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Decode-and-Forward Relaying for Cooperative NOMASystems with Direct LinksDOI:10.1109/TWC.2018.2873999

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Citation for published version (APA):Liu, H., Ding, Z., Kim, K. J., Kwak, K. S., & Poor, H. V. (2018). Decode-and-Forward Relaying for CooperativeNOMA Systems with Direct Links. IEEE Transactions on Wireless Communications.https://doi.org/10.1109/TWC.2018.2873999

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1

Decode-and-Forward Relaying for Cooperative NOMASystems with Direct Links

Hongwu Liu, Member, IEEE, Zhiguo Ding, Senior Member, IEEE,Kyeong Jin Kim, Senior Member, IEEE, Kyung Sup Kwak, Member, IEEE,

and H. Vincent Poor, Fellow, IEEE

Abstract—This paper investigates a cooperative non-orthogonal multiple access (NOMA) system, in which a basestation communicates with two far users with the aid of adecode-and-forward (DF) relay. Three cooperative relayingschemes, namely, the fixed relaying (FR), selective DF withcoordinated direct and relay transmission (SDF-CDRT), andincremental-selective DF (ISDF) relaying are proposed toenhance the outage performance for the two far users byutilizing both the direct and relay links. Taking into accountthe received signal-to-noise ratio (SNR) events at the relay, theSDF-CDRT scheme adaptively forms an orthogonal transmissionbranch with respect to the direct link or keeps silent to reduceerror propagation. Besides considering the relay detectionresults, the ISDF scheme further exploits the limited feedbackof the received SNR events from two users, so that errorpropagation can be avoided and unnecessary relaying can bereduced. Analytical expressions for the outage probabilitiesand average throughputs of the paired users are derived inthe closed-form for the three cooperative relaying schemes.Asymptotic expressions for the outage probabilities are derivedin the high SNR region. It is shown that the FR and SDF-CDRTschemes achieve a diversity order of one for both users, while theISDF scheme achieves a diversity order of two for both users.The superior system performance achieved by the proposedschemes over those of the existing methods is verified by MonteCarlo simulations.

Index Terms—Non-orthogonal multiple access, decode-and-forward, selective relaying, outage probability.

I. INTRODUCTION

Due to its superior spectral efficiency and its capabilityto support massive connectivity, non-orthogonal multiple ac-cess (NOMA) has been envisioned as a promising multipleaccess candidate for fifth generation (5G) networks [1]–[3].In particular, power-domain NOMA can serve multiple usersfor demanding large-scale heterogeneous traffic by using the

H. Liu is with the School of Information Science and Electrical En-gineering, Shandong Jiaotong University, Jinan 250357, China (e-mail:hong.w.liu@hotmail.com).

Z. Ding is with the School of Electrical and Electronic Engineering, TheUniversity of Manchester, Manchester M13 9PL, United Kingdom (e-mail:zhiguo.ding@manchester.ac.uk).

K. J. Kim is with Mitsubishi Electric Research Laboratories, Cambridge,MA 02139 USA (e-mail: keyong.j.kim@hotmail.com).

K. S. Kwak is with the Department of Information and Communica-tion Engineering, Inha University, Incheon 22212, South Korea (e-mail:kskwak@inha.ac.kr).

H. V. Poor is with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08544 USA (e-mail: poor@princeton.edu).

This work was supported in part by the U.S. National Science foundationunder Grant CNS-1702808. The work of Z. Ding was supported by the UKEPSRC under grant number EP/P009719/2 and by H2020-MSCA-RISE-2015under grant number 690750.

same time/frequency/ code resource but with different powerlevels [3], [4]. As a special case of NOMA, multiple usersuperposition transmission (MUST) has been proposed for3GPP long term evolution (LTE) [5], [6]. To exploit chan-nel conditions of different users opportunistically, users withpoorer channel qualities are allocated more transmit power inNOMA systems, while users with better channel qualities areless allocated. In this way, both poorer-channel and better-channel users can decode their messages successfully at thecost of applying successive interference cancellation (SIC) atbetter-channel users. It has shown that NOMA is more powerefficient than conventional orthogonal multiple access (OMA)when users are properly grouped according to their channelqualities or quality of service (QoS) requirements [7]–[10].

Recently, cooperative NOMA schemes incorporating re-laying have been proposed in the literature to strengthensystem performances. The first cooperative NOMA schemewas proposed in [2], where a user with strong channelconditions was used as a relay beside decoding its ownmessage via SIC. In [11], a dedicated multi-antenna amplify-and-forward (AF) relay was introduced to aid transmissionsfrom a base station (BS) to NOMA users. Considering perfectand imperfect knowledge of channel state information (CSI),the performances of cooperative NOMA systems with an AFrelay under Nakagami-m fading were studied in [12] and [13],respectively. A partial relay selection (RS) was proposed in[14], where several AF relay criteria were proposed to assistthe BS-to-users transmissions. Furthermore, the authors in[15] proposed a two-stage AF RS scheme, which achieves adiversity order the same as the number of the relay nodes. Thetwo-stage RS schemes have also been investigated for NOMAdecode-and-forward (DF) systems to decrease outage proba-bility and obtain spatial diversity in [15], [16]. To enhancecell edge coverage, multiple antennas have been employed atthe DF relay and users for cooperative NOMA systems [17].Based on the user-aided relay and dedicated relay, several full-duplex relaying schemes have been proposed for cooperativeNOMA DF systems in [18]–[21]. In [22] and [23], cooperativeNOMA transmissions have been combined with simultaneouswireless information and power transfer.

When direct links between the BS and users are non-negligible, it is worth pointing out that coordinated trans-missions incorporating direct links can significantly enhancethe performance of cooperative NOMA systems [14], [18],[19], [24], [25]. Direct links exist widely in not only theoverlapping of macro and micro cells [14], [24], but alsodevice-to-device (D2D) communications [19], [25]. Compared

2

to the conventional NOMA-DF scheme adopted in [15]–[17],the coordinated direct and relay transmission (CDRT) schemeachieves the improved outage performance and increasedergodic sum rate [24]. However, in coordinated transmis-sions, a main challenge is to acquire side information forinterference cancellation, which may result in huge overheaddue to massive connectivity in cooperative NOMA systems.Another issue in the current CDRT schemes is that the usersare paired based on channel quality, so that the far useralways achieves a low-rate transmission. Currently, the CDRTschemes designed for aiding two far users can be recognizedas the novel contribution. It is also worthwhile to point outthat the existences of direct links are commonly assumed inmultiple access relay channels (MARCs) for downlink anduplink transmissions [26]–[31]. In [26], the authors proposeda reverse compute-and-forward (CF) relaying scheme for thedowlink MARCs under an error-free and capacity-limitedsouce-to-relay channel. For the uplink MARCs with solidrelay-to-destination links, the CF relaying scheme for twouser access was investigated in [27] and lately extended tothe scenario with multiuser and multiple relays in [28].

On the other hand, when direct links are available incooperative systems, adaptive relaying schemes such as theselective relaying and incremental relaying can be applied toreduce error propagation resulted from the detection failureat the relay [32]–[36]. As a simple way to reduce errorpropagation for cooperative systems, the selective DF (SDF)relaying makes a decision whether it operates in the DFmode or keeps silent based on a received signal-to-noiseratio (SNR) threshold at the relay, so that the end-to-endperformance can be improved [32], [33]. By exploiting thelimited SNR feedback from the destination, the authors in[34] proposed the incremental relaying with a significantimprovement of spectrum efficiency over the conventionalfixed relaying (FR) and SDF relaying schemes. The outageprobability and end-to-end performance of the incrementalrelaying have been investigated in [35] and [36], respectively.In [37], the authors proposed the incremental-selective DF(ISDF) relaying by combining incremental relaying and SDFrelaying, which achieved an improved system performanceover that of the simple incremental DF relaying. Several hybridrelaying schemes that combine SDF, AF, and incrementalrelaying have been investigated in [38]–[40]. However, tothe best knowledge of the authors, selective relaying andincremental relaying have not been applied for cooperativeNOMA systems taking into account the superposition in thepower domain for multiple access.

In this paper, we consider a cooperative NOMA system withone BS communicating with two far receivers with the aidof an intermediate relay. Different from existing works onMARCs where the CF relaying is applied, we assume thatthe relay operates in the DF mode and investigate cooperativerelaying taking into account both the direct and relay links.The main contributions of this paper are summarized asfollows.

• In the presence of direct links, three cooperative relayingschemes, i.e., the FR, SDF with CDRT (SDF-CDRT),and ISDF schemes, are individually investigated for the

cooperative NOMA system. Compared to most existingcooperative NOMA schemes incorporating a single directlink to the near user [18], [19], [24], [25], our workconsiders the direct links from the BS to two far users,so that two far users with different QoS requirementscan be paired with each other and get benefit fromcooperative relaying. In contrast to cooperative NOMAsystems where the direct links from the BS to the farusers are unavailable, the proposed cooperative relayingschemes significantly improve the outage performance bytaking advantages of both the direct and relay links.

• To reduce error propagation resulted from the detectionfailures at the relay, the SDF-CDRT scheme is designedto switch the relaying on or off depending on the receivedSNRs at the relay. When the received SNRs at the relayare high enough to ensure the correct detection, theSDF-CDRT scheme forms an orthogonal branch withrespect to the direct link transmissions, so that only linearcombination is applied at each receiver and the decodingcomplexity can be reduced in such a case. When thereceived SNRs cannot ensure the correct detection forthe superposition, the relay keeps silent and the BS startsa new transmission block, which takes the advantagesprovided by the SDF relaying scheme [32], [33]. Basedon limited information feedback of the received SNRevents from two users, the ISDF scheme switches therelaying on or off depending on the combinations ofthe received SNR events at the relay and two users.Accordingly, the unnecessary relaying can be reduced bythe ISDF scheme over the SDF scheme.

