BEP and Throughput Analysis of IncrementalSelective Relaying in DS-CDMA Systems

Hela Hakim1,2, Hatem Boujemaa1 and Wessam Ajib2

1Carthage University, Higher School of Communication of Tunis, Tunisia2Department of Computer Sciences, Universite de Quebec a Montreal, Canada

Emails: hela.hakim@gmail.com, boujemaa.hatem@supcom.rnu.tn and ajib.wessam@uqam.ca

Abstract—In this paper, we derive exact form expressions forthe Bit Error Probability (BEP) and throughput of cooperativeDirect Sequence-Code Division Multiple Access (DS-CDMA)systems using incremental selective relaying which combinesselective relaying with incremental relaying protocols in thepresence of multipath propagation. The derived results are validfor any multipath intensity profile of the channel and any pathdelays. They also consider the correlation of the multipath gains.Simulation results along with analytical studies of BEP andthroughput prove that the combination of incremental relayingwith selective relaying in cooperative DS-CDMA systems improvesignificantly the throughput performance and can achieve themaximum possible spatial diversity when it is required bydestination.

I. INTRODUCTION

Cooperative Communication is an innovative technique tocreate spatial diversity without any reliance on multiple an-tennas. It consists in introducing relay node(s) to increasediversity order by relaying the transmitter signal to the receiver[1]–[3]. This paper focuses on digital relaying mode wherethe relay decodes the received message from the source andthen forwards the decoded message to the destination. Inincremental relaying (IR), the process of relay selection andcooperative transmission is done only if the signal-to-NoiseRatio (SNR) in source-destination link is below a predeter-mined threshold γt. Otherwise, no cooperative transmissionwill be performed and the source transmits the next data [7].In case where the SNR in source-destination link is belowγt, the system uses the selective relaying protocol (SR) whichconsists in building a set of “reliable” relays, defined as therelays having an SNR in source-relay link beyond γt. Onlyone “reliable” relay is selected to cooperate. We consider theselection of the relay having the largest SNR in the relay-destination link. The motivation behind the use of SR is toprevent relays which receive the source signal with an SNRbelow γt to participate in cooperation process since they arelikely not able to correctly decode the source signal and thusthey can not forward a correct information to destination.The selection of one relay among “reliable” relays avoid thatthe system squanders many time slots (a time slot for each“reliable” relay) to transmit one symbol. The combination ofincremental relaying and several relaying schemes has beenstudied in earlier works and their performances have beenanalyzed over Rayleigh fading environments in the presence ofa single path propagation [4]-[7]. Namely in [4], Chen et. all

have derived closed-form expressions for error probability ofincremental-selective digital relaying scheme which combinesthe incremental digital relaying and the selective digital re-laying. [5] proposes a new incremental relaying transmissiontechnique in conjunction with selective digital relaying, andprovides its performance in terms of outage probability andbit error probability. In [6], the authors have derived a closed-form expression for the end-to-end (e2e) bit-error rate (BER)of incremental opportunistic relaying scheme which com-bines incremental relaying scheme and opportunistic relayingscheme. The performance of the incremental-relay-selectiondecode-and-forward technique over independent non-identicalRayleigh fading channels is derived in terms of average bit er-ror probability (BEP), outage probability and average channelcapacity in [7]. In [8], the author derived the BEP for selectivedigital relaying assuming multipath propagation. However, thecombination with incremental relaying is not considered andthroughput performance is not studied. Besides, the relayselection in [8] is different from the one considered in ourwork.

All previous works dealing with the combination of incre-mental relaying with other relaying schemes consider Rayleighfading environments and single path channels. In this paper,we consider cooperative DS-CDMA systems and multipathpropagation channels. We analyze e2e BEP and throughputperformances of incremental selective relaying (ISR). The con-sidered cooperative system uses incremental relaying protocolin conjunction with selective relaying. We assume that thedestination knows the SNR of RΘ −D links through trainingsequences, where Θ is the set of “reliable” relays and RΘ

denotes the relays in the set Θ. Based on the collected SNRinformation, the destination selects one relay among Θ. Thederived results are valid for any multipath intensity profileof the channel, any path delays and take into account thecorrelation between path gains.

The remainder of this paper is organized as follows. Insection II, we present the system model. In sections III andIV, we derive e2e BEP and throughput expressions of DS-CDMA cooperative system using ISR in the presence ofmultipath propagation. In section V, we present simulationand theoretical results. Section VI draws concluding remarksand a summary of our findings.

