Adaptive transmission in MIMO AF relay networks with ...834781/FULLTEXT01.pdf · Adaptive transmission in MIMO AF relay networks with orthogonal space-time block codes over Nakagami-m

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Adaptive transmission in MIMO AF relay networkswith orthogonal space-time block codes overNakagami-m fadingHoc Phan1*, Trung Q Duong1, Hans-Jürgen Zepernick1 and Lei Shu2

Abstract

In this article, we apply different adaptive transmission techniques to dual-hop multiple-input multiple-outputamplify-and-forward relay networks using orthogonal space-time block coding over independent Nakagami-mfading channels. The adaptive techniques investigated are optimal simultaneous power and rate (OSPR), optimalrate with constant power (ORCP), and truncated channel inversion with fixed rate (TCIFR). The expressions for thechannel capacity of OSPR, ORCP, and TCIFR, and the outage probability of OSPR, and TCIFR are derived based onthe characteristic function of the reciprocal of the instantaneous signal-to-noise ratio (SNR) at the destination. Forsufficiently high SNR, the channel capacity of ORCP asymptotically converges to OSPR while OSPR and ORCPachieve higher channel capacity compared to TCIFR. Although TCIFR suffers from an increase in the outageprobability relative to OSPR, it provides the lowest implementation complexity among the considered schemes.Along with analytical results, we further adopt Monte Carlo simulations to validate the theoretical analysis.

Keywords: adaptive transmission, amplify-and-forward, orthogonal space–time block coding, optimal simultaneouspower and rate, optimal power with constant rate, truncated channel inversion with fixed rate

1. IntroductionDuring the last decade, multiple-input multiple-output(MIMO) techniques have attracted great attention as away of improving spectral efficiency and reliability inwireless communications. MIMO systems with orthogo-nal space-time block coding (OSTBC) transmission areconsidered as a means of providing full diversity gainand linear decoding complexity [1-3]. In recent years,the combination of MIMO systems using OSTBC trans-mission with relay networks has been significantly con-sidered (see, e.g., [4-9] and the references therein). In[4,5], the outage probability and symbol error rate (SER)performance of MIMO decode-and-forward (DF) relaynetworks with OSTBC transmission over Rayleigh fadingchannels were investigated, respectively. In [6], the SERof MIMO systems in which the source employs OSTBCtransmission to transmit the signal to the destinationthrough the help of semi-blind amplify-and-forward(AF) relays over Rayleigh fading channels was derived.

In [7], the authors investigated the bit error rate (BER)performance of MIMO channel state information (CSI)-assisted AF relay networks with OSTBC transmissionover Rayleigh fading channels. More recently, by takingthe direct link between source and destination intoaccount, the SER and outage probability of MIMO CSI-assisted AF relay cooperative networks with OSTBCtransmission over Rayleigh fading channels were investi-gated [8]. Furthermore, closed-form expressions for theoutage probability and the SER of dual-hop MIMO CSI-assisted AF relay networks with OSTBC transmissionhave been derived for independent and correlated Naka-gami-m fading channels in [9,10], respectively.Although MIMO relay networks with OSTBC trans-

mission have received a lot of research efforts, all of theabove mentioned contributions concentrate on coopera-tive communications with constant transmission rateand power. For adaptive transmission, depending on thelink quality provided by the fading channels, the systemwill adapt its transmission power, transmission rate,coding rate/scheme, modulation scheme, or the arbitrarycombination of these techniques to the fluctuations

* Correspondence: [email protected] Institute of Technology, SE-371 79 Karlskrona, SwedenFull list of author information is available at the end of the article

Phan et al. EURASIP Journal on Wireless Communications and Networking 2012, 2012:11http://jwcn.eurasipjournals.com/content/2012/1/11

© 2012 Phan et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons AttributionLicense (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,provided the original work is properly cited.

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induced by the fading channels in order to enhance thespectral efficiency [11-17]. In particular, a cooperativerelay network where the source employs constant trans-mission power and adapts transmission rate through M-ary quadrature amplitude modulation (M-QAM) hasbeen studied in terms of outage probability, SER, andspectral efficiency for Rayleigh fading channels [13]. Thecombination of opportunistic incremental relaying withadaptive modulation deployed in cooperative relay net-works was analyzed in [14]. This scheme has beenshown to guarantee a specific BER performance leveland improve both spectral efficiency and outage prob-ability. In [15], approximations for the channel capacityof opportunistic cooperative multiple relay networksover Rayleigh fading channels under optimal simulta-neous power and rate (OSPR), optimal rate with con-stant power (ORCP), and truncated channel inversionwith fixed rate (TCIFR) were investigated. The upperbounds of channel capacity for AF cooperative systemsover Rayleigh fading channels under adaptive transmis-sion were derived in [16]. Furthermore, the use of differ-ent adaptive schemes in AF multi-hop relaying networksover Nakagami-m fading environments was studied in[17] wherein the achievable channel capacity was evalu-ated by using the characteristic function (CHF) of thereciprocal of the instantaneous SNR at the destination.In [18], the Shannon channel capacity of the maximumratio combining (MRC) receiver over h-μ fading chan-nels and adaptive transmission has been investigated.Upper bounds for the capacity of different adaptivetransmission techniques for an AF system with bestrelay selection over Rayleigh fading channels have beenreported in [19]. The study of [20] has presented a fra-mework for practical application of adaptive transmis-sion to MIMO systems to enhance the transmission rateof broadband wireless systems.Most efforts that have been made for the utilization of

adaptive transmission focus on traditional cooperativerelay networks over Rayleigh or Nakagami-m fadingchannels. To the best of the authors’ knowledge, there isno previous study considering adaptive transmission forMIMO AF relay networks with OSTBC transmission.Although OSTBC transmission provides diversity gainand decoding simplicity, it decreases transmission rateas compared to other kinds of space-time block codes(STBC). Therefore, it is beneficial to employ adaptivetransmission schemes for MIMO AF relay networkswith OSTBC transmission in order to enhance its spec-tral efficiency. In this article, we therefore analyze theperformance of these systems over independent, identi-cally distributed (i.i.d.) and independent, non-identicallydistributed (i.n.i.d.) Nakagami-m fading channels. Ourkey contributions can be summarized as follows. Weinvestigate the performance of MIMO CSI-assisted AF

