Outage Performance Analysis of Multicarrier Relay Selection for Cooperative Networks
OOutage Performance Analysis of Multicarrier RelaySelection for Cooperative Networks
Shuping Dang, Justin P. Coon, Gaojie Chen and David E. Simmons
Department of Engineering Science, University of Oxford, Oxford, UK, OX1 3PJEmail: { shuping.dang, justin.coon, gaojie.chen, david.simmons } @eng.ox.ac.uk Abstract —In this paper, we analyze the outage performanceof two multicarrier relay selection schemes, i.e. bulk and per-subcarrier selections, for two-hop orthogonal frequency-divisionmultiplexing (OFDM) systems. To provide a comprehensive anal-ysis, three forwarding protocols: decode-and-forward (DF), fixed-gain (FG) amplify-and-forward (AF) and variable-gain (VG) AFrelay systems are considered. We obtain closed-form approxima-tions for the outage probability and closed-form expressions forthe asymptotic outage probability in the high signal-to-noise ratio(SNR) region for all cases. Our analysis is verified by MonteCarlo simulations, and provides an analytical framework formulticarrier systems with relay selection.
Keywords — Multicarrier relay selection, parallel fading channel,cooperative systems, outage performance.
I. I
NTRODUCTION
Cooperative communications and relay-assisted systemshave attracted a large amount of attention since they wereproposed and comprehensively analyzed in [1]. In such anetwork, relays are employed as intermediate communicationnodes to assist the transmission between source and desti-nation, so that the communications can be maintained whenthe direct transmission link between source and destinationundergoes deep fading [2], [3]. It has been proved that withthe help of relays and proper forwarding protocols, the networkcoverage expansion, energy efficiency, system reliability andquality of service (QoS) can be significantly enhanced [4]–[7]. In particular, relay selection can further enhance thesystem performance and obtain an extra diversity [8]. A varietyof selection schemes have been proposed and analyzed fordifferent types of relays and channel conditions [9]–[14].Meanwhile, multicarrier communications, especially or-thogonal frequency-division multiplexing (OFDM), is alsoproposed and utilized in relay-assisted networks, which is ca-pable of providing a better system performance and bandwidthefficiency over frequency selective channels [15], [16]. Butrelay selection in multicarrier systems is not straightforward,because this is a two-tier selection/allocation problem involv-ing relay selection and subcarrier allocation. In [17], two relayselection schemes for OFDM systems, i.e. bulk selection (asingle relay is selected for transmission on all subcarriers) andper-subcarrier selection (selection is treated independently foreach subcarrier, thus, potentially, leading to transmission viamultiple relays) are proposed and analyzed.However, all previous works on multicarrier relay selectionare carried out based on an assumption that the outage con- dition regarding each subcarrier is treated individually, whichcorresponds to the block fading channel model when a useris allocated a contiguous block of carriers that lie within acoherence bandwidth. To be more realistic, we should considerthe parallel fading channel model that general OFDM systemsare well approximated by. The parallel fading channel modelis utilized to model a fading channel consisting of a finitenumber of flat independent and identically distributed (i.i.d.)fading subchannels [18]. The most unique property of the par-allel fading channel relative to the conventional block fadingchannel is the definition of outage probability. Considering thecoding over the parallel fading channel, the outage probabilityis defined as the probability that the mutual information ofall subchannels is smaller than a target transmission rate[19]. However, the mathematical tractability of the outageprobability over the parallel fading channel is rather poor,because the distribution of the summation of a finite number ofrandom variables cannot be derived in a generic closed-formexpression [20]. In [21], an oversimplified case of a parallelfading channel with only two subchannels is analyzed and anintegral formula for the outage probability is given. Also, upperand lower bounds as well as an approximation for the outageprobability for any number of subchannels are determined in[18] and [22], respectively.On the other hand, to the best of the authors’ knowledge,the work considering multicarrier relay selection for two-hop OFDM systems approximated by parallel fading channelmodel has not been fully treated. To fill this gap, we analyzethis scenario in a Rayleigh fading condition. Note, the methodapplied in this paper can be easily tailored to analyze