On The Performance of Two-Way Amplify-and-Forward Relay Networks

Wireless Pers CommunDOI 10.1007/s11277-010-0061-1

Capacity Analysis of Opportunistic CooperativeNetworks under Adaptive Transmission over RayleighFading Channels

Hyung Yun Kong · Vo Nguyen Quoc Bao

© Springer Science+Business Media, LLC. 2010

Abstract In this paper, opportunistic cooperative amplify-and-forward networks in con-junction with three different adaptive policies, namely optimal simultaneous power and rateadaptation (OPRA), constant power with optimal rate adaptation (OPA) and truncated chan-nel inversion with fixed rate (TIFR), are investigated and compared in terms of Rayleighchannel capacity where the source adapts its rate and/or power level according to channelconditions while the best relay simply amplifies and then forwards the received signals. Fur-thermore, the effect of diversity combining on the network is studied by investigating twocases of maximal ratio combining (MRC) and selection combining (SC) equipped at thedestination. To this end, the mathematically tractable form of the upper and lower bound ofthe end-to-end effective signal-to-noise ratio (SNR) is provided and then used to derive theclosed-form expression of the Shannon capacity. Our results are verified through compari-son with Monte Carlo simulations in some representative scenarios where we also illustratethat, among them, for an arbitrary number of cooperative relays, OPRA slightly outperformsORA, which, in turn, outperforms TIFR.

Keywords Cooperative diversity · Adaptive transmission · Amplify-and-forward ·Shannon capacity · Maximal ratio combining · Selection combining

1 Introduction

It is well known that the concept of cooperative relaying has recently drawn a great attentionin the research community due to its potential in improving the reliability of data transmission

This paper was presented in part at the IEEE PIMRC 2009, Tokyo, Japan, Sep. 2009.

H. Y. Kong (B) · V. N. Q. BaoUniversity of Ulsan, Ulsan, South Koreae-mail: [email protected]

V. N. Q. Baoe-mail: [email protected]; [email protected]

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H. Y. Kong, V. N. Q. Bao

and in extending coverage of wireless networks in a cost-effective manner (see, e.g.[1,2] and the references cited therein). It is based on the broadcast nature of the wirelessmedium and enables wireless nodes, which are in the communication range of a source, torelay the source information toward the destination. Several cooperation strategies with dif-ferent relaying techniques, including amplify-and-forward (AF), decode-and-forward (DF),and selective relaying (SR), have been proposed and investigated in terms of outage probabil-ity and bit error probability. Furthermore, cooperative transmissions have been categorizedinto space-time coded cooperation [3,4], repetition-based cooperation [5–8] and opportunis-tic relay selection cooperation [9–12].

Recently, optimum resource allocation has emerged as an important research topic toimprove the performance of cooperative networks (e.g., see [13–17]). Most of the abovementioned studies, however, focus on the optimum power allocation and/or relay locationoptimization with assumption that accurate channel state information (CSI) of all links inthe network are available at each node. Furthermore, these systems are effectively designedfor the worst-case channel conditions, resulting in insufficient utilization of the full channelcapacity. However, this can be ameliorated through the use of adaptive transmission.

Adaptive transmission is a prominent and powerful technique by which wireless communi-cation systems can take advantage of the time-varying nature of wireless channels to enhancespectral efficiency without having to sacrifice error rate performance. The key idea is thatthe transmitter balances on real-time the link budget by adaptively adjusting its transmittingpower, its symbol transmission rate, its constellation size as well as its coding rate/scheme,or any combination of these parameters [18–22].

Recently, this concept has gained new actuality in the context of cooperative/relayingwireless communication networks [23–28]. In particular, in [23], Hasna investigated the per-formance of both DF and AF single-relay networks under Rayleigh type of fading for threedifferent adaptive transmission schemes. In [24,25], the use of constant-power, rate-adap-tive M-QAM transmission for amplify-and-forward multi-relay cooperative networks wasproposed and investigated in terms of outage probability, achievable spectral efficiency anderror rate. It is shown that the system with adaptive discrete rate M-QAM can approach theShannon capacity within 5 dB for both independent and identically distributed (i.i.d.) andindependent but not identically distributed (i.n.d.) Rayleigh fading channels. More recently,in [26], Kalansuriya et. al. have extended their previous work in [24,25] for i.i.d. Nakagami-mfading channels. In [27], Hwang et. al. proposed an incremental opportunistic relaying withvariable-rate transmission. Also in this paper, the authors showed that this scheme providesa certain improvement of the spectral efficiency and outage probability while satisfying therequired bit error rate (BER) performance. In [28], Tyler et. al. proposed the use of AF rep-etition-based cooperative systems with adaptive transmission techniques in which only thesource adapts its rate and/or power level according to the changing channel conditions, whilethe cooperative relays amplify and then forward the signals in their own time slots. How-ever, this cooperation strategy suffers from one disadvantage—a loss in spectral efficiencyas multiple orthogonal channel allocations (equal number of relays plus one) are required totransmit one data packet.

