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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TGCN.2016.2646819, IEEETransactions on Green Communications and Networking

Energy-efficiency Maximization for CooperativeSpectrum Sensing in Cognitive Sensor Networks

Meng Zheng, Member, IEEE, Lin Chen, Member, IEEE, Wei Liang, Member, IEEE, Haibin Yuand Jinsong Wu, Senior Member, IEEE

Abstract—Spectrum sensing is the prerequisite of opportunisticspectrum access in cognitive sensor networks (CSNs) as itsreliability determines the success of transmission. However,spectrum sensing is an energy-consuming operation that needsto be minimized for CSNs due to resource limitations. Thispaper considers the case where the cognitive sensors cooperativelysense a licensed channel by using the CoMAC-based cooperativespectrum sensing (CSS) scheme to determine the presence ofprimary users. Energy efficiency (EE), defined as the ratio of theaverage throughput to the average energy consumption, is a veryimportant performance metric for CSNs. We formulate an EE-maximization problem for CSS in CSNs subject to the constrainton the detection performance. In order to address the non-convexand non-separable nature of the formulated problem, we firstfind the optimal expression for the detection threshold and thenpropose an iterative solution algorithm to obtain an efficient pairof sensing time and the length of the modulated symbol sequence.Simulations demonstrate the convergence and optimality of theproposed algorithm. It is also observed in simulations that thecombination of the CoMAC-based CSS scheme and the proposedalgorithm yields much higher EE than conventional CSS schemeswhile guaranteeing the same detection performance.

Index Terms—Cognitive radio, spectrum sensing, multiple-access channel, analog computation, energy efficiency.

I. INTRODUCTION

W ITH respect to the significant growth in request forwireless services and the scarcity of the spectrum

resources, cognitive radio (CR) technology which allows thesecondary users (SUs) to access the vacant licensed spectrumof the primary users (PUs) in an opportunistic way, hasbeen proposed [1]-[2]. Spectrum sensing functions have beenfrequently considered as important components in existing CRapproaches, such as [3]-[5]. However, due to the presenceof noises and fading of wireless channels, hidden terminals,obstacles, and so on, spectrum sensing of individual nodes

This work was supported by Natural Science Foundation of China undergrant (61233007, 61304263, 61673371), Liaoning Provincial Natural ScienceFoundation of China (2014020081), Youth Innovation Promotion Association,CAS (2015157). The work of Lin Chen was supported by French AgenceNationale de la Recherche (ANR) under the grant Green-Dyspan (ANR-12-IS03). The work of Jinsong Wu was supported by ERANet LAC ProjectELAC2015/T10-0761 and CONICYT FONDEF ID16I10466. (CorrespondingAuthor: Haibin Yu and Wei Liang)

Meng Zheng, Wei Liang and Haibin Yu are with State Key Laboratory ofRobotics, 110016 Shenyang, China, and with Key Laboratory of NetworkedControl Systems, Chinese Academy of Sciences, 110016 Shenyang, China.(email: zhengmeng 6, weiliang, [email protected]).

Lin Chen is with Laboratoire de Recherche en Informatique (LRI), Univer-sity of Paris-Sud XI, 91405 Orsay, France. (email: [email protected]).

Jinsong Wu is with Department of Electrical Engineering, Universidad deChile, Santiago, 8370451, Chile, and with School of Software, Central SouthUniversity, Changsha, 410083, China. (email: [email protected]).

cannot achieve high detection accuracies [6]-[7]. In contrast,cooperative spectrum sensing (CSS) exploits a parallel fusionsensing architecture in which independent secondary users(SUs) transmit their sensing data to a fusion center in dif-ferent time slots based on a time division multiple access(TDMA) scheme [8]-[15] or a random access scheme [16]-[17], due to the extremely limited bandwidth of the commoncontrol channel1. The fusion center then makes a final softor hard decision regarding the presence or absence of PUs.The cooperative gains in spectrum sensing performance havebeen extensively demonstrated in existing works [6]-[17].However, the drawback of CSS schemes is that both theenergy consumption and the reporting time grow linearlywith the number of cooperative SUs. Our recent work [18]proposed a time-efficient CSS scheme that takes advantageof the computation over multiple-access channel (CoMAC)method [19] to accelerate the CSS process via merging thereporting and the computation steps of CSS, which noticeablyreduces the reporting delay in conventional CSS schemes.However, the issue of high energy consumption still existsdue to the fact that each SU has to send a long sequence ofencoded symbols to the fusion center during the CSS process.

In this paper, we are interested in the problem of CSS incognitive sensor networks (CSNs) [20] that are referred toas wireless networks of low-power radios gaining secondaryspectrum access according to the CR paradigm previouslydiscussed. CSNs can be particularly used in industrial sensornetworks [21] which have recently been considered as anopportunity to realize reliable and low-cost remote monitoringsystems for smart grids [22], and their performance has beeninvestigated for various application domains of smart gridsin [23]. The cognitive sensors (CSs) of CSNs are typical-ly powered by batteries, which must be either replaced orrecharged when or before out of energy. However, it is notalways possible to renew or recharge the powers of CS nodes,such as those in unapproachable or hazard zones, which wouldsimply be discarded once their energy sources are depleted.

Motivated by the above discussion, this paper adopts theCoMAC-based CSS scheme due to its high time efficiencyfor CSNs. To reduce the energy consumption of the CoMAC-based CSS scheme, we propose the joint optimization ofthe sensing time, the detection threshold and the length ofsymbol sequence to maximize the energy efficiency (EE) ofCSNs. By considering all possible scenarios between PUs

1Notice that, in some cases, such as those in [7], a dedicated control channelis not mandatory for reporting the sensing results in the conventional CSSschemes.



and CSs, we define the EE of the CSN as the ratio of theaverage throughput to the average energy consumption. Dueto the complicated expression of the EE, the formulated EE-maximization problem is non-convex and non-separable withrespect to the optimization variables and thus difficult to solvefor global optimality. To address this issue, we first find anoptimal expression for the detection threshold by rigorousproof and then propose an iterative solution algorithm (ISA)to obtain an efficient pair of the sensing time and the lengthof the modulated symbol sequence. Specifically, for a givenlength of symbol sequence, we prove the quasi-convexity ofthe EE-maximization subproblem in terms of the sensing timeand propose a bisection-based solution algorithm to solve it.Then, provided the newly computed sensing time, we find theoptimal length of symbol sequence to the EE-maximizationsubproblem by using exhaustive search. The above procedurerepeats until convergence. The major contributions of thispaper are summarized as follows:• For the CoMAC-based CSS scheme, we for the first time

formulate an EE-maximization problem using the sensingtime, the detection threshold and the length of symbolsequence as variables to jointly maximize the EE of CSNswhile giving adequate protection to the PUs.