978-1-4673-1881-5/12/$31.00 ©2012 IEEE

II. SYSTEM MODEL

A. Scenario of the combined digital relaying protocol

We consider a source (S), a destination (D) and M potentialrelays Ri. We assume that the destination knows the SNR ofRΘ−D links, where Θ is the set of RΘ, the “reliable” relays.Based on the collected SNR information, the destinationselects one relay belonging to Θ. The derived results are validfor any multipath intensity profile of the channel, any pathdelays and take into account the correlation between pathgains.

Fig. 1. Phase 1: S broadcasts a signal to D while relays listen

A two-phase digital relaying protocol is considered. In thefirst phase (1st time slot) the source broadcasts its signalas shown in Fig. 1. If the destination has a received SNRabove a threshold value γt, it feeds back this informationinforming the potential relays that cooperation is not requiredand then the source transmits the next data during the 2nd

time slot. Otherwise, the destination selects the relay belongingto Θ having the largest SNR in RΘ − D links. In thesecond phase (2nd time slot), the selected relay forwardsdata to the destination. Then, the destination uses a Rakereceiver to estimate the transmitted symbol from the sourceand the selected relay. Finally, it combines these estimatesusing maximum ratio combining (MRC).

Let LXY , f lXY and τ l

XY the number of paths, the complexgain and the delay of the path l of the X-Y link. The noise atY is an additive white gaussian noise with variance NXY .

The mth correlation of the Rake receiver at Y during thekth symbol period can be written as [8]

zmXY (k) = sk

LXY∑l=1

f lXY (kTs)q(τm

XY − τ lXY ) + nm(k), (1)

where sk is the k-th transmitted symbol, nm(k) is a term due tonoise, q(τ) = (g�g)(τ), � denotes the convolution operationand g(t) is the shaping filter.

III. E2E BEP ANALYSIS OF THE SYSTEM

In this section, we derive the e2e BEP at D for BinaryPhase Shift Keying (BPSK) modulation and Rayleigh fadingchannels. The e2e BEP at D can be written as

Pe,D = P (γSD < γt)Pndiv(e|γSD < γt) + P (γSD ≥ γt)×Pdirect(e|γSD ≥ γt), (2)

where Pndiv(e|γSD ≥ γt) is the average conditional BEPat the destination given that γSD ≥ γt, i.e., the destinationneeds a cooperative diversity and Pdirect(e|γSD < γt) is theaverage BEP at the destination given that γSD < γt, i.e., thedestination relies only on the direct transmission. Next, wederive each term of (2). The probability density function (PDF)of γXY is [8]:

pγXY(x) =

LXY∑j=1

π(j)XY

β(j)XY

exp(− x

β(j)XY

), if x ≥ 0, (3)

where

β(j)XY = λ

(j)XY

EX

NXY, (4)

and

π(j)XY =

∏1≤k≤LXY

k �=j

λ(j)XY

λ(j)XY − λ

(k)XY

, (5)

where λ(j)XY is the jth eigenvalue of the matrix√

QXY E(fXY f†XY )√

QXY . The probability that γXY < γt

is given by

P (γXY < γt) =LXY∑j=1

π(j)XY [1 − exp(− γt

β(j)XY

)]. (6)

The conditional probability Pdirect(e|γSD ≥ γt) can bewritten as

Pdirect(e|γSD ≥ γt) =∫ ∞

γt

Q(√

2x)pγSD|γSD≥γtdx, (7)

where Q(x) = 1√2π

∫ +∞x

e−t2/2dt and

pγSD|γSD≥γt=

{pγSD

(x)

ΛSD(γt), if x ≥ γt

0, o.w.(8)

where ΛXY (γt) = P (γXY ≥ γt). Using integration by partsand an adequate variable substitution we obtain

Pdirect(e|γSD ≥ γt) =LSD∑j=1

π(j)SD

ΛSD(γt)

[Q(√

2γt)e− γt

β(j)SD

−√√√√ β

(j)SD

1 + β(j)SD

Q

(√2γt(1 +

1

β(j)SD

)

)⎤⎦ . (9)

On the other hand, Pndiv(e|γSD < γt) can be written as

Pndiv(e|γSD < γt) =∑Θ

P (Θ)Pndiv(e|γSD < γt,Θ), (10)

where

P (Θ) =∏i∈Θ

LSRi∑j=1

π(j)SRi

exp(− γt

β(j)SRi

)∏k∈Θ

LSRk∑j=1

π(j)SRk

×[1 − exp(− γt

β(j)SRk

)]. (11)

