06775012(1)

1178 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 4, APRIL 2014

Rate Adaptation and Power Allocation for CognitiveRadio Networks with HARQ-Based Primary System

Moon-Gun Song, Young-Jin Kim, Eun-Yeong Park, and Gi-Hong Im, Senior Member, IEEE

Abstract—In this paper, we propose a protocol for the co-existence of primary and secondary systems over block-fadingchannels. In the protocol, the primary system employs a hybridautomatic repeat request (HARQ). When the primary systemretransmits the data signal, the secondary system serves as a relayfor the retransmission of the primary system and simultaneouslytransmits its data signal. To efficiently accomplish the protocol,we analyze the average throughput of the primary and secondarysystems by using the long-term average throughput (LAT). Weformulate an optimization problem to maximize the LAT of thesecondary system. The constraint of the optimization problemis that the LAT of the primary system with secondary systemis not less than that of the primary system alone. Through theoptimization problem, we obtain the closed-form solutions of thetransmission rate of the secondary system and the fraction ofthe transmit power for relaying the primary system’s data signaland transmitting the secondary system’s data signal. Numericalresults show that the primary system does not lose the averagethroughput, and rather achieves an additional throughput gainby adjusting the fraction of the transmit power of the secondarysystem.

Index Terms—Cognitive radio, cooperative communication,hybrid automatic repeat request, outage probability, long-termaverage throughput.

I. INTRODUCTION

COGNITIVE radio (CR) has attracted considerable atten-tion because of its potential to efficiently utilize scarce

radio spectrum resources [1]–[7]. In a CR-based network, anunlicensed system, referred to as a secondary system, canshare the licensed bands dedicated to a primary (licensed)system. There are three basic operational models used to im-plement the CR networks: overlay, underlay, and cooperationmodels [8]. In the overlay model, the secondary system sensesthe temporal spectrum holes in the licensed bands, and thenuses these holes to avoid interference to the primary system.Although this approach can improve the spectral efficiency,the improvement depends on the accuracy of spectrum sensing[2], [3]. In the underlay model, the secondary system coexistswith the primary system and can simultaneously use thelicensed bands while the primary system occupies the bands.In this case, the secondary system works under the interference

Manuscript received February 18, 2013; revised July 30, 2013 and February12, 2014. The editor coordinating the review of this paper and approving itfor publication was L. K. Rasmussen.

This work was supported by the Basic Science Research Program throughthe National Research Foundation of Korea (NRF), funded by the Ministryof Science, ICT and Future Planning (No. 2011-0013807).

The authors are with the Department of Electrical Engineering, PohangUniversity of Science and Technology (POSTECH), Pohang 790-784, Korea(e-mail: {knights, yjcom, pey0531, igh}@postech.ac.kr).

Digital Object Identifier 10.1109/TCOMM.2014.021714.130140

constraint of the primary system to preserve the quality-of-service (QoS) requirement of the primary system [5]. Inthe cooperation model, the secondary system cooperates withthe primary system and improves the QoS of the primarysystem by using cooperative diversity. As compensation forcooperation, the secondary system can access the licensedbands of the primary system [6], [7].

In wireless communication systems, the automatic repeatrequest (ARQ) and hybrid ARQ (HARQ) have been consid-ered to be principal techniques to provide high reliability tousers. The ARQ is a simple retransmission technique basedon acknowledgment (ACK) for reliability of transmission. TheHARQ is a retransmission technique that employs forward-error-correction (FEC) code to improve reliability [28]. Thereare two widely-used HARQ schemes inducing the buffer tostore and combine the received information: HARQ withchase combining (HARQ-CC) and HARQ with incrementalredundancy (HARQ-IR) [9]. If a destination fails to decodethe coded data packet of the source, the destination requestsretransmission through a non-acknowledgment (NACK) sig-nal. In the HARQ-CC scheme, the source sends the samecoded data packet for (re)transmissions. The destination storesthe coded data packets for the (re)transmission and combinesthem before decoding. In the HARQ-IR scheme, the sourcetransmits different parity of the coded data packet for each(re)transmission, and the destination accumulates the pari-ties of the coded data packet during the (re)transmission.HARQ-IR generally achieves higher performance and datarates than HARQ-CC [10]. There are several prior studiesrelated to the outage probability and throughput analysisof wireless communication systems with HARQ [10]–[13].With the quasi-static channel assumption, [11] investigated thedelay-limited throughput and scheduling optimization of thedownlink transmission in a direct link. The authors of [12]derived a closed-form expression of the outage probability forthe decode-and-forward (DF) cooperative system with HARQ-IR. On the other hand, utilizing the block-fading channelassumption, the authors of [10] analyzed the long-term averagethroughput (LAT) with a limited number of retransmissions. In[13], the outage probability of a DF cooperative relay schemewith HARQ-IR was derived, and the diversity gain caused bythe cooperation and HARQ-IR was described with a high SNRassumption over Rayleigh fading channels.

Recently, studies have been conducted to investigate theperformance of the CR network with a retransmission-basedor HARQ-based primary system [8], [16]–[19], [24]–[27].The authors of [16] proposed a framework where the sec-ondary system shares the licensed band of the primary system