• By observing that the PU protection constraint is alwaystight in the optimal condition, we achieve a closed-formexpression for the optimal detection threshold. Further,by eliminating the optimal detection threshold, we arriveat a simplified but equivalent form of the original EE-maximization problem.

• We propose an ISA to the simplified EE-maximizationproblem. In spite of the non-convex and non-separablenature of the EE-maximization problem, it is shown insimulations that the proposed ISA is not sensitive to theinitial value of variables and always converges to the samemaximum EE.

The rest of this paper is organized as follows. SectionII reviews the related works of this paper. Section III firstgives out the system model and then briefly illustrates theCoMAC-based CSS scheme. In Section IV, the EE expressionof the CoMAC-based CSS scheme is presented and an EE-maximization problem subject to the constraints of protectingPUs is formulated. In Section V, we present the ISA tosolve the EE-maximization problem. Simulation results andconclusions are provided in Section VI and Section VII,respectively.

II. RELATED WORK

In the literature, there are various energy-efficient CSSschemes [20]-[32] to reduce the energy consumption duringthe CSS process. Maleki et al. [20] proposed a censoring-basedCSS scheme for CSNs and optimized the censoring thresholdsto minimize the energy consumption while considering theconstraints on the detection accuracy. Deng et al. [24] reducedsensing energy consumption via partitioning the set of the SUsinto several subsets and activating one subset at certain period.Najimi et al. [25]-[26] proposed an algorithm to divide thesecond users into subsets. Only the subset that has the lowest

cost function (while the cost function is represented by thetotal energy consumption) and guarantees the desired detectionaccuracy is selected while the other subsets enter a low powermode. Unlike previous works [20]-[26] whose optimizationobjectives were to minimize the energy consumptions ofspectrum sensing, Monemian et al. [27] and Najimi et al. [28]proposed the sensor selection method for CSS in CSNs withthe aim of energy balancing or lifetime maximization. Feng etal. [29] proposed an energy-aware utility function to strike abalance between energy consumption and system throughput.The utility function was maximized by optimizing the sensingtime subject to the constraints of sufficient protection for PUs.Huang et al. [30] proposed a novel metric-average sensingenergy efficiency measured in [dB/Joule/SU] to evaluate thetradeoff between sensing gain and energy consumption. Themetric was then maximized by optimizing the number ofSUs. Peh et al. [31] for the first time defined the EE as theaverage successfully transmitted data normalized by the energyconsumption, and optimized the sensing parameters in orderto maximize the EE of cognitive radio networks. Althunibatet al. [32] proposed an objection-based CSS scheme whichhighly reduced the number of reporting SUs via arranging oneSU to broadcast its local decision among the whole networkand maximized the EE of the objection-based CSS scheme byoptimizing the selection of the broadcasting SU. More relatedworks can be found in a recent survey by Althunibat et al.[33].

Based on the the spirits of conventional energy-efficient CSSschemes [16]-[25], this paper proposes to maximize EE ofthe CSS scheme while preserving PU protection constraints.There are mainly three differences between this paper andconventional CSS schemes [20]-[32].• First, this paper performs an EE-maximization for the

CoMAC-based CSS scheme which is a soft decision innature, while [20]-[32] optimize the sensing parametersof CSS schemes that are based on the hard decision fusionrule (“K-out-of-N”).

• Second, the EE expression of the studied CSS scheme ismuch more complicated than metrics of the existing ones,which leads the formulated EE-maximization difficult tosolve for global optimality due to its non-convex and non-separable nature.

• Third, unlike the solution methods in conventional CSSschemes [20]-[32], this paper proposes to decouple theEE-maximization problem via mathematical decomposi-tion into two separated subproblems and to solve eachof them iteratively until convergence. One subproblemis the sensing time optimization problem that is shownquasi-convex and solved by a bisection-based solutionalgorithm. The other subproblem is the sequence lengthoptimization problem that is in nature an integer program-ming and solved by one-dimensional exhaustive search.

III. SYSTEM MODEL AND COMAC-BASED COOPERATIVESPECTRUM SENSING

We consider a single-hop infrastructure CSN with M CSs,one fusion center, one control channel and one licensed



Fig. 1. The time frame architecture for the periodic spectrum sensing.

channel with bandwidth W Hz. It is assumed that all CSs arehomogeneous. The time period is partitioned into frames andall nodes are synchronized with the fusion center2. Each frameis designed with the periodic spectrum sensing for the CSN.Fig. 1 shows that the frame structure consists of three phases:a sensing phase, a reporting phase, and a transmission phase.In the sensing phase, all cooperative CSs perform spectrumsensing via energy detection. In the reporting phase, the localsensing data at each CS is reported to the fusion center inthe form of a sequence of power encoded symbols. Then,the fusion center makes the global decision with respect toreceived local sensing results according to a given detectionthreshold and broadcasts the global decision over the controlchannel at the end of the reporting phase. The CSN selects anarrow band unlicensed spectrum as the control channel whichis contention-free but may suffer a flat-fading process. Thetime for decision fusion and broadcast at the fusion center isfixed and we set the time as one reporting time unit (or mini-slot) for simplicity in the following analysis. The transmissionphase is slotted. In the transmission phase, the transmissions ofCSs are scheduled by a TDMA-MAC protocol if the PUs aredetected absent by the end of the reporting phase. Specifically,at the beginning of the transmission phase, the schedule ofdata transmission is centrally performed and broadcast by thefusion center over the control channel. After retrieving theschedule, each CS transmits their data to the fusion center inthe allocated time slot and falls asleep for the rest of time.Let τs and τt denote the lengths of the sensing phase and thetransmission phase, respectively, and let τ denote one mini-slotlength in the reporting phase. Then one frame length is givenby T = τs +2τ +τt . It is possible that PUs are inactive duringthe spectrum sensing time, but later become active during thefollowing transmission time. However, the probability of thisevent is negligible due to low spectrum utilization rate of PUsand short duration of the periodic spectrum sensing. To makethe analysis tractable as well as focus on the intrinsic featureof the studied problem, we make an assumption to limit thestate of the PUs to be fixed within one frame. For notational

2As the studied CSN is fully centralized, all nodes synchronize to thefusion center when they join the network. After the joining process, thesynchronization among all nodes can be achieved from the periodic broadcastof global decision by the fusion center.

convenience, we drop the time index and consider an arbitraryframe.

