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JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, XXXXX 1

On Physical Layer Security: Weighted FractionalFourier Transform based User Cooperation

Xiaojie Fang, Student Member, IEEE, Ning Zhang, Member, IEEE, Shan Zhang, Member, IEEE,Dajiang Chen, Member, IEEE, Xuejun Sha, Member, IEEE, and Xuemin (Sherman) Shen, Fellow, IEEE

Abstract—In this paper, we propose a novel user coopera-tion scheme based on weighted fractional Fourier transform(WFRFT), to enhance the physical (PHY) layer security of wire-less transmissions against eavesdropping. Specifically, instead ofdissipating additional transmission power for friendly jamming,by leveraging the features of WFRFT, the information bearingsignal of cooperators can create an identical “artificial noise”effect at the eavesdropper while causing no performance degra-dation on the legitimate receiver. Further, to form the cooperationset in an autonomous and distributed manner, we model WFRFT-based PHY-layer security cooperation problem as a coalitionalgame with non-transferable utility. A distributed merge-and-split algorithm is devised to facilitate the autonomous coalitionformation to maximize the security capacity while accounting forthe cooperation cost in terms of power consumption. We analyzethe stability of the proposed algorithm and also investigate howthe network topology efficiently adapts to the mobility of inter-mediate nodes. Simulation results demonstrate that the WFRFT-based user cooperation scheme leads to a significant performanceadvantage, in terms of secrecy ergodic capacity, compared withthe conventional security-oriented user cooperation schemes, suchas relay-jamming and cluster-beamforming.

Index Terms—Physical layer security, weighted fractionalFourier transform (WFRFT), secrecy capacity, coalitional gametheory.

I. INTRODUCTION

ENSURING information security is of paramount impor-tance for wireless communications. Due to the broadcast

nature of radio propagation, any receiver within the coverrange can listen and analyze the transmission without be-ing detected, which makes wireless networks vulnerable toeavesdropping attacks. Traditionally, cryptographic algorithm-s/protocols implemented at upper layers of the open systemsinterconnection (OSI) protocol stack, have been widely used toprevent information disclosure to unauthorized users [1]. How-ever, the layered design architecture with transparent physical

X. Fang is with the Department of Electronics and Information Engineering,Harbin Institute of Technology, Harbin 150001, China and the Department ofElectrical and Computer Engineering, University of Waterloo, Waterloo, ON,N2L 3G1, Canada. (e-mail: [email protected]; [email protected]).

N. Zhang, S. Zhang and X. Shen are with the Department of Electricaland Computer Engineering, University of Waterloo, Waterloo, ON, N2L 3G1,Canada (e-mail: [email protected]; zhangshan [email protected];[email protected]).

D. Chen is with the School of information and software engineering,University of Electronic Science and Technology of China, Chengdu, 611731,China and the Department of Electrical and Computer Engineering, Universityof Waterloo, Waterloo, ON, N2L 3G1, Canada. (e-mail: [email protected]).

X. Sha is with the Department of Electronics and Information Engineering,Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]).

Manuscript received June 1, 2017.

layer leads to a loss in security functionality [2], especially forwireless communication scenarios where a common physicalmedium is always shared by both legitimate and non-legitimateusers. Moreover, the cryptographic protocols can only offer acomputational security [3]. The encryption algorithms couldbe compromised when the adversary has powerful computingcapability, e.g. when quantum computing is available, most ofthe applied cryptography can be broken [4]. Most importantly,it is difficult to design an applicable secret key distribution andmanagement protocol to satisfy the dynamic nature of wirelessscenarios [5].

As an alternative, exploiting physical (PHY) layer char-acteristics for secure transmission has become an emerginghot topic in wireless communications [6–9]. The pioneeringwork by Wyner in [6] introduced the concept of “secrecycapacity” as a metric for PHY-layer security. Let X , Y , andZ be the input of the source, the outputs of the destination,and the eavesdropper, respectively. Then, secrecy capacity isdefined by max[I(Y ;X)− I(Z;X)], where I(·; ·) representsthe mutual information. Specifically, the secrecy capacitycharacterizes the maximum rate at which message can besecurely delivered. It is pointed out that perfect securityis in fact possible without the aid of an encryption keyswhen the source-eavesdropper (wire-tap) channel is a degradedversion of the source-destination (main) channel [6]. Takinguncertainty of wireless channels into consideration, the studyon information-theoretic security is subsequently generalizedto the Gaussian wiretap channels in [7] and the fading channelsin [8], respectively. Triggered by the proliferation of multipleantenna systems, considerable research efforts towards devel-oping multiple-input multiple-output (MIMO) techniques forPHY-layer security are devoted recently [10–12]. However,the feasibility of traditional PHY-layer security paradigmsis hampered by the limitations of practical applications, i.e.secure information exchange can not be guaranteed when thewire-tap channel is stronger than the main channel, for whichthe secrecy capacity is typically zero under this scenario.

To overcome the above issues and improve attainable se-crecy capacity, a multitude of sophisticated signal processingtechniques have been developed to explore the extra degreesof freedom of MIMO systems for PHY-layer security. In [13–15], security-oriented beamforming is employed, whereby thetransmitter constructs the transmission signal dedicated forthe destination, leading to a secrecy capacity improvement.In [16–18], a sort of specific signals with nulls directedtowards the destination, termed as “artificial noise”, are gen-erated to eliminate information leakage. In [19], the optimal




power allocation/sharing problem between the informationsignal and the artificial noise is studied. Due to the resourcelimitations of wireless terminals, user cooperation is furtherproposed as a practical strategy to harvest the multiple an-tenna gains [20–22]. The work in [14, 20] investigate thecooperative beamforming design problem by exploring theoptimal relay selection for secrecy capacity maximization,under both decode-and-forward (DF) and amplify-andforward(AF) modes, respectively. In [21, 23], friendly jammers aredeployed for generating “artificial noise” to defend againsteavesdropping attacks. Generally, the ultimate objective ofbeamforming or “artificial noise” is to boost the secrecyrate of the main channel by decreasing the leakage rate ofthe wiretap channel. However, most of existing beamformingtechniques could be compromised to powerful adversaries thatequipped with high-performance antenna (arrays). Typically,only quadratic antenna resources are required for eavesdrop-pers to have a competitive chance at undermining the secretcommunication of the main channel [24]. Although inserting“artificial noise” can mitigate the information leakage, itincurs the cost of wasting valuable transmission power, sincea certain amount of the transmission energy has to be assignedto the “artificial noise”. To warp up, we pose the followingquestion: is it possible that the information bearing signalcan be designed as the “artificial noise” to confound theeavesdropper while imposing no impact on the legitimatereceiver? One potential solution to this question is Weightedfractional Fourier Transform (WFRFT). WFRFT, known asa novel time-frequency analyzing tool, has drawn gradualattention in wireless communications recently. The concep-t of WFRFT-based communication system is pioneeringlyproposed by [25], where WFRFT is recognized as a hybridmodulation scheme of weighted single-carrier (SC) and multi-carrier (MC) modulation. Explicitly, the compatible processof SC and MC modulation makes it feasible for a WFRFT-based system to adjust its the signal distribution to adaptto different communication scenarios. With this favorableproperty, WFRFT has manifested itself a promising solutionfor interference suppression [26–30].