• For the FR, SDF-CDRT, and ISDF schemes, the closed-form expressions for the outage probabilities are derivedfor both two users. To highlight the impact of the systemparameters on the outage performance, asymptotic outageprobabilities achieved at both two users are derived forthe three proposed cooperative relaying schemes, respec-tively. It is shown that the FR and SDF-CDRT schemesachieve a diversity order of one for both two users,whereas the ISDF scheme provides a diversity gain orderof two for the two users’ case. Taking into account theadditional time slots consumed for relaying, the analyticalresults for the average throughputs are derived for the FR,SDF-CDRT, and ISDF schemes, respectively.

The rest of this paper is organized as follows: Section IIpresents the system model, the FR scheme, and the SDF-CDRT scheme; Section III presents the ISDF scheme. SectionIV derives the outage probabilities for the three proposedcooperative relaying schemes and conducts asymptotic outageperformance analyses; Section V gives simulation results toverify the superior performance achieved by the proposedcooperative relaying schemes and Section VI summarizes thepaper.

II. SYSTEM MODEL AND SIMPLE RELAYING SCHEMES

We consider a downlink cooperative NOMA system con-sisting of a BS, a DF relay, and two far users. The systemmodel is depicted in Fig. 1, in which the two far users are

3

!

"#$%&

U'

U(

hs'

hsrhr'

hr(

hs(

Fig. 1. System model of the cooperative NOMA system with a DF relay.

denoted by U1 and U2, respectively. We assume that all thenodes are equipped with a single antenna and work in thehalf-duplex mode. In the considered system, the BS and relaycan be the transmitters of a macro cell and a micro cell,respectively, and we assume that the direct links from theBS to two users are non-negligible. The considered systemmodel can also be applied for D2D communications [25], inwhich the relay can be a device in the proximity of the twousers and we also assume that the direct links from the BSto the two users exist. The channels from the BS to the relay,from the BS to Ui, and from the relay to Ui are denotedby hsr, hsi, and hri, respectively, with i = 1 and 2. Allthe channels are modeled as independent but non-identicallydistributed (i.n.i.d.) complex Gaussian random variables (RVs)with zero means and variances βsr, βsi, and βri for hsr, hsi,and hri, respectively. Moreover, all the channels are assumedto be block fading, i.e., a channel keeps constant during onetransmission block and can vary from one transmission blockto another. The additive white Gaussian noises (AWGNs) at allthe receivers are assumed to have zero mean and variance σ2,while the information signal for Ui is denoted by xi satisfyingE{xi} = 0 and E{|xi|2} = 1.

In cooperative NOMA systems, users can be ordered bytheir channel qualities, which in turn determines the decodingorder at the receiver side and results in a huge overhead forthe case with a large number of users. Another user pairingcriterion is to order users according to their QoS requirements[15], [16], which enables grouping users even if they havestatistically the same channel qualities. Moreover, the CSIacquirements at the transmitters can be significantly reducedif QoS-based user pairing is applied [15]. In this paper, wealso apply QoS-based user pairing and assume that user U1 isa low target rate device but requiring a timely service, whileuser U2 is more delay-tolerated than user U1 but needs a higherthroughput.

In the considered system, each downlink transmission blockcan be accomplished within two time slots. In the first timeslot, the BS broadcasts the superposed signal xb ,

√α1x1 +√

α2x2 to all the other nodes, where αi is the power allocationcoefficient, which satisfies

∑2i=1 αi = 1 and αi > 0. The

received signals at the relay and Ui are respectively given by

yr = hsr

(√α1Psx1 +

√α2Psx2

)+ nr and (1)

y(1)i = hsi

(√α1Psx1 +

√α2Psx2

)+ n

(1)i , (2)

where Ps is the transmit power, nr and n(1)i are the AWGNs

at the relay and Ui, respectively. Since we assume that the

TABLE IRECEIVED SNR EVENTS AND DETECTION RESULTS AT THE RELAY

Received SNR event Detection result

ε1 := (γx1r ≥ γth,1) ∩ (γx2

r ≥ γth,2) Correct x1 and x2

ε2 := (γx1r ≥ γth,1) ∩ (γx2

r < γth,2) Correct x1 and incorrect x2

ε3 := γx1r < γth,1 Incorrect x1 and x2

QoS requirements of U1 are much less demanding than thoseof U2, the decoding order at the relay is always from U1 to U2

[15]. Therefore, the received SNRs at the relay for decodingx1 and x2 can be respectively expressed as

γx1r =

α1ρ|hsr|2

α2ρ|hsr|2 + 1and γx2

r = α2ρ|hsr|2, (3)

where ρ , Ps/σ2 is the transmit SNR.

A. FR Scheme

In this subsection, we investigate the conventional FRscheme which utilizes both the direct and relay link trans-missions to enhance the detection performance at the receiverside. After correct detecting x1 and x2, the relay regeneratesand forwards the superposition xb, so that the received signalat Ui in the second time slot can be expressed as

y(2)i = hri

(√α1Prx1 +

√α2Prx2

)+ n

(2)i , (4)

where Pr is the relaying transmit power and n(2)i is the AWGN

at Ui in the second time slot. It is worthwhile to point out thatthe above signal model strictly relies on the correct detectionat the relay. When the source-to-relay link suffers deep fading,the relay may not always detect x1 and x2 correctly. In TableI, three detection results and corresponding received SNRevents at the relay are presented, where γth,i , 2Ri − 1 isthe required SNR threshold for correct detecting xi with Ri

denoting the target transmission rate of Ui. In the FR scheme,each user applies maximum ratio combining (MRC) to recoverits desired signal. The MRC processing and the end-to-endreceived SNRs at U1 and U2 are discussed as follows.

1) Event ε1: For each user, MRC is applied to (2) and (4)at the receiver side to recover its desired signal. When theevent ε1 occurs, the relay can correctly forward xb, so thatthe end-to-end received SNR at U1 for detecting x1 is givenby

γx1

1, fr(ε1) =α1ρ(|hs1|2 + |hr1|2)

α2ρ(|hs1|2 + |hr1|2) + 1, (5)

where hr1 , √χhr1 with χ , Pr/Ps. At U2, MRC is also

applied to (2) and (4) for detecting x1. Then, SIC is performedfor detecting x2. The end-to-end received SNRs at U2 fordetecting x1 and x2 can be respectively expressed as

γx1

2, fr(ε1) =α1ρ(|hs2|2 + |hr2|2)

α2ρ(|hs2|2 + |hr2|2) + 1and (6)

γx2

2, fr(ε1) = α2ρ(|hs2|2 + |hr2|2). (7)

4

2) Event ε2: When the event ε2 occurs, we have γx1

1,fr(ε2) =γx1

1,fr(ε1) and γx1

2, fr(ε2) = γx1

2,fr(ε1) due to the correct forward-ing of

√α1x1 by the relay. However, the relay forwards the

incorrect√α2x2 in this case, so that the end-to-end received

SNR for U2 to detect x2 becomes

γx2

2, fr(ε2) =α2ρ|hs2|2

α2ρ|hr2|2 + 1. (8)

Compared to (7), we can see that the end-to-end received SNRin (8) becomes smaller.

3) Event ε3: When the event ε3 occurs, neither x1 nor x2

is correctly forwarded by the relay, so that the end-to-endreceived SNRs at U1 and U2 degrade to

γx1

1, fr(ε3) =α1ρ|hs1|2

α2ρ|hs1|2 + ρ|hr1|2 + 1, (9)

γx1

2, fr(ε3) =α1ρ|hs2|2

α2ρ|hs2|2 + ρ|hr2|2 + 1, and (10)

γx2

2, fr(ε3) =α2ρ|hs2|2

ρ|hr2|2 + 1. (11)

The expressions in (5)-(11) show that the end-to-end receivedSNRs are heavily affected by the detection results at therelay. When the events ε2 and ε3 occur, error propagation isunavoidable, which decreases the end-to-end received SNRsat U1 and U2. In order to reduce error propagation when theevents ε2 and ε3 occur and to avoid SIC when ε1 occurs, wepropose the SDF-CDRT scheme in the next subsection.

B. SDF-CDRT Scheme

Based on a certain received SNR threshold at the relay, theSDF relaying is a simple way to reduce error propagationfor cooperative relaying systems [33]. In this section, wepropose an SDF-CDRT scheme for the considered cooperativeNOMA systems. In the SDF-CDRT scheme, when the eventε1 occurs, the relay generates a new superposition such that anorthogonal structure is formed with respect to the direct andrelay transmissions. Consequently, linear combining is appliedat each receiver to recover the desired signal and SIC canbe avoided in such a case. Moreover, when ε2 or ε3 occurs,the relay keeps silent, so that error propagation resulted fromthe failure detection at the relay can be avoided and spectralefficiency can be improved by a new block transmission viathe BS [33].

In the SDF-CDRT scheme, the BS broadcasts xb to the relayand two users in the first time slot. The operation procedure ofthe SDF-CDRT scheme in the second time slot and the end-to-end received SNRs at U1 and U2 are described as follows.