In the rest of this section, we will briefly introduce theCoMAC-based CSS scheme.

A. Spectrum Sensing

The local spectrum sensing problem at each CS, say i (i =1,2, . . . ,M ), can be formulated as a binary hypothesis betweenthe following two hypothesis:

H0 : yi(n) = ui(n), n = 1,2, ...,NH1 : yi(n) = hi(n)s(n)+ui(n), n = 1,2, ...,N,

(1)

where H0 and H1 denote the PUs are absent and present onthe licensed channel, respectively. N = τs f s denotes the num-ber of samples, where f s represents the sampling frequencyof CSs. yi(n) represents the received signal at the CS i. s(n) isthe PU signal whose power is σ2

s . The channel gain |hi(n)| isassumed Rayleigh-distributed with the same variance σ2

h . ui(n)represents the circular symmetric complex Gaussian noise withmean 0 and variance σ2

u . Assume that s(n), ui(n), and hi(n)are independent of each other, and the average received SNRat each CS is given as γ =

σ2h σ2

sσ2

u. For spectrum sensing, CS i

then uses the average of energy content in received samplesas the test statistic for energy detector, which is given by

ψi(y) =1N

N

∑n=1|yi(n)|2.

With data fusion, the global test statistic used for final decisionat the fusion center is then represented as

T alls (y) =

M

∑i=1

Ti(y), (2)

where Ti(y) = giψi(y) is the local statistic of the CS i andgi ≥ 0 is the weighting factor associated with CS i. Without

loss of generality, we assumeM∑

i=1gi = 1.

B. Reporting Phase

Conventionally, in order to compute T alls (y), the fusion

center has to collect Ti(y) from CSs successively. We observethat the transmission of Ti(y) and the computation of T all

s (y)in conventional CSS schemes are separate in the time domain.Such separation-based computation schemes are generally in-efficient as a complete reconstruction of individual Ti(y) at thefusion center is unnecessary to compute T all

s (y). In contrast,this paper proposes to merge the process of Ti(y) transmissionand T all

s (y) computation via exploiting the CoMAC scheme,which takes only one time unit of the reporting phase.

In the reporting phase, CS i transmits a distinct complex-valued symbol sequence at a transmit power that dependson the the value of Ti(y). Specifically, the transmit power ofCS i is Pi(Ti(y)) =

Pmax(Ti(y)−xmin)xmax−xmin

, where Pmax represents thetransmit power constraint on each CS and [xmin, xmax] (xmin <xmax) denotes the sensing range in which the CSs are able toquantify values. For example, a low-power temperature sensor[34] operates in a typical sensing range [−55oC,130oC]. For



notation convenience, we define αarit =Pmax

xmax−xmin. Each CS

then independently generates a sequence of random transmitsymbols with length L. Let Si := (Si[1],Si[2], ...,Si[L])T withSi[m] = eiθi[m](m = 1,2, ...,L) denote the sequence of CSi, where i denotes the imaginary unit and θi[m]i,m arecontinuous random phases that are independent identicallyand uniformly distributed on [0,2π). Then the m-th transmitsymbol of CS i takes the form

Γi[m] =

√Pi(Ti(y))|Hi[m]|

Si[m], (3)

where |Hi[m]| denotes the channel magnitude of an indepen-dent complex-valued flat fading process between CS i andthe fusion center. We assume the channel gain Hi[m] doesnot change within one frame length. Therefore, we haveHi[m] = Hi, m = 1,2, . . . ,L. It has been reported in [35] thatthe channel magnitude |Hi| at the transmitter is sufficientto achieve the same performance as with full channel stateinformation. Notice that, the channel inversion in (3) may leadto significant energy consumption especially when the fadingis severe. Therefore, we adopt a weighed sum of data fusionby setting

gi =|Hi|2

M∑

i=1|Hi|2

.

In practical systems |Hi|,giM can be obtained by the fusioncenter through channel training and estimation, and togetherwith the global decision is broadcast by the fusion centerperiodically. In doing so, small weights will be assigned to theCSs who suffer severe fading and their transmit powers can besignificantly saved. When gi =

1M (i = 1,2, . . . ,M), this work

becomes the CoMAC-based CSS scheme in [18]. Without lossof generality, we take [18] as an example and study its EE-maximization problem in the rest of this paper.

Concurrent transmission of CSs yields the output at thefusion center

Y =M

∑i=1

√Pi(Ti(y))Si +U, (4)

where Y := (Y [1],Y [2], ...,Y [L])T ∈ CL represents the outputvector at the fusion center, and U := (U [1],U [2], ...,U [L])T ∈CL represents an independent stationary complex Gaussiannoise vector.

By estimating the energy of signal (4), i.e., ∥Y∥22, the fusion

center computes an unbiased and consistent estimate fL(T(y))of the overall test statistic T all

s (y) for spectrum sensing, whereT(y) = [T1(y),T2(y), . . . ,TM(y)]. Specifically, fL(T(y)) is givenas follows [18]:

fL(T(y)) :=1

Mαarit

(∥Y∥2

2L−σ2

u

)+ xmin. (5)

To make the final decision, the fusion center comparesfL(T(y)) with a threshold εs. The PUs are estimated to beabsent if fL(T(y))< εs, or present otherwise.

The detection probability and the false alarm probability ofthe CoMAC-based CSS scheme are given by [18]

Pd = Q

Mαarit(εs− (1+ γ)σ2

u)√

Ω1(M−1+2σ2u )+σ4

uL +

Mα2arit σ4

u (2γ+1)2Wτs

(6)

Pf = Q

Mαarit(εs−σ2

u)√

Ω0(M−1+2σ2u )

L +Mσ4

u α2arit

2Wτs

(7)

where Q(x) is the complementary distribution of the stan-dard Gaussian, Ω0 = Mαarit(σ2

u −xmin) and Ω1 = Mαarit((γ +1)σ2

u − xmin).In this paper, we do not model the impact of wireless

fast fading (e.g., Rayleigh or Rician fading), which enablesus to gain insights into the EE-maximization problem whilemaintaining the solution to the formulated problem sufficientlytractable. The extension of the model in (3) to capture wirelessfading will be considered in our future works. Relevant resultspublished in the recent work [35] which combats fast fading byequipping the fusion center with multiple antennas and somestatistical channel knowledge will be useful for these furtherstudies.