relay networks with OSTBC transmission for the adap-tive schemes of OSPR, ORCP, and TCIFR over Naka-gami-m fading channels. Specifically, we deriveanalytical expressions for the channel capacity of thethree adaptive MIMO AF relay networks with OSTBCbased on the CHF of the reciprocal of the instantaneousSNR at the destination. Furthermore, we present expres-sions of the outage probability for the considered coop-erative relay networks with OSPR and TCIFR whereintransmission will be suspended as long as the instanta-neous SNR falls below an optimal value. For the ORCPscheme, however, there is no need to evaluate the out-age probability as the source constantly keeps transmis-sion regardless of the value of the instantaneous SNR.The remaining parts of this article are organized as

follows. In Section 2, we present the system and channelmodel, describing fundamental concepts of MIMO CSI-assisted AF relay networks with OSTBC transmission. InSection 3, we derive the CHF of the reciprocal of theinstantaneous SNR for the considered cooperative relaynetworks with OSTBC and adpative transmission. Thederivation of the channel capacity and the outage prob-ability of these cooperative relay networks are presentedin Section 4. We then derive the channel capacity of theinvestigated systems with ORCP in Section 5. In Section6, the channel capacity and the outage probability of thecooperative relay networks using TCIFR are derived.Analytical results and Monte Carlo simulations alongwith further discussions are presented in Section 7.Finally, in Section 8, conclusions of this study are given.Notation: Throughout this article, we will use the fol-

lowing notations. The Frobenius norm of a vector ormatrix is denoted as ||·||F. The probability density func-tion (PDF) and the cumulative distribution function(CDF) of a random variable (RV) X are denoted as fX(·)and FX(·), respectively. Then, FX(·) and ΨX(·) stand forthe moment generating function (MGF) and the CHF ofan RV X, respectively. In addition, CN (0, N0) representsan additive white Gaussian noise (AWGN) RV with zeromean and variance N0. We denote Γ(n) as the gammafunction [21, eq. (8.310.1)] and Γ(n, x) as the incompletegamma function [21, eq. (8.350.2)]. Furthermore, Ei(·)indicates the exponential integral function [22, eq.(6.9.2.25)], and Kn(·) is the nth order modified Besselfunction of the second kind [21, eq. (8.432.1)]. Finally,R{·} is the real part of a complex expression.

2. System and channel modelIn this article, we consider a dual-hop relay networkwith OSTBC transmission consisting of an n1-antennasource S, a single-antenna relay R, and an n2-antennadestination D (see Figure 1). Suppose that the directcommunication link between S and D is not applicabledue to severe shadowing. To achieve spatial diversity,


Page 2 of 14

the source utilizes OSTBC consisting of M symbols, s1,..., sM, which are chosen from a particular modulationconstellation. An OSTBC matrix C has the size of n1 ×N, where N is the block-length. The transmit power ofeach symbol is denoted as Ps and the code rate of anOSTBC is defined as Rc = M/N. In addition, all channelsare assumed to experience mutually i.i.d. or i.n.i.d.quasi-static Nakagami-m fading. Accordingly, the fadingcoefficients are constant during a transmission blockand change independently for every block. We alsoassume that the relay utilizes CSI-assisted AF mode toforward the signal to the destination. The transmissionfrom source S to destination D stretches over two hops.During the first hop, source S transmits an OSTBCmatrix C to relay R. The signal received at R is thereforegiven by

r = h1C + nR (1)

where h1 = [h1,1, . . . , h1,n1 ] is the row vector of thefading channel coefficients of the link from S to R andnR = [nR, 1,..., nR, N] denotes the row vector of the com-plex AWGN at relay R. Furthermore, h1, l, l = 1, 2, ...,n1, represents the fading channel coefficient of the linkfrom the lth antenna of S to R and nR, t, t = 1, 2,..., N,denotes the AWGN with zero mean and variance N0 atR during the tth symbol period.In the second hop, relay R first amplifies the signal

received from source S with an amplifying gain b andthen forwards the amplified signal to destination D.Accordingly, the signal received at destination D can beexpressed as

Y = βhT2r + WD (2)

where h2 = [h2,1, . . . , h2,n2 ] is the row vector of thefading channel coefficients from R to D and WD is then2 × N AWGN matrix at D whose entries representcomplex Gaussian RVs CN (0, N0). In particular, h2, l, l

= 1, 2, ..., n2, denotes the fading channel coefficient ofthe link from R to the lth antenna of D. The channelpower gain gi, l = |hi, l|

2 follows the gamma distributionof which the PDF and the CDF, respectively, are givenby

fgi,l (x) =α

mi,l

i,l

�(mi,l)xmi,l−1 exp(−αi,lx) (3)

Fgi,l(x) = 1 − �(mi,l, αi,lx)�(mi,l)

(4)

where αi,l =mi,l

�i,l, mi, l denotes the fading severity para-

meter of the Nakagami-m channel, and Ωi, l = E{gi, l} isthe average channel power, i Î {1, 2}, l Î {1, 2,..., ni}.Assuming that CSI in terms of the channel coefficient

vector h1 is perfectly known to relay R, the amplifyinggain b can be derived based on the constraint that thetransmit power of R for the second hop is set to beequal to the transmit power of S used in the first hop.In case of high SNR, b can be approximated asβ2 = (||h1||2F)−1. Due to the orthogonality associatedwith OSTBCs, maximum-likelihood (ML) decoding atthe destination reduces to symbol-wise decoding. Thismeans that decoding can be performed on a symbol-by-symbol basis with the related instantaneous SNR persymbol at destination D given as [7]

γ = ρ||h1||2F||h2||2F

||h1||2F + ||h2||2F(5)

where ρ =ρ0

n1Rc, and ρ0 =

Ps

N0represents the average

SNR of the source. For the sake of brevity let us denoteγ1 = ||h1||2F, and γ2 = ||h2||2F. The instantaneous SNR in(5) can then be rewritten as

Figure 1 System model of the considered dual-hop MIMO AF relay network with OSTBC and adaptive transmission.


