Throughput Analysis of Wireless Powered Cognitive Radio Networks with Compressive Sensing and Matrix Completion

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by  Zhijin Qin

1
Throughput Analysis of Wireless Powered
Cognitive Radio Networks with Compressive
Sensing and Matrix Completion
Zhijin Qin, Yuanwei Liu, Yue Gao, Maged Elkashlan, and Arumugam Nallanathan
6 Abstract—Inthispaper,weconsideracognitiveradionetwork the recent research of WPT. In [4], a new hybrid network
1 in which energy constrained secondary users (SUs) can harvest architectureis designedto enablechargingmobileswirelessly
0 energy from the randomly deployed power beacons (PBs). A in cellular networks. For cooperative systems, new power
2 new frame structure is proposed for the considered network.
allocation strategies are proposed in a cooperative networks
A wireless power transfer (WPT) model and a compressive
n spectrum sensing model are introduced. In the WPT model, a wheremultiplesourcesanddestinationsarecommunicatedby
a new WPT scheme is proposed, and the closed-form expressions an energy harvesting relay [5]. For non-orthogonal multiple
J
for the power outage probability are derived. In compressive access(NOMA)networks,in[6],anewcooperativesimultane-
5 spectrum sensing model, two scenarios are considered: 1) Single
ouslywirelessinformationandpowertransferNOMAprotocol
2 SU, and 2) Multiple SUs. In the single SU scenario, in order to
is proposed with considering the scenario where all users are
reduce the energy consumption at the SU, compressive sensing

] techniquewhich enablessub-Nyquistsamplingisutilized.In the randomly deployed.
T
multiple SUs scenario, cooperative spectrum sensing (CSS) is Along with improving energy efficiency through energy
.I performed with adopting low-rank matrix completion technique harvesting, cognitive radio (CR) technique can improve the
s to obtain the complete matrix at the fusion center. Throughput
spectrumefficiencyandcapacityofwirelessnetworksthrough
c optimizations of the secondary network are formulated into
[ dynamic spectrum access [7]. Therefore, in order to design
two linear constrained problems, which aim to maximize the

throughput of single SU and the CSS networks, respectively. networks which are both spectrum and energy efficient, the
1
Three methods are provided to obtain the maximal throughput secondary users (SUs) in CR networks can be equipped
v
8 ofsecondarynetworkbyoptimizingthetimeslotsallocationand with the energy harvesting capability. For the SUs powered
the transmit power. Simulation results show that: 1) Multiple
7 by energy harvested from wireless radio frequency, high
SUs scenario can achieve lower power outage probability than
5 sampling rate is difficult to be achieved. To overcome this
single SU scenario; and 2) The optimal throughput can be
6
improved by implementing compressive spectrum sensing in the issue,compressivesensing(CS),whichwasinitially proposed
0
proposed frame structure design. in [8], is introduced to wideband spectrum sensing in [9]
.
1
to reduce the power consumption at SUs. As the spectrum
0 Keywords: Compressive sensing, low-rank matrix completion,
of interest is normally underutilized in reality [10], [11],
6 spectrumsensing,sub-Nyquistsampling,wirelesspowertransfer.
1 it exhibits a sparse property in frequency domain, which
: makes sub-Nyquist sampling possible by implementing the
v
i I. INTRODUCTION CS technique at SUs. In addition, when dealing with ma-
X ENERGY efficiency and spectrum efficiency are two crit- trices containing limited available entries, low-rank matrix
r completion (MC) [12] was proposed to recover the complete
ical issues in designingwireless networks.Recentdevel-
a
matrix. As the sparse property of received signals can be
opments in energy harvesting provides a promising technique
transformed into low-rank property of the matrix constructed
to improve the energy efficiency in wireless networks. Dif-
in the cooperative networks, in [13], the authors proposed
ferent from harvesting energy from traditionalenergy sources
to apply joint sparsity recovery and low-rank MC to reduce
(e.g., solar, wind, water, and other physical phenomena) [1],
the sensing and transmission requirements and improve the
the emerging wireless power transfer (WPT) further under-
sensing performance in CR networks. In order to improve
pins the trend of green communications by harvesting en-
robustness against channel noise, a denoised algorithm was
ergy from radio frequency (RF) signals [2]. Inspiring by the
proposed in [14] for compressive spectrum sensing at single
great convenience offering by WPT, several works have been
SUandlow-rankMCbasedspectrumsensingatmultipleSUs.
