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IEEE TRANSACTIONS ON COMMUNICATIONS 1

Wireless Powered Cognitive Radio Networks withCompressive Sensing and Matrix Completion

Zhijin Qin, Yuanwei Liu, Yue Gao, Maged Elkashlan, and Arumugam Nallanathan

Abstract—In this paper, we consider cognitive radio networksin which energy constrained secondary users (SUs) can harvestenergy from the randomly deployed power beacons (PBs). Anew frame structure is proposed for the considered networks.In the considered network, a wireless power transfer (WPT)model is proposed, and the closed-form expressions for thepower outage probability are derived. Additionally, in order toreduce the energy consumption at SUs, sub-Nyquist samplingare performed at SUs. Subsequently, compressive sensing andmatrix completion techniques are invoked to recover the originalsignals at the fusion center, by utilizing the sparsity propertyof spectral signals. Throughput optimizations of the secondarynetworks are formulated into two linear constrained problems,which aim to maximize the throughput of a single SU andthe whole cooperative network, respectively. Three methodsare provided to obtain the maximal throughput of secondarynetworks by optimizing the time slots allocation and the transmitpower. Simulation results show that the maximum throughputcan be improved by implementing compressive spectrum sensingin the proposed frame structure design.

Keywords: Compressive sensing, matrix completion, spectrumsensing, sub-Nyquist sampling, wireless power transfer.

I. INTRODUCTION

ENERGY efficiency and spectrum efficiency are two crit-ical issues in designing wireless networks. Recent devel-

opments in energy harvesting provides a promising techniqueto improve the energy efficiency in wireless networks. Dif-ferent from harvesting energy from traditional energy sources(e.g., solar, wind, water, and other physical phenomena) [1],the emerging wireless power transfer (WPT) further under-pins the trend of green communications by harvesting en-ergy from radio frequency (RF) signals [2]. Inspiring by thegreat convenience offering by WPT, several works have beenstudied to investigate the performance of different kinds ofenergy constraint networks [3]–[7]. Two practical receiverarchitectures, namely a time switching receiver and a powersplitting receiver [8], were proposed in a multi-input and multi-output (MIMO) system in [3], which laid a foundation inthe recent research of WPT. In [4], a new hybrid networkarchitecture is designed to enable charging mobiles wirelesslyin cellular networks. In [5], three wireless power transferschemes were proposed for secure device-to-device (D2D)

Manuscript received May 7, 2016; revised August 18, 2016; acceptedOctober 24, 2016. Date of current version Oct 26, 2016. Part of this work waspresented at the IEEE Globecom Conference, San Diego, CA, USA, 2015.

Z. Qin, Y. Liu, Y. Gao and M. Elkashlan are with Queen Mary Universityof London, London, United Kingdom. (email:z.qin; yuanwei.liu; yue.gao;[email protected])

A. Nallanathan is with King’s College London, London, United Kingdom.(email: [email protected])

communication scenarios. For cooperative systems, new powerallocation strategies are proposed in a cooperative networkswhere multiple sources and destinations are communicated byan energy harvesting relay in [6]. For non-orthogonal multipleaccess (NOMA) networks, a new cooperative simultaneouslywireless information and power transfer NOMA protocol isproposed with considering the scenario where all users arerandomly deployed in [7].

Along with improving energy efficiency through energy har-vesting, wideband cognitive radio (CR) technique can improvethe spectrum efficiency and capacity of wireless networksthrough dynamic spectrum access [9]. More particularly, thesecondary users (SUs) in CR networks (CRNs) are normallyenergy constrained and powered by battery. Therefore, SUs arenormally critical to energy consumption in order to increasethe battery life of SUs. In order to design networks with highspectrum efficiency and energy efficiency, it is necessary forthe SUs in CRNs to be equipped with the energy harvestingcapability. However, for the SUs powered by energy harvestedfrom wireless RF, high sampling rate is difficult to be achievedin wideband CRNs. To overcome this issue, compressive sens-ing (CS), which was initially proposed in [10], is introducedto wideband spectrum sensing in [11] to reduce the powerconsumption at SUs. As the spectrum of interest is normallyunderutilized in reality [12], [13], it exhibits a sparsity propertyin frequency domain, which makes sub-Nyquist samplingpossible by implementing the CS technique at SUs. In addi-tion, when dealing with matrices containing limited availableentries, low-rank matrix completion (MC) [14] was proposedto recover the complete matrix. As the sparsity property ofreceived signals can be transformed into low-rank property ofthe matrix constructed in the cooperative networks, the authorsin [15] proposed to apply joint sparsity recovery and low-rankMC to reduce the requirements on sensing and transmissionwithout degrading the sensing performance in CRNs. In orderto improve robustness against channel noise, a denoised algo-rithm was proposed in [16] for compressive spectrum sensingat single SU and low-rank MC based spectrum sensing atmultiple SUs.

A. Related Work

Some throughput optimization work has been recently de-veloped in wireless powered communication networks. The au-thors in [17] considered the throughput maximization problemfor both battery-free and battery-deployed cases by optimizingthe time slots for energy harvesting and data transmission.In addition, recent work [18]–[20] on the CRNs powered by




energy harvesting mainly focuses on the spatial throughputoptimization under various constraints. The authors in [18]considered CRNs with an energy-harvesting SU with infinitebattery capacity. The goal is to design an optimal spectrumsensing policy that maximizes the expected throughput subjectto an energy causality constraint and a collision constraint.In order to improve both energy efficiency and spectral effi-ciency, the authors in [19] considered a similar network modeland the stochastic optimization problem is formulated intoa constrained partially observable Markov decision process.At the beginning of each time slot, a SU needs to determinewhether to remain idle so as to conserve energy, or to executespectrum sensing to acquire knowledge of the current spectrumoccupancy state. The throughput is maximized by optimizingthe spectrum sensing policy and the detection threshold. Theauthors in [20] consider an energy constraint RF-poweredCRN by optimizing the pair of the sensing duration and thesensing threshold to maximize the average throughput of thesecondary network. More practically, the authors in [21] pro-posed to optimize the throughput of energy harvesting sensornetworks by further considering the constraint on hardwarememory. Furthermore, the implementation of the proposedscheme has been realized, which motives us to bring thewireless powered networks from theory to practice.

B. Motivations and Contributions

The aforementioned work has played a vital role and laidsolid foundation for developing new strategies for frame struc-ture design. However, in [17], spectrum sensing is not consid-ered in the frame structure design. In addition, in [18]–[20], theproposed frame structure designs mainly aim to maximize thethroughput by optimizing the threshold and time slots. Whenconsidering the energy efficiency and spectrum efficiency, itis meaningful to introduce sub-Nyquist sampling to reducethe energy consumption at SUs in wireless powered CRNs.Difficult from [22], in this paper, a new frame structure isproposed for the SUs in wireless powered CRNs with differentbehaviours (active and inactive which are to be introducedlater in Section II). With invoking CS and MC techniques,throughput is optimized at the level of an individual SU andthe whole cooperative network, respectively.

The summarized contributions of this paper are illustratedas follows:• We propose a new frame structure for the wireless

powered CRNs, in which a new WPT model and thesub-Nyquist sampling are considered.