Page 3 of 14

γ = ργ1γ2

γ1 + γ2(6)

It is noted that the tractable form of the instantaneousSNR in (6), which is expressed as the harmonic mean ofg1 and g2, is utilized for deriving the performance ofdual-hop MIMO AF relay networks with OSTBC andadaptive transmission in the sequel.

3. Derivation of the CHF of the reciprocal of theinstantaneous SNRGenerally, the PDF of the instantaneous SNR, g, isrequired to quantify the channel capacity of the consid-ered relay networks with adaptive transmission [12,16].However, due to the complex mathematical expressionfor the PDF, fg(g), rather intractable analytical expres-sions are obtained for the channel capacity with thisapproach. Hence, in this article, we utilize a treatmentdeployed in [17] that takes advantage of the harmonicmean form of the instantaneous SNR by using the CHFof its reciprocal. In view of (6), to derive the CHF ofγ −1 = ρ−1(γ −1

1 + γ −12 ), let us denote

X1 = γ −1i , i = 1, 2 (7)

Y = γ −1 = ρ−1(X1 + X2) (8)

The CHF of Y, ΨY (s), can be obtained from its corre-sponding MGF, FY (s), by substituting s = −jω wherej =

√−1. Due to the fact that gi, i = 1, 2, are mutuallyindependent, the CHF of Y can be obtained as

Y(ω) = �X1

(−jω

ρ

)�X2

(−jω

ρ

)(9)

In addition, the PDF of Y can be represented via itscorresponding CHF as follows:

fY(y) =1

2π

∞∫−∞

Y(ω) exp(−jωy)dω (10)

Depending on whether the fading channels are identi-cally distributed, the obtained CHF of g-1 has differentforms, resulting in different performance expressions foreach case. We now continue to derive the MGF andCHF of Y based on its respective PDF for i.i.d. and i.n.i.d. Nakagami-m fading channels.

3.1. MGF and CHF for i.i.d. Nakagami-m fading channelsIn this case, the statistics of all channels within eachhop are identical, given as mi,1 = · · · = mi,ni = mi andαi,1 = · · · = αi,ni = αi. To obtain the MGF and CHF of g-1, we require the PDF of gi, i = 1, 2, which is given by[23]

fγi (γ ) =α

nimii

�(nimi)γ nimi−1 exp(−αiγ ) (11)

Given the relationship, fT−1 (t) = t−2fT(t−1), for an RVT, the PDF of Xi is obtained as

fXi (x) =α

nimi1

�(n1mi)

(1x

)nimi+1

exp(−αi

x

)(12)

The MGF of Xi, i = 1, 2, can be derived from the PDFgiven in (12) by using the transformation

�Xi(s) =∫ ∞

−∞fXi(x) exp(−sx)dx (13)

and is obtained, in view of [21, eq. (3.471.9)], as

�Xi(s) =(2αis)

nimi

2

�(nimi)Knimi (2

√αis)

(14)

From (9) and (14), we finally obtain the CHF of thereciprocal of the instantaneous SNR for the consideredrelay networks in the presence of i.i.d. Nakagami-m fad-ing channels as

Y(ω) =4α

n1m1

21 α

n2m2

22

�(n1m1)�(n2m2)

(−jω

ρ

)n1m1 + n2m2

2 Kn1m1

(2

√−jα1ω

ρ

)Kn2m2

(2

√−jα2ω

ρ

) (15)

3.2. MGF and CHF for i.n.i.d. Nakagami-m fading channelsIn practical systems, the antennas of the source and thedestination are often located asymmetrically. This leadsto non-identical fading statistics among the source-relayhop and the relay-destination hop. Therefore, in thisarticle, we also consider i.n.i.d. Nakagami-m fadingchannels. For this case, we utilize [24, eq. (6)] in orderto derive the PDF of gi and eventually obtain

fγi (γ ) =ni∑l=1

mi,l∑k=1

ηi,l(k)αki,l

�(k)γ k−1 exp(−αi,lγ ) (16)

where the weighting coefficients hi, l(k) are determinedfrom [24, eq. (7)] as

ηi,l(k) =mi,l∑t1=k

t1∑t2=k

· · ·tni −3∑

tni−2=k

[(−1)m1−mi,l�

nir=1(αi,r)

mi,r

(αi,l)k

(mi,l + mi,1 + U(1 − l) − t1 − 1)!(mi,1+U(1−l) − 1)!(mi,l − t1)!

× (αi,l − αi,1+U(1−l))t1−mi,l−mi,1+U(1−l)(tni−2 + mi,ni−1+U(ni−1−l) − k − 1)!

(mi,ni−1+U(ni−1−l) − 1)!(tni−2 − k)!

× (αi,l − αi,ni−1+U(ni−1−l))k−tni−2−mi,ni −1+U(ni−1−l)ni−3∏w=1

(tw + mi,w+1+U(w+1−l) − tw+1 − 1)!

(mi,w+1+U(w+1−l) − 1)!(tw − tw+1)!

×(αi,l − αi,w+1+U(w+1−l))tw+1−tw−mi,w+1+U(w+1−l)]

(17)

with mi =∑ni

l=1 mi,l and U(x) being the unit step func-tion. Similarly to obtaining (12), the PDF of Xi can bewritten as


Page 4 of 14

fXi(x) =ni∑

l=1

mi,l∑k=1

ηi,l(k)αki,l

�(k)x−k−1 exp

(−αi,l

x

)(18)

Again, using [21, eq. (3.471.9)] with several manipula-tions, the MGF of Xi can be derived as

�Xi(s) = 2ni∑l=1

mi,l∑k=1

ηi,l(k)α

k2i,l

�(k)s

k2Kk(2

√αi, lS)

(19)

By substituting (19) into (9), the CHF of the reciprocalof the instantaneous SNR, Y = g-1, for i.n.i.d. Nakagami-m fading channels is written as

Y (ω) = 4n1∑l=1

m1,l∑k=1

n2∑p=1

m2,l∑q=1

η1,l(k)η2,p(q)α

k21,l α

q22,p

�(k)�(q)