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SONG et al.: RATE ADAPTATION AND POWER ALLOCATION FOR COGNITIVE RADIO NETWORKS WITH HARQ-BASED PRIMARY SYSTEM 1179

and ensures a specified target rate for the primary systemby estimating the throughput loss from the ARQ feedbackthrough an information-theoretic approach. In [17], a trans-mission policy of the secondary system was proposed tomaximize its throughput under a limited throughput loss ofthe retransmission-based primary system. In [8], under boththe peak and the average transmit power constraints in thesecondary system, the channel average rate of the secondarysystem was analyzed with a constraint on the interferencepower to the HARQ-based primary system when the chan-nel state information (CSI) of the interference channel wasperfectly or imperfectly available at the secondary system.The authors of [24] analyzed the throughput of the secondarysystem with repetition time diversity and HARQ-IR protocolsunder the interference constraint to the primary system wherethe secondary system is provided with perfect CSI or imperfectCSI. The authors of [25] solved the optimization problemof joint power allocation and relay selection for a cluster-based multi-hop CR network to minimize the total powerconsumption. The authors of [18] proposed a cooperate-and-access spectrum-sharing protocol. In this protocol, if thesecondary system detects the NACK signal from the primarydestination (PD), the secondary source (SS) serves as a relayfor the primary system and provides cooperative diversity.As a reward for cooperative diversity, the secondary systemaccesses the spectrum band. The authors of [27] proposedthe backward interference cancellation method in which thesecondary system buffers the outage data signal caused by theinterference from the primary system until the data signal ofthe primary system is available at the secondary system andthe interference can be canceled. The authors of [19] proposeda protocol where the secondary system operates as a relay forthe retransmission of the primary system, and simultaneouslyuses the spectrum band to transmit its data signal by exploitingsuperposition coding. However, in [8], [17], [24], [25] and[27], the coexistence of the primary and secondary systemscauses a throughput loss in the primary system. In [18], thesecondary system corrupts the ACK signal of the PD, whichcauses a latency increase in the primary system. Furthermore,the abovementioned studies considered quasi-static channelsthat are not changed during the retransmission process. If thechannels suffer from block-fading, the quasi-static channelmodels are not available and instantaneous CSI is not per-mitted for the SS. The authors of [26] proposed the spectrum-leasing strategy with the fading temporal correlation and theretransmission by exploiting locally available information.However, the primary system should recognize the existenceof the secondary system and adjust the frame structure forcooperation.

In this paper, we consider the coexistence of the HARQ-based primary and secondary systems over block-fading chan-nels. Because the SS cannot obtain the instantaneous CSI, wepropose the coexistence protocol based on Alamouti coding[20] rather than the superposition coding of [19]. In theprotocol, if the PD fails to decode the data signal of theprimary source (PS), the SS transmits both its data signal andthe data signal of the PS, which was decoded in the initialtransmission of the PS. We also consider the optimizationproblem to find the optimal transmission rate and the fraction

of the transmit power for the PD and the secondary destination(SD) to maximize the average throughput of the secondarysystem, subject to the constraint that the average throughput ofthe primary system with the secondary system is not less thanthat of the primary system alone. To solve the optimizationproblem, we asymptotically derive the outage probabilities ofthe primary and secondary systems using only the statisticsof the channels. On the basis of the outage probabilities, theaverage throughput of the primary and secondary systems isanalyzed through their LATs, which are widely employed tomeasure the throughput performance of a system with HARQ[10], [22]. From the LATs of the primary and secondarysystems, the closed-form solutions are derived with a simpleapproximation approach introduced in [23]. By adjusting thefraction of the transmit power, the secondary system canprovide additional throughput gain to the primary system.

A. Related Works

In [30], the secondary system coexists with the ARQ-based primary system. The authors considered a block-fadingscenario, but each fading block contains more than twotransmission time slots. The secondary system overhears theACK/NACK signals of the primary to evaluate the channelquality of the primary system. As the channel quality ishigh, the secondary system have more opportunity to use thespectrum resource. Although the secondary system takes intoaccount the channel quality of the primary system, the inter-ference from the secondary system affects the performance ofthe primary system. In [19], if the initial transmission of theprimary system fails and the secondary system successfullydecodes the data of the primary system, the secondary systemuses the spectrum resource of the primary system. Both theunderlay transmission scheme without CSI at the transmitter(CSIT) and the overlay transmission scheme with CSIT areproposed by using superposition coding in quasi-static fadingchannels where the data signal experiences the same channelgain during (re)transmissions. The authors of [31] extendedthe work of [19], considering multiple relays for the secondarysystem. The outage probability and the average throughput ofthe primary and secondary systems are analyzed in quasi-staticfading channels. Due to the interference, the primary systemaccumulates different mutual information for retransmissionwhether the secondary system and its relays are active or not.Hence, the average throughput of the primary system withsecondary system is lower than that of the primary systemwithout secondary system. We extend the overlay scheme of[19] for block-fading channels. In block-fading channels, thechannel gain is static for each data block, but it independentlychanges for retransmission of the data block. We also employAlamouti coding rather than superposition coding, and thusthe proposed scheme does not require CSIT. The authorsof [29] proposed the overlay transmission scheme where theprimary system without retransmission process coexists withthe secondary system employing two transmit antennas toexploit Alamouti coding. The protocol consists of two phases,and all channels are static during the two phases. In the phase1, the secondary system transmits its own data through oneantenna and receives the data of the primary system through


PS PD

SS

SD

PS PD

SS

SD

,2pp dsh

,2sp dsh

,1pp dsh

,1sp dsh

,1sp ssh

,2ss dsh

,2ps dsh

Fig. 1. Protocol of the cognitive radio network with HARQ-based primarysystem.

the other antenna. In the phase 2, if the secondary systemsuccessfully decodes the data of the primary system for thephase 1, the secondary system transmits the Alamouti pair ofits own data and the data of the primary system, i.e., Alamouticoding is applied at block level. In our scheme, Alamouticoding is applied at symbol level by the data of the primarysystem during initial transmission and retransmission. Weexploits Alamouti coding to maintain the average throughputof the primary system.

B. Organization

The remainder of this paper is organized as follows. InSection II, we describe the protocol and the system modelof the CR network with the HARQ-based primary system.In Section III, we derive the LATs of the primary and thesecondary systems. In Section IV, we formulate and solve anoptimization problem based on the LATs to maximize the LATof the secondary system. Section V provides the numericalresults and Section VI presents the conclusions of this study.

II. SYSTEM MODEL AND PROBLEM STATEMENT

A. Protocol Description

We consider a cognitive network in which a secondarysystem coexists with a HARQ-based primary system overblock-fading channels. The network consists of a pair of PSand PD and a pair of SS and SD as shown in Fig. 1. Theprotocol works as follows.

