Multi-relay MIMO Systems With OSTBC Over ... - Semantic Scholar

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 8, OCTOBER 2013

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Multi-relay MIMO Systems With OSTBC Over Nakagami-m Fading Channels Ehsan Soleimani-Nasab, Student Member, IEEE, Michail Matthaiou, Member, IEEE, and Mehrdad Ardebilipour

Abstract—We investigate the performance of a dual-hop amplifyand-forward (AF) multi-relay system over independent and identically distributed (i.i.d.) Nakagami-m fading channels, where all nodes in the system are equipped with multiple antennas, and space–time block codes are used in both transmissions. By assuming that the number of source–relay spatial subchannels is equal to the number of relay–destination subchannels, new analytical expressions for the most important figures of merit, namely, outage probability, symbol error probability (SEP), and average channel capacity, are derived for arbitrary signal-to-noise ratios (SNRs) and for arbitrary values of the m parameter. We also present simplified expressions in the high-SNR regime that enable us to quantify the system performance in terms of diversity order and coding gain. Two different scenarios have been considered. First, each relay and destination combine the received signal, and then, the harmonic mean of the source–relay and relay–destination channels is computed. After that, the relay with the best harmonic mean is selected. In the second scenario, each relay and destination combine the received signal, and then, the relay with the best source–relay SNR is selected. After that, the harmonic mean of the source-to-selected-relay channel and selected-relay-to-destination channel is computed. For both considered scenarios, some special cases of interest (e.g., Nakagami-0.5 and Rayleigh) are examined. Our results explicitly demonstrate that the first scheme has higher diversity order than the second scheme, whereas the coding gain of the second scheme is always greater for the multi-relay case. Index Terms—Amplify-and-forward (AF), orthogonal space– time block codes (OSTBC), outage probability.

I. I NTRODUCTION

U

TILIZING cooperative diversity among multiple-relay nodes can offer significant performance enhancement and improve the coverage of communication systems. The basic cooperative relaying techniques are amplify-and-forward (AF) and decode-and-forward (DF) schemes. More specifically, in AF relaying schemes, the relay amplifies the received signal from the source and forwards it to the destination (without Manuscript received September 14, 2012; revised December 17, 2012, January 16, 2013, and March 9, 2013; accepted April 21, 2013. Date of publication May 7, 2013; date of current version October 12, 2013. This paper was presented in part at the IEEE International Conference on Communications, Budapest, Hungary, June 2013. The work of M. Matthaiou was supported by the VINN Excellence Center Chase. The review of this paper was coordinated by Prof. X. Wang. E. Soleimani-Nasab and M. Ardebilipour are with the Faculty of Electrical and Computer Engineering, K.N. Toosi University of Technology, Tehran 1431714191, Iran (e-mail: [email protected]; mehrdad@eetd. kntu.ac.ir). M. Matthaiou is with the Department of Signals and Systems, Chalmers University of Technology, 412 96 Gothenburg, Sweden (e-mail: michail. [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2013.2262009

explicitly decoding or demodulating the messages or symbols) [1], [2]. For this reason, AF relaying systems have low hardware requirements and implementation cost, thereby representing an elegant solution from a practical viewpoint. Therefore, we hereafter elaborate on the performance of AF relaying schemes. The Nakagami-m distribution often gives the best fit to reallife data from land mobile and indoor mobile radio links [3], [4]. For this reason, the performance assessment of AF relaying in Nakagami-m fading channels has been a very active area of research. In this context, in [5], a comprehensive framework was proposed for the analysis of cooperative dual-hop wireless systems over generalized fading channels based on an AF relaying mechanism with blind and semi-blind relays [6]. In [7], the outage and bit error probabilities of a dual-hop AF relay network over independent and identically distributed (i.i.d.) Nakagami-m fading channels were deduced, by considering the variable-gain relaying protocol, which leads to the so-called harmonic mean end-to-end signal-to-noise ratio (SNR) criterion. In [8], an opportunistic relaying method was introduced, in which a single relay is selected, which is based on the best end-to-end instantaneous SNR criterion, and then forwards the message to the destination. In [9], a closed-form expression for the outage probability of the harmonic mean of dual-hop singlerelay systems was derived, where the Nakagami-m parameter m should be an integer. In [10], an analytical expression for the outage probability over Nakagami-m fading channels for channel-state-information (CSI)-assisted single-relay systems was proposed. A selection combiner with AF relays has been studied in [11], where an upper bound on the outage probability was derived. In [12], the work in [9] was extended to the multiple-relay case. A closed-form expression for the symbol error probability (SEP) at high SNRs was also derived. Moreover, in [13], the exact outage probability of dual-hop AF relaying over Nakagami-m fading channels was investigated. The analysis was done only for one relay in the network, and analytical expressions were presented neither for the SEP nor for the average channel capacity. In [14], a new tighter upper bound was proposed for the end-to-end SNR of multihop AF relaying, which was used to derive the outage probability and SEP of the system over Rayleigh fading channels. In [15], a closed-form expression was derived for the ergodic capacity of dual-hop Nakagami-m fading channels. In [16], the high-SNR outage probability and the SEP of dual-hop selection-combining (SC) AF relaying over Nakagami-m fading channels were investigated. In [17], the work in [16] was extended to the case of the N th best relay selection, using the upper bound of the harmonic mean. The main disadvantage of these works is that they either investigated a single-relay system

0018-9545 © 2013 IEEE

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[6], [10], [15], did not use the exact form of the harmonic mean of dual-hop relaying systems [11], [14], [16], [17], or they assumed the special case of integer m for multi-relay systems [7]–[9], [12]. In addition, in all previous works, all nodes in the system are assumed to be equipped with a single antenna. This precludes the beneficial use of multiple-input–multiple-output (MIMO) technology in conjunction with relaying systems. However, it has been theoretically shown that the performance of relaying systems can be significantly enhanced by exploiting the benefits offered by MIMO technology [18], [19]. On these lines, orthogonal space–time block codes (OSTBC) can be used to achieve full diversity in MIMO AF relaying. In [20], different adaptive transmission techniques were applied to dual-hop MIMO AF relay networks using OSTBC over independent Nakagami-m fading channels, where the relay was equipped with a single antenna. The moment-generating function (mgf) and the transmission rate were derived in closedform. In [21], closed-form expressions were derived for the end-to-end outage probability and the SEP of MIMO dual-hop relaying systems over correlated Nakagami-m fading channels, where only one antenna was utilized at the relay. In [22], the asymptotic SEP performance of a downlink cooperative communication system, where all nodes in the system are equipped with multiple antennas, was analyzed. In [23], a multi-relay MIMO cooperative system with OSTBC was analyzed, where no SC was performed. Closed-form expressions for the outage probability and the symbol error rate of a single-relay MIMO AF dual-hop system can be found in [24]. In both [23] and [24], the fading parameter m was assumed to be an integer. In [25], four joint relay and antenna selection strategies for dual-hop AF MIMO relay networks were proposed. In particular, the outage probability was derived for integer values of m, whereas the SEP was derived only from high SNRs. Most recently, exact closed-form expressions for the probability density function (pdf) and the cumulative distribution function (cdf) of the end-to-end SNR of opportunistic dual-hop AF relaying over Nakagami-m and Rician fading channels have been derived in [26]. Although new analytical expressions for single-relay systems were derived therein, the most important differences between this paper and [26] are as follows: First, in [26], all relay nodes and the destination deploy a single antenna. Second, only numerical results for SC were obtained, whereas all analytical results were limited to integer values of m. Finally, although some numerical results for the SEP and average channel capacity are illustrated, analytical expressions for these important performance metrics were not derived. Motivated by the aforementioned limitations of [26], we herein pursue a detailed performance analysis of dual-hop AF relaying systems employing OSTBC. In this light, we derive new analytical expressions for the outage probability, the SEP, and the average channel capacity of multi-antenna SC relaying networks over Nakagami-m fading channels, which hold for arbitrary values of the m parameter. In particular, the contributions of this paper can be summarized as follows. • We consider a general multi-relay dual-hop configuration, where all nodes are equipped with multiple antennas, and OSTBC are used in the transmissions from the source and relay nodes. Note that this is a very complicated setup that