IV. PROBLEM FORMULATION

In this section, we study the relationship between energyconsumption and achievable throughput of the CSN. Thereare four possible scenarios between the activities of PUs andCSs.

Scenario 1: PUs are absent and CSs detect them correctly.In this scenario, the data of CSs are transmitted after spectrumsensing. The energy cost C1 and the throughput F1 of Scenario1 are given as follows:

C1 = M(Esτs +LEtν)+Et(T −2τ− τs) (8)

F1 =T −2τ− τs

TC0 (9)

where Es and Et denote the sensing power and the transmitpower, respectively. ν denotes the symbol duration. C0 is thethroughput of the CSN if CSs are allowed to continuouslyoperate over the licensed channel in the case of H0.

Scenario 2: PUs are absent and CSs detect them as present(false alarm happens). In this scenario, the transmissions ofCSs are refrained and thus the throughput F2 is 0. The energycost C2 of Scenario 2 is given as follows:

C2 = M(Esτs +LEtν). (10)

Scenario 3: PUs are present and CSs detect them successful-ly. In this scenario, the transmissions of CSs are also refrained.Both the energy cost C3 and the throughput F3 of Scenario 3are the same as those of Scenario 2, i.e., C3 =C2 and F3 = F2.

Scenario 4: PUs are present and CSs fail to detect them. Inthis scenario, the transmissions of CSs are allowed due to themis-detection. The energy cost C4 of Scenario 4 is the same asthat of Scenario 1, i.e., C4 =C1. Let C0 denote the throughputof the CSN in the case of H1. Then the throughput F4 ofScenario 4 is given as follows:

F4 =T −2τ− τs

TC0. (11)



Let Pr(Hi) (i=0,1) denote the probability of Hi. The prob-abilities of Scenarios 1-4 are given by P1 = Pr(H0)(1−Pf ),P2 = Pr(H0)Pf , P3 = Pr(H1)Pd and P4 = Pr(H1)(1− Pd).Then, the average energy cost within a frame can be con-structed as

C(τs,εs,L)

=M(Esτs +LEtν)+(P1 +P4)(T −2τ− τs)Et . (12)

Similar to [3], we assume Pr(H1)< 0.5 (say less than 0.3, e.g.,0.2 is selected in Section VI), Pf < 0.5 and Pd > 0.5, whichcan represent most of typical CR scenarios. Otherwise, it isnot economically advisable to explore the secondary usage ofthe licensed band. As a result, we have P4 ≪ P1. Then, weapproximate C(τs,εs,L) as

C(τs,εs,L)

=M(Esτs +LEtν)+Pr(H0)(1−Pf )(T −2τ− τs)Et . (13)

The average throughput of the CSN can be calculated by

F(τs,εs,L) =T −2τ− τs

T(C0P1 +C0P4). (14)

The operation of the CSN during the secondary scenarioexperiences interference from the primary user, and hence,we have C0 < C0. Under the consideration of C0 < C0 andP4 ≪ P1, the first part of F(τs,εs,L) dominates the secondpart of F(τs,εs,L). Thus, F(τs,εs,L) can be approximated by

F(τs,εs,L) =T −2τ− τs

TC0P1. (15)

Finally, the EE of the CoMAC-based CSS schemeξ (τs,εs,L) is defined as the ratio of the average throughputto the average energy consumption, i.e.,

ξ (τs,εs,L) =F(τs,εs,L)C(τs,εs,L)

. (16)

The EE is widely considered as a comprehensive metric thatis able to represent the overall performance of a CSN, sinceit jointly takes into account the achievable throughput, theoverall energy consumption and the sensing accuracy.

The objective of this paper is to identify the optimal sensingparameters (τs,εs,L) such that the EE of the CoMAC-basedCSS scheme is maximized while the PUs are sufficientlyprotected. Let ω denote a pre-specified threshold that is closeto but less than 1. For a given ω , the EE-maximization problemcan be formulated as follows:

maxτs,εs,L

ξ (τs,εs,L) (17a)

s.t. Pd(τs,εs,L)≥ ω, (17b)0≤ τs ≤ T −2τ, (17c)εs ≥ 0, (17d)Llower ≤ L≤ Lmax, (17e)

where Llower and Lmax denote the lower bound and the upperbound of L, respectively. Llower is given according to therequired approximation accuracy (such as Llower = 200 for theaccuracy level of 10−2 [18]). Taking IEEE 802.15.4 physicallayer whose one symbol time is ν = 16µs (symbol rate =62.5 kbaud) as an example [36], we calculate the maximum

number of symbols Lmax during one reporting mini-slot byLmax =

τν = 625.

Due to the integer nature of variable L and the complexexpressions of Pd and Pf in (6) and (7), problem (17) isobviously non-convex, and a high degree of coupling existsamong optimization variables τs,εs,L.

V. SPECTRUM SENSING OPTIMIZATION

In this section, we will first find the optimal detectionthreshold εs and then propose an iterative solution algorithm(ISA) to find an efficient pair of sensing time τs and sequencelength L.