• During initial transmission, the PS transmits a data signal;the PD, SS and SD receive it.

• If the PD fails to decode the data signal, it sends anNACK signal to report the failure.

• After the SS and SD overhear the NACK signal of the PD,the SD decodes the data signal of the PS for the initialtransmission. If decoding is successful, the SD sends anACK signal to the SS.

• If the SS successfully decodes the data signal, it is readyto cooperate with the primary system.

• During retransmission, while the PS transmits the ad-ditional parity of the data signal, the SS transmits theAlamouti coded parity of the PS data signal and its owndata signal simultaneously.

In the protocol, we assume that the SS perfectly estimates thestatistics of the channels of the SS-PD, SS-SD and PS-PDlinks from the reporting channels. We also assume that allnodes perfectly decode the ACK and NACK signals and thatthe ACK/NACK feedback delay is ignored. There are a varietyof channel estimation techniques [14], [15], and we assume

that all the receivers perfectly estimate the instantaneous CSIsof the channels corresponding to them. We also assume thatthe reporting signals are always successfully decoded.

B. Transmitted and Received Signals

All the links among the PS, PD, SS and SD are representedas normalized Rayleigh channels with block-fading, i.e., thechannel gains in the initial transmission and retransmissionare independent. Let hab,l be the channel gain of the A-B link in the lth (re)transmission. The probability densityfunction (PDF) and cumulative density function of |hab,l|2 arerespectively given by

P|hab,l|2 (t) = e−t, F|hab,l|2 (t) = 1− e−t. (1)

The PS encodes the information bits to codeword vector xwith a coding rate based on the feedback channel quality. Weassume that a continuous modulation scheme and a capacityachieving channel coding are used. Also, xp,l is the lth part ofcodeword x for the lth (re)transmission, and the total numberof (re)transmissions of the PS is limited to 2. When the PStransmits data signal vector xp,1 for the initial transmission,the PD and the pair of the SS and SD receive the signalvectors. The received signal vectors of the PD, SS and SDare respectively given by

ypd,1 =√P loss

pspdPphpspd,1xp,1 + npd,1, (2)

yss,1 =√P loss

psssPphpsss,1xp,1 + nss,1, (3)

ysd,1 =√P loss

pssdPphpssd,1xp,1 + nsd,1, (4)

where Pp denotes the transmit power of the PS, and P losspspd

,P loss

psssand P loss

pssdare the path losses of the PS-PD, PS-SS and

PS-SD links, respectively. Also, npd,1, nss,1 and nsd,1 are thecomplex additive white Gaussian noise (AWGN) vectors at thePD, SS and SD, respectively, and their elements are distributedas CN (0, 1). If the PD fails to decode the received signals,the PS transmits another part of codeword x.

In the retransmission process, the SS can cooperate withthe PS when both the SS and SD successfully decode the datasignal of the PS. The transmit signal of the SS is given by

xs,2 =√1− αx̃p,2 +

√αxs, (5)

where xs is the data signal for the SD, and x̃p,2 isthe Alamouti code pair of xp,2, defined as x̃p,2 =[−x2∗

p,2,x1∗p,2,−x4∗

p,2,x3∗p,2, · · · ], where xi

p,2 is the ith elementof xp,2 and (·)∗ is the complex conjugate operator [20], [21].We assume perfect synchronization between the primary andsecondary system for the distributed Alamouti code. We alsoassume that the PD recognizes the activity of the SS to decodethe data signal applied by the distributed Alamouti code. And,α is the fraction of the transmit power allocated to the datasignal for the SD. The received signal vector of the PD withcooperation in the retransmission is expressed as

ypd,2=√P loss

pspdPphpspd,2xp,2+

√P loss

sspdPshsspd,2xs,2+npd,2, (6)

where P losssspd

is the path loss of the SS-PD link, and npd,2 is thecomplex AWGN vector with CN (0, 1) in the retransmission.


CP(α,Rp) =bp

T P[Ipspd,1

≥Rp]+2T P

[Ipspd,1

<Rp] (P [Ipspd,1 ≥ Rp

]+(1− P

[Ipsss,1 ≥ Rp

]P[Ipssd,1 ≥ Rp

])P[Ipspd,1

< Rp ≤ Ipspd,1+ Ipspd,2

]+P[Ipsss,1 ≥ Rp

]P[Ipssd,1 ≥ Rp

]P

[Ipspd,1 < Rp ≤ Ipspd,1 + Iαpssspd,2

])=

Rp

1 + P outpspd,1

(Rp)

{(1−P out

pspd,2(Rp)

)+(1− P out

psss,1 (Rp))(

1−P outpssd,1 (Rp)

)(P out

pspd,2(Rp)−P out

pspd,2(α,Rp)

)}.

(13)

The SD also receives the data signal and interference from theSS and PS. The received signal vector of the SD is given by

ysd,2=√P loss

sssdPshsssd,2xs,2+

√P loss

pssdPphpssd,2xp,2+nsd,2, (7)

where P losssssd

is the path loss of the SS-SD link in the retransmis-sion, and npd,2 is the complex AWGN vector with CN (0, 1)at the SD.

If the SS or SD fails to decode the data signal of the PS,the SS cannot participate in the cooperation and the receivedsignal vector of the PD in the retransmission is given by

ypd,2 =√P loss

pspdPphpspd,2xp,2 + npd,2. (8)

C. Problem Statement

To efficiently design the protocol, we find the optimal αand transmission rate Rs of the SS to maximize the averagethroughput of the secondary system. Additionally, the averagethroughput of the primary system with the secondary systemshould be no less than that of the primary system alone.

III. LONG-TERM AVERAGE THROUGHPUT

To obtain the optimal α and the transmission rate of the SS,we should analyze the outage probability and the throughputperformance of the primary and secondary systems. LAT isdefined as CLAT � E[b]

E[T ] , where b is the number of successfullydecoded information bits and T is the number of channelsused. In this section, we derive the LAT of the primary andsecondary systems based on their outage probabilities.