• In the WPT model, we adopt a bounded WPT schemewhere each SU selects a beacon (PB) nearby with thestrongest channel to harvest energy. The closed-formexpressions for the power outage probability is derived.With the sub-Nqyust sampling performed at each SU, CSand low-rank MC techniques are utilized at the remoteFC to recovery the original signals.

• Throughput optimisation of the proposed frame structuresare formulated into two linear constrained problems withthe purpose of maximising the throughput of a singleSU and the whole cooperative networks, respectively. The

formulated problems are solved by using three differentmethods to obtain the maximal achievable throughput,respectively.

• We show that the proposed frame structure design out-performs the traditional one in terms of throughput. It isnoted that the multiple SUs scenario can achieve betteroutage performance than the single SU scenario.

C. Organisations

The rest of this paper is organised as follows. Section IIdescribes the considered WPT model and spectrum sensingmodel based on the proposed frame structure. Section IIIpresents the throughput optimization for the single SU withapplying CS technique. Section IV provides the throughputoptimization at the cooperative network level with adoptingMC technique. Section V shows the numerical analyses ofthe considered network model with the purpose of optimizingthroughput of single SU and the whole cooperative network,respectively. Section VI concludes this paper.

II. NETWORK MODEL

A. Network Description

We consider a CRN, where SUs are energy constrained. Thewhole spectrum of interest can be divided into I channels.A channel is either occupied by a primary user (PU) orunoccupied. Meanwhile, there is no overlap between differentchannels. The number of occupied channels is assumed tobe K, where K ≤ I. Each SU performs sensing on thewhole spectrum. It is assumed that all SUs keep quiet asforced by protocols, e.g., at the media access control layerduring spectrum sensing period. Thus the received signalsfrom active PUs with channel noise. As shown in Fig. 1,for each SU, it is assumed that the sensing and transmissioncan only be scheduled by utilising energy harvested fromPBs. The spatial topology of all PBs are modelled usinghomogeneous poisson point process (PPP) Φp with densityλp. Without loss of generality, we consider that a typical SUis located at the origin in a two-dimensional plane. Each SUis equipped with a single antenna and has a correspondingreceiver with fixed distance. Each PB is furnished with Mantennas and maximal ratio transmission (MRT) is employedat PBs to perform WPT to the energy constrained SU. Once thespectrum holes are identified, each SU is assigned a spectrumhole by the FC to start data transmission. Here, we assume thatthere are enough number of spectrum holes to be allocated toSUs in a CSS network as the wideband spectrum sensing isperformed at each SU. Additionally, it is assumed that thetime of each frame is T . The proposed framework structurefor the considered network with J (J ≥ 1) SUs is shown inFig. 1. The J spatially distributed SUs can be divided into twosets: active SUs and inactive SUs. Active SUs refer to SUsthat can complete the spectrum sensing task with harvestedenergy. Inactive SUs refer to SUs without enough energy toconduct spectrum sensing. The frame structures for active SUsand inactive SUs are as follows:

1) Active SUs: As outlined by the blue oval in Fig. 1,the frame period for an active SU includes four time




slots: 1) energy harvesting time slot, in which each SUharvests the energy from PBs during the α1T period,with α1 being the fraction of energy harvesting in oneframe period; 2) spectrum sensing time slot, in whicheach SU performs sub-Nyquist sampling by applying CStechniques. The compressed measurements are then sentto a remote powerful FC during the βT period by usingthe harvested energy during the α1T period; 3) energyharvesting time slot for data transmission, in whicheach SU harvests the energy from PBs during the α2Tperiod, with α2 being the fraction of energy harvestingin one frame period. During this time slot, the originalsignals are recovered and the spectrum occupancies aredetermined by adopting energy detection at the FC. Sub-sequently, the obtained sensing decisions are sent backto corresponding SUs; and 4) data transmission slot, inwhich each SU performs data transmission during the(1− α1 − β − α2)T period. By moving the burden ofspectrum recovery and decision making from energy-constrained SUs to a powerful FC, energy consumptionat the SU can be reduced significantly. Additionally,SUs have a higher opportunity to performing energyharvesting;

2) Inactive SUs: Before performing spectrum sensing, eachparticipating SU compares energy harvested during thefirst time slot EH1 with ES , where ES is the energyconsumption for spectrum sensing at the second timeslot. If EH1

is greater than ES , the SU is determined asactive and would continue performing spectrum sensing.If EH1

is less than ES , the SU would switch to energyharvesting model again and wait for the decision onspectrum occupancies from the FC before starting datatransmission. Therefore, as outlined by the red oval inFig. 1, the frame structure of inactive SUs only includestwo time slots: (α1 + β + α2)T for energy harvestingand the rest for data transmission. In the case of CSS,only measurements from active SUs are collected atthe FC. The signals received by SUs exhibit a sparsityproperty that yields a matrix with low-rank propertyat the FC. Therefore, the full information of spectrumoccupancies can be obtained by adopting the low-rankMC technique. Once the complete matrix is recoveredat the FC, the final decisions on spectrum occupanciescan be determined. Subsequently, the FC sends back thebinary decision with the allocated spectrum hole to eachSU in the CSS network.

B. Wireless Power Transfer Model

Different from the previous research works [5], [23], weconsider a bounded power transfer model by assuming aminimum allowed distance d0 between the selected PB andpowered SU, which is used to avoid the singularity caused byproximity between PBs and SUs [24], [25]. If PBs are reallyclose to the SU, the harvested energy would mathematicallygo to infinity [4]. It is assumed that there is no batterystorage energy for future use. Therefore, all the harvestedenergy during energy harvesting time slots is used to perform

spectrum sensing and data transmission in the current frameperiod [4], [6], [26]. It is also assumed that all required energyof SUs are harvested from PBs. Additionally, all PBs areassumed to operate in the same specific frequency band forsimplicity. The operation band of PBs is assumed to be isolatedfrom that of the spectrum sensing and data transmission.

In this work, we adopt a power transfer scheme where eachSU only selects one PB with the strongest channel1 The energyharvesting channels are assumed to be quasi-static Rayleighfading channels, where the channel coefficients are constantfor each transmission block but vary independently betweendifferent blocks. At each SU, the energy harvested from theselected PB in the first and the third time slots can be obtainedas follows:

EH1= maxp∈Φp,‖dp‖≥d0

‖hp‖2L (dp)

ηPpγT, (1)

where γ is the ratio of the time used for energy harvestingto the total time of a frame, η is the power conversionefficiency at the SU, Pp is the transmit power of PBs. Here,due to the WPT channels are modeled as Rayleigh fadingwith zero mean and unit variance, the entries of hp ∈ CM×1

are independent complex Gaussian distributed with zero meanand unit variance employed to capture the effects of small-scale fading between PBs and the SU, which is because of theMRT enabled power transmission from PB with M antennas.L (dp) = Ad−ξp is the power-law path-loss exponent. Thepath-loss function depends on the distance dp, a frequencydependent constant A, and an environment/terrain dependentpath-loss exponent ξ > 2.

For the active SUs, based on (1), the maximum harvestedpower PH1

which can be utilized for sensing at the SU isgiven by

PH1=EH1

βT= maxp∈Φp,‖dp‖≥d0

‖hp‖2L (dp)

ηPpα1

β, (2)

where EH1 refers to the energy harvested in the first time slot.As energy can only be stored in the current frame, the

total energy for data transmission is the sum of the remainingenergy in the second slot and the energy harvested in the thirdtime slot, which is given by

ET2 = (EH1 − ES) + EH2 , (3)

where ES = PsβT with Ps being the power consumption ofspectrum sensing.