In this paper, we aim to investigate the superiority andpracticability of WFRFT from the PHY-layer security per-spective. To the best of our knowledge, this is the first workthat utilizes the WFRFT theory to facilitate the PHY-layersecurity through cooperation. The WFRFT signal, in fact, is aweighted combination of multiple different signals bearing theidentical information. More specifically, WFRFT operationswith different parameters result in distinct outputs that canonly be recovered through the corresponding inverse-WFRFToperations, i.e. the orthogonality of each sub-signal has tobe guaranteed. By exploring those features, we attempt togeneralize beamforming signal and artificial noise to enhancethe secrecy without any impact on the destination. Moreover,we consider user cooperation for further secrecy enhance-ment, where each intermediate user decodes and forwardsthe source message with WFRFT techniques. Pre-equalizationis employed such that the signal combination of multiplerelays can be reliably decoded by the destination. Taking thewireless channel uncertainty into account, the eavesdropper

only receives a degraded version of the transmitted signal-s, even if the wiretap channel is much stronger than themain channel. The reason is that interference between sub-signals occurs when their orthogonality is broken, leading toan “artificial noise” effect. To stimulate and select suitablecooperators, a distributed coalitional game strategy is furtherproposed. As selfish and rational player, each intermediateuser is rewarded according to its contribution. To find thedistributed solution, we proposed a Merge-and-Split basedalgorithm that allows each user to autonomously join or leavea coalition. The proposed algorithm is proved to convergeto a Nash-stable coalitional structure. Simulation results areprovided to evaluate the performance of the WFRFT-basedPHY-layer security cooperation scheme.

The remainder of this paper is organized as follows: Math-ematical precepts of WFRFT are presented in Section II.The system model and formulation of the WFRFT-based usercooperation for PHY-layer security are given in Sections IIIand IV, respectively. In Section V, the proposed distributedcoalitional game is formulated and solved. Simulation resultsare provided in Section VI. Finally, Section VII concludes thispaper.

II. WEIGHTED FRACTIONAL FOURIER TRANSFORM

Weighted Fractional Fourier Transform, a generalization ofthe Fourier Transform (FT), is a promising techinique forsignal and image processing. In this section, we introduce thebasic definitions and properties of WFRFT, based on whichWFRFT-based PHY-layer security technique will be developedin the subsequent sections.

Definition 1: Given an arbitrary complex vector X :=[x0, ..., xN−1] ∈ CN , the α-order WFRFT of X, Fα(X), isdefined by1:

Fα(X) = FαM [X] =M−1∑l=0

Al(α)T4lM X, (1)

where T represents the WFRFT kernel (or basic operator),while M ∈ Z+;M ≥ 4 indicates the number of the ba-sic operators involved. Moreover, the weighting coefficients{Al(α)}M−1l=0 are generated by

Al(α) =1

M

1− exp [−2πj(α− l)]1− exp [−2πj(α− l)/M ]

. (2)

In particular, we refer to M -WFRFT as α-order WFRFToperator consisting of M basic transforms, and denote it byFαM . For briefness, we define Wα

M =∑M−1l=0 Al(α)T

4lM as the

α-order M -WFRFT matrix.Definition 2: The WFRFT kernel T is a finite recursive

combination of the state functions of the conventional FT andsatisfies the following axioms:• Unitary axiom: FαM

[F−αM [X]

]= X, {Wα

M}H = W−αM ;• Boundary axiom: F0

M [X] = X, F1M [X] = DFT (X);

• Additive axiom: Fα+βM [X] =FαM

[FβM [X]

]=FβM [FαM [X]].

1For better understanding, this paper defines the WFRFT in a discrete form,the continuous WFRFT definition can be found in [31].




Without loss of generality, we utilize the classical-WFRFT(CWFRFT) as the basic operator of M -WFRFT. In whatfollows, we will provide the relevant basic theorems.

Let I denote an N×N identify matrix and F denote anN -point normalized Fourier matrix with (n, k)-th elementsatisfying [F]n,k = (1/

√N) · exp(−j2πnk/N).

Definition 3: The α-order CWFRFT of an arbitrary complexvector X := [x0, ..., xN−1] ∈ CN is defined by

Fαc (X) = (wα0 I + wα1 F + wα2 F2 + wα3 F3)X, (3)

where the coefficients {wαp }3p=0 = 14

3∑k=0

exp{−2πj(α−p)k

4

}.

Property 1: The CWFRFT has a periodicity of 4:

Fα+4c (X) = Fα

c (X), (4)

Property 2: When a ∈ Z, the CWFRFT degenerates intothe conventional FT:

Fαc (X) = Fα · X, (5)

Property 3: For real a and b, the additive property holds:

Fα+bc (X) = F a

c

(F bc (X)

)= F b

c (F ac (X)) , (6)

For proofs of these properties, please refer to Appendix A. �Theorem 1: Let CWFRFT be the basic operator of M -

WFRFT, and M -WFRFT degrades to CWFRFT when M = 4.Proof: Let T = (wα0 I +wα1 F +wα2 F2 +wα3 F3) be the basic

operator, and set M = 4. With Property 2, we have

Fα4 [X] =3∑l=0

Al(α)TlX =3∑l=0

Al(α)FlX. (7)

In addition, when M = 4, {Al(α)}3l=0 we have:

Al(α) =1

4

3∑k=0

exp

{−2πj(α− l)k

4

}. (8)

Thus, the proof is completed. �Theorem 2: Let α4 and αM be the parameters of 4-WFRFT

and M -WFRFT, respectively. Then, Wα44 = Wα

M

M holds forαM = M

4 α4, and vice versa.Proof: Please refer to Appendix B. �By aforementioned theorems and properties, it is true that

FαM satisfies all of the axioms in Definition 2, when T =∑4l=0Al(α)Fl.Theorem 3: Wα

M · WβM = I, iff α + β = 0 and the

weighting coefficients {Al(α)}M−1l=0 and {Al(β)}M−1l=0 are notcontaminated.

Proof: Please refer to Appendix C. �Remark 1: M -WFRFT can split the input signal into M sig-

nal components with weights {Al(α)}M−1l=0 , which is similarto the functionality of resource allocation.

Remark 2: Theorem 2 provides an unified decoding strategyfor M -WFRFT. The basic 4-WFRFT demodulator can be usedto demodulate any M -WFRFT signal properly. Therefore,no matter how does the transmitter structure change, thereceiver can remain unchanged. Theorem 3 suggests thatthe orthogonality in signal components directly determinesthe decoding performance, which is the most fundamentalproperty leveraged by the WFRFT-based PHY-layer securityscheme for information privacy preserving.

Fig. 1. The cooperative multiuser relay system model.

III. SYSTEM MODEL

As depicted in Fig. 1, we consider a cooperative systemconsisting of one source (S), one destination (D), and Ntrustful intermediate nodes ({Ri}Ni=1), in the presence of aneavesdropper (E) who attempts to eavesdrop on the sourceinformation. There is no direct link between S and D [14, 20],e.g., the direct link between source and destination (S → D)is blocked by an obstacle. S aims to transmit information se-curely to D, with the aid of intermediate nodes. All nodes areequipped with one single antenna and operate in a half-duplexmode. The system operation is performed in a two-phasefashion. In the first phase, the source broadcasts its signal tothe N trustful intermediate nodes among which the nodes thatcan successfully decode the source signal form a collaborationset R.2 In the second phase, the intermediate nodes performWFRFT operation to forward a converted version of thesource message to D, to boost the secrecy rate of the S-Dlink. In particular, only those that form the optimal coalitionwith best security performance are selected to forward theirreceived signals.3 To concentrate on the intermediate nodescollaboration, we assume the links S → Ri,∀Ri ∈ R aresecure, similar to [20, 21, 32, 33].