A. Detection Threshold OptimizationLemma 1: Suppose an optimal solution of problem (17)

exists, denoted as (τ∗s ,ε∗s ,L∗), where ∗ symbol denotes op-timality. Then, we have

ε∗s (ω) =(1+ γ)σ2u+√

Ω1(M−1+2σ2u )+σ4

uL∗ +

Mα2arit σ4

u (2γ+1)2Wτ∗s

Q−1(ω)

Mαarit. (18)

Proof: We first give out the partial derivative ofξ (τs,εs,L) with εs,

∂ξ (τs,εs,L)∂εs

=−∂Pf

∂εs∆, (19)

where ∆ = T−2τ−τsTC2(τs,εs,L)

C0 Pr(H0)M(Esτs+LEtν). According to(7), we conclude that Pf is monotonic decreasing in εs, i.e.,∂Pf∂εs

< 0, which further implies that ∂ξ (τs,εs,L)∂εs

> 0.For any given τs and L, we can choose a detection

threshold ε0 according to (6) such that Pd(ε0,τs,L) = ω .We may also choose a detection threshold ε1 < ε0 such thatPd(ε1,τs,L) > Pd(ε0,τs,L) and Pf (ε1,τs,L) > Pf (ε0,τs,L).In doing so, we have ξ (ε1,τs,L) < ξ (ε0,τs,L). Therefore,the optimal solution to problem (17) (ε∗s ,τ∗s ,L∗) mustbe achieved when the equality constraint in (17b) isfulfilled, i.e., Pd(ε∗s ,τ∗s ,L∗) = ω . By expanding thePd(ε∗s ,τ∗s ,L∗), we achieve L∗Mαarit(ε∗s − (1 + γ)σ2

u ) =

Q−1(ω)

(√L∗Ω1(M−1+2σ2

u )+L∗σ4u +

(L∗)2Mα2arit σ4

u (2γ+1)2Wτ∗s

),

which further completes the proof after mathematicalsimplification.

Remark 1: From (18), we conclude that ε∗s (ω) is indepen-dent with the active probability of primary users Pr(H1), butan increasing function of the received SNR γ at each CS.With the increase of γ and ε∗s (ω), the false alarm probabilityPf will decrease accordingly, which further improves the EEof the studied CSS scheme.

From (7), we know that ε∗s (ω) has to fulfill the conditionε∗s (ω) > σ2

u in order to make Pf < 0.5, which should be thecase for most CR scenarios. To this end, we get τs > τmin

s byconsidering (18), where

τmins =

Mα2aritσ4

u (2γ +1)

2W[(

γMαarit σ2u

−Q−1(ω)

)2− Ω1(M−1+2σ2

u )+σ4u

L

] . (20)



Further, in order to guarantee τmins ≥ 0, we require L ≥ Lmin,

where

Lmin (21)

=max

(⌈(Ω1(M−1+2σ2

u )+σ4u )(−Q−1(ω)

)2

γ2M2α2aritσ4

u

⌉,Llower

).

Plugging (18) into problem (17), we then achieve its equiv-alent form under the condition τmin ≤ T −2τ

max ξ (τs,L) := ξ (τs,εs,L)|εs=ε∗s (ω) (22a)

s.t. τmin < τs ≤ T −2τ, (22b)Lmin ≤ L≤ Lmax, (22c)

where ξ (τs,L)=(T−2τ−τs)C0 Pr(H0)(1−Pf (τs,L;ε∗s (ω)))/T

M(Esτs+LEt ν)+Et Pr(H0)(T−2τ−τs)(1−Pf (τs,L;ε∗s (ω))) .Therefore, the feasibility of problem (22) is guaranteed.

B. Iterative Solution Algorithm

Instead of directly solving the two-variable optimizationproblem (22), we propose the ISA that decouples problem(22) into two single-variable suboptimal problems and solvethem iteratively until convergence.

1) Sensing time optimization: For a given L, the firstsuboptimization problem of (22) is given as

max ξ (τs) := ξ (τs,L,εs)|L=L,εs=ε∗s (ω) (23a)

s.t. τmin < τs ≤ T −2τ. (23b)

Lemma 2: F(τs) := F(τs,εs,L)|L=L,εs=ε∗s (ω) is strictly con-cave in τs when τs falls in the domain defined by (23b).The proof of Lemma 2 will be given in Appendix A.

Similarly, we can prove that C(τs) :=C(τs,εs,L)|L=L,εs=ε∗s (ω) is also strictly concave in τs whenτmin < τs ≤ T − 2τ . We transform problem (23) into anequivalent form

minτmin<τs≤T−2τ

ϕ(τs) =−F(τs)

C(τs). (24)

Definition 1: [37] A function Q : Rn 7→ R is called quasi-convex if its domain and all its sublevel sets Sα = x ∈domQ|Q(x)≤ α, for all α ∈ R, are convex.

According to Definition 1, we get the sublevel set of ϕ(τs)

Sα = τs|τmin < τs ≤ T −2τ, fα(τs)≤ 0, (25)

where fα(τs) =−F(τs)−αC(τs).Taking the second derivative of fα(τs) with respect to τs,

we get

f ′′α(τs) =−Pr(H0)

(C0

T+αEt

)(2P′f (τs; L,ε∗s )

−(T −2τ− τs)P′′f (τs; L,ε∗s )).

According to the proof of Lemma 2, we conclude that f ′′α(τs)>0 when C0

T +αEt ≥ 0 . Therefore, ϕ(τs) is quasi-convex in τsand as a result, problem (24) is a quasi-convex optimizationproblem with respect to τs for α ≥ l (l = −C0

T Et).

We propose a simple bisection-based algorithm to problem(24) by solving a series of convex feasibility problems. Let ϕ ∗

denote the optimal value of the quasi-convex problem (24). Ifthe convex feasibility problem

find τs (26a)s.t. fα(τs)≤ 0, (26b)

τmin < τs ≤ T −2τ, (26c)

is feasible, then we have ϕ ∗ ≤ α . Conversely, if the problem(26) is infeasible, then we can conclude ϕ ∗ > α . Thus, we cancheck whether the optimal value of a quasi-convex problem isless than or more than a given value α by solving the convexfeasibility problem (26).

When α ≥ 0, fα(τs)≤ 0 always holds. Thus, we concludethat the interval [l, u] with u= 0 contains the optimal value ϕ ∗.We then solve the convex feasibility problem at its midpointα = l+u

2 , to determine whether ϕ ∗ is in the lower or upperhalf of the interval, and update the interval accordingly. Thisproduces a new interval, which also contains ϕ ∗, but has halfthe width of the initial interval. This is repeated until thewidth of the interval is small enough. The above procedureis summarized in Algorithm 1.

The interval [l, u] is guaranteed to contain ϕ ∗, i.e., wehave l ≤ ϕ ∗ ≤ u at each step. In each iteration, the intervalis bisected, so the length of the interval after k iterations is2−k(u− l), where u− l is the length of the initial interval. Itfollows that exactly ⌈log2 ((u− l)/ε)⌉ iterations are requiredbefore the algorithm terminates. Each step involves solving theconvex feasibility problem (26). Efficient solution methods,such as interior-point methods [37], could be then employedto solve problem (26) in polynomial time.