A. Long-term Average Throughput of the Primary System

For comparison, we derive the LAT of the primary systemwhen the secondary system is absent. In the (re)transmission,the mutual information is accumulated on the basis of theHARQ-IR at the PD. The LAT of the primary system is givenby

CP,Conv(Rp)

=bp(P[Ipspd,1≥Rp

]+P[Ipspd,1<Rp≤Ipspd,1 + Ipspd,2

])T P[Ipspd,1

≥Rp]+2T P

[Ipspd,1

<Rp]

=Rp

(1− P out

pspd,2(Rp)

)1 + P out

pspd,1(Rp)

, (9)

where Ipspd,l is the mutual information of the PS-PD linkin the lth (re)transmission, which is defined as Ipspd,l =log2(1 + P loss

pspdPp|hpspd,l|2). Also, Rp is the transmission rate

of the PS. The data signal of the PS in each (re)transmission

contains bp bits corresponding to T channel symbols, thusRp =

bp

T . And, P outpspd,1

(Rp) denotes the outage probability ofthe PS-PD link with respect to transmission rate Rp in the nth(re)transmission. The outage probability is expressed as

P outpspd,n

(Rp) = P

[n∑

l=1

Ipspd,l < Rp

]. (10)

We now analyze the LAT of the primary system whenthe secondary system coexists with the primary system. Ifboth the SS and the SD succeed in decoding the data signalin the initial transmission of the PS, the SS transmits xs,2

and mutual information Iαpspd,2is accumulated at the PD. The

mutual information is expressed as

Iαpspd,2

= log2

(1+

P losspspd

Pp

∣∣hpspd,2

∣∣2 +(1−α)P losssspd

Ps

∣∣hsspd,2

∣∣21+αP loss

sspdPs

∣∣hsspd,2

∣∣2).

(11)

Thus, the outage probability of the PD in the retransmissionis given by

P outpspd,2

(α,Rp) = P

[Ipspd,1 + Iαpspd,2

< Rp

]. (12)

If the SS or the SD fails to decode the data signal in theinitial transmission of the PS, mutual information Ipspd,2 isaccumulated at the PD.

In the case of a primary system that coexists with asecondary system, cooperation is considered when the SS andthe SD successfully decode the data signal of the PS in theinitial transmission. The LAT of the primary system is givenby (13), where Ipsss,1 and Ipssd,1 are the mutual informationof the PS-SS and PS-SD links in the initial transmission ofthe PS, respectively, which are given by Ipsss,1 = log2(1 +P loss

psssPp|hpsss,1|2), and Ipssd,1 = log2(1+P loss

pssdPp|hpssd,l|2). And,

P outpsss,1(Rp) and P out

pssd,1(Rp) are the outage probabilities of theselinks, which are expressed as

P outpsss,1(Rp)=P

[Ipsss,1<Rp

], P out

pssd,1(Rp)=P[Ipssd,1<Rp

].

(14)

B. Long-term Average Throughput of the Secondary System

To derive the LAT of the secondary system, we assumethat the interference from the data signal of the PS is perfectlycanceled at the SD. This assumption is allowed in the proposedprotocol because the SS transmits when the SD succeedsin decoding the data signal of the PS during the initial


CS,t(α,Rs) =bs(P[Ipspd,1

< Rp]P[Ipsss,1 ≥ Rp

]P[Ipssd,1 ≥ Rp

]P[Iαsssd,2 ≥ Rs

])T P[Ipspd,1 ≥ Rp

]+ 2T P

[Ipspd,1 < Rp

]=RsP

outpspd,1

(Rp)(1− P out

psss,1 (Rp))(

1− P outpssd,1 (Rp)

) (1− P out

sssd,2 (α,Rs))

1 + P outpspd,1

(Rp). (18)

transmission [19]. Hence, the effective received signal vectorof the SD is given by

ysd,2 =√P loss

sssdPshsssd,2

√αxs + nsd,2. (15)

From the effective received signal, we can derive the LAT ofthe secondary system. In the case of the secondary system,because the SS does not always transmit its data signal,we consider two types of LATs. One is the LAT for thetransmission time of the SS, and the other is the LAT forthe total time duration. LAT CS,u of the secondary system forthe transmission time is simply expressed as

CS,u(α,Rs) =bsP[Iαsssd,2≥Rs

]T = Rs

(1−P out

sssd,2 (α,Rs))

=Rse− 2Rs −1

P losssssd

Psα , (16)

where Iαsssd,2 denotes the mutual information of the SS-SD link,and Iαsssd,2 = log2(1+ P loss

sssdPsα|hsssd,2|2). The data signal of

the SS for the SD contains bs bits corresponding to T channelsymbols, and Rs = bs

T . The outage probability P outsssd,2(α,Rs)

of the SS-SD link is given by

P outsssd,2 (α,Rs) = P

[Iαsssd,2 < Rs

]. (17)

In the case of the LAT for the total time duration, we considerthe transmission opportunity of the SS. When both the SS andSD decode the data signal of the PS in the initial transmission,the SS can participate in the transmission. Thus, LAT CS,t ofthe secondary system for the total time duration is given by(18).

IV. RATE ADAPTATION AND POWER ALLOCATION

In the previous section, we analyzed the LATs of theprimary and secondary systems. On the basis of these LATs,we now formulate the optimization problem to maximizethe LATs of the secondary system subject to the conditionthat the LAT of the primary system with the secondarysystem is not less than that of the primary system withoutthe secondary system. Because the primary system is notresponsible for maximizing the throughput of the secondarysystem, we assume that Rp is given for the QoS of the primarysystem during the optimization. To consider the transmissionopportunity of the secondary system, the objective of theoptimization problem should be based on CS,t. However, withRp given, we can discover that CS,t ∝ CS,u and the two LATshave the same optimal α and Rs. Thus, for simplicity weformulate the optimization problem based on CS,u as follows.

argmaxα,Rs

CS,u (α,Rs) , (19a)

s.t. Rs ≥ 0, 0 ≤ α ≤ 1, (19b)

CP(α,Rp) ≥ CP,Conv(Rp). (19c)

A. Constraint Simplification

With Rp given, constraint (19c) depends only on parameterα. We now simplify the constraint as inequalities of α.