Without loss of generality, a quasi-static block Rayleighfading environment is considered, and the thermal noise ischaracterized by a complex Gaussian variable with zero meanand variance σ2

n. The fading coefficients of link S → Ri,S → E, Ri → D, Ri → E, (∀Ri ∈ R) are denoted by his,hse, hid, hie, respectively. Similar to [14] and [15], the CSI ofboth main and wiretap channels are assumed available, whichis a typical assumption in the PHY-layer security literature.Moreover, we consider the worst case where the eavesdropperknows everything about the signal transmission in the linkS → Ri → D, including the knowledge of WFRFT, theencoding paradigm at S, the decoding, re-encoding and theforwarding (DEF) protocols at Ri, and decoding method atD. Moreover, all intermediate nodes are considered to havethe ability to perform 4- WFRFT (De)-modulation operation.In what follows, we analytically show how WFRFT can becooperatively performed among multiple intermediate nodesfor PHY-layer security enhancement.

2Node Ri ∈ R, ∀i ∈ {1, 2, ..., N} if Cisec ≥ R0, where Cisec is thesecrecy capacity of link S → Ri and R0 is the transmission rate of S.

3The strategy of choosing optimal intermediate nodes for the two-phasecooperation is explicitly explained in Section IV.




Throughout this paper, [x]+ , max{x, 0}; E[·] is theexpectation operator; log(·) denotes the base-2 logarithm; ||·||denotes the Euclidean norm; |·| represents the cardinality ofa set; x? picks the optimal value of x; and superscripts (·)∗,(·)T and (·)H represent the conjugate, transpose and conjugatetranspose, respectively.

IV. PHY-LAYER SECURITY VIA WFRFT-BASED USERCOOPERATION

Following the existing PHY-layer security literature, we usesecrecy capacity to evaluate the performance of a PHY-layersecurity scheme. Basically, secrecy capacity upper bounds themaximum achievable rate of link S →R→ D while leakingno information to E.

Consider an N intermediate nodes network N := {Ri|i =1, ..., N}, during the broadcast phase, the transmitter contrivesto form an initial coalition pool R ⊂ N , which includes allnodes that can decode the transmission securely. Assume thatartificial noise injection strategy has been performed, then Rcan be generated by:

∀Ri ∈R

s.t. Cisec ≥ R0; Cisec =[Cisr − Cse

]+,

Cisr = log

(1 +

P0||his||2

σ2n

);

Cse = log

1 +P0||hse||2∑

Rj∈NPj ||hje||2 + σ2

n

,

Ri 6= Rj ; Ri, Rj ∈N ,

(9)

where R0 is the source transmission rate; Cisr and Csecharacterize the channel capacity of link S → Ri and S → E,respectively. P0 and Pj denote the transmission power of thesource and the intermediate node (jammer) Rj , respectively.Note that artificial noise based PHY-layer security is a welldeveloped technique, following the existing research results,R can be reliably obtained. In particular, the artificial noiseinjection strategy of (9) can be achieved by a set of iterationprocedures expressed by Algorithm 1.

In the collaboration phase, the intermediate nodes in Rcooperatively relay the source message to D using DEFprotocol. In particular, each intermediate node performs as abasic operator of WFRFT, and only the optimal combinationR? will be selected to re-encode and transmit their decodedoutcome to D. For an arbitrary combination C the signaltransmitted from the Ri(Ri ∈ C) is given by,

yi = λiAi(α)F4i|C|4 [X] , (10)

where λi represents the pre-equalization factor. Denotey = [y1, y2, ...y|C|]

T , hrd = [h1d, h2d, ...h

|C|d ] and hre =

[h1e, h2e, ...h

|C|e ]. Note that, hid and hie represent the complex

channel coefficients from node Ri to D and E, respectively,where i ∈ {1, 2, ..., |C|}. Then, the received signals rD and

Algorithm 1 Artificial noise injection Algorithm for the securebroadcast phaseStage I: Initial State

Initially, each node in N reports its current states to thenetwork, including available power consumption, channelstate information, etc..Stage II: Initial Coalition Pool FormationLOOP

Phase 1:S checks the network status and selects Ri as thetemporary relay node.

Phase 2:S calculates Cisr and Cse to check whether node Risatisfies the constraint Cisec ≥ R0

Phase 3:IF Cisec ≥ R0 THENS add node Ri into the initial coalition pool R;Meanwhile, S send the source message to Ri andother intermediate nodes Ri 6= Rj ;Rj ∈ Ngenerates the artificial noise to selectively degradeeavesdropper’s reception.

ELSEbreak;

Untill All nodes in N have been checked, S completes theconstruction of the initial coalition pool R

rE at D and E in the collaboration phase can be writtenrespectively:

rD = hrdy + n0 =

|C|∑i=0

hidλiAi(α)F4i|C|4 [X] + nrd

rE = hrey + n0 =

|C|∑i=0

hieλiAi(α)F4i|C|4 [X] + nre.

(11)

where nrd and nre are the thermal noise at D and E, respec-tively. Based upon Theorem 3, when λ := [λ1, λ2..., λ|C|]

T

is an amplitude&phase pre-equalization vector satisfying λi =

(hid)∗/∣∣∣∣hid∣∣∣∣2, the cooperation can facilitate the reception at

D. The achievable rate at D can be expressed as follows:

Crd = log

(1 +

∑|C|i=0 ||Ai(α)||2 P0

σ2n

). (12)

Whereas at the eavesdropper E, the orthogonality betweenthe signal components from different intermediate nodes arecontaminated. Assume that β-order 4-WFRFT is performedby E,4 then the signal to interference plus noise ratio (SINR)γire at Ri is a straightforward result of [34]:

γire =

∣∣∣∣κi∣∣∣∣2 ∣∣∣∣hieλiAi(α)∣∣∣∣2 P0

(1− ||κi||2) ||hieλiAi(α)||2 P0 + σ2n

, (13)

4According to Definition 1, the received signal at E is the mixture of |C|different 4-WFRFT signals. There is no way for E to rule out the mutual-interference, when the orthogonality of the signal components is damaged.




where κi = w( 4i|C|+β)

0 determines the information leakage fromRi to E while w0 is the weighting coefficient of 4-WFRFT.Then the achievable rate at E can be expressed by [14]:

Cre = log

1 +

|C|∑i=0

γire

. (14)

Combining (12) and (14), the secrecy capacity withWFRFT-based cooperation is given by:

Csec = [Crd − Cre]+

=

[log

(1 +

∑|C|i=0 ||Ai(α)||2 P0

σ2n

)

− log

1+

|C|∑i=0

∣∣∣∣κi∣∣∣∣2 ∣∣∣∣hieλiAi(α)∣∣∣∣2 P0

(1−||κi||2) ||hieλiAi(α)||2P0+σ2n

+

(15)The objective of employing cooperation is to maximize

the overall secrecy capacity. However, as rational players, theintermediate nodes might not cooperate unconditionally. Tostimulate cooperation, an appropriate amount of reward can bepaid for the intermediate nodes. With this regard, the problemof selecting suitable cooperators can be formulated as follows:

R? = argmaxC⊆R

[ηCsec −

∑Ri∈C

ξiPis

]s.t. P is ≤ P ia,∀Ri ∈ C and Csec > R0,

(16)

where η is the profit per secrecy rate, ξi is the cooperationcost per unit power consumption of Ri. P is and P ia representthe total power consumption and the available power budgetof Ri during each transmission.