otherchannel conditions.The main contributions of this paper are summarized infra: • We obtain closed-form generic approximations foroutage probabilities of two-hop OFDM systemswhen applying bulk and per-subcarrier relay selectionschemes. • We derive closed-form approximations for the out-age probability of decode-and-forward (DF), fixed-gain (FG) amplify-and-forward (AF) and variable-gain(VG) AF relay systems with bulk and per-subcarrierrelay selection schemes. • We obtain closed-form asymptotic expressions for theoutage probability with different selection schemes athigh SNR and also derive diversity gains. a r X i v : . [ c s . I T ] A ug he rest of this paper is organized as follows. In SectionII, the system model and fundamentals are given. Then,the outage performance is analyzed in Section III. Specificapplications including DF, FG AF and VG AF relay systemsare investigated in Section IV. The analysis is numericallyverified by Monte Carlo simulations and further discussed inSection V. Finally, Section VI concludes the paper.II. S YSTEM M ODEL
A. Parallel fading channel and outage probability
In this paper, we consider a two-hop OFDM system with K orthogonal subcarriers and M relays gathered in a relaycluster, the size of which is relatively small compared to thedistance between source and destination. Therefore, K parallelfading subchannels are constructed at each hop, and for eachsubcarrier, there are M interfering channels. We denote thesets of subcarriers and relays as K = { , , . . . , K } and M = { , , . . . , M } . Also, we assume the entire two-hopOFDM system operates in a half-duplex protocol and the directtransmission link between source and destination does notexist due to deep fading. As a result, two orthogonal timeslots are required for one complete transmission from sourceto destination via relay(s). Here, we denote i.i.d. Rayleighchannel coefficients in the first and second hops by h ( m, k ) and h ( m, k ) ∀ m ∈ M and k ∈ K . The channel gain | h i ( m, k ) | , where i ∈ { , } , is exponentially distributed withthe mean of µ i . Therefore, its probability density function(PDF) and cumulative distribution function (CDF) are f | h i | ( x ) = e − x/µ i /µ i ⇔ F | h i | ( x ) = 1 − e − x/µ i . (1)Subsequently, the end-to-end SNR transmitted on the k thsubcarrier and forwarded by the m th relay is denoted as γ ( m, k ) . Accordingly, the outage probability over the parallelfading channel can be defined as P out ( s ) = P (cid:40) K (cid:88) k =1 ln(1 + γ ( m k , k )) < s (cid:41) (2)where P {·} denotes the probability of the enclosed; m k isthe index of the relay selected to forward the k th subcarrier;we also let s = 2 ξ for the convenience of the followinganalysis, where ξ is the mutual information outage threshold .In addition, we assume all noise statistics are assumed to bezero-mean, complex Gaussian random variables with variance N / per dimension, from which the noise power can beexpressed by N . B. Forwarding protocols
We assume equal bit and power allocation schemes areapplied and the average transmit power per subcarrier atsource and relay is the same, denoted by P t . Therefore, theinstantaneous end-to-end SNR of the k th subcarrier forwardedby the m th relay using a DF protocol is written as γ DF ( m, k ) = ¯ γ min (cid:0) | h ( m, k ) | , | h ( m, k ) | (cid:1) , (3)where ¯ γ = P t /N . Similarly, for FG AF relaying that isblind to all channel conditions and is only able to amplify the The factor ‘2’ comes from the fact that two time slots are required for onecomplete transmission in two-hop systems. Fig. 1. Illustration of (a) bulk, (b) per-subcarrier relay selection schemes forsingle source, single destination and M clustered relays with K subcarriers.Note, K = (cid:80) Mm =1 k m for k m ∈ N , ∀ ≤ m ≤ M . received signal by a fixed gain, the instantaneous end-to-endSNR is written as γ F G ( m, k ) = ¯ γ | h ( m, k ) | | h ( m, k ) | ¯ γµ + ¯ γ | h ( m, k ) | + 1 . (4)For VG AF relaying which is able to estimate channel con-ditions and amplify accordingly, the instantaneous end-to-endSNR is thereby γ V G ( m, k ) = ¯ γ | h ( m, k ) | | h ( m, k ) | ¯ γ | h ( m, k ) | + ¯ γ | h ( m, k ) | P t + 1 . (5) C. Relay selection schemes1) Bulk selection:
By the bulk selection scheme, the sourceonly selects one out of M relays by which all subcarriers areforwarded according to the selection criterion L bulk = arg max m ∈M (cid:40) K (cid:88) k =1 ln(1 + γ ( m, k )) (cid:41) , (6)where L bulk is the set (of cardinality one) denoting the onlyselected relay. This selection scheme is easy to implementand will not involve an overcomplicated coordination protocol,because there is only one selected relay. However, it is obviousthat the outage performance cannot be optimized in this case.