To the best of the authors’ knowledge, there is no research work concerning the Shannoncapacity of the adaptive transmission schemes in opportunistic cooperative networks. Ourcontribution in this paper is threefold.

– We first present a new and generalized statistics result for the end-to-end SNR of opportu-nistic cooperative networks. Due to the relatively complicated statistics of AF cooperativerelaying [29], we shall resort to using the approximate approach for evaluating the system

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Capacity Analysis of Opportunistic Cooperative Networks

capacity. More specifically, the probability density function (PDF) of the upper and lowerbounds of the end-to-end SNR is derived and expressed in a tractable form. Beside, weconsider both optimal maximal-ratio combining and selection combining at the destina-tion.

– Second, based on the statistical PDF, we provide a capacity analysis over Rayleigh fad-ing channel for AF opportunistic cooperative networks under adaptive transmission. Inparticular, the capacity of the system under three different adaptive transmission policies,namely optimal simultaneous power and rate adaptation, constant power with optimalrate adaptation and channel inversion with fixed rate is studied.

– Third, we investigate the impact of adaptive transmission policies on opportunistic coop-erative networks showing that for the same fading conditions, OPRA offers the highestcapacity, whereas TIFR achieves the lowest capacity, although these differences are notsignificant. By comparing the performance of a system that uses selection combiner toone installed optimum maximal ratio combiner, it is shown that the performance lossdue to using a less complex combiner at the destination is not substantial. Furthermore,the advantage of opportunistic cooperative networks over repetition-based cooperativenetworks [28] under adaptive transmission is confirmed, as expected.

The remainder of this paper is organized as follows. In Sect. 2, the system model under inves-tigation and the upper and lower bound of the end-to-end SNR expressed in a mathematicallytractable form are provided. In Sect. 3, the optimum and sub-optimum adaptation policiesfor AF opportunistic cooperative relaying networks are studied in terms of Rayleigh fadingcapacity. Finally, numerical results are given in Sect. 4 and conclusions are drawn in Sect. 5.

2 System Model

Consider a distributed wireless cooperative network in which the source node S commu-nicates with the destination node D with the help of N relay nodes, R1, . . . , Rk, . . . , RN ,employing amplify-and-forward protocol as illustrated in Fig. 1. Each node is equipped withsingle antenna and operates in half-duplex mode. All transmissions are assumed orthogonaleither in time or in frequency.

Fig. 1 Opportunistic cooperative network with source adaptive transmission

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Let us denote the channel gains corresponding to the links of the S → D, S → Rk andRk → D as hSD, hSRk and hRkD, respectively. Due to Rayleigh fading, the channel pow-ers, denoted by α0 = |hSD|2, α1,k = |hSRk |2 and α2,k = |hRkD|2, are exponential randomvariables whose means are λ0, λ1,k and λ2,k , respectively.

To facilitate the explanation, we assume that the cooperative transmission under consid-eration takes place in two time-slots. In the first time slot, the source broadcasts its symbol,s, with an average transmitted power per symbol Ps to all the relays and the destination.The source adopts the distributed relay selection scheme where the “best” relay is chosen apriori according to the distributed timer fashion with minimum signaling overhead and minoradditional complexity as shown in [10,11]. Specifically, the relay with shortest timer expiresfirst and thus becomes the transmitter (also denoted as the “best” relay) in the second timeslot while other relays discard the received signals after receiving this node transmission.Since AF relaying is employed, the best relay, indexed by k∗, simply amplifies the receivedsignals with a fixed gain G and forwards the resultant signal to D. With the assumption thatRk∗ transmits the same amount of power as S and the additive white Gaussian noise at all ter-minals has zero mean and variance of N0, the system model can be described mathematicallyby the following set of equations as follows:

ySD = √PshSDs + nSD (1a)

ySRk∗ = √PshSRk∗ s + nSRk∗ (1b)

yRk∗ D = √PshRk∗ D(GySRk∗ ) + nRk∗ D (1c)

where G =√

1/[Ps |hSRk∗ |2 + N0

]. nSD and nSRk∗ as well as nRk∗ D are the additive noise.

Let us define the effective instantaneous SNRs for S → D, S → Rk and Rk → D links asγ0 = Ps |hSD|2/N0, γ1,k = Ps |hSRk |2/N0 andγ2,k = Ps |hRkD|2/N0, respectively. Thus, theinstantaneous dual-hop SNR of the “best” relay at the destination is γk∗ = arg max1≤k≤N γk

[11] where γk with k = 1, . . . N can be shown to be [30,31]

γk = γ1,kγ2,k

γ1,k + γ2,k + 1(2)

As can be seen from (2), the constant 1 in the denominator is generally insignificant giventhat the considered SNR range is sufficiently high. Therefore, by dropping this term, anapproximation of γk , which facilitates our analysis, can be found as

γk ≈ γ1,kγ2,k

γ1,k + γ2,k(3)