The optimal cooperator set can be determined such thatthe best secrecy performance is achieved at the lowest cost.Intuitively, to find R?, an exhaustive search can be ap-plied. However, finding the optimal cooperator set via anexhaustive search is a non-deterministic polynomial (NP)-complete problem. The possible cooperation structures willgrow exponentially with the number of intermediate nodesin R. Moreover, the complexity further increases becauseof the fact that the overall PHY-layer secrecy rate Csec andthe system cost

∑Ri∈C ξiP

is varies for different coalitions.

Therefore, it is desirable to have a highly efficient means toform a stable cooperation.

V. COALITION FORMATION AND SOLUTION

In this section, we model the user cooperation among theintermediate nodes as a coalitional game [35–37] to select theoptimal cooperators for the WFRFT-based PHY-layer securityscheme.

A. Coalition Formation

Definition 4: The coalitional form of an N -player gamewith non-transferable utility (NTU) is defined by (N , V ),where N := {Ri|i =, 1, ..., N} represents the set of playersand V (S) personifies the value of coalition S;S ⊂N .

In this paper, the intermediate nodes are regarded as theplayers and coalition S denotes the group of intermediatenodes that cooperatively relay the source message via a |S|-WFRFT mechanism. Without loss of generality, consider anarbitrary coalition S := {R1, R2, ..., R|S|} ⊆ N and itscoalition value is defined as follows:

V (S) = {φ(S) ∈ R|S|| ∀Ri ∈ S, φi(S) = [vi(S)− ui(S)]+

for P is ≤ P ia, and φi(S) = −∞ otherwise.},(17)

where vi(S) = CSsec−CS−Ri

sec denotes the PHY-layer secrecycapacity gain for node Ri ∈ S given by (15) under theconstraint P is ≤ P ia while ui(S) = ξiP

is is the payoff

that quantifies the cost when node Ri ∈ S. Generally, (17)evaluates the benefits obtained from cooperation, in terms ofthe PHY-layer secrecy gain, taking into account the cost orpayoff to stimulate the intermediate nodes.

In the WFRFT-based PHY-layer security cooperationscheme, the source S acts as the “buyer” who pays an amountof reward for the “service” provided by the intermediatenodes Ri ∈ N , in order to achieve a secure informationexchange with D. Note that, only the optimal coalition R?

will be chosen for data transmission. Therefore, there is acompetition among the intermediate nodes to participate intothe coalition for the reward. Subsequently, the WFRFT-basedPHY-layer security cooperation problem can be formulated bya coalitional game as follows:

Proposition 1: The WFRFT-based PHY-layer security coali-tional game is a (N , V ) NTU game.

proof: Proposition 1 is an immediate result of the fact thatthe coalition value of coalition S ⊆N given in (17) relies onhow the players in the coalition are structured. Specifically,the coalition value of the game is a set of benefit vectors,φ(S) ∈ R|S|, whose element φi(S) indicates the benefits thatcan obtain by absorbing Ri member of S. Consequently, theWFRFT-based PHY-layer security coalitional game has a non-transferable utility. �

Definition 5: The coalitional game with NTU is super-additive if and only if:

V (S) ⊇ {φ(S) ∈ R|S||φi(S1) ∈ V (S1); φj(S2) ∈ V (S2)

∀S1,S2 ⊂N ;S1 ∪ S2 = S;S1 ∩ S2 = ∅}.(18)

Proposition 2: The WFRFT-based PHY-layer security coali-tional game (N , V ) is a non-superadditive game.

proof: Consider two disjoint coalitions S1 and S2, satis-fying S = S1 ∪ S2. The non-superadditive of the WFRFT-based PHY-layer security coalitional game can be proved bychecking:

V (S) + {φ(S) ∈ R|S||φi(S1) ∈ V (S1);φj(S2) ∈ V (S2)}.(19)

Consider a special case in which coalition S1 achievesan extreme high secrecy rate while coalition S2 is withlow secrecy rate. This case can be typically found whenplayers in S1 and S2 are at proximate of the destination andeavesdropper, respectively. Assume Rj ∈ S2 is with positivecoalition value φj(S2). Clearly, when S1 and S2 are merged,CS1+Rjsec − CS1

sec <= 0, which means the overall secrecy rate




of the coalition is degraded. In this case, φj(S1 ∪ S2) < 0,i.e. Rj’s coalition value would be 0 in the merged coalitionS. However, in the original coalition S2, Rj’s coalition valueis positive, which does not belong to V (S). Therefore, theWFRFT-based PHY-layer security coalitional game is non-superadditive. �

The non-superadditive property implies that the coalitionswith more participants may not be always beneficial. Thatis because the benefits provided by a coalition, in terms ofsecrecy capacity, would be limited by the cost associated. Inthis regard, the coalition with huge number of intermediatenodes are not optimal and will not be formed. Instead, inthe proposed scenario, multiple disjoint coalitions will formand the one with best secrecy performance is selected as theultimate coalition set. To this end, a distributive algorithm isproposed to form the disjoint coalitions, which is elaboratedin the following subsection.

B. Coalition AlgorithmIn this subsection, we design an algorithm to achieve

autonomous coalition formation for the WFRFT-based PHY-layer security cooperation problem.

Definition 6: A coalitional structure is referred to as acollection, if L := {S1,S2, ...SL| ∀i, j ∈ 1, 2, ...L, i 6=j,Si ∩ Sj = ∅ and Si ∪ Sj ⊆ N }. Typically, a collectionthat satisfies

⋃li=1 Si = N , is referred to as a partition of N .

Note that, a partition is not necessarily formed.For an N -players coalitional game, there exists 2N − 1

nonempty coalitions which can form BN different collections,where BN is the N -th Bell number, given by:

BN =

N−1∑i=0

(N − 1

i

)Bi, for N ≥ 1 B0 = 1. (20)

Let Sl be an arbitrary coalition, and a player’s action is todecide whether to be a member of Sl according to its statusin the group, namely its contribution to the group in term ofsecrecy rate gain, and its expected payoff. Since the players arerational, the designation of the players into suitable coalitionsis performed according to following criteria.

Definition 7: Consider two collections L :={S1,S2, ...SL} and P := {S∗1,S

∗2, ...S

∗K} that partition the

same set R ⊆ N . Then, we use the symbol � to representtheir comparison relation. Specifically, L � P implies thatthe way L partitions R outperforms the way P partitions R.