1) Event ε1: When the event ε1 occurs, the relay generatesa new superposition

xr =√α2x1 −

√α1x2 (12)

for its transmission in the second time slot. Then, the receivedsignal at Ui in the second time slot is given by

y(2)i = hri

(√α2Prx1 −

√α1Prx2

)+ n

(2)i . (13)

For each user, the received signals from both the first andsecond time slots can be rewritten in a matrix form as y

(1)i

hsi

√Ps

y(2)i

hri

√Pr

=

[ √α1

√α2√

α2 −√α1

]︸ ︷︷ ︸

H

[x1

x2

]+

n(1)i

hsi

√Ps

n(2)i

hri

√Pr

, (14)

where H is the equivalent power allocation matrix. Differentfrom the 2× 2 orthogonal design for space-time block coding(STBC) [41], where H contains the complex space-time chan-nel coefficients weighted by ±1, the proposed power allocationmatrix is a 2 × 2 real matrix, which avoids the conjugate ofthe signal in the power-domain. In general, H can be in anyform of the 2× 2 rotation or reflection, i.e.,

H =

[cos θ − sin θsin θ cos θ

]or H =

[cos θ sin θsin θ − cos θ

], (15)

where θ ∈ (0, π/2). Thus, the power allocation can beinterpreted as a rotation or reflection of [x1, x2]

T . It can beshown that H is an orthogonal matrix with a unit norm foreach of its row, so that it not only satisfies the power constraintα21 + α2

2 = 1, but also provides the orthogonality for theproposed SDF-CDRT scheme. Without loss of generality, weadopt the reflection form (14) in the sequential. To recoverthe desired singal, each user applies linear combining to itsreceived signals from both the first and second time slots.Specifically, U1 and U2 apply the linear combinations as

x1 = y(1)1

√α1hr1 + y

(2)1

√α2hs1

= hs1hr1

√Px1 +

√α1hr1n

(1)1 +

√α2hs1n

(2)1 and (16)

x2 = y(1)2

√α2hr2 − y

(2)2

√α1hs2

= hs2hr2

√Px2 +

√α2hr2n

(1)2 −

√α1hs2n

(2)2 (17)

for detecting x1 and x2, respectively. Based on (16) and (17),the end-to-end received SNRs at U1 and U2 for detecting x1

and x2 can be respectively expressed as

γx1

1,sdf−cdrt(ε1) =ρ|hs1|2|hr1|2

α1|hr1|2 + α2|hs1|2and (18)

γx2

2,sdf−cdrt(ε1) =ρ|hs2|2|hr2|2

α1|hs2|2 + α2|hr2|2. (19)

2) Event ε2 or ε3: When the event ε2 or ε3 occurs, therelay cannot correctly regenerate xr. In such a case, the relaykeeps silent and the BS needs to start a new superpositiontransmission [33]. At each receiver, the detection is based onthe received signal in the first time slot. Thus, when εi (i =2, 3) occurs, the end-to-end received SNRs at U1 and U2 arerespectively given by

γx1

1,sdf−cdrt(εi) =α1ρ|hs1|2

α2ρ|hs1|2 + 1, (20)

γx1

2,sdf−cdrt(εi) =α1ρ|hs2|2

α2ρ|hs2|2 + 1, and (21)

γx2

2,sdf−cdrt(εi) =α1ρ|hs1|2

α2ρ|hs1|2 + 1. (22)

5

TABLE IIPRELIMINARY DECISION RULE FOR THE ISDF RELAYING

U1

Decision U2

ε1 ε2 ε3

ε1 Silent x2 xb

ε2 x1 xb xb

Compared to the conventional FR scheme, the SDF-CDRTscheme forms an orthogonal transmission branch with respectto the direct link when ε1 occurs. When either ε2 or ε3occurs, the SDF-CDRT scheme stops relaying, so that errorpropagation can be avoided. However, when ε1 occurs, theSDF-CDRT scheme always operates in the DF mode nomatter what the direct links’ qualities are, which results inunnecessary relaying if Ui can recover its desired signal viathe direct link transmission. Motivated by this, we propose theISDF relaying scheme in the next section.

III. ISDF RELAYING

By exploiting the limited SNR feedback from the receiverto the BS and relay, the incremental relaying can significantlyimprove reception reliability over the fixed and selectiverelaying schemes for point-to-point communication systems[34]–[36]. However, due to the superposition in the power-domain for multiple access, the feedback of the received SNRsin the considered system is much more complicated thanthat of a point-to-point communication system. In the casethat two users are grouped for multiple access, there are sixcombinations of the received SNR events with respect to thedetection results at U1 and U2 in delay-limited transmissionsbased on the proposed ISDF relaying scheme.

In the ISDF scheme, the BS broadcasts a superposition xb inthe first time slot. Then, the received SNR events at U1 and U2

are fed back to the relay, based on which the relay makes a pre-liminary decision on whether it forwards a signal/superpositionor not. If the preliminary decision is to forward, the relayfurther checks whether its received SNRs are high enoughto correctly regenerate the required signal. Only when therequired signal/superposition can be regenerated correctly, therelay forwards the signal/superposition; otherwise, the relaykeeps silent and the BS starts a new transmission [34]–[36].

The preliminary decision rule at the relay is presented inTable II, which is designed to forward a signal to either U1,U2, or both if it is necessary. In Table II, the received SNRevents at U1 and U2 are defined as

ε1 := γx11 ≥ γth,1,

ε2 := γx11 < γth,1,

ε1 := (γx12 ≥ γth,1) ∩ (γx2

2 ≥ γth,2),

ε2 := (γx12 ≥ γth,1) ∩ (γx2

2 < γth,2), andε3 := γx1

2 < γth,1, (23)

where γx11 is the received SNR at U1 in the first time slot for

detecting x1, which is given by

γx11 =

α1ρ|hs1|2

α2ρ|hs1|2 + 1. (24)

Moreover, γx12 and γx2

2 are the received SNRs at U2 in thefirst time slot for detecting x1 and x2, respectively, which arerespectively given by

γx12 =

α1ρ|hs2|2

α2ρ|hs2|2 + 1and (25)

γx22 = α2ρ|hs2|2. (26)

After a preliminary decision is made, the relay furtherevaluates its received SNRs and makes a formal decision onwhether it operates in the DF relaying mode or keeps silentaccording to the formal decision rule provided in Table III.Note that Table III is extended from Table II by taking intoaccount the received SNR events at the relay. For example,when the events {ε2, ε2} occur, the preliminary rule is tomake the relay forward xb according to Table II. In sucha case, the relay further checks which one of {ε1, ε2, ε3}occurs to make the formal decision. If ε1 occurs, the relaycan forward xb. However, if ε2 occurs, the relay can onlyforward x1 due to its failure detection of x2. As such, whenε3 occurs, the relay keeps silent even if the events {ε2, ε2}occur. At the receiver side, we assume that each user canacquire the indication knowledge of the forwarded signal, i.e.,which signal of {x1, x2, xb} is forwarded, so that MRC can beperformed adaptively at each receiver. The end-to-end receivedSNRs achieved by the ISDF scheme are discussed as follows.

1) Events {ε1, ε1, ε2}: According to Table III, when{ε1, ε1, ε2} occurs, the relay only forwards x2 with its fulltransmit power to help U2. For U1, its end-to-end receivedSNR is determined by its received signal in the first time slot,which is characterized by

γx1

1,isdf(ε1, ε1, ε2) ≥ γth,1. (27)

At U2, x1 is detected based on the received signal in the firsttime slot. Since U2 can detect x1 successfully when ε2 occurs,we have

γx1

2,isdf(ε1, ε1, ε2) ≥ γth,1. (28)

After subtracting the detected x1 from the received signal ofthe first time slot, MRC is applied to the remaind signal andthe received signal in the second time slot for recovering x2.Thus, the end-to-end received SNR for detecting x2 at U2 isgiven by

γx2

2,isdf(ε1, ε1, ε2) = ρ(α2|hs2|2 + |hr2|2). (29)

2) Events {ε1, ε1, ε3}: When {ε1, ε1, ε3} occurs, the relayforwards xb in the second time slot. Since ε1 occurs inthis case, U1 detects x1 based on the received signal in thefirst time slot. Thus, the end-to-end received SNR at U1 fordetecting x1 satisfies

γx1

1,isdf(ε1, ε1, ε3) ≥ γth,1. (30)

At U2, MRC is applied to detect x1 based on the receivedsignals from the two time slots. After then, SIC is performedfor detecting x2. The end-to-end received SNRs at U2 fordetecting x1 and x2 can be respectively expressed as

γx1

2,isdf(ε1, ε1, ε3) =α1ρ(|hs2|2 + |hr2|2)

α2ρ(|hs2|2 + |hr2|2) + 1and (31)

6

TABLE IIIFORMAL DECISION RULE AT THE RELAY

Relay’s SNR

Decision Feedback

{ε1, ε1} {ε1, ε2} {ε1, ε3} {ε2, ε1} {ε2, ε2} {ε2, ε3}

ε1 Silent x2 xb x1 xb xb

ε2 Silent Silent x1 x1 x1 x1

ε3 Silent Silent Silent Silent Silent Silent

TABLE IVFEEDBACK AND SIGNALING REQUIRED BY THE RELAYING SCHEMES

Relaying scheme Received SNR event feedback Signaling for detecting

FR Null One bit for indicating either the BSor the relay is transmitting

SDF-CDRT Relay: one bit for indicating ε1 or non-ε1event

One bit for indicating either xb or xr

is transmitted

ISDF U1: one bit for indicating ε1 and ε2; U2:two bits for indicating ε1, ε2, and ε3

