capacity, R=1.5 (T=7), 1 =

2

PDF relaying [Analysis]RDF relaying [Analysis]

1 = 0.1 (10dB)

1 = 0.01 (20dB)

Fig. 3. Effective ergodic capacity analysis vs the number of relays M forthe P-DF and R-DF relaying schemes. The threshold and the mean SNR perhop are T = 7, and 1/1 = 1/2 = 10 and 20 dB, respectively.

in the case of R-DF with 1 = 0.01 (20dB), no additionalimprovement is achieved for M > 4 by increasing the numberof relays. This is because the diversity gain achieved throughrelay selection is limited to consideration only of the Rm-Dlink in the R-DF mode, as mentioned in (2). If there are |VT |relay candidates1 whose S-Rm link SNRs are better than thethreshold, the R-DF method selects the one relay which hasthe best channel on the Rm-D link from those candidates,i.e., m VT . The selected relay, Rm , therefore, has thebest channel gain for the Rm-D link. However, this does notguarantee that the channel gain of the S-Rm link for theselected relay is also the best compared to that of the otherrelays. The end-to-end link quality through the selected relaymay not be good, even though the Rm -D link is the best,which means that the overall link quality is dominated by thequality of the S-Rm link for the R-DF scheme. Consequently,the effective ergodic capacity of the R-DF scheme can bebounded to a certain value even though the number of relaysincreases.

In contrast, the effective ergodic capacity of the P-DFmethod improves as the number of relays increases, similar tothe average SNR in (5). In the P-DF relaying scenario, both theS-Rm and Rm-D links are considered simultaneously whenchoosing the best relay. Therefore, the selection diversity ordercan be increased by increasing the number of relays in the P-DF.

Fig. 4 depicts the effective ergodic capacity depending onthe relative channel quality between the S-Rm and Rm-Dlinks. Let us define the relative channel quality ratio as =2/1 for given 1. This term then determines whether ornot the S-Rm link is better than Rm-D: Lower values (below1) mean better Rm-D link quality; higher values (above1) indicate better S-Rm link quality. Actually, can be alsoexpressed by the path-loss exponent and distance. When the

1|X | stands for the cardinality of a set X : number of elements in the set



12 9 6 3 0 3 6 9 120

0.5

1

1.5

2

2.5

3

3.5

4

SR link / RD link mean SNR ratio (10 log10

(2/

1)) [dB]

Endt

oE

nd E

rgod

ic C

apac

ity [b

its/s

/Hz]

EndtoEnd Effective Ergodic Capacity, R=1.5 (T=7), M=4

PDF relaying [Analysis]RDF relaying [Analysis]

SR link is betterRD link is better

1 = 0.1 (10dB)

1 = 0.01 (20dB)

Fig. 4. Effective Ergodic capacity analysis vs. relative channel quality ratio, defined as = 2/1. The term indicates whether or not the S-Rm linkis better than the Rm-D link for given 1. The threshold and the number ofrelays are T = 7, and M = 4, respectively.

signal power is reduced by long-term fading with the path-loss exponent , and the link distances of S-Rm and Rm-Dare d1 and d2, respectively, the ratio can be described as

=(

d2d1

). Therefore, this result can be also interpreted

the performance with respect to the relay position.Of interest in the result inferred from the figure is that

P-DF is more sensitive to this disparity between the S-Rm and Rm-D SNR values. As mentioned above, the P-DFrelaying scheme reflects both the S-Rm and Rm-D links whenchoosing the path, whereas the R-DF scheme considers onlythe Rm-D link when selecting the relay. Therefore, it is clearthat the P-DF method is more deeply affected by the SNRdifference between the two links. In addition, the effectiveergodic capacity of the R-DF scheme maintains almost thesame value in the interval < 1 for the high SNR regime.This is because the overall link quality for R-DF is notaffected by the increase in Rm-D quality for the fixed S-Rm SNR value, 1, since the capacity is primarily dominatedby the S-Rm link, as mentioned above; in Section III-B it wasshown that the average SNR R-DFk converges to 1/1 as thequality of the Rm-D links improves (2 0). Consequently,when the channel qualities for the S-Rm and Rm-D links

are different, consideration should be given to which type ofrelaying scheme is better for that environment. In addition,it is better to use the R-DF scenarios when the SNRs of theS-Rm and Rm-D links vary frequently, as in the case of amobile relay environment, for example.

VI. CONCLUSION

We evaluated the performance of proactive and reactiveopportunistic decode-and-forward relaying schemes from theviewpoints of the end-to-end average SNR and the effectiveergodic capacity with respect to outage events, and numericaland our simulation results verified the correctness of ouranalysis for arbitrary link SNRs. The derived average SNR

and capacity are expressed by the number of relays, a pre-determined threshold to prevent outage, and the arbitraryaverage SNR of the S-Rm and Rm-D links. From theseresults, it is possible to determine which opportunistic relayingscheme is better in terms of SNR and effective capacity forall relaying systems with arbitrary parameters. In general, theproactive scheme is proven to outperform the reactive one interms of general and effective ergodic capacity, whereas theoutage probabilities of both schemes are the same.

REFERENCES

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