2) Per-subcarrier selection:
The per-subcarrier selectionscheme selects L relays from M relays in a per-subcarriermanner and ≤ L ≤ min( M, K ) . Therefore, the relay se-lected to forward the k th subcarrier is individually determinedby l ps ( k ) = arg max m ∈M ln(1 + γ ( m, k )) = arg max m ∈M γ ( m, k ) , (7)where l ps ( k ) is the set of the selected relay correspondingto the k th subcarrier. Then, this selection process will berepeatedly applied for all subcarriers and finally L relaysare selected. The set of all L selected relays is denoted by L ps = (cid:83) Kk =1 { l ps ( k ) } . Note, it is allowed that l ps ( k ) = l ps ( n ) for k (cid:54) = n , i.e. one relay is allowed to assist in forwardingtwo or more subcarriers at the same time. Obviously, thisselection scheme is optimal in terms of outage performance,but this optimality will result in a high system complexity[23]. For illustration purposes, these two selection schemesare illustrated in Fig. 1.II. O UTAGE P ERFORMANCE A NALYSIS
A. Bulk selection
According to (2) and (6), the a posteriori outage probabilitywith bulk selection can be defined by P out ( s ) = P (cid:40) max m ∈M (cid:40) K (cid:88) k =1 ln(1 + γ ( m, k )) (cid:41) < s (cid:41) = M (cid:89) m =1 P (cid:40) K (cid:88) k =1 ln(1 + γ ( m, k )) < s (cid:41) ( a ) = ( F I ( s )) M , (8)where F I ( s ) := P (cid:110)(cid:80) Kk =1 ln(1 + γ ( m, k )) < s (cid:111) , ∀ m ∈ M and ∀ k ∈ K ; (a) is valid, because all parallel subchannels areassumed to be i.i.d.Assume the CDF and PDF of γ ( m, k ) are F γ ( s ) and f γ ( s ) ,respectively. Now following the Proposition 1 proved in [22],we also make a hypothesis that f γ ( s ) can be expanded as apower series about zero as f γ ( s ) = s q ( g + g s + O ( s )) , (9)where q is a non-negative integer representing the inherentsubchannel diversity; g , g are non-zero functions of ¯ γ andsatisfy the condition g (¯ γ ) = o ( g (¯ γ )) when ¯ γ → ∞ . Thisis a common assumption applicable to most two-hop channelcases [24]. Then, F I ( s ) can be written as [22] F I ( s ) = S ( K, s, q ) Kq +1 ( S ( K, s, q ) − S ( K, s, q )) Kq × f γ (cid:18) S ( K, s, q ) − S ( K, s, q ) S ( K, s, q ) (cid:19) K + O (cid:18) γ K ( q +1)+2 (cid:19) . (10)The coefficient S x ( K, s, q ) is defined as S x ( K, s, q ) := q (cid:88) j =0 (cid:18) qj (cid:19) ( − j (cid:88) a p ∈ N , ∀ ≤ p ≤ q (cid:80) qp =0 a p = K − (cid:18) K − a , a , . . . , a q (cid:19) × (cid:34) q (cid:89) p =0 (cid:18)(cid:18) qp (cid:19) ( − p (cid:19) a p (cid:35) · (cid:34) πi (cid:90) C e sz d z (cid:81) Kk =0 ( z − β k ) (cid:35) , (11)where β k is the k th element in B = (0 , q + 1 , . . . , q + 1 (cid:124) (cid:123)(cid:122) (cid:125) a terms , q, . . . , q (cid:124) (cid:123)(cid:122) (cid:125) a terms , . . . , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) a q terms , q + 1 + x − j ) (12)and C is the contour which encloses all poles of the integrand.It should be noted that πi (cid:82) C e sz d z (cid:81) Kk =0 ( z − β l ) is equivalent tothe inverse Laplace transform of / (cid:81) Kk =0 ( z − β k ) , which canbe expressed by the closed form as follows [25]: πi (cid:90) C e sz d z (cid:81) Kk =0 ( z − β k ) = K (cid:88) k =0 e β k s (cid:81) ≤ n ≤ Kn (cid:54) = k ( β k − β n ) . (13)However, this closed-form expression given above is only validif all poles of the integrand on the are simple (first-order poles),i.e. β n (cid:54) = β k , for n (cid:54) = k . Closed-form expressions exist for the cases with higher-order poles, but cannot be written as ageneral form. This is the reason why we still keep the integralform in (11) for generality. Another reason for keeping theintegral form is because this integral