Algorithm 1 Bisection-based solution algorithm1: Input: l, u, Llower, Lmax, T , τ , C0, ω , γ , M, Et , Es, ν , W ,

σ2u , αarit , Pr(H0) and Pr(H1) and tolerance ε > 0.

2: Initialization: m← 0, l(m)← −C0T Et

, u(m)← 0, τs(m)← 0.3: repeat4: α(m)← (l(m)+u(m))/2.5: Solve the convex feasibility problem (26) with α(m).6: If problem (26) with α(m) is feasible7: u(m+1)← α(m);8: τs(m+1)← x. % save the feasible solution9: else

10: l(m+1)← α(m).11: EndIf12: m← m+1.13: until u(m)− l(m)≤ ε .14: Output: τs = τs(m).

2) Sequence length optimization: For a given τs, the secondsuboptimization problem of (22) is given as

max ξ (L) := ξ (τs,L,εs)|τs=τs,εs=ε∗s (ω) (27a)

s.t. Lmin ≤ L≤ Lmax. (27b)

No closed-form solution for L is available for problem (27),and an exhaustive search over all feasible L is required. Dueto the integer nature of L, it is not computationally expensiveto perform exhaustive search for an optimal L. Note that the



uniqueness of the optimal L is not guaranteed. In the case ofmultiple optimal solutions, the one with the smallest value ispreferable, since energy could be saved by shrinking the lengthof the sequence.

3) Global algorithm and convergence analysis: Combiningthe steps for solving the two suboptimization problems ofproblem (22), we summarize the proposed ISA in Algorithm2.

Algorithm 2 Iterative Solution Algorithm (ISA)

1: Input: Llower, Lmax, T , τ , C0, ω , γ , M, Et , Es, ν , W , σ2u ,

αarit , Pr(H0) and Pr(H1).2: Initialization: n← 0, L(0)← Lmax.3: repeat4: Update τmin(n) and Lmin(n) according to (20) and (21),

respectively.5: Given L(n), find τs(n) that solves problem (23) by

Algorithm 1.6: τs(n+1)← τs(n).7: Given τs(n+1), find L(n) using exhaustive search, i.e.,

L(n) = arg maxs.t. (27b)

ξ (L).

8: L(n+1)← L(n).9: n← n+1.

10: until ξ (τs(n),L(n))≤ ξ (τs(n−1),L(n−1)).11: Output: τs = τs(n), L = L(n).

Next, we analyze the convergence and the optimality of theproposed ISA. Let ξ ∗ denote the optimum of problem (22).Let (τs(n),L(n)) denote the value of variables (τs,L) after then-th iteration in ISA.

Theorem 1: Given an initial feasible L(0) fulfilling con-straint (27b), ξ (τs(n),L(n)) in Algorithm 2 converges toa steady value ξ ≤ ξ ∗.

Proof: From Algorithm 2, we can conclude that theobjective function of problem (22), ξ (τs,L), is nondecreasingat each iteration, i.e., ∀ n,

ξ (τs(n),L(n))≤ ξ (τs(n+1),L(n))≤ ξ (τs(n+1),L(n+1)).

It is also obvious that the feasible domain of problem (22)is closed and ξ (τs,L) is continuous on the feasible domain,which implies that ξ (τs,L) is bounded from above. Thistogether with the monotonicity proves the convergence ofξ (τs(n),L(n)). This completes the proof of Theorem 1.

Let (τs, L) denote a solution to ξ = ξ (τs,L). From the proofof Theorem 1, we find that the convergence point (τs, L) iswith the following property: at (τs, L), ξ (τs, L)≥ ξ (τs, L) forall τs and ξ (τs, L) ≥ ξ (τs,L) for all L. In other words, at τs,ξ (τs, L) is the largest across the dimension of τs, and at L,ξ (τs, L) is the largest across the dimension of L. Although wecannot rigorously prove the global optimality of the ISA, wehave found from the simulations in Section VI that the ISA isinsensitive to the initial value of L(0) and always convergesto the same maximum point, i.e., ξ = ξ ∗.

TABLE IPARAMETER VALUES USED IN THE SIMULATION

Parameter Valuethe number of CSs M [10,20]the lower bound of L, Llower 200the upper bound of L, Lmax 625the transmit power Et 3 Wattthe sensing power Es 0.1 Wattthe symbol duration ν 16 µsthe average received SNR at each CS γ [−16, 0] dBthe lower bound on the detection probability ω 0.9the occurrence probability of H1 Pr(H1) 0.2the occurrence probability of H0 Pr(H0) 0.8the bandwidth of the licensed band W 3 MHzthe sampling frequency fs 6 MHzthe CSN throughput when PUs are inactive C0 6.6582 bits/sec/Hzthe CSN throughput when PUs are active C1 6.6137 bits/sec/Hzthe length of one time frame T 300 msthe variance of noise σ2

u 1the coefficient of transmit power coding αarit 10the length of one mini-slot τ 10 ms

5 10 15 20 253.8

4

4.2

4.4

4.6

4.8

5

5.2

Spectrum sensing time τs (ms)

Ene

rgy

effic

ienc

y

ξ(L = Lmin)ξ(L = Lmin)

ξ(L = Lmin+Lmax

2 )

ξ(L = Lmin+Lmax

2 )

ξ(L = Lmax)ξ(L = Lmax)

Fig. 2. EE for the CSN: γ =−16dB and M = 10 (Fixed L)

VI. PERFORMANCE EVALUATION

This section presents the simulation results to verify theeffectiveness of this work. All simulated parameters are sum-marized in Table I, representing values describing typical CRand CSN scenarios [3] [6] [20]. First, we are interested inthe exact EE of the CoMAC-based CSS scheme, ξ (τs,εs,L),which is defined as

ξ (τs,εs,L) =F(τs,εs,L)C(τs,εs,L)

, (28)

where C(τs,εs,L) and F(τs,εs,L) are defined in (12) and(14), respectively. Fig. 2 shows that both ξ (τs,εs,L) of (16)and ξ (τs,εs,L) of (28) achieve their respective maximumsat the same spectrum sensing times of about 2.5ms fordifferent values of (εs, L) (i.e., εs = ε∗s (τs,L;ω) and L =

Lmin,(Lmin+Lmax)

2 ,Lmax. Fig. 3 shows that both ξ (τs,εs,L)and ξ (τs,εs,L) decrease with the increase of L and thusachieve their maximum at the same symbol sequence length ofLmin = 294 for different values of (εs,τs) (i.e., εs = ε∗s (τs,L;ω)and τs = 5ms,50ms,100ms), which again demonstrates theapproximation operation to ξ (τs,εs,L) does not harm theoptimality of τs,εs,L.