Based on (1) and (3), the corresponding power for datatransmission is given by

PT2 =

maxp∈Φp,‖dp‖≥d0

‖hp‖2L (dp)

ηPp (α1 + α2)− Psβ

1− α1 − β − α2.

(4)

1Note that the strongest PB is not necessarily to be the nearest one,because of the instantaneous effect of small-scale fading is also taken intoconsiderations. to harvest energy. The other PBs would turn to sleep mode tosave the energy waste. The motivation behind using this scheme is that it iscapable of achieving more energy efficient power transfer compared to otherpossible schemes (e.g., use all of PBs for power transfer).




Active SU

Fusion center

Active SU

Inactive SU

... ...

Inactive SU

Data Transmission

Energy Harvesting

( )1 21 Tα β α− − −

T

Energy Harvesting

Spectrum Sensing

Data Transmission

1Tα Tβ

Energy Harvesting

2Tα

T

( )1 21 Tα β α− − −

Active SU

( )1 2 Tα β α+ +

Samples reporting & decision feedback Decision feedback onlyPower beacon

Fig. 1: Proposed frame structure design with energy harvesting, spectrum sensing and data transmission.

For the inactive SUs, the harvested energy can be given by

EH3 = maxp∈Φp,‖dp‖≥d0

‖hp‖2L (dp)

ηPp(α1 + β + α2)T.

(5)

Based on (5), the maximum transmit power at the inactiveSU can be expressed as

PH3 =EH3

(1− α1 − β − α2)T. (6)

C. Spectrum Sensing Model with Sub-Nyquist SamplingAt the jth SU (SUj) in the considered network, the received

signals can be expressed as:

rj = hj ∗ s + nj, (7)

where s ∈ Cn×1 refers to the transmitted primary signals intime domain, and hj ∈ Cn×1 is the channel gain betweenthe transmitter and receiver, and nj ∼ CN (0, σ2In) refers toAdditive White Gaussian Noise (AWGN) with zero mean andvariance σ2.

Based on the Nyquist sampling theory, the sampling rateis required to be no less than twice of the signal bandwidth.However, for wideband spectrum sensing, high sampling rateis difficult to achieve and also leads to high energy consump-tion which is challenging for energy-constrained SUs. It isnoticed that the transmitted signal s exhibits a sparsity prop-erty in frequency domain as a large percentage of spectrumis normally underutilized in practice. This sparsity propertyenables sub-Nyquist sampling at SUs without loss any signalinformation. The compressed measurements collected at SU jcan be given by

xj = Φjrj = ΦjF−1j rfj = Θj (hfjsf + nfj) , (8)

where Φj ∈ CΛ×n (Λ < n) is the measurement matrix utilizedto collect the compressed measurements xj ∈ CΛ×1, and sf ,hfj and nfj refer to the frequency representations of transmitprimary signals, channel coefficients, and the AWGN receivedat SU j . Additionally, Θj = ΦjFj−1, where Fj−1 is inversediscrete Fourier transform (IDFT) matrix. The compressionratio is defined as κ = Λ

n , (0 ≤ κ ≤ 1).After the compressed measurements are collected, SUs send

them to a remote FC to perform signal recovery by an error-free reporting channel. By adopting such a powerful FC,energy-constrained SUs can get rid of signal recovery processand continue harvesting more energy from PSs. At the FC, thecompressed measurements X can be expressed as

X = Θvec (Rf ) = Θ (vec (HfSf ) + Nf ) , (9)

where Rf = [rf1, rf2, . . . , rfJ] which is in size of n × J ,

and vec (Rf) =[rTf1, . . . , r

TfJ

]T∈ C(nJ)×1. Sf ∈ Cn×J ,

Hf ∈ Cn×n and Nf ∈ Cn×J refer to the matrix constructedby transmit primary signals, channel coefficients and AWGNin frequency domain. In addition, the measurement matrix is adiagonal matrix Θ = diag (Θ1,Θ2, . . . ,ΘJ) in size of Ω×Nwith Ω = Λ × J and N = n × J . After the compressedmeasurements are collected at the FC, signal recovery can beformulated as a convex optimization problem. The considerednetwork becomes a single node case when J = 1. Thenthe existing algorithm for CS can be utilized to recover theoriginal signals. In the cooperative networks (J > 1), existingalgorithms for low-rank MC can be implemented to completethe matrix. It has been proved that exact signal recovery canbe guaranteed if the number of available measurements areno less than a minimum bound. With enough number of




measurements, exact signals are obtained at the FC. Subse-quently, energy detection is adopted to determine spectrumoccupancies, in which the decisions are made by comparingthe energy of recovered signal with a threshold [27] definedas

λ =(σ2s + σ2

)(1 +

Q−1(Pd)√

n/2

). (10)

Once the sensing decisions are determined at the FC, theywould be sent back to the participating SUs in the CSSnetwork by the reporting channel to start the data transmissionperiod.

III. THROUGHPUT OPTIMIZATION OF A SINGLESECONDARY USER

In this section, the closed form expressions are derivedfor the power outage probability of spectrum sensing anddata transmission, respectively. With the CS technique imple-mented, performance analysis with the target of maximizingthroughput of each individual active SU locally is provided.

A. Power Outage Probability Analysis

We assume there exists a threshold transmit power, belowwhich the spectrum sensing in the second slot or the data trans-mission in the fourth slot cannot be scheduled. We introducepower outage probability, i.e., probability that the harvestedenergy is not sufficient to perform spectrum sensing or carryout the data transmission at a certain desired quality-of-service(QoS) level. In practical scenarios, we expect a constant powerfor the data transmission. Therefore, we also denote the powerthreshold Ps as the sensing power in the second slot and Ptas the transmit power of the SU, respectively. The followingtheorem provides the exact analysis for the power outageprobability at the single SU scenario.

Theorem 1. The power outage probability of spectrum sensingP outs in the second time slot and the power outage probabilityof data transmission P outt in the fourth time slot can beexpressed in closed-form as follows:

P outζ = exp

−πλpδµδζ

M−1∑m=0

Γ(m+ δ, µζd

ξ0

)m!

, (11)

where ζ ∈ (s, t), µs = βPsηPpAα1

, µt = Pt(1−α1−β−α2)+PsβηPpA(α1+α2) ,

δ = 2/ξ, and Γ(·, ·) is the upper incomplete Gamma function.

Proof. See Appendix A.

Remark 1. The derived results in Theorem 1 indicates thatthe power outage probability is a strictly monotonic increasingfunction of λp.

B. Compressive Spectrum Sensing

In the scenario of CS based spectrum sensing, the originalsignal sf of each individual SU can be obtained respectivelyby solving the l1 norm optimization problem as follows:

min ‖sf‖1 s.t. ‖Θj · hfjsf − xj‖22 ≤ εj, (12)

where εj is the error bound related to the noise level. Thisoptimization problem can be solved by many existing algo-rithms for CS, such as by adopting many existing algorithmssuch as CoSaMP [28], SpaRCS [29], etc. The computationalcomplexity of solving the optimization problem in (12) isO(n3), which is mainly determined by the size of signals

(n) to be recovered. However, this process is proposed to beperformed at a powerful FC rather than the energy-constrainedSUs. By doing so, energy consumption at SUs can be signif-icantly reduced.