has been scarcely investigated in the literature. We focus on the popular Nakagami-m fading model and relax the assumption of integer m values. Our analysis requires that the number of source–relay spatial MIMO subchannels is equal to the number of relay–destination subchannels since the analysis of the general case is a challenging mathematical problem. New analytical expressions for the outage probability, the SEP, and the average channel capacity are derived. We elaborate on two relay selection techniques, namely, the maximum of the harmonic mean and the harmonic mean of the maximum of source-torelay SNRs. In the first scheme, the source–relay and relay–destination channels are separately combined using OSTBC. The orthogonal property of OSTBC allows us to do independent maximum-likelihood (ML) decoding and transform a MIMO channel into a scalar singleinput–single-output channel. Subsequently, the relay selection is made after computing the harmonic mean of the source–relay and relay–destination channels [25], [26]. In the second scheme, after applying OSTBC at the first hop, the relay with the strongest SNR among the firsthop channels is selected. Then, OSTBC at the second-hop are applied. Finally, the harmonic mean of the source-toselected-relay and selected-relay-to-destination channels is computed, which yields the effective end-to-end SNR. • To obtain some additional insights into the impact of system parameters, such as fading parameters, the number of relays, and the number of antennas, we elaborate on the asymptotically high-SNR regime. In this asymptotic case, we investigate the generic SEP parameterization in terms of diversity order and coding gain [27]. One important conclusion is that the diversity order of the first scheme is greater than that of the second scheme, whereas the coding gain of the second scheme is always greater for the multirelay case. We also particularize the presented results to the worst case of Nakagami-0.5 and to the case of Rayleigh fading channels. The remainder of this paper is organized as follows. Section II introduces the system model. In Section III, we pursue an exact and high-SNR analysis for the first scenario and derive analytical expressions for the outage probability and SEP. Moreover, an upper bound on the harmonic mean is investigated along with the average channel capacity. In Section IV, a similar performance analysis for the second scenario is presented. Finally, Section V presents a set of numerical results, whereas Section VI concludes this paper. II. S YSTEM M ODEL AND FADING S TATISTICS We consider a cooperative relay system with L + 2 users: one acting as the Ns antenna source S, one acting as the Nd antenna destination D, and L acting as the relay nodes, which deploy Nr1 transmit antennas and Nr2 receive antennas (see Fig. 1). Moreover, Ps and Pr are the transmitted powers from the source and that from each relay, respectively; hSi Rk, l is the channel coefficient between the ith antenna of the source and the lth receive antenna of relay k (i.e., the Si → Rk, l link); and hRk, j Dn is the channel coefficient between the jth transmit antenna of relay k and the nth antenna of the destination

SOLEIMANI-NASAB et al.: MULTI-RELAY MIMO SYSTEMS WITH OSTBC OVER NAKAGAMI-m FADING CHANNELS

Fig. 1.

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Schematic illustration of the system under consideration.

(i.e., the Rk, j → Dn link). Additionally, N0 denotes the noise power at relay k and the destination. Hence, the instantaneous SNRs for the Si → Rk, l and Rk, j → Dn links are γSi Rk, l = Ps |hSi Rk, l |2 /N0 and γRk,j Dn =Pr |hRk,j Dn |2 /N0 , respectively. The transmitted symbols from the source and relays are encoded by the OSTBC matrices C1 and C2 , which are of size Ns × T1 for the source and Nr2 × T2 for each relay, respectively, while T1 and T2 are the block lengths of the transmitted signal from the source and relays, respectively. Since M symbols are transmitted over T1 and T2 time intervals, the code rates of the source and relay OSTBC matrices are Rc1 = M/T1 and Rc2 = M/T2 [28], respectively. Δ

Δ

Finally, H1k = [hS1 Rk , . . . , hSNs Rk ] ∈ C Nr1 ×Ns and H2k = [hRk,1 D , . . . , hRk,Nr D ] ∈ C Nd ×Nr2 , where hSi Rk ∀i = 1, 1 . . . , Ns is the channel vector between the ith antenna of the source and Nr1 antennas of the kth relay, whereas hRk, j D ∀j = 1, . . . , Nr1 is the channel vector between the jth antenna of the kth relay and Nd antennas of the destination. Omitting explicit details and following the general methodology of [20], [21], and [24], the relay gain for the kth selected relay is Gk = 1/(Ps H1k 4F + N0 H1k 2F )1/2 , where AF defines the Frobenius norm of matrix A. Then, the end-toΔ end SNR γe2e = γ1k γ2k /(γ1k + γ2k + 1) can be tightly upper bounded via the harmonic mean criterion, [22, Eq. 3] Y =

γ1k γ2k 1 = γ1k + γ2k y1k + y2k

(1)

where γ1k=s H1k 2F , γ2k=r H2k 2F , s=Ps /(Ns Rc1 N0 ), r = Pr /(Nr2 Rc2 N0 ), y1k = 1/γ1k , and y2k = 1/γ2k .1 As stated previously, in this paper, we assume that the amplitude of both links follows the Nakagami-m distribution, where m ≥ 0.5 represents the Nakagami-m fading parameter [3]. Thus, the corresponding SNRs are i.i.d. Gamma random variables, whose shape parameter is m and scale parameter is Ω/m, where Ω

is the average SNR per symbol. In this case, the pdf of both γSi Rk, l and γRk, j Dn is given by αm γ m−1 −αγ e (2) Γ(m) ∞ Δ where α = m/Ω, and Γ(n) = 0 e−t tn−1 dt denotes the gamma function [29, Eq. 8.310.1]. In addition, the cdf of γSi Rk, l and γRk, j Dn can be expressed as fγ (γ) =

Fγ (γ) = 1 −

(3)

∞ where Γ(b, x) = x e−t tb−1 dt is the upper incomplete gamma function [29, Eq. 8.350.2]. As was outlined earlier, we are particularly interested in two different relay selection schemes, which have been well investigated in the open literature. In the following, we investigate the first scheme based on the maximum of the harmonic mean. III. R ELAY S ELECTION BASED ON THE M AXIMUM OF H ARMONIC M EAN In this first scheme, by applying an OSTBC transmission scheme, the harmonic mean of the source–relay and relay–destination channels is computed. After that, the relay with the best harmonic mean is selected. In the subsequent analysis, the distribution of the harmonic mean SNR will be particularly important. The following result introduces a novel expression for this figure of merit. Note that, due to mathematical limitations, the subsequent analysis requires that the number of source–relay spatial subchannels is equal to the number of relay–destination subchannels, i.e., Ns Nr1 = Nr2 Nd , and that s = r = . Note that both assumptions have been widely used in the related literature [21]–[23]. Proposition 1: Assuming Ns Nr1 = Nr2 Nd and s = r = , the pdf of the harmonic mean SNR in (1) is given by √ 2mNs Nr1 α 2 mNs Nr1 − 2 γ 2 mNs Nr1 − 2 π 3 1 [Γ(mNs Nr1 )]2 2 mNs Nr1 −2 3

fY (γ) = 1 Note that a scaling factor of 2 is missing from the exact harmonic mean definition in (1); we, henceforth, denote the end-to-end SNR in (1) as the harmonic mean SNR, without significant loss of generality.

Γ(m, αγ) Γ(m)

×e

− 2αγ

1

3

3

W mNs Nr1 , mNs Nr1 2

2

4αγ

(4)

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where Wυ,μ (z) is the Whittaker hypergeometric function [29, Eq. 9.220.4]. Proof: See Appendix A. With the pdf expression in our hands, we can now work out the mgf of the harmonic mean SNR. Using [30, Eq. 07.45.26.0006.01] and [29, Eq. 9.31.5] and after some manipulations, the pdf of the harmonic mean SNR in (1) can be alternatively rewritten as √ πα fY (γ) = 2mN N −3 s r1 2 [Γ(mNs Nr1 )]2 mNs Nr1 − 12 2, 0 4αγ × G1, 2 (5) mNs Nr1 − 1, 2mNs Nr1 − 1

terms of the more tractable hypergeometric functions, which are also more amenable to mathematical manipulations. Corollary 1: The pdf, the cdf, and the mgf of the harmonic mean SNR in (1) are given, respectively, in (8)–(10), shown at the bottom of the page, where p Fq (·) represents the generalized hypergeometric function with p, q ∈ Z [29, Eq. 9.1]. Proof: By substituting the Meijer’s G function in (5)–(7) into [30, Eq. 07.34.26.0004.01], the pdf, the cdf, and the mgf of the harmonic mean can be rewritten as (8)–(10), respectively. Note that, in the general case, mNs Nr1 is not necessarily an integer. In the case of m being an integer, we can introduce an infinitely small perturbation term ε, such that m + ε becomes a noninteger. By doing so, we can avoid the singularities caused by the gamma functions appearing in (9) and (10).