If Θ contains more than two relays, we have

Pndiv(e|γSD < γt,Θ) =∑q∈Θ

Pndiv(e|I)P (RSelΘ = q),(12)

where I = {γSD < γt,Θ, RSelΘ = q} and RSelΘ is theactivated relay in Θ. The probability P (RSelΘ = q) is givenby

P (RSelΘ = q) =∏k∈Θk �=q

P (γRqD > γRkD). (13)

The obtained expression of P (RSelΘ = q) is given by

P (RSelΘ = q) =

∏k∈Θk �=q

LRqD∑j=1

π(j)RqD

LRkD∑l=1

π(l)RkD

β(l)RkD

⎡⎣β

(l)RkD −

β(j)RqDβ

(l)RkD

β(j)RqD + β

(l)RkD

⎤⎦ .(14)

If the selected relay belonging to Θ (RSelΘ) decodesincorrectly the received signal, it forwards an erroneous signalleading to error propagation event. The BEP at the destinationdue to error propagation is denoted Pprop,D(e|I) whereas theBEP at the destination given that RSelΘ forwarded a correctlydecoded signal is Pcoop,D(e|I).

Pndiv(e|I) = PRSelΘ(e|I)Pprop,D(e|I)+(1 − PRSelΘ(e|I))Pcoop,D(e|I), (15)

where PRSelΘ(e|I) is the BEP at RSelΘ. Pprop,D(e|I) can bebounded by the worst case value i.e., Pprop,D(e|I) ≈ 1

2 as in[13] and [14].

Following the same methodology to obtain the conditionalprobability Pdirect(e|γSD < γt) in (9), we obtain the BEP atRSelΘ,

PRSelΘ(e|I) =LSRSelΘ∑

j=1

π(j)SRSelΘ

ΛSRSelΘ(γt)

[Q(√

2γt)e− γt

β(j)SRSelΘ

−√√√√ β

(j)SRSelΘ

1 + β(j)SRSelΘ

Q

(√2γt(1 +

1

β(j)SRSelΘ

)

)⎤⎦ .(18)

The conditional probability Pcoop,D(e|I) is given by

Pcoop,D(e|I) =∫ ∞

0

∫ γt

0

Q(√

2(x + u))pγSD|γSD<γt(x)

×pγRSelΘD(u)dxdu. (19)

The conditional PDF pγSD|γSD<γtis given by

pγSD|γSD<γt(x) =

{pγSD

(x)

ΥSD(γt), if 0 ≤ x < γt

0, o.w.(20)

where ΥSD(γt) = P (γSD < γt). Let F denotes the integral

F =∫ γt

0

Q(√

2(x + u))pγSD|γSD<γt(x)dx. (21)

Using integration by parts, we obtain the expression of F in(16). The SNR of RSelΘ − D is given by

γRSelΘD = maxi∈Θ

{γRiD}. (22)

Let {l(Θ, i, p)}|Θ|−1p=1 be the set of relays indices which belong

to Θ and different from i. The PDF of γRSelΘDis given

in [8, eq. (36)] where εn,|Θ| = (εn,|Θ|(1), . . . εn,|Θ|(|Θ| −1)) is the binary representation of 0 ≤ n ≤ 2|Θ|−1 −1, ξ(n) =

∑|Θ|−1p=1 εn,|Θ|(p) and 1

αnikm1...m|Θ|−1=

1

β(k)RiD

+∑|Θ|−1

p=1εn,|Θ|(p)

β(mp)Rl(Θ,i,p)D

. To determine the expression of

Pcoop,D(e|I) in (19), we use the following result which canbe proved using integration by parts

Ψ(a, b, α) =∫ ∞

0

Q(√

au + b)1α

exp(−u

α)du =

Q(√

b) −√

11 + 2

aα

exp(b

aα)Q

(√b(1 +

2aα

)

). (23)

Using (16) and (23), we find the expression of Pcoop,D(e|I)given in (17), where 1

α′nijkm1...m|Θ|−1

= 1αnikm1...m|Θ|−1

−1

β(j)SD

. By substituting equations (17) and (18) in (15), we

obtain the expression of Pndiv(e|I). By substituting the re-sult equation and (14) in (12), we obtain the expressionof Pndiv(e|γSD<γt,Θ). Finally, by substituting the resultequation and (11) in (10), we obtain the expression ofPndiv(e|γSD<γt

). The expressions of the conditional proba-bility Pndiv(e|γSD<γt,Θ) for the cases where Θ contains asingle relay and Θ is an empty set can be straightforwardlyobtained using similar derivations. By substituting the obtainedexpressions of Pndiv(e|γSD<γt

) and Pdirect(e|γSD ≥ γt) in(2), we obtain the expression of Pe,D.

