www.ietdl.org Published in IET Communications Received on 29th December 2009 Revised on 21st May 2010 doi: 10.1049/iet-com.2009.0840

ISSN 1751-8628

Moment generating function-based performance evaluation of amplify-andforward relaying in N∗Nakagami fading channels H. I˙lhan1 I˙. Altunbas¸1 M. Uysal2 1

Department of Electronics and Communication Engineering, Istanbul Technical University, Maslak, Istanbul 34469, Turkey ¨ zyeg˘in University, 34662, Istanbul, Turkey Faculty of Engineering, O E-mail: [email protected] 2

Abstract: In this study, the authors investigate the error rate performance of amplify-and-forward relaying over N∗ Nakagami fading channels. This is a recently introduced channel model that involves the product of N Nakagami-m-distributed random variables. Employing the moment generating function approach, the authors derive symbol error rate expressions for a single-relay system under instantaneous power scaling (IPS) and average power scaling (APS) factors at the relay node, that is, variable and ﬁxed gains. The results achieved by the authors demonstrate that the achievable diversity order is a function of Nakagami fading parameter (m) and degree of cascading (N ). An identical diversity order is obtained under both scaling factors when the relay is close to the destination. When the relay is close to the source, IPS becomes advantageous over APS. Monte –Carlo simulations are further provided to conﬁrm the analytical results.

1

Introduction

Cooperative communication techniques [1 – 3] have emerged as a powerful alternative to MIMO (multi-input multioutput antennas) communications for wireless applications in which size and power limitations exclude the possibility of using multiple co-located antennas. In a cooperative communication system, network nodes help each other via relaying information and realise spatial diversity advantages in a distributed manner. With its indisputable advantages and wide range of potential applications areas, cooperative communication has received much attention and extensively studied over the last few years, see, for example, [1 – 12] and the references therein. A key assumption in majority of these works is the underlying fading channel model which is characterised by Rayleigh distribution because of multipath fading. When the channel suffers from both long-term (i.e. shadowing) and short-term (i.e. multipath) fading, composite multipath/ IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

shadowing channel models are considered in which shadowing effect is typically modelled by log-normal fading [13]. A recent composite channel model so-called generalised-K fading is further considered in [14, 15]. This leads to more tractable mathematical expressions in the performance analysis. For example, performance results for dual-hop transmissions over generalised-K fading channels can be found in [16, 17]. Classical Rayleigh fading captures the propagation characteristics of cellular radio systems. This statistical model has been originally developed under the assumption of a wireless communication scenario with a stationary base station antenna above rooftop level and a mobile station at street level. In contrast to this cellular scenario, in intervehicular communication and mobile ad hoc applications, transmitter and receiver antennas are close to the ground level and experience fading generated from independent groups of scatterers around the mobile terminals. This requires the deployment of ‘cascaded’ 253

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www.ietdl.org fading channels to realistically model such scenarios [18 – 20]. Such cascaded channels can be also used when the transmission media contains keyholes [21] and diffracting wedges such as street corners and/or rooftops [19]. They can be further considered for modelling the composite fading. Cascaded Rayleigh fading channel, which involves the product of N independent Rayleigh-distributed random variables, is presented in [22]. For N ¼ 2, this reduces to double Rayleigh fading model which has been considered in [23, 24]. In addition, double Rayleigh fading channel model has been used for keyhole channel modelling of MIMO communications [21, 25]. Based on Nakagami-m fading, cascaded Nakagami (also named as N∗ Nakagami) channel is proposed in [26]. Special case with N ¼ 2 is studied in [27]. Cascaded Weibull and cascaded generalised-K channels are further introduced and investigated in [28, 29], respectively. In a recent paper [30], we have investigated the error rate performance of a cooperative intervehicular system over a double Nakagami (i.e. N ¼ 2) fading channel. A Chernoff bound on the pair-wise error probability is derived in [30] assuming average power scaling (APS) constraint, that is, ﬁxed gain, at the relay node and a union bound on error rate is presented. In this paper, we extend our analysis for N∗ Nakagami case which subsumes double Nakagami in [30] as a special case. We derive an exact symbol error rate (SER) expression based on the moment generating function (MGF) approach assuming instantaneous power scaling (IPS) constraint, that is, variable gain, as well as APS. The derived expressions give insight into the achievable diversity orders for various relay locations and channel parameters. The rest of the paper is organised as follows: In Section 2, we describe the relay-assisted transmission model and fading channel under consideration. In Section 3, we derive the MGF expressions over N∗ Nakagami channel for IPS and APS constraints. In Section 4, we present Monte – Carlo simulation results to verify the analytical results. Finally, we conclude in Section 5.

2

System model

We consider a single-relay scenario (as shown in Fig. 1) where source, relay and destination nodes operate in half-duplex mode and are equipped with a single pair of transmit and receive antennas. In Fig. 1, aSD , aSR and aRD represent

Figure 1 Relay-assisted cooperative system 254 & The Institution of Engineering and Technology 2011

source-to-destination (S D), source-to-relay (S R) and relay-to-destination (R D) links’ complex fading coefﬁcients whose magnitudes hSD ¼ |aSD|, hSR ¼ |aSR| and hRD ¼ |aRD| follow cascaded Nakagami-m distribution, respectively. These magnitudes are assumed to be the product of statistically independent, but not necessarily identically distributed Nakagami-m random variables [26]. NSD NSR hSD,l1 , hSR = Pl2 =1 hSR,l2 Speciﬁcally, we have, hSD = Pl1 =1 NRD and hRD = Pl3 =1 hRD,l3 where NSD , NSR and NRD are the number of random variables (which we deﬁne as ‘degree of cascading’) for S D, S R and R D links, respectively. Here, hSD,l1 , l1 ¼ 1, 2, . . . , NSD; hSR,l2 , l2 ¼ 1, 2, . . . , NSR and hRD,l3 , l3 ¼ 1, 2, . . . , NRD’s are Nakagamim-distributed variables with probability density function (pdf) [13] m 2ml l ml 2 2ml −1 h h exp − fhl (hl ) = (E[h2l ])ml G(ml ) l E[h2l ] l

(1)

where the corresponding subscripts SD, SR and RD and numerical subscripts 1 – 3 are dropped for convenience. In (1), ml , l ¼ 1, 2, . . . , N is a parameter describing the fading severity given by ml = E[h2l ]/E[(h2l − E[h2l ])2 ] ≥ 0.5. Taking E[h2l ] = 1, one can normalise the power of the fading process to unity. Here, E[.] denotes the expectation operator and G(.) is the Gamma function [31]. The system model under consideration builds upon socalled receive diversity cooperation protocol [1, 6]. In this protocol, transmission is divided into two phases. During the ﬁrst transmission phase, the source node broadcasts to the relay and the destination nodes. In the second transmission phase, the source node is silent and the relay node forwards the signal received during the ﬁrst phase to the destination. The received signals at the relay and the destination are, respectively, given by ESR hSR x + nR = ESD hSD x + nD1

rR = rD1

(2) (3)

where x denotes M-PSK (phase shift keying) modulation symbol broadcasted from the source. In (2) and (3), ESR and ESD represent the average energies available at the relay and the destination which take into account possibly different path loss and shadowing effects in S R and S D links, respectively. nR and nD1 are independent samples of zero-mean complex Gaussian random variables with variance N0/2 per dimension. The relay node multiplies the received signal rR by a scaling factor of b = 1/ E[|rR |2 ] to restrict the energy at its output. Based on the availability of channel state information of S R link at the relay node, either IPS or APS constraint can be employed. In the former, the expectation is carried only over the noise term, whereas in the latter, the expectation is with respect to noise and fading terms. IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

www.ietdl.org where VgSD (s) and VgSRD (s) are the MGFs of gSD and gSRD , respectively. VgSD (s) is given by

Therefore the scaling factor takes the form of ⎧ ⎪ ⎨ 1/ E [|rR |2 ] = 1/ ESR |hSR |2 + N0 , nR b= ⎪ ⎩ 1/ E [|rR |2 ] = 1/ ESR + N0 , nR ,hSR

1

for IPS VgSD (s) =

for APS

After scaling by b, the relay forwards the resulting signal to the destination. After proper normalisations [3], the received signal can be written as rD2

ESR = hSR hRD x + nD2 2 (b ERD )−1 + h2RD

(4)

exp(sgSD )fgSD (gSD ) dgSD

0

(9)

Inserting (7) in (9) and using the closed-form solution [31, Eq. 7.813.1], we have ⎛

⎞

NSD SD 1 1 NSD ,1 ⎜ Pl1 =1 ml1 G1,NSD ⎝− VgSD (s) = N s gSD SD P SD G(mSD ) l1 =1

⎟ ⎠

,...,mSD m1 ,mSD N 2

l1

SD

(10) where ERD represents the average energy available at the destination node and nD2 is conditionally zero-mean complex Gaussian random variable with variance N0/2 per dimension. At the destination node, the received signals rD1 and rD2 are fed to a maximum likelihood detector.