×(−jω

ρ

)k + q2 Kk

(2

√−jα1,lω

ρ

)Kq

(2

√−jα2,pω

ρ

)(20)

In the following sections, we utilize the CHF of thereciprocal of the instantaneous SNR to analyze the per-formance of dual-hop MIMO AF relay networks usingOSTBC with different adaptive transmission schemes.Specifically, the source will adapt its transmission rateand/or power to the variations of the channel coeffi-cients. Moreover, for implementing adaptive transmis-sion, it is required that the instantaneous SNR isperfectly measured at the destination and is then sentback to the source through a feedback channel. Thisfeedback channel is assumed to be error free with negli-gible delay and therefore enables the source to timelyperform the transmission rate and/or power adaptation.Recall that the Shannon capacity of a fading channel

determines the theoretical upper bound on the maxi-mum transmission rate with an arbitrarily small errorprobability. Adaptive transmission has been known as ameans of achieving this bound [12]. For dual-hopMIMO AF relay networks using OSTBC where adapta-tion is only implemented at the source, the channelcapacity of different adaptive schemes is analyzed in thesequel.

4. Optimal simultaneous power and rateadaptationIn this section, we investigate the channel capacity andthe outage probability of the considered MIMO AFrelay network with OSTBC for the case of adaptivetransmission with OSPR. Accordingly, in response tothe instantaneous SNR fed back from the destination,the source will adapt its transmission power and

transmission rate in an optimal way, i.e., maximizing thechannel capacity subject to the average transmit powerconstraint. In order to conserve transmission power dur-ing deep fades, the transmission will be suspendedunder such channel conditions. This mode of operationremains as long as the instantaneous SNR, g, that is fedback from the destination to the source, is below theoptimal cutoff SNR, g0, for OSPR.Specifically, given an average transmit power con-

straint, the channel capacity of a system with OSPR,COSPR (bits/second), is defined as [12]

COSPR =B2

∞∫γ0

log2

(γ

γ0

)fγ (γ )dγ (21)

where B (Hz) denotes the channel bandwidth. Notethat the factor 1/2 in (21) accounts for the fact thatcommunication from source S to destination D throughthe help of relay R occupies two time slots. In order tosolve the integral given in (21), we apply a change ofvariable as t = g-1 together with (10) and [21, eq.(8.212.1)], yielding

COSPR =B2

∫ 1/γ0

0log2

(1

γ0t

)fY(t)dt

=B

4π ln 2

∞∫−∞

Y(ω)jω

[C + ln(jω) − ln(γ0) + Ei

(jω

γ0

)]dω

(22)

where C = 0.577215 is Euler’s constant [21, p. xxxii].Furthermore, making use of [17, eq. (9)], the formula-tion (22) can be refined as

COSPR =B

2π ln 2

π/2∫0

�{

Y(tan(θ))j tan(θ)

[C + ln(j tan(θ)) − ln(γ0) + Ei

(j tan(θ)

γ0

)]}sec2(θ)dθ (23)

where sec(·) denotes the secant function, i.e., sec(·) =1/cos(·). To the best of the authors’ knowledge, noclosed-form solution is available for this integration.However, we now can readily evaluate the performanceof the considered relay network by simply using thenumerical integration method as in [17].As far as the optimal cutoff SNR, g0, for OSPR is con-

cerned, it must satisfy the average transmit power con-straint

J =

∞∫γ0

(1γ0

− 1γ

)fγ (γ )dγ = 1 (24)

The optimal cutoff SNR, g0, can be found by numeri-cally solving (24) for g0. Again, utilizing the change ofvariable t = g-1 for the integral in (24) along with (10)and [17, eq. (9)], the average transmit power constraintis obtained as


Page 5 of 14

J =

1/γ0∫0

(1γ0

− t)

fY(t)dt =1

2π

∞∫−∞

Y(ω)

⎡⎢⎢⎣

γ0 − jω − γ0 exp(−jω

γ0

)γ0ω2

⎤⎥⎥⎦ dω

=1π

π/2∫0

�

⎧⎪⎪⎨⎪⎪⎩Y(tan(θ))

γ0 − j tan(θ) − γ0 exp(−j tan(θ)

γ0

)γ0 tan (θ)2

⎫⎪⎪⎬⎪⎪⎭sec2(θ)dθ = 1

(25)

As the transmission in OSPR is suspended for theduration of the instantaneous SNR, g, being below theoptimal cutoff SNR, g0, the related outage probability,Po,OSPR, can be defined as the probability of the event g<g0 and may hence be formulated as

Po,OSPR =

γ0∫0

fγ (γ )dγ = 1 −∞∫

γ0

fγ (γ )dγ (26)

Also, by using [17, eq. (9)], the outage probability canbe given by

Po,OSPR = 1 − 1π

π/2∫0

�{

Y(tan(θ))j tan(θ)

[1 − exp

(−j tan(θ)γ0

)]}sec2(θ)dθ (27)

4.1. Channel capacity and outage probability for i.i.d.Nakagami-m fading channelsBy substituting the CHF expression of Y from (15) intothe general formulae of Co,OSPR given in (23), the analy-tical expression for the channel capacity of dual-hopMIMO AF relay networks with OSTBC transmissionemploying OSPR over i.i.d. Nakagami-m fading channelsis obtained as

COSPR =2Bα

n1m1

21 α

n2m2

22

π ln 2�(n1m1)�(n2m2)

π/2∫0

�{

1j tan(θ)


(j tan(θ)

γ0

)].

