Cooperative Secure Resource Allocation in Cognitive Radio Networks with Guaranteed Secrecy Rate for Primary Users

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Cooperative Secure Resource Allocation inCognitive Radio Networks with

Guaranteed Secrecy Rate for Primary UsersNader Mokari, Saeedeh Parsaeefard, Hamid Saeedi, and Paeiz Azmi

Abstract—In this paper, we introduce a new cooperativeparadigm for secure communication in cognitive radio networks(CRNs) where secondary users (SUs) are allowed to accessthe spectrum of primary users (PUs) as long as they preservethe secure communication of PUs in the presence of maliciouseavesdroppers. To do so, the SU transmission is divided into twohops: at first hop, the SU transmitter sends the information toa relay set and the SU receiver acts as a friendly jammer todisturb the overhearing of eavesdroppers and at the second hop,one of the relays is selected to pass the information to the SUreceiver and the SU transmitter acts as a friendly jammer forthe PU. In this new setup, the time duration for each hop, thepower transmissions of all nodes in CRN, and relay selectionat the second hop are allocated in such a way that the secrecyrate of the SU is maximized subject to the minimum requiredPU’s secrecy rate. From primary service perspective, this trans-forms the possibly disturbing secondary service activities intoa beneficial network element. We investigate instantaneous andergodic resource allocation problems for perfect and imperfectchannel state information (CSI). Since these problems are non-convex, we propose a solution based on decomposition of mainoptimization problem into three subproblems related to thepower allocation, time allocation, and relay selection. We showthat the power allocation problem can be transformed into ageneralized geometric programming (GGP) model via the so-called scaled algorithm and it can be solved very efficiently.Simulation results indicate that in terms of the secondary secrecyrate, the proposed setup outperforms the conventional setup inwhich the secrecy rate of the PU is not guaranteed.

Index Terms—Cognitive radio networks, ergodic and instan-taneous resource allocation, generalized geometric programming(GPP), secure communication.

I. INTRODUCTION

SPECTRUM sharing through cognitive radio networks(CRNs) is a promising approach to increase the spectrum

efficiency for next generation of wireless communicationnetworks [1] where the unlicensed/secondary users (SUs) areallowed to access the spectrum of primary users (PUs) subjectto maintaining the quality of service (QoS) of PUs. Onecommon model in this context is the underlay approach where

Manuscript received May 22, 2013; revised September 16, 2013; acceptedNovember 5, 2013. The associate editor coordinating the review of this paperand approving it for publication was R. Zhang.

The authors are with the Department of Electrical and Computer En-gineering, Tarbiat Modares University, P. O. Box 14115-194, Tehran, Iran(e-mail: {nader.mokari, parsaeefard, hsaeedi, pazmi}@modares.ac.ir). Thecorresponding author is H. Saeedi.

This work was supported Iran Telecommunications Research Center (ITRC)under research grant T/500/19232-90/12/28.

Digital Object Identifier 10.1109/TWC.2013.010214.130929

the SUs can simultaneously utilize the licensed spectrum ofPUs if the resulting interference on the PUs’ receivers is keptunder a predefined threshold.

Similar to any wireless network, security against overhear-ing of the third parties, referred to as eavesdroppers, is one ofthe important issues in CRNs. Recently, physical layer securityintroduced by [2], is drawing a lot of attentions in whichthe objective is to maximize the secrecy rate defined as theachievable rate from the transmitter to the legitimate receiverminus the rate overheard by eavesdropper. Obviously, whenthe channel gain between transmitter and its correspondingreceiver is less than the channel gain between transmitter andeavesdropper, the secrecy rate is equal to zero.

For non-cognitive networks, achieving a non-zero secrecyrate is studied from different aspects in non-cooperative [3] aswell as cooperative frameworks including cooperative relaying[4], cooperative jamming [5], and jointly cooperative jammingand relaying [6]. Cooperative jamming, also known as friendlyjamming, creates interference by legitimate network nodes,transmitting noise [7], [8] or codewords [9], [10], so as toimpair the eavesdroppers ability to decode the confidentialinformation, and thus, increase secure communication ratesbetween each legitimate transmitter and receiver. This prob-lem in cognitive case has been considered in [8], [11]–[17].Information theoretic aspect of secrecy rate are addressed in[11], [12] where the effect of trustworthy SUs to increase thesecrecy rate of PU is investigated. Resource allocation (RA)problems to maximize the secondary secrecy rate underlayapproach in different MIMO transmission modes are investi-gated in [13]–[15]. A similar study in a cooperative relayingframework has been proposed in [16]. The effect of friendlyjammer in the underlay cognitive radio network was studiedin [8]. In [17], the secrecy rate of PU is maximized in MIMOchannels subject to the minimum required shannon rate of SU.

In RA problems associated to [13]- [17], the objectiveis to provide secure transmission for either primary or sec-ondary users subject to the imposed constraints by PUs andin particular, interference threshold constraint in underlayschemes. However, in CRNs, secure communication for bothPUs and SUs is of high importance and previously proposedsettings do not accommodate secure communications for bothprimary and secondary users. In this paper, we propose acooperative paradigm for secure communication in CRNs inwhich secure communications for both primary and secondaryservices are simultaneously provided. This goal is achieved by

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taking advantage of the interference caused by the secondaryuser activity to reduce the primary service overhearing bythe eavesdroppers. From primary service perspective, thistransforms the possibly disturbing secondary service activitiesinto a beneficial network element.

The RA problem for the proposed setup is written as anoptimization problem with the objective of maximizing theSU’s secrecy rate subject to guaranteeing a given PU’s secrecyrate. It can be seen that the feasibility set of this problemhighly depends on the channel gains, referred to as channelstate information (CSI) between network nodes, i.e., the PU,the SU and eavesdroppers, as well as the required primarysecrecy rate. Consequently, there is a good chance that theRA problem is not feasible meaning that the secondary secrecyrate is zero.

To make the problem feasible, we propose to expandthe feasibility set by deploying relays within the secondarynetwork. Then, at any primary service transmission period,the transmission of SU is done in two hops. In the first hop,the secondary transmitter (ST) sends the information to the setof relays and the secondary receiver (SR) acts as a friendlyjammer to interfere the overhearing of eavesdroppers. In thesecond hop, one of the relays is selected to transmit theinformation to the SR and the ST acts as a friendly jammer.This setup can be considered as a joint cooperative jammingand relaying scheme where the RA problem includes powerallocation of all nodes (i.e., the ST, the SR and relays), relayselection for the second hop, and time allocation for each hop.We show that the expansion of the feasibility set results inhigher chance of having a non-zero secondary secrecy ratewhile maintaining a given primary secrecy rate.

The proposed RA problem is non-convex and we apply thescaled algorithm in [18] to transform it into a convex one withrespect to each set of variables. We show that this transforma-tion can be represented as a generalized geometric programing(GGP) problem which can be solved very efficiently usingexisting approaches such as interior-point algorithms [19]. Weconsider two cases of RA problems: Instantaneous resourceallocation (IRA) and ergodic resource allocation (ERA). In theformer case, we assume the availability of perfect CSI betweenany transmitter and receiver within the network. Consequently,for each new set of CSI values, the IRA problem has to besolved [20]. In the latter case, allocations are made basedon the long term channel distribution information (CDI).Apparently, ERA exhibits a less computational complexitycompared to that of IRA. However, the drawback of ERAis that we can guarantee a secrecy rate for PUs only inaverage sense not instantaneously, meaning that there existsthe probability that the secrecy rate of PU is below thanits predefined threshold called outage probability of primarysecrecy rate. To deal with this issue, we introduce a modifiedERA problem where the outage probability of primary secrecyrate can be kept below any value of interest.

The last challenge for our setup is the assumption ofavailability of perfect values of CSI between different nodesof the network is not realistic, mainly due to the existence ofmalicious eavesdroppers which are not supposed to cooperatewith SUs and PUs to provide the CSI values. We approachthis challenge by considering imperfect values for CSI and

propose the robust counterparts of the RA problems. For theIRA problem, we apply the worst case robust optimization toguarantee the PU’s secrecy rate under any condition of error.For The ERA problem, we show that the marginal channeldistribution can be used to tackle the uncertainty [21], [22].

An important aspect of the proposed paradigm is that replac-ing the conventional interference threshold constraint by theprimary secrecy rate constraint not only does not decrease thesecondary secrecy rate with respect to the conventional case,but can also provide significantly higher secondary secrecyrate.

The rest of this paper is organized as follows. In Section II,the system model is discussed in details. In Section III, the RAproblem is introduced and the solution of IRA is presented.Section IV includes two cases of ERA followed by SectionV, where the imperfect CSI is considered for both IRA andERA. Section IV provides simulation results and Section IIVconcludes the paper.

II. NETWORK SETUP

A. System Model

We consider an interference limited CRN in which thereexist a primary network with single transmitter and receiver, atrustworthy secondary network, and a set of eavesdroppingmalicious nods i.e., E = {1, · · · , E}, which attempt tooverhear the primary and secondary messages. The primarytransmitter (PT) wants to send confidential data to its cor-responding receiver in its own available spectrum B. Theprimary network allows the ST to access its spectrum as longas the secrecy rate between PT and primary receiver (PR) ishigher than a predefined threshold donated by CPT→PR

min .In our system model, we assume decode and forward (DF)

relaying strategy where the relay nodes are assumed to operatein half-duplex mode, i.e., they do not transmit and receivesimultaneously in the same frequency band. Accordingly, thetransmission between the secondary transmitter and receiveroccurs in two hops: in the first hop, the ST transmits data tothe selected relay node; and in the second hop, the selectedrelay node sends data to the SR.