IV. THROUGHPUT ANALYSIS OF THE SYSTEM

Throughput is the amount of data successfully delivered pertime unit. The throughput of the three protocols is given by

Thx =R(1 − P x

e,D)E(Tx)

, (24)

where R (bits/s/Hz) is the target transmission rate andE(Tx), x ∈ {IR, SR, ISR} is the expected number of timeslots and P x

e,D, x ∈ {IR, SR, ISR} is the average e2e BEPat D for IR, SR and ISR, respectively.

1) Throughput Analysis of IR: For IR, the average e2e BEPP IR

e,D is given by

P IRe,D = P (γSD ≥ γt) (1 − Pdirect(e|γSD ≥ γt))

+P (γSD < γt)(1 − Pcoop,D(e|γSD < γt)), (25)

where Pdirect(e|γSD ≥ γt) is given in (9), P (γSD ≥ γt) =1 − P (γSD < γt), P (γSD < γt) is given by (6) andPcoop,D(e|γSD < γt) is given by (17).

The expected number of time slots is given by

E(TIR) =P (γSD < γt) × 2 + P (γSD ≥ γt) × 1. (26)

2) Throughput Analysis of SR: For SR, the average e2eBEP PSR

e,D is given by

PSRe,D = Γ(γt)(1− Pdirect(e)) + (1− Γ(γt))(1− Pcoop,D(e)),

(27)

F =LSD∑j=1

π(j)SD

1 − ΛSD(γt)

⎡⎣Q(

√2u) − Q(

√2(γt + u))e

−γt

β(j)SD −

exp( u

β(j)SD

)√1 + 1

β(j)SD

(Q

(√2(1 +

1

β(j)SD

)u

)− Q

(√2(1 +

1

β(j)SD

)(γt + u)

))⎤⎦ .

(16)

Pcoop,D(e|I) =∑i∈Θ

LRiD∑k=1

π(k)RiD

β(k)RiD

LRl(Θ,i,1)D∑m1=1

π(m1)Rl(Θ,i,1)D

. . .

LRl(Θ,i,|Θ|−1)D∑m|Θ|−1=1

π(m|Θ|−1)

Rl(Θ,i,|Θ|−1)D

2|Θ|−1−1∑n=0

(−1)ξ(n)αnikm1...m|Θ|−1

LSD∑j=1

π(j)SD

1 − ΛSD(γt)

×

⎡⎢⎢⎣Ψ(2, 0, αnikm1...m|Θ|−1) − e

−γt

β(j)SD Ψ(2, 2γt, αnikm1...m|Θ|−1) −

α′nijkm1...m|Θ|−1

αnikm1...m|Θ|−1

√β

(j)SD+1

β(j)SD

[Ψ

(2(1 +

1

β(j)SD

), 0, α′nijkm1...m|Θ|−1

)

−Ψ

(2(1 +

1

β(j)SD

), 2(1 +1

β(j)SD

)γt, α′nijkm1...m|Θ|−1

)]].

(17)

where Γ(γt) =M∏i=1

P (γSRi< γt) and is given by

Γ(γt) =M∏i=1

⎡⎣LSRi∑

j=1

π(j)SRi

[1 − exp(− γt

β(j)SRi

)]

⎤⎦ . (28)

Pdirect(e) is given by [8, eq. (18)]. Pcoop,D(e) can be writtenas

Pcoop,D(e) =∑Θ

P (Θ)Pcoop,D(e|Θ), (29)

where P (Θ) is given in equation (11). If Θ contains more thantwo relays, we have

Pcoop,D(e|Θ, RSelΘ) =∫ ∞

0

∫ ∞

0

Q(√

2(γ + β))pγSD(γ)

×pγRSelΘD(β)dγdβ. (30)

The expression of Pcoop,D(e|Θ) is given in [8, (30)], where

Δnijkm1...m|Θ|−1 = Ω(β(j)SD)

β(j)SDαnikm1...m|Θ|−1

β(j)SD − αnikm1...m|Θ|−1

+Ω(αnikm1...m|Θ|−1)α2

nikm1...m|Θ|−1

αnikm1...m|Θ|−1 − β(j)SD

, (32)

where Ω(x) = 12

[1 −

√x

x+1

].