form can be efficientlyevaluated by computer-based simulations using the residuetheorem [26].As a result, the a posteriori outage probability by bulkselection as defined in (8) can be determined by P bulkout ( s ) = ( F I ( s )) M = S ( K, s, q ) M ( Kq +1) ( S ( K, s, q ) − S ( K, s, q )) MKq × f γ (cid:18) S ( K, s, q ) − S ( K, s, q ) S ( K, s, q ) (cid:19) MK + O (cid:18) γ MK ( q +1)+2 (cid:19) . (14)Furthermore, we can derive the asymptotic outage probabilityat high SNR by power series and obtain P bulkout ( s ) ∼ ˜ P bulkout ( s ) = (cid:0) S ( K, s, q ) g K (cid:1) M . (15)Note, because of the condition g (¯ γ ) = o ( g (¯ γ )) when ¯ γ →∞ , the diversity order can be derived from (15) by G bulkd = − lim ¯ γ →∞ log P bulkout ( s )log ¯ γ = M K ( q + 1) . B. Per-subcarrier selection
According to (2) and (7), the a posteriori outage probabilitywhen per-subcarrier selection is employed can be defined by P out ( s ) = P (cid:40) K (cid:88) k =1 ln(1 + max m ∈M γ ( m, k )) < s (cid:41) . (16)Denote Ψ( m, k ) = max m ∈M γ ( m, k ) . We can obtain the CDFand PDF of Ψ( m, k ) as F Ψ ( s ) = ( F γ ( s )) M ⇔ f Ψ ( s ) = M ( F γ ( s )) M − f γ ( s ) . (17)Following the assumption given in (9), f Ψ ( s ) can be furtherexpressed by f Ψ ( s ) = M g M ( q + 1) M − s M ( q +1) − + M (cid:34) g M − g ( q + 1) M − + ( M − g M − g ( q + 1) M − ( q + 2) (cid:35) s M ( q +1) + O (cid:16) s M ( q +1)+1 (cid:17) . (18)Therefore, if we denote q (cid:48) = M ( q + 1) − g (cid:48) = Mg M ( q +1) M − g (cid:48) = M (cid:104) g M − g ( q +1) M − + ( M − g M − g ( q +1) M − ( q +2) (cid:105) (19) f Ψ ( s ) can also be written as f Ψ ( s ) = s q (cid:48) ( g (cid:48) + g (cid:48) s + O ( s )) , (20)which also aligns with the form given in (9). This resultindicates that we can similarly employ Proposition 1 derivedin [22] to analyze the outage performance of per-subcarrierrelay selection over the parallel fading channel, and the onlymodification is to replace f γ ( · ) with f Ψ ( · ) . Hence, the outageerformance of per-subcarrier relay selection over the parallelfading channel as defined in (16) can be calculated by P psout ( s ) = S ( K, s, q (cid:48) ) Kq (cid:48) +1 ( S ( K, s, q (cid:48) ) − S ( K, s, q (cid:48) )) Kq (cid:48) × f Ψ (cid:18) S ( K, s, q (cid:48) ) − S ( K, s, q (cid:48) ) S ( K, s, q (cid:48) ) (cid:19) K + O (cid:18) γ K ( q (cid:48) +1)+2 (cid:19) . (21)Furthermore, we can derive the asymptotic outage probabilityat high SNR by power series and obtain P psout ( s ) ∼ ˜ P psout ( s ) = S ( K, s, q (cid:48) ) g (cid:48) K . (22)Similarly to the case for bulk selection, we can also obtainthe diversity of per-subcarrier selection from (22) to be G psd = M K ( q + 1) as expected.IV. A PPLICATIONS
A. DF relay networks
According to (1) and (3), we can derive the PDF of theend-to-end SNR γ DF ( m, k ) for DF relay networks by f DFγ ( s ) = 1¯ γ (cid:18) µ + 1 µ (cid:19) e − s ¯ γ (cid:16) µ + µ (cid:17) , (23)which can be expanded by power series as ¯ γ → ∞ tothe standard expanded form given in (9), and thus can besubstituted into (14) and (21) to yield the approximated outageprobability with bulk and per-subcarrier selections over theparallel fading channel. B. FG AF and VG AF relay networks