The secondary transmitter enjoys this opportunity to trans-mit messages securely to the secondary receiver, where thesecondary network consists of a ST and its corresponding SR,and a set of intermediate nodes i.e., R = {1, · · · , R}.

The intermediate nodes help the ST to transmit the datainto the SR as a relay set, as shown in Fig. 1. Accordingly,the transmission between the ST and the SR occurs in twohops:

• First hop: Transmission from the ST to relays withduration T1 where the SR acts as a friendly jammerfor the primary service to interfere with eavesdropper’soverhearing.

• Second hop: Transmission from one selected relay tothe SR with duration T2 where T = T1 + T2 is thetransmission period of the primary service and the STacts as a friendly jammer for the PT to decrease theeavesdroppers rate.

For both hops, the transmit power of the PT is fixed to PPT. Themaximum power of the ST and SR are equal to PST

max and PSRmax,

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MOKARI et al.: COOPERATIVE SECURE RESOURCE ALLOCATION IN COGNITIVE RADIO NETWORKS WITH GUARANTEED SECRECY RATE . . . 3

Fig. 1. System model of cooperative CRN with secure transmission.

TABLE INOTATIONS OF CSI VALUES BETWEEN EACH TRANSMITTER AND

RECEIVER IN OUR SETUP.Symbol DescriptionhPT→PR The CSI between PT and PRhPT→e The CSI between PT and eavesdropper ehPT→r The CSI between PT and relay rhPT→ST The CSI between PT and SThPT→SR The CSI between PT and SR

hST→PR The CSI between ST and PRhST→e The CSI between ST and eavesdropper ehST→r The CSI between ST and relay rhST→PT The CSI between ST and PThST→SR The CSI between ST and SR

hSR→PR The CSI between SR and PRhSR→e The CSI between SR and eavesdropper ehSR→r The CSI between SR and relay rhSR→PT The CSI between SR and PThSR→ST The CSI between SR and ST

hr→PR The CSI between relay r and PRhr→e The CSI between relay r and eavesdropper ehr→PT The CSI between relay r and PThr→SR The CSI between relay r and SR

respectively and pRelaymax = [p1max, · · · pR

max] denotes the vector ofmaximum transmit power of all relay nodes where prmax is themaximum transmit power of relay r.

Throughout this paper, the superscripts 1 and 2 are utilizedfor any parameters in the first and second hops and m → n isused to denote a correspondence between a transmitter namedm and a receiver named n. Accordingly, for transmissionfrom transmitter m to receiver n, γi

m→n and cim→n denotethe corresponding SINR and secrecy rate where superscripti ∈ {1, 2} shows the transmission occurs in hop i. We alsoassume that N0B is the white gaussian noise power overbandwidth B which is equal for all users in CRN. Also, hm→n

denotes the CSI between transmitter m and receiver n whichis assumed to be fixed during one transmission period. Forthe case of imperfect CSI, hm→n, hm→n and hm→n show theexact, estimated and error value of the CSI between transmitterm and receiver n. When the CSI is perfect, hm→n = hm→n.The corresponding gain notations are summarized in Table I.

B. First hop

At the first hop, the SINR of the PR is computed as

γ1PT→PR(p

1) =PPThPT→PR

N0B + I1PR, (1)

where p1 = [p1ST, p1SR] in which p1ST is the transmit power

of the ST and p1SR is the transmit power of the SR at thefirst hop when it acts as a jammer for eavesdroppers; andI1PR is the induced interference in the PR, which is equal toI1PR = I1SR→PR+I1ST→PR = p1SRhSR→PR+p1SThST→PR. Similarly,SINR for the first hop at eavesdropper e is equal to

γ1PT→e(p

1, e) =PPThPT→e

N0B + I1PT→e, (2)

where I1PT→e = I1SR→e + I1ST→e = p1SRhSR→e + p1SThST→e. Inthis hop, the secrecy rate of PU is equal to

c1PT→PR(p1) = min

e

{c1PT→PR(p

1, e)

}, (3)

where

c1PT→PR(p1, e) =

T1

T1 + T2×[

log2(1 + γ1PT→PR(p

1))− log2(1 + γ1PT→e(p

1, e))

]+.

Simultaneously in the secondary network, the ST sends thedata to all relay nodes and the SINR of relay r is

γ1ST→r(p

1, r) =p1SThST→r

N0B + I1r, ∀r ∈ R, (4)

where I1r = I1SR→r+ I1PT→r = pSRhSR→r+PPThPT→r, and theSINR received at the eavesdropper e is equal to

γ1ST→e(p

1, e) =p1SThST→e

N0B + I1ST→e

, (5)

where I1ST→e = I1SR→e + I1PT→e = p1SRhSR→e + PPThPT→e.Therefore, the secrecy rate of secondary user is

cST→r(p1, r) = min

e∈E

{cST→r(p

1, e)

},

where

cST→r(p1, r, e) =

T1

T1 + T2× (6)[

log2(1 + γ1ST→r(p

1, r)) − log2(1 + γ1ST→e(p

1, e))

]+.

C. Second hop

In this phase, the SINR of the PR is equal to

γ2PT→PR(p

2, r) =PPThPT→PR

N0B + I2PR(r), (7)

where p2 = [p2ST, p2r] in which p2ST is the transmit power of the

ST in the second phase and p2r = [p21, · · · , p2R] is the vector

of transmit powers of relay nodes where p2r is the transmitpower of relay r at the second hop, and I2PR(r) = I2ST→PR +I2r→PR = p2SThST→PR + p2rhr→PR, for all r ∈ R. The SINR ateavesdropper e is equal to

γ2PT→e(p

2, r, e) =PPThPT→e

N0B + I2PT→e(r), (8)

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where I2PT→e(r) = I2ST→e + I2r→e = p2SThST→e + p2rhr→e, forall r ∈ R. Now, the secrecy rate at the second hop from thePT to the PR is given by

c2PT→PR(p2, r) = min

e

{c2PT→PR(p

2, r, e)

}, (9)

where

c2PT→PR(p2, r, e) =

T2

T1 + T2×

[log2(1 + γ2

PT→PR(p2, r))− log2(1 + γ2

PT→e(p2, r, e))

]+.

Consequently, the secrecy rate of the PU is obtained as

cPT→PR(p, r) = c1PT→PR(p1, r) + c2PT→PR(p

2, r), (10)

where p = [p1,p2]. At this hop, in the secondary network,the relays send the message from the ST to the SR and theSINR of the SR from relay r is

γ2r→SR(p

2, r) =prhr→SR

N0B + I2SR, (11)

where I2SR = I2ST→SR + I2PT→SR = p2SThST→SR + PPThPT→SR.Also, the eavesdropper SINR from relay r is

γ2r→e(p

2, r, e) =prhr→e

N0B + I2r→e

, ∀r ∈ R, (12)

in which I2r→e = I2ST→e+ I2PT→e = p2SThST→e+PPThPT→e forall r ∈ R and

c2r→SR(p2, r) = min

e∈E

{c2r→SR(p

2, r, e)

},

where

c2r→SR(p2, r, e) =

T2

T1 + T2× (13)[

log2(1 + γ2r→SR(p

2, r)) − log2(1 + γ2r→e(p

2, r, e))

]+.

Finally, the secondary secrecy rate will be

cST→r→SR(p, r) = min

{c1ST→r(p

1), c2r→SR(p2)

}. (14)

Similar to other works in literature, in this paper we assumethat the CSI values between different nodes of the network areavailable to the secondary transmitter to be used in allocatingthe resources. We then consider the case where such CSIvalues are imperfect and derive corresponding secrecy rates.We also assume that eavesdroppers use single-user decoding,i.e., while decoding primary user data, secondary user data isconsidered as noise and vise versa.

III. INSTANTANEOUS RESOURCE ALLOCATION PROBLEM

AND ITS SOLUTION

A. The RA Problem

From the setup of Section II, the secondary secrecy ratedepends on the following parameters which are selected fromtheir corresponding sets:1) T1 and T2 chosen from T =

{T1, T2| T1 > 0, T2 >

0, T1 + T2 = T}

which is the set of time intervals for thefirst and second hops;

2) The transmit power of nodes in two hops i.e., p1 and p2,picked up from the set P =

{p| 0 � p � pmax

}where1

pmax = [PSTmax, PSR

max, pRelaymax ];

3) The relay r which is deployed in the second hop to transmitthe information to the SR for which the corresponding set isdenoted by ϕ where ϕ =

{ρ| ρ.1T = 1

}in which ρ =

[ρ1, · · · , ρR] and ρr = {0, 1} for all relay nodes, implying thatonly one relay is selected for transmission to the SR. Now,the RA problem of the secondary network may be written as

maxΞ=T ∪ϕ∪P

R∑r=1

ρrcST→r→SR(p, r), (15)

s.t. C1 :

R∑r=1

ρrcPT→PR(p, r) ≥ CPT→PRmin ,

where Ξ = T ∪ϕ∪P is the vector of optimization variables.In the sequel, prior to solving (15), we provide a discussionon its feasibility to show the effect of defining Ξ on secrecyrate of both PU and SU.