The performance metric of spectrum sensing can be mea-sured by the probability of detection Pd and probability offalse alarm Pf . For a target probability of detection, Pd, withwhich the PUs are sufficiently protected, the threshold can bedetermined by (10) accordingly if the number of samples nis fixed. As a result, Pf for the spectrum sensing with singleSU can be derived as follows:

Pf =Q

(Q−1

(Pd)√σ2

s + σ2

σ2+

√n

2

σ2s

σ2

), (13)

where σ2 and σ2s refer to the noise power and signal power,

respectively.Assuming the energy cost per sample es in spectrum sensing

is fixed, the energy consumption of spectrum sensing isproportional to the number of collected samples as given by:

n =βTPses

. (14)

In fact, the number of collected samples n is determinedby the sensing time slot. In this case, the energy consumedby reporting collected measurements and receiving decisionresults between SUs to the FC is ignored. Substituting (14)into (13), we obtain

Pf =

(1−Q

(Q−1

(Pd)(

1 +σ2s

σ2

)+

√βTPs2es

σ2s

σ2

)).

(15)

C. Throughput Analysis

By considering the power outage probability, the throughputof each individual SU in a CRN can be expressed as

τ =(1− P outs

)×(1− P outt

)× (1− Pf )

× (1− α1 − β − α2) τt, (16)

where τt = log2

(1 + Pt

N0

)is the throughput for the data

transmission slot, and N0 refers to the AWGN level in the datatransmission channel. Here we simplify the data transmissionprocess by not considering the fading, which means thethroughput τt is only determined by the transmit signal-to-noise-radio (SNR).

When implementing CS technique to achieve sub-Nyquistsampling rate at an SU, it has been proven that the exactsignal recovery can be guaranteed if the number of collectedmeasurements satisfies Λ ≥ C · K log (n/K), where C issome constant depending on each instance [30]. If the signal isrecovered successfully by Λ samples, the achieved Pf wouldbe the same as that no CS technique is implemented with n




samples. Therefore, the necessary sensing time slot to achievethe same Pf can be reduced from βT to κβT with theCS technique adopted, where κ is the compression ratio atSUs. Replacing P outs , P outt and Pf in (16) by (11) and (15)respectively, the full expression of the throughput with CSimplemented can be expressed as follows:

τcs =∏

ζ=s,t

1− exp

− πλpδ(µcsζ

)δ M−1∑m=0

Γ(m+ δ, µcsζ d

ξ0

)m!

×

(1−Q

(Q−1

(Pd)(

1 +σ2s

σ2

)+

√βTPs2es

σ2s

σ2

))

× (1− α1 − κβ − α2)× log2

(1 +

PtN0

). (17)

If there is no CS technique implemented at SUs, the timeslot fraction for spectrum sensing follows the condition that0 ≤ β ≤ 1. When the CS technique is implemented at SUs,the time slot for spectrum sensing follows the condition thatCK log (n/K) ≤ κβTPs

es≤ n. By combining these two condi-

tions, the constraint for β becomes es(CK log(n/K))κTPs

≤ β ≤ 1.With the implementation of CS, the β in (11) is replaced byκβ.

Furthermore, the throughput can be maximized by solvingthe following problem:

(P0) : maxα1,β,α2,Pt

τcs (18)

s.t. C1 : 0 ≤ α1 ≤ 1,

C2 :es (CK log (n/K))

κTPs≤ β ≤ 1,

C3 : α2,min ≤ α2 ≤ 1,

C4 : 0 ≤ 1− α1 − κβ − α2 ≤ 1,

C5 : Pt,min ≤ Pt ≤ Pt,max,

where α2,min refers to the minimum time slot fraction for thethird time slot. Pt,min and Pt,max refer to the allowed mini-mum and maximum power levels for data transmission period.Constraint C1 bounds the time slot for energy harvesting forthe first time slot. Constraint C2 highlights the minimum timeslot for compressive spectrum sensing. The lower bound of βis used to make sure the compressed sample is sufficient toguarantee exact signal recovery. Constraint C3 ensures that thethird time slot is at least enough for signal recovery at the FCand data transmission between the SU and FC. Constraint C4

guarantees that the last time slot of a frame period is utilizedfor data transmission. Finally, constraint C5 limits the transmitpower level for data transmission. It is noticed that (27) is aconstrained nonlinear optimization problem and the objectivefunction is very complex. However, the constraints are linear,which motives us to solve the optimization problem by thefollowing three methods:

1) Grid search: The grid search algorithm for solving (27)can be described in Algorithm 1. The grid search al-gorithm can find out the global optimal value if thestep size ∆i (i = 1, 2, 3, 4) for α1, β, α2, Pt are smallenough. However, the computational complexity wouldgreatly increase if the step is set to be small enough.

Algorithm 1 Grid search1: Initialization: ∆1, ∆2, ∆3, ∆4, and τtemp = ∅.2: for all α1 ∈ (0 : ∆1 : 1), β ∈

(es(CK log(n/K))

κTPs: ∆2 : 1

),

α2 ∈ (α2,min : ∆3 : 1), Pt ∈ (Pt,min : ∆4 : Pt,max) do3: while 0 ≤ 1− α1 − κβ − α2 ≤ 1 do4: τtemp = [τtemp, τcs] where τcs is expressed as (17).5: end while6: end for7: Return τmax = τtemp.

2) fmincon: fmincon is a toolbox in MATLAB which isefficient but it may return a local optimal value.

3) Random sampling: A set S = v1, v2, · · · , vZ thatsatisfies the conditions in (27) is generated randomly,where vi = (α1, β, α2, Pt) (i ∈ 1, 2, · · · ,Z) is a tupleof generated random samples, and Z is the number ofgenerated tuples. The accuracy of this method dependson number of tuples Z generated for calculation. Thismethod is efficient for solving (27) as it does not relyon advanced optimization techniques and the method ofgenerating (α1, β, α2, Pt) is efficient.

IV. THROUGHPUT OPTIMIZATION OF MULTIPLESECONDARY USERS

In this section, the closed-form expressions for the poweroutage probability of data transmission are deprived for boththe active and inactive SUs, respectively. With the imple-mentation of MC technique, throughput of the CSS networkwith multiple SUs including the active and inactive ones isoptimized.

A. Power Outage Probability Analysis

In this subsection, we provide power outage probabilityanalysis for spectrum sensing and data transmission for bothactive and inactive SUs. In stochastic geometry networks, weusually focus on the average networks performance. As such,we first evaluate the average number of potential active users.

Corollary 1. Based on Theorem 1, we obtain the averagenumber of potential active users as follows:

Jact = J

1− exp

−πλpδµδs

M−1∑m=0

Γ(m+ δ, µsd

ξ0

)m!

.

(19)

Proof. Note that the condition that a typical single user canbe active is that its harvested energy is capable of supportspectrum sensing. Otherwise, it has to keep silent during thesensing slot. As such, the probability to become an active useris equal to the non-outage probability for spectrum sensing.Following the similar procedure as [31], we can obtain theaverage number of active users as (19).