3

On the other hand, VgSRD (s) can be calculated as 1 1 VgSRD (s) = =

Derivation of SER

(M−1)p/M Vgend

0

sin2 (p/M) s=− du sin2 u

gSRD =

for IPS

⎪ ⎪ ⎩

for APS

gSR gRD , gRD + ESR /N0 + 1

⎞ ⎟ ⎟ ⎠

m1 ,m2 ,...,mSR N

SR

⎛

− N RD ⎜ N ,0 gRD RD ×G0,NRDRD⎜ m l3 ⎝g RD l =1 RD 3

⎞ ⎟ ⎟dgSR dgRD ⎠

,...,mRD m1 ,mRD N 2

RD

(11) Substituting (10) and (11) into (8), we have ⎛

⎞ 1 B N ,1 ⎠ Vgend (s) = AG1,NSDSD ⎝− s mSD , mSD ,..., mSD 1 1

Noting that hSD , hSR and hRD follow cascaded Nakagami-m distribution, the pdf of corresponding instantaneous SNR is given by [26] − N 1 N ,0 ⎝g G0,N ml f g (g ) = N g gPl =1 G(ml ) l =1

1 NSR NRD RD Pl2 =1 G(mSR l2 )Pl3 =1 G(ml3 )

1 1 exp(sgSRD ) 0 0 gSR gRD ⎛ − NSR g NSR ,0 ⎜ SR SR ⎜ m × G0,NSR ⎝ SR l =1 l2 g 2 SR SR

(5)

(6)

⎛

exp(sgSRD )fgSR (gSR )fgRD (gRD )dgSR dgRD

×

where Vgend (s) is the MGF of instantaneous endto-end signal to noise ratio (SNR) gend . Let the instantaneous SNRs of S D, S R and R D links SD = h2SD ESD /N0 , gSR = h2SR g SR = be given by gSD = h2SD g RD = h2RD ERD /N0 . gend is then h2SR ESR /N0 and gRD = h2RD g obtained as the summation of instantaneous SNRs of direct and relaying links, that is, gend ¼ gSD + gSRD where ⎧ gSR gRD ⎪ , ⎪ ⎨ gRD + gSR + 1

0

1 1

Based on the MGF approach, the exact SER for M-PSK is given by [13] 1 Ps = p

0

× 0

×

⎞ ⎠

0

1

2

NSD

1 1 exp(sgSRD ) gSR gRD

N ,0 G0,NSRSR (C gSR | − , mSR ,..., mSR mSR N 1 2

)

SR

× G0,NRDRD (K gRD | − mRD , mRD ,..., mRD )dgSR dgRD N

(7)

,0

1

2

NRD

m1 ,m2 ,...,mN

(12)

where the subscripts SD, SR and RD are dropped for p,t (.|.), p, t, u, v [ N , p ≤ v, t ≤ u convenience and Gu,v is the Meijer’s G-function [31]. The MGF of instantaneous end-to-end SNR can be calculated as Vgend (s) = VgSD (s)VgSRD (s) IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

(8)

N

N

N

SR RD SD SR RD where A = 1/(Pl1 =1 G(mSD l1 )Pl2 =1 G(ml2 )Pl3 =1 G(ml3 )), B = N

N

N

SD SR RD mSD gSD , C = Pl2 =1 mSR gSR and K = Pl3 =1 mRD Pl1 =1 l1 / l2 / l3 / RD . To the best of our knowledge, a closed-form solution g for (12) is unfortunately not available for the general case, yet this two-fold integral can be numerically evaluated through commercially available mathematics software such as Mathematica, Matlab and Maple. In the following, we

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www.ietdl.org investigate some representative cases to seek closed-form expressions.

respect to gSR leads to Vgend (s)

3.1 Case I: relay close to source

⎛

When the relay is very close to the source node, SNR in S R link is much larger than that in R D link, that is, gSR ≫ gRD . Therefore under IPS constraint, we have gSRD ¼ gSRgRD/( gRD + gSR + 1) ≃ gRD . In this case, using the closed-form solution [31, Eq. 7.811.4], the inner integration in (12) with respect to gSR yields

⎞

1

B N ,1 = AG1,NSDSD ⎝− s

⎠

,mSD ,...,mSD mSD N 1 2

1 −1 1,NSR × gRD GNSR ,1 − 0

1−mSR ,1−mSR ,...,1−mSR 1 NSR 2 g C(ESR /N0 ) RD 0

⎛

⎞ 1 NSD ,1 ⎝ B SR ⎠ G(ml2 )G1,NSD − =A s mSD ,mSD ,...,mSD l =1 NSR 2

1

1

N

,0

0

,0

2

NRD

(13) Further using [31, Eq. 7.813.1], we obtain ⎛

⎞ 1 NSD ,1 ⎝ B ⎠ G(mSR Vgend (s) = A l2 )G1,NSD − s mSD ,mSD ,...,mSD l =1 NSR 2

1

⎛

2

⎞ 1 K N ,1 ⎠ × G1,NRDRD ⎝− s mRD ,mRD ,...,mRD 1

2

(16)

NRD

Similar to (14), (17) can be alternatively obtained in terms of the hypergeometric pFq function.

NSD

1

2

Further using [33, Eq. 21], we obtain (see (17))

− RD g−1 RD exp(sgRD )G0,NRD (K gRD |mRD ,mRD ,...,mRD ) dgRD N

×

2

s

× G0,NRDRD (K gRD |− mRD ,mRD ,...,mRD )dgRD 1

Vgend (s)

SD

NSD

(14)

NRD

Replacing (14) and (17) in (5), we obtain the exact SER expressions, respectively, under IPS and APS constraints. These expressions can be also used for diversity gain evaluation. In Table 1, we present the (asymptotic) diversity orders achieved for various values of NSD, NSR , mSD mSR l2 = 1, 2, . . . , NRD , l1 , l1 = 1, 2, . . . , NSD , l2 , NSR and mRD l3 , l3 = 1, 2, . . . , NRD . These are obtained numerically from the slope of the SER–SNR function (i.e. 2log Ps/log SNR) in high SNR region for values of m and N under consideration. It can be concluded from observations in Table 1 that the maximum diversity order is given by SD SD RD RD RD min(mSD 1 , m2 , . . . , mNSD ) + min(m1 , m2 , . . . , mNRD ) for SD IPS constraint and by min(mSD 1 , m2 , . . . ,

Note that (14) can be written in terms of hypergeometric pFq function using [32, Eq. (07.34.26.0004.01)] for (mj 2 mk) Z, j = k, j, k [ {1, 2, . . . , NSD (or NRD)}. For the double Nakagami case (NSD ¼ NSR ¼ NRD ¼ 2) without the constraint on (mj 2 mk), it can be also written in terms of hypergeometric U-function [31] using the results from [32, Eq. (07.34.03.0719.01)]

SD RD B m1 K m1 − Vgend (s) = − s s B SD SD SD × U m1 , 1 + m1 − m2 , − s K RD RD RD × U m1 , 1 + m1 − m2 , − s

(15)

SR SR RD RD RD min(mSR 1 , m2 , . . . , mNSR , m1 , m2 , . . . , mNRD ) for APS constraint.