B. Feasibility Condition

As mentioned before, one concern for secrecy rate is thatit might be zero depending on the value of hPT→PR andhPT→e. Now, we want to show how by extending the setof optimization variables, we can increase the chance that(15) is feasible, meaning that the primary secrecy rate is non-zero and greater than CPT→PR

min . In line with existing literatureon interference limited networks, the following discussion onfeasibility is based on the assumption of high SINR at the PRand the SR [23].

For the case that there is no SU in the network, (15) isfeasible, if the following optimization problem has a solution[24]

min0≤ξ

ξ (16)

s.t. C1 : ξ ≥ CPT→PRmin − (log2

PPThPT→PR

PPThPT→e), ∀e ∈ E ,

which is a linear programming problem, and it has a solution ifλmin(M) > 1 where λmin is the smallest eignevalue of matrixM which is an E × E diagonal matrix whose eth elementis hPT→PR

2CPT→PRmin hPT→e

which only depends on the CSI values and

CPT→PRmin .If SU can access the spectrum via underlay approach i.e.,

the interference to the PR is less than a given value Γ,the primary secrecy rate is equal to log2 (1 +

PPThPT→PRN0B+Γ ) −

log2 (1 +PPThPT→e

N0B+ΓhST→ehST→PR

). For high SINR regime, it can be

approximated by log2hPT→PRhPT→e

− log2N0B+

ΓhST→ehST→PR

N0B+Γ . Therefore,(15) is feasible when the following optimization problem hasa solution

min0≤ξ

ξ (17)

s.t. ξ ≥ CPT→PRmin − (log2

hPT→PR

hPT→e+ log2

N0B + ΓhST→e

hST→PR

N0B + Γ).

1The symbol � represents the element-wise comparison.

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MOKARI et al.: COOPERATIVE SECURE RESOURCE ALLOCATION IN COGNITIVE RADIO NETWORKS WITH GUARANTEED SECRECY RATE . . . 5

As (17) is also a linear programming problem, it has asolution when λmin(MN) > 1 where N is an E ×E diagonal

matrix whose eth element isN0B+

ΓhST→ehST→PR

N0B+Γ . For this case, whenhST→e/hST→PR > 1, e.g., the CSI between ST and PR is largerthan that between ST and eavesdropper, the feasibility set isenlarged compared to the feasibility set of (16). The abovediscussions can be extended to the case when there is a relayset in the network as shown in the next proposition.

Proposition 1: When relay nodes are used in the networkand if the time durations of the two hops are equal and setto T/2, the primary secrecy rate is the same as (10) withT1 = T2 = T/2. In this case, when the noise is negligible andif hr→e

hr→PR> 1 and hSR→e

hSR→PR> 1, (15) is feasible if F1, defined

below, is a non-empty set

F1 ={∃r ∈ R|λmin((M1)

2N1Nr2) > 22CPT→PR

min

}(18)

where M1, N1 and Nr2 are E×E diagonal matrices whose eth

elements are hPT→PRhPT→e

, hST→e

hST→PRand hr→e

hr→PR, respectively.

Proof: See Appendix A.Note that hr→e

hr→PR> 1 and hST→e

hST→PR> 1 correspond to the case

where the jamming effect of relay nodes and SR is beneficialfor the CRN, e.g., the interference of SR and relay r on theset of eavesdropper nodes is greater than that on the PR.

For the case when T1 and T2 are not necessarily equal (thesetup of this paper), the set defined in (18) is transformed into

F2 ={∃r ∈ R|λmin(MN1

1N2r2 ) > 2CPT→PR

min

}(19)

where N11 and N2r

2 are E×E diagonal matrices whose eth ele-ments are ( hST→e

hST→PR)

T1T and (hPT→PR

hPT→e)

T2T , respectively. Obviously,

introducing the relay set in both cases when hr→e

hr→PR> 1 and

hST→e

hST→PR> 1, leads to the expansion of feasibility set compared

to that for the underlay approach. Now, consider the case inwhich hr→e

hr→PR> 1 and hST→e

hST→PR≈ 1, i.e., interference caused by

of ST on both PR and eavesdropper are on the same order.As such, (18) and (19) are transformed into

F1 ={∃r ∈ R|λmin

(M1

√Nr

2

)> 2CPT→PR

min

}(20)

and

F2 ={∃r ∈ R|λmin(MN2r

2 ) > 2CPT→PRmin

}. (21)

When T2/T > 1/2, the feasibility region from (21) is largerthan that of (20). This comparison shows that by adjustingthe time duration of each hop, the feasibility set size of theoptimization problem is increased. The non-emptiness of F2

can be shown by an approach similar to Proposition 1.The above analysis on the feasibility set of (15) shows

that by introducing new optimization variables to the RAproblem, new degree of freedom is added to the feasible setand therefore the chance that secrecy rate of CPT→PR

min can besupported under a given channel condition is increased. Inparticular, for the setup of this paper, the feasibility set islarger than the setup where there is no relay in the network,e.g., [12]–[15] or there is a relay with fixed time durationfor each hop e.g., [16]. Also, the chance of having non-zerosecondary secrecy rate is increased.

TABLE IIALGORITHM I

Step1: Initialize Lmax, and set l = 0,Step2: Initialize p0 and ρ0 and T 0

1 ,Step3: Repeat:Step4: Find a power allocation with ρ = ρl andT1 = T l

1, using the algorithm proposed in subsection III.C.1,Step5: Find a relay selection with P = P l andT1 = T l

1, using the algorithm proposed in subsection III.C.2,Step6: Find a time allocation with P = P l andρ = ρl, using the algorithm proposed in subsection III.C.3,Step7: l = l + 1, until ‖P l − P l−1‖ < ε or l = Lmax.

C. The Iterative Algorithm

It can be seen that (15) is a non-convex optimizationproblem with respect to Ξ. To solve the problem, we uti-lize the iterative algorithm introduced by [25] where theoptimization variables are divided into independent sets ofvariables. Then, corresponding to each set of variables, thenew optimization problem is solved. For example, for ourproblem, we have three sets of optimization variables: 1) P ,2) T , 3) ϕ. The optimization problem can be decomposed intothree subproblems: 1) Power allocation subproblem, 2) Relayselection subproblem, 3) Time allocation subproblem. Theiterative algorithm to solve these subproblems is summarizedin Table II. In this algorithm, l is the current iteration numberand the superscript l indicates that the associated variable isobtained after the lth iteration. In [25], it has been shown thatthe iterative algorithms converges to a near optimal solutionof (15) if each subproblem can be solved optimally. Thesesubproblems can be either convex or transformed into a convexoptimization problem.

1) Power allocation problem: Assuming a fixed value forT and ϕ, the power allocation problem is obtained as

maxP

R∑r=1

cST→r→SR(p, r), (22)

s.t.R∑

r=1

cPT→PR(p, r) ≥ CPT→PRmin .

Although (22) has only one set of optimization variables,it is still a non-convex optimization problem. To transform itinto a convex optimization problem, we use the following twosteps: 1) Introducing a tight lower bound of secrecy rate basedon SCALE algorithm [18] 2) Utilizing exponential auxiliaryvariables to transform (22) to a standard GP model.

Tight lower bound on secrecy rate:

In order to find the solution, we use the following tightlower bound log2(1 + z) ≥ α log2 z + β where α = z

1+z ,β = log2(1 + z) − z

1+z log2(z) and z is the SINR ofpervious iteration. Note that this lower bound is tight atz = z. Hence, we can introduce a lower-bound of the secrecyrate as cST→r(p

1, r) ≥ cST→r((p1, r)) and cr→SR(p

2, r) ≥

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cr→SR(p2, r), where

cST→r(p1, r) = min

e∈E

T1

T1 + T2× (23)[

αST→r log2(γ1ST→r(p

1, r)) + βST→r

−αST→e log2(γ1ST→e(p

1, e))− βST→e

]+,

and

cr→SR(p2, r) = min

e∈E

T2

T1 + T2× (24)[

αr→SR log2(γ2r→SR(p

2, r)) + βr→SR

−αr→e log2(γ2ST→e(p

2, r, e)− βr→e

]+.

Consequently, (14) is transformed into

cST→r→SR(p, r) ≥ min

{cST→r(p

1, r), cr→SR(p2, r)

}.

The same procedure can be applied to obtain a convexlower bound for primary secrecy rate as cPT→PR(p, r) ≥cPT→PR(p, r) where

cPT→PR(p, r) = c1PT→PR(p1) + c2PT→PR(p

2, r), (25)

in which c1PT→PR(p1) = mine∈E

{c1PT→PR(p

1, e)

}, and

c2PT→PR(p2), r) = mine∈E

{c2PT→PR(p

2, r, e)

}where

c1PT→PR(p1, e) =

T1

T1 + T2×[

α1PT→PR log2(γ

1PT→PR(p

1, e)) + β1PT→PR −

α1PT→e log2(γ

1PT→e(p

1, e))− β1PT→e

]+,

and

c2PT→PR(p2, r, e) =

T2

T1 + T2×[

αPT→PR log2(γ2PT→PR(p

2, r)) + β2PT→PR

−α2PT→e log2(γ

2PT→e(p

2, r, e))− β2PT→e

]+.

Since the introduced lower bound is derived based on thedifference of two logarithmic functions, we should demon-strate that this lower bound holds throughout iterations. Thisis shown in the next proposition.

Proposition 2: In each iteration, the introduced lower boundholds.