Furthermore, for tractable analysis, (3) can be upper- and lower-bounded as [30,31]

1

2min(γ1,k; γ2,k)

�= γ Lk ≤ γk < γ U

k�= min(γ1,k; γ2,k) (4)

Since γ1,k and γ2,k are exponentially distributed random variables with hazard rates μ1,k =1/γ̄1,k = 1/(Psλ1,k) and μ2,k = 1/γ̄2,k = 1/(Psλ2,k), respectively. Making use the factthat the minimum of two independent exponential random variables is again an exponentialrandom variable with a hazard rate equals to the sum of the two hazard rates [32, p. 195, eq.(6.82)], i.e., μk = μ1,k +μ2,k = γ̄1,k+γ̄2,k

γ̄1,k γ̄2,k. For brevity, by introducing γ̄k = K

μk= K γ̄1,k γ̄2,k

γ̄1,k+γ̄2,k

and from (4), we have

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fγk (γ ) = 1

γ̄ke−γ /γ̄k (5a)

Fγk (γ ) =γ∫

0

fγk (γ )dγ =1 − e−γ /γ̄k (5b)

where K equals 1 or 1/2 associated with the cases of upper and lower bound in (4), respec-tively. Under the assumption that all links are subject to independent fading, order statisticsgives the cumulative distribution function (CDF) of γk∗ as

Fγk∗ (γ ) = Pr (γ1 < γ, . . . , γk < γ, . . . , γN < γ ) =N∏

k=1

Fγk (γ ) (6)

Hence, the joint PDF of γk∗ is given by differentiating (6) with respect to γ .

fγk∗ (γ ) = ∂ Fγk∗ (γ )

∂γ=

N∑

k=1

⎡

⎢⎢⎣ fγk (γ )

N∏

j=1j �=k

Fγ j (γ )

⎤

⎥⎥⎦ (7)

Substituting (5) into (7) (and after some manipulation) gives us [33]

fγk∗ (γ ) =N∑

k=1

(−1)k−1N∑

n1,...,nk=1n1<···<nk

1

χke− γ

χk (8)

where χk =(∑k

l=1 γ̄ −1nl

)−1.

At the destination, diversity combining techniques, i.e. maximal ratio combining and selec-tion combining in which two copies (one from the source and the other from the best relay)of the same information-bearing signal are skillfully combined, are employed to increase theoverall SNR.

2.1 Maximal Ratio Combining

In the absence of interference, MRC is considered as the optimal combining scheme in senseof instantaneous SNR (regardless of fading statistics) but comes at the expense of complexitysince it requires knowledge of all channel-fading parameters. As per the rule of MRC, thecombined instantaneous SNR at the output of the maximal ratio combiner at the destinationis given by

γ� = γ0 + γk∗ (9)

Assuming the independence of γ0 and γk∗ , the moment generating function (MGF) of γ�

can be written as

Mγ� (s) = Mγ0(s)Mγk∗ (s) (10)

where Mγ0(s) and Mγk∗ (s) are MGF functions of γ0 and γk∗ , respectively. Using the defini-tion of the MGF as Mγ (s) = E{esγ } [34, p. 4, eq. (1.2)] where E(.) is the statistical averageoperator, we have

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Mγ0(s) = (1 − sγ̄0)−1 (11a)

Mγk∗ (s) =N∑

k=1

(−1)k−1N∑

n1,...,nk=1n1<···<nk

(1 − sχk)−1 (11b)

where γ̄0 = E{γ0} = Psλ0. Employing the partial fraction expansion of the MGF [35, Ch.9], (10) can be shown that

Mγ� (s) = (1 − sγ̄0)−1

N∑

k=1

(−1)k−1N∑

n1,...,nk=1n1<···<nk

(1 − sχk)−1

=N∑

k=1

(−1)k−1N∑

n1,...,nk=1n1<···<nk

(1 − sγ̄0)−1(1 − sχk)

−1

=N∑

k=1

(−1)k−1N∑

n1,...,nk=1n1<···<nk

m(γ̄0, χk) (12)

where m(γ̄0, χk) is defined as

m(γ̄0, χk) ={

γ̄0γ̄0−χk

(1 − sγ̄0)−1 + χk

χk−γ̄0(1 − sχk)

−1, χk �= γ̄0

(1 − sγ̄0)−2, χk = γ̄0

Finally, the PDF of γ� is determined by the inverse Laplace transforms of Mγ� (s) asfollows:

f mrcγ�

(γ ) =N∑

k=1

(−1)k−1N∑

n1,...,nk=1n1<···<nk

p(γ̄0, χk) (13)

with

p(γ̄0, χk) =

⎧⎪⎨

⎪⎩

(γ̄0

γ̄0−χk

)1γ̄0

e− γ

γ̄0 +(

χkχk−γ̄0

)1χk

e− γ

χk , χk �= γ̄0

γ

γ̄ 20

e− γ

γ̄0 , χk = γ̄0

2.2 Selection Combining

Although optimum performance offered by MRC is highly desirable, practical systems oftensacrifice some performance in order to reduce their complexity. Instead of using MRC whichrequires exact knowledge of the channel state information, SC is often used in practice becauseof its reduced complexity relative to the optimum MRC scheme. Selection combiner onlyselects the best signal out of two replicas for further processing and neglects the remainingones. This reduces the computational costs and may even lead to a better performance thanMRC, because channels with very low SNR cannot be accurately estimated and contributemuch noise [36]. Mathematically speaking, the combined instantaneous SNR at the outputof the selection combiner is written by