α ,...,α

1 p s where Gr, p, q [x|β1 ,...,βq ] denotes the Meijer’s G function [29, Eq. 9.301]. By using [30, Eq. 07.34.21.0003.01], the cdf of the harmonic mean SNR in (1) can be expressed as √ παγ FY (γ) = 2mN N −3 s r1 2 [Γ(mNs Nr1 )]2 0, mNs Nr1 − 12 2, 1 4αγ × G2, 3 . (6) mNs Nr1 − 1, 2mNs Nr1 − 1 − 1

Finally, the mgf of the harmonic mean SNR in (1) can be derived from [30, Eq. 07.34.22.0003.01] as follows: √ πα φY (s) = 2mN N −3 s r1 2 [Γ(mNs Nr1 )]2 s 0, mNs Nr1 − 12 2, 1 4α × G2, 2 . (7) s mNs Nr1 − 1, 2mNs Nr1 − 1 Although (5)–(7) are given in terms of Meijer’s G functions, which are standard built-in functions in most software packages (e.g., MAPLE and MATHEMATICA), they provide limited physical insights. For this reason, (5)–(7) can be written in

A. Exact SNR Analysis In practical scenarios, where the phases of the relays’ signals are not aligned up to certain accuracy, beamforming gains may no longer be attained, and in fact, destructive interference may occur at the destination. In this case, it is meaningful that only one relay participates in the cooperative transmission; the technique is widely known as selective relaying or opportunistic relaying [8], [12], [25]. In this case, the relay with the highest harmonic mean will be selected among all relays. Using order statistics, the cdf of this SC SNR can be expressed as [31] FSC (γ) =

L

fY (γ)

i=1

=

√ παγ 3 G2, 2, 3 2mNs Nr1 −3 2 [Γ(mNs Nr1 )]2 L 4αγ 0, mNs Nr1 − 12 × . mNs Nr1 − 1, 2mNs Nr1 −1, −1 (11)

Referring to (11), we can now provide the link of our results with some previous results. For instance, for single-antenna

√ 2mNs Nr −1 1 2αmNs Nr1 γ mNs Nr1 −1 4αγ 22mNs Nr1 +1 πα2mNs Nr1 γ fY (γ) = F N ; − 0.5;1 − mN + 1 1 s r 1 Γ(mNs Nr1 )mNs Nr1 [Γ(mNs Nr1 )]2 2mNs Nr1 Γ (−mNs Nr1 ) 4αγ × (8) 1 F1 mNs Nr1 + 0.5; mNs Nr1 + 1; − Γ (−mNs Nr1 + 0.5) mNs Nr1 2mNs Nr1 22mNs Nr1 +1 √π αγ 2 αγ 4αγ FY (γ) = + 2 F2 mNs Nr1 , 0.5;1 − mNs Nr1 , mNs Nr1 + 1; − Γ(mNs Nr1 + 1) [Γ(mNs Nr1 )]2 Γ(−mNs Nr1 )Γ(2mNs Nr1 ) 4αγ × 2 F2 2mNs Nr1 , mNs Nr1 + 0.5; mNs Nr1 + 1, 2mNs Nr1 + 1; − Γ(2mNs Nr1 + 1)Γ(0.5 − mNs Nr1 ) (9) √ 2αmNs Nr1 4α 22mNs Nr1 +1 πα2mNs Nr1 φY (s) = + 2 F1 mNs Nr1 , 0.5; 1 − mNs Nr1 ; − s (s)mNs Nr1 [Γ(mNs Nr1 )]2 (s)2mNs Nr1 Γ(−mNs Nr1 )Γ(2mNs Nr1 ) 4α × (10) 2 F1 2mNs Nr1 , mNs Nr1 + 0.5; mNs Nr1 + 1; − Γ(0.5 − mNs Nr1 ) s


systems (i.e., Ns = Nr1 = Nr2 = Nd = 1), (11), by using [30, Eqs. 07.34.03.0002.01, 7.34.03.0612.01, and 07.33.03.0047.01], simplifies to m−1

1 α n 2αm − 2αγ FSC (γ) = 1 − e γ m+n Γ(m)m n! n=0 ⎞ L n m−1

n m−1 2αγ ⎠ × (12) Kj−k+1 k j j=0 k=0

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Proof: See Appendix B. It is now interesting to examine the convergence of the infinite series in (82) in Appendix B, which represents an alternative expression for the cdf of the SC SNR and has been used to derive (15). To do so, we resort to the use of convergence tests [29, Eq. 0.22]. The series in (82) can be written as S=

∞

c1, k γ k+mNs Nr1 −1 +

k=0

where Kv (·) is the vth-order modified Bessel function of the second kind [29, Eq. 8.432], [31]. Note that, in [10], [12], [13], and [25], the authors worked with γe2e = γ1 γ2 /(γ1 + γ2 + 1) instead of γe2e = γ1 γ2 /(γ1 + γ2 ), which makes (12) a tight lower bound for [26, Eq. 17]. For single-antenna and singlerelay systems (L = Ns = Nr1 = Nr2 = Nd = 1), (12) is a tight lower bound for [12, Eq. 5], [10, Eqs. 5 and 14], and [13, Eq. 2]. The outage probability is the probability that the output SNR falls below a normalized threshold γth . Using (11), the outage probability of the first approach can be readily obtained as Pout (γth ) = FSC (γth ).

(13)

The pdf of the SC can be derived by differentiating (11) with respect to γ and, thereafter, using [30, Eq. 07.34.21.0003.01] L √ πα fSC (γ) = L 2mN −3 2 [Γ(mN )]2 mN − 12 2, 0 4αγ × G1, 2 mN − 1, 2mN − 1 L−1 0, mN − 12 2, 1 4αγ × γG2, 3 mN − 1, 2mN − 1, −1 (14) where N = Nr1 Ns . Since the Laplace transform of (14) is almost impossible to derive, the mgf of (14) cannot be analytically determined. Hence, we follow a different line of reasoning, as suggested by the following proposition. Proposition 2: The mgf of the relay selection SNR, which is based on the maximum of the harmonic mean, is equal to ⎞ ⎛ ∞ ∞ L

L Γ(ς + 1) j, k ⎠ (15) ⎝ ψ j ϕk φSC (s) = ρL l sςj, k j=0 l=0

k=0

where Δ

ϕ0 =AL 0

j 1 Ak (kl − j +k) ϕj−k ϕj = jA0 k! Δ

∀j ≥ 1

k=1

Δ

ψ0 =B0L−l

j

Bk (kL−kl−j + k)ψj−k Δ 1 ∀j ≥ 1 ψj = jB0 k!

ςj, k =j +k+mNs Nr1 (2L − l). Note that Ak and Bk are defined in (83) in Appendix B.

c2, k γ k+2mNs Nr1 −1 (16)

k=0

where Δ

c1, k =

Ak k!

Δ

c2, k =

Bk . k!

(17)

Since the sum of two convergent series is also convergent, it is sufficient to prove that each of series in (16) is convergent. To do so, by using the ratio test, the convergence radius of the infinite series can be obtained as c1, k = lim (k+1)(ν1 +k)(ν2 +k) → ∞ R1 = lim k→∞ c1,k+1 k→∞ 4α(τ +k)(0.5+k) c2,k (k+1)(ω1 +k)(ω2 +k) → ∞. = lim R2 = lim k→∞ c2,k+1 k→∞ 4α(σ1 + k)(σ2 +k) (18) Therefore, the series in (82) converges absolutely. Note that the convergence speed of (82) is significantly high. This is mainly due to the factorial terms in the denominator, which remarkably accelerate the convergence speed of the series. Having the mgf in our hands, we can now work out the SEP of the selection scheme under consideration. On this basis, we recall that the SEP of many digital communication systems [e.g., binary phase-shift keying (BPSK) and M -ary pulse amplitude modulation (PAM)] can be expressed as [4] ∞ Pe =

bQ(

cγγ)fSC (γ)dγ

(19)

0 Δ

where is the average SNR of the fading channel, Q(x) = √ γ ∞ 2 1/ 2π x e−t /2 dt is the Gaussian Q-function, and the constants b and c depend on the type of the modulation. For the case under consideration, the following corollary will be particularly useful: Corollary 2: If the pdf of the relay selection SNR, which is based on the maximum of the harmonic mean, can be expressed as fSC (γ) = aγ t , where a is a generic constant, then the SEP can be written as 2t Γ(t + 1.5) (cγ)−(t+1) . Pe = ab √ π(t + 1)

(20)

Proof: By substituting the pdf of the SC into (19) and using the Q-function definition [4, Eq. 4.1], we have

k=1

Δ

∞

ab Pe = √ 2π

∞ ∞ 0

√

cγγ

γ t e−

u2 2

dudγ.