Proof: See Appendix II.In the sequel, we show that the power allocation problem

can be transformed into a standard form of GP based on theabove introduced lower bound for secrecy rate.

Standard GP modelIn a standard form GP, the objective function must be posyn-

omial (and it must be minimized); the equality constraintscan only have the form of a monomial equal to one, and the

inequality constraints can only have the form of a posynomialless than or equal to one [19]. The main motivation of utilizingGP modelling in our problem is its efficiency and existence offast algorithms to solve the optimization problems involvinga large number of constraints and variables. Moreover, inaddition to the efficiency of GP, the corresponding algorithmsare not sensitive to initial points and converge to the optimalsolution.

To transform our problem into standard form of GP, referredto as GP modelling, we define a set of new variables andexpress the problem based on them accordingly. For the firsthop, let us rewrite the primary SINR as follows:

γ1PT→PR(p

1) =1

z1PT→PR + a1SR→PRp1SR + a1ST→PRp

1ST, (26)

where z1PT→PR = N0B/(PPThPT→PR), a1ST→PR =hST→PR/(PPThPT→PR), a1SR→PR = hSR→PR/(PPThPT→PR)and we consider y1PT→PR = 1

z1PT→PR+a1

SR→PRp1SR+a1

ST→PRp1ST

. For

γ1PT→e, we again have

γ1PT→e(p

1, e) =1

z1PT→e + a1SR→ep1SR + a1ST→ep

1ST, (27)

where z1PT→e = N0B/(PPThPT→e), a1ST→e =hST→e/(PPThPT→e), a1SR→e = hSR→e/(PPThPT→e),y1PT→e = z1PT→e + a1SR→epSR + a1ST→ep

1ST. Now, based

on new variables, the lower bound of primary secrecy rate istransformed into

c1PT→PR(y1, e) = (28)

1PT→PR log2(y

1PT→PR)

+1PT→e log2(y

1PT→e) + log2(Γ

1PT→PR(e)),

where 1PT→PR = T1

T1+T2×α1

PT→PR, 1PT→e =

T1

T1+T2×α1

PT→e

and Γ1PT→PR(e) = exp

(T1

T1+T2(β1

PT→PR − β1PT→e)

).

At the second hop, similar to the first hop, γ2PT→PR can be

rewritten as follows:

γ2PT→PR(p

2, r, e) =1

z2PT→PR + a2r→PRp2r + a2ST→PRp

2ST

, (29)

where z2PT→PR = N0B/(PPThPT→PR), a2ST→PR =hST→PR/(PPThPT→PR), a2r→PR = hr→PR/(PPThPT→PR)and we consider y2PT→PR = 1

z2PT→PR+a2

ST→PRp2ST+a2

r→PRp1r

. Atsecond hop, the primary eavesdropper SINR can be writtenas follows:

γ2PT→e(p

2, r, e) =1

z2PT→e + a2r→ep1r + a2ST→ep

2ST

, (30)

where z2PT→e = N0B/(PPThPT→e), a2ST→e =hST→e/(PPThPT→e), a2r→e = hr→e/(PPThPT→e) and weconsider y2PT→e = z2PT→e + a2r→ep

2r + a2ST→ep

2ST.

At the second hop, based on the new variables, the lowerbound of primary secrecy rate is transformed into

c2PT→PR(y2, e, r) = (31)

2PT→PR log2(y

2PT→PR(r)) +2

PT→e log2(y2PT→e(r))

+ log2(Γ2PT→PR(e)),

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MOKARI et al.: COOPERATIVE SECURE RESOURCE ALLOCATION IN COGNITIVE RADIO NETWORKS WITH GUARANTEED SECRECY RATE . . . 7

where 2PT→PR = T2

T1+T2×α2

PT→PR, 2PT→e =

T2

T1+T2×α2

PT→e

and Γ2PT→PR(e) = exp

(T2

T1+T2(β2

PT→PR − β2PT→e)

).

The primary secrecy rate is then obtained as

cPT→PR(y, r, e) = (32)

1PT→PR log2 y

1PT→PR +1

PT→e log2 y1PT→e +

2PT→PR log2 y

2PT→PR(r) +2

PT→e log2 y2PT→e(r)

+ log2

(Γ1

PT→PR(e)Γ2PT→PR(e)

).

Similar to primary secrecy rate, we use the new variables totransform the secondary SINR and secrecy rate. At first hop,the secondary SINR can be defined as

γ1ST→r = y1STy

1ST→r (33)

where y1ST = p1ST, y1ST→r = 1z1

ST→r+p1SRhSR→r/hST→r

, z1ST→r =

(N0B + PPThPT→r)/hST→r.The secondary eavesdropper SINR can be defined as

γ1ST→e =

y1ST

y1ST→e

, (34)

where y1ST→e = z1ST→e+p1SRhSR→e/hST→e, z1ST→e = (N0B+PPThPT→e)/hST→e.

Based on the new variables, the lower bound of secondarysecrecy rate at first hop is

cST→r(y, r, e) = (35)

1ST→r log2(y

1STy

1ST→r) +1

ST→e log2(y1ST→e

y1ST)

+ log2(Γ1ST→r(e)),

where 1ST→r = T1

T1+T2αST→r, 1

ST→e = T1

T1+T2αST→e and

Γ1ST→r(e) = exp

(T1

T1+T2(β1

ST→r − β1ST→e)

).

At the second hop,

γ2r→SR = y2r y

2r→SR, (36)

where y2r = p2r , y2r→SR = 1z2

r→SR+p2SThST→r/hSR→r

, z2r→SR =

(N0B + PPThPT→r)/hr→SR.The secondary eavesdropper SINR is obtained as

γ2r→e =

y2ry2r→e

, (37)

where y2r→e = z2r→e + p2SThST→e/hr→e, z2r→e = (N0B +PPThPT→e)/hr→e.

Finally, the lower bound of secondary secrecy rate atsecond hop is cr→SR(y, r, e) = 2

r→SR log2(y2ry

2r→SR) +

2r→e log2(

y2r→ey2

r) + log2(Γ

2r→SR(e)), where 2

r→SR =T2

T1+T2α2r→SR, 2

r→e = T2

T1+T2α2r→e and Γ2

r→SR(e) =

exp

(T2

T1+T2(β2

r→SR − β2r→e)

).

Now we obtain new forms for constraints in the first andsecond hop based on the new variables. Let us considerp1ST = y1ST and p1SR =

(1

y1ST→r

− z1ST→r

)hST→r/hSR→r. Now,

by some mathematical manipulation y1ST→e can be obtainedbased on y1ST and y1ST→r as

C1 : y1ST→e =

[z1ST→e +

(1

y1ST→r

−

z1ST→r

)hST→rhSR→e

hSR→rhST→e

]y1

ST→r≤1

z1ST→r

where [x]a = x for any

x if statement a holds and otherwise [x]a = 0.In GP modelling, all the equality constraints should be

monomial functions but C1 is still a posynomial function.To take care of this, one can form a relaxation of theconsidered problem by replacing the equality constraint withan inequality constraint. Consequently, we can replace C1 withf1(y, e, r) ≤ 1 where

f1(y, e, r) =

⎧⎪⎨⎪⎩

1y1

ST→ez1ST→e +

1y1

ST→e

(1

y1ST→r

− z1ST→r

)hST→rhSR→e

hSR→rhST→e, if y1ST→r ≤ 1

z1ST→r

,

0, o.w,

The same process can be applied to other constraints andwe have

C2 : y1PT→PR =1

z1PT→PR+a1

SR→PR

(1

y1ST→r

−z1ST→r

)hST→rhSR→r

+a1ST→PRy1

ST

,

C3 : y1PT→e = z1PT→e + a1SR→e

(1

y1ST→r

− z1ST→r

)hST→r

hSR→r+

a1ST→ey1ST

f2(y, e, r) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

y1PT→PR

(z1PT→PR + a1

SR→PR

(1

y1ST→r

− z1ST→r

)hST→rhSR→r

+ a1ST→PRy

1ST

), if y1

ST→r ≤ 1z1ST→r

,

0, o.w,

f3(y, r, e) =

⎧⎪⎨⎪⎩

1y1

PT→ez1PT→e +

1y1

PT→ea1SR→e

(1

y1ST→r

− z1ST→r

)hST→rhSR→r

+ 1y1

PT→ea1ST→ey

1ST, if y1

ST→r ≤ 1z1ST→r

,

0, o.w,

At the second hop, we define p2r = y22 andp2ST =

(1

y2r→SR

− z2r→SR

)hST→r/hSR→r which leads to

C4 : y2r→e = z2r→e +(

1y2

r→SR− z2r→SR

)hST→rhST→e

hSR→rhr→e,

C5 : y2PT→PR = 1

z2PT→PR+a2

SR→PR

(1

y2r→SR

−z2r→SR

)hST→rhSR→r

+a2r→PRy2

r

andC6 : y2PT→e = z2PT→e+a1SR→e

(1

y2r→SR

−z2r→SR

)hST→r

hSR→r+a2r→ey

2r .