The constraint is expressed as

CP(α,Rp)− CP,Conv(Rp)

=Rp

(1− P out

psss,1 (Rp))(

1− P outpssd,1 (Rp)

)1 + P out

pspd,1(Rp)

·(P out

pspd,2(Rp)− P out

pspd,2(α,Rp)

)≥ 0. (20)

From the Appendix A, (20) is simplified as

αl < α ≤ αu, (21)

where αl and αu are given by

αl =1

2Rp−C̄p,1

(1−

P losspspd

Pp

P losssspd

Ps

), (22)

αu =1

2Rp−C̄p,1

{1−

P losspspd

Pp

P losssspd

Ps

(1− e

2Rp−C̄p,1−1

P losspspd

Pp

)}, (23)

and the expected value C̄p,1 of Ipspd,1 is expressed as

C̄p,1 =e

1

P losspspd

Pp

ln 2

{E1

(1

P losspspd

Pp

)− E1

(2Rp

P losspspd

Pp

)

−Rpe− 2

Rp

P losspspd

Pp ln 2

}, (24)

from the Appendix B. The exponential integral function E1(x)

is defined as E1(x) =∫∞x

e−t

t dt.

B. Solving the Optimization Problem

The optimization problem is rewritten as

(α∗, R∗s ) = argmax

α,Rs

CS,u (α,Rs) , (25a)

s.t. Rs ≥ 0, 0 ≤ α ≤ 1, (25b)

αl < α ≤ αu, (25c)

where α∗ and R∗s are the optimal solutions of (25). Because

Rp > C̄p,1 and 1

2Rp−C̄p,1< 1, αl < 1. Hence, the set of αs

that satisfy (25b) and (25c) is not empty.The first order partial derivative of CS,u(α,Rs) with respect

to α is given by

∂

∂αCS,u (α,Rs) =

2Rs − 1

P losssssd

Psα2Rse

− 2Rs −1

P losssssd

Psα . (26)

Because ∂∂αCS,u(α,Rs) > 0 for all α > 0, CS,u(α,Rs) is

maximized when α is the upper bound value, i.e., α∗ isexpressed as

α∗ = min (1, αu) . (27)


The first and second order partial derivatives of CS,u(α,Rs)with respect to Rs are respectively obtained as

∂

∂RsCS,u (α,Rs) =

1

P losssssd

Psαe− 2Rs −1

P lossss sd

Psα

·(P loss

sssdPsα−Rs ln 2e

Rs ln 2), (28)

∂2

∂R2sCS,u (α,Rs) =

2Rs ln 2e− 2Rs −1

P losssssd

Psα(P loss

sssdPsα)2

·{(

Rs ln 2eRs ln 2 − P loss

sssdPsα)− P loss

sssdPsα (1 +Rs ln 2)

}.

(29)

There is one solution Rs =W (P loss

sssdPsα)

ln 2 for a given α suchthat ∂

∂RsCS,u(α,Rs) = 0, where W (t) denotes the Lambert

W function defined as W (t) = x such that t = xex. When

Rs =W (P loss

sssdPsα)

ln 2 , ∂2

∂R2sCS,u(α,Rs) < 0. Hence, R∗

s is obtainedas

R∗s =

W (P losssssd

Psα∗)

ln 2, (30)

for a given α∗.

C. Reward to the Primary System

To provide a reward to the primary system for resourcesharing, the constraint of α can be adjusted as

αl < α ≤ αu − αr, (31)

where αr is the reserve parameter for the reward to the primarysystem. For the feasibility of (25c), αr is limited to

0 ≤ αr < αu − αl. (32)

When αr = 0, the average throughput of the primary systemwith the secondary system is almost the same as that of theprimary system without the secondary system. If αr increases,the SS uses more transmit power to relay the data signal of theprimary system, and thus the primary system that coexists withthe secondary system can achieve a higher average throughputthan the primary system alone.

V. NUMERICAL RESULTS

There exist the primary and secondary systems that consistof PS, PD, SS, and SD. The number of (re)transmissions of thePS is limited to two. Simplified path loss models are used, i.e.,P loss

pspd= (d0/dpspd

)γ , P losspsss

= (d0/dpsss)γ , P loss

pssd= (d0/dpssd)

γ ,P loss

sssd= (d0/dsssd)

γ and P losssspd

= (d0/dsspd)γ , where the

reference distance d0 = 10 m and the path loss exponentγ = 3. Also, dpspd

, dpsss , dpssd , dsssd and dsspddenote the

distances of the PS-PD, PS-SS, PS-SD, SS-SD and SS-PDlinks, respectively. The distance of the PS-PD link is 35 m(dpspd

= 35 m). The distances of the SS-SD and SS-PD linksare 10 m (dsssd = dsspd

= 10 m), and the SS and SD arelocated 26.81 m from the PS (dpsss = dpssd = 26.81 m).The normalized block-fading Rayleigh channel is consideredfor each link as the small-scale channel model. We assumethat the transmit power of the SS is 30 dB (Ps = 30 dB)and the noise power of each receiver is 1. For comparison,a genie-aided scheme is considered where the SS knows the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Ave

rage

thro

ughp

ut(b

pcu)

Conventional ( Pp = 4 0 dB)Pr oposed (Pp = 40 dB)Conventional (Pp = 30 dB)Pr oposed (Pp = 30 dB)Conventional (Pp = 20 dB)Pr oposed (Pp = 20 dB)

α

feasible

Fig. 2. Average throughput of the primary systems of the conventional andproposed schemes versus α for Pp = 20, 30 and 40 dB when Rp = 2 bpcu.

instantaneous CSIs of all the links for the (re)transmission.For efficient evaluation of performance, throughput is definedas the transmission rate over the number of (re)transmissionsuntil the data signal is successfully decoded. If the decodingof the data signal fails for the retransmission, the throughputis zero.