Two bits for indicating which one of{x1, x2, xb} is transmitted

γx2

2,isdf(ε1, ε1, ε3) = α2ρ(|hs2|2 + |hr2|2). (32)

3) Events {ε1, ε2, ε1}: In this case, the relay forwards x1

with its full transmit power to aid U1. At U1, MRC is appliedfor detecting x1 with the end-to-end received SNR given by

γx1

1,isdf(ε1, ε2, ε1) =ρ(α1|hs1|2 + |hr1|2)

α2ρ|hs1|2 + 1. (33)

Due to ε1, U2 detects its desired messages based on itsreceived signal in the first time slot. The end-to-end receivedSNRs at U2 satisfy

γx1

2,isdf(ε1, ε2, ε1) ≥ γth,1 and (34)

γx2

2,isdf(ε1, ε2, ε1) ≥ γth,2. (35)

4) Events {ε1, ε2, ε2}: When {ε1, ε2, ε2} occurs, xb isforwarded by the relay and MRC is applied at both users.The end-to-end received SNR at U1 for detecting x1 is givenby

γx1

1,isdf(ε1, ε2, ε1) =α1ρ(|hs1|2 + |hr1|2)

α2ρ(|hs1|2 + |hr1|2) + 1. (36)

Since that ε2 already occurs in the first time slot, the end-to-end received SNR for U2 to detect x1 with the aid of MRCalso satisfies

γx1

2,isdf(ε1, ε2, ε2) ≥ γth,1. (37)

Moreover, the end-to-end received SNR for U2 to detect x2 isthe same as that of (32).

5) Events {ε1, ε2, ε3}: Since xb is forwarded by the relay,the end-to-end received SNR at U1 is the same as that of (36).At U2, the end-to-end received SNRs for detecting x1 and x2

are the same as those of (31) and (32), respectively.6) Events {ε2, ε1, ε3}: In this case, the relay can only

forward x1 taking into account ε2. Due to the event ε1, we

have γx1

1,isdf(ε2, ε1, ε3) ≥ γth,1. At U2, the end-to-end receivedSNRs for detecting x1 and x2 can be respectively derived as

γx1

2,isdf(ε2, ε1, ε3) =ρ(α1|hs2|2 + |hr2|2)

α2ρ|hs2|2 + 1and (38)

γx2

2,isdf(ε2, ε1, ε3) = α2ρ|hs2|2. (39)

7) Events {ε2, ε2, ε1}: Only x1 is forwarded in this case.The end-to-end received SNR at U1 is the same as that of(33). Moreover, the end-to-end received SNRs at U2 are char-acterized by γx1

2,isdf(ε2, ε2, ε1) ≥ γth,1 and γx2

2,isdf(ε2, ε2, ε1) ≥γth,2.

8) Events {ε2, ε2, ε2}: In this case, the relay can forwardonly x1 due to ε2, so that the end-to-end received SNR at U1

can be enhanced by using MRC as that of (33). At U2, the end-to-end received SNRs are based on its received signal in thefirst time slot, which are characterized by γx1

2,isdf(ε2, ε2, ε2) ≥γth,1 and γx2

2,isdf(ε2, ε2, ε2) < γth,2.9) Events {ε2, ε2, ε3}: In this case, the relay forwards x1

with its full transmit power. Both U1 and U2 apply MRCto enhance the received SNRs. Thus, the end-to-end receivedSNR at U1 for detecting x1 has the same form as that of (33).At U2, the end-to-end received SNRs for detecting x1 and x2

are the same as those of (38) and (39), respectively.10) Events for relay to keep silent: For all the scenarios in

which the relay keeps silent regarding Table III, the end-to-endreceived SNRs at U1 and U2 are the same as those achievedby the direct link transmissions in the first time slot, whichare characterized by the corresponding events {ej , ek}.

It is worthwhile to point out that all the 18 combinations ofthe received SNR event {εi, εj , εk} belong to the above tencases. Moreover, these events can be classified into three types,i.e., outage event, non-outage event, and undetermined event assummarized in Table VII in Appendix B, which will be used tofacilitate the outage probability analysis for the ISDF scheme.

7

For the proposed FR, SDF-CDRT, and ISDF schemes, the re-quired feedback for the received SNR and control signaling fordetection processing are summarized in Table IV. Especially,we can see that the required feedback and signaling for theFR and SDF-CDRF schemes are small and realistic. Althoughthe received SNR events have 18 combinations in the ISDFscheme, the required control signaling at each receiver is stillsmall, i.e., only the indicating information for the transmitted{x1, x2, xb} is needed for the SIC/MRC processing.

IV. OUTAGE PROBABILITY ANALYSES

In this section, we perform the outage probability analysesfor the FR, SDF-CDRT, and ISDF schemes, respectively. Theclosed-form expressions are derived for the outage probabil-ities achieved by the FR, SDF-CDRT, and ISDF schemes,respectively. Then, asymptotic outage performance analysesfor the FR, SDF-CDRT, and ISDF schemes are respectivelyderived in the high transmit SNR region.

A. FR Scheme

For the conventional FR scheme, each receiver always com-bines the received signals from the two time slots for detectingits desired message irrespective of the decision results made atthe relay. Accordingly, error propagation due to the incorrectdetections at the relay cannot be avoided. With respect to thethree types of the received SNR events at the relay, the outageprobabilities achieved by the FR scheme at U1 and U2 can berespectively expressed as

P frout,1 =

3∑i=1

Pr(εi)A1(εi) and (40)

P frout,2 =

3∑i=1

Pr(εi)(A2(εi) +A3(εi)), (41)

where A1(εi) , Pr{γx1

1,fr(εi) < γth,1}

, A2(εi) ,Pr{γx1

2,fr(εi) < γth,1}

, and A3(εi) , Pr{γx1

2,fr(εi) ≥γth,1, γ

x2

2,fr(εi) < γth,2}

.Since we assumed that all the channel gains follow i.n.i.d.

exponential distributions, the probabilities of the received SNRevents at the relay can be obtained as

Pr{ε1} = e−ξmaxβsr , (42)

Pr{ε2} =[e−

ξ1βsr − e−

ξ2βsr

]+, and (43)

Pr{ε3} = 1− e−ξ1βsr , (44)

where ξ1 =γth,1

ρ(α1−α2γth,1), ξ2 =

γth,2

α2ρ, ξmax , max(ξ1, ξ2),

and [x]+ , max(x, 0). After some mathematical manipula-tions, the terms A1, A2, and A3 are derived in Table V, whereβi , χβi for i = 1 and 2. Based on the result in TableV and (42)-(44), the closed-form evaluations for the outageprobabilities of U1 and U2 can be readily conducted via (40)and (41).

Theorem 1. For the FR scheme, as ρ → ∞, asymptotic outageprobabilities at U1 and U2 are respectively given by

P fr,∞out,1 =

ξ1βsr

χβr1γth,1βs1(α1 − α2γth,1) + χβr1γth,1

and (45)

P fr,∞out,2 =

ξ1βsr

χβr2γth,1βs2(α1 − α2γth,1) + χβr2γth,1

+[ξ2 − ξ1]

+

βsr

χβr2γth,2χβr2γth,2 + βs2

. (46)

Proof: Applying the fact that e−x → 1− x as x → 0 to(42)-(44) and the terms in Table IV, the asymptotic expressionsfor (40) and (41) can be readily derived as (45) and (46),respectively.

Defining the diversity order achieved by Ui by

di = − limρ→∞

log(P frout,i)

log(ρ)(47)

and based on (45) and (46), it can be shown that both U1 andU2 achieve a diversity order of one when the FR scheme isapplied. Considering that relaying occurs in each transmissionblock, the average throughput achieved by the FR scheme forUi (i = 1, 2) can be expressed as follows:

Rfri =

1

2Ri

(1− P fr

out,i

), (48)

where the factor 12 is resulted from half-duplex relaying.

B. SDF-CDRT scheme

In the SDF-CDRT scheme, the operation at the relay de-pends on its own received SNR events, which also results indifferent end-to-end received SNRs at U1 and U2. Based onthe derived end-to-end received SNRs at U1 and U2 and thereceived SNR events at the relay, the outage probabilities atU1 and U2 can be respectively expressed as

P sdf−cdrtout,1 =

3∑i=1

Pr{εi}B1(εi) and (49)

P sdf−cdftout,2 =

3∑i=1

Pr{εi}B2(εi), (50)

where B1(εi) , Pr{γx1

1,sdf−cdrt(εi) < γth,1}

, B2(ε1)

, Pr{γx2

2,sdf−cdrt(ε1) < γth,2}

, and B2(εi) , Pr{γx1

2,sdf−cdrt(εi) < γth,1}

+ Pr{γx1

2,sdf−cdrt(εi) ≥ γth,1,

γx2

2,idf(εi) < γth,2}

for i = 2 and 3. After some mathematicalmanipulations (see Appendix A), B1 and B2 are derived asfollows:

B1(ε1) = 1− 2γth,1ρ

√α1α2

βs1βr1

e−(

α2βr1

+α1βs1

)γth,1

ρ

×K1

(2γth,1ρ

√α1α2

βs1βr1

), (51)

B1(ε2) = B1(ε3) = 1− e−ξ1βs1 , (52)

8

TABLE VAi FOR THE OUTAGE PROBABILITY EXPRESSION OF THE FR SCHEME

A1(ε1) =

1 + βs1e

− ξ1βs1

βr1−βs1− βr1e

− ξ1βr1

βr1−βs1, βr1 = βs1

1− e− ξ1

βs1(1 + ξ1

βs1

), βr1 = βs1

A1(ε2) = A1(ε1)