Again in order to use the relaxation method, we define thefollowing functions

f4(y, r, e) =

⎧⎪⎨⎪⎩

1y2

r→ez2r→e +

1y2

r→e

(1

y2r→SR

− z2r→SR

)hST→rhST→e

hSR→rhr→e, if y2ST→r ≤ 1

z2r→SR

,

0, o.w,

f5(y, r, e) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

y2PT→PR

(z2PT→PR + a2

SR→PR

(1

y2r→SR

− z2r→SR

)hST→rhSR→r

+ a2r→PRy

2r

), if, y2

ST→r ≤ 1z2r→SR

,

0, o.w,

f6(y, r, e) =

⎧⎪⎨⎪⎩

1y2

PT→ez2PT→e +

1y2

PT→ea1SR→e

(1

y2r→SR

− z2r→SR

)hST→r

hSR→r+ 1

y2PT→e

a2r→ey2r , ify2ST→r ≤ 1

z2r→SR

,

0, o.w,

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8 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION

Followed by the above new variables and constraints, thepower allocation problem is changed to the following opti-mization problem

maxy,π

R∑r=1

πr, (38)

s.t. cr→SR(y, r, e) ≥ πr, ∀r, e,cST→r(y, r, e) ≥ πr, ∀r, e,R∑

r=1

cPT→PR(y, r, e) ≥ CPT→PRmin , ∀e ∈ E

fi(y, r, e) ≤ 1, ∀r, e, i = 1, . . . , 6,

which is a standard GP and can be solved very efficiently vianumerical methods as in [26] and [27].

2) Relay Selection: For a fixed value of P and T , the relayselection problem can be stated as

maxϕ

R∑r=1

ρrcST→r→SR(p, r), (39)

s.t.R∑

r=1

ρrcPT→PR(p, r) ≥ CPT→PRmin ,

which is a binary linear programming with respect to ϕ andcan be solved efficiently via numerical approaches and CVXtool. In addition to our approach, there are different suboptimalapproaches for relay selection [16] which can be applied toour problem as follows:

• Maximum Primary Secrecy Rate (MPSR) where the relaywhich leads to the maximum rate for PUs is selected fromthe following problem

r∗ = argmaxr

{cPT→PR(p, r)} (40)

• Minimum Interference on Primary (MIP) where the re-lay which induces the minimum interference to PUs isselected from the following problem

r∗ = argminr

{prhr→PR} (41)

3) Time Allocation: For a fixed value of P and φ, the timeallocation problem can be stated as follows

maxT

R∑r=1

ρrcST→r→SR(p), (42)

s.t.R∑

r=1

ρrcPT→PR(p, r) ≥ CPT→PRmin ,

which is a linear programming problem with respect to Tand can be solved efficiently via numerical approaches andCVX tool. Another approach to reduce the computationalcomplexity is to set T1(n) = ΔT for n = 1, · · · , N whereΔT = T/N . For each value of T1(n) and T2(n) = T−T1(n),the optimal value of power and the selected relay are derivedfrom the proposed iterative algorithms of subsections III.C.1and II.C.2.

IV. ERGODIC RESOURCE ALLOCATION (ERA)

As discussed before, in IRA, the optimization problem issolved for any new channel realization through updating theLagrangian multipliers. This imposes a high computationalcomplexity. To reduce such overhead, ergodic resource allo-cation can be deployed in which the objective function is op-timized in average sense. In this case, Lagrangian multipliersdo not need to be updated for different channel realization.Generally in ERA, the constraints are also satisfied in averagesense. In this respect in the next subsection, we formulatethe ERA problem corresponding to the IRA problem (15) inwhich the average secondary secrecy rate is maximized subjectto keeping the average primary secrecy rate above a giventhreshold. The main drawback of this setting is that it onlyguarantees the primary secrecy rate in long term sense. In otherwords, the probability that the instantaneous primary secrecyrate is above the threshold is only fifty percent. To take careof this drawback, in Subsection IV-B, we propose a modifiedERA problem in which the probability of primary secrecy rateoutage is kept above a desired threshold. This ERA problemis referred to as the Probabilistic ERA (P-ERA) while we callthe first one the Non-Probabilistic ERA (NP-ERA).

A. Non-probabilistic ERA (NP-ERA)

Assuming that the long term CDI’s are available, the NP-ERA is given by

maxΞ

R∑r=1

ρrEh

{cST→r→SR(p)

}, (43)

s.t.R∑

r=1

ρrEh

{cPT→PR(p, r)

}≥ CPT→PR

min .

Following the same arguments as Section III, we divideabove problem into three subproblems. The correspondingpower allocation problem of (43), can be transformed intothe standard GP model as

maxy,π

R∑r=1

πr, (44)

s.t. Eh

{cr→SR(y(h), r, e)

}≥ πr, ∀r, e,

Eh

{cST→r(y(h), r, e)

}≥ πr, ∀r, e,

R∑r=1

Eh

{cPT→PR(y(h), r, e)

}≥ CPT→PR

min , ∀e,

∀fi(y(h), r, e) ≤ 1, ∀r, e, ∀h, andi = 1, . . . , 6,

The time allocation and relay selection problems for ERAare very similar to the IRA case have thus been omitted.

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MOKARI et al.: COOPERATIVE SECURE RESOURCE ALLOCATION IN COGNITIVE RADIO NETWORKS WITH GUARANTEED SECRECY RATE . . . 9

B. Probabilistic ERA (P-ERA)

By substituting the constraint of Problem (43) with itsprobabilistic version, we obtain the following ERA problem

maxΞ

R∑r=1

ρrEh

{cST→r→SR(p)

}, (45)

s.t. Pr

{ R∑r=1

ρrcPT→PR(p, r) ≤ CPT→PRmin

}≤ ζ,

where 0 < ζ < 1 is a predefined threshold of outage probabil-ity of primary secrecy rate.Via the probabilistic constraint in(45) for any value of CSI, the instantaneous primary secrecyrate is guaranteed to be above the threshold with a desiredprobability.

Due to the inclusion of the probabilistic constraint, thisproblem can not be solved directly by conventional meth-ods. Therefore, we propose to replace the probabilistic con-

straint with Eh

{R∑

r=1ρrcPT→PR(p, r)

}≥ C

PT→PRmin (l), where

CPT→PRmin (l) is an auxiliary variable to be explained later. The

new problem is then solved using Algorithm I. If the resultingoptimization variables satisfy the probabilistic constraint, thesevariable are treated as the solution of Problem (45). Toconverge to a feasible solution, we propose a novel approachin which this goal is achieved through an iterative algorithm,referred to as Algorithm II. At the lth iteration of this algo-rithm, the following problem is solved using Algorithm I:

maxΞ

R∑r=1

ρrEh

{cST→r→SR(p)

}, (46)

s.t. Eh

{ R∑r=1

ρrcPT→PR(p, r)

}≥ C

PT→PRmin (l),

where CPT→PRmin (l) = θ(l)C

PT→PRmin (l − 1) in which θ(l) ∈ [0, 1]

is a scaling parameter whose value is assigned later. Ininitialization step i.e., l = 0, we set C

PT→PRmin (l = 0) =

ωCPT→PRmin where ω is an arbitrary positive large number. From

the obtained parameters, the following outage probability ofprimary secrecy rate is derived

OU(l) = Pr

{cPT→PR(p, r

∗) ≤ CPT→PRmin

}. (47)

Let D(l) = |OU(l)− ζ|. If D(l) ≤ ε, where ε is an arbitrarypositive small value, the proposed iterative algorithm stops. Ateach iteration, we set θ(l) = (1−D)Θ where Θ is an arbitrarynumber greater than 1. This iterative algorithm, referred to asAlgorithm II is summarized in Table III. The value of OU(l)can be obtained based on the following proposition.

Proposition 3: For the given solution set of (46), (47) can

TABLE IIIALGORITHM II

Step1: Initialize the auxiliary variable, CPT→PRmin (l = 1)

Step2: Generate the new optimization problem at the lth

iteration from (46),Step2.1: Find the appropriate power, relay and time from

section III,Step3: Compute the outage probability of secrecy rate at thelth iteration, OU (l) from (47),Step4: Compute D(l) = |OU (l)− ζ|,

Step4.1: If D(l) ≤ ε, then go to Step5,Step4.2: Compute θ(l) = 1−D(l) and adjust the auxiliary

variables

as CPT→PRmin (l) = θ(l)C

PT,→PRmin (l− 1), then go to Step2,

Step5: End.

be obtained from

OU(l) = Pr

{cPT→PR(p, r

∗) ≤ CPT→PRmin

}(48)

= 1−E∏

e=1

(1−

∫ +∞

0

x1−T2T2

T2×

[ ∫ +∞

0

Fγ1PT→ST

(T1

√2CPT→PR

min

x(y + 1)− 1)fγ1

PT→e(y)dy

]

×[ ∫ +∞

0

(y′ + 1)fγ2PT→ST

( T2√x(y′ + 1)− 1)fγ2

PT→edy′

]dx

)Proof: See Appendix C.

In the sequel, we discuss convergence behavior and speedfor Algorithm II. The initial value of C

PT→PRmin (l) is assumed

to be very large, making OU(l = 0) pretty close to 1. Hence,D(l = 0) is definitely larger than ε. Since θ(l) ∈ [0, 1] and

CPT→PRmin (l) = θ(l)C

PT→PRmin (l− 1), C

PT→PRmin (l), OU(l) and D(l)

are decreasing functions with respect to l. Meanwhile θ(l)is an increasing function with respect to l. Therefore, whenl → ∞, θ(l) tends to 1, implying the convergence of theiterative algorithm.