Remark 2. The derived result in Corollary 1 indicates thatthe average active number is a strictly monotonic increasingfunction of λp. As such, we can increase the value of Jact byincreasing λp.




1) Power Outage Probability of Spectrum Sensing: In theconsidered CSS networks, we assume that the number ofactive participating SUs is J1. In order to guarantee thereare at least J1 number of active SUs to collect enough num-ber measurements for exact recovery, the constraint Jmin ≤J1 ≤ Jact should be satisfied. Here, Jmin is defined asthe minimal number of active SUs that can guarantee exactrecovery. By considering the energy consumption for spectrumsensing and the minimal number of measurements requiredfor exact recovery, the constraint on J1 can be rewritten asCµ2νKlog6ν ≤ κβTPsJ1

es≤ nJact. As a result, the constraint

on J1 can be transformed into the constraint on β as follows:

C6 :es(Cµ2νklog6ν

)ρTPsJ1

≤ β ≤ 1, (20)

where Cµ2νklog6ν is the lower bound of the number ofobserved measurements at the FC to guarantee the exact matrixrecovery [32]. Here, ν = max (n, J) and µ = O

(√log ν

).

With the constraint on β in (20), by considering the averagenetwork performance, the power outage probability of sensingcan be regarded as zero for simplicity. This behavior can beexplained as follows: 1) For active SUs, it is assumed that theharvested energy from the first time slot is enough for them totake measurements for spectrum sensing. Therefore, the poweroutage probability of spectrum sensing for active SUs are zero;and 2) For the inactive SUs, they will not perform spectrumsensing and only transmit data during the last time slot.

2) Power Outage Probability of Data Transmission: Forthose active SUs, the power outage probability of data trans-mission is same that of the single SU scenario in (11).For those inactive SUs, as all the time slots before datatransmission are used for energy harvesting, the power outageprobability is different from (11). The following theoremprovides exact analysis for the power outage probability ofdata transmission for both active and inactive SUs in CSSnetworks.

Theorem 2. The power outage probability of data transmis-sion at the active and inactive SUs in the fourth time slot canbe expressed in closed-form as follows:

P outψ = exp

−πλpδµδψ

M−1∑m=0

Γ(m+ δ, µψd

ξ0

)m!

(21)

where ψ ∈ (a, i), µa = Pt(1−α1−β−α2)+PsβηPpA(α1+α2) , and

µi = Pt(1−α1−β−α2)ηPpA(α1+β+α2) .

Proof. Based on (6), the power outage probability of datatransmission at the active SUs is the same as P outt as givenin (11) by replacing µa → µζ .

Similarly, based on (6), the power outage probability of datatransmission at the inactive SUs can be expressed as follows:

P outi = Pr PH3 < Pt (22)

= Pr

max

p∈Φp,‖dp‖≥d0

‖hp‖2dp−ξ

< µi

.

Following the similar procedure as (A.5) and applying µi →µs, we can obtain P outi in (21).

The proof is completed.

B. Matrix Completion Based Cooperative Spectrum Sensing

As the spectrum is normally underutilized in practice, thetransmitted signals exhibit a sparsity property in frequencydomain, which can be transformed into a low-rank propertyof the matrix X at the FC. When only the J1 active SUssend compressed samples to the FC, the matrix with collectedmeasurements is incomplete at the FC. The exact matrixSf can be obtained by solving the following low-rank MCproblem

min rank (Sf ) s.t. ‖Θvec (HfSf )− vec (X)‖22 ≤ ε,(23)

where rank (·) is the rank function of a matrix whose value isequal to the number of nonzero singular values of the matrix,and ε refers to the noise level. However, the problem in (23)is NP-hard due to the combinational nature of the functionrank (·). It has been proved that the nuclear norm is the bestconvex approximation of the rank function over the unit ballof matrixes with norm no less than one [33]. Therefore, (23)can be replaced by the following convex formulation:

min ‖Sf‖∗ s.t. ‖Θvec (HfSf )− vec (X)‖22 ≤ ε. (24)

The problem in (24) can be solved by multiple existing MCsolvers, such as singular value decomposition [34], nuclearnorm minimization [14], etc.

Once the exact matrix is recovered, energy detection canbe used to determine the spectrum occupancy. In the CSSnetworks, as all the participating SUs send the collectedsamples to the FC, the data fusion is considered. Supposethe channel coefficients from the PUs to each participatingSUs are known. When the channel coefficients are unknown,the weighting factor associated with the jth SU is set to begj = 1√

J. By using the maximal ratio combining, probability

of false alarm of the CSS networks Qf becomes

Qf = Q

(Q−1

(Pd)√

1 +σ2s

Jσ2H +

√n

2J

σ2s

σ2H

), (25)

where H =J∑j=1

|hj|22 refers to the channel coefficients for all

the participating SUs in the considered CSS networks.

C. Throughput Analysis

By applying the MC technique at the FC, β in (21) shouldbe replaced by κβ. In addition, the throughput of consideredCSS networks with multiple SUs can be expressed as

τmc =J∏j=1

(1− P outψ (j)

)× (1−Qf )

× (1− α1 − κβ − α2) τt, (26)

where κ is defined as the compression ratio at active SUs.ψ = a refers to the J1 number of active SUs and ψ = irefers to the rest of inactive SUs in the considered networks.If no MC technique is implemented at the FC and each SU issupposed to take samples at Nyquist rate, all the participatingSUs will send samples to the FC. Then the throughput of the




considered networks can be given by replacing P outψ (j) in (26)with P outa (j) and κ = 1. The constraints for the multiplenodes case are the same as that for the single node case exceptthe condition for β given in (20).

The throughput can be maximized by solving the followingproblem:

(P1) : maxα1,β,α2,Pt

τmc (27)

s.t. C1 : 0 ≤ α1 ≤ 1,

C3 : α2,min ≤ α2 ≤ 1,

C4 : 0 ≤ 1− α1 − κβ − α2 ≤ 1,

C5 : Pt,min ≤ Pt ≤ Pt,max,

C6 :es(Cµ2νklog6ν

)ρTPsJ1

≤ β ≤ 1.

Constraints C1 and C4 bound the time slot allocated forenergy harvesting in first time slot and data transmission.Constraint C3 ensures that the third time slot is enough formatrix recovery at the FC and data transmission between theSU and FC. Constraint C5 limits the power level for datatransmission. Constraint C6 highlights the minimum time slotfor compressive spectrum sensing, in which the lower boundof β is used to make sure that the number of compressedsamples is sufficient to guarantee exact recovery at the FC. Itis noticed that the structure of (P1) is similar as (P0). Thuswe can use the similar methods to obtain the near optimalthroughput for the CSS network.

V. NUMERICAL RESULTS

In the simulation, we set the frame period to be T = 1 s,the transmit power of PBs to be Pp = 43 dBm, the number ofantennas of PB to be M = 32, the carrier frequency for powertransfer to be 900 MHz, and the energy conversion efficiencyof WPT to be η = 0.8. In addition, it is assumed that thetarget probability of detection Pd is 90%. In this paper, welet d0 ≥ 1 m to make sure the path loss of WPT is equal orgreater than one. In the following part, SNR =

σ2s

σ2 refers to theSNR in sensing channels. Regarding the considered Rayleighfading model, the parameters are set as zero mean and unitvariance, which is corresponding to our theoretical analysis.