300 320 340 360 380 400 420

3.5

4

4.5

5

Sequence length L

Ene

rgy

effic

ienc

y

ξ(τs = 5ms)ξ(τs = 5ms)

ξ(τs = 50ms)ξ(τs = 50ms)

ξ(τs = 100ms)ξ(τs = 100ms)

Fig. 3. EE for the CSN: γ =−16dB and M = 10 (Fixed τs)

−16 −14 −12 −10 −8 −6 −4 −2 00

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

SNR,γ (dB)

Opt

imal

sen

sing

tim

e (m

s)

M=10 (2D exhausive search)M=10 (ISA)M=15 (2D exhausive search)M=15 (ISA)M=20 (2D exhausive search)M=20 (ISA)

Fig. 4. Optimal sensing time that maximizes the EE.

−16 −14 −12 −10 −8 −6 −4 −2 0200

250

300

350

400

450

500

SNR,γ (dB)

Opt

imal

seq

uenc

e le

ngth

M=10 (2D exhausive search)M=10 (ISA)M=15 (2D exhausive search)M=15 (ISA)M=20 (2D exhausive search)M=20 (ISA)

Fig. 5. Optimal symbol sequence that maximizes the EE.

−16 −14 −12 −10 −8 −6 −4 −2 01

2

3

4

5

6

7

SNR,γ (dB)

Ene

rgy

effic

ienc

y

M=10 (ISA)M=10 (Conven−CSS)M=15 (ISA)M=15 (Conven−CSS)M=20 (ISA)M=20 (Conven−CSS)

M=15M=10

M=20

Fig. 6. Maximum EE comparison for various γs.

Fig. 4 and Fig. 5 show the optimal τs and L at differentvalues of γ , respectively. The problem (22) is solved by usingthe ISA, and the obtained solutions are compared with thosedetermined by a 2-dimensional (2D) exhaustive search over τsand L. One important observation in Fig. 4 is that the optimalsensing time monotonically decreases in the SNR γ and thenumber of CSs M. This stems from the fact that larger γ andM need fewer signal examples to maintain the same sensingaccuracy than smaller γ and M. Unlike Fig. 4, Fig. 5 showsthat the optimal sequence length is not monotonic in γ andM, which highlights the necessity of the extensive search inproblem (27). It can be observed from Figs. 4 and 5 thatthe proposed ISA solutions coincide with the global optimalvalues for τs and L in all cases. Because of this consistency, inthe following simulations, we will only consider the proposedresults.

Fig. 6 compares the maximum EE of this work with thatof conventional CSS schemes (denoted by ”Conven-CSS”)[20] [25] [27] [29]-[31] when γ and M vary, respectively.In conventional CSS schemes, the local statistic of each CShas to be sent to the fusion center in different time slots,as a result of which the reporting delay during the reportingphase of conventional CSS grows linearly with M. The “K-out-of-M” fusion (the same Pf is assumed) is adopted by thefusion center. For notational convenience, the maximum EE ofconventional CSS schemes is denoted by ξcon, which is givenby

ξcon =(T − (M+1)τ− τs)C0 Pr(H0)(1−Pf )/T

M(Esτs +Etτ)+Pr(H0)(1−Pf )(T − (M+1)τ− τs)Et.

For fair comparison, all the parameters in ξcon are selectedfrom Table 1. It is clearly shown in Fig. 6 that the EE ofISA is much higher than that of conventional CSS schemes,regardless of γ and M. The maximum performance gain (i.e.,ξ/ξcon) reaches up to 3.1070 which occurs when γ =−16 dBand M = 20. Finally, we find that the steady state of EE arriveswhen γ=-8dB in Fig. 6, which is consistent with the resultsin Fig. 4 and Fig. 5 since the steady states of both optimal τsand L also arrive when γ=-8dB.

Fig. 7 presents the comparison result between the maximum



0.05 0.1 0.15 0.2 0.25 0.31

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Pr(H1)

Ene

rgy

effic

ienc

y

M=10 (ISA)M=10 (Conven−CSS)M=15 (ISA)M=15 ((Conven−CSS))M=20 (ISA)M=20 ((Conven−CSS))

M=10 M=15 M=20

Fig. 7. Maximum EE comparison for various Pr(H1)s.

EE of this work with that of conventional CSS schemesunder different Pr(H1)s. Without loss of generality, we fixγ =−16dB and vary Pr(H1) from 0.05 to 0.3. Similar to Fig.6, the EE of ISA is again much higher than that of conventionalCSS schemes in Fig. 7, regardless of Pr(H1) and M. It isalso obvious in Fig. 7 that the EEs of all the compared CSSschemes decrease with the increase of Pr(H1), which impliesthe fact that the loss in throughput performance due to theincreasing occurrence of PUs dominates the energy saving inthe transmission phase of CSS schemes.

VII. CONCLUSION

This paper has formulated a novel EE-maximization prob-lem for the CoMAC-based soft decision CSS scheme in CSNs.In order to solve the EE-maximization problem, we first havefound the optimal expression for the detection threshold andthen proposed an ISA to find an efficient pair of sensing timeand the length of the modulated symbol sequence. The ISAdecouples the EE-maximization problem into two subproblems(i.e., the sensing time optimization and the sequence lengthoptimization) and solves them iteratively until convergence.For a given length of symbol sequence, we have provedthe quasi-convexity of the sensing time optimization problemand proposed a bisection-based solution algorithm to solve it.Further, provided the newly computed sensing time, we havefound the optimal length of symbol sequence to the sequencelength optimization problem via exhaustive search.

Extensive simulations have demonstrated the effectivenessof this work despite of the heuristic decomposition method bythe proposed ISA. By comparing this work with conventionalCSS schemes, we have shown a large performance gain in EEdue to this work.