×(

−j tan(θ)ρ

)n1m1 + n2m2

22∏

i=1

Knimi

(2

√−jαi tan(θ)

ρ

)⎫⎪⎬⎪⎭ sec2(θ)dθ

(28)

The average transmit power constraint for solving theoptimal cutoff SNR, g0, in (28) is found by substituting(15) into (25) as

J =4α

n1m1

21 α

n2m2

22

π�(n1m1)�(n2m2)

π/2∫0

�

⎧⎪⎪⎨⎪⎪⎩


γ0

)γ0 tan (θ)2

×(

−j tan(θ)ρ

)n1m1 + n2m2

22∏

i=1

Knimi

(2

√−jαi tan(θ)

ρ

)⎫⎪⎬⎪⎭ sec2(θ)dθ = 1

(29)

Substituting (15) in (27), the expression for the outageprobability is given by

Po,OSPR = 1 − 4α

n1m1

21 α

n2m2

22

π�(n1m1)�(n2m2)

π/2∫0

�

⎧⎪⎪⎨⎪⎪⎩

1 − exp(−j tan(θ)

γ0

)j tan(θ)

×(

−j tan(θ)ρ

)n1m1 + n2m2

22∏

i=1

Knimi

(2

√−jαi tan(θ)

ρ

)⎫⎪⎬⎪⎭ sec2(θ)dθ

(30)

4.2. Channel capacity and outage probability for i.n.i.d.Nakagami-m fading channelsBy substituting (20) into (23), the expression for thechannel capacity of dual-hop MIMO AF relay systemswith OSTBC transmission using OSPR over i.n.i.d.Nakagami-m fading channels is given by

COSPR =2B

π ln 2

n1∑l=1

m1,l∑k=1

n2∑p=1

m2,l∑q=1

η1,l(k)η2,p(q)αk21,lα

q22,p

�(k)�(q)

×π/2∫0

�{

1j tan(θ)


(j tan(θ)

γ0

)]

×(−j tan(θ)

ρ

)k + q2 Kk

(2

√−jα1,l tan(θ)

ρ

)Kq

⎛⎝2

√−jα2,p tan(θ)

ρ

⎞⎠⎫⎪⎪⎬⎪⎪⎭ sec2(θ)dθ

(31)

The optimal cutoff SNR, g0, in (31) is the numericalsolution of the respective average transmit power con-straint, which is achieved by substituting (20) into (25),as

J =4π

n1∑l=1

m1,l∑k=1

n2∑p=1

m2,l∑q=1


q22,p

�(k)�(q)

×π/2∫0

�

⎧⎪⎪⎨⎪⎪⎩(−j tan(θ)

ρ

)k + q2


γ0

)γ0 tan (θ)2

×Kk

(2


ρ

)Kq

⎛⎝2


ρ

⎞⎠⎫⎬⎭ sec2(θ)dθ = 1

(32)

Similarly to the case of i.i.d. Nakagami-m fading chan-nels, by substituting (20) into (27), the expression forthe outage probability is found as

Po,OSPR = 1 − 4π

n1∑l=1

m1,l∑k=1

n2∑p=1

m2,l∑q=1


q22,p

�(k)�(q)

×π/2∫0

�

⎧⎪⎪⎨⎪⎪⎩


γ0

)j tan(θ)

(−j tan (θ)

ρ

)k + q2

×Kk

(2


ρ

)Kq

⎛⎝2


ρ

⎞⎠⎫⎬⎭ sec2(θ)dθ

(33)

5. Optimal rate adaptation with constant transmitpowerIn this section, we derive an analytical expression for thechannel capacity of ORCP in the context of the consid-ered relay networks. For the case of OSPR, the sourceonly adapts its transmission rate in response to the

channel states asB2

log2(1 + γ ). The instantaneous SNR

at the destination is also provided to the source througha feedback channel. Compared to the fixed rate systems,wherein the transmission rate of the source is designedin advance to operate efficiently in specific level of chan-nel quality, the ORCP scheme can take advantage of the


Page 6 of 14

variations of fading channels because it constantlyadapts the transmission rate to the channel condition[25]. In addition, as the source adapts only its transmis-sion rate but not the power, the implementation ofORCP is less complex as compared to OSPR. Since thetransmission remains at arbitrary value of the instanta-neous SNR, no outage event occurs for ORCP scheme.The channel capacity, CORCP, of ORCP can be calcu-

lated as [12]

CORCP =B2

∞∫0

log2(1 + γ )fγ (γ )dγ (34)

which basically represents the average channel capa-city of a flat-fading channel with respect to the distribu-tion of the instantaneous SNR at the destination.As with OSPR, using the change of variable, t = g-1,

the integral of CORCP can be expressed as

CORCP =B2

∞∫0

log2

(1 +

1t

)1t2

fγ

(1t

)dt =

B2

∞∫0

log2

(1 +

1t

)fY(t)dt (35)

From (10) and [17, eq. (9)], we have

CORCP =B

4π ln 2

∞∫−∞

Y(ω)ln(jω) + exp(jω)Ei(jω) + C

jωdω

=B

2π ln 2

π/2∫0

�{Y(tan(θ))

ln(j tan(θ)) + exp(j tan(θ))Ei(j tan(θ)) + C

j tan(θ)

}sec2(θ)dθ

(36)

where sec(·) = 1/cos(·), and ω = tan(θ). In fact, theintegral of the real part of a complex expression givenin (36) can be numerically solved by mathematics soft-ware packages.

5.1. Channel capacity for i.i.d. Nakagami-m fadingchannelsThe channel capacity of ORCP for i.i.d. Nakagami-mfading channels can be calculated by substituting theCHF of the reciprocal of the instantaneous SNR givenin (15) into (36), yielding

CORCP =2Bα

n1m1

21 α

n2m2

22

π ln 2�(n1m1)�(n2m2)

π/2∫0

�{

ln(j tan(θ)) + exp(j tan(θ))Ei(j tan(θ)) + Cj tan(θ)

×(

−j tan(θ)ρ

)n1m1 + n2m2

22∏

i=1

Kni,mi

(2

√−jαi tan(θ)

ρ

)⎫⎪⎬⎪⎭ sec2(θ)dθ

(37)

5.2. Channel capacity for i.n.i.d. Nakagami-m fadingchannelsSimilarly, the channel capacity for i.n.i.d. Nakagami-mfading channels is obtained by substituting (20) into (36)

CORCP =2B

π ln 2

n1∑l=1

m1,l∑k=1

n2∑p=1

m2,l∑q=1

η1,l(k)η2,p(q)α

k21,l α

q22,p

�(k)�(q)

×π/2∫0

�{

ln(j tan(θ)) + exp(j tan(θ))Ei(j tan(θ)) + Cj tan(θ)

×(−j tan(θ)

ρ

)k + q2 Kk

(2


ρ

)Kq

⎛⎝2


ρ

⎞⎠⎫⎪⎪⎬⎪⎪⎭ sec2(θ)dθ

(38)

These above integrals given in (37) and (38) can benumerically solved with the help of standard mathe-matics software packages.