γ� = max {γ0, γk∗ } (14)

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From the elementary theory of order statistics, we can obtain the joint PDF for γ� [32, p.194, eq. (6-79)] as follows:

f scγ�

(γ ) = Fγ0(γ ) fγk∗ (γ ) + fγ0(γ )Fγk∗ (γ )

=N∑

k=1

(−1)k−1N∑

n1,...,nk=1n1<···<nk

[1

γ̄0e− γ

γ̄0 + 1

χke− γ

χk −(

1

γ̄0+ 1

χk

)e−γ

(1γ̄0

+ 1χk

)]

(15)

3 Capacity Analysis

Adaptive transmission can be performed on a frame-by-frame basis subject to an averagepower constraint. To facilitate the analysis, the following three idealized assumptions areemployed: (i) instantaneous feedback (no latency) (ii) the feedback channel is error-free(iii) perfect channel state information available at the destination as well as the source. Theabove assumptions allow us to take an analytical approach and to derive the closed-form forShannon capacity of adaptive transmission for opportunistic cooperative networks.

Before getting in detail the system capacity, we emphasize that the PDFs given in (13)and (15) are expressed under a mathematically tractable form which renders them convenientfor analytical capacity evaluation. Furthermore, based upon the same approach discussed in[21], we can now evaluate the system capacity under the three different adaptive transmissionpolicies over Rayleigh fading channels in the following subsections.

3.1 Optimal Simultaneous Power and Rate Adaptation

Since the equivalent end-to-end SNR, γ� , is tracked perfectly by the destination and thensent back to the source via an error-free delayless feedback link, it therefore allows thesource to adapt both its power and rate to the actual channel condition. More specifically,the source allocates high power levels and rates for good channel conditions (γ� large), andlower power levels and rates for unfavorable channel conditions (γ� small). The achievablechannel capacity of opportunistic cooperative networks under OPRA is given by [21, Eq. 7]

Copra = B

2

+∞∫

γc

log2

(γ

γc

)fγ� (γ )dγ (16)

where B is the channel bandwidth in Hz and the ratio 1/2 in (16) is included to reflect that thesource-to-destination information transmission via relays will occupy two time slots. γc isthe optimal cutoff SNR threshold below which data transmission over the network is halted.In this case, the power adaptation is given by

Ps(γ�)

Ps={ 1

γc− 1

γ�, if γ� ≥ γc,

0, if γ� < γc

(17)

Note that the optimal cutoff threshold can be determined by using the total average powerconstraint [21, Eq. 8] as

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+∞∫

γc

(1

γc− 1

γ

)fγ� (γ )dγ = 1 (18)

3.1.1 Maximal Ratio Combining

Inserting (13) into (18) and after some manipulations yields

+∞∫

γc

(1

γc− 1

γ

)fγ� (γ )dγ =

N∑

k=1

(−1)k−1N∑

n1,...,nk=1n1<···<nk

mrc(γ̄0, χk) = 1 (19)

with

mrc(γ̄0, χk) =

⎧⎪⎪⎨

⎪⎪⎩

γ̄0γc

e− γc

γ̄0 −E1

(γcγ̄0

)

γ̄0−χk+

χkγc

e− γc

χk −E1

(γcχk

)

χk−γ̄0, χk �= γ̄0

(1 + γc

γ̄0− 1

γ̄0

)e− γc

γ̄0 , χk = γ̄0

With the current form of (19), in general, no closed-form solution of γc is available andthus γc has to be computed numerically where En(x) is the exponential integral of order ndefined by En(x) = ∫ +∞

1 t−ne−xt dt, x > 0 [37, p. 228, eq. (5.1.4)].Substituting (13) into (16), we obtain the channel capacity with MRC in terms of the

integral Jn(μ) = ∫ +∞1 tn−1 ln te−μt dt [21, eq. (70)] as follows:

Cmrcopra = B

2 ln 2

N∑

k=1

(−1)k−1N∑

n1,...,nk=1n1<···<nk

cmrcopra(γ̄0, χk) (20)

with

cmrcopra(γ̄0, χk) =

⎧⎨

⎩

γcγ̄0−χk

J1

(γcγ̄0

)+ γc

χk−γ̄0J1

(γcχk

), χk �= γ̄0

(γcγ̄0

)2J2

(γcχk

), χk = γ̄0

According to the operation mode of optimal simultaneous power and rate adaptation, thesystem will stop transmitting when γ� < γc. Mathematically speaking, the system suffersan outage probability, equal to the probability of the end-to-end SNR falls below the cutoffSNR, given by