A. Numerical Results on Optimizing Throughput of SingleUser

In this subsection, simulation results of the optimizedthroughput of single SU are demonstrated after the derivedpower outage probability in (11) and Pf in (15) are verifiedby Monte Carlo simulations.

Fig. 2 plots the power outage probability of spectrum sens-ing versus density of PBs λp with different power thresholdPs. The black solid and dash curves are used to representthe analytical results with d0 = 1 m and d0 = 1.5 m,respectively, which are both obtained from (11). Monte Carlosimulations are marked as “•” to verify our derivation. Thefigure shows the precise agreement between the simulationand analytical curves. One can be observed is that as densityof PBs increases, the power outage probability dramatically

Density of PBs10-3 10-2 10-1

Pow

er o

utag

e pr

obab

ility

10-3

10-2

10-1

100

Analytical d0=1.0 m

Analytical d0=1.5 m

Simulation

Ps=10 dBm

5 dBm0 dBm

Fig. 2: Power outage probability of spectrum sensing versusdensity of PBs with M = 32, Pp = 43 dBm, α1 = 0.25,α2 = 0.2, and β = 0.25.

decreases, which is also corresponding to the remark we obtainfrom Theorem 1. This is because the multiuser diversity gainis improved with increasing number of PBs when chargingwith WPT. The figure also demonstrates that the outage occursmore frequently as the power threshold Ps and the radius ofd0 increase.

Fig. 3 plots the probability of false alarm Pf versus SNRin sensing channels with different compression ratios κ. It isobserved that Pf decreases with higher SNR, which wouldimprove the throughput of secondary network. In this case,the sparsity level is set to 12.5%, which means 12.5% ofthe spectrum of interest is occupied by PUs. The blacksolid curve is used to represent the analytical result whichis obtained from (15) with enough protection to PUs beingprovided (Pd = 90% is defined in the IEEE 802.22 standardabout the cognitive radio [35]). Monte Carlo simulations withcompression ratio κ = 100% are marked as “” to verify ourderivation, which represents the scenario without CS techniqueimplemented. The figure shows precise agreement betweenthe simulation results with κ = 100% and analytical curves,which is a benchmark for the following comparison. When κ isreduced to 50%, it is noticed that Pf is still well matched withthe analytical result, which means the performance of spectrumsensing is not degraded when only 50% of the samples arecollected at the SU. When κ is further reduced to 25%, Pfis increased, which means the signal recovery is not exactany more. As a result, wrong decisions would be made forspectrum occupancies, which leads to a higher Pf . Throughputof the CSS network would be degraded correspondingly asthe chance to access vacant channels is reduced. Actually,the minimal compression ratio guaranteeing successful signalrecovery is dependent on the sparsity level of signals tobe recovered and the number of samples, when signals aresampled at or above Nyquist sampling rates. This is provedby Candes in [10] and out of the scope of this paper.

Fig. 4 plots the achieved throughput τcs versus the lowerbound of time allocated to the third slot α2. Here α2 is




SNR (dB)-20 -15 -10 -5

Pro

babi

lity

of fa

lse

alar

m

0

0.2

0.4

0.6

0.8

1Analyticalκ=100%κ=50%κ=25%

Fig. 3: Performance comparison between theoretic results fortraditional energy detection and the compressive spectrumsensing under different SNR levels in sensing channels anddifferent compression ratio κ, Pd = 90%, and sparsity level= 12.5%.

reserved for signal recovery at the FC and data transmissionbetween the SU and the FC. In this figure, several observationsare drawn as follows: 1) The maximal throughput achieved bygrid search method is slight higher than that of fmincon methodas fmincon relies on the initial input and may return a localoptimal value. However, the accuracy of grid search method isdependent on the step sizes; 2) The random sampling methodachieves lower throughput than grid search and fmincon meth-ods, which demonstrates the benefits of the presented gridsearch and fmincon methods. When the generated sets increasefor random sampling, the achieved throughput get closer to theoptimal but the computational complexity is much increase; 3)It is seen that the optimal value of time assigned to the thirdslot α2 always equals to the lower bound α2,min. This gives asign that the throughput can be improved if the time slot for thesignal recovery at the FC and data transmission between SUsand the FC is reduced. In other words, the energy harvestingshould be done mainly in the first time slot α1 to reduce thepower outage probability in the following spectrum sensingslot if the signal recovery and data transmission between theSU and FC can be promised.

Fig. 5 presents the throughput comparison between theproposed frame structure and the traditional one that doesnot adopting compressive spectrum sensing and remote FCto perform signal recovery. Without implementing the FC,SUs have to perform signal recovery by themselves andcannot harvest energy again after sampling. Meanwhile, inorder to make decisions on spectrum occupancies, each SUhas to perform energy detection locally, which introducesmore energy consumption. Here, for SUs employing traditionalframe structure, the time slot consumed for decision makingis set to α2,min to simplify the comparison. The time slotrequired for conduct spectrum sensing at Nyquist rate is setto βT with β = 0.2. In Fig. 5, we can see that the achievedthroughput is improved with decreasing compression ratio κ.

α2,min

(T)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Thr

ough

put (

bits

/s/H

z)

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22fmincongrid searchrandom sampling

Fig. 4: Throughput of single SU τcs versus lower bound ofthe third time slot α2,min, SNR = −10 dB, and κ = 100%.

α2,min

(T)0 0.1 0.2 0.3 0.4 0.5

Thr

ough

put (

bits

/s/H

z)

0

0.1

0.2

0.3

0.4

0.5

Proposed frame κ=100%Proposed frame κ=40%Proposed frame κ=20%Traditional frame

Fig. 5: Throughput comparison between the proposed framestructure and the traditional one versus lower bound of thethird time slot α2,min, SNR = −10 dB, and κ = 100%, 40%and 20%.

This benefits from that more time slot becomes available forenergy harvesting and data transmission when compressionratio κ decreases. It can be also noted that the achievedthroughput with the traditional frame is lower than that withthe proposed frame structure, even though the compressionratio κ is set to 100%. This performance degradation is causedby that the third time slot cannot be utilized for energyharvesting, as no FC is implemented for energy detection inthe traditional frame structure. In a summary, we can concludethat the proposed frame structure outperforms the traditionalone in terms of achieved throughput.

Fig. 6 plots the achieved throughput τcs versus lower boundof the third time slot α2,min and compression ratio κ whensolving the problem in (27). It shows that the achieved max-imal throughput increases with decreasing κ and increasingα2,min. This behavior can be explained as follows: as κ de-creases, the number of samples Λ to be collected for spectrum




0.60.4α

2,min

0.201

0.5κ

0.5

0.6

0.3

0.4

0.2

0.10

Thr

ough

put (

bit/s

/Hz)

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Fig. 6: Optimized throughput of single SU τcs versus lowerbound of the third time slot α2,min and compression ratio κ,and SNR = −10 dB in sensing channels.

sensing at an SU is reduced as the signal in original size ofN × 1 can be recovered from less number of measurementsby utilizing the CS technique. When the time slot assignedfor spectrum sensing is reduced, the energy consumption forspectrum sensing is reduced. As a result, the time whichcan be assigned for data transmission is increased. Therefore,the throughput of the secondary network is improved. Byoptimizing the transmission power Pt, the energy harvestedin the current frame period would be fully utilized and themaximal throughput can be achieved accordingly. It should benoted that when the compression ratio κ is set to 100% and thelower bound of the third time slot α2,min is zero, the achievedthroughput can be regarded as that of the traditional framestructure design without considering sub-Nyquist sampling.