A1(ε3) = 1− βs1e− ξ1

βs1

ρβr1ξ1+βs1

A2(ε1) =

1 + βs2e

− ξ1βs2

βr2−βs2− βr2e

− ξ1βr2

βr2−βs2, βr2 = βs2

1− e− ξ1

βs2(1 + ξ1

βs2

), βr2 = βs2

A2(ε2) = A2(ε1)

A2(ε3) = 1− βs2e− ξ1

βs2

ρβr2ξ1+βs2

A3(ε1) =

[βr2

(e− ξ1

βr2 −e− ξ2

βr2

)βr2−βs2

+βs2

(e− ξ2

βs2 −e− ξ1

βs2

)βr2−βs2

]+

, βr2 = βs2

[e− ξ1

βs2

(1+ ξ1

βs2

)− e

− ξ2βs2

(1+ ξ2

βs2

) ]+, βr2 = βs2

A3(ε2) =

e− ξ1

βr2 +βs2

(e− ξ1

βr2 −e− ξ1

βs2

)βr2−βs2

− βs2e− ξ2

βs2

α2βr2ρξ2+βs2, βr2 = βs2, ξ2 > ξ1

βr2e− ξ1

βr2

βr2−βs2− βr2βs2(1+α2ρξ2)e

− ξ1(α2βr2ρξ2+βs2)+ξ2(βr2−βs2)

βr2βs2(1+α2ρξ2)

(βr2−βs2)(α2βr2ρξ2+βs2), βr2 = βs2, ξ2 ≤ ξ1

(βs2+ξ1)e− ξ1

βs2

βs2− e

− ξ2βs2

1+α2ρξ2, βr2 = βs2, ξ2 > ξ1

ξ2(1+α2ρ(βs2+ξ1))e− ξ1

βs2

βs2(1+α2ρξ2), βr2 = βs2, ξ2 ≤ ξ1

A3(ε3) =βs2e

− ξ1βs2

βr2ρξ1+βs2− βs2e

− ξ2βs2

βr2ρξ2+βs2

B2(ε1) = 1− 2γth,2ρ

√α1α2

βs2βr2

e−(

α2βr2

+α1βs2

)γth,2

ρ

×K1

(2γth,2ρ

√α1α2

βs2βr2

), and (53)

B2(ε2) = B2(ε3) = 1− e−ξ1βs2 +

[e−

ξ1βs2 − e−

ξ2βs2

]+. (54)

Theorem 2. For the SDF-CDRT scheme, asymptotic outageprobabilities at U1 and U2 are respectively given by

P sdf−cdrt,∞out,1 =

γth,1ρ

(α2

χβr1+

α1

βs1

)+ξ1([ξ2 − ξ1]

+ + ξ1)

βsrβs1(55)

and

P sdf−cdrt,∞out,2 =

γth,2ρ

(α2

χβr2+

α1

βs2

)+([ξ2 − ξ1]

+ + ξ1)2

βsrβs2. (56)

Proof: As x → 0, applying the fact that e−x → 1 − xand xK1(x) → 1 to (42)-(44) and B1 and B2, the asymptotic

expressions for (49) and (50) can be readily derived as (55)and (56), respectively.

With the obtained asymptotic expressions for the outageprobabilities, it can be shown that both U1 and U2 achieve adiversity order of one when the SDF-CDRT scheme is applied.Since a relaying transmission occurs only when ε1 happens inthe SDF-CDRT scheme, whereas a new transmission blockstarts when ε2 or ε3 occurs, the average throughput achievedby the SDF-CDRT scheme for Ui (i = 1, 2) can be expressedas follows:

Rsdf−cdrti =

1

2Ri

(1− Pr{ε1}Bi(ε1)

)+Ri

(1−

3∑k=2

Pr{εk}Bi(εk)

). (57)

C. ISDF schemeSince the relaying is switched on or off depending on not

only the received SNR event feedback from two users, but also

9

TABLE VIPi AND Pj FOR THE OUTAGE PROBABILITY EXPRESSIONS OF THE ISDF SCHEME

P1 =

βs1γth,1

(1−e

− ξ1βs1

)−βr1ρξ1

(1−e

−γth,1

βr1ρ)

βs1γth,1−βr1ρξ1, βs1 = ρβr1ξ1

γth,1

1− e− ξ1

βs1(1 + ξ1

βs1

), βs1 = ρβr1ξ1

γth,1

P2 =

βr1

(1−e

− ξ1βr1

)−βs1

(1−e

− ξ1βs1

)βr1−βs1

, βs1 = βr1

1− e− ξ1

βs1(1 + ξ1

βs1

), βs1 = βr1.

P1 =

βr2e

− ξ1βs2

(1−e

α2(ξ1−ξ2)

βr2

)βr2−α2βs2

−α2βs2

(e− ξ1

βs2 −e− ξ2

βs2

)βr2−α2βs2

, βr2 = α2βs2

e− ξ1

βs2 − e− ξ2

βs2(1 + ξ2−ξ1

βs2

), βr2 = α2βs2,

P2 =

1−e

− ξ1βr2 +

βs2e− ξ1

βs2

1−e

ξ1βs2

− ξ1βr2

βr2−βs2

+

βr2

e

ξ1βr2

− ξ2βr2

− ξ1βs2 +e

− ξ1βr2 −e

− ξ2βr2 −e

− ξ1βs2

βr2−βs2

, βr2 = βs2

1− e− ξ1

βs2 − ξ1βs2

e− ξ2

βs2 , βr2 = βs2,

P3 =

e− ξ1

βs2 + βs2e− ξ2

βs2

βr2−βs2− βr2e

ξ1βr2

− ξ2βr2

− ξ1βs2

βr2−βs2, βr2 = βs2

e− ξ1

βs2 − e− ξ2

βs2(1 + ξ2−ξ1

βs2

), βr2 = βs2,

P4 = 1− e− ξ1

βs2

the received SNRs at the relay, the outage events at U1 and U2

occur regarding the combinations of {εi, εj , εk}. Noting thatnon-outage events always occur when ε1 happens at U1, theoutage probability achieved by the ISDF scheme at U1 can beexpressed as

P isdf1,out=

3∑i=1

3∑k=1

Pr{εi}Pr{ε2, εk, γ

x1

u1,isdf(εi, εj , εk)<γth,1

}=

3∑i=1

3∑k=1

Pr{εi}Pr{εk}Pr{ε2,γ

x1

u1,isdf(εi,εj ,εk)<γth,1

}.

(58)

In (58), we have applied the fact that the event εk is inde-pendent of ε2. Similarly, the outage probability at U2 can beexpressed as

P isdf2,out =

3∑i=1

2∑j=1

3∑k=2

Pr{εi}Pr{εj}Pr{εk, γ

x1

u2,isdf(εi, εj , εk)

< γth,1}+ Pr{εi}Pr{εj}Pr

{εk, γ

x1

u2,isdf(εi, εj , εk)

≥ γth,1, γx2

u2,isdf(εi, εj , εk) < γth,2

}. (59)

In (58) and (59), the probabilities Pr{εj} and Pr{εk} can bederived as

Pr{ε1} = e−ξ1βs1 , (60)

Pr{ε2} = 1− e−ξ1βs1 , (61)

Pr{ε1} = e−ξmaxβs2 , (62)

Pr{ε2} =[e−

ξ1βs2 − e−

ξ2βs2

]+, and (63)

Pr{ε3} = 1− e−ξ1βs2 . (64)

Theorem 3. For the ISDF scheme, the achieved outageprobabilities at U1 and U2 are respectively given by

P isdfout,1 = P1

(Pr{ε1}Pr{ε1}+ Pr{ε2}

)+ P2 Pr{ε1}

×(1− Pr{ε1}

)+ Pr{ε3}Pr{ε2} (65)

and

P isdfout,2 = P1 Pr{ε1}Pr{ε1}+ P2 Pr{ε1}

+P3 Pr{ε1}Pr{ε2}+ P4 Pr{ε2}+Pr{ε2}Pr{ε2}+ Pr{ε3}(1− Pr{ε1}), (66)

where Pi (i = 1, 2) and Pj (j = 1, 2, 3, 4) are given in TableVI.

Proof: See Appendix B.The expressions in Theorem 3 show that the outage prob-

ability at each user consists of two parts, i.e., one part dueto the Undetermined SNR events and another part due to theOutage SNR events, as marked in Table VII in Appendix B.When ρ → ∞, the Undetermined SNR events in Table VIItend to be non-outage. Thus, asymptotic expressions for theoutage probabilities are mainly determined by the asymptoticvalues of the part corresponding to the Outage SNR events,

10

which are respectively provided as

P isdf,∞out,1 =

ξ21βsrβs1

(67)

and

P isdf,∞out,2 =

{[ξ2−ξ1]

+([ξ2−ξ1]++ξ1)

βsrβs2+ ξ1(ξ2−ξ1)

χβs2βr2, χβr2 =βs2

[ξ2−ξ1]+([ξ2−ξ1]

++ξ1)βsrβs2

+ ξ1ξ2β2s2

, χβr2=βs2

. (68)

Based on (67), it can be seen that U1 achieves a diversityorder of two. As such, when ξ2 > ξ1, it can be seen that U2

also achieves a diversity order of two. Compared to the FRand SDF-CDRT schemes, the ISDF scheme achieves a greaterdiversity order for both users.