(21)

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√ The integration region of (21), i.e., cγγ ≤ u < ∞, 0 ≤ γ < ∞, is equivalent to the following area 0 ≤ u < ∞, 0 ≤ γ ≤ u2 /cγ. Using this observation and by exchanging the order of integrations in (21), the SEP can be rewritten as u2

ab Pe = √ 2π

∞ c¯γ 0

γ t dγe−

u2 2

du.

(22)

0

By solving the inner integral using [29, Eq. 2.01.1], we have ab √ Pe = (t + 1) 2π

∞

u2(t+1) − u2 e 2 du. (c¯ γ )t+1

(23)

0

Using [32, Eq. 18.77], the proof is completed. Substituting (86) in (19) and, thereafter, using (20), the SEP can be derived as follows: ⎞ ⎛ ∞ ∞ L ςj,k −1

L ψ ϕ 2 Γ(ς + 1/2) j k ⎠. ⎝ √j,k Pe = bρL ςj,k l (c¯ γ ) π j=0 l=0

To examine how the fading parameters m and Ω and the system parameters L and N = Ns Nr1 affect the SEP, we take the first derivative of (25) with respect to these parameters. By using [30, Eq. 06.05.20.0001.01] and after some simple manipulations, the first derivative of (25) with respect to m can be obtained as mN L 2m N L2L−1 Γ(mN L + 0.5) ∂Pe∞ = √ ∂m γ Ω π [Γ(mN + 1)]L c¯ 2m exp(1) × ψ(mN L + 0.5)−ψ(mN + 1)+ln (30) c¯ γ Ω

k=0

(24) B. High-SNR Analysis Since the exact results earlier are relatively hard to evaluate and provide limited physical insights, we now focus on the high-SNR regime. Corollary 3: At high SNRs, the SEP of the relay selection scheme, which is based on the maximum of the harmonic mean, can be expressed as (25) Pe∞ = (Gc γ¯ )−Gd + o γ¯ −Gd where the diversity order and coding gain are, respectively (26) Gd =mNs Nr1 L

− mN 1N L s r1 c 2mNs Nr1 L+L−1 Γ(mNs Nr1 L + 1/2) √ . Gc = α π [Γ(mNs Nr1 + 1)]L

where ψ(x) is Euler’s digamma function [29, Eq. 8.360.1]. At high SNRs, γ¯ goes to infinity; hence, the logarithmic function yields always a negative value (for finite values of m), which scales faster than the digamma function in (30). Therefore, the expression in (30) is always negative; hence, by increasing m, the SEP will decrease. Likewise, it is obvious that, by increasing Ω, the SEP will be reduced. Using similar methodology as for (30), we can also show that the SEP is a monotonically decreasing function of L. Finally, the first derivative of (25) with respect to N can be expressed as mN L 2m mL2L−1 Γ(mN L + 0.5) ∂Pe∞ = √ ∂N γ Ω π [Γ(mN + 1)]L c¯ 2m × ψ(mN L + 0.5) − ψ(mN + 1) + ln . (31) c¯ γ Ω Once more, we can see that the SEP is a decreasing function of N . We can now investigate the asymptotic outage probability: Corollary 4: In the low outage regime, the outage probability of the relay selection scheme, which is based on the maximum of the harmonic mean, can be expressed as mNs Nr1 L αγth 2L ∞ Pout (γth ) = [Γ(mNs Nr1 + 1)]L mN N L + o γth s r1 . (32)

Proof: The proof follows easily by using (28). To examine how the fading parameters m and Ω and the Proof: To compute the SEP at high SNRs, we first substi- system parameters L and N affect the outage probability, we tute the dominant term of [29, Eq. 9.14.1] in [29, Eq. 9.303], take the first derivative of (32) with respect to these parameters. By using [30, Eq. 07.34.03.0002.01] and after some simple such that the cdf in (11) is simplified as manipulations, the first derivative of (32) with respect to m is mNs Nr1 L mN N L αγ 2L mN L ∞ FSC (γ)= + o γ s r1 . ∂Pout mγth exp(1) 2L N L (γth ) L = [Γ(mNs Nr1 +1)] L ∂m Ω [Γ(mN (28) + 1)] mγth exp(1) By differentiating (28) with respect to γ, then the asymptotic × ln − ψ(mN + 1) . (33) Ω pdf can be obtained according to (27)

fSC (γ) =

mNs Nr1 L α γ mNs Nr1 L−1 L + 1)] +o γ mNs Nr1 L−1 . (29)

mNs Nr1 L2L [Γ(mNs Nr1

Substituting (29) in (19) and using (20), the proof is completed.

In the low outage regime, γth → 0; therefore, the argument of the logarithmic function is always less than one (for finite values of m). In addition, ψ(x) has a positive value when x ≥ 1.5, which holds true for Nakagami-m fading with m ≥ 0.5. Hence, (33) is always a negative function of m; thus, the outage probability is monotonically decreasing with respect to m. In addition, it is obvious that, by increasing Ω, the outage


probability will be decreased. The first derivative of (32) with respect to N can be written as mN L ∞ αγth ∂Pout 2L mL (γth ) = ∂N [Γ(mN + 1)]L mγth × ln − ψ(mN + 1) . (34) Ω Again, for the same reason as in (33), the outage probability is a monotonically decreasing function of N . Finally, the first derivative of (32) with respect to L can be expressed as ⎛ mN ⎞ mγth 2 ∞ mN L L Ω (αγth ) 2 ∂Pout (γth ) ⎟ ⎜ = ln ⎝ ⎠ ∂L Γ(mN + 1) [Γ(mN + 1)]L mN L which is evidently negative for γth → 0 and m finite. Thus, the outage probability is a decreasing function of L. C. Upper Bound Analysis

3727

Since, in this method, the relay with the strongest upper bounded harmonic mean should be selected, the effective endto-end SNR can be expressed as γSC = max (γup ). k=1,...,L

(38)

Considering (38), the cdf of the end-to-end SNR can be written as αγ mNs Nr1 L αγ 2 L e− L FˆγSC (γ)= +o γ mNs Nr1 L L [Γ(mNs Nr1 +1)] (39) which can very easily lead to the corresponding lower bound on the outage probability in (32). Next, by differentiating the cdf in (39) with respect to γ, the pdf of the end-to-end SNR can be written as mNs Nr1 L α mNs Nr1 L2L ˆ fγSC (γ) = L [Γ(mNs Nr1 + 1)] αγ ×γ mNs Nr1 L−1 e− L + o γ mNs Nr1 L−1 . (40)

It has been shown in [11] that XY /(X + Y ), where X and Y are any arbitrary nonnegative random variables, can be tightly upper bounded by min(X, Y ), With this result in our hands, we can now derive simplified results for the SEP and average channel capacity. If we define the upper bound of the harmonic mean in (1) as γup = min(γ1k , γ2k )2 and consider the independence between the fading distributions of both links, the cdf of γup can be written as [37]3

Using [38, Eq. 3.7.2.5] and [29, Eq. 9.303], the asymptotic SEP of the relay selection scheme, which is based on the maximum of the upper bound of harmonic mean, can be expressed as

L−1 mNs Nr1 L Γ(mNs Nr1 L + 0.5) α 2 ∞ ˆ Pe = √ (cγ/2)mNs Nr1 L π [Γ(mNs Nr1 + 1)]L + o γ −mNs Nr1 L . (41)

Fˆγup (γ) = Pr (min(γ1k , γ2k ) ≤ γ) = 1 − Pr(γ1k ≥ γ, γ2k ≥ γ) ⎤ ⎤⎡ ⎡ Γ mNd Nr2 , αγ Γ mNs Nr1 , αγ ⎦⎣ ⎦. =1 − ⎣ Γ(mNs Nr1 ) Γ(mNd Nr2 )

Note that (41) represents a lower bound on the exact SEP in (24). The expression in (41) implies that the diversity order and coding gain are, respectively, equal to (26) and (27). This is anticipated since the SEP lower bound becomes, by definition, exact at high SNRs. Another important figure of metric is channel capacity, which represents the maximum transmission rate when the probability of error is infinitely small. The average channel capacity for the upper bound approach is given in the following proposition: Proposition 3: At high SNRs, the average channel capacity (in bits/s/Hz) of the relay selection scheme, which is based on the maximum of the upper bound of the harmonic mean, can be expressed as

(35) Note that (35) represents a lower bound on the exact outage probability in (13). Since an exact SNR analysis is tedious, we now focus on the high-SNR regime. We now invoke the infinite series representation of Γ(m, x) according to [29, Eq. 8.354.1] ∞

xm+k −x . (36) Γ(m, x) = Γ(m) 1 − e Γ(m + k + 1) k=0

By assuming Ns Nr1 = Nd Nr2 and substituting (36) into (35) and keeping only the dominant term as the SNR grows large, we end up with Fˆγup (γ) = 2e−