studied to investigate the performance of different kinds of
energy constraint networks [3]–[6]. Two practical receiver
architectures, namely a time switching receiver and a power A. Related works
splitting receiver, were proposed in a multi-input and multi-
Somethroughputoptimizationworkshavebeenrecentlyde-
output (MIMO) system in [3], which laid a foundation in
velopedinwirelesspoweredcommunicationnetworks.Theau-
Z.Qin,Y.Liu,Y.GaoandM.ElkashlanarewithQueenMaryUniversity thorsin[15]consideredthethroughputmaximizationproblem
of London, London, United Kingdom. (email:{z.qin; yuanwei.liu; yue.gao; forbothbattery-freeandbattery-deployedcasesbyoptimizing
maged.elkashlan}@qmul.ac.uk)
the time slots for energy harvesting and data transmission. In
A.Nallanathan iswithKing’sCollegeLondon,London,UnitedKingdom.
(email:[email protected]) addition, recent work [16]–[18] on the CR networks powered
2
byenergyharvestingmainlyfocusesonthespatialthroughput the FC to perform cooperative spectrum sensing (CSS).
optimization under various constraints. The authors in [16] When the signal recovery process is performed at the
considered CR networks with an energy-harvesting SU with remote FC, energy harvesting can be performedagain at
infinite battery capacity. The goal is to determine an optimal the SUs locally in the third time slot.
spectrum sensing policy that maximizes the expected total Throughputoptimizationsoftheproposedframestructure
•
throughput subject to an energy causality constraint and a areformulatedinto two linear constrainedproblemswith
collisionconstraint.Inordertoimprovebothenergyefficiency the purpose of maximizing the throughput of a single
andspectralefficiency,theauthorsin[17]consideredasimilar SUandthewholecooperativenetworks,respectively.The
network model and the stochastic optimization problem is formulated problems are solved by using three different
formulated into a constrained partially observable Markov methods to obtain the maximal achievable throughput
decision process. At the beginning of each time slot, a SU respectively.
needs to determine whether to remain idle so as to conserve Simulation results show that the proposed frame struc-
•
energy, or to execute spectrum sensing to acquire knowledge ture design outperforms the traditional one in terms of
of the current spectrum occupancy state. The throughput is throughput. It is noted that the multiple SUs scenario
maximized by the design of a spectrum sensing policy and a can achieve better outage performance than the single
detection threshold. The authors in [18] consider an energy SU scenario.
constraint RF-powered CR network by optimizing the pair of
the sensing duration and the sensing threshold to maximize C. Organizations
the average throughputof the secondary network.
The rest of this paper is organized as follows. Section II
describes the considered WPT model and spectrum sensing
B. Motivations and contributions model based on the proposed frame structure. Section III
presents the throughput analysis of single SU scenario with
The aforementioned works have played a vital role and
applying CS technique. Section IV provides the throughput
laid solid foundation for fostering new strategies for frame
analysisofmultipleSUswithadoptingMCtechnique.Section
structure design. However, in [15], spectrum sensing is not
V shows the numerical analyses of the considered network
consideredin the frame structure design. In addition,in [16]–
modelwith the optimized throughputof single SU and multi-
[18], the proposed frame structure designs mainly aim to
ple SUs, respectively. Section VI concludes this paper.
maximizethethroughputbyoptimizingthethresholdandtime
slots. When considering the energy efficiency and spectrum
efficiency,it is meaningfulto introducesub-Nyquistsampling II. NETWORK MODEL
to reduce the energyconsumptionat SUs in wireless powered A. Network description
CR networks.Inthispaper,weproposea newframestructure We consider a CR network, where SUs are energy con-
design for wireless powered CR networks with implementing strained. The whole spectrum of interest can be divided
sub-Nyquist sampling at SUs. The CS and MC techniques into I channels. A channel is either occupied by a primary
are adopted to perform the signal recoveryat a remote fusion user (PU) or unoccupied. Meanwhile, there is no overlap
center (FC) for making decision on spectrum occupancy. To betweendifferentchannels.Thenumberofoccupiedchannels
the best of our knowledge,this is the first paper on the frame is assumed to be K, where K I. Each SU is supposed
structure design employing the sub-Nyquist sampling rates to perform sensing on the who≤le spectrum. It is assumed
based spectrum sensing in wireless powered CR networks. that all SUs keep quiet as forced by protocols, e.g., at the
The summarized contributions of this paper are illustrated media access control layer during spectrum sensing period.