We investigate the average throughput for the proposedprotocol where the HARQ-based primary system coexists withthe secondary system. In Fig. 2, we compare the averagethroughputs of the primary systems for the proposed schemeand the conventional scheme where the HARQ-based primarysystem alone exists for Pp = 20, 30 and 40 dB when Rp = 2bpcu. The cross point of the average throughput lines for theproposed scheme and the conventional scheme is the optimalα of (19), and the region below the cross point is the feasibleregion of (19c). As α increases, the average throughput of theprimary system for the proposed scheme decreases because thefraction of the transmit power for the cooperation declines andthe interference to the PD increases. Because the slope of theaverage throughput for the proposed scheme is more gradualas Ps increases, the mismatch between α∗ and the optimal αof (19) does not seriously affect the average throughput of theprimary system when Ps is high.

In Fig. 3, we describe solutions α∗ and R∗s of (25) versus

Pp when Rp = 2 bpcu. In the region of low Ps, α∗ and R∗s

are almost matched with the optimal α and Rs obtained byan exhaustive search. Although the mismatch exists in theregion of high Ps, the average throughput of the primary andsecondary systems with α∗ and R∗

s is almost the same asthat of the primary and secondary systems with the optimalα and Rs as shown in Fig. 4. Furthermore, α∗ and R∗

s areobtained from the closed-form solutions of (25) instead of anexhaustive search. When Rp = 2 bpcu, the average throughputof the secondary system declines as Pp increases. Note that theopportunity for the secondary system to access the spectrumband decreases because the outage probability of the primarysystem in the initial transmission decreases, and because α∗

and R∗s decreases. Compared to the genie-aided scheme, the

secondary system with α∗ and R∗s achieves approximately


20 22 24 26 28 30 32 34 36 38 400.2

0.3

0.4

0.5

0.6

0.7

Pp (dB)

20 22 24 26 28 30 32 34 36 38 405

5.4

5.8

6.2

6.6

7

Rs(b

pcu

)

- exhaustive- proposed

Rs - exhaustiveRs - proposed

αα

α

Fig. 3. Comparison of the α and Rs parameters obtained by the optimizationproblem and exhaustive search when Rp = 2 bpcu.

20 22 24 26 28 30 32 34 36 38 400

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Pp (dB)

Ave

rage

thro

ughput

(bpcu

)

Primary - conv., genie, exhaustive

Primary - proposed ( r = 0)

Primary - proposed ( r = 0.1)

Secondary - genie

Secondary - exhaustive

Secondary - proposed ( r = 0)

Secondary - proposed ( r = 0.1)

αα

αα

Fig. 4. Average throughput of the primary and secondary systems versus Ppwhen Rp = 2 bpcu.

90% of the average throughput of the genie-aided scheme.From Fig. 4, we also find out that the secondary system canprovide additional average throughput gains to the primarysystem by using αr in the region of low Pp when Rp = 2bpcu.

We now investigate the average throughput of the primaryand secondary systems versus Rp when Pp = 30 dB. Figure5 shows the average throughput of the primary systems forthe proposed scheme and the conventional scheme for Rp =1, 2 and 3 bpcu when Pp = 30 dB. The cross point of theaverage throughput lines is the optimal α of (19), and theregion below the cross point is the feasible region. When Rp

is high, the primary system requires more mutual informationaccumulated at the PD in the retransmission to succeed indecoding. Thus, the mismatch of α∗ and the optimal α of (19)causes a decrease in the average throughput of the primarysystem when Rp is high.

In Fig. 6, we compare α∗ and R∗s and the optimal α and

Rs obtained by an exhaustive search when Pp = 30 dB. Inthe region of low Rp, α∗ and R∗

s are almost matched withthe optimal α and Rs. On the other hand, in the region

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Ave

rage

thro

ughput

(bpcu

)

Conventional (Rp = 3 bpcu)

Proposed (Rp = 3 bpcu)





α

feasible

Fig. 5. Average throughput of the primary systems of the conventional andproposed schemes versus α for given Rp = 1, 2 and 3 bpcu when Pp = 30dB.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

Rp (bpcu)

0 0.5 1 1.5 2 2.5 3 3.5 44

4.8

5.6

6.4

7.2

8

Rs(b

pcu

)

- exhaustive- proposed

Rs - exhaustiveRs - proposed

αα

α

Fig. 6. Comparison of the α and Rs parameters obtained by the optimizationproblem and exhaustive search when Pp = 30 dB.

of high Rp, α∗ and R∗s are overestimated because of the

approximation. This causes the loss of the average throughputof the primary system for the proposed scheme as comparedwith the conventional scheme when Rp is high. However,as shown in Fig. 7, the loss of average throughput can bemitigated or the primary system can achieve the gain ofaverage throughput by using αr. As αr increases, the averagethroughput of primary system is enhanced when Pp = 30 dBand Rp = 4 bpcu. When αr > 0.01, the average throughputof the primary system of the proposed scheme is higher thanthat of the primary system of the conventional system. InFig. 8, although both α∗ and R∗

s decline as Rs increases, theaverage throughput of the secondary system increases. This isdue to the fact that the opportunity for the secondary systemto use the spectrum band increases because of an increasein the outage probability of the primary system in the initialtransmission. The average throughput of the secondary systemwith α∗ and R∗

s is approximately 90% of that of the genie-aided scheme when Pp = 30 dB.