αγ

(αγ)mNs Nr1 + o γ mNs Nr1 . mNs Nr1 Γ(mNs Nr1 + 1) (37)

2 When X > 0 and Y > 0, clearly, XY /(X + Y + 1) ≤ XY /(X + Y ). This inequality becomes an equality when X and Y grow large with no bound (a scenario corresponding to high SNRs). Hence, in summary, we have XY /(X + Y + 1) ≤ XY /(X + Y ) ≤ min(X, Y ). Interestingly, this tight upper bound is practically the same as the end-to-end SNR of relaying systems using the DF protocol [33]–[36]. 3 Note that, in the subsequent analysis, we use the hat symbol to denote all metrics that are based on the upper bound.

mNs Nr1 2L−1 1 Cˆ ∞ = ln 2 [Γ(mNs Nr1 + 1)]L LmNs Nr1 L−1 γ¯ −mNs Nr1 L + 1, 1, 1 1, 3 ×G3, 2 . 1, 0 Lα

(42)

Proof: See Appendix C. Note that the analytical relationship in (42) represents an upper bound for the average channel capacity of the SC scheme described in Section III-A. D. Special Cases Here, we particularize the earlier reported results to some practical cases of interest. We begin with the case of Nakagami-0.5 fading.

3728


a) Nakagami-0.5 fading channels: This type of Nakagami-m has significant importance in wireless systems when high quality of service is needed [39]. For this worst-case scenario, when we set L = 1 and Ns = Nr1 = Nd = Nr2 = 1 in (13), we have 4αγth 2, 1 4αγth 0, 0 G Pout (γth ) = √ −0.5, 0, −1 π 2, 3 (a) 4αγth 0 1,1 4αγth G = √ −0.5, −1 π 1, 2 4αγth 1 (b) 1 = √ G1,1 0.5, 0 π 1, 2

4αγth (c) = 1 − erfc (43) √ ∞ 2 where erfc(x) = 2/ π x e−t dt is the complementary error function. Note that (a), (b), and (c) are, respectively, obtained by using [30, Eq. 07.34.03.0002.01], [29, Eq. 9.31.5], and [30, Eq. 06.27.26.0005.01]. By considering the relation between the Q function √ and the complementary error function Q(x) = 1/2erfc(x/ 2), the exact outage probability can be alternatively expressed as

2αγth Pout (γth ) = 1 − 2Q 2 (44) which coincides with [40, Eq. 18]. b) Rayleigh fading channels (m = 1): The Rayleigh distribution is frequently used to model multipath fading with no direct line-of-sight path [4]. For the Rayleigh case, when we set L = 1 and Ns = Nr1 = Nd = Nr2 = 1, (13) can be further simplified according to √ αγth 2, 1 4αγth 0, 0.5 G2, 3 Pout (γth ) = 2 π 0, 1, −1 √ αγth 2, 1 4αγth 0, 0.5 =1 − 2 π G2, 3 −1, 1, 0 √ αγth 2,0 4αγth 0.5 (d) G1, 2 = 1−2 π −1, 1 2αγth αγth − 2αγ th (e) e =1−2 K1 . (45)

complexity. In fact, in this second scheme, the relay chooses the best channel based on the source–relay channels, whereas in the first scheme, the destination chooses the best end-toend harmonic mean, which requires full instantaneous CSI of all links. This requirement renders the implementation of such schemes laborious. In addition, CSI may be outdated at the time of selection since it takes some time for the destination to acquire the estimates of all source–relay channels. Note that, to the best of the authors’ knowledge, a comparison between the performance of dual-hop multi-antenna AF relaying systems employing the maximum end-to-end SNR relay selection method and those employing the maximum source-to-relay selection method does not exist in the literature. In this approach, each relay, after applying an OSTBC transmission scheme, computes the source–relay SNR. The relay with the best SNR is selected and forwards the signal to the destination. Then, the harmonic mean of the source-to-selectedrelay channel and selected-relay-to-destination channel is computed. Mathematically speaking, the relay selection scheme can be formulated as γSC = max γ1k . The cdf of this selection combiner can be written as ⎞ ⎛ L Γ mNs Nr1 , αγ ⎠. ⎝1 − FSC (γ) = Γ(mN N ) s r 1 i=1

IV. R ELAY S ELECTION BASED ON S OURCE - TO -R ELAY S IGNAL - TO -N OISE R ATIO In a conventional relay selection scheme, the relay with the strongest harmonic mean of the two instantaneous hop SNRs is selected. However, several previous works, such as [12], [26], [42], investigated relay selection based on the first-hop fading characteristics. Note that, although this scheme is suboptimal compared with the first scheme, it incurs lower implementation

(47)

Considering (36), the cdf in (47) can be expanded as ⎛ ⎜ FSC (γ) = ⎝e

− αγ

∞

k=0

αγ

⎞L

mNs Nr1 +k

Γ(mNs Nr1 + k + 1)

⎟ ⎠ .

(48)

Regarding the convergence of the infinite series in (48), where (48) stems from (36), by substituting (36) in (3), the cdf for Nakagami-m fading channels can be rewritten as Fγ (γ) = e−

αγ

∞

k=0

∞

(αγ)m+k = ck γ m+k m+k Γ(m + k + 1) k=0

where c k = e− Δ

Note that (d) and (e) are obtained by using [30, Eq. 07.34.03.0001.01] and [30, Eq. 03.04.26.0010.01], respectively. Note that (45) coincides with [41, Eq. 12].

(46)

k=1,...,L

αγ

∞

k=0

αm+k . Γ(m + k + 1)m+k

(49)

By using the ratio test, the convergence radius of the infinite series can be obtained as ck m + k + 1 → ∞. (50) = lim R = lim k→∞ ck+1 k→∞ −1 α Therefore, the series converges for all γ < ∞. Using [29, Eq. 0.314], the cdf of the selection combiner becomes k+mNs Nr1 L ∞

αγ − αγ L FSC (γ) = e βk (51) k=0


where

By using multinomial coefficients [32, Eq. 3.17], we have that 1

Δ

β0 = Δ

βl =

3729

[Γ(mNs Nr1 + 1)]L L Γ(mNs Nr + 1) 1

l

k=1

FSC (γ) = 1 +

(kL − l + k)βl−k ∀l ≥ 1. Γ(mNs Nr1 + k + 1)

L

L j=1

By differentiating (51) with respect to γ and after some manipulations, the SC pdf can be written as ∞ α − αγ L fSC (γ) = e βk (k + mNs Nr1 L) k=0 k+mNs Nr1 L

k+mNs Nr1 L−1 ∞ αγ αγ α − L βk . × k=0 (52) Since a statistical characterization based on the exact harmonic mean is a challenging mathematical problem, we henceforth work on the upper bound of the harmonic mean.

×

i1 =0

×

1−e

∞

ξk

k=0

=e

− αγ L

∞

βk

k=0

−e

− αγ (L+1)

αγ

∞

k=0

Δ

αγ

u+1

v+1 +e

∞

k=0

ηk

− αγ

αγ

ξk

αγ

u+1

u+1

Δ

(53) Δ

where v = k + mNs Nr1 L − 1, u = k + mNd Nr2 − 1, ξl = Δ 1/Γ(mNd Nr2 + l + 1), and ηl = (αγ/)mNs Nr1 L lk=0 βk / ξl−k . Based on (53), the outage probability for this second approach is Pôut (γth ) = Fˆγup (γth ).

(54)

Note that, when mNs Nr1 is an integer, by using [29, Eq. 8.352.2], (47) can be simplified as ⎞L ⎛ mNs Nr1 −1 i

αγ (αγ) ⎠ FSC (γ) = ⎝1 − e− . (55) i!i i=0

αγ

j

it

t=1

e−

j ij =0

αγ

j

.

(56)

it ! t=1

Finally, the cdf of the end-to-end SNR can be written as

Fˆγup (γ) = 1 +

L

L

j

j=1

mNs Nr1 −1

(−1)

j

mNs Nr1 −1

mNd Nr2 −1

αγ

...

i1 =0 j

ij =0

it +i

t=1

e−

j i=0

k=0

mNs Nr1 −1

...

×

By using the SC method in (46), the best relay for transmitting signal in the current time slot is selected. As aforementioned, in this second scheme, the harmonic mean of the source-to-selected-relay and selected-relay-to-destination channels is defined as the end-to-end SNR. Since γup = min(γ1l , γ2l ), where l is the index of the selected relay in (46), the cdf of the effective end-to-end SNR, γup , by using [29, Eq. 0.316] is given by v+1

∞

αγ αγ − L Fˆγup (γ) = 1 − 1 − e βk − αγ

(−1)j

mNs Nr1 −1

A. Upper Bound Analysis

j

i!