Pmrco = Pr (γ� < γc) =

γc∫

0

f mrcγ�

(γ )dγ

=N∑

k=1

(−1)k−1N∑

n1,...,nk=1n1<···<nk

po(γ̄0, χk) (21)

with

po(γ̄0, χk) =

⎧⎪⎨

⎪⎩

(γ̄0

γ̄0−χk

) (1 − e

− γcγ̄0

)+(

χkχk−γ̄0

) (1 − e

− γcχk

), χk �= γ̄0

1 −(

1 + γcγ̄0

)e− γc

γ̄0 , χk = γ̄0

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3.1.2 Selection Combining

Following a similar procedure, we can also evaluate the optimal cutoff threshold for the caseof selection combining. In particular, we substitute (15) in (18) and then numerically solvethe resultant equation for γc given by1

+∞∫

γc

(1

γc− 1

γ

)f scγ�

(γ )dγ =N∑

k=1

(−1)k−1N∑

n1,...,nk=1n1<···<nk

sc(γ̄0, χk) = 1 (22)

with

sc(γ̄0, χk) = 1

γc

[e− γc

γ̄0 + e− γc

χk − e−γc

(1γ̄0

+ 1χk

)]

−[

1

γ̄0E1

(γc

γ̄0

)+ 1

χkE1

(γc

χk

)−(

1

γ̄0+ 1

χk

)E1

(1

γ̄0+ 1

χk

)]

Substituting (15) into (16) and then using the first order of Jn(μ) yields a closed-formcapacity formula as

Cscopra = B

2 ln 2

N∑

k=1

(−1)k−1N∑

n1,...,nk=1n1<···<nk

×[

γc

γ̄0J1

(γc

γ̄0

)+ γc

χkJ1

(γc

χk

)−(

γc

γ̄0+ γc

χk

)J1

(γc

γ̄0+ γc

χk

)](23)

Similarly, since no data is sent when γ� < γc, the cooperative network suffers an outage,namely

Psco = Pr (γ� < γc) =

γc∫

0

f scγ�

(γ )dγ

=N∑

k=1

(−1)k−1N∑

n1,...,nk=1n1<···<nk

[1 − e

− γcγ̄0 − e

− γcχk + e

−γc

(1γ̄0

+ 1χk

)](24)

3.2 Optimal Rate Adaptation with Constant Transmit Power

Under the condition of optimal rate adaptation with constant transmit power, the capacityof opportunistic cooperative network with an arbitrary number of relays subject to Rayleighfading is given by [21, Eq. (29)]

Cora = B

2

+∞∫

0

log2(1 + γ ) fγ� (γ )dγ (25)

Note that (25) also denotes the achievable capacity with CSI at the destination only. There-fore, it is interesting to mention at the viewpoint of hardware complexity that although the

1 Equation 19 and 22 can be straightforwardly done with the help of the Matlab function fzero inoptimization toolbox.

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system with CSI at the destination can achieve the same channel capacity but require lowercomplexity as opposed to that with ORA.


The channel capacity with MRC can be simply derived by making use the integral In(μ) =∫ +∞0 tn−1 ln(1 + t)e−μt dt, μ > 0, n = 1, 2, . . . [21, eq. (78)] in conjunction with (13) and

(25) as follows:

Cmrcora = B

2 ln 2

N∑

k=1

(−1)k−1N∑

n1,...,nk=1n1<···<nk

cmrcora (γ̄0, χk) (26)

with

cmrcora (γ̄0, χk) =

⎧⎪⎨

⎪⎩

I1(1/γ̄ 0)γ̄0−χk

+ I1(1/χk )χk−γ̄0

, χk �= γ̄0

I2(1/γ̄ 0)γ̄ 2

0, χk = γ̄0


Analogous to (26), from (15) and (25), the channel capacity with SC is now

Cscora = B

2 ln 2

N∑

i=1

(−1)i−1N∑

n1,...,ni =1n1<···<ni

[I1(1/γ̄ 0

)

γ̄0+I1

(1/χi)

χi−(

1

γ̄0+ 1

χi

)I1

(1

γ̄0+ 1

χi

)]

(27)

3.3 Truncated Channel Inversion with Fixed Rate

In order to maintain a constant received SNR at the destination, the source adapts its transmitpower based on the actual channel fading, i.e., it inverts the channel fading. As previouslydiscussed in [18,21], with channel inversion with fixed rate (CIFR), the equivalent channelbetween the source and destination appears to the encoder/decoder as a time-invariant addi-tive white Gaussian noise (AWGN) channel and the related channel capacity is given by [21,Eq. (29)]

Ccifr = B

2log2

⎛

⎜⎜⎜⎝

1 + 1+∞∫

0

fγ�(γ )

γdγ

⎞

⎟⎟⎟⎠

(28)

This technique has the advantage of maintaining a fixed data rate over the channel regard-less of channel conditions, however, results in a large capacity loss compared with otheradaptation techniques because a large amount of transmitted power is used to compensatefor the deep channel fades.