as follows: Thus the received signals only contain the signals of active
We propose a new frame structure for wireless powered PUs and channel noise. As shown in Fig. 1, for each SU,
•
CR networks which includes four time slots: energy it is assumed that the sensing and transmission can only be
harvesting,spectrumsensing, energyharvesting and data scheduled by utilizing energy harvested from power beacons
transmission. We introduce a WPT model and a spec- (PBs). The spatial topology of all PBs are modeled using
trum sensing model with sub-Nyquist sampling for the homogeneous poisson point process (PPP) Φ with density
p
considered networks. λ . Without loss of generality, we consider that a typical SU
p
In the WPT model, we propose a new bounded WPT is located at the origin in a two-dimensional plane. Each SU
•
scheme where each SU selects a PB nearby with the is equipped with a single antenna and has a corresponding
strongest channel to harvest energy. We derive closed- receiver with fixed distance. Each PB is furnished with M
form expressions for the power outage probability. antennas and maximal ratio transmission (MRT) is employed
In the spectrum sensing model, sub-Nyquist sampling is at PBs to perform WPT to the energy constrained SU. The
•
performedat each SU to reduce the energy consumption energy harvesting channels are assumed to be quasi-static
duringspectrumsensingperiod.We considertwoscenar- fading channels where the channel coefficients are constant
ios:singleSU scenarioandmultipleSUsscenario.Inthe for each transmission block but vary independently between
single SU scenario, we adopt CS recovery algorithm to different blocks. The spectrum of interest is wideband and
obtain the original signal. In the multiple SUs scenario, each SU performs wideband spectrum sensing to discover
we utilize MC algorithm to obtain complete matrix at spectrumholesfordatatransmission.Oncethespectrumholes
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are identified, SUs can start data transmission. It is assumed thatthereisnobatterystorageenergyforfutureuseandallthe
that the time of each frame is T. In the considered networks, harvestedenergyduringenergyharvestingtimeslotsisusedto
single SU and multipleSUs scenariosare analyzedto achieve performspectrumsensinganddatatransmissioninthecurrent
different throughputtargets. frame period.
1) Single SU scenario: in the considered spectrum sensing WeadoptapowertransferschemewheretheSUselectsthe
network with single SU, a frame period at a single SU PB with the strongest channel to harvest energy.At each SU,
includes four time slots as outlined in the blue oval in the energyharvested from the selected PB in the first and the
Fig.1:1)energyharvestingtimeslot,inwhicheachSU third time slots can be obtained as follows:
harvests the energy from PBs during the α1T period, E = max h 2L(d ) ηP γT, (1)
H p p p
wfraitmheαp1ebrieoidn;g2th)espferacctrtuiomn osefnesninegrgytimhearvsleostt,inigniwnhoicnhe p∈Φp,kdpk≥d0nk k o
where γ is the ratio of the time used for energy harvesting to
eachSUperformssub-NyquistsamplingbyapplyingCS
the total time of a frame,η is the powerconversionefficiency
techniques.Thecompressedmeasurementsarethensent at the SU, P is the transmit power of PBs. Here, h is a
p p
to a remotepowerfulFC duringtheβT periodbyusing M 1 vector, whose entries are independent complex Gaus-
×
the harvested energy during the α T period; 3) energy C
1 sian distributed with zero mean and unit variance employed
harvestingtimeslotfordatatransmission,inwhicheach
to capture the effects of small-scale fading between PBs and
SUharveststheenergyfromPBsduringtheα2T period, the SU. L(d )=Ad ξ is the power-law path-loss exponent.
with α being the fraction of energy harvesting in one p −p
2 Thepath-lossfunctiondependsonthedistanced ,afrequency
p
frame period.As the spectrumis typicallyunderutilized
dependent constant A, and an environment/terrain dependent
inpractice,signalsreceivedateachSUexhibitsasparse
path-loss exponent ξ 2. All the channel gains are assumed
property. In this time slot, the original signals can be ≥
to be independent and identically distributed (i.i.d.).