αγ

(j+1)

.

(57)

it ! t=1

Note that (57) is an exact closed-form relation for the cdf of the end-to-end SNR γup , whereas (53) includes an infinite series. As before, we now present a new analytical expression for the SEP of the second approach, which represents a lower bound on the exact SEP. Proposition 4: The SEP of the upper bound of the harmonic mean of γ1l and γ2l is given in (58), shown at the bottom of the next page. Proof: To derive the SEP for the second method, we first have to derive the pdf of the end-to-end SNR. To this end, by differentiating (53) with respect to γ, we get the pdf that is given in (59), shown at the bottom of the next page. Similar to the analysis earlier and by using (20), we can obtain (58), by using [38, Eq. 3.7.2.5]. We finally investigate the average channel capacity, via the following proposition. Proposition 5: The average channel capacity (in bits/s/Hz) of the upper bound of the harmonic mean of γ1l and γ2l is given in (60), shown at the bottom of the next page. Proof: Considering (59) and (88) and then by using (90), the proof follows trivially. B. High-SNR Analysis To simplify the previous exact results, we now derive asymptotic high-SNR approximations. Note that, for simplicity, we assume that Ns Nr1 = Nd Nr2 . The SEP at high SNRs is derived in the following corollary. Corollary 5: At high SNRs, the SEP of the upper bound of the harmonic mean of γ1l and γ2l can be expressed as ! Pê∞ =

(Gc1 γ)−Gd + o(γ −Gd ), (Gc2 γ)−Gd + o(γ −Gd ),

L>1 L=1

(61)

3730


Since γup = min(γ1l , γ2l ), the cdf of the end-to-end SNR can be approximated at high SNRs as

where the diversity and coding gains are defined as follows: Gd = mNs Nr1

Gc1

c = 2α

Gc2

c = 2α

(62)

Γ(mNs Nr1 + 0.5) √ 2 πΓ(mNs Nr1 + 1) Γ(mNs Nr1 + 0.5) √ πΓ(mNs Nr1 + 1)

− mN 1N s

r1

(63)

r1

.

(64) Pê∞

(αγ)mNs Nr1 L [Γ(mNs Nr1 + 1)]L mNs Nr1 L

Pê =

+o γ

(66)

L = 1.

1

Proof: Using [30, Eq. 06.06.06.0004.02], (47) becomes

FˆSC (γ) =

L>1

Similar to the earlier results in (20), the SEP becomes

− mN 1N s

⎧ αγ mNs Nr 1 ⎪ ⎨ () + o γ mNs Nr1 , Γ(mN N +1) s r 1 Fˆγup (γ) = αγ mNs Nr1 mN N ⎪ ⎩ 2( ) s r1 , Γ(mNs Nr +1) + o γ

mNs Nr1 L

.

(65)

⎧ 2α mN ⎪ ) Γ(mN +0.5) ⎨ ( c¯γ √ + o γ −mN , 2 πΓ(mN +1) = 2α mN ⎪ ⎩ ( c¯γ √) Γ(mN +0.5) + o γ −mN , πΓ(mN +1)

L>1

(67)

L = 1.

Finally, the asymptotic SEP is obtained as in (61). Note that the second term in (61) and (66) is considered only when L = 1. Referring to (61), we can infer that the diversity gain is mNs Nr1 , which reveals that the second scheme has

v+2 ∞ ∞

2v Γ(v + 1.5) βk αv+1 L2v+1 Γ(v + 2.5) βk α 2αL √ √ − 2 F1 v + 1; v + 1.5; v + 2; − (c¯ γ )v+1 c¯ γ c¯ γ (v + 1) π π

k=0

× 2 F1 v + 2; v + 2.5; v + 3; −

2αL c¯ γ

k=0

∞

ξk αu+1 2u Γ(u + 1.5) 2α √ + 2 F1 u + 1; u + 1.5; u + 2; − c¯ γ (c¯ γ )u+1 π k=0

∞

2u+1 Γ(u + 2.5) ξk αu+2 2α √ − 2 F1 u + 2; u + 2.5; u + 3; − (u + 1)(c¯ γ )u+2 c¯ γ π k=0

∞

2u Γ(u + 1.5) ηk αu+1 2α(L + 1) √ − 2 F1 u + 1; u + 1.5; u + 2; − (c¯ γ )u+1 c¯ γ π k=0

∞

(L + 1)2u+1 Γ(u + 2.5) ηk αu+2 2α(L + 1) √ + 2 F1 u + 2; u + 2.5; u + 3; − (u + 1)(c¯ γ )u+2 c¯ γ π

(58)

k=0

fˆγup (γ) = e−

αγ

L

∞

α k=0

βk (v + 1)

αγ

k+mNs Nr1 L−1

∞

αγ α − Le− L βk k=0

αγ

v+1 +e

− αγ

∞

α k=0

ξk (u + 1)

αγ

u

u+1 u+1 u ∞ ∞ ∞

αγ αγ αγ α α − αγ α − αγ (L+1) − αγ (L+1) (u + 1)ηk − e ξk + (L + 1)e ηk −e (59) k=0

Cˆ =

k=0

k=0

∞ ∞ γ¯ −v, 1, 1 ¯ −v − 1, 1, 1 ¯ −u, 1, 1 βk 1, 3 γ 1, 3 γ − + G ξk (u + 1)G3, 2 1, 0 Lv+1 αL 1, 0 Lv+1 3, 2 αL α 1, 0 k=0 k=0 k=0 ∞

∞ ∞

¯ −u−1, 1, 1 γ¯ −u, 1, 1 γ¯ −u−1, 1, 1 ηk (u+1) 1, 3 ηk 1, 3 γ 1, 3 − − + ξk G3, 2 G G 1, 0 1, 0 α (L+1)u+1 3, 2 α(L+1) 1, 0 (L+1)u+1 3, 2 α(L+1) 1 2 ln 2

k=0

∞

βk (v + 1)

3 G1, 3, 2

k=0

k=0

(60)


lower diversity order than the first scheme. Regarding the coding gain, we have the following result. Corollary 6: When L > 1, the coding gain of the second scheme is greater than that of the first scheme. Proof: By using (27) and (67), it suffices to show that Gc < Gc1 for L > 1. After some simple manipulations, we have −mNs Nr1 L L−1 Γ(mNs Nr L + 0.5) Gc 1 = 22L−1 π 2 . Gc1 [Γ(mNs Nr1 + 0.5)]L (68) Using [32, Eqs. 25.9 and 25.10], we can rewrite (68) as −mNs Nr1 L Gc Gc1 [Γ(mNs Nr1 )]L Γ(2mNs Nr1 L) = 2L Γ(mNs Nr1 L) [Γ(2mNs Nr1 )]L L−1

Γ 2mNs Nr1 +

= 2L LmNs Nr1 L i=0 L−1

Γ mNs Nr1 +

i=0 L

=2 L

i L

[Γ(mNs Nr1 )]L

[Γ(2mNs Nr1 )]L

Fig. 2. Outage probability of multi-antenna multi-relay systems over Nakagami-m fading channels (first approach) (γ0 = 3, m = 0.55, and Ω = 1).

a) Nakagami-0.5 fading channels: By taking L = 1 and Ns = Nr1 = Nr2 = Nd = 1 and using (47), the outage probability can be rewritten, for the case of arbitrary m, as follows: ⎞2 ⎛ Γ m, αγth ⎠ (71) Pout (γth ) = 1 − ⎝ Γ(m)

mNs Nr1 L

L−1

2mNs Nr1 +

×

i L

3731

i=0

i L

− 1 · · · mNs Nr1 +

(2mNs Nr1 − 1)L · · · (mNs Nr1 )L

i L

. (69)

Since L ≥ 2 and m ≥ 0.5, it is obvious that (69) is always greater than one; hence, Gc1 /Gc > 1. Similar to the earlier approach, by taking the first derivative of (67) with respect to the fading and system parameters, it can be conjectured that, by increasing m, Ns , Nr1 , and Ω, the SEP will be decreased. It can be seen that the SEP is independent of L when L > 1. This is due to the fact that, in this approach, the diversity gain of the first and second hops is mNs Nr1 L and mNs Nr1 , respectively. Note that these findings are consistent with those in [25, Eq. 36]. However, the minimum diversity of the two hops determines the diversity order of the system. We can now turn our attention to the outage probability. Corollary 7: In the low outage regime, the outage probability of the upper bound of the harmonic mean of γ1l and γ2l can be expressed as ⎧ mNs Nr1 (αγth )mNs Nr1 ⎨ , L>1 mNs Nr +o γth 1 Γ(mNs Nr1 +1) ∞ Pôut (γth ) = mNs Nr mN N 1 s r 2(αγ ) th 1 ⎩ , L = 1. +o γth Γ(mN N +1)mNs Nr1 s

r1

(70) Proof: The proof follows easily by using (66). As shown, the outage probability is independent of L when L > 1. Similar to the high-SNR results earlier, the outage probability is a monotonically decreasing function of m, Ω, Ns , and Nr1 . C. Special Cases We now particularize the results of Section IV-A to some simplified scenarios of practical importance.