APPENDIX APROOF OF LEMMA 2

Proof: First, we take the second derivative of F(τs) withrespect to τs

F′′(τs) =

C0 Pr(H0)(2P′f − (T −2τ− τs)P

′′f )

T. (29)

It is evident from (29) that the concavity of F(τs) can bedeclared if Pf (τs; L,ε∗s ) is monotonic decreasing and convexin τs.

Then, taking derivative of Pf (τs; L,ε∗s ) with respect to τsyields

P′f (τs; L,ε∗s ) =−

1√2π

exp(−Θ2(τs)

)Θ′(τs) (30)

with Θ(τs) =Q−1(ω)

√A+ B

τs +C√D+ E

τs

, where A= LΩ1(M−1+2σ2u )+

Lσ4u , B =

L2Mσ4u α2

arit (2γ+1)2W , C = LMαaritγσ 2

u , D = LΩ0(M−1+

2σ2u ) and E =

L2Mσ4u α2

arit2W .

Further, skipping the tedious deduction we expand Θ′(τs)as follows

Θ′(τs) =

Q−1(ω)(AE−BD)

2(Aτs +B)32√

Dτs +E︸︷︷︸Φ1(τs)

+EC

2τ2s

(D+ E

τs

) 32︸︷︷︸

Φ2(τs)

. (31)

It is trivial to verify that AE−BD > 0, which together withQ−1(ω) < 0 (as ω is close to 1) proves that Θ′(τs) > 0.Combining (30) and (31), we conclude that P

′f (τs; L,ε∗s ) < 0,

i.e., Pf (τs; L,ε∗s ) is monotonic decreasing in τs.

When ε∗s (ω) > σ2u (τs > τmin), Q−1(ω)

√A+ B

τs+C ≥ 0

holds and we thus have Θ(τs)> 0. As both Φ1(τs) and Φ2(τs)are monotonic decreasing in τs, Θ′(τs) is also monotonicdecreasing in τs. This, together with Θ(τs) ≥ 0, implies thatP′f (τs; L,ε∗s ) is increasing in τs, i.e., P

′′f (τs; L,ε∗s ) > 0. So far,

we have proved that F′′(τs) in (29) is negative, which implies

the strict concavity of F(τs).

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Meng Zheng (M’14) received the B.S. degree ininformation and computing science and the M.S.degree in operational research and cybernetics fromNortheastern University, Shenyang, China, in 2005and 2008, respectively, and the Ph.D. degree inmechatronic engineering from University of ChineseAcademy of Sciences, Beijing, China, in 2012. From2010 to 2012, he was a visiting student with theFraunhofer Institute for Telecommunication, Hein-rich Hertz Institute, Berlin, Germany. He is currentlyan Associate Professor with Shenyang Institute of

Automation, Chinese Academy of Sciences. His research interests includewireless ad hoc and sensor networks, cognitive radio networks, and greencommunications.

Lin Chen (S’07-M’10) received his B.E. degreein Radio Engineering from Southeast University,China, in 2002 and the Engineer Diploma fromTelecom ParisTech, Paris, in 2005. He also holdsa M.S. degree of Networking from the University ofParis 6. He currently works as associate professor inthe department of computer science of the Universityof Paris-Sud. He serves as Chair of IEEE SpecialInterest Group on Green and Sustainable Networkingand Computing with Cognition and Cooperation,IEEE Technical Committee on Green Communica-

tions and Computing. His main research interests include modeling and controlfor wireless networks, distributed algorithm design and game theory.

Wei Liang (M’11) received the Ph.D. degree inmechatronic engineering from Shenyang Instituteof Automation, Chinese Academy of Sciences,Shenyang, China, in 2002. She is currently a Profes-sor with Shenyang Institute of Automation, ChineseAcademy of Sciences. As a primary participant/aproject leader, she developed the WIA-PA and WIA-FA standards for industrial wireless networks, whichare specified by IEC 62601 and IEC 62948, re-spectively. Her research interests include industrialwireless sensor networks and wireless body area

networks. Wei Liang received the International Electrotechnical Commission1906 Award in 2015 as a distinguished expert of industrial wireless networktechnology and standard.



Haibin Yu received his B.S. and M.S. degrees in in-dustrial automation from the Northeastern Universi-ty, Shenyang, China, in 1984 and 1987, respectively,and Ph.D. degree in control theory and control en-gineering from Northeastern University, Shenyang,China, in 1997. He has been with Shenyang Instituteof Automation, Chinese Academy of Sciences sinceApril 1993. He is currently a Professor and alsothe director of the institute. His primary researchinterests are in industrial communication and controlsystems, and industrial wireless sensor networks. In

2011, he was elected as an ISA Fellow for his contributions in industrialcommunication technologies.

Haibin Yu has also been very active in serving professional communities. Heis the Deputy Editor-in-Chief of Information and Control. He is also the Vice-Chairman of China National Technical Committee for Automation Systemsand Integration Standardization, the Deputy Director of Chinese Associationof Automation, the council members of China Instrument and Control Society.He is also one of the expert members for Intelligent Manufacturing Fund ofthe Ministry of Science and Technology of the People’s Republic of China.

Jinsong Wu (SM’11) received his Ph.D. in De-partment of Electrical and Computer Engineering,Queens University at Kingston, Canada in 2006.He was the Founder and Founding Chair of theTechnical Committee on Green Communications andComputing (TCGCC). He is the co-founder andfounding Vice-Chair of the Technical SubCommitteeon Big Data (TSCBD). He is the Founder and Editorof IEEE Series on Green Communication and Com-puting Networks in IEEE Communications Maga-zine. He is an Area Editor in IEEE Transactions on

Green Communications and Networking. He has served as was Editor in theIEEE Journal of Selected Areas on Communications (JSAC) Series on GreenCommunications and Networking, the leading Guest Editor in Special Issue onGreen Communications, Computing, and Systems in IEEE Systems Journal,Associate Editor in Special Section on Big Data for Green Communicationsand Computing, IEEE Access. He was the leading Editor and a co-authorof the comprehensive book, entitled Green Communications: TheoreticalFundamentals, Algorithms, and Applications, published by CRC Press inSeptember 2012. He is currently an Associate Professor in Department ofElectrical Engineering, Universidad de Chile, Santiago, Chile.

energy-efﬁciency maximization for cooperative spectrum … · elac2015/t10-0761 and conicyt...

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