In the ISDF scheme, a relaying transmission does notoccur when {ε1, ε1} happens. In such a subcase, the cor-responding average throughput achieved at U1 is given byR1 Pr{ε1}Pr{ε1}. Moreover, an additional relaying trans-mission actually occurs only when either ε1 or ε2 happensaccording to Table III, whereas a new transmission block startswhen ε3 happens. In such a subcase, the corresponding outageprobability is the first two terms on the righthand side of(65). Therefore, the average throughput achieved by the ISDFscheme for U1 can be expressed as follows:

Risdf1 = R1 Pr{ε1}Pr{ε1}+

1

2R1

(1− Pr{ε1}Pr{ε1}

)×(1− P1

(Pr{ε1}Pr{ε1}+ Pr{ε2}

)− P2 Pr{ε1}

(1− Pr{ε1}

)). (69)

As such, the average throughput achieved by the ISDF schemefor U2 is given by

Risdf1 =R2 Pr{ε1}Pr{ε1}+

1

2R2

(1− Pr{ε1}Pr{ε1}

)×(1− P1 Pr{ε1}Pr{ε1} − P2 Pr{ε1} − P3

Pr{ε1}Pr{ε2} − P4 Pr{ε2} − Pr{ε2}Pr{ε2}). (70)

V. SIMULATION RESULTS

This section provides Monte Carlo simulation results toverify the correctness of the analytical results and evaluate theoutage performance achieved by the proposed schemes. In thefigures for the simulation results, the curves corresponding tothe conventional FR schemes are denoted by “FR, Conv.” andthe curves corresponding to the DF relaying without directlinks are denoted by “DF w/o DL”. For simplicity, we setχ = 1 assuming equal power allocation at the source and therelay.

The outage probability versus the transmit SNR is investi-gated in Fig. 2, where we consider two scenarios. For scenario1, we set βs1 = βs2 = 0.8, and the remained βij = 1. Forscenario 2, we set all βij = 1. The power allocation coefficientis set as α1 = 0.8 for both scenarios. In Fig. 2(a) and Fig.2(b), the correctness of the analytical outage probability isverified by simulation results. The correctness of the asymp-totic expressions for all the considered schemes is also verifiedin Fig. 2(a). It is shown that the ISDF scheme achieves the

5 10 20 30 4010

−3

10−2

10−1

100

Out

age

prob

abili

ty

SNR (dB)

Ana, U1, FR, Conv.

Ana, U1, SDF-CDRT

Ana, U1, ISDF

Ana, U2, FR, Conv.

Ana, U2, SDF-CDRT

Ana, U2, ISDF

Sim, U1, FR, Conv.

Sim, U1, SDF-CDRT

Sim, U1, ISDF

Sim, U2, FR, Conv.

Sim, U2, SDF-CDRT

Sim, U2, ISDF

Asymptotic expressions

(a) Scenario 1 (R1 = 2 bps/Hz, R2 = 4 bps/Hz).

0 5 10 15 20 25 3010

−4

10−3

10−2

10−1

100

SNR (dB)

Out

age

prob

abili

ty

Ana, U1, FR, Conv.

Ana, U1, SDF-CDRT

Ana, U1, ISDF

Ana, U2, FR

Ana, U2, SDF-CDRT

Ana, U2, ISDF

Sim, U1, FR, Conv.

Sim, U1, SDF-CDRT

Sim, U1, ISDF

Sim, U2, FR, Conv.

Sim, U2, SDF-CDRT

Sim, U2, ISDF

(b) Scenario 2 (R1 = 2 bps/Hz, R2 = 5 bps/Hz).

Fig. 2. Outage probability versus SNR.

smallest outage probabilities for U1 and U2, respectively, in thewhole SNR region. The SDF-CDRT scheme also achieves thesmaller outage probabilities for U1 and U2 over those of theconventional FR scheme, respectively. In Fig. 2(a), at the 10−2

outage probability level, the ISDF and SDF-CDRT schemesrespectively achieve about 9 dB and 4 dB SNR gains overthat of the conventional FR scheme for U1. The similar resultscan be observed for U2 at the 10−2 outage probability level.Thus, the superior outage performance of the ISDF scheme isverified by Fig. 2. By applying selective relaying, the SDF-CDRT scheme not only prevents error propagation comparedto the conventional FR scheme, but also avoids SIC when theBS-to-relay link is good enough (when ε1 occurs). Due to errorpropagation at the relay, the conventional FR scheme achievesthe highest outage probability in Fig. 2(a) and Fig. 2(b).

In Fig. 3, we investigate the impact of the target rate on theoutage probability, where we set ρ = 30 dB and α1 = 0.8. InFig. 3(a), we fix R1 = 2 bps/Hz and increase R2 from 2 to 10bps/Hz. As it is shown in Fig. 3(a), the SDF-CDRT and ISDFscheme achieve the smaller outage probabilities than those ofthe conventional FR scheme for U1 and U2, respectively. Forall the considered relaying schemes, the outage probabilityof U2 increases by increasing R2 and approaches one in thehigh target rate region. Also, the outage probability of U1

achieved by the SDF-CDRT scheme increases by increasing

11

2 4 6 8 1010

−4

10−3

10−2

10−1

100

Targeted rate R2 (bps/Hz)

Out

age

prob

abili

ty

Ana, U1, FR, Conv.

Ana, U1, SDF-CDRT

Ana, U1, ISDF

Ana, U2, FR, Conv.

Ana, U2, SDF-CDRT

Ana, U2, ISDF

Sim, U1, FR, Conv.

Sim, U1, SDF-CDRT

Sim, U1, ISDF

Sim, U2, FR, Conv.

Sim, U2, SDF-CDRT

Sim, U2, ISDF

(a) Fixed R1 = 2 bps/Hz in scenario 1.

0.4 2 4 6 810

−4

10−3

10−2

10−1

100

Targeted rate R2 (bps/Hz)

Out

age

prob

abili

ty

Ana, U1, FR, Conv.

Ana, U1, SDF-CDRT

Ana, U1, ISDF

Ana, U2, FR, Conv.

Ana, U2, SDF-CDRT

Ana, U2, ISDF

Sim, U1, FR, Conv.

Sim, U1, SDF-CDRT

Sim, U1, ISDF

Sim, U2, FR, Conv.

Sim, U2, SDF-CDRT

Sim, U2, ISDF

(b) R1 = 0.4R2 in scenario 2.

Fig. 3. Outage probability versus target rate.

R2. In contrast, the outage probability of U1 achieved by theconventional FR scheme keeps constant irrespective of thevariation of R2, while the outage probability of U1 achieved bythe ISDF scheme almost keeps constant by increasing R2. InFig. 3(b), we set R1 = 0.4R2 and investigate the impact of thetarget rate on the outage performance for different schemes.Note that when both R1 and R2 increase, the affordable outageperformances of the different schemes can be observed. For U1

and U2, Fig. 3(b) shows that the outage probabilities achievedby all the schemes first increase by increasing the target rateand then jump to one as the target rate reaches a large value.It is shown that the ISDF scheme achieves the smallest outageprobabilities for U1 and U2, respectively. The SDF-CDRTscheme also achieves the smaller outage probability for U1

than that of the conventional FR scheme. However, for U2,the SDF-CDRT scheme achieves the higher outage probabilitythan that of the conventional FR scheme in the low targetregion. In the middle and high target rate regions, the SDF-CDRT scheme achieves the smaller outage probability thanthat of the conventional FR scheme, which is preferred by ahigher QoS requirement.

To see the details of the impact of the power allocation onthe outage performance, we investigate the outage probabilityversus α1 in Fig. 4, where we set ρ = 30 dB, R1 = 2 bps/Hz,

0.5 0.6 0.7 0.8 0.9 110

−5

10−4

10−3

10−2

10−1

100

α1

Out

age

prob

abili

ty

Ana, U1, FR, Conv.

Ana, U1, SDF-CDRT

Ana, U1, ISDF

Ana, U2, FR, Conv.

Ana, U2, SDF-CDRT

Ana, U2, ISDF

Sim, U1, FR, Conv.

Sim, U1, SDF-CDRT

Sim, U1, ISDF

Sim, U2, FR, Conv.

Sim, U2, SDF-CDRT

Sim, U2, ISDF

Fig. 4. Outage probability versus power allocation coefficient.

and R2 = 5 bps/Hz in scenario 2. It is shown in Fig. 4 thatthe ISDF scheme achieves the smallest outage probabilities forU1 and U2 in the whole α1 region. For U2, the SDF-CDRTscheme achieves the smaller outage probability than that of theconventional FR scheme. However, for U1, the conventionalFR scheme achieves the smaller outage probability than thatof the SDF-CDRT scheme in the high α1 region.