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.5

1

1.5

2

2.5

3

r

Ave

rage

thro

ughput

(bpcu

)

Primary - conv., genie, exhaustivePrimary - proposedSecondary - genieSecondary - exhaustiveSecondary - proposed

α

Fig. 7. Average throughput of the primary and secondary systems versus αrwhen Pp = 30 dB.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

Rp (bpcu)

Ave

rage

thro

ughput

(bpcu

)

Primary - conv., genie, exhaustivePrimary - proposed ( r = 0)Primary - proposed ( r = 0.1)Secondary - genieSecondary - exhaustiveSecondary - proposed ( r = 0)Secondary - proposed ( r = 0.1)

αα

αα

Fig. 8. Average throughput of the primary and secondary systems versusRp when Pp = 30 dB.

VI. CONCLUSIONS

We proposed a protocol for the coexistence of the HARQ-based primary and secondary systems over block-fading chan-nels. In the protocol, if the PD fails to decode the data signalof the PS in the initial transmission, the SS serves as arelay for the primary system by using Alamouti coding, andsimultaneously transmits its data signal in the retransmissionof the primary system. We analyzed the average throughput ofthe primary and secondary systems through their LATs. Wealso obtained the closed-form solutions of the transmissionrate and the fraction of the transmit power of the SS. Theclosed-form solutions maximize the LAT of the secondarysystem subject to the constraint that the LAT of the primarysystem with the secondary system is not less than that of theprimary system alone. By adjusting the fraction of the trans-mit power, the secondary system can provide the additionalaverage throughput gain to the primary system.

APPENDIX A

(20) can be rewritten as

P outpspd,2

(Rp)− P outpspd,2

(α,Rp) ≥0. (33)

To simplify (33), we obtain the distributions of Ipspd,1, Ipspd,2

and Iαpspd,2. From the PDF and cumulative distribution function

(CDF) of |hpspd,l|2, the PDFs and CDFs of both Ipspd,1

andIpspd,2 are given by

PIpspd,l(t) =

ln 2

P losspspd

Pp2te

− 2t−1

P losspspd

Pp ,

FIpspd,l(t) = 1− e

− 2t−1

P losspspd

Pp , for l = 1, 2. (34)

With the assumption of P losssspd

Ps � 1, Iαpspd,2is asymptotically

given by

Iαpspd,2

≈ log2

(1+

P losspspd

Pp

∣∣hpspd,2

∣∣2 +(1−α)P losssspd

Ps

∣∣hsspd,2

∣∣2αP loss

sspdPs

∣∣hsspd,2

∣∣2)

= log2

(1

α+

1

α

P losspspd

Pp

P losssspd

Ps

∣∣hpspd,2

∣∣2∣∣hsspd,2

∣∣2). (35)

This assumption is reasonable when the SS is located close tothe PD. The CDF of Iαpspd,2

is expressed as

FIαpspd,2

(t) =P

[Iαpspd,2

< t]= P

⎡⎢⎣∣∣hpspd,2

∣∣2∣∣hsspd,2

∣∣2 <2t − 1

α

1α

P losspspd

Pp

P losssspd

Ps

⎤⎥⎦

=

∫ ∞

0

∫ 2t− 1α

1α

P losspspd

Pp

P losssspd

Ps

y

0

e−xe−ydxdy

=

∫ ∞

0

⎛⎜⎜⎝1− e

− 2t− 1α

1α

P losspspd

Pp

P losssspd

Ps

y⎞⎟⎟⎠ e−ydy

=1− 1

α

P losspspd

Pp

P losssspd

Ps

1

2t − 1α

(1− P loss

pspdPp

P losssspd

Ps

) . (36)

As shown in Fig. 9, the serious mismatch between the distri-bution obtained by simulation and the derived CDF of Iαpspd,2

exists at the tail part. This is because the domain of FIαpspd,2

(t)

is limited such that 2t > 1α (1−

P losspspd

Pp

P losssspd

Ps) due to the denominator.

Utilizing the PDFs and CDFs of Ipspd,1and Ipspd,2

, the firstterm of (33), P out

pspd,2(Rp), is calculated as

P outpspd,2

(Rp) =P[Ipspd,1 + Ipspd,2 < Rp

]=

∫ Rp

0

∫ Rp−x

0

PIpspd,1(x)PIpspd,2

(y)dydx

=

∫ Rp

0

FIpspd,2(Rp − x)PIpspd,1

(x) dx

=

∫ Rp

0

(1− e

− 2Rp−x−1

P losspspd

Pp

)PIpspd,1

(t) dx. (37)


1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ips pd,2

CD

F

Simulation (P lossps pd

Pp = 20 dB, P lossss pd

Ps = 20 dB)

Approx. (P lossps pd


Ps = 20 dB)



Ps = 30 dB)



Ps = 30 dB)



Ps = 40 dB)



Ps = 40 dB)

α

Fig. 9. CDF of mutual information Iαpspd,2when P loss

sspdPs = 20 dB, 30 dB

and 40 dB, and P losspspd

Pp = 20 dB.

The second term of (33), P outpspd,2

(α,Rp), is calculated as

P outpspd,2

(α,Rp)

=P

[Ipspd,1 + Iαpspd,2

< Rp

]=

∫ Rp

0

∫ Rp−x

0

PIpspd,1(x)PIα

pspd,2(y)dydx

=

∫ Rp

0

FIαpspd,2

(Rp − x)PIpspd,1(x) dx

=

∫ Rp

0

⎛⎜⎝1− 1

α

P losspspd

Pp

P losssspd

Ps

1

2Rp−x − 1α

(1− P loss

pspdPp

P losssspd

Ps

)⎞⎟⎠

· PIpspd,1(x) dx, (38)

where PIαpspd,2

(y) denotes the PDF of Iαpspd,2. Hence, (33) is

expressed as

P outpspd,2

(Rp)− P outpspd,2

(α,Rp)

=

∫ Rp

0

⎛⎜⎝ 1

α

P losspspd

Pp

P losssspd

Ps

1

2Rp−x− 1α

(1− P loss

pspdPp

P losssspd

Ps

)−e− 2

Rp−x−1

P losspspd

Pp

⎞⎟⎠

· PIpspd,1(x) dx. (39)

However, it is difficult to obtain the closed-form of (39).By using an approximation approach introduced in [23], wereplace the random variable Ipspd,1 with the expected valueover the fading as follows.