In order to achieve a better channel capacity, a modified version of this policy, termedtruncated channel inversion with fixed rate (TIFR) [21], in which the transmission is sus-pended in unfavorable channel conditions, γ� < γc, is proposed. Specifically, the power

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allocation strategy at the source obeys the following rule

Ps(γ�)

Ps=

⎧⎪⎨

⎪⎩

1

γ�

+∞∫γc

fγ� (γ )

γdγ

, if γ� ≥ γc

0, if γ� < γc

(29)

The related capacity in this adaptive policy, obtained by maximizing over all possible γc,is

Ctifr = maxγc

B

2log2

⎛

⎜⎜⎜⎝

1 + 1+∞∫γc

fγ�(γ )

γdγ

⎞

⎟⎟⎟⎠

(1 − Po) (30)

where Po is the outage probability previously provided in (21) and (24) for the case of MRCand SC, respectively.


The channel capacity with the truncated-inversion policy can be obtained by inserting (13)in (30) as

Cmrctifr = B

2log2

⎛

⎜⎜⎜⎜⎜⎜⎝

1 + 1N∑

k=1(−1)k−1

N∑

n1,...,nk=1n1<···<nk

cmrctifr (γ̄0, χk)

⎞

⎟⎟⎟⎟⎟⎟⎠

(1 − Pmrco ) (31)

with cmrctifr (γ̄0, χk) is given by

cmrctifr (γ̄0, χk) =

⎧⎨

⎩

E1(γc/γ̄0)γ̄0−χk

+ E1(γc/χk )χk−γ̄0

, χk �= γ̄0

1γ̄0

e− γc

γ̄0 , χk = γ̄0


Likewise, from (15) and (30), the capacity induced by the TIFR is

Csctifr = B

2log2

⎛

⎜⎜⎜⎜⎜⎜⎝

1 + 1N∑

k=1(−1)k−1

N∑

n1,...,nk=1n1<···<nk

csctifr(γ̄0, χk)

⎞

⎟⎟⎟⎟⎟⎟⎠

(1 − Psco ) (32)

with csctifr(γ̄0, χk) is expressed in terms of E1(.) as

csctifr(γ̄0, χk) = 1

γ̄0E1

(γc

γ̄0

)+ 1

χkE1

(γc

χk

)−(

1

γ̄0+ 1

χk

)E1

(1

γ̄0+ 1

χk

)(33)

Note that it is also possible to obtain channel capacity of the CIFR policy for both casesof MRC and SC by setting γc = 0 in (31) and (32), respectively.

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0 2 4 6 8 10 12 14 16 18 200.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Average SNR [dB]

Op

tim

al C

uto

ff T

hre

sho

ld (γ

c)

Optimal Simultaneous Power and Rate Adaptation

N = 2 MRC (Upper)N = 2 MRC (Lower)N = 4 MRC (Upper)N = 4 MRC (Lower)N = 2 SC (Upper)N = 2 SC (Lower)N = 4 SC (Upper)N = 4 SC (Lower)

Fig. 2 Optimal cutoff threshold versus average SNR for OPRA

4 Numerical Results and Discussion

In this section, we validate our analysis by comparing with simulation. Numerical results arepresented for N = 2 and 4 with λ0 = 1, {λ1,k}N

k=1 = 2, and {λ2,k}Nk=1 = 3.

In Figs. 2 and 3, we investigate the optimal cutoff threshold for OPRA and TIFR, definedas the value that maximizes the channel capacity, as a function of average SNR in dB. It canbe observed from Fig. 2 that the optimal cutoff threshold for OPRA increases as the numberof relays increases. Furthermore, the curves also reveal that for a fixed number of relays, theincrement of average SNR makes the optimal cutoff threshold increase, however, all curvesare bounded between [0 1] as reported in [21]. In Fig. 3, we compare the optimal thresholdvalue for TIFR versus the average SNR for two cases of network topologies, i.e., N = 2 and4. Contrary to the behavior of the optimal cutoff threshold of OPRA, which always varieswithin [0 1], this value increases linearly with the average SNR as well as with number ofrelays. However, note that in both figures, for a fixed number of relays and average SNR,there is a distinct gap between the upper and lower curve of optimal cutoff threshold. It isinteresting to note that the behaviour of these gaps is completely different at high and lowSNR regime. Specifically, with OPRA, the gap tends to diminish as average SNR goes fromlow to high regime; however, with TIFR, the contrary result is obtained. For completeness,it should also be mentioned that the use of MRC increases the optimal cutoff threshold ascompared with SC in the same network condition.