6. Truncated channel inversion with fixed rateWith TCIFR, the source only adapts its transmit powerto provide a constant instantaneous SNR at the destina-tion while keeping the transmission rate fixed. In otherwords, it inverts the channel response as long as chan-nel conditions are better than a specific level. Hence,the fading channels appear as time-invariant and a fixedtransmission rate can be maintained by changing thetransmit power properly regardless of the channel con-ditions. Therefore, TCIFR incurs the least implementa-tion complexity among OSPR and ORCP schemes. ForTCIFR, transmission will be suspended during periodsof deep fading. Specifically, as long as the instantaneousSNR is greater than the optimal cutoff SNR, gc, TCIFRis active.The channel capacity, CTCIFR, of TCIFR can be deter-

mined as follows [12]:

CTCIFR =B

2log2(1 + K−1)(1 − Po,TCIFR) (39)

where the outage probability of TCIFR, Po,TCIFR = Pr{g<gc}, and the average of the reciprocal of the instanta-neous SNR, K, are given by, respectively

Po,TCIFR = 1 −∞∫

γc

fγ (γ )dγ (40)

K =∫ ∞

γc

1γ

(γ )dγ (41)

and gc denotes the optimal cutoff SNR for TCIFR. Theoptimal cutoff SNR, gc, in (41) is chosen such that a par-ticular value of the outage probability is provided whilechannel capacity is maximized. It can be obtained bynumerically solving ∂CTCIFR/∂gc = 0. That is,

∂CTCIFR/∂γc =1

γcK(1 + K)(1 − Po,TCIFR) − ln

(1 +

1K

)= 0 (42)


Page 7 of 14

The expression for the outage probability of TCIFRmay be reformulated in terms of the CHF of Y by repla-cing g0 given in (27) with gc and can then be written as

Po,TCIFR = 1 − 1π

π/2∫0

�{

Y(tan(θ))j tan(θ)

[1 − exp

(−j tan(θ)γc

)]}sec2(θ)dθ (43)

Again, using the change of variable, t = g-1, togetherwith (10) and [17, eq. (9)], term K can be obtained as

K =∫ 1/γc

0tfY (t)dt

=1π

π/2∫0

�{

Y (tan(θ))

tan (θ)2γc

[(j tan(θ) + γc) exp

(−j tan(θ)γc

)− γc

]}sec2(θ)dθ

(44)

6.1. Channel capacity and outage probability for i.i.d.Nakagami-m fading channelsBy substituting (15) into (44), term K in (39) and (42)can be determined as

K =4α

n1m1

21 α

n2m2

22

π�(n1m1)�(n2m2)

π/2∫0

�

⎧⎪⎪⎨⎪⎪⎩

(j tan(θ) + γc) exp(−j tan(θ)

γc

)− γc

tan (θ)2γc

×(

−j tan(θ)ρ

)n1m1 + n2m2

22∏

i=1

Knimi

(2

√−jαi tan(θ)

ρ

)⎫⎪⎬⎪⎭ sec2(θ)dθ

(45)

Similarly, substituting (15) in (43), the outage prob-ability, Po,TCIFR, is

Po,TCIFR = 1 − 4α

n1m2

21 α

n2m2

22

π�(n1m1)�(n2m2)

π/2∫0

�

⎧⎪⎪⎨⎪⎪⎩


γc

)j tan(θ)

(−j tan(θ)

ρ

)n1m1 + n2m2

2

×Kn1m1

(2

√−jα1 tan (θ)

ρ

)Kn2m2

(2

√−jα2 tan(θ)

ρ

)}sec2 (θ) dθ

(46)

Both integral expressions given in (45) and (46) can benumerically solved by mathematics software packages.The respective channel capacity, CTCIFR, is obtained bysubstituting (45) and (46) in (39).

6.2. Channel capacity and outage probability for i.n.i.d.Nakagami-m fading channelsThe expression for K in (39) and (42) is obtained, bysubstituting (20) into (44), as

K =4π

n1∑l=1

m1,l∑k=1

n2∑p=1

m2,1∑q=1

η1,l(k)η2,p(q)α

k21,lα

q22,p

�(k)�(q)

×π/2∫0

�

⎧⎪⎪⎨⎪⎪⎩

(j tan(θ) + γc) exp(−j tan(θ)

γc

)γc

tan (θ)2γc

(−j tan(θ)ρ

)k + q

2

×Kk

(2


ρ

)Kq

⎛⎝2


ρ


(47)

Substituting (20) in (43), the outage probability, Po,TCIFR, is determined as

Po,TCIFR = 1 − 4π

n1∑l=1

m1,l∑k=1

n2∑p=1

m2,l∑q=1


q22,p

�(k)�(q)

×π/2∫0

�

⎧⎪⎪⎨⎪⎪⎩

[1 − exp

(−j tan(θ)γc

)]j tan(θ)

(−j tan (θ)

ρ

)k + q2

×Kk

(2


ρ

)Kq

⎛⎝2


ρ


(48)

The above expressions in (47) and (48) can be solvednumerically. The respective channel capacity, CTCIFR, isobtained by substituting (47) and (48) in (39).It is noted that the obtained expressions for the out-

age probability and channel capacity given in (28), (30),(31), (33), (37), (38), and (45)-(48) are represented interms of one-dimensional integrals with definite limits.These expressions can readily be solved by standardmathematics software packages such as Mathematica. Itis noted that for single antenna relay networks, theobtained expressions for the system performance aregiven in integral forms in [17].