Note that the speed of convergence can be controlled viaΘ. In fact for larger values of Θ, the speed of convergence ofalgorithm is increased which leads to the reduction of ergodicrate. This is mainly due to the fact that for larger values ofΘ, the algorithm jumps over some feasible solutions that maysatisfy the probabilistic constraints while they are not global oreven strong local optimums. Therefore, a compromise betweenthe complexity of the RA problem and the performance of thesecondary network can be obtained via Θ.

V. IMPERFECT CHANNEL STATE INFORMATION

In this section, we consider a situation where the CSIvalues between the PU, the SU, relays and eavesdroppersare imperfectly known. We then propose a robust approachto solve the corresponding IRA and ERA problems.

A. Imperfect CSI in IRA

To model the imperfect CSI, the actual value of CSI isconsidered as the sum of the nominal value of the CSI (theestimated value of the CSI by users) and an additive error[28], e.g.,

hPT→E = hPT→E + hPT→E. (49)

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10 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION

where hPT→E, hPT→E, and hPT→E are 1×E vectors represent-ing the exact, nominal and additive error of CSI values wheretheir eth elements are denoted by hPT→e, hPT→e and hPT→e

2.For the IRA problem, we apply the worst-case optimizationtheory as it can preserve the primary secrecy rate under anycondition of error where the error is assumed to be bounded ina closed region called uncertainty region [28], [29]. Usually,uncertainly regions for imperfect CSI values are defined bygeneral norm functions such as, e.g., [28]

hPT→E ∈ RPT→E = (50)

{hPT→E

∣∣ ‖hPT→E − hPT→E‖ ≤ εPT→E},hST→E ∈ RST→E (51)

= {hST→E

∣∣ ‖hST→E − hST→E‖ ≤ εST→E},hr→E ∈ Rr→E (52)

= {hr→E

∣∣ ‖hr→E − hr→E‖ ≤ εr→E}, ∀r ∈ R,

hSR→E ∈ RSR→E (53)

= {hSR→E

∣∣ ‖hSR→E − hSR→E‖ ≤ εSR→E},

where ‖x‖ is the general norm function. Through the worstcase robust optimization theory, the allocated power vectoris derived such that in the worst case condition of error inthe uncertainty region, the worst case secrecy rate of PU ispreserved and the minimum secrecy rate of SU is achieved.Mathematically, the worst case robust counterpart of (15) maybe expressed as

maxQ

minH

R∑r=1

ρrcST→r→SR(eq1

, r), (54)

C1 :

R∑r=1

ρr cPT→PR(eq, r) ≥ CPT→PR

min , ∀h ∈ H,

where h is the vector of all CSIs in our set up and His the set of all the uncertainty regions defined in (50)-(53). Note that (54) is a cumbersome optimization problemand its computational complexity is directly dependent onthe definition of norm functions. To simplify the problemand reduce the computational complexity, we utilize the D-norm approach introduced in [30] where each uncertaintyregion is transformed into the bounded interval, e.g., hPT→e ∈[hPT→e − εPT→e, hPT→e + εPT→e] where εPT→e is the boundof uncertainty region represented by D-norm. Following bythe worst case robust optimization, the lower bound of theuncertainty region is considered for each CSI between eachtransmitter and each eavesdropper. Thus, the robust powerallocation problem is transformed into the nominal powerallocation problem, e.g., (15), except that all the CSIs betweentransmitters and eavesdroppers are modified.

B. Imperfect CSI for ERA

In NP-ERA with imperfect CSI, the marginal distributionfor each channel is required [21], [22]. Again, the uncertainparameters can be rewritten as (49). The marginal fading

2Similar notations are considered for other imperfect CSI values in thissection.

distribution for each eavesdropper conditioned on the esti-mated value is non-zero mean complex Gaussian randomvariable denoted by hPT→E|hPT→E ∼ CN (hPT→E, σ

2PT→E)

where σ2PT→E is the prediction of error variance [22]. The

same can be applied to (51)-(53). Now, Algorithm I can beapplied, except that the Lagrange function is solved based onthe conditional means of the CSI values given their estimates,instead of the actual CSI values [22]. The same can also beapplied for the P-ERA using Algorithm II, except that at eachiteration, conditional means of CSI values given their estimatesare used instead of the actual CSI values.

VI. SIMULATION RESULTS

In this section, we provide simulation results to evaluatethe performance of the proposed schemes for perfect andimperfect CSI for both IRA and ERA. We assume that allthe nodes in the network are placed in the circle with thediameter 5 Km and hm→n = ι/dςm→n where dm→n is thedistance between transmitter m and receiver n and ι is thefading coefficient and 1 ≤ ς ≤ 4 where ι is taken from anormalized Rayleigh distribution. Maximum power of the ST,SR, PT, and relays are set to 20 Watt and N0B = 1. We alsoset CPT→PR

min = 2 Bit/Sec/Hz and R, the number of relay nodes,to 15 unless otherwise stated.

A. Performance Comparison between the Proposed Paradigmand Conventional Underlay Approach

In this paper, our objective is to maximize the secrecy rateof the secondary user subject to guaranteeing a given secrecyrate for primary user as opposed to conventional case wheresuch maximization is subject to the interference thresholdconstraint. Meanwhile, it is important to see whether using thenew constraint causes the secondary secrecy rate to decreasecompared to the conventional case. To be able to make a faircomparison, we consider the following framework. For bothscenarios, we assume that the CSI value corresponding to anytransmitter and receiver pair is equal for both cases. CSI valuesare picked up randomly from a normalized Rayleigh distribu-tion. For a given set of CSI values, we fix the interferencethreshold and solve the conventional problem. We then obtainthe maximized secondary secrecy rate. For such a setting, wealso obtain the resulting primary secrecy rate. In Fig. 2, wehave reported the values of the resulting primary secrecy ratesversus hST→PR

hPT→PRfor different values of interference threshold in

IRA case.Now for different values of primary secrecy rate reported

in Fig. 2, we solve the proposed IRA problem and obtainthe corresponding secondary secrecy rate. We define η as theratio of the secondary secrecy rate of the proposed schemeto that of conventional scheme. In Fig. 3, we have plottedη versus hST→PR

hPT→PR. As can be seen in Fig. 3, the value of η

is always greater than or equal to 1, implying that the newconstraint always provides a superior secondary secrecy rate.This superiority is more pronounced for smaller values ofhST→PRhPT→PR

. In other words, we have been able to maintain thesecrecy rate of PU, and yet such a benefit has not come atany cost to the secondary secrecy rate. Moreover, as can beseen in Fig. 2, increasing the amount of tolerable interference

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MOKARI et al.: COOPERATIVE SECURE RESOURCE ALLOCATION IN COGNITIVE RADIO NETWORKS WITH GUARANTEED SECRECY RATE . . . 11

0 10 20 30 400

1

2

3

4

5

6

7

hST−PR

/hPT−PR

Prim

ary s

ecre

cy ra

te [B

it/Se

c/Hz]

Q=0.1N0B

Q=.5N0B

Q=N0B

Q=2N0B

Fig. 2. The primary secrecy rate resulting from the conventional problemversus hST→PR

hPT→PRfor different values of Q in IRA.

0 10 20 30 401

2

3

4

5

6

hST−PR

/hPT−PR

η

Q=2N0B

Q=N0B

Q=0.5N0B

Q=0.1N0B

Fig. 3. Comparing the secondary secrecy rate of the proposed paradigmfor IRA and the conventional underlay approach versus hST→PR

hPT→PRfor different

values of Q.

threshold may cause the secrecy rate of PU to considerablydecrease and even tend to zero. In such cases, changing theinterference threshold constraint to the one proposed by uscan guarantee a desired primary secrecy rate while providinga secondary secrecy rate equal or better than the conventionalcase. Figs. 4 and 5 demonstrate the same trend for secrecyrate of PU and SU for ERA problems.

B. Effect of System Parameters on the SU’s Secrecy Rate

In this part, we investigate the effect of system parameters(e.g., perfect and imperfect CSI values, the value of CPT

min andthe number of relays) on the performance of CRNs for bothIRA and ERA.

1) SU’s Secrecy Rate in IRA Problem: We demonstrate theeffect of increasing the value of CPT

min on SU’s secrecy ratefor IRA in Fig. 6 versus the number of relays. In this figure,R = 0 indicates the case where there is no relay node in thenetwork and the SU directly transmits to its correspondingreceiver. Clearly, by increasing the number of relays anddecreasing the value of CPT

min, the secondary secrecy rate isincreased. Comparing the secrecy rate of SU for R = 0 andR = 1 reveals that introducing the set of relay node in CRNconsiderably increases the secrecy rate of SU while maintainsthe minimum required secrecy rate of the PU for differentvalues of CPT

min. This can be associated to the feasibility setof the IRA problem. As shown in Section II.B, the feasibilityset of the IRA problem expands by decreasing the value of

0 10 20 30 400

1

2

3

4

5

6

hST−PR

/hPT−PR

Prim

ary s

ecre

cy ra

te [B

it/Se

c/Hz]

Q=0.1N0B

Q=.5N0B

Q=N0B

Q=2N0B

Fig. 4. The primary secrecy rate resulting from the conventional problemversus hST→PR

hPT→PRfor different values of Q in ERA.

0 10 20 30 400.5

1

1.5

2

2.5

3

3.5

4

hST−PR

/hPT−PR

η

Q=2N0B

Q=N0B

Q=0.5N0B

Q=0.1N0B

Fig. 5. Comparing the secondary secrecy rate of the proposed paradigmfor ERA and the conventional underlay approach versus hST→PR

hPT→PRfor different

values of Q.