3.2 Case 2: relay close to destination When the relay is very close to the destination, SNR in S R link is much smaller than that in R D link, that is, gSR ≪ gRD . Therefore under the IPS constraint, we have gSRD ¼ gSRgRD/(gRD + gSR + 1) ≃ gSR . Integrating of (12) over gRD and gSR and using the closed-form solutions [31, Eq. 7.811.4] and [31, Eq. 7.813.1], respectively, the MGF is given by ⎛

⎞ 1 B NSD ,1 ⎝ ⎠ G(mRD Vgend (s) = A l3 )G1,NSD − s mSD ,mSD ,...,mSD l =1 N RD 3

Under APS constraint, we have gSRD ¼ gSRgRD/ ( gRD + ESR/N0 + 1) ≃ gSRgRD/(ESR/N0). Using closedform solution [31, Eq. 7.813.1], integration in (12) with ⎛

Vgend (s) =

⎞

1

B N ,1 AG1,NSDSD ⎝− s

,mSD ,...,mSD mSD N 1 2

×

256 & The Institution of Engineering and Technology 2011

1

⎛

⎞

1

C N ,1 G1,NSRSR ⎝− s

2

NSD

⎠

(18)

,mSR ,...,mSR mSR N 1 2

SR

⎛

⎞

1

⎠G NSR +NRD ,1 ⎝− CK (ESR /N0 ) 1,NSR +NRD

SD

mSD NSD ) +

s

⎠

(17)

,mSR ,...,mSR ,mRD ,mRD ,...,mRD mSR N N 1 2 1 2 SR

RD

IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

www.ietdl.org Table 1 Diversity orders achieved for various combinations of channel parameters SD SR SR RD RD NSD mSD 1 , . . . , mNSD NSR m1 , . . . , mNSR NRD m1 , . . . , mNRD

Diversity orders APS and gSR ≫ gRD

IPS and gSR ≫ gRD

APS/IPS and gRD ≫ gSR

1

1

1

2

1

3

3

4

3

1

1

1

2

1

1

2

2

3

1

1

1

2

2

1, 3

2

2

3

1

2

1

1

3

3, 4, 5

3

5

3

1

1

2

2, 1

1

3

2

4

2

1

2

2

3, 1

2

4, 5

3

6

3

1

2

2

4, 5

3

8, 7, 6

6

8

6

1

1

3

3, 4, 5

1

2

3

3

4

1

2

3

3, 1, 4

2

5, 6

3

7

3

1

2

3

3, 4, 2

3

5, 6, 4

4

6

4

2

2, 3

1

2

1

3

4

5

4

2

2, 1

1

2

2

5, 4

3

5

3

2

2, 4

1

2

3

3, 5, 6

4

5

4

2

3, 2

2

1, 2

1

3

3

5

3

2

2, 1

2

3, 1

2

5, 6

2

6

2

2

1, 1

2

2, 2

2

1, 1

2

2

3

2

3, 1

2

3, 2

3

4, 6, 3

3

4

3

2

2, 3

3

3, 2, 4

1

3

4

5

4

2

3, 2

3

1, 3, 4

2

6, 3

3

5

3

2

3, 1

3

3, 1, 4

3

5, 2, 5

2

3

2

3

2, 1, 3

1

2

1

3

3

4

3

3

2, 3, 4

1

3

2

5, 2

4

4

5

3

2, 3, 1

1

2

3

5, 4, 2

3

3

3

3

2, 1, 3

2

3, 1

1

4

2

5

2

3

1, 2, 3

2

4, 1

2

6, 5

2

6

2

3

1, 2, 3

2

3, 2

3

4, 5, 6

3

5

3

3

2, 3, 4

3

3, 2, 4

1

1

3

3

4

3

2, 3, 3

3

7, 5, 4

2

2, 5

4

4

6

3

1, 2, 3

3

4, 6, 7

3

5, 7, 8

5

6

5

3

1, 1, 1

3

1, 1, 1

3

3, 3, 3

2

4

2

3

1, 1, 1

3

2, 2, 2

3

1, 1, 1

2

2

3

3

2, 2, 2

3

1, 1, 1

3

1, 1, 1

3

3

3

3

2, 2, 2

3

4, 4, 4

3

2, 2, 2

4

4

6

3

4, 4, 4

3

1, 1, 1

3

2, 2, 2

5

6

5

IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

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www.ietdl.org For NSD ¼ NSR ¼ NRD ¼ 2, (18) can be further written in terms of the hypergeometric U-function as SD SR B m1 C m1 − Vgend (s) = − s s B SD SD SD × U m1 , 1 + m1 − m2 , − s C SR SR SR × U m1 , 1 + m1 − m2 , − s

† Scenario 1: m SD ¼ 1, m SR ¼ 1, m RD ¼ 3 and NSD ¼ NSR ¼ NRD ¼ 3.

(19)

Under APS constraint, we again have gSRD ¼ gSRgRD/ (gRD + ESR/N0 + 1) ≃ gSR . Therefore the MGF is identical to (18). Replacing (18) in (5), we obtain the exact SER expression under IPS and APS constraints for this relay location. In Table 1, we present the diversity orders achieved for various combinations of channel parameters. From these observations, the maximum diversity order turns out to be the same for both IPS and APS constraints and is given by SD SD SR SR SR min(mSD 1 , m2 , . . . , mNSD ) + min(m1 , m2 , . . . , mNSR ).

4

mechanisms. In Fig. 2, we consider the following scenarios based on the combinations of channel parameters:

Numerical results

In this section, we present Monte– Carlo simulations to conﬁrm the derived analytical results. We consider singlerelay scenario and assume 4-PSK modulation. In Figs. 2 and 3, we present the SER performance of the cooperative scheme against ESD/N0 when the relay is very close to the source under APS constraint. Speciﬁcally, we assume that the average SNR in S R link is 30 dB larger than that in R D link and ESD ¼ ERD . This can be justiﬁed in practice through the deployment of power control

† Scenario 2: m SD ¼ 2, NSD ¼ NSR ¼ NRD ¼ 3.

m SR ¼ 1,

m RD ¼ 1

and

† Scenario 3: m SD ¼ 2, NSD ¼ NSR ¼ NRD ¼ 3.

m SR ¼ 4,

m RD ¼ 2

and

† Scenario 4: m SD ¼ 4, NSD ¼ NSR ¼ NRD ¼ 3.

m SR ¼ 1,

m RD ¼ 2

and

In Fig. 3, we consider the following scenarios: † Scenario 5: m SD ¼ 1, m SR ¼ 2, m RD ¼ 1 and NSD ¼ NSR ¼ NRD ¼ 1. † Scenario 6: m SD ¼ 1, m SR ¼ 2, m RD ¼ 1 and NSD ¼ NSR ¼ NRD ¼ 2. † Scenario 7: m SD ¼ 1, m SR ¼ 2, m RD ¼ 1 and NSD ¼ NSR ¼ NRD ¼ 3. It is observed from both ﬁgures that the derived analytical SER expressions (dash lines) provide a perfect match to the simulation results (solid lines). Since diversity orders, by the deﬁnition, are for asymptotically high SNR, we extend our analytical results to very low SER values while the simulation results are limited to at most 1026 because of computation time. From Fig. 2, we observe that

Figure 2 SER performance of the cooperative scheme when the relay is very close to the source under APS constraint 258 & The Institution of Engineering and Technology 2011

IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

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Figure 3 SER performance of the cooperative scheme when the relay is very close to the source under APS constraint the diversity orders are given by 2, 3, 4 and 5, respectively, for scenarios 1–4 which conﬁrms our earlier observation SD SD SR SR mSR of min(mSD 1 , m2 , . . . , mNSD ) + min(m1 , m2 , . . . , NSR , RD RD RD m1 , m2 , . . . , mNRD ). From Fig. 3, we observe that diversity orders are 2 and identical for scenarios 5–7 again conﬁrming our analytical ﬁndings.