7. Numerical resultsIn this section, we present numerical examples for theperformance metrics derived above. Monte Carlo simu-lations are provided together with analytical results inorder to verify our analysis. Importantly, in all testedscenarios, there is very close agreement between theanalytical and simulated curves. This confirms the cor-rectness of the analysis presented in this article.Firstly, let us examine the channel capacity of the con-

sidered relay networks with adaptive transmissionunder-going i.i.d. Nakagami-m fading channels. Thenumber of antennas at source S and destination D areselected as n1 = n2 = 1, 2, 3, and the fading severityparameter is set as m = 2. The optimal cutoff SNR forOSPR and TCIFR can be found by numerically solving(25) and (42), respectively, that is presented in Table 1.Figure 2 depicts the channel capacity per unit band-

width, i.e., COSPR/B, CORCP/B, and CTCIFR/B versus aver-age SNR. It can be seen from Figure 2 that OSPR

Table 1 Optimal cutoff SNR g0 and gc for i.i.d. Nakagami-m fading channels and different number of antennas

n1 = n2 = 1 n1 = n2 = 2 n1 = n2 = 3

SNR (dB) g0 gc g0 gc g0 gc0 0.1069 0.1338 0.1086 0.1293 0.1090 0.1267

4 0.2010 0.2706 0.2140 0.2715 0.2196 0.2713

8 0.3489 0.5267 0.3852 0.5475 0.4007 0.5569

12 0.5408 0.7237 0.6018 1.0557 0.6227 1.0898

16 0.7302 1.7913 0.7888 1.9736 0.8058 2.0639

20 0.8663 3.2490 0.9035 3.6980 0.9125 3.9181


Page 8 of 14

achieves a slight improvement in capacity as comparedto ORCP, and this improvement decreases as the aver-age SNR, r0, increases. The reason for this is that, forboth OSPR and ORCP, the source tends to transmitwith constant power in the high SNR regime. Moreimportantly, as the number of antennas increases, thechannel capacity improvement of OSPR relative toORCP tends to be negligible. This is due to the fact thatthe instantaneous SNR approaches infinity more rapidlyfor larger number of antennas when the average SNRconverges to infinity. However, a system with OSPRadapts both its transmission rate and power simulta-neously, thereby incurring higher implementation com-plexity as compared to the one using ORCP.Furthermore, TCIFR suffers the largest reduction in

channel capacity in comparison with OSPR and ORCP.In fact, with specific average transmit power constraint,TCIFR must consume more power to compensate forbad channel conditions in order to maintain a constantvalue of instantaneous SNR at the destination, causingconsiderable reduction in channel capacity relative toother schemes. Unlikely, OSPR and ORCP take advan-tage of favorable fading channel conditions by deployinghigher transmission rate and/or consuming more trans-mit power. As expected, when the number of antennasincreases, the effects of severe fading can be reduced,and thus this capacity reduction diminishes significantly.In practice, due to the limitation of the number ofantennas for each terminal, perfectly eliminating theimpact of fading by exploiting spatial diversity is almost

0 2 4 6 8 10 12 14 16 18 200.0

0.5

1.0

1.5

Cha

nnel

cap

acity

per

uni

t ban

dwid

th (b

it/s/

Hz)

SNR (dB)

Analysis OSPR ORCP TCIFR

Simulation OSPR ORCP TCIFR n

1=n

2=1, 2, 3

Figure 2 Channel capacity per unit bandwidth versus average SNR of the source, r0, of MIMO AF relay networks with OSTBCtransmission over i.i.d. Nakagami-m fading channels using different adaptive schemes. (Fading severity parameter m = 2, average channelpower gain Ω = 0.25).


Page 9 of 14

impossible. Since TCIFR provides the lowest implemen-tation complexity, it is needed to consider a tradeoffbetween channel capacity and complexity.Figure 3 shows the channel capacity per unit band-

width given in (39), CTCIFR/B, versus gc, with constantvalues for r0 = 5, 10, 15 dB. It can be seen that thereexists a specific optimal cutoff SNR of gc at which thechannel capacity of the considered networks with TCIFRis maximized. Also, as expected from Table 1, the valuesof the optimal cutoff SNR, g0, are smaller than those forthe optimal cutoff SNR, gc, of TCIFR at the same averageSNR. This observation implies that maintaining a con-stant value of the instantaneous SNR in TCIFR consumesmore power than maximizing channel capacity in OSPRwith the same fading condition. In other words, fadingconditions for activating OSPR is likely less favorable

than TCIFR, causing considerable reduction in channelcapacity and outage probability of TCIFR scheme.In addition, Figure 4 plots the outage probability of

the considered relay networks with OSPR and TCIFR.Note that for these schemes, the transmission processwill be suspended once the instantaneous SNR fallsbelow the optimal cutoff SNR g0 and gc, respectively.The achievable values of these optimal metrics impose asubstantial impact on the channel capacity and the out-age probability. As expected, one can see from Figure 4that the outage probability of OSPR outperforms that ofTCIFR considerably. Especially, when the number ofantennas increases this outage probability improvementbecomes more significant.Figure 5 shows the channel capacity per unit band-

width versus average SNR of the considered relay

-10 -8 -6 -4 -2 0 2 4 6 8 100.0

0.4

0.8

1.2

ρ0=10dB

ρ0=15dB

n1=n2=1

n1=n2=2

n1=n2=3

Cutoff SNR (dB)

Cha

nnel

cap

acity

per

uni

t ban

dwid

th (b

it/s/

Hz)

ρ0=5dB

Figure 3 Channel capacity per unit bandwidth versus gc of MIMO AF relay networks with OSTBC transmission over i.i.d. Nakagami-mfading channels using TCIFR. (Fading severity parameter m = 2, average channel power gain Ω = 0.25).