In Fig. 4, the closed-form channel capacity given in (20), (26), and (31) is presented andcompared for the network with two relays, N = 2. In general, from Fig. 4 and other numeri-cal experiments not included here, we notice that, as expected, TIFR gives the worst channelcapacity; OPRA gives the best channel capacity and ORA has a channel capacity quality inbetween the others. However, as can be observed, with no power adaptation at the source, the

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0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

40

45

50Channel Inversion with Fixed Rate

Average SNR [dB]

Op

tim

al C

uto

ff T

hre

sho

ld (γ

c)N = 2 MRC (Upper)N = 2 MRC (Lower)N = 4 MRC (Upper)N = 4 MRC (Lower)N = 2 SC (Upper)N = 2 SC (Lower)N = 4 SC (Upper)N = 4 SC (Lower)

Fig. 3 Optimal cutoff threshold versus average SNR for TIFR

ORA achieves nearly the highest capacity at high SNRs. This observation can be explainedby using the standpoint of a tradeoff between capacity and complexity.

In Fig. 5, we investigate the effects of number of relays on the network capacity per unitbandwidth. It can be seen that increasing number of relays makes the normalized capacityslightly increase. It is due to the fact that regardless of number relays involved in the coopera-tive transmission only two time slots is always required for the data communication betweenthe source and the destination.

Analogous to Figs. 4, 5, 6 and 7 illustrate the channel capacity and effect of number ofrelays for the same network topologies but with selection combining at the destination. It isworth noting that the conclusion remarks obtained for the case of MRC are also reached forthe case of SC.

Figure 8 compares the effect of combining techniques (MRC and SC) on the system capac-ity. The capacity per unit bandwidth curves confirm that, under same channel conditions, thecapacity of systems employing MRC receiver is always better as compared with equivalentsystems using SC by around 1 dB. Also in Fig. 8, the Shannon capacity of opportunistic coop-erative network under three adaptive transmission policies with 4 relays in both i.i.d and i.n.d.channels is examined. The results are based on the assumption that λ0 = 1, {λ1,k}N

k=1 = 2,and {λ2,k}N

k=1 = 3 for i.n.d. channels and λ0 = {λ1,k}Nk=1 = {λ2,k}N

k=1 = 1 for i.i.d. channels.It is clearly seen that the performance of DF relaying systems under i.n.d. channels is betterthan that under i.i.d. channels. Furthermore, the simulated and analytical results are in goodagreement.

The corresponding outage probability for the optimum adaptation and truncated channelinversion policies are plotted in Fig. 9 for the case of MRC (the results for the case of SC areomitted for brevity). As we can see, the benefit of the maximization of the capacity for TIFRsystems takes place at the cost of increasing probability of outage.

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0 2 4 6 8 10 12 14 16 18 200.5

1

1.5

2

2.5

3

3.5

4

Average SNR [dB]

Cap

acit

y p

er U

nit

Ban

dw

idth

C/B

[B

it/S

ec/H

z]

OPRA (Upper) AnalysisOPRA (Upper) SimulationOPRA (Lower) AnalysisOPRA (Lower) SimulationORA (Upper) AnalysisORA (Lower) AnalysisORA (Upper) SimulationTIFR (Upper) AnalysisTIFR (Upper) SimulationTIFR (Lower) AnalysisTIFR (Lower) Simulation

Fig. 4 Channel capacity per unit bandwidth for opportunistic cooperative networks under different adaptationpolicies with MRC

0 2 4 6 8 10 12 14 16 18 200.5

1

1.5

2

2.5

3

3.5

4

4.5

Average SNR [dB]

Cap

acit

y p

er U

nit

Ban

dw

idth

C/B

[B

it/S

ec/H

z]

N = 2 OPRA (Upper)N = 2 ORA (Upper)N = 2 TIFR (Upper)N = 4 OPRA (Upper)N = 4 ORA (Upper)N = 4 TIFR (Upper)

Fig. 5 Effect of number of relays on channel capacity per unit bandwidth for opportunistic cooperativenetworks under different adaptation policies with MRC

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0 2 4 6 8 10 12 14 16 18 200.5

1

1.5

2

2.5

3

3.5

4

Average SNR [dB]

Cap

acit

y p

er U

nit

Ban

dw

idth

C/B

[B

it/S

ec/H

z]OPRA (Upper) AnalysisOPRA (Upper) SimulationOPRA (Lower) AnalysisOPRA (Lower) SimulationORA (Upper) AnalysisORA (Lower) AnalysisORA (Upper) SimulationTIFR (Upper) AnalysisTIFR (Upper) SimulationTIFR (Lower) AnalysisTIFR (Lower) Simulation

Fig. 6 Channel capacity per unit bandwidth for opportunistic cooperative networks under different adaptationpolicies with SC

0 2 4 6 8 10 12 14 16 18 200.5

1

1.5

2

2.5

3

3.5

4

Average SNR [dB]

Cap

acit

y p

er U

nit

Ban

dw

idth

C/B

[B

it/S

ec/H

z]

N = 2 OPRA (Upper)N = 2 ORA (Upper)N = 2 TIFR (Upper)N = 4 OPRA (Upper)N = 4 ORA (Upper)N = 4 TIFR (Upper)

Fig. 7 Effect of number of relays on channel capacity per unit bandwidth for opportunistic cooperativenetworks under different adaptation policies with SC

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0 2 4 6 8 10 12 14 16 18 200.5

1

1.5

2

2.5

3

3.5

4

4.5

Average SNR [dB]

Cap

acit

y p

er U

nit

Ban

dw

idth

C/B

[B

it/S

ec/H

z]

OPRA (Upper) − SCORA (Upper) − SCTIFR (Upper) − SCOPRA (Upper) − MRCORA (Upper) − MRCTIFR (Upper) − MRC

i.i.d.

i.n.d.