B. Numerical Results on Optimizing Throughput of Coopera-tive Spectrum Sensing Networks

In the case of optimizing the throughput of the CSS net-works, the total number of participating SUs is set be to J =500, including both the active and inactive SUs. Comparing theformat of (P1) and (P0), we can see that both of them are linearconstrained. Therefore, similar as (P0), the grid search methodcan be applied to obtain the optimal throughput but with non-negligible complexity, especially for the case of optimizingthroughput of the whole cooperative network. The fminconmethod can be adopted to obtain the sub-optimal throughputefficiently. In the following simulations, the fmincon methodis utilized to solve the optimization problem (P1). In addition,as aforementioned in the introduction part, many algorithmshave been proposed for the low-rank MC based cooperativespectrum sensing. With Pd = 90%, the detection performancewith different compression ratios is presented in the authors’previous work in [16], which would not be demonstrated hereagain to reduce redundancy. In the following simulations,how the achieved throughput is influenced by parameters,such as the number of active SUs J1, compression ratio κ

Density of PBs10-3 10-2 10-1

Pow

er o

utag

e pr

obab

ility

10-4

10-3

10-2

10-1

100

Pout single SU

Pout active SU

Average Pout whole network

Pout inactive SU

Fig. 7: Power outage probability comparison for single SUand multiple SUs with versus density of PBs, Ps = 0 dBm,α1 = 0.25, α2 = 0.20, and β = 0.25.

and the lower bound of the third time slot α2,min, would bedemonstrated.

Fig. 7 plots the power outage probability Pout versus densityof PBs with different power threshold Ps. In this case, thepower outage probability here is for the whole system, whichcan be calculated as Pout = 1 − (1− P sout) × (1− P tout).Both the single SU scenario and multiple SUs scenario areillustrated in the figure. It can be observed that as densityof PBs increases, the power outage probability dramaticallydecreases, which is caused by that the multiuser diversity gainis improved with increasing number of PBs when chargingwith WPT. We can also observe that the Pout of multipleSUs scenario is lower than that of single SU scenario. This isbecause the power outage probability of spectrum sensing isalways zero in multiple SUs scenario, which in turn lower thepower outage probability of the whole system. For the multipleSUs scenario, the Pout of the active SUs, inactive SUs andthe average Pout of the CSS networks are all presented in thefigure. It is noted the averaged Pout falls between the Pout ofactive SUs and inactive SUs, which as we expected.

Fig. 8 plots the throughput τmc averaged on per SU withdifferent number of active SUs J1 and different compressionratios κ at each active SU. In this case, it is assumed thatthe exact matrix completion can be guaranteed when thenumber of active SUs J1 is in the range of 50 to 500, whichrefers to that the compression ratio κ changes from 10% to100%. It shows that the average throughput achieves the bestperformance when the number of active SUs is set to be theminimal possible number J1 = 50 in comparison with thatof case J1 = J . This benefits from the increasing number ofinactive SUs when the number of active SUs J1 decreases from500 to 50. With increasing number of inactive SUs, energyconsumption for spectrum sensing can be alleviated and moreenergy can be harvested for data transmission. It is furthernoticed that the achieved throughput is increased with thecompression ratio κ decreasing from 75% to 20%.This benefitsfrom that the required time slot for sub-Nyquist sensing is




50 100 150 200 250 300 350 400 450 500

J1

1.08

1.085

1.09

1.095

1.1

1.105

1.11

1.115

Thr

ough

put (

bits

/s/H

z)

κ=20%

κ=50%κ=75%

Fig. 8: Optimized throughput averaged on per SU of multipleSUs τmc versus number of active SUs J1 and compressionratio κ, SNR = −10 dB in sensing channels, and α2,min =0.05.

10

α2,min

0.2

0.5

0.5

Opt

imal

thro

ughp

ut (

bits

/s/H

z)

0.4

κ

1

0.6

1.5

0.8

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Fig. 9: Optimized throughput averaged on per SU of multipleSUs τmc versus lower bound α2,min and compression ratio κ,and SNR = −10 dB in sensing channels.

reduced when the compression ratio κ is decreased.Fig. 9 plots the achieved throughput averaged on per SU

versus different compression ratio κ and lower bound for thethird time slot α2,min. In this case, the number of activeSUs is set to J1 = 300. The achieved throughput shownin Fig. 9 is based on the condition that the exact matrixrecovery can be guaranteed with the given compression ratioκ. As shown in the figure, the achieved throughput increaseswith decreasing compression ratio κ and α2,min. The minimalcompression ratio guaranteing the exact matrix recovery isnondeterministic, which is dependent on the rank order of thematrix to be recovered and the size of matrix to be recovered.If the rank of matrix is fixed, the larger network size J ,the lower minimal compression ratio which can guarantee theexact matrix recovery.

VI. CONCLUSIONS

In this paper, a wireless powered cognitive radio (CR) net-work has been considered. In the considered networks, whileprotecting the primary users, we proposed a new frame struc-ture including energy harvesting, spectrum sensing, energyharvesting and data transmission. In the considered networkwith the proposed frame structure, closed-form expressionsin terms of power outage probability was derived for theproposed wireless power transfer (WPT) scheme. Additionally,sub-Nyquist sampling was performed at secondary users (SUs)to reduce the energy consumption during spectrum sensing.The compressive sensing and matrix completion techniqueswere adopted at a remote fusion center to perform the signalrecovery for detection making on spectrum occupancy. Byoptimizing the four time slots, throughput of an individualSU and the whole cooperative networks were maximized,respectively. Simulation results showed that the throughput canbe improved by adopting the proposed new frame structuredesign. We conclude that by carefully tuning the parametersfor different time slots and transmit power, WPT can be usedalong with sub-Nyquist sampling to provide a high qualityof throughput performance for CR network, with significantlyenergy computation reduction at power-limited SUs.

APPENDIX A: PROOF OF THEOREM 1

Based on (2), the power outage probability of spectrumsensing can be expressed as

P outs = Pr PH1≤ Ps

= EΦp

∏p∈Φp,‖dp‖≥d0

Pr‖hp‖2 ≤ dξpµs

= EΦp

∏p∈Φp,‖dp‖≥d0

F‖hp‖2(dξpµs

), (A.1)

where F‖hp‖2 is the cumulative density function (CDF) of‖hp‖2. As aforementioned in Section II, each channel elementof hp follows Rayleigh fading, which is independent complexGaussian distributed with zero mean and unit variance. As aconsequence, ‖hp‖2 follows a chi-squared distribution, whichis given by

F‖hp‖2 (x) = 1− Γ (M,x)

Γ (M). (A.2)

Given the fact that M is an integer value, with the aid of [36,Eq. (8.832.2)], (A.2) can be expressed as

F‖hp‖2 (x) = 1− e−x(M−1∑m=0

xm

m!

). (A.3)

Applying the moment generating function, we rewrite (A.1)as

P outs = exp

−λp ∫R2

(1− F‖hp‖2

(dξpµs

))ddp

. (A.4)




Then changing to polar coordinates and substituting (A.3) into(A.4), we obtain

P outs = exp

[−2πλp

M−1∑m=0

µms∫∞d0dpmξ+1e−µsd

ξpddp

m!