In Fig. 4, we return our attention to SER performance under IPS constraint and still assume the case when the relay is very close to the source with the aforementioned scenarios 1–4. Comparison of Figs. 2 and 4 reveals that diversity orders under IPS are higher. This is expected because it takes the advantage of channel state information (CSI) of the S R link available

Figure 4 SER performance of the cooperative scheme when the relay is very close to the source under IPS constraint IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

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www.ietdl.org at the relay. Speciﬁcally, we observe diversity orders of 4, 3, 4 and 6 for scenarios 1–4 which conﬁrms our earlier observation SD SD RD RD RD of min(mSD 1 , m2 , . . . ,mNSD ) + min(m1 , m2 , . . . , mNRD ). As earlier noted, we emphasise that we keep our SER and SNR values in a wide range to be able to demonstrate the diversity orders. As the degree of cascading increases,

convergence to the asymtotical diversity order becomes slower and is observed for very high SNR values. In Fig. 5, we consider the case when the relay is very close to the destination and compare the performance of APS and IPS constraints. Speciﬁcally, we assume that the average SNR

Figure 5 SER performance comparison of APS and IPS constraints when the relay is very close to the destination

Figure 6 Effect of relay location on the SER performance under APS constraint when mSD ¼ 2, mSR ¼ mRD ¼ 1 for scenario 2 260 & The Institution of Engineering and Technology 2011

IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

www.ietdl.org

Figure 7 Effect of relay location on the SER performance under APS constraint when mSR ¼ 4, mSD ¼ mRD ¼ 2 for scenario 3 in S R link is assumed 30 dB smaller than that in R D link and ESD ¼ ESR . In this relay location, the advantage of IPS over APS disappears. Recall that end-to-end SNRs for both cases under this relay location become the same. Therefore diversity orders are identical for both constraints, speciﬁcally 2 and 6 for scenarios 1 and 3, respectively, SD conﬁrming earlier observation of min(mSD 1 , m2 , . . . , SD SR SR SR mNSD ) + min(m1 , m2 , . . . , mNSR ).

ﬁxed gain based on the availability of channel knowledge at the relay. Our results demonstrate that a diversity order of SD SD RD RD RD min(mSD 1 , m2 , . . . , mNSD ) + min(m1 , m2 , . . . , mNRD ) is available under IPS constraint when the relay is located close to the source. Under APS constraint, a diversity SD SD SR SR order of min(mSD 1 , m2 , . . . , mNSD ) + min(m1 , m2 , . . . , SR RD RD RD mNSR , m1 , m2 , . . . , mNRD ) is achieved under the same location. When the relay is close to the destination, an SD SD identical diversity order of min(mSD 1 , m2 , . . . , mNSD ) + SR SR SR min(m1 , m2 , . . . , mNSR ) is observed for both constraints. Monte – Carlo simulations are further provided for various values of channel parameters and show excellent agreement with the derived analytical results.

In Figs. 6 and 7, we examine the effect of relay location on the SER performance. We assume APS constraint and compare the performance for ESR/ERD ¼ 230, 210, 0, 10 and 30 dB. In Fig. 6, we consider scenario 2 with m SD ¼ 2, m SR ¼ m RD ¼ 1 and observe that diversity order remains the same. Best performance is obtained for ESR/ ERD ¼ 230 dB while worst performance is obtained for ESR/ERD ¼ 30 dB. Speciﬁcally, we observe a performance difference of 6.5 dB between these two locations at the SER of 1024. In Fig. 7, we consider scenario 3 with m SR ¼ 4, m SD ¼ m RD ¼ 2 and observe that diversity orders for different relay locations differ from each other for m SR = m RD.

[1] LANEMAN J.N., WORNELL G.W.: ‘Distributed space-time coded protocols for exploiting cooperative diversity in wireless networks’, IEEE Trans. Inf. Theory, 2003, 49, (10), pp. 2415 – 2525

5

[2] SENDONARIS A. , ERKIP E. , AAZHANG B. : ‘User cooperation diversity part I: system description’, IEEE Trans. Commun., 2003, 51, (11), pp. 1927 – 1938

Conclusions

In this paper, we have derived SER expressions through MGF approach for a single-relay system with amplify-andforward relaying over N∗ Nakagami fading channels. We have considered both IPS and APS factors at the relay node which correspond to, respectively, variable gain and IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

6

References

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[5] YIU S., SCHOBER R., LAMPE L.: ‘Decentralized distributed space-time trellis coding’, IEEE Trans. Wirel. Commun., 2007, 6, (11), pp. 3985– 3993

[18] KOVACS I.Z.: ‘Radio channel characterization for private mobile radio systems: mobile-to-mobile radio link investigations’. PhD thesis, Aalborg University, September 2002

[6] OCHIAI H., MITRAN P., TAROKH V.: ‘Variable-rate two-phase collaborative communication protocols for wireless networks’, IEEE Trans. Inf. Theory, 2006, 52, (9), pp. 4299–4313

[19] ANDERSEN J.B. : ‘Statistical distributions in mobile communications using multiple scattering’. URSI General Assembly Proc., session CBF, August 2002

[7] TSIFTSIS T.A.: ‘Performance of wireless multihop communications systems with cooperative diversity over fading channels’, Int. J. Commun. Syst., 2008, 21, (5), pp. 559–565

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[8] MHEIDAT H., UYSAL M.: ‘Impact of receive diversity on the performance of amplify-and-forward relaying under APS and IPS power constraints’, IEEE Commun. Lett., 2006, 10, pp. 468– 470 [9] MAHAM B., HJORUNGNES A.: ‘Symbol error rate of amplifyand-forward distributed space-time codes over Nakagami-m fading channel’, IET Electron. Lett., 2009, 45, (3), pp. 174–175 [10] DUONG T.Q., ZEPERNICK J. , BAO V.N.Q.: ‘Symbol error probability of hop-by-hop beamforming in Nakagami-m fading’, IET Electron. Lett., 2009, 45, (20), pp. 1042– 1044 [11] SURAWEERA H., LOUIE R.H.Y., LI Y., KARAGIANNIDIS G.K., VUCETIC B.: ‘Two hop amplify-and-forward transmission in mixed Rayleigh and Rician fading channels’, IEEE Commun. Lett., 2009, 13, (4), pp. 227– 229 [12] SURAWEERA H., KARAGIANNIDIS G.K., SMITH P.J.: ‘Performance analysis of the dual-hop asymmetric fading channel’, IEEE Trans. Wirel. Commun., 2009, 8, (6), pp. 2783 – 2788 [13] SIMON M.K., ALOUINI M.S.: ‘Digital communication over fading channels’ (John Wiley & Sons, 2005, 2nd edn.) [14] SHANKAR P.M.: ‘Error rates in generalized shadowed fading channels’, Wirel. Pers. Commun., 2004, 28, (4), pp. 233– 238 [15]

BITHAS P.S., SAGIAS N.C., MATHIOPOULOS P.T., KARAGIANNIDIS G.K.,

RONTOGIANNIS A.A. : ‘On the performance analysis of digital communications over generalized-K fading channels’, IEEE Commun. Lett., 2006, 5, (10), pp. 353 – 355