In order to achieve a rational and stable coalition, a mul-titude of comparison orders, based on both coalition utili-ty and individual payoff, have been introduced in existingworks[35, 36]. In particular, a comparison order, such asthe utilitarian order, is coalition rational if L � P ⇔∑Li=1 V (Si) ≥

∑Ki=1 V (S∗i ). On the contrary, a comparison

order is regarded as individual rational, such as the Paretoorder, if:

L � P ⇔{∀Ri ∈R, φi(L) ≥ φi(P)∃Ri ∈R, φi(L) > φi(P),

}(21)

which indicates that L is superior to P iff at least one playercan achieve more benefits by changing the partition structurefrom L to P , without degrading other players’ benefits.

Algorithm 2 Merge-and-Split Coalition Formation Algorithmfor WFRFT-based Collaborative PHY-layer SecurityStage I: Initial State

Initially, S broadcasts the information to nearby nodes.Suppose that there exists an initial set R, consisting ofL suitable nodes. Set the initial partition state as L :={S1,S2, ...SL}.

Stage II: Coalition Formation AlgorithmPhase 1: Neighbor nodes discovering

Each node in R reports its available power consumptionto the network, and checks the network status to findcandidate partners.

Phase 2: Dynamic coalition constructionSet intermediate nodes into cooperating mode, and initial-ize the collaborator list of each node as NULL.LOOP

a) L′ = Merge(L), merge any disjoint collisions in Lthat meets the Merge rule.b) L′′ = Split(L), split any collisions in L that meetsthe Split rule.C) Update the collaborator list of each node.

Untill The partition converges to a stable statute, that is,no player has an incentive to deviate from its currentcoalition.

Stage III: Cooperative Secure TransmissionThe collision S? that provides the highest utility, whichachieves the best secrecy performance at the lowest cost,will be chosen for data transmission.

As defined in (17), the proposed scheme mainly concernswhether the secrecy gain compensates for the reward payed tosimulate players into the coalition. Therefore, in this work, weuse the individual value order, i.e., the Pareto order, to checkwhich partition is more desirable. The coalitions are dynami-cally formed via an iterative merge-and-split operations:• Merge Rule: Merge any disjoint collections{S1,S2, ...Sl} whenever

⋃li=1 Si � {S1,S2, ...Sl},

that is, small coalitions desire to merge into a largercoalition if merging yields a superior partition on R.

• Split Rule: Split any collection⋃li=1 Si whenever

{S1,S2, ...Sl} �⋃li=1 Si, thus, a larger coalition desires

to split into multiple disjoint collections if splitting yieldsa superior partition on R.

According to the above rules, the WFRFT-based PHY-layersecurity coalitional game can be solved through a three-stagealgorithm, given in Algorithm 2. Details are elaborated asfollows:• Initial Stage: The source S broadcasts its information to

nearby nodes. Following (9), an initial set R, consistingof L suitable nodes, is selected and we assume all suitablenodes in R are non-cooperative in the beginning. Asa result, the partition structure is initialized as L :={S1,S2, ...SL}, where Si (i = 1, 2, ...L) includes oneintermediate node only.

• Coalition Forming Stage: The partition structure isadjusted adaptively based on the iterative merge-and-splitrules. Each individual player in R first checks the over-




all network status for potential cooperators in vicinity.Then, the Max-Pareto order based iterative merge-and-split operations continues until a stable partition structureis reached. Particularly, the merge (or split) operationis valid if the two following conditions are satisfied: 1)all the other players in the coalition believe that theirindividual payoffs can be improved when at least oneplayer is absorbed into (or removed from) the coalition;and 2) one or more players believe that they can achievemore benefits when being (or not being) a member of thecoalition.

• Cooperative Transmission Stage: After the stable parti-tion structure is reached, each coalition in the collectionreports its individual utility, i.e. the coalition value givenby (17). The specific coalition that provides the optimalperformance will be chosen for secure date transmission.Note that, if no coalition in the ultimate partition structurecan meet the source’s secrecy demand, no transmissionis allowed by any of the players (intermediate nodes).Under this scenario, the solution for the WFRFT-basedcoalitional game is ∅, and no secure communication linkis available from S to D.

C. Stability analysis

With the proposed algorithm, a network partition structureconsisting of multiple disjoint collisions is obtained. Wenow analyze the stability of the resulting network structure.The concept of Nash-stable is used to evaluate the obtainednetwork structure.

Definition 8: A partition structure L := {S1,S2, ...SL}is Nash-stable iff ∀Ri ∈ R, L � L′, where L′ :={S1,S2, ..., {Si −Ri}, ..., {Sj ∪Ri}, ...,SL}; and Si 6= Sj[38].

A partition structure is stable in the sense that no playerRi in the game has an incentive to leave its current coalitionSi to any other coalitions Sj . Furthermore, according toDefinition 8, a stable partition implies that no player Riin current partition structure would be beneficial by joininganother coalition while convincing other players that theirutilities would not be reduced, i.e.:

@Ri ∈ Si;φi(Si) > φi({Sj ∪Ri})s.t. ∀Rj ∈ Si;Rj 6= Ri, φj(Si) ≥ φj({Si −Ri}),

and ∀Rj ∈ Sj , φj(Sj) ≥ φj({Sj ∪Ri}).(22)

Proposition 3: For the WFRFT-based PHY-layer securitycoalitional game, the proposed Algorithm 2 converges to aNash-stable coalitional structure L?.

Proof: We can prove it based on the Merge and Split Rule.According to the definition of Pareto order, any intermediatenode Ri ∈ R is able to join any coalition only if noneof the original nodes in that coalition could be adverselyaffected. Start with an arbitrary coalitional structure L, if thereexists any node Ri that prefers to switch to another coalition,i.e.{{Si−Ri}, {Sj ∪Ri}, {Sl}l 6=i,j} � {Si,Sj , {Sl}l 6=i,j},then the current structure L is not stable. As a consequence,node Ri will be moved from Si to Sj via an iterativemerge-and-split operations (i.e., line 15-19 of Algorithm 2).

Fig. 2. The secrecy performance of WFRFT-based cooperative system versusnumber of intermediate nodes in the case of the eavesdropper gaining the fullknowledge of WFRFT.

The iterative merge-and-split operations stops until each nodesatisfies with its current situation. In other words, no one hasan incentive to leave its current coalition. As a result, an Nash-stable coalitional structure L? will be formed. �

VI. SIMULATION RESULTS AND DISCUSSIONS

In this section, simulation results are provided to evaluatethe performance of the WFRFT-based PHY-layer securitycooperation scheme. The network is set up as follows. Thesource is located at the origin of a 2.5Km × 2.5Km squarewith the destination, eavesdropper, and intermediate usersrandomly deployed in the region. We consider a block quasi-static fading channel modeled by hi,k = ejφi,k

√κ/dµi,k with

µ the propagation loss exponent, κ the path loss constant, di,kand φi,k the Euclidean distance and the phase offset betweenterminals i and k, respectively. Particularly, the propagationloss µ is set as 3 and the path loss constant κ is fixed at 1during the simulation. Moreover, the maximum transmit powerof each intermediate node is set to 20 mW, and the noisevariance is set to −70 dBm. Without loss of generality, weset parameter ξi ∈ [0, 1] and η = 1. 20000 independent trialswith randomly generated network scenarios are performed toobtain the average results.