Fig. 8 Channel capacity per unit bandwidth for opportunistic cooperative networks under different adaptationpolicies with MRC and SC over i.i.d and i.n.d. channels

0 5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR [dB]

Ou

tag

e P

rob

abili

ty

OPRA Analysis (Upper)OPRA Simulation (Upper)OPRA Analysis (Lower)OPRA Simulation (Lower)TIFR Analysis (Upper)TIFR Simulation (Upper)TIFR Analysis (Lower)TIFR Simulation (Lower)

N = 2

N = 4

Fig. 9 Outage probability of the ORPA and TIFR for opportunistic cooperative networks (MRC)

In Fig. 10, we perform numerical investigation and a channel capacity comparison oftwo schemes, namely repetition-base cooperative relaying [28] and opportunistic coopera-tive relaying. It is shown that the channel capacity of repetition-based cooperative networkis worse than that of the opportunistic networks. This observation is reasonable due to the

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0 5 10 15 20 25 300

1

2

3

4

5

6

Average SNR [dB]

Ch

ann

el C

apac

ity

per

Un

it B

and

wid

th C

/B [

Bit

/Sec

/Hz]

OPRA (Upper)OPRA (Lower)ORA (Upper)ORA (Lower)TIFR (Upper)TIFR (Upper)

OpportunisticCooperative

Network

Repetition−basedCooperative

Network

N = 2

Fig. 10 Channel capacity per unit bandwidth for opportunistic cooperative networks and repetition-basecooperative networks under different adaptation policies (MRC)

fact that in repetition-based cooperative networks, multiple time slots are required to real-ize orthogonal channelization which normally equals to number of relays. Meanwhile, foropportunistic cooperative relaying networks, the number of required time slots for a completedata communication is always two irrespective of number of relays.

5 Conclusion

The paper has investigated and compared opportunistic cooperative networks under threedifferent adaptive transmission policies: OPRA, ORA and TIFR. We presented a methodto derive the channel capacity of opportunistic cooperative networks in which two com-bining schemes, i.e., MRC and SC, are considered. The analysis is applicable for gen-eral cases, including independent identically distributed and independent but not identi-cally distributed Rayleigh fading channels. Numerical results were provided to show agood agreement between analytical and simulated curves, as such, validating the accu-racy of our analytical approach and illustrating the impact of adaptive transmission poli-cies on the opportunistic cooperative networks. It has been shown that among them, OPRAprovides the best capacity while TIFR suffers the largest capacity penalty relative to theother policies. Furthermore, the results also confirm the advantage of opportunistic cooper-ative networks over repetition-based cooperative networks in terms of channel capacity, asexpected.

Acknowledgment This research was supported by Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No.2009-0073895).

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Author Biographies

Hyung Yun Kong received the Ph.D. and M.E. degrees in Elec-trical Engineering from Polytechnic University, Brooklyn, New York,USA, in 1996 and 1991, respectively. And he received B.E. in Electri-cal Engineering from New York Institute of Technology, New York in1989. Since 1996, he was with LG electronics Co., Ltd. in multime-dia research lab developing PCS mobile phone systems and LG chair-man’s office planning future satellite communication systems from1997. Currently he is an associate professor in Electrical Engineering atUniversity of Ulsan, Ulsan, Korea. He performs several govern-ment projects supported by ITRC (Information Technology ResearchCenter), KOSEF (Korean Science and Engineering Foundation), etc.His research area includes high data rate modulation, channel coding,detection and estimation, cooperative communications, and sensor net-work. He is member of IEEK, KICS, KIPS, IEICE and IEEE.

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Vo Nguyen Quoc Bao received the B.E. and the M.S. degrees inElectronics and Telecommunications Engineering from Ho Chi MinhCity University of Technology, Vietnam, in 2002 and 2005, respec-tively. From December 2002 to March 2007, he was a lecturer atElectronics Engineering Department, Posts and TelecommunicationInstitute of Technology, Vietnam. He is currently working toward thePh.D. degree in the Department of Electrical Engineering, Universityof Ulsan, Korea. His major research interests are modulation and cod-ing techniques, MIMO system, digital signal processing, cooperativecommunications.

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On The Performance of Two-Way Amplify-and-Forward Relay Networks

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