[16] WU L., LIN J. , NIU K., HE Z.: ‘Performance of dual-hop transmissions with ﬁxed gain relays over generalized-K fading channels’. IEEE ICC’09, Dresden, Germany, pp. 1 – 5 [17] DATSIKAS C.K., PEPPAS K.P., LAZARAKIS F.I., TOMBRAS G.S.: ‘Error rate performance analysis of dual-hop relaying transmissions over 262 & The Institution of Engineering and Technology 2011

[21] CHIZHIK D., FOSCHINI G.J., VALENZUELA R.A.: ‘Keyholes, correlations, and capacities of multielement transmit and receive antennas’, IEEE Trans. Wirel. Commun., 2002, 1, (2), pp. 361– 368 [22] KARAGIANNIDIS G.K., TSIFTSIS T.A., SAGIAS N.C.: ‘A closed-form upper-bound for the distribution of the weighted sum of Rayleigh variates’, IEEE Commun. Lett., 2005, 9, (7), pp. 589– 591 [23] ERCEG V., FORTUNE S.J., LING J., RUSTAKO A., VALENZUELA R. : ‘Comparisons of computer-based propagation prediction tool with experimental data collected in urban microcellular environments’, IEEE J. Sel. Areas Commun., 1997, 15, (4), pp. 677– 684 [24] UYSAL M.: ‘Maximum achievable diversity order for cascaded Rayleigh fading channels’, IET Electron. Lett., 2005, 41, (23), pp. 1289 – 1290 [25] CHIZHIK D., FOSCHINI G.J., VALENZUELA R.A.: ‘Capacities of multi-element transmit and receive antennas: correlations and keyholes’, IET Electron. Lett., 2000, 36, (13), pp. 1099 – 1100 [26] KARAGIANNIDIS G.K., SAGIAS N.C., MATHIOPOULOS P.T. : ‘N ∗ Nakagami: a novel stochastic model for cascaded fading channels’, IEEE Trans. Commun., 2007, 55, (8), pp. 1453 – 1458 [27] SHIN H., LEE J.H.: ‘Performance analysis of space-time block codes over keyhole Nakagami-m fading channels’, IEEE Trans. Veh. Technol., 2004, 53, pp. 351– 362 [28] SAGIAS N.C., TOMBRAS G.S. : ‘On the cascaded weibull fading channel model’, J. Franklin Inst., 2007, 344, pp. 1 – 11 [29] TRIGUI I., LAOURINE A., AFFES S. , STEPHENNE A.: ‘On the performance of cascaded generalized K fading channels’. IEEE GLOBECOM 2009, pp. 1 – 5 IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

www.ietdl.org [30] ILHAN H., UYSAL M., ALTUNBAS I.: ‘Cooperative diversity for intervehicular communication: performance analysis and optimization’, IEEE Trans. Veh. Technol., 2009, 58, pp. 3301 – 3310 [31] GRADSHTEYN I.S., RYZHIK M.I.: ‘Table of integrals, series and products’ (Academic Press, 1994, 7th edn.)

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[32] Wolfram Research, Inc.: ‘The Wolfram functions site [Online]’, 2010. Available at http://functions.wolfram.com [33] ADAMCHIK V.S., MARICHOV O.I.: ‘The algorithm for calculating integrals of hypergeometric type functions and its realizations in REDUCE system’. Proc. Int. Conf. Symbolic and Algebraic Computation, Tokyo, Japan, 1990

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ISSN 1751-8628

Moment generating function-based performance evaluation of amplify-andforward relaying in N∗Nakagami fading channels H. I˙lhan1 I˙. Altunbas¸1 M. Uysal2 1

Department of Electronics and Communication Engineering, Istanbul Technical University, Maslak, Istanbul 34469, Turkey ¨ zyeg˘in University, 34662, Istanbul, Turkey Faculty of Engineering, O E-mail: [email protected] 2

Abstract: In this study, the authors investigate the error rate performance of amplify-and-forward relaying over N∗ Nakagami fading channels. This is a recently introduced channel model that involves the product of N Nakagami-m-distributed random variables. Employing the moment generating function approach, the authors derive symbol error rate expressions for a single-relay system under instantaneous power scaling (IPS) and average power scaling (APS) factors at the relay node, that is, variable and ﬁxed gains. The results achieved by the authors demonstrate that the achievable diversity order is a function of Nakagami fading parameter (m) and degree of cascading (N ). An identical diversity order is obtained under both scaling factors when the relay is close to the destination. When the relay is close to the source, IPS becomes advantageous over APS. Monte –Carlo simulations are further provided to conﬁrm the analytical results.

1

Introduction

Cooperative communication techniques [1 – 3] have emerged as a powerful alternative to MIMO (multi-input multioutput antennas) communications for wireless applications in which size and power limitations exclude the possibility of using multiple co-located antennas. In a cooperative communication system, network nodes help each other via relaying information and realise spatial diversity advantages in a distributed manner. With its indisputable advantages and wide range of potential applications areas, cooperative communication has received much attention and extensively studied over the last few years, see, for example, [1 – 12] and the references therein. A key assumption in majority of these works is the underlying fading channel model which is characterised by Rayleigh distribution because of multipath fading. When the channel suffers from both long-term (i.e. shadowing) and short-term (i.e. multipath) fading, composite multipath/ IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

shadowing channel models are considered in which shadowing effect is typically modelled by log-normal fading [13]. A recent composite channel model so-called generalised-K fading is further considered in [14, 15]. This leads to more tractable mathematical expressions in the performance analysis. For example, performance results for dual-hop transmissions over generalised-K fading channels can be found in [16, 17]. Classical Rayleigh fading captures the propagation characteristics of cellular radio systems. This statistical model has been originally developed under the assumption of a wireless communication scenario with a stationary base station antenna above rooftop level and a mobile station at street level. In contrast to this cellular scenario, in intervehicular communication and mobile ad hoc applications, transmitter and receiver antennas are close to the ground level and experience fading generated from independent groups of scatterers around the mobile terminals. This requires the deployment of ‘cascaded’ 253

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www.ietdl.org fading channels to realistically model such scenarios [18 – 20]. Such cascaded channels can be also used when the transmission media contains keyholes [21] and diffracting wedges such as street corners and/or rooftops [19]. They can be further considered for modelling the composite fading. Cascaded Rayleigh fading channel, which involves the product of N independent Rayleigh-distributed random variables, is presented in [22]. For N ¼ 2, this reduces to double Rayleigh fading model which has been considered in [23, 24]. In addition, double Rayleigh fading channel model has been used for keyhole channel modelling of MIMO communications [21, 25]. Based on Nakagami-m fading, cascaded Nakagami (also named as N∗ Nakagami) channel is proposed in [26]. Special case with N ¼ 2 is studied in [27]. Cascaded Weibull and cascaded generalised-K channels are further introduced and investigated in [28, 29], respectively. In a recent paper [30], we have investigated the error rate performance of a cooperative intervehicular system over a double Nakagami (i.e. N ¼ 2) fading channel. A Chernoff bound on the pair-wise error probability is derived in [30] assuming average power scaling (APS) constraint, that is, ﬁxed gain, at the relay node and a union bound on error rate is presented. In this paper, we extend our analysis for N∗ Nakagami case which subsumes double Nakagami in [30] as a special case. We derive an exact symbol error rate (SER) expression based on the moment generating function (MGF) approach assuming instantaneous power scaling (IPS) constraint, that is, variable gain, as well as APS. The derived expressions give insight into the achievable diversity orders for various relay locations and channel parameters. The rest of the paper is organised as follows: In Section 2, we describe the relay-assisted transmission model and fading channel under consideration. In Section 3, we derive the MGF expressions over N∗ Nakagami channel for IPS and APS constraints. In Section 4, we present Monte – Carlo simulation results to verify the analytical results. Finally, we conclude in Section 5.