We first evaluate the optimality of using WFRFT for PHY-layer security. Fig. 2 shows the secrecy performance of the pro-posed approach versus the number of intermediate nodes underthe scenario where the eavesdropper gains the full knowledgeof WFRFT. In particular, ∆α = β − α, where β and αrepresents the WFRFT parameters utilized by the eavesdropperand destination, respectively. Unlike other existing literaturewhere the secrecy of the parameter α directly affects thetransmission security [39], Fig. 2 shows that in our proposedscheme secure transmission can still be guaranteed even ifthe eavesdropper’s receiver operates identically with that ofthe destination. More specifically, when N is small, thereexists a slight difference in secrecy performance as ∆α varies.




Fig. 3. The secrecy performance versus the available transmission powerfor WFRFT-based cooperation, beamforming and artificial noise systems.

Fig. 4. Secrecy performance comparison of the proposed scheme with artificialnoise, beamforming, optimal relay selection and joint relay-jamming schemes.

Fig. 5. One snapshot of the final network structure resulting from the proposedcoalitional game with N = 10 randomly deployed intermediate nodes.

However, as N increases, the gap between different curvesbecomes negligible. The reason is that, with the network scaleenlarged, the number of “qualified helper” increases. As aresult, the message energy is allocated onto more sub-signalsthat will selectively degrade the eavesdropper’s reception dueto the uncertainty of wireless channels, leading to secrecyperformance enhancement.

Fig. 3 shows the secrecy performance versus the availabletransmission power for WFRFT-based cooperation, beamform-ing and artificial noise schemes. It can be seen that theattainable ergodic secrecy capacity of these three schemesall increase proportionately with the achievable transmissionpower of intermediate nodes. However, as the achievable trans-mission power increases, WFRFT-based cooperation scheme

Node

Node

Fig. 6. Self-adaption of Algorithm 2 in handling the mobility of network’stopology as Node 6 moves towards Node 2.

outperforms both beamforming and artificial noise schemes.This is because the secrecy of the WFRFT-based cooperationscheme comes from the mutual-interference between the sig-nal components from different intermediate nodes. Therefore,with more achievable transmission power, the intermediatenodes bear more information exchange with D. However,the eavesdropper could not benefit from the increment ofthe intermediate nodes’ power consumption. As a result, theintermediate nodes can provide better chance to form WFRFT-based cooperation with improved secrecy capacity.

In Fig. 4, we compare the WFRFT-based cooperationscheme with beamforming, artificial noise, optimal relay s-election and joint relay-jamming schemes, in terms of thesecrecy performance. It can be seen that, as the network size




N increases, the WFRFT-based cooperative system providessignificant secrecy rate gains, reaching up to 121.6% and149.5% at N = 14, compared with the artificial noise andbeamforming schemes, respectively. Note that artificial noisescheme outperforms the proposed scheme at the beginning.This due to the fact that, under the small network scalescenario, there is less freedom for the WFRFT-based schemeand beamforming scheme to optimize their transmission strate-gies. As a result, the artificial noise scheme achieves a bettersecrecy performance. However, as more intermediate nodes aredeployed, the likelihood of finding an optimal coalition withimproved secrecy for the WFRFT-based cooperation schemeincreases. Besides, it is noticed that both beamforming andartificial noise insertion used in Fig. 4 have been formedas the coalition game. Therefore, a simplified relay selectionprocedure have already been involved. That’s why they exhibitthe similar secrecy tendency as the optimal relay selection andjoint relay-jamming schemes. More importantly, the secrecyof the WFRFT-based cooperation scheme relies on the mutual-interference between the signal components from differentintermediate nodes. Therefore, the WFRFT-based cooperationscheme outperforms the artificial noise scheme since no extrapower consumption is required to generate non-informationbearing signal.

Next, we evaluate the feasibility of the proposed coalitionalgame strategy. We start with a simple case consisting ofN = 10 randomly deployed intermediate nodes. Fig. 5 showsone snapshot of the network structure obtained from theproposed coalitional game. 4 independent coalitions are havebeen formed in the depicted network. In particular, the coali-tion formation process follows the iterative merge-and-splitrule given by Algorithm 2. For example, node 6, located faraway from any other nodes, can not find suitable partners forcooperation. Coalition {1,3,5,9} is formed due to the fact thatthe individual utility V ({1, 5, 3, 9}) = [0.37, 0.61, 0.26, 1.03]which is an apparent improvement compared with the ze-ro utilities obtained by acting alone (in the proximity ofeavesdropper). Moreover, the coalition {4,8,10} is selectedas the output for its optimal performance. In general, Fig. 5illustrates how the distributed intermediate nodes self-organizeinto multiple disjoint coalitions using Algorithm 2, to improvethe PHY-layer secrecy of the network.

Fig. 6 shows the performance of Algorithm 2, consideringthe node mobility in the network. It shows the individualutilities of nodes {2,4,6,7,8,10} as node 6 moves towards node2 for 2.5 km. As shown in Fig. 5, when node 6 moves towardsnode 2, it gets closer to D, and hence, its utility increases atfirst. Meanwhile, the utilities of other nodes remain unchangedas long as node 6’s moving distance is less than 0.4 km, sinceno alteration is made to the network topology. After that, node6 is absorbed as a new member of collision {4,8,10}, andthe utilities of all four nodes increases significantly. This isbecause the new collision improves secrecy performance withlower power cost. Further, when the movement distance ofnode 6 reaches 0.8 km, it begins to move away from its currentpatenters and the cost in coalition {4,6,8,10} increases. As aresult, at 1.3 km, node 6 will split from coalition {4,7,8,10}to guarantee the performance of the network. When node

Fig. 7. Performance comparison in terms of secrecy ergodic capacity betweenAlgorithm 2, opportunistic selection and exhaustive searching.

6 moves around 1.8 km, it would be beneficial for node 6to join collision {2,7}, however, collision {4,8,10} will stillbe the final option due to its superior performance. Theseresults illustrates how the network topology is dynamicallyself-organized using Algorithm 2.

Fig. 7 compares different selection schemes, in terms ofsecrecy ergodic capacity, with respect to the network size N .It can be seen the proposed coalitional game scheme achievesbetter secrecy performance, compared with the opportunisticmethod [22]. The performance gain increases with the networksize N . However, there is a performance gap between theoptimal exhaustive search solution. This is because that theproposed coalitional game is solved by the distributed merge-and-split algorithm that only provides locally optimal solutionswithin the discovering range of a collision. Nevertheless,compared to the exhaustive search solution, the proposedalgorithm still achieves a comparable performance with asecrecy degradation less than 5% at N = 10. Moreover, albeitthe performance loss, the proposed algorithm provides a morefeasible solution in handling the user cooperation. Particularly,for the network size N ≥ 10, finding the optimal collisionsby exhaustive searching will be computationally infeasible.

VII. CONCLUSION

In this paper, we have proposed a WFRFT-based cooper-ation scheme to enhance PHY-layer security in an energy-efficient way. Different from conventional “artificial noise”schemes that requires extra jamming signals to disrupt theeavesdroppers reception, based on WFRFT, the proposedscheme can create an identical “artificial noise” effect with theinformation bearing signal to confound the eavesdropper whileimposing no impact on the legitimate receiver. To maximizethe secrecy of the network efficiently, we have proposed aWFRFT-based PHY-layer security cooperation scheme basedon NTU coalitional game and developed a distributed merge-and-split algorithm to autonomously construct the coalition.The proposed algorithm is stable and can efficiently adapt




to dynamic network structure change, such as the mobilityof intermediate nodes. Simulation results demonstrate thatthe WFRFT-based user cooperation scheme can achieve sig-nificant performance advantage, in terms of secrecy ergodiccapacity, compared with conventional PHY-layer security-oriented user cooperation schemes, such as relay-jamming andcluster-beamforming.