CPTmin as well as increasing the number of relay. This leads to

increasing the secrecy rate of the SU [24].We also study the effect of expanding the uncertainty

region on the secondary secrecy rate of SUs in the IRA incase of imperfect CSI. In this simulation, we assume thatthe bound of uncertainty regions for (50) - (53) are equaland normalized to the value of estimated CSI values, e.g.,ε = εPT→e% = ‖hPT→e−hPT→e‖

hPT→e. In Fig. 7, the effect of

increasing the value of ε on reduction of SU’s secrecy rateis demonstrated. Obviously, with increasing the value of ε,the secondary secrecy rate is decreased. This is because basedon the worst case robust optimization theory, the SU acts veryconservatively against the uncertainty in the CSI between thePU and each eavesdropper and tries to keep the minimumrequired primary secrecy rate in the maximum extent. Conse-quently, the SU and relay nodes allocate their transmit powerin such a way that the rate of each eavesdropper is suppressedand reaching their own maximum secrecy rate is not a priority.This fact is also supported by comparing rate reduction of theSU under different values of CPT

min. For larger value of CPTmin,

the SU’s rate reduction is larger compared to smaller valuesof CPT

min.2) SU’s Secrecy Rate in ERA Problem: The effect of

increasing the value of CPTmin for ERA problem is demonstrated

in Fig. 8. In the simulations of this part, we set σPT→E = 0.1for NP-ERA and P-ERA with imperfect CSI. Fig. 8 showsthat by increasing the value of CPT

min, the SU’s secrecy rate

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12 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Total number of relays

Sec

onda

ry S

ecre

cy R

ate

[B

it/Se

c/H

z]

CminPT−PR= 0.5

CminPT−PR= 1

CminPT−PR= 2

CminPT−PR= 2.5

CminPT−PR= 3

Fig. 6. The effect of increasing the value of CPTmin on the SU’s secrecy rate

for IRA for different number of relays.

0% 20% 40% 60% 80% 100%0

1

2

3

4

5

6

ε

Sec

onda

ry S

ecre

cy R

ate

[Bi

t/Sec

/Hz]

CminPT−PR=0.5

CminPT−PR=1

CminPT−PR=2

CminPT−PR=3

Fig. 7. The secondary secrecy rate versus bound of uncertainty region fordifferent values of CPT

min.

is dramatically decreased. Comparing NP-ERA and P-ERA,we can see that while P-ERA is very desirable from thePU’s perspective, it provides a lower rate for the SU. On theother hand in NP-ERA, the SU enjoys a higher rate while theoutage probability for primary secrecy rate is increased to fiftypercent.

Fig. 9 demonstrates the secondary secrecy rate versus thenumber of relay nodes in CRNs for NP-ERA and P-ERA. Forall schemes of ERA, by increasing the number of relay nodes,the SU’s secrecy rate is increased. Clearly, for both NP-ERAand P-ERA, when R = 0, the secrecy rate is too small. Onthe other hand, by introducing one relay node in the network,the secrecy rate of SU is increased considerably which showsthe benefit of the proposed introduced cooperative paradigmin CRNs.

C. Effect of Relay Selection Methods on the Secondary Se-crecy Rate

We plot the secondary secrecy rate for different relayselection algorithms in Figs. 10 and 11 for both IRA and ERA,respectively. Again, when R = 0, the secondary secrecy rate isvery small for both IRA and ERA. However, for all methods,by increasing the number of relays, the SU can experiencea larger value of secrecy rate as expected. Interestingly, the

1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

CminPT−PR

Sec

onda

ry S

ecre

cy R

ate

[Bi

t/Sec

/Hz]

NP−ERA NP−ERA & Imperfect CSI P−ERA & ζ=0.2P−ERA & ζ=0.2 & Imperfect CSI P−ERA & ζ=0.1P−ERA & ζ=0.1 & Imperfect CSI

Fig. 8. The effect of increasing the value of CPTmin on the SU’s secrecy rate

for ERA.

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

Total Number of Relays

Sec

onda

ry S

ecre

cy R

ate

[B

it/Se

c/H

z]

NP−ERA NP−ERA & Imperfect CSI

P−ERA & ζ=0.2

P−ERA & ζ=0.2 & Imperfect CSI

P−ERA & ζ=0.1

P−ERA & ζ=0.1 & Imperfect CSI

Fig. 9. The effect of increasing the number of relays on the secondarysecrecy rate for ERA.

performance of MPSR is very close to the optimal relayselection. It means that via MPSR with less computationalcomplexity, we attain the near optimal solution which isvery desirable from practical implementation perspective. Incontrast, the performance of MIP approach is not convincingfrom the SU’s perspective.

D. Effect of Time Allocation on the SU’s Secrecy Rate

In Figs. 12 and 13, the SU’s secrecy rate versus hPT→e

hPT→PRis

shown for both IRA and ERA, respectively in two modes: 1)The time intervals of two hops are equal: T1 = T2 = T/2,referred to as symmetric time allocation, 2) The time intervalsof T1 and T2 are obtained from Section III. C. 3, referred toas the asymmetric time allocation. From Figs. 12 and 13, itis obvious that the asymmetric time allocation has a betterperformance in terms of achievable secondary secrecy ratecompared to that of the symmetric time allocation for IRA,NP-ERA and P-ERA. When the channel gain between the PUand each eavesdropper increases, the asymmetric time allo-cation considerably increases the SU’s secrecy rate comparedto that for the symmetric time allocation. This was alreadypredicted in Section III. B, since by choosing asymmetric timeallocation, we are in fact expanding the feasibility set.E. Iterative Algorithm and its Convergence Behavior

In Figs. 14 and 15, we study the effect of the value of Θ onthe performance of iterative algorithm. Fig. 14 demonstrates

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MOKARI et al.: COOPERATIVE SECURE RESOURCE ALLOCATION IN COGNITIVE RADIO NETWORKS WITH GUARANTEED SECRECY RATE . . . 13

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

Total Number of Relays

Sec

onda

ry S

ecre

cy R

ate

[B

it/Se

c/H

z]

IRA & Optimal Relay Selection IRA & Maximum Primary Secrecy Rate IRA & Minimum Interference on Primary

Fig. 10. The effect of relay selection algorithm on the SU’s rate for IRA.

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

Total Number of Relays

Sec

onda

ry S

ecre

cy R

ate

[B

it/Se

c/H

z]

NP−ERA & Optimal Relay Selection NP−ERA & Maximum Primary Secrecy Rate NP−ERA & Minimum Interference on Primary

Fig. 11. The effect of relay selection algorithm on the SU’s rate for ERA.

that increasing the value of Θ reduces the SU’s secrecy ratein P-ERA for any value of ζ while it reduces the convergencetime of iterative algorithm as shown in Fig. 15.

These two figures highlight the effect of Θ on the trade-off between optimality and convergence time for iterativealgorithm in CRNs. When a smaller convergence time to reachthe solution is required (e.g., in highly dynamic situation suchas fast moving users in CRNs), the larger value of Θ isappealing at the cost of reducing SU’s secrecy rate. For thecase of slow moving users, we can use a smaller value for Θto reach a larger utility.

VII. CONCLUSION

In this paper, we proposed a novel cooperative paradigm forsecure communication in cognitive radio networks where wesimultaneity provide secure communications for both primaryand secondary services. The proposed setting is differentfrom previously proposed schemes where maximizing thesecondary secrecy rate is only subject to maintaining a certainlevel quality of service for primary users via the interferencethreshed constraint. In the proposed IRA and ERA problems,transmit power, relay and time duration of two hops are chosento maximize the secondary secrecy rate while preserving theprimary secrecy rate. By considering imperfect channel stateinformation, we then proposed the robust counterparts of thethe proposed IRA and ERA problems and investigated theeffect of uncertain parameters on the performance of thesystem. The proposed idea in fact transforms the possibly

0 0.2 0.4 0.6 0.80

1

2

3

4

5

hPT−e

/hPT−PR

Sec

onda

ry S

ecre

cy R

ate

[Bi

t/Sec

/Hz]

Asymmetric Time Allocation Symmetric Time Allocation

Fig. 12. The SU’s secrecy rate versus hPT→ehPT→PR

for symmetric and asymmetrictime allocation in IRA.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.5

1

1.5

2

2.5

3

hPT−e

/hPT−PR

Sec

onda

ry S

ecre

cy R

ate

[B

it/Se

c/H

z]

NP−ERA & Asymmetric Time Allocation NP−ERA & Symmetric Time Allocation P−ERA & ζ=0.2 & Asymmetric Time Allocation P−ERA & ζ=0.2 & Symmetric Time Allocation

Fig. 13. The SU’s secrecy rate versus hPT→ehPT→PR

for symmetric and asymmetrictime allocation in ERA.

disturbing secondary service activities to a beneficial networkelement.