recovered based on the collected measurements. The
For the active SUs, based on (1), the maximum sensing
sensingdecisionsaresentbacktothecorrespondingSU
power at the SU is given by
attheendofthethirdtimeslot;and4)datatransmission
slot, in which each SU performs data transmission P = EH1 = max h 2L(d ) ηPpα1. (2)
during the (1−α1−β−α2)T period. H1 βT p∈Φp,kdpk≥d0nk pk p o β
2) Multiple SUs scenario: as shown in Fig. 1, in the
As energy can only be stored in the current frame, the
considered networks with multiple SUs, named as co-
total energyfor data transmission is the sum of the remaining
operative spectrum sensing, SUs are spatially randomly
energyinthesecondslotandtheenergyharvestedinthethird
distributed. The total number of participating SUs is
time slot, which is given by
J(J 1) in the CSS networks. Before performing
≥
spectrum sensing, each participating SU compare the ET2 =(EH1 −ES)+EH2, (3)
harvested energy E with the energy consumption for
H1 where P is the sensing power consumed at an SU.
spectrum sensing E =P βT at the first time slot. If s
S H1 Based on (1) and (3), the corresponding power for data
E isgreaterthanE ,theSUwouldcontinueperform-
H1 S transmission is given by
ing spectrum sensing. This kind of SUs are named as
activeSUs.TheframestructureofactiveSUsissameas max h 2L(d ) ηP (α +α ) P β
p p p 1 2 s
aEthgSaat,inothfaensidnSgwUlaeiwStoUfourladtshdsewedsiectccrihibsietodonaeobnnoevsrgep.yeIcfhtraEurmHve1osticiscnlugepsmasntochdiaeensl PT2 = p∈Φp,kdpk≥d0nk 1k−α1−βo−α2 − (4.)
fromtheFCbeforestartingdatatransmission.Therefore,
For those inactive SUs, the harvested energy before data
the framework structure of inactive SUs would only
transmission is given by
includes two time slots: (α +β+α )T for energy
1 2
harvesting and the rest for data transmission. In the E = max h 2L(d ) ηP (α +β+α )T.
case of CSS, only measurements from active SUs are H3 p∈Φp,kdpk≥d0nk pk p o p 1 2 (5)
collected at the FC. The signals received at SUs exhibit
a sparsity property that yields a low-rank matrix of Based on (5), the maximum sensing power at the SU can
compressedmeasurementsattheFC. Therefore,the full be expressed as
informationofspectrumoccupanciescanbeobtainedby
E
adopting low-rank MC methods. PH3 = (1 α Hβ3 α )T. (6)
1 2
− − −
B. Wireless power transfer model C. Spectrum sensing model with sub-Nyquist sampling
Weconsideraboundedpowertransfermodelwithaprotec- In the considered spectrum sensing model, when J =1, it
tionzonewitharadiusd0,whichmeansthatnoPBisallowed becomes a single node scenario. At the jth SU (SUj) in the
to exist in this zone. If PBs are really close to the SU, the considered network, the received signals can be expressed as:
harvested energy would mathematically go to infinity [19]. It
r (t)=h (t) s(t)+n (t), (7)
is assumed that the SU is battery-free [6], [20], which means j j j
∗
4
a1T bT a2T (1- a-1b- a 2)T (a1+b a+ 2)T (1- a-1b- a 2)T
Energy Spectrum Energy Data Energy Data
Harvesting Sensing Harvesting Transmission Harvesting Transmission
T T
Active SU Inactive SU
... ...
Fusion center
Active SU Active SU
Power beacon
Samples reporting &
decision feedback
Decision feedback only Inactive SU
Fig. 1: Proposed frame structure design with energy harvesting, spectrum sensing and data transmission.
where s(t) Cn 1 refers to the transmitted primary signals transmitted signals and AWGN. In addition, the measurement
×
∈
in time domain, and h (t) is the channel gain between the matrix is a diagonal matrix Θ = diag(Θ ,Θ ,...,Θ ) in
j 1 2 J
transmitter and receiver, and n (t) (0,σ2I ) refers to size of Ω N with Ω = Λ J and N = n J. After
j n
∼ CN × × ×
AdditiveWhite Gaussian Noise (AWGN) with zeromean and the compressed measurements are collected at the FC, signal
variance σ2. recoverycanbeformulatedasaconvexoptimizationproblem.
Based on the Nyquist sampling theory,the sampling rate is As aforementioned,the considered network becomes a single
requiredto be atleast twice of the bandwidth.Itleads to high node case when J = 1. Then the existing algorithm for
energy cost which would be challenging to the energy con- CS can be utilized to recover the original signals. In the
strainedSUsinaCRnetwork.Itisnoticedthatthetransmitted cooperative networks (J