For the future work, we will consider a more general casewith the multiple source-destination pairs. In addition, wewill also consider how to perform cooperation for PHY layersecurity without CSI [40, 41].

APPENDIX APROOF OF THE PROPERTIES OF CWFRFT

A. Proof of Property 1

Let a ∈ R, b ∈ R and a 6= b. Following the definition ofCWFRFT in (3), if F a

c (X) = F bc (X), there must be {wap =

wbp}3p=0, i.e.

3∑k=0

exp

{−2πj(a− p)k

4

}≡

3∑k=0

exp

{−2πj(b− p)k

4

}for p = 0, 1, 2, 3.

(23)then, the equality of (23) holds when(

exp

(−2πjak

4

)− exp

(−2πjbk

4

))3

k=0

= 0 (24)

Therefore, a− b = 4n, n ∈ Z. �

B. Proof of Property 2

According to the Periodicity property of CWFRFT,Fα+4c (X) = Fα

c (X), Property 2 can be proved if

F 〈α〉4c (X) = F〈α〉4 · X, (25)

where 〈·〉4 the denotes modulo-4 calculation. Let {α, l} ∈{0, 1, 2, 3}, it is easy to verify that

wαl =

{1 (l = α)0 (l 6= α).

(26)

It is certain that Fαc (X) = Fα · X, for α ∈ {0, 1, 2, 3}. Thus

we complete the proof. �

C. Proof of Property 3

Straightforwardly, Property 3 can be proved if the coeffi-cients of F a+b

c (·), F ac (F b

c (·)) and F bc (F a

c (·)) are equivalent,namely, all of the following equations are satisfied.

wa+b0 = wa0wb0 + wa1w

b3 + wa2w

b2 + wa3w

b1


b0 + wa2w

b3 + wa3w

b2


b1 + wa2w

b0 + wa3w

b3


b2 + wa2w

b1 + wa3w

b0

(27)

The equality of (27) can be easily verified through some basicalgebraic manipulations. �

APPENDIX BPROOF OF THEOREM 2

Theorem 2 is proved by induction. For briefness, we use{Ak,l(αk)}k−1l=0 to denote the weighting coefficients of k-WFRFT.

Step 1 : (Base case) We prove that Theorem 2 holds forM = 5, i.e. α5 = 5

4α4, when Wα44 = Wα5

5 .By (1) and Theorem 1, we have

Wα4

4 =3∑

n=0

A4,n(α4)Fn, (28)

and

Wα5

5 =4∑l=0

A5,l(α5)W4l54

=4∑l=0

A5,l(α5)3∑

n=0

A4,n(4l

5)Fn.

(29)

Let Φn =∑4l=0A5,l(α5)A4,n( 4l

5 ), then we have Wα5

5 =∑3n=0 ΦnFn. Explicitly, Wα5

5 = Wα44 iff Φn = A4,n(α4),

by which the relationship between α4 and α5 can be proved.When n = 0: Let ∆l = A5,l(α5)A4,0( 4l

5 ), where l =0, 1, 2, 3. According to Property 2, we have A4,0(0) = 1, andtherefore

∆0 = A5,0(α5) =1

5

1− exp [−2πj(α5)]

1− exp [−2πj(α5)/5]. (30)

By (2), the following equation holds4∑l=1

∆l =1− 5 exp (−8πjα5/5) + 4 exp (−2πjα5/5)

20[1− exp (−2πjα5/5)](31)

Substituting (30) and (31) into (29) yields:

Φ0 =1− exp (−8πjα5/5)

4[1− exp (−2πjα5/5)]. (32)

Moreover, since

A4,0(α4) =1− exp (−2πjα4)

4[1− exp (−2πjα4/4)],

it holds that if A4,0(α4) = Φ0, there must be α5 = 54α4.

The same results can also be obtained for the scenariosn = 1, 2 and 3, and the proof is omitted here.

Step 2: (Inductive step) Suppose that Theorem 2 holdsfor M = k (k > 5). i.e., if Wα4

4 = Wαk

k , there must beαk = k

4α4. Then, based on this assumption, we prove thatTheorem 2 holds for M = k + 1 (k > 5).

To prove the relationship between αk and α4, we first provethat αk+1 = k+1

k αk is true for Wαk

k = Wαk+1

k+1 . According to(1), Wαk+1

k+1 can be expressed by

Wαk+1

k+1 =k∑l=0

Ak+1,l(αk+1)W4l

k+1

4 . (33)

With the aforementioned assumption, by substituting Wα44

with Wαk

k , (33) can be rewritten as:

Wαk+1

k+1 =4∑l=0

Ak+1,l(αk+1)Wkl

k+1

k (34)




Since αk+1 = k+1k αk, using (1), we obtain the following

constraint:k∑l=0

Ak+1,l(αk+1)

k−1∑n=0

Ak,n(kn

k + 1)T

4nk =

k−1∑n=0

Ak,n(αk)T4nk (35)

Denote Φn =∑kl=0Ak+1,l(αk+1)Ak,n( kn

k+1 ). Then, theequality of (35) holds iff Φn = Ak,n(αk).

Similar to Step 1, we consider the scenario n = 0 for ex-ample. Denote ∆l = Ak+1,l(αk+1)Ak,n( kn

k+1 ). When n = 0,Ak,0(0) = 1, we have :

∆0 = Ak+1,l(αk+1) =1

k + 1

1− exp [−2πjαk+1]

1− exp[−2πjαk+1

k+1

] . (36)

whereas for {∆l}kl=1, it holds that

k∑l=1

∆l =1− (k + 1) exp (

−2πkjαk+1

k+1) + k exp (−2πjαk+1)

k(k + 1)[1− exp (

−2πjαk+1

k+1)] (37)

With (36) and (37), we obtain:

Φ0 =1− exp (−2πkjαk+1

k+1 )

k[1− exp (−2πjαk+1

k+1 )] . (38)

According to the constraint Φ0 = Ak,0(αk), the followingequation holds:

1− exp (−2πkjαk+1

k+1 )

k[1− exp (−2πjαk+1

k+1 )] =

1− exp [−2πjαk]

k[1− exp

(−2πjαk

k

)] . (39)

Therefore, it is true that αk+1 = k+1k αk. Then, by substituting

αk with k4α4, we obtain that αk+1 = k+1

4 α4.Thus, we complete the proof. �

APPENDIX CPROOF OF THEOREM 3

Theorem 3 is proved by contradiction. Suppose that theequality of Wα

M · WβM = I holds under the assumption

α + β 6= 0 or {Al(α)}M−1l=0 (or/and {Al(β)}M−1l=0 ) are con-taminated. Typically, {Al(α)}M−1l=0 is contaminated, implyingthat the weighting coefficients are not generated properly, i.e.:

Al(α) =1

M


+ ∆l. (40)

The proof can be completed by considering the following twocases.

Case 1: Suppose that α+β = η. According to the additionaxiom given in Definition 2, it holds that Wα

M ·WβM = Wη

M=I, which contradicts the boundary axiom.