2

System model

We consider a single-relay scenario (as shown in Fig. 1) where source, relay and destination nodes operate in half-duplex mode and are equipped with a single pair of transmit and receive antennas. In Fig. 1, aSD , aSR and aRD represent

Figure 1 Relay-assisted cooperative system 254 & The Institution of Engineering and Technology 2011

source-to-destination (S D), source-to-relay (S R) and relay-to-destination (R D) links’ complex fading coefﬁcients whose magnitudes hSD ¼ |aSD|, hSR ¼ |aSR| and hRD ¼ |aRD| follow cascaded Nakagami-m distribution, respectively. These magnitudes are assumed to be the product of statistically independent, but not necessarily identically distributed Nakagami-m random variables [26]. NSD NSR hSD,l1 , hSR = Pl2 =1 hSR,l2 Speciﬁcally, we have, hSD = Pl1 =1 NRD and hRD = Pl3 =1 hRD,l3 where NSD , NSR and NRD are the number of random variables (which we deﬁne as ‘degree of cascading’) for S D, S R and R D links, respectively. Here, hSD,l1 , l1 ¼ 1, 2, . . . , NSD; hSR,l2 , l2 ¼ 1, 2, . . . , NSR and hRD,l3 , l3 ¼ 1, 2, . . . , NRD’s are Nakagamim-distributed variables with probability density function (pdf) [13] m 2ml l ml 2 2ml −1 h h exp − fhl (hl ) = (E[h2l ])ml G(ml ) l E[h2l ] l

(1)

where the corresponding subscripts SD, SR and RD and numerical subscripts 1 – 3 are dropped for convenience. In (1), ml , l ¼ 1, 2, . . . , N is a parameter describing the fading severity given by ml = E[h2l ]/E[(h2l − E[h2l ])2 ] ≥ 0.5. Taking E[h2l ] = 1, one can normalise the power of the fading process to unity. Here, E[.] denotes the expectation operator and G(.) is the Gamma function [31]. The system model under consideration builds upon socalled receive diversity cooperation protocol [1, 6]. In this protocol, transmission is divided into two phases. During the ﬁrst transmission phase, the source node broadcasts to the relay and the destination nodes. In the second transmission phase, the source node is silent and the relay node forwards the signal received during the ﬁrst phase to the destination. The received signals at the relay and the destination are, respectively, given by ESR hSR x + nR = ESD hSD x + nD1

rR = rD1

(2) (3)

where x denotes M-PSK (phase shift keying) modulation symbol broadcasted from the source. In (2) and (3), ESR and ESD represent the average energies available at the relay and the destination which take into account possibly different path loss and shadowing effects in S R and S D links, respectively. nR and nD1 are independent samples of zero-mean complex Gaussian random variables with variance N0/2 per dimension. The relay node multiplies the received signal rR by a scaling factor of b = 1/ E[|rR |2 ] to restrict the energy at its output. Based on the availability of channel state information of S R link at the relay node, either IPS or APS constraint can be employed. In the former, the expectation is carried only over the noise term, whereas in the latter, the expectation is with respect to noise and fading terms. IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

www.ietdl.org where VgSD (s) and VgSRD (s) are the MGFs of gSD and gSRD , respectively. VgSD (s) is given by

Therefore the scaling factor takes the form of ⎧ ⎪ ⎨ 1/ E [|rR |2 ] = 1/ ESR |hSR |2 + N0 , nR b= ⎪ ⎩ 1/ E [|rR |2 ] = 1/ ESR + N0 , nR ,hSR

1

for IPS VgSD (s) =

for APS

After scaling by b, the relay forwards the resulting signal to the destination. After proper normalisations [3], the received signal can be written as rD2

ESR = hSR hRD x + nD2 2 (b ERD )−1 + h2RD

(4)

exp(sgSD )fgSD (gSD ) dgSD

0

(9)

Inserting (7) in (9) and using the closed-form solution [31, Eq. 7.813.1], we have ⎛

⎞

NSD SD 1 1 NSD ,1 ⎜ Pl1 =1 ml1 G1,NSD ⎝− VgSD (s) = N s gSD SD P SD G(mSD ) l1 =1

⎟ ⎠

,...,mSD m1 ,mSD N 2

l1

SD

(10) where ERD represents the average energy available at the destination node and nD2 is conditionally zero-mean complex Gaussian random variable with variance N0/2 per dimension. At the destination node, the received signals rD1 and rD2 are fed to a maximum likelihood detector.

3

On the other hand, VgSRD (s) can be calculated as 1 1 VgSRD (s) = =

Derivation of SER

(M−1)p/M Vgend

0

sin2 (p/M) s=− du sin2 u

gSRD =

for IPS

⎪ ⎪ ⎩

for APS

gSR gRD , gRD + ESR /N0 + 1

⎞ ⎟ ⎟ ⎠

m1 ,m2 ,...,mSR N

SR

⎛

− N RD ⎜ N ,0 gRD RD ×G0,NRDRD⎜ m l3 ⎝g RD l =1 RD 3

⎞ ⎟ ⎟dgSR dgRD ⎠

,...,mRD m1 ,mRD N 2

RD

(11) Substituting (10) and (11) into (8), we have ⎛

⎞ 1 B N ,1 ⎠ Vgend (s) = AG1,NSDSD ⎝− s mSD , mSD ,..., mSD 1 1

Noting that hSD , hSR and hRD follow cascaded Nakagami-m distribution, the pdf of corresponding instantaneous SNR is given by [26] − N 1 N ,0 ⎝g G0,N ml f g (g ) = N g gPl =1 G(ml ) l =1

1 NSR NRD RD Pl2 =1 G(mSR l2 )Pl3 =1 G(ml3 )

1 1 exp(sgSRD ) 0 0 gSR gRD ⎛ − NSR g NSR ,0 ⎜ SR SR ⎜ m × G0,NSR ⎝ SR l =1 l2 g 2 SR SR

(5)

(6)

⎛

exp(sgSRD )fgSR (gSR )fgRD (gRD )dgSR dgRD

×

where Vgend (s) is the MGF of instantaneous endto-end signal to noise ratio (SNR) gend . Let the instantaneous SNRs of S D, S R and R D links SD = h2SD ESD /N0 , gSR = h2SR g SR = be given by gSD = h2SD g RD = h2RD ERD /N0 . gend is then h2SR ESR /N0 and gRD = h2RD g obtained as the summation of instantaneous SNRs of direct and relaying links, that is, gend ¼ gSD + gSRD where ⎧ gSR gRD ⎪ , ⎪ ⎨ gRD + gSR + 1

0

1 1

Based on the MGF approach, the exact SER for M-PSK is given by [13] 1 Ps = p

0

× 0

×

⎞ ⎠

0

1

2

NSD

1 1 exp(sgSRD ) gSR gRD

N ,0 G0,NSRSR (C gSR | − , mSR ,..., mSR mSR N 1 2

)

SR

× G0,NRDRD (K gRD | − mRD , mRD ,..., mRD )dgSR dgRD N

(7)

,0

1

2

NRD

m1 ,m2 ,...,mN

(12)

where the subscripts SD, SR and RD are dropped for p,t (.|.), p, t, u, v [ N , p ≤ v, t ≤ u convenience and Gu,v is the Meijer’s G-function [31]. The MGF of instantaneous end-to-end SNR can be calculated as Vgend (s) = VgSD (s)VgSRD (s) IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

(8)

N

N

N

SR RD SD SR RD where A = 1/(Pl1 =1 G(mSD l1 )Pl2 =1 G(ml2 )Pl3 =1 G(ml3 )), B = N

N

N

SD SR RD mSD gSD , C = Pl2 =1 mSR gSR and K = Pl3 =1 mRD Pl1 =1 l1 / l2 / l3 / RD . To the best of our knowledge, a closed-form solution g for (12) is unfortunately not available for the general case, yet this two-fold integral can be numerically evaluated through commercially available mathematics software such as Mathematica, Matlab and Maple. In the following, we

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www.ietdl.org investigate some representative cases to seek closed-form expressions.

respect to gSR leads to Vgend (s)