P outpspd,2

(Rp)− P outpspd,2

(α,Rp)

≈ 1

α

P losspspd

Pp

P losssspd

Ps

1

2Rp−C̄p,1− 1α

(1−P loss

pspdPp

P losssspd

Ps

)−e− 2

Rp−C̄p,1−1

P losspspd

Pp ≥0.

(40)

Thus, α is bounded by

α ≤ 1

2Rp−C̄p,1

{1−

P losspspd

Pp

P losssspd

Ps

(1− e

2Rp−C̄p,1−1

P losspspd

Pp

)}, (41)

where C̄p,1 denotes the expected value of Ipspd,1.

From the limitation of the domain of FIαpspd,2

(t), α isbounded by

α >1

2Rp−C̄p,1

(1−

P losspspd

Pp

P losssspd

Ps

). (42)

APPENDIX B

The expected value C̄p,1 is obtained as

C̄p,1 =

∫ Rp

0

tPIpspd,1(t) dt =

∫ Rp

0

tln 2

P losspspd

Pp2te

− 2t−1

P losspspd

Pp dt

=

∫ 2Rp

P losspspd

Pp

1

P losspspd

Pp

e1

P losspspd

Pp

ln 2e−x ln

(P loss

pspdPpx)dx

= −e1

P losspspd

Pp

ln 2e−x ln

(P loss

pspdPpx)∣∣∣∣∣∣

2Rp

P losspspd

Pp

1

P losspspd

Pp

+e

1

P losspspd

Pp

ln 2

∫ 2Rp

P losspspd

Pp

1

P losspspd

Pp

1

xe−xdx

=e

1

P losspspd

Pp

ln 2

{E1

(1

P losspspd

Pp

)− E1

(2Rp

P losspspd

Pp

)

−Rpe− 2

Rp

P losspspd

Pp ln 2

}, (43)

where E1(·) denotes the exponential integral function, whichis defined as E1(x) =

∫∞x

e−t

t dt.

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[28] D. Kim, Y. Choi, S. Jin, K. Han, and S. Choi, “A MAC/PHY cross-layer design for efficient ARQ protocols,” IEEE Commun. Lett., vol. 12,no. 12, pp. 909–911, Dec. 2008.

[29] V. A. Bohara, S. H. Ting, Y. Han, and A. Pandharipande, “Interference-free overlay cognitive radio network based on cooperative space timecoding,” in Proc. 2010 International Conference on Cognitive RadioOriented Wireless Networks and Communications, pp. 1–5.

[30] J. C. F. Li, W. Zhang, A. Nosratinia, and J. Yuan, “Opportunistic spec-trum sharing based on exploiting ARQ retransmission in cognitive radionetworks,” in Proc. 2010 IEEE Global Telecommunications Conference.

[31] S. B. Mafra, R. D. Souza, J. L. Rebelatto, E. MG Fernandez, andH. Alves, “Cooperative overlay secondary transmissions exploiting pri-mary retransmissions,” EURASIP J. Wireless Commun. and NetworkingJul. 2013.

Moon-Gun Song received the B.S. and M.S.degrees in electronic and electrical engineeringfrom Pohang University of Science and Technology(POSTECH), Gyeongbuk, Korea, in 2008 and 2010,respectively. He is currently working toward thePh.D. degree at Communications Research Labora-tory, POSTECH. His current research interests aredynamic spectrum access, cognitive radio systems,and MIMO systems.

Young-Jin Kim received the B.S. and Ph.D degreesin electronic and electrical engineering from PohangUniversity of Science and Technology (POSTECH),Gyeongbuk, Korea, in 2007 and 2013, respectively.He is currently a senior researcher at SamsungElectronics Co., Ltd., Suwon, Gyeonggi, Korea.His current research interests are MIMO systems,spectrum sharing, and cognitive radio networks.

Eun-Yeong Park received the B.S. degree in elec-tronic and electrical engineering from Pohang Uni-versity of Science and Technology (POSTECH),Gyeongbuk, Korea, in 2013. She is currently work-ing toward the Ph.D. degree at the CommunicationsResearch Laboratory, POSTECH. Her current re-search interests are cognitive radio systems.

Gi-Hong Im (M’87-SM’94) was with AT&T BellLaboratories, Holmdel, NJ, where he was respon-sible for the design and implementation of high-speed digital transmission systems for loop plant,local area network, and broadband access applica-tions (1990-1996). He authored or co-authored morethan twenty standards contributions to standardsorganizations such as ANSI T1E1.4, ETSI, IEEE802.9, ANSI X3T9.5, and the ATM Forum. Thesecontributions led to the adoption of three AT&Tproposals for new standards for high-speed LANs

and broadband access. In 1995, he was appointed as Distinguished Memberof Technical Staff at AT&T Bell Laboratories. Since 1996, he has been withPOSTECH as a professor. From 1996 to 2000, he was a Bell LaboratoriesTechnical Consultant. From 2002 to 2003, he was a visiting vice presidentof Samsung Electronics, where he worked on 4G wireless communicationsystems. His current research interests include signal processing and digitalcommunications with applications to high-speed digital transmission systems.

Dr. Im received the 1996 Leonard G. Abraham Prize Paper Award fromthe IEEE Communications Society for the paper entitled “Bandwidth-efficientdigital transmission over unshielded twisted pair wiring,” published in theIEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, the 2000 LGAward from LG Electronics, and the 2005 National Scientist Award from theKorean government. From 2004 to 2010, he served as an editor for the IEEETRANSACTIONS ON COMMUNICATIONS and IEEE COMMUNICATIONS LET-TERS. From 2010 to 2013, he served as a division editor for the Journal ofCommunications and Networks. He has published over 100 papers in IEEEJournals and Conferences, and has been granted 29 U.S. patents.

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