Case 2: Suppose that

Al(α) =1

M


+ ∆l,

Al(β) =1

M

1− exp [−2πj(β − l)]1− exp [−2πj(β − l)/M ]

,

(41)

which means that only {Al(α)}M−1l=0 is contaminated.For briefness, we only consider the case where M = 4

(similar results can also be obtained for M 6= 4). According

to the addition axiom, we have Wα4 · Wβ

4 = Wα+β4 . Since

Wα+β4 =

∑3n=0Al(α+ β)Fn, it holds that Wα+β

4 = I, whenA0(α + β) = 1 and {Al(α + β) = 0}3l=1 = 0. SubstitutingAl(α) and Al(β) into (27), when α+ β = 0, we have:

A0(0) = 1 +∑3

l=0∆lA〈4−l〉4(β) = 1

A1(0) = 0 +∑3

l=0∆lA〈5−l〉4(β) = 0

A2(0) = 0 +∑3

l=0∆lA〈6−l〉4(β) = 0

A3(0) = 0 +∑3

l=0∆lA〈3−l〉4(β) = 0

(42)

Then, the equality of (42) is achieved when {∆l = 0}3l=0,which contradicts the assumption.

Similarly, we can obtain the same results for the scenariosthat only {Al(β)}M−1l=0 is contaminated or both {Al(α)}M−1l=0

and {Al(β)}M−1l=0 are contaminated. Thereby, we concludethat the constraints in Theorem 3 must be true to achieveWαMWβ

M = I, and thus the proof is complete.

ACKNOWLEDGMENT

This work was completed during Xiaojie Fang’s visit to theBroadband Communications Research Group (BBCR), Uni-versity of Waterloo and was supported by the National BasicResearch Program of China under Grant 2013CB329003, theNational Nature Science Foundation of China under Grant61671179 and the Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada.

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Xiaojie Fang (IEEE S’14) received his B.Sc. andM.Sc. degrees from Department of Electronics andInformation Engineering, Harbin Institute of Tech-nology in 2010 and 2012, respectively. He was avisiting scholar in the Broadband CommunicationsResearch Group (BBCR), Department of Electricaland Computer Engineering, University of Waterloo,Waterloo, ON, Canada. He is currently pursuing thePh.D. degree with the Department of Electronicsand Information Technology, Harbin Institute ofTechnology. His current research interests include

physical layer security, coding and modulation theory.

Ning Zhang (IEEE S’12-M’16) received the Ph.Ddegree from University of Waterloo in 2015. Hereceived his B.Sc. degree from Beijing Jiaotong Uni-versity and the M.Sc. degree from Beijing Universityof Posts and Telecommunications, Beijing, China,in 2007 and 2010, respectively. From May 2015to Apr. 2016, he was a postdoc research fellow atBBCR lab in University of Waterloo. Since May2016, he has been a postdoc research fellow atWireless Computing lab at University of Toronto. Heis now an associate editor of International Journal

of Vehicle Information and Communication Systems and a guest editor ofMobile Information System. He is the recipient of the Best Paper Award atIEEE Globecom 2014 and IEEE WCSP 2015. His current research interestsinclude sensor networks, next generation wireless networks, software definednetworking, green communication, and physical layer security.

Shan Zhang (IEEE S’13-M’16) received her Ph.D.degree in Department of Electronic Engineeringfrom Tsinghua University and B.S. degree in Depart-ment of Information from Beijing Institute Technol-ogy, Beijing, China, in 2016 and 2011, respectively.She is currently a post doctoral fellow in Depart-ment of Electronical and Computer Engineering,University of Waterloo, Ontario, Canada. Her re-search interests include resource and traffic manage-ment for green communication, intelligent vehicularnetworking, and software defined networking. Dr.

Zhang received the Best Paper Award at the Asia-Pacific Conference onCommunication in 2013.




Dajiang Chen (IEEE M’15) is currently a PostDoctoral Fellow at the University of Waterloo, Wa-terloo, ON, Canada. He is also a Post DoctoralFellow in the School of information and softwareEngineering at University of Electronic Science andTechnology of China. He received the B.Sc. de-gree in 2005 and the M.Sc. degree in 2009 fromNeijiang Normal University and Sichuan University,respectively, and the Ph.D. degree in information andcommunication engineering from UESTC in 2014.His current research interest includes Information

Theory, Channel coding, and their applications in Wireless Network Security,Wireless Communications and other related areas.

Xuejun Sha (IEEE M’09) is a Professor in Commu-nication Research Center (CRC) at the Departmentof Electronics and Information Engineering, Harbin,China. He joined CRC as a Teaching Assistant in1993, and became a Lecturer and a Professor in1994 and 2000, respectively. He received his Ph.D.degree in Communications and Electronics Systemsfrom Harbin Institute of Technology, in 1995. From1997 to 1998, he was a postdoctoral fellow inKorea University, Seoul, Korea. His main researchinterests include communication network, wireless

ad-hoc network, and broadband wireless access.

Xuemin (Sherman) Shen (IEEE M’97-SM’02-F’09) received the B.Sc.(1982) degree from DalianMaritime University (China) and the M.Sc. (1987)and Ph.D. degrees (1990) from Rutgers University,New Jersey (USA), all in electrical engineering.He is a University Professor and the AssociateChair for Graduate Studies, Department of Electricaland Computer Engineering, University of Waterloo,Canada. Dr. Shens research focuses on resourcemanagement in wireless networks, wireless networksecurity, social networks, smart grid, and vehicular

ad hoc and sensor networks. He was an elected member of IEEE ComSocBoard of Governor, and the Chair of Distinguished Lecturers SelectionCommittee. Dr. Shen served as the Technical Program Committee Chair/Co-Chair for IEEE Globecom16, Infocom14, IEEE VTC10 Fall, and Globecom07,the Symposia Chair for IEEE ICC10, the Tutorial Chair for IEEE VTC’11Spring and IEEE ICC08, the General Co-Chair for ACM Mobihoc15, theChair for IEEE Communications Society Technical Committee on Wire-less Communications, and P2P Communications and Networking. He alsoserves/served as the Editor-in-Chief for IEEE Network, Peer-to-Peer Net-working and Application, IET Communications, and Editor-in-Chief for IEEEInternet of Things Journal; a Founding Area Editor for IEEE Transactionson Wireless Communications; an Associate Editor for IEEE Transactions onVehicular Technology, Computer Networks, and ACM/Wireless Networks. Dr.Shen received the Excellent Graduate Supervision Award in 2006, and theOutstanding Performance Award in 2004, 2007, 2010, and 2014 from theUniversity of Waterloo, the Premiers Research Excellence Award (PREA)in 2003 from the Province of Ontario, Canada, and the DistinguishedPerformance Award in 2002, 2007, 2016 from the Faculty of Engineering,University of Waterloo. Dr. Shen is a registered Professional Engineer ofOntario, Canada, an IEEE Fellow, an Engineering Institute of Canada Fellow,a Canadian Academy of Engineering Fellow, a Royal Society of CanadaFellow, and a Distinguished Lecturer of IEEE Vehicular Technology Societyand Communications Society.

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JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX ......JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, XXXXX 1 On Physical Layer Security: Weighted Fractional Fourier Transform based User