which coincides with [11, Eq. 5]. For the worst-case scenario m = 0.5, the outage probability in (71) simplifies as ⎤2 ⎡ γth Γ 0.5, 2Ω ⎦ √ Pout (γth ) = 1 − ⎣ π & 2 γth = 1 − erfc 2Ω & 2 γth (b) =1−4 Q . Ω

(a)

(72)

Note that (a) and (b) are obtained by using [30, Eq. 06.06.03.0004.01] and [30, Eq. 06.27.26.0016.01]. b) Rayleigh fading channels (m = 1): For the Rayleigh case, when we set L = 1 and Ns = Nr1 = Nr2 = Nd = 1 in (47), the outage probability reduces to Pout (γth ) = 1 − e−2

αγth

(73)

which coincides with [43, Eq. 22]. V. N UMERICAL R ESULTS To verify the analytical results, we compare them against numerical simulations. For the sake of brevity in interpretation, we assume that Nr1 = Nr2 and Ns = Nd and full-rate OSTBC and, for simplicity, neglect rate penalty when there are more than two antennas (i.e., Rc1 = Rc2 = 1), In Fig. 2, the exact analytical expression (85) for the outage probability of the first method, along with the high-SNR approximation in (32) and the lower bound result in (39), are shown. Note that all curves are plotted against the SNR, where SNR = γ0 /γth , with γ0 being a fixed threshold value. As shown, the performance

3732


Fig. 3. Outage probability of multi-antenna multi-relay systems over Nakagami-m fading channels (first approach) (γ0 = 3, m = 0.55, and Ω = 0.5).

improves when L, Ns , or Nr1 increase. In addition, higher coding gain is obtained by increasing Ns or Nr1 , which is in agreement with (32). In addition, the lower bound and highSNR results coincide with the exact results across a wide SNR range, thereby representing an efficient means to approximate the outage probability. Note that, although the given results are provided in terms of infinite series, only some tens of terms are required to obtain accurate results. For instance, the analytical curve is obtained by considering 30 terms of the infinite series in (85). This is mainly due to the factorial terms, which remarkably accelerate the convergence speed of the series. Fig. 3 investigates the interesting scenario when L + Ns Nr1 = 6 is kept fixed, for different combinations of L and Ns Nr1 . The graph demonstrates that the best performance is attained for L = Nr1 = 3 and Ns = 1; this is expected since, when we want to maximize LNs Nr1 subject to L + Ns Nr1 = c0 , where c0 is fixed, the optimal value is L = Ns Nr1 = c0 /2. As such, the configuration L = Nr1 = 3, Nr1 = 1 yields the largest diversity order. Moreover, we can see that the L = 5 and Ns = Nr1 = 1 configuration yields larger coding gain compared with the L = Ns = 1, Nr1 = 5 configuration, whereas both of them have the same diversity order. By increasing the SNR, the distance between the L = Nr1 = 3, Ns = 1 and the other two curves will increase. Fig. 4 investigates the SEP of the first method in (24) along with the high-SNR expression in (25) and the lower bound result in (41) for different values of the m parameter (assuming BPSK modulation such that b = 1 and c = 1). As can be seen, by increasing m, the SEP reduces systematically. Moreover, the diversity order for the m = 0.55, m = 1.1, and m = 1.65 cases is 2.2, 4.4, and 6.6, respectively. In addition, it can be seen that the relative distance between the curves reduces for higher values of m. This implies that its impact becomes increasingly less pronounced, which agrees with [7]. Fig. 5 shows the average channel capacity of the upper bound of the first method in (42) versus the SNR. As we can see, the average channel capacity will increase if we increase the number of relays or number of antennas. More importantly, we observe that the L = 2, Ns = Nr1 = 2 configuration yields

Fig. 4. SEP of multi-antenna multi-relay systems over Nakagami-m fading channels (first approach) (BPSK, L = 2, Ns = 2, Nr1 = 1, and Ω = 0.5).

Fig. 5. Average channel capacity of multi-antenna multi-relay systems over Nakagami-m fading channels (first approach) (m = 0.8 and Ω = 0.5).

Fig. 6. Outage probability of multi-antenna multi-relay systems over Nakagami-m fading channels (first and second approaches) (γ0 = 3, L = 2, Ns = 2, Nr1 = 1, and Ω = 0.5).

higher capacity than the L = 4, Ns = 2, Nr1 = 1 configuration. This implies that the number of antennas has a greater impact on the average capacity than the number of relays. This conclusion is in line with the results of [44]. In Fig. 6, the outage probability of both methods is shown. The analytical lower bound curve for the second method has been generated via (54), whereas the high-SNR curve is based


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were able to parameterize the system performance in terms of diversity order and coding gain and draw useful insights into the implications of the model parameters on the system performance. It was analytically shown that the first scheme yields higher diversity order than the second scheme, although the latter has lower implementation complexity. A PPENDIX A P ROOF OF P ROPOSITION 1 To investigate the pdf of the harmonic mean, we first need to work out the mgf of y1k in (1). By taking the Laplace transform of (2), the mgf of both γSi Rk, l and γRk, j Dn can be written as Fig. 7. SEP of multi-antenna multi-relay systems over Nakagami-m fading channels (second approach) (BPSK, m = 0.55, and Ω = 0.3).

on (66). As can be seen, the lowest outage probability is obtained for the first approach when m = 1.1, whereas the highest outage probability corresponds to the second approach when m = 0.55. Another important observation is that for the same value of m, the first method yields lower outage probability than the second approach. This confirms the superiority of the first method against the second method. Note that the two middle curves have the same diversity order (i.e., Gd = 2.2); however, the coding gain of the second method, with m = 1.1, is higher than that of the first method when m = 0.55. Finally, Fig. 7 compares the SEP lower bound of the second method in (58) against the high-SNR result in (67). The diversity order of L = Ns = Nr1 = 1; L = 2, Ns = Nr1 = 1; L = Ns = 2, Nr1 = 1; L = 2, Ns = 1, Nr1 = 3; and L = 2, Ns = Nr1 = 4 configurations is 0.55, 0.55, 1.1, 1.65, and 2.2, respectively. This validates the independence of the diversity order with respect to the number of relays L. This phenomenon is actually quite intuitive. Considering (67), the diversity orders of the first and second hops are mNs Nr1 L and mNs Nr1 , respectively. However, the minimum diversity of two hops determines the diversity order of the system. When L > 1 and Ns = Nr1 = 1, no significant reduction in the SEP is observed by increasing L. This is due to the fact that, when L > 1 in (66), the diversity order is independent of L, and only an increased coding gain accounts for a SEP reduction. To overcome this problem, increasing the number of antennas, i.e., Ns and Nr1 , is a meaningful choice that confirms the importance of MIMO systems. VI. C ONCLUSION We investigated the performance of a dual-hop AF multirelay system over i.i.d. Nakagami-m fading channels, where all nodes are equipped with multiple antennas while both transmissions employ OSTBC. Two methods were presented in which the harmonic mean of source–relay and relay–destination channels and SC were combined. We derived new analytical expressions for the outage probability, the SEP, and the average channel capacity of the aforementioned schemes, whereas the derived expressions apply for arbitrary values of m. Simplified high SNRs results were also deduced. By doing so, we

φγSi Rk, l (s) = φγRk, j Dn (s) =

αm . (s + α)m

(74)

In general, it is known (see, e.g., [31]) that the sum of N i.i.d. Gamma random variables with shape parameter k and scale parameter θ is also a Gamma random variable with parameters kN and θ. Therefore, for the source-to-relay-k and relay-k-todestination channels, the SNR mgf and pdf read as follows: φγ1k (s) =

αmNs Nr1 (s s + α)mNs Nr1

(75)

φγ2k (s) =

αmNd Nr2 (r s + α)mNd Nr2

(76)

fγ1k (γ) =

αmNs Nr1 γ mNs Nr1 −1 − αγ e s s Nr1 Γ(mNs Nr1 ) mN s

fγ2k (γ) =

αmNd Nr2 γ mNd Nr2 −1 − αγ e r . d Nr2 Γ(mNd Nr2 ) mN r

(77)

Supposing that x is a random variable and that z = 1/x, the pdf of z is fZ (z) = z −2 fX (z −1 ). Therefore, since y1k = 1/γ1k and y2k = 1/γ2k , the pdfs of y1k and y2k are as follows: αmNs Nr1 z −mNs Nr1 −1 e− s z α

fy1k (z) =

mNs Nr1

Γ(mNs Nr1 )s

αmNd Nr2 z −mNd Nr2 −1 e− r z α

fy2k (z) =

mNd Nr2

.