Page 10 of 14

networks using OSPR, ORCP, and TCIFR in the pre-sence of i.n.i.d. Nakagami-m fading channels. The num-ber of antennas at source S and destination D areselected as n1 = n2 = 1, 2, 3 while the fading severityparameters m and average channel power gains Ω are asfollows:

n1 = n2 = 1 : {(m1,1, �1,1)} = {(2, 0.2)}{(m2,1, �2,1)} = {(3, 0.3)}

n1 = n2 = 2 : {(m1,i, �1,i)}2i=1 = {(1, 0.25); (2, 0.35)}

{(m2,i, �2,i)}2i=1 = {(2, 0.3); (1, 0.4)}

n1 = n2 = 3 {(m1,i, �1,i)}3i=1 = {(2, 0.25); (3, 0.3); {2, 0.4}}

{(m2,i, �2,i)}3i=1 = {(3, 0.4); (2, 0.2); (4, 0.35)}

The optimal cutoff SNR for OSPR and TCIFR is pre-sented in Table 2. Again, it can be seen from Figure 5

that the channel capacity of OSPR and ORCP outper-forms that of TCIFR for sufficiently large values of theaverage SNR. More importantly, OSPR obtains the bestperformance at the expense of high implementationcomplexity. Figure 6 depicts the channel capacity per

unit bandwidth given in (39),CTCIFR

B, versus gc, at con-

stant values of r0 = 5, 10, 15 dB. It can be also seenthat the channel capacity obtains the maximum value atone optimal value of gc The outage probability versusaverage SNR for OSPR and TCIFR is shown in Figure 7.As can be seen, OSPR outperforms TCIFR in terms ofoutage probability. The best outage performance can beobtained for the case of n1 = n2 = 3.

0 2 4 6 8 10 12 14 16 18 2010-6

10-5

10-4

10-3

10-2

10-1

100

Simulation OSPR, n

1=n

2= 1

TCIFR, n1=n

2= 1

OSPR, n1=n

2= 2

TCIFR, n1=n

2= 2

OSPR, n1=n

2= 3

TCIFR, n1=n

2= 3

Analysis OSPR TCIFR

SNR (dB)

Out

age

prob

abili

ty

Figure 4 Outage probability versus average SNR of the source, r0, of MIMO AF relay networks with OSTBC transmission over i.i.d.Nakagami-m fading channels using either OSPR or TCIFR. (Fading severity parameter m = 2, average channel power gain Ω = 0.25).


Page 11 of 14

8. ConclusionsWe have analyzed the performance of the three adap-tive transmission schemes of OSPR, ORCP, and TCIFRapplied to MIMO CSI-assisted AF cooperative relaynetworks with OSTBC. Our analysis is based on both

the i.i.d. and i.n.i.d. Nakagami-m fading channels,which generalizes a wide class of multi-path fadingenvironments. Closed-form expressions for the MGFsof the reciprocal of the instantaneous SNR are derivedand then utilized to evaluate the performance metrics.We present Monte Carlo simulations, which are in avery close agreement with analytical results, to validateour analysis. We also show that for sufficiently highSNR the channel capacity of OSPR and ORCP arealmost the same and better than that of TCIFR for theconsidered scenarios. Regarding the practical imple-mentation, ORCP is less complex to set up as com-pared to OSPR. It is also seen that for low SNR, thechannel capacity of TCIFR outperforms that of theORCP scheme. Moreover, it can be seen that the out-age probability of OSPR is significantly lower than thatof TCIFR; however, TCIFR offers the least complexityfor practical implementation.

0 2 4 6 8 10 12 14 16 18 200.0

0.5

1.0

1.5

2.0C

hann

el c

apac

ity p

er u

nit b

andw

idth

(bit/

s/H

z)

SNR (dB)

Analysis OSPR ORCP TCIFR

Simulation OSPR ORCP TCIFR

n1=n

2=1, 2, 3

Figure 5 Channel capacity per unit bandwidth versus average SNR of the source, r0, of MIMO AF relay networks with OSTBCtransmission over i.n.i.d. Nakagami-m fading channels using different adaptive schemes.

Table 2 Optimal cutoff SNR g0 and gc for i.n.i.d.Nakagami-m fading channels and different number ofantennas

n1 = n2 = 1 n1 = n2 = 2 n1 = n2 = 3

SNR (dB) g0 gc g0 gc g0 gc0 0.1054 0.1301 0.1297 0.1609 0.1318 0.1535

4 0.2011 0.2658 0.2450 0.3278 0.2620 0.3281

8 0.3527 0.5226 0.4210 0.6416 0.4648 0.6710

12 0.5492 0.9860 0.6306 1.2065 0.6850 1.3071

16 0.7397 1.8056 0.8069 2.2186 0.8450 2.4823

20 0.8731 3.2986 0.9125 4.1195 0.9319 4.7573


Page 12 of 14

-10 -8 -6 -4 -2 0 2 4 6 8 100.0

0.4

0.8

1.2

1.6

ρ0=10dB

ρ0=15dB

n1=n2=1

n1=n2=2

n1=n2=3

Cutoff SNR (dB)

Cha

nnel

cap

acity

per

uni

t ban

dwid

th (b

it/s/

Hz)

ρ0=5dB

Figure 6 Channel capacity per unit bandwidth versus gc of MIMO AF relay networks with OSTBC transmission over i.n.i.d. Nakagami-mfading channels using TCIFR.

0 2 4 6 8 10 12 14 16 18 2010-6

10-5

10-4

10-3

10-2

10-1

100

Simulation OSPR, n

1=n

2= 1

TCIFR, n1=n

2= 1

OSPR, n1=n

2= 2

TCIFR, n1=n

2= 2

OSPR, n1=n

2= 3

TCIFR, n1=n

2= 3

SNR (dB)

Out

age

prob

abili

ty

Analysis OSPR TCIFR

Figure 7 Outage probability versus average SNR of the source, r0, of MIMO AF relay networks with OSTBC transmission over i.n.i.d.Nakagami-m fading channels using either OSPR or TCIFR.


Page 13 of 14

Author details1Blekinge Institute of Technology, SE-371 79 Karlskrona, Sweden 2GraduateSchool of Information Science and Technology, Osaka University, Osaka,Japan

Competing interestsThe authors declare that they have no competing interests.

Received: 13 August 2011 Accepted: 12 January 2012Published: 12 January 2012

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doi:10.1186/1687-1499-2012-11Cite this article as: Phan et al.: Adaptive transmission in MIMO AF relaynetworks with orthogonal space-time block codes over Nakagami-mfading. EURASIP Journal on Wireless Communications and Networking 20122012:11.

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