APPENDIX APROOF OF PROPOSITION 1

When T1 = T2 = T/2, C1 in (15) is transformed into1/2[log2(

PPThPT→PRN0B+I1

PR) − log2(

PPThPT→e

N0B+I1PR→e

) + log2(PPThPT→PRN0B+I2

PR) −

log2(PPThPT→PR

N0B+I2PR→e

) ≥ Cmin, and for the case of high SINR, i.e.,N0B << 1, it is simplified to

hPT→PR

hPT→e

N0B + I1PR→e

N0B + I1PR× hPT→PR

hPT→e

N0B + I2PR→e

N0B + I2PR≥ 22Cmin ,

(A.1)which can be rewritten as

(hPT→PR

hPT→e)2 × χ1 × χ2 ≥ 22Cmin , (A.2)

where

χ1 =I1PR→e

I1PR=

hST→e

hST→PR×

1 +p1

SRhSR→e

p1SThST→e

1 +p1

SRhSR→PR

p1SThST→PR

, (A.3)

and

χ2 =I2PR→e

I2PR=

hST→e

hST→PR×

1 +p2rhr→e

p2SThST→e

1 +p2rhr→PR

p2SThST→PR

. (A.4)

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14 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION

1 2 3 4 5 6 70.5

1

1.5

2

2.5

3

Θ

Seco

ndary

Secre

cy R

ate [

Bit/S

ec/Hz

]

P−ERA & ζ=0.3 P−ERA & ζ=0.2 P−ERA & ζ=0.1

Fig. 14. Effect of the value of Θ on the SU’s secrecy rate.

2 4 6 8 10 12 14 160

0.02

0.04

0.06

0.08

0.1

l

D(l)

Θ=1Θ=2Θ=4Θ=7

Fig. 15. Effect of the value of Θ on convergence speed of Algorithm II.

To simplify (A.2), let us consider χ1 = hSR→e

hSR→PRand χ2 =

hr→e

hr→PR . This case corresponds to the condition that the friendlyjamming effects of ST, SR and relay r are beneficial for PR,i.e., the imposed interferences of ST, SR and relay r on theeavesdropper e is larger than that of PR. Now, we have

(hPT→PR

hPT→e)2

hSR→e

hSR→PR

hr→e

hr→PR≥ 22Cmin . (A.5)

(15) is feasible, if there exists one relay node that satisfies(A.5). Consequently, the feasibility region in (18) is derived.

APPENDIX BPROOF OF PROPOSITION 2

To prove the lower bound of secrecy rate, let z1l andz2l be the legitimate and eavesdropper SINR at iteration l ,respectively. Accordingly, the secrecy rate can be obtained asfollows

Cl =

[log(1 + z1l )− log(1 + z2l )

]+. (B.1)

Fact 1: Since Cl is greater than or equal to zero, we havez1l ≥ z2l . Therefore, the secrecy rate can be rewritten as

Cl = log(1 + z1l )− log(1 + z2l ). (B.2)

The lower bound of secrecy rate based on scale algorithm isdefined as

Cl = α1l log(z

1l ) + β1

l − α2l log(z

2l )− β2

l , (B.3)

where α1l =

z1l−1

1+z1l−1

, β1l = log(1 + z1l−1)−

z1l−1

1+z1l−1

log(z1l−1),

α2l =

z2l−1

1+z2l−1

and β2l = log(1+z2l−1)−

z2l−1

1+z2l−1

log(z2l−1). Now,we want to prove that

Cl ≥ Cl. (B.4)

By substituting (B.2) and (B.3) into (B.4), we have

log(1 + z1l )− log(1 + z2l ) ≥ (B.5)z1l−1

1 + z1l−1

log(z1l ) + log(1 + z1l−1)−

z1l−1

1 + z1l−1

log(z1l−1)−z2l−1

1 + z2l−1

log(z2l )

− log(1 + z2l−1) +z2l−1

1 + z2l−1

log(z2l−1).

With some mathematical manipulation, (B.5) is transformedinto [

log(1 + z1l

z1l

z1l−1

1+z1l−1

)− log(1 + z2l

z2l

z2l−1

1+z2l−1

)

]− (B.6)

[log(

1 + z1l−1

z1l−1

z1l−1

1+z1l−1

)− log(1 + z2l−1

z2l−1

z2l−1

1+z2l−1

)

]≥ 0.

Fact 2: For such an iterative algorithm, based on Theorem 2of [25], Cl ≥ Cl−1 and, then consequently we have z1l ≥ z1l−1

and z2l ≤ z2l−1. Based on Fact 1, the first and second term of(B.6) is no-negative. Moreover, Based on Fact 2, the first termis greater than or equal of the second term. Consequently, thelower bound always holds.

APPENDIX CPROOF OF PROPOSITION 3

To prove the equality of (48), let X1, Y1 , X2 and Y2 befour independent random variables and let Z1 = 1+X1

1+Y1, Z2 =

1+X2

1+Y2, and W = ZT1

1 ZT22 = W1W2. The CDF of Z1 is given

by

FZ1(z1) = Pr{1 +X1

1 + Y1< z1} (C.1)

= EY

{Pr

{X1 < z1(1 + Y1)− 1|Y1

}}

=

∫ +∞

0

FX1(z1(y + 1)− 1)fY1(y)dy.

Similar to (C.1), the CDF of Z2 is obtained as

FZ2(z2) =

∫ +∞

0

FX2(z2(y + 1)− 1)fY2(y)dy. (C.2)

From (C.1) and (C.2), the CDF of W1 and W2 can be obtainedas FW1 (w) = Pr{ZT1

1 < w} = FZ1(T1√w)} and FW2 (w) =

Pr{ZT22 < w} = FZ2(

T2√w)}. Now, the CDF of W is yielded

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MOKARI et al.: COOPERATIVE SECURE RESOURCE ALLOCATION IN COGNITIVE RADIO NETWORKS WITH GUARANTEED SECRECY RATE . . . 15

as

FW (w) = Pr{W1W2 < w} = (C.3)∫ +∞

0

FW1(w

x)fW2(x)dx =

∫ +∞

0

FZ1(T1

√w

x)x

1−T2T2

T2fZ2(

T2√x)dx =

∫ +∞

0

x1−T2T2

T2

[ ∫ +∞

0

FX1 (T1

√w

x(y + 1)− 1)fY1(y)dy

][ ∫ +∞

0

(y′ + 1)fX2(T2√x(y′ + 1)− 1)fY2(y

′)dy′]

By rewriting OU(l) as

OU(l) = Pr

{mine

{cPT→PR(p, r

∗, e) < CPT→PRmin

}}

= 1−E∏

e=1

Pr

{cPT→PR(p, r

∗, e) ≥ CPT→PRmin

}

(C.4)

= 1−E∏

e=1

(1− Pr

{(1 + γ1

PT→ST

1 + γ1PT→e

)T1

×(1 + γ2

PT→ST

1 + γ2PT→e

)T2

≤ 2CPT→PRmin

}).

and using the above equality the proposition is proved.

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Nader Mokari is a Ph.D. student at Tarbiat ModaresUniversity, Tehran, Iran. His main research interestsinclude wireless communications, radio resource al-location, secure communication, and spectrum shar-ing techniques. He is a student member of IEEE.

Saeedeh Parsaeefard (S’09) received the B.Sc.and M.Sc. degrees from Amirkabir University ofTech1nology (Tehran Polytechnic), Tehran, Iran, in2003 and 2006, respectively, and the Ph.D. degreein electrical and computer engineering from Tar-biat Modares University, Tehran, in 2012. She iscurrently a Post-Doctoral Research Fellow with theTelecommunication and Signal Processing Labora-tory in the Department of Electrical and ComputerEngineering at the McGill University, Canada. FromNovember 2010 to October 2011, she was a Visiting

Ph.D. Student with the Department of Electrical Engineering, University ofCalifornia, Los Angeles, CA, USA. Her current research interests include theapplications of robust optimization theory and game theory on the resourceallocation and management in wireless networks.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

16 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION

Hamid Saeedi (S’01-M’08) received the B.Sc. andM.Sc. degrees from Sharif University of Technology,Tehran, Iran, in 1999 and 2001, respectively, andthe Ph.D. degree from Carleton University, Ottawa,ON, Canada, in 2007, all in electrical engineering.In 2008-2009, he was a postdoctoral fellow with theDepartment of Electrical and Computer Engineering,University of Massachusetts, Amherst, MA, USA.In 2010, he joined the Department of Electrical andComputer Engineering, Tarbiat Modares University,Tehran, Iran, where he is now an Assistant Profes-

sor. His research interests include coding and information theory, wirelesscommunications, and cognitive radio networks.

Dr. Saeedi is the recipient of some awards including Carleton UniversitySenate Medal for Outstanding Academic Achievement, a Natural Sciencesand Engineering Council of Canada (NSERC) Industrial Research and De-velopment Fellowship, and an Ontario Graduate Scholarship.

Paeiz Azmi was born in Tehran, Iran, on April17, 1974. He received the B.Sc., M.Sc., and Ph.D.degrees in electrical engineering from Sharif Univer-sity of Technology, Tehran, Iran, in 1996, 1998, and2002, respectively. Since September 2002, he hasbeen with the Electrical and Computer EngineeringDepartment of Tarbiat Modares University, Tehran,Iran, where he became an associate professor onJanuary 2006 and he is currently a full professor.

From 1999 to 2001, Prof. Azmi was with theAdvanced Communication Science Research Labo-

ratory, Iran Telecommunication Research Center (ITRC), Tehran, Iran. From2002 to 2005, he was with the Signal Processing Research Group at ITRC.Prof. Azmi is a senior member of IEEE.

His current research interests include modulation and coding techniques,digital signal processing, wireless communications, and estimation and detec-tion theories.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.