(78)

Γ(mNd Nr2 )r

Since U = y1k + y2k , the mgf of U is the multiplication of the mgfs of y1k and y2k . Therefore, at first, we should derive the mgfs of y1k and y2k . Using [38, Eq. 2.2.2.1], the mgf of y1k = 1/γ1k and y2k = 1/γ2k can be obtained as 2 φy1k (s) =

mNs2Nr1

Γ(mNs Nr1 ) 2

φy2k (s) =

αs s

αs r

KmNs Nr1

& αs 2 s

KmNd Nr2

& αs 2 . r

mNd2Nr2

Γ(mNd Nr2 )

(79)

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Setting U = y1k + y2k , we can then easily work out the mgf of U as follows: m

4(αs) 2 (Ns Nr1 +Nd Nr2 )

φU (s) =

mNs Nr

1

mNd Nr

2

Γ(mNs Nr1 )Γ(mNd Nr2 )s 2 r 2 & & αs αs ×KmNs Nr1 2 KmNd Nr2 2 . s r

l=0

(80)

The pdf of U can then be derived from the definition of inverse Laplace transformation [45, Eq. 3.16.13.6] and [30, Eq. 07.45.16.0001.01] or by using [46, Eq. 5.16.59]. This inverse Laplace transformation requires the orders and arguments of the two Bessel functions in (80) to be equal or that Ns Nr1 = Nr2 Nd , s = r = . Then, we have fU (u) =

√

2mNs Nr1 α 2 mNs Nr1 − 2 3

π

2

1

[Γ(mNs Nr1 )] ×e

− 2α u

3 1 2 mNs Nr1 − 2

k=0

(85) By differentiating (85) with respect to γ, the pdf of the SC can be obtained according to ⎞ ⎛ ∞ ∞ L

L ⎝ ςj, k ψj ϕk γ ςj, k −1 ⎠ . (86) fSC (γ) = ρL l j=0 l=0

k=0

Taking the Laplace transform of (86), the mgf in (15) can be obtained.

u− 2 mNs Nr1 − 2 3

1

W mNs Nr1 , mNs Nr1 2

Using the definition of binomial coefficients and [29, Eq. 0.314], the cdf of the selection combiner becomes ⎞ ⎛ ∞ ∞ L

L ⎝ ψj ϕk γ j+k+2mN L−mN l ⎠ . FSC (γ) = ρL l j=0

2

4α u

A PPENDIX C P ROOF OF P ROPOSITION 3

.

(81)

Using (81), we deduce the pdf of the harmonic mean in (4).

The channel capacity is defined (in bits/s/Hz) according to the standard formula [4] Δ

C= A PPENDIX B P ROOF OF P ROPOSITION 2 By substituting [29, Eq. 9.14.1] into [29, Eq. 9.303], the cdf can be alternatively written as ∞

L

Ak γ k+mN −1 + Bk γ k+2mN −1 L FSC (γ) = (ργ) k! k=0 (82) where N = Ns Nr1 , and k+mN −1 4α k (τ )k (0.5)k Ak = λ(−1) (ν1 )k (ν2 )k √ πα 8 Δ ρ = 2mN 2 [Γ(mN )]2 k+2mN −1 (σ1 )k (σ2 )k 4α Δ Bk = μ(−1)k (ω1 )k (ω2 )k 2 [Γ(mN )] Δ λ=√ πΓ(mN + 1) Γ(−mN )Γ(2mN ) Δ μ= Γ −mN + 12 Γ(2mN + 1)

1 E {log2 (1 + γSC )} 2

(87)

where E{·} denotes expectation, and the factor of 1/2 accounts for the fact that transmission occupies two time slots. For the upper bound approach in Section III-C, the upper bounded channel capacity can be expressed as Δ 1 Cˆ = 2

∞ log2 (1 + γ)fˆγSC (γ)dγ.

(88)

0

By using [47, Eq. 11], we can rewrite (88) as ∞ ∞ 1, 1 − 1,0 −px v v 1,2 ln(1+x)e x dx = x G2,2 x G0,1 px dx. 1, 0 0

Δ

0

0

(89) By using [30, Eq. 07.34.21.0011.01] and then using [29, Eqs. 9.31.2 and 9.31.5], (89) can be obtained as ∞ (83)

3 ln(1 + x)e−px xv dx = p−v−1 G1, 3, 2

1 −v, 1, 1 . p 1, 0

(90)

0

By substituting (40) into (88) and using (90), the average channel capacity is obtained as in (42).

Δ

where we can define (a)n = Γ(a + n)/Γ(a) and Δ

τ = mN

R EFERENCES

Δ

ν1 = − mN + 1 Δ

ν2 = mN + 1 Δ

σ1 = 2mN Δ

σ2 = mN + 0.5 Δ

ω1 = mN + 1 Δ

ω2 = 2mN + 1.

(84)

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Ehsan Soleimani-Nasab (S’11) was born in Kerman, Iran, in 1984. He received the B.Sc. degree in electrical engineering from the Iran University of Science and Technology, Tehran, Iran, in 2006 and the M.Sc. degree in communication systems from K.N. Toosi University of Technology, Tehran, in 2009, where he is currently working toward the Ph.D. degree. From April to October 2012, he was with the Department of Signals and Systems, Chalmers University of Technology, Gothenburg, Sweden, working as a Visiting Researcher. His research interests include signal processing for wireless communications, cooperative communications, multiple-input–multipleoutput systems, and cognitive radio networks.

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Michail Matthaiou (S’05–M’08) was born in Thessaloniki, Greece in 1981. He received the Diploma degree (5 years) in electrical and computer engineering from the Aristotle University of Thessaloniki, Greece, in 2004. He then received the M.Sc. (with distinction) in communication systems and signal processing from the University of Bristol, U.K., and the Ph.D. degrees from the University of Edinburgh, U.K., in 2005 and 2008, respectively. From September 2008 through May 2010, he was with the Institute for Circuit Theory and Signal Processing, Munich University of Technology (TUM), Germany, working as a Postdoctoral Research Associate. In June 2010, he joined Chalmers University of Technology, Sweden, as an Assistant Professor, and in 2011, he was awarded the Docent title. His research interests span signal processing for wireless communications, random matrix theory and multivariate statistics for MIMO systems, and performance analysis of fading channels. Dr. Matthaiou is the recipient of the 2011 IEEE ComSoc Young Researcher Award for the Europe, Middle East, and Africa Region and a co-recipient of the 2006 IEEE Communications Chapter Project Prize for the best M.Sc. dissertation in the area of communications. He was an Exemplary Reviewer for IEEE C OMMUNICATIONS L ETTERS for 2010. He has been a member of Technical Program Committees for several IEEE conferences such as ICC, GLOBECOM, etc. He currently serves as an Associate Editor for the IEEE T RANSACTIONS ON C OMMUNICATIONS and the IEEE C OMMUNICATIONS L ETTERS and was the Lead Guest Editor of the special issue on “Large-scale multiple antenna wireless systems” of the IEEE J OURNAL ON S ELECTED A REAS IN C OMMUNICATIONS. He is an associate member of the IEEE Signal Processing Society SPCOM and SAM technical committees.

Mehrdad Ardebilipour was born in Iran in February 1954. He received the B.Sc. degree in electrical engineering from K.N. Toosi University of Technology, Tehran, Iran, in 1977; the M.Sc. degree in electrical engineering from Tarbiat Modarres University, Tehran, in 1989; and the Ph.D. degree from the University of Surrey, Surrey, U.K., in 2001. Since 2001, he has been an Assistant Professor with K.N. Toosi University of Technology, where he directed the Department of Communications Engineering for six years. He is currently the Director of the Spread Spectrum and Wireless Communications Research Laboratory, K.N. Toosi University of Technology. His current research interests include cognitive radios, cooperative communications, ad hoc and sensor networks, multipleinput–multiple-output communications, orthogonal frequency-division multiplexing, game theory, and cross-layer design for wireless communications.