To investigate the impact of the relay’s location on theoutage performance, we also consider a geometric locationmodel as depicted in Fig. 5(a), in which the BS is locatedat the center of a unit circle and two far users (U1 and U2)are located on the edge of the circle. For the comparison withthe MUST scheme, we assume that two near users (U3 andU4) are located near the BS. In the geometric location model,we set the path loss model by βij = d−τ

ij , where the pathloss exponent is set as τ = 3 and dij is the distance betweeni ∈ {s, r} and j ∈ {r, 1, 2, 3, 4}. For the simulation resultsin Fig. 5(b), we set ρ = 30 dB, R1 = 2 bps/Hz, R2 = 5bps/Hz, θ1 = 15◦, and θ2 = −15◦. In the simulation, therelay moves away from the BS along the vertical direction.For the comparison purpose, the outage probabilities at U1

and U2 achieved by the DF relay without direct links andthe direct link transmission using the MUST scheme (U1 andU2 are paired in terms of their target transmission rates) arealso presented. It is shown that the ISDF scheme achievesthe smallest outage probabilities for U1 and U2, respectively.As the distance dsr increases from 0 to 0.9, the outageprobabilities achieved by the ISDF scheme first decreases andthen increases. Moreover, as dsr approaches zero, the ISDFscheme becomes the MUST scheme with the retransmissionprotocol, which is described in Table II. However, this figureshows that the optimal location of the relay is not dsr = 0,so that the ISDF scheme can effectively decrease the outageprobability even compared with MUST with the retransmissionprotocol. In the small and middle dsr regions, it is shownthat the conventional FR scheme achieves the smaller outageprobabilities than those of the SDF-CDRT scheme for U1 andU2, respectively. However, when the relay is located moreclose to the users, the SDF-CDRT scheme achieves the smalleroutage probabilities than those of the conventional FR schemefor U1 and U2, respectively. For the SDF-CDRT scheme, the

12

(a) The geometric location model.

0 0.1 0.3 0.5 0.7 0.9

10−4

10−3

10−2

10−1

100

dsr

Out

age

prob

abili

ty

U1, ISDF

U2, ISDF

U1, SDF-CDRT

U2, SDF-CDRT

U1, FR, Conv.

U2, FR, Conv.

U1, DF w/o DL

U2, DF w/o DL

U1, MUST

U2, MUST

(b) Outage probability versus relay location.

Fig. 5. The impact of the relay location on the outage probability.

achieved outage probability at U1 increases by increasing dsr,whereas the achieved outage probability at U2 first decreasesand then increases by increasing dsr. For the SDF-CDRTscheme, the achieved outage probabilities also first decreaseand then increase by increasing dsr. However, the variationsare not dramatic compared to those of the ISDF and SDF-CDRT schemes. Moreover, all the considered schemes withdirect links achieve the better outage performance over thoseof the “DF w/o DL” and MUST for the paired U1 and U2.

Considering the MUST scheme that pairs a far user anda near user in terms of their channel qualities, we comparethe outage probability achieved by the proposed ISDF andMUST schemes in Fig. 6. The target transmission rates for U1,U2, U3, and U4 are set as 1.2 bps/Hz, 2.4 bps/Hz, 2 bps/Hz,and 3 bps/Hz, respectively. The distances from the BS to thetwo near users (U3 and U4) are set by ds3 = ds4 = 0.5,as depicted in Fig. 5(a). The relay is located by dsr = 0.3referring to the results in Fig. 5(b). We assume that channelquality based pairing and QoS requirement based pairing areapplied in scheme 1 and scheme 2, respectively. That is, inscheme 1, U1 and U2 are paired and the ISDF is deployedwith the aid of the relay; U3 and U4 are paired in terms oftheir target transmission rates. In contrast, MUST is adoptedin scheme 2, i.e., the far user U1 is paired with the near user

0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

Out

age

prob

abili

ty

U1, scheme 1

U2, scheme 1

U3, scheme 1

U4, scheme 1

U1, scheme 2

U2, scheme 2

U3, scheme 2

U4, scheme 2

U1, MUST-RT

U2, MUST-RT

(a) Outage probability versus SNR.

0 5 10 15 20 25 300

0.5

1.0

1.5

2.0

2.5

3.0

SNR (dB)

Ave

rage

thro

ughp

ut (

bps/

Hz)

Ana, U1,scheme 1

Sim, U2,scheme 1

Ana, U3,scheme 1

Sim, U4,scheme 1

Ana, U1,scheme 2

Sim, U3,scheme 2

Ana, U2,scheme 2

Sim, U4,scheme 2

(b) Average throughput versus SNR.

Fig. 6. Performance comparison between the ISDF and MUST schemes.

U3, meanwhile U2 is paired with U4. For the two users in eachpair, we assume that α1 = 0.8 and α2 = 0.2 are applied forthe power allocation.

It is shown in Fig. 6(a) that scheme 1 achieves lower outageprobabilities for U1, U2, and U4 than those of scheme 2. Onlyfor U3 do the two schemes achieve the same outage probabili-ties. When scheme 2 is applied, the outage probabilities of bothU2 and U4 are both equal to one, which indicates that MUSTcannot support a higher rate transmission for the paired U2 andU4. However, scheme 1 achieves a lower outage probabilityfor U2 than those of U3 and U4 in the middle and high SNRregions, even if the far user U2 has a relatively higher targettransmission rate. Note that scheme 1 degenerates to MUSTwith retransmission (MUST-RT) when dsr = 0 regardingTable II. The outage probabilities achieved by MUST-RT forthe paired U1 and U2 are also presented in Fig. 6(a). It isshown that MUST-RT achieves larger outage probabilities forthe paired U1 and U2 than those of scheme 1. Under the samesimulation parameters, the average throughputs achieved byscheme 1 and scheme 2 are plotted in Fig. 6(b). The results inFig. 6(b) clearly show that scheme 1 achieves non-zero averagethroughputs for all the users in the whole SNR region, whereasscheme 2 achieves zero throughputs for the paired U2 and U4.For the far user U2 with a relatively higher target transmission

13

rate, scheme 1 achieves a remarkable average throughput in thewhole SNR region. The similar results are also observed forU4. Although scheme 2 achieves a larger average throughputfor U1 in the middle SNR region than scheme 1, obviouslyscheme 1 achieves a much larger sum rate than scheme 2.Therefore, the proposed ISDF scheme provides better rateperformance than the MUST scheme, especially when the faruser has demanding QoS requirements.

VI. CONCLUSIONS

In this paper, we have investigated the FR, SDF-CDRT,and ISDF schemes for the cooperative NOMA system withnon-negligible direct links from the BS to two far users. Toreduce error propagation resulted from the incorrect detectionsat the relay, the SDF-CDRT and ISDF schemes take intoaccount the received SNRs at the relay and operate in the DFmode adaptively. When the relay can detect its received signalcorrectly, the SDF-CDRT can avoid SIC at the receiver sideby forming an orthogonal transmission branch with respect tothe direct link. For the ISDF scheme, unnecessary relayingcan be reduced by exploiting the limited feedback of thereceived SNR events from the two users. For the consideredthree cooperative relaying schemes, the outage probabilitieshave been derived in closed-form for the two users along withthe asymptotic expressions for the outage probabilities anddiversity orders. Simulation results have verified the superioroutage performance achieved by the proposed cooperativerelaying schemes over those of the cooperative DF relayingwithout direct links and MUST without relaying.

APPENDIX A: A DERIVATION FOR B1 AND B1

In this part, we evaluate the terms B1 and B2. The termB1(ε1) can be rewritten as

B1(ε1) = Pr

{Y <

2α1α2γth,1ρ

}, (A.1)

where the RV Y is defined as

Y , 2Y1Y2

Y1 + Y2(A.2)

with Y1 , α2|hs1|2 and Y2 , α1|hr1|2 following i.n.i.d.exponential distributions. Since that Y is the harmonic meanof two i.n.i.d exponential RVs, the cumulative distributionfunction (CDF) of Y can be expressed as [42]

FY (y) = 1− y√α1α2βs1βr1

e− y

2

(1

α2βr1+ 1

α1βs1

)

×K1

y√α1α2βs1βr1

, (A.3)

where K1(·) is the first order modified Bessel function of thesecond kind [43, Eq. (8.432)]. With the obtained (A.3), B1(ε1)can be evaluated as

B1(ε1) = 1− 2γth,1ρ

√α1α2

βs1βr1

e−(

α2βr1

+α1βs1

)γth,1

ρ

×K1

(2γth,1ρ

√α1α2

βs1βr1

). (A.4)

Similarly, we can evaluate B2(ε1) as

B2(ε1) = 1− 2γth,2ρ

√α1α2

βs2βr2

e−(

α2βr2

+α1βs2

)γth,2

ρ

×K1

(2γth,2ρ

√α1α2

βs2βr2

). (A.5)

For B1(ε2) and B1(ε3), it can be easily shown that B1(ε2) =

B1(ε3) = 1 − e−ξ1βs1 . For B2(ε2) and B2(ε3), we have

B2(ε2) = B2(ε3), which can be evaluated by

B2(εi) = Pr{|hs2|2 < ξ1}+ Pr{|hs2|2 ≥ ξ1, |hs2|2 < ξ2}

= 1− e−ξ1βs2 +

[e−

ξ1βs2 − e−

ξ2βs2

]+. (A.6)

APPENDIX B: A PROOF OF THEOREM 3

The end-to-end received SNR events at U1 and U2 can besummarized in Table VI. Based on the results in Table VI,the undetermined SNR events at U1 can be classified into twotypes, i.e., {(33), ε2} and {(36), ε2}. Regarding {(33), ε2} and{(36), ε2}, we define

P1 , Pr{

ρ(α1|hs1|2+|hr1|2)α2ρ|hs1|2+1 < γth,1, ε2

}and (B.1)

P2 , Pr{

α1ρ(|hs1|2+|hr1|2)α2ρ(|hs1|2+|hr1|2)+1

< γth,1, ε2

}, (B.2)

respectively. Since that all the channel gains follow exponen-tial distributions, P1 and P2 can be respectively evaluated as

P1 = Pr{|hs1|2 < ξ1

(1− ρ|hr1|2

γth,1

), |hs1|2 < ξ1

}=

βs1γth,1

(1−e

− ξ1βs1

)−βr1ρξ1

(1−e

−γth,1

βr1ρ)

βs1γth,1−βr1ρξ1, βs1 = ρβr1ξ1

γth,1

γ(2, ξ1

βs1

), βs1 = ρβr1ξ1

γth,1

(B.3)

and

P2 = Pr{|hs1|2 + |hr1|2 < ξ1, |hs1|2 < ξ1

}=

βr1

(1−e

− ξ1βr1

)−βs1

(1−e

− ξ1βs1

)βr1−βs1

, βs1 = βr1

γ(2, ξ1

βs1

), βs1 = βr1.