By (1) and (4), we can similarly derive the PDF of the end-to-end SNR γ F G ( m, k ) for FG AF relay networks by [27] f F Gγ ( s ) = e − s ¯ γµ (cid:34) γµ )¯ γ µ µ K (cid:32) (cid:115) s (1 + ¯ γµ )¯ γ µ µ (cid:33) + 2¯ γµ (cid:115) s (1 + ¯ γµ )¯ γ µ µ K (cid:32) (cid:115) s (1 + ¯ γµ )¯ γ µ µ (cid:33)(cid:35) , (24)where K x ( · ) is the x th order modified Bessel function of thesecond kind.Similarly, by (1) and (5), the PDF of the end-to-end SNR γ V G ( m, k ) for VG AF relay networks is given by [27] f V Gγ ( s ) = e − s ¯ γ (cid:16) µ + µ (cid:17) (cid:34) sµ )¯ γ µ µ K (cid:32) (cid:115) s (1 + sµ )¯ γ µ µ (cid:33) + 2¯ γ (cid:18) µ + 1 µ (cid:19) (cid:115) s (1 + sµ )¯ γ µ µ K (cid:32) (cid:115) s (1 + sµ )¯ γ µ µ (cid:33)(cid:35) . (25)Although (24) and (25) cannot be expanded by power seriesto the form as given in (14), these two PDFs can still besubstituted into (14) and (21) to yield the approximated outageprobability with a divergent error at low SNR (see Corollary3 in [22]). −5 −4 −3 −2 −1 γ (dB) O u t age P r obab ili t y DF NumericalDF ApproxFG AF NumericalFG AF ApproxVG AF NumericalVG AF Approx
M=3 ;K=3 M=2 ;K=2
Fig. 2. Bulk selection case: numerical results and analytical approximationswith different system configurations of M and K . −5 −4 −3 −2 −1 γ (dB) O u t age P r obab ili t y DF NumericalDF ApproxFG AF NumericalFG AF ApproxVG AF NumericalVG AF Approx
M=2; K=2M=3; K=3
Fig. 3. Per-subcarrier selection case: numerical results and analyticalapproximations with different system configurations of M and K . V. N
UMERICAL R ESULTS
To verify our analysis in Section III and Section IV, wecarry out Monte Carlo simulations and present the numericalresults in this section. To simplify simulations, without los-ing generality, we normalize the two-hop system by letting µ = µ = 1 and s = 2 (i.e. ξ = 1 as the mutual informationoutage threshold). Then, we can plot the relation among ¯ γ ,analytical approximations for outage probability (as given in(14) and (21)) and numerical outage probabilities for differentcombinations of M and K . We present the simulation resultsfor bulk selection and per-subcarrier selection in Fig. 2 andFig. 3, respectively. From these two figures, it is clear that theproposed approximations for the outage probability with bulkand per-subcarrier selections over the parallel fading channelhave been verified to be effective to approximate the outageperformance of DF, FG AF and VG AF relay systems at highSNR. Note, although DF, FG AF and VG AF relay systemshave the same diversity gain determined by M K ( q + 1) , theconvergence rate of FG AF case to the asymptotic region issmaller due to different correction terms. Another minor points that the similarity of outage performance between DF andVG AF relay systems in the high SNR region as valid overblock fading channels [24], can still be found valid over theparallel fading channel. In addition, by comparing Fig. 2 andFig. 3, we can find that the performance difference betweenbulk and per-subcarrier selections is not so significant as thatin block fading channels [28]. This is because the outageevent over the parallel fading channel depends on the mutualinformation over all subcarriers, instead of each individualsubcarrier. VI. C ONCLUSION
In this paper, we analyzed multicarrier relay selection fortwo-hop OFDM systems and obtained closed-form approxi-mations for the outage probabilities when applying bulk andper-subcarrier relay selections. It has been numerically shownthat the derived approximations for DF, FG AF and VG AFrelay networks are valid and able to effectively approximate theexact outage performance at high SNR. Meanwhile, we alsoderived the generic asymptotic expressions for the outage prob-abilities at high SNR when applying bulk and per-subcarrierrelay selections. All these results provide an insight into theoutage performance of multicarrier relay systems over parallelfading channels. A
CKNOWLEDGMENT
This work was supported by the SEN grant (EPSRC grantnumber EP/N002350/1) and the grant from China ScholarshipCouncil (No. 201508060323).R
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