].

(A.5)

Applying [36, Eq. (3.381.9)] to calculate the integral, weobtain (11).

Similarly, based on (4), the power outage probability of datatransmission P outt can be expressed as follows:

P outt = Pr PT2≤ Pt

= Pr

max

p∈Φp,‖dp‖≥d0

‖hp‖2dp−ξ

≤ µt

. (A.6)

Following the similar procedure as (A.5) and applying µt →µs, we obtain P outt in (11).

The proof is completed.

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Zhijin Qin (S’13, M’16) received her doubleB.Sc degree from Beijing University of Posts andTelecommunications, Beijing, China and the Ph.D.degree in electronic engineering from Queen MaryUniversity of London, London, U.K., in 2012 and2016, respectively. She is currently a research as-sociate with Department of Computing, ImperialCollege London, London, U.K. She was awardedthe best paper at Wireless Technology Symposium,April 2012, London, U.K.

Her research interests include low-power wide-area network in IoT, fog networking, compressive spectrum sensing and non-orthogonal multiple access in 5G network. She has served as a TPC memberfor IEEE conferences such as ICC’16 and VTC’15 and VTC’14.

Yuanwei Liu (S’13, M’16) received his Ph.D. de-gree in electrical engineering from the Queen MaryUniversity of London, UK in 2016. Before that, hereceived his M.S. degree and B.S. degree from theBeijing University of Posts and Telecommunicationsin 2014 and 2011, respectively. He is currently thePost-Doctoral Research Fellow with the Departmentof Informatics, Kings College London, U.K.

His research interests include non-orthogonalmultiple access, millimeter wave, massive MIMO,Hetnets, D2D communication, cognitive radio, and

physical layer security. He received the Exemplary Reviewer Certificate ofthe IEEE Wireless Communication Letter in 2015. He has served as a TPCmember for many IEEE conferences such as GLOBECOM and ICC.

Yue Gao (S’03-M’07-SM’13) received his Bach-elor degree from Beijing University of Posts andTelecommunications in China in 2002, and his MScand PhD degrees in telecommunications and mi-crowave antennas from Queen Mary University ofLondon (QMUL) in United Kingdom in 2003 and2007, respectively. Prior to his current role as aSenior Lecturer (Associate Professor), he worked asResearch Assistant and Lecturer (Assistant Profes-sor) in the School of Electronic Engineering andComputer Science at QMUL after his PhD degree.

He is currently leading the Whitespace Machine Communication (WMC)Lab to develop theoretical research into practice in the interdisciplinary areaamong antennas, signal processing and spectrum sharing for cyber-physicalsystem (CPS), machine-to-machine (M2M) communications and internet ofthings (IoT) applications. Dr Gao has authored and co-authored over 100 peer-reviewed journal and conference papers, 2 best paper awards, 2 patents and2 licensed works to companies, and 1 book chapter. He is a Senior Memberof the IEEE and the principal investigator for TV white space testbed projectfunded by the Engineering and Physical Sciences Research Council (EPSRC),and a number of projects funded by companies. He has served as the SignalProcessing for Communications Symposium Co-Chair for IEEE ICCC 2016,and is serving Publicity Co- Chair for GLOBECOM 2016, Cognitive Radioand Networks Symposia Co- Chair for IEEE GLOBECOM 2017, and theGeneral Co-Chair of the IEEE WoWMoM 2017.

Maged Elkashlan received the Ph.D. degree inElectrical Engineering from the University of BritishColumbia, Canada, 2006. From 2007 to 2011, hewas with the Wireless and Networking Technolo-gies Laboratory at Commonwealth Scientific and In-dustrial Research Organization (CSIRO), Australia.During this time, he held an adjunct appointmentat University of Technology Sydney, Australia. In2011, he joined the School of Electronic Engineeringand Computer Science at Queen Mary University ofLondon, UK. He also holds visiting faculty appoint-

ments at Beijing University of Posts and Telecommunications, China. Hisresearch interests fall into the broad areas of communication theory, wirelesscommunications, and statistical signal processing for NOMA and Hetnets.

Dr. Elkashlan currently serves as Editor of IEEE TRANSACTIONS ONWIRELESS COMMUNICATIONS, IEEE TRANSACTIONS ON VEHICULARTECHNOLOGY, and IEEE COMMUNICATIONS LETTERS. He also serves asLead Guest Editor for the special issue on “Green Media: The Future ofWireless Multimedia Networks” of the IEEE WIRELESS COMMUNICATIONSMAGAZINE, Lead Guest Editor for the special issue on “Millimeter WaveCommunications for 5G” of the IEEE COMMUNICATIONS MAGAZINE, GuestEditor for the special issue on “Energy Harvesting Communications” ofthe IEEE COMMUNICATIONS MAGAZINE, and Guest Editor for the specialissue on “Location Awareness for Radios and Networks” of the IEEEJOURNAL ON SELECTED AREAS IN COMMUNICATIONS. He received BestPaper Awards at the IEEE International Conference on Communications(ICC) in 2016 and 2014, the International Conference on Communicationsand Networking in China (CHINACOM) in 2014, and the IEEE VehicularTechnology Conference (VTC-Spring) in 2013.

Arumugam Nallanathan (S’97-M’00-SM’05) isProfessor of Wireless Communications in the De-partment of Informatics at King’s College London(University of London). He served as the Headof Graduate Studies in the School of Natural andMathematical Sciences at King’s College London,2011/12. He was an Assistant Professor in the De-partment of Electrical and Computer Engineering,National University of Singapore from August 2000to December 2007. His research interests include5G Wireless Networks, Molecular Communications,

energy harvesting and cognitive radio networks. He published nearly 300technical papers in scientific journals and international conferences. He isa co-recipient of the Best Paper Award presented at the IEEE InternationalConference on Communications 2016 (ICC’2016) and IEEE InternationalConference on Ultra-Wideband 2007 (ICUWB 2007). He is an IEEE Distin-guished Lecturer. He has been selected as a ’Thomson Reuters Highly CitedResearcher’ in 2016.

He is an Editor for IEEE Transactions on Communications and IEEE Trans-actions on Vehicular Technology. He was an Editor for IEEE Transactionson Wireless Communications (2006-2011), IEEE Wireless CommunicationsLetters and IEEE Signal Processing Letters. He served as the Chair forthe Signal Processing and Communication Electronics Technical Commit-tee of IEEE Communications Society, Technical Program Co-Chair (MACtrack) for IEEE WCNC 2014, Co-Chair for the IEEE GLOBECOM 2013(Communications Theory Symposium), Co-Chair for the IEEE ICC 2012(Signal Processing for Communications Symposium), Co-Chair for the IEEEGLOBECOM 2011 (Signal Processing for Communications Symposium),Technical Program Co-Chair for the IEEE International Conference on UWB2011 (IEEE ICUWB 2011), Co-Chair for the IEEE ICC 2009 (WirelessCommunications Symposium), Co-chair for the IEEE GLOBECOM 2008(Signal Processing for Communications Symposium) and General Track Chairfor IEEE VTC 2008. He received the IEEE Communications Society SPCEoutstanding service award 2012 and IEEE Communications Society RCCoutstanding service award 2014.

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