3.1 Case I: relay close to source

⎛

When the relay is very close to the source node, SNR in S R link is much larger than that in R D link, that is, gSR ≫ gRD . Therefore under IPS constraint, we have gSRD ¼ gSRgRD/( gRD + gSR + 1) ≃ gRD . In this case, using the closed-form solution [31, Eq. 7.811.4], the inner integration in (12) with respect to gSR yields

⎞

1

B N ,1 = AG1,NSDSD ⎝− s

⎠

,mSD ,...,mSD mSD N 1 2

1 −1 1,NSR × gRD GNSR ,1 − 0

1−mSR ,1−mSR ,...,1−mSR 1 NSR 2 g C(ESR /N0 ) RD 0

⎛

⎞ 1 NSD ,1 ⎝ B SR ⎠ G(ml2 )G1,NSD − =A s mSD ,mSD ,...,mSD l =1 NSR 2

1

1

N

,0

0

,0

2

NRD

(13) Further using [31, Eq. 7.813.1], we obtain ⎛

⎞ 1 NSD ,1 ⎝ B ⎠ G(mSR Vgend (s) = A l2 )G1,NSD − s mSD ,mSD ,...,mSD l =1 NSR 2

1

⎛

2

⎞ 1 K N ,1 ⎠ × G1,NRDRD ⎝− s mRD ,mRD ,...,mRD 1

2

(16)

NRD

Similar to (14), (17) can be alternatively obtained in terms of the hypergeometric pFq function.

NSD

1

2

Further using [33, Eq. 21], we obtain (see (17))

− RD g−1 RD exp(sgRD )G0,NRD (K gRD |mRD ,mRD ,...,mRD ) dgRD N

×

2

s

× G0,NRDRD (K gRD |− mRD ,mRD ,...,mRD )dgRD 1

Vgend (s)

SD

NSD

(14)

NRD

Replacing (14) and (17) in (5), we obtain the exact SER expressions, respectively, under IPS and APS constraints. These expressions can be also used for diversity gain evaluation. In Table 1, we present the (asymptotic) diversity orders achieved for various values of NSD, NSR , mSD mSR l2 = 1, 2, . . . , NRD , l1 , l1 = 1, 2, . . . , NSD , l2 , NSR and mRD l3 , l3 = 1, 2, . . . , NRD . These are obtained numerically from the slope of the SER–SNR function (i.e. 2log Ps/log SNR) in high SNR region for values of m and N under consideration. It can be concluded from observations in Table 1 that the maximum diversity order is given by SD SD RD RD RD min(mSD 1 , m2 , . . . , mNSD ) + min(m1 , m2 , . . . , mNRD ) for SD IPS constraint and by min(mSD 1 , m2 , . . . ,

Note that (14) can be written in terms of hypergeometric pFq function using [32, Eq. (07.34.26.0004.01)] for (mj 2 mk) Z, j = k, j, k [ {1, 2, . . . , NSD (or NRD)}. For the double Nakagami case (NSD ¼ NSR ¼ NRD ¼ 2) without the constraint on (mj 2 mk), it can be also written in terms of hypergeometric U-function [31] using the results from [32, Eq. (07.34.03.0719.01)]

SD RD B m1 K m1 − Vgend (s) = − s s B SD SD SD × U m1 , 1 + m1 − m2 , − s K RD RD RD × U m1 , 1 + m1 − m2 , − s

(15)

SR SR RD RD RD min(mSR 1 , m2 , . . . , mNSR , m1 , m2 , . . . , mNRD ) for APS constraint.

3.2 Case 2: relay close to destination When the relay is very close to the destination, SNR in S R link is much smaller than that in R D link, that is, gSR ≪ gRD . Therefore under the IPS constraint, we have gSRD ¼ gSRgRD/(gRD + gSR + 1) ≃ gSR . Integrating of (12) over gRD and gSR and using the closed-form solutions [31, Eq. 7.811.4] and [31, Eq. 7.813.1], respectively, the MGF is given by ⎛

⎞ 1 B NSD ,1 ⎝ ⎠ G(mRD Vgend (s) = A l3 )G1,NSD − s mSD ,mSD ,...,mSD l =1 N RD 3

Under APS constraint, we have gSRD ¼ gSRgRD/ ( gRD + ESR/N0 + 1) ≃ gSRgRD/(ESR/N0). Using closedform solution [31, Eq. 7.813.1], integration in (12) with ⎛

Vgend (s) =

⎞

1

B N ,1 AG1,NSDSD ⎝− s

,mSD ,...,mSD mSD N 1 2

×

256 & The Institution of Engineering and Technology 2011

1

⎛

⎞

1

C N ,1 G1,NSRSR ⎝− s

2

NSD

⎠

(18)

,mSR ,...,mSR mSR N 1 2

SR

⎛

⎞

1

⎠G NSR +NRD ,1 ⎝− CK (ESR /N0 ) 1,NSR +NRD

SD

mSD NSD ) +

s

⎠

(17)

,mSR ,...,mSR ,mRD ,mRD ,...,mRD mSR N N 1 2 1 2 SR

RD

IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

www.ietdl.org Table 1 Diversity orders achieved for various combinations of channel parameters SD SR SR RD RD NSD mSD 1 , . . . , mNSD NSR m1 , . . . , mNSR NRD m1 , . . . , mNRD

Diversity orders APS and gSR ≫ gRD

IPS and gSR ≫ gRD

APS/IPS and gRD ≫ gSR

1

1

1

2

1

3

3

4

3

1

1

1

2

1

1

2

2

3

1

1

1

2

2

1, 3

2

2

3

1

2

1

1

3

3, 4, 5

3

5

3

1

1

2

2, 1

1

3

2

4

2

1

2

2

3, 1

2

4, 5

3

6

3

1

2

2

4, 5

3

8, 7, 6

6

8

6

1

1

3

3, 4, 5

1

2

3

3

4

1

2

3

3, 1, 4

2

5, 6

3

7

3

1

2

3

3, 4, 2

3

5, 6, 4

4

6

4

2

2, 3

1

2

1

3

4

5

4

2

2, 1

1

2

2

5, 4

3

5

3

2

2, 4

1

2

3

3, 5, 6

4

5

4

2

3, 2

2

1, 2

1

3

3

5

3

2

2, 1

2

3, 1

2

5, 6

2

6

2

2

1, 1

2

2, 2

2

1, 1

2

2

3

2

3, 1

2

3, 2

3

4, 6, 3

3

4

3

2

2, 3

3

3, 2, 4

1

3

4

5

4

2

3, 2

3

1, 3, 4

2

6, 3

3

5

3

2

3, 1

3

3, 1, 4

3

5, 2, 5

2

3

2

3

2, 1, 3

1

2

1

3

3

4

3

3

2, 3, 4

1

3

2

5, 2

4

4

5

3

2, 3, 1

1

2

3

5, 4, 2

3

3

3

3

2, 1, 3

2

3, 1

1

4

2

5

2

3

1, 2, 3

2

4, 1

2

6, 5

2

6

2

3

1, 2, 3

2

3, 2

3

4, 5, 6

3

5

3

3

2, 3, 4

3

3, 2, 4

1

1

3

3

4

3

2, 3, 3

3

7, 5, 4

2

2, 5

4

4

6

3

1, 2, 3

3

4, 6, 7

3

5, 7, 8

5

6

5

3

1, 1, 1

3

1, 1, 1

3

3, 3, 3

2

4

2

3

1, 1, 1

3

2, 2, 2

3

1, 1, 1

2

2

3

3

2, 2, 2

3

1, 1, 1

3

1, 1, 1

3

3

3

3

2, 2, 2

3

4, 4, 4

3

2, 2, 2

4

4

6

3

4, 4, 4

3

1, 1, 1

3

2, 2, 2

5

6

5

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www.ietdl.org For NSD ¼ NSR ¼ NRD ¼ 2, (18) can be further written in terms of the hypergeometric U-function as SD SR B m1 C m1 − Vgend (s) = − s s B SD SD SD × U m1 , 1 + m1 − m2 , − s C SR SR SR × U m1 , 1 + m1 − m2 , − s

† Scenario 1: m SD ¼ 1, m SR ¼ 1, m RD ¼ 3 and NSD ¼ NSR ¼ NRD ¼ 3.