(B.4)

Then, regarding Table VII, the outage probability in (58) canbe expressed as

P isdfout,1 = P1

(Pr{ε1}Pr{ε1}+ Pr{ε2}Pr{ε1}

+Pr{ε2}Pr{ε2}+ Pr{ε2}Pr{ε3})

+P2

(Pr{ε1}Pr{ε2}+ Pr{ε1}Pr{ε3}}

)+Pr{ε3}

(Pr{ε2}Pr{ε1}+ Pr{ε2}Pr{ε2}

+Pr{ε2}Pr{ε3})

= P1

(Pr{ε1}Pr{ε1}+ Pr{ε2}

)+P2 Pr{ε1}

(1− Pr{ε1}

)+ Pr{ε3}Pr{ε2}, (B.5)

which proves the case for U1.According to Table VII, the undetermined SNR events

at U2 can be classified into four types, i.e., {(29), ε2},

14

TABLE VIIRECEIVED SNR EVENTS IN THE ISDF SCHEME

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{(31), (32), ε3}, {(32), ε2}, and {(38), (39), ε3}. With respectto these four types of the undetermined SNR events, we define

P1 , Pr{ρ(α2|hs2|2 + |hr2|2) < γth,2, ε2}, (B.6)

P2 , Pr{

α1ρ(|hs2|2+|hr2|2)α2ρ(|hs2|2+|hr2|2)+1

< γth,1, ε3

}+Pr

{α1ρ(|hs2|2+|hr2|2)

α2ρ(|hs2|2+|hr2|2)+1≥ γth,1,

α2ρ(|hs2|2 + |hr2|2) < γth,2, ε3

}, (B.7)

P3 , Pr{α2ρ(|hs2|2 + |hr2|2) < γth,2, ε2}, and (B.8)

P4 , Pr{

ρ(α1|hs2|2+|hr2|2)α2ρ|hs2|2+1 < γth,1, ε3

}+Pr

{ρ(α1|hs2|2+|hr2|2)

α2ρ|hs2|2+1 ≥ γth,1,

α2ρ|hs2|2 < γth,2, ε3

}, (B.9)

respectively. Taking into that all the channel gains follow ex-ponential distributions, P1, P2, P3, and P4 can be respectivelyevaluated as those in Table VI. As such, the outage probabilityat U2 can be expressed as (66).

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Hongwu Liu received the Ph.D. degree from South-west Jiaotong University in 2008. From 2008 to2010, he was with Nanchang Hangkong University.From 2010 to 2011, he was a Post-Doctoral Fellowwith the Shanghai Institute of Microsystem andInformation Technology, Chinese Academy of Sci-ence. From 2011 to 2013, he was a Research Fellowwith the UWB Wireless Communications ResearchCenter, Inha University, South Korea. Since 2014,he has been an Associate Professor with ShandongJiaotong University. From 2017 to 2018, he was a

Research Professor with the Department of Information and CommunicationEngineering, Inha University. His research interests include MIMO signalprocessing, cognitive radios, cooperative communications, wireless secrecycommunications, and 5G networks.

Zhiguo Ding (S’03-M’05) received his B.Eng inElectrical Engineering from the Beijing Universityof Posts and Telecommunications in 2000, and thePh.D. degree in Electrical Engineering from Impe-rial College London in 2005. From Jul. 2005 toApr. 2018, he was working in Queen’s UniversityBelfast, Imperial College, Newcastle University andLancaster University. Since Apr. 2018, he has beenwith the University of Manchester as a Professorin Communications. From Oct. 2012 to Sept. 2018,he has also been an academic visitor in Princeton

University.Dr. Ding’s research interests are 5G networks, game theory, cooperative and

energy harvesting networks and statistical signal processing. He is serving asan Editor for IEEE Transactions on Communications, IEEE Transactions onVehicular Technology, and Journal of Wireless Communications and MobileComputing, and was an Editor for IEEE Wireless Communication Letters,IEEE Communication Letters from 2013 to 2016. He received the best paperaward in IET ICWMC-2009 and IEEE WCSP-2014, the EU Marie CurieFellowship 2012-2014, the Top IEEE TVT Editor 2017, IEEE Heinrich HertzAward 2018 and the IEEE Jack Neubauer Memorial Award 2018.

16

Kyeong Jin Kim (SM’11) received the M.S. degreefrom the Korea Advanced Institute of Science andTechnology (KAIST) in 1991 and the M.S. andPh.D. degrees in electrical and computer engineeringfrom the University of California, Santa Barbara,CA, USA, in 2000. From 1991 to 1995, he was aResearch Engineer with the Video Research Center,Daewoo Electronics, Ltd., Korea. In 1997, he joinedthe Data Transmission and Networking Laboratory,University of California, Santa Barbara. After re-ceiving his degrees, he joined the Nokia Research

Center and Nokia Inc., Dallas, TX, USA, as a Senior Research Engineer,where he was an L1 Specialist, from 2005 to 2009. From 2010 to 2011,he was an Invited Professor at Inha University, Korea. Since 2012, he hasbeen a Senior Principal Research Staff with the Mitsubishi Electric ResearchLaboratories, Cambridge, MA, USA. His research include transceiver design,resource management, scheduling in the cooperative wireless communicationssystem, cooperative spectrum sharing system, physical layer secrecy system,and device-to-device communications.

Dr. Kim served as an Editor of the IEEE COMMUNICATIONS LETTERSand INTERNATIONAL JOURNAL OF ANTENNAS AND PROPAGATION. Healso served as a Guest Editor of the EURASIP JOURNAL ON WIRELESSCOMMUNICATIONS AND NETWORKING: SPECIAL ISSUE ON COOPERATIVECOGNITIVE NETWORKS and IET COMMUNICATIONS: SPECIAL ISSUE ONSECURE PHYSICAL LAYER COMMUNICATIONS. He currently serves as anEditor of the IEEE TRANSACTIONS ON COMMUNICATIONS and a leadingguest Editor of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICA-TIONS: SPECIAL ISSUE ON HARDWARE FRIENDLY SPATIAL MODULATIONIN EMERGING WIRELESS SYSTEMS.

Kyung Sup Kwak received the Ph.D. degree fromthe University of California at San Diego andworked for Hughes Network Systems, and IBM Net-work Analysis Center, USA. Since then he has beenwith the School of Information and CommunicationEngineering, Inha University, Korea as a profes-sor, and served the dean of the Graduate Schoolof Information Technology and Telecommunicationsand the director of UWB Wireless CommunicationsResearch Center, a IT research center, Korea since2003. In 2006, he served as the president of Korean

Institute of Communication Sciences (KICS), and in 2009, the president ofKorea Institute of Intelligent Transport Systems (KITS). In 1993, he receivedEngineering College Achievement Award from Inha University, and a serviceaward from the Institute of Electronics Engineers of Korea (IEEK), in 1996and 1999 he received distinguished service awards from the KICS. He receivedthe LG Paper Award in 1998, and Motorola Paper Award in 2000. Hereceived official commendations for UWB radio technology research anddevelopment from Minister of Information & Communication, Prime Minister,and President of Korea in 2005, 2006, and 2009, respectively. In 2007, hereceived Haedong Paper Award and in 2009, Haedong Scientific Award ofresearch achievement. In 2008, he was elected for Inha Fellow Professor (IFP)and now for Inha Hanlim Fellow Professor. He published more than 200 peer-reviewed journal papers and served as TPC and Track chairs/organizing chairsfor several IEEE related conferences. His research interests include multipleaccess communication systems, mobile & UWB radio systems, future IoT,Wireless Body Area Network: Nano Network and Molecular Communications.

H. Vincent Poor (S’72-M’77-SM’82-F’87) receivedthe Ph.D. degree in EECS from Princeton Universityin 1977. From 1977 until 1990, he was on the facultyof the University of Illinois at Urbana-Champaign.Since 1990 he has been on the faculty at Princeton,where he is currently the Michael Henry Strater U-niversity Professor of Electrical Engineering. During2006 to 2016, he served as Dean of Princeton’sSchool of Engineering and Applied Science. Hehas also held visiting appointments at several otheruniversities, including most recently at Berkeley and

Cambridge. His research interests are in the areas of information theoryand signal processing, and their applications in wireless networks, energysystems and related fields. Among his publications in these areas is the recentbook Information Theoretic Security and Privacy of Information Systems(Cambridge University Press, 2017).

Dr. Poor is a member of the National Academy of Engineering and theNational Academy of Sciences, and is a foreign member of the ChineseAcademy of Sciences, the Royal Society, and other national and internationalacademies. He received the Marconi and Armstrong Awards of the IEEECommunications Society in 2007 and 2009, respectively. Recent recognitionof his work includes the 2017 IEEE Alexander Graham Bell Medal, HonoraryProfessorships at Peking University and Tsinghua University, both conferredin 2017, and a D.Sc. honoris causa from Syracuse University also awardedin 2017.