The expected number of time slots is given by

E(TSR) = Γ(γt) × 1 + (1 − Γ(γt)) × 2. (33)

3) Throughput Analysis of ISR: For ISR, the average e2eBEP P ISR

e,D is given by

P ISRe,D = P (γSD ≥ γt) (1 − Pdirect(e|γSD ≥ γt))

+P (γSD < γt)Γ(γt) (1 − Pdirect(e|γSD < γt))+(1 − Γ(γt)) (1 − Pcoop,D(e|γSD < γt)) P (γSD < γt).(34)

The expected number of time slots is given by

E(TISR) = [P (γSD ≥ γt) + P (γSD < γt)Γ(γt)] × 1P (γSD < γt)(1 − Γ(γt)) × 2. (35)

V. NUMERICAL AND SIMULATION RESULTS

In this section, we provide numerical and simulation resultsin terms of BER and throughput performances for BPSKmodulation. We allocate the same power to the source and theactivated relay, i.e. EX = Eb/2, where Eb is the transmittedenergy per bit. All the paths of a given link have equal averagepower and i.i.d gains.

0 2 4 6 8 10 12 14 16 18 2010

−5

10−4

10−3

10−2

10−1

100

Eb/N

0 (dB)

BE

R

Theo: Direct transmission, η=Tc/4

Theo: Direct Transmission, η=Tc

Theo: Incremental relaying+SR, η=Tc/4

Theo: Incemental relaying+SR, η=Tc

Sim: Direct Transmission, η=Tc/4

Sim: Direct transmission, η=Tc

Sim: Incremental relaying+SR, η=Tc/4

Sim: Incremental relaying+SR, η=Tc

Fig. 2. Effect of time delay spacing on BER, L=2, M=2, γt = 6 dB

For a link X-Y, let LXY and τ lXY be the number of paths

and the delay of the path l, respectively. Fig. 2 studies theeffect of time delay spacing η = τ2

XY − τ1XY on the BEP for

LXY = L = 2, M = 2 and γt = 6 dB. We observe that thediversity decreases as path delays decrease. At high SNR, theBER performance tends to have a diversity order of 1. This

Pcoop,D(e|Θ)=LSD∑j=1

π(j)SD

∑i∈Θ

LRiD∑k=1

π(k)RiD

β(k)RiD

LRl(Θ,i,1)D∑m1=1

π(m1)Rl(Θ,i,1)D

. . .

LRl(Θ,i,|Θ|−1)D∑m|Θ|−1=1

π(m|Θ|−1)

Rl(Θ,i,|Θ|−1)D

2|Θ|−1−1∑n=0

(−1)ξ(n)Δnijkm1...m|Θ|−1 ,(31)

is because at high SNR, the destination will rarely need anyretransmission from the relay, thus no cooperative transmissionwill be performed and hence the system will have a diversityorder equal to 1. Finally, we observe a perfect agreementbetween analytical BEP results and simulations curves.

0 5 10 15 200.4

0.5

0.6

0.7

0.8

0.9

1

Eb/N

0 (dB)

Th

rou

gh

pu

t

Sim: IRSim: SRSim: ISRTheo: IRTheo: SRTheo: ISR

Fig. 3. Throughput comparison for η = Tc, L=2, M=2 and γt = 6 dB

0 5 10 15 2010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N

0

BE

R

Direct transmissionIncremental RelayingIncremental relaying+SRSR

Fig. 4. BER comparison for η = Tc, L=2, M=2 and γt = 6 dBFig. 3 and Fig. 4 compare throughput and BER perfor-

mances of IR, SR and the combined ISR protocol, respectively,for R = 1 bit/s/Hz. We observe that ISR provides signif-icantly higher throughput compared to SR or IR exclusivelywithout deteriorating the BER performance mainly at mediumSNR. At low SNR, performances of the combined protocolin terms of BER is confused with those of SR, while at highSNR they tend to those of IR. This is because at low SNR,the SNR of S-D link is often below γt so relaying process is

controlled by the SR protocol only. At high SNR, the SNRof the links between S and relays are often beyond γt, hence,only IR protocol controls the system.

VI. CONCLUSION

In this paper, we studied BEP and throughput performancesof cooperative DS-CDMA systems using incremental relayingin conjunction with best relay selection in the presence ofmultipath propagation. The derived results are valid for anymultipath intensity profile, any path delays, and take intoaccount the correlation between path gains. Throughput perfor-mance analysis shows that the combination between selectiveand incremental relaying significantly improves the systemthroughput without deteriorating BER performance.

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