(19)

Under APS constraint, we again have gSRD ¼ gSRgRD/ (gRD + ESR/N0 + 1) ≃ gSR . Therefore the MGF is identical to (18). Replacing (18) in (5), we obtain the exact SER expression under IPS and APS constraints for this relay location. In Table 1, we present the diversity orders achieved for various combinations of channel parameters. From these observations, the maximum diversity order turns out to be the same for both IPS and APS constraints and is given by SD SD SR SR SR min(mSD 1 , m2 , . . . , mNSD ) + min(m1 , m2 , . . . , mNSR ).

4

mechanisms. In Fig. 2, we consider the following scenarios based on the combinations of channel parameters:

Numerical results

In this section, we present Monte– Carlo simulations to conﬁrm the derived analytical results. We consider singlerelay scenario and assume 4-PSK modulation. In Figs. 2 and 3, we present the SER performance of the cooperative scheme against ESD/N0 when the relay is very close to the source under APS constraint. Speciﬁcally, we assume that the average SNR in S R link is 30 dB larger than that in R D link and ESD ¼ ERD . This can be justiﬁed in practice through the deployment of power control

† Scenario 2: m SD ¼ 2, NSD ¼ NSR ¼ NRD ¼ 3.

m SR ¼ 1,

m RD ¼ 1

and

† Scenario 3: m SD ¼ 2, NSD ¼ NSR ¼ NRD ¼ 3.

m SR ¼ 4,

m RD ¼ 2

and

† Scenario 4: m SD ¼ 4, NSD ¼ NSR ¼ NRD ¼ 3.

m SR ¼ 1,

m RD ¼ 2

and

In Fig. 3, we consider the following scenarios: † Scenario 5: m SD ¼ 1, m SR ¼ 2, m RD ¼ 1 and NSD ¼ NSR ¼ NRD ¼ 1. † Scenario 6: m SD ¼ 1, m SR ¼ 2, m RD ¼ 1 and NSD ¼ NSR ¼ NRD ¼ 2. † Scenario 7: m SD ¼ 1, m SR ¼ 2, m RD ¼ 1 and NSD ¼ NSR ¼ NRD ¼ 3. It is observed from both ﬁgures that the derived analytical SER expressions (dash lines) provide a perfect match to the simulation results (solid lines). Since diversity orders, by the deﬁnition, are for asymptotically high SNR, we extend our analytical results to very low SER values while the simulation results are limited to at most 1026 because of computation time. From Fig. 2, we observe that

Figure 2 SER performance of the cooperative scheme when the relay is very close to the source under APS constraint 258 & The Institution of Engineering and Technology 2011

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Figure 3 SER performance of the cooperative scheme when the relay is very close to the source under APS constraint the diversity orders are given by 2, 3, 4 and 5, respectively, for scenarios 1–4 which conﬁrms our earlier observation SD SD SR SR mSR of min(mSD 1 , m2 , . . . , mNSD ) + min(m1 , m2 , . . . , NSR , RD RD RD m1 , m2 , . . . , mNRD ). From Fig. 3, we observe that diversity orders are 2 and identical for scenarios 5–7 again conﬁrming our analytical ﬁndings.

In Fig. 4, we return our attention to SER performance under IPS constraint and still assume the case when the relay is very close to the source with the aforementioned scenarios 1–4. Comparison of Figs. 2 and 4 reveals that diversity orders under IPS are higher. This is expected because it takes the advantage of channel state information (CSI) of the S R link available

Figure 4 SER performance of the cooperative scheme when the relay is very close to the source under IPS constraint IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

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www.ietdl.org at the relay. Speciﬁcally, we observe diversity orders of 4, 3, 4 and 6 for scenarios 1–4 which conﬁrms our earlier observation SD SD RD RD RD of min(mSD 1 , m2 , . . . ,mNSD ) + min(m1 , m2 , . . . , mNRD ). As earlier noted, we emphasise that we keep our SER and SNR values in a wide range to be able to demonstrate the diversity orders. As the degree of cascading increases,

convergence to the asymtotical diversity order becomes slower and is observed for very high SNR values. In Fig. 5, we consider the case when the relay is very close to the destination and compare the performance of APS and IPS constraints. Speciﬁcally, we assume that the average SNR

Figure 5 SER performance comparison of APS and IPS constraints when the relay is very close to the destination

Figure 6 Effect of relay location on the SER performance under APS constraint when mSD ¼ 2, mSR ¼ mRD ¼ 1 for scenario 2 260 & The Institution of Engineering and Technology 2011

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Figure 7 Effect of relay location on the SER performance under APS constraint when mSR ¼ 4, mSD ¼ mRD ¼ 2 for scenario 3 in S R link is assumed 30 dB smaller than that in R D link and ESD ¼ ESR . In this relay location, the advantage of IPS over APS disappears. Recall that end-to-end SNRs for both cases under this relay location become the same. Therefore diversity orders are identical for both constraints, speciﬁcally 2 and 6 for scenarios 1 and 3, respectively, SD conﬁrming earlier observation of min(mSD 1 , m2 , . . . , SD SR SR SR mNSD ) + min(m1 , m2 , . . . , mNSR ).

ﬁxed gain based on the availability of channel knowledge at the relay. Our results demonstrate that a diversity order of SD SD RD RD RD min(mSD 1 , m2 , . . . , mNSD ) + min(m1 , m2 , . . . , mNRD ) is available under IPS constraint when the relay is located close to the source. Under APS constraint, a diversity SD SD SR SR order of min(mSD 1 , m2 , . . . , mNSD ) + min(m1 , m2 , . . . , SR RD RD RD mNSR , m1 , m2 , . . . , mNRD ) is achieved under the same location. When the relay is close to the destination, an SD SD identical diversity order of min(mSD 1 , m2 , . . . , mNSD ) + SR SR SR min(m1 , m2 , . . . , mNSR ) is observed for both constraints. Monte – Carlo simulations are further provided for various values of channel parameters and show excellent agreement with the derived analytical results.

In Figs. 6 and 7, we examine the effect of relay location on the SER performance. We assume APS constraint and compare the performance for ESR/ERD ¼ 230, 210, 0, 10 and 30 dB. In Fig. 6, we consider scenario 2 with m SD ¼ 2, m SR ¼ m RD ¼ 1 and observe that diversity order remains the same. Best performance is obtained for ESR/ ERD ¼ 230 dB while worst performance is obtained for ESR/ERD ¼ 30 dB. Speciﬁcally, we observe a performance difference of 6.5 dB between these two locations at the SER of 1024. In Fig. 7, we consider scenario 3 with m SR ¼ 4, m SD ¼ m RD ¼ 2 and observe that diversity orders for different relay locations differ from each other for m SR = m RD.

[1] LANEMAN J.N., WORNELL G.W.: ‘Distributed space-time coded protocols for exploiting cooperative diversity in wireless networks’, IEEE Trans. Inf. Theory, 2003, 49, (10), pp. 2415 – 2525

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[2] SENDONARIS A. , ERKIP E. , AAZHANG B. : ‘User cooperation diversity part I: system description’, IEEE Trans. Commun., 2003, 51, (11), pp. 1927 – 1938

Conclusions

In this paper, we have derived SER expressions through MGF approach for a single-relay system with amplify-andforward relaying over N∗ Nakagami fading channels. We have considered both IPS and APS factors at the relay node which correspond to, respectively, variable gain and IET Commun., 2011, Vol. 5, Iss. 3, pp. 253– 263 doi: 10.1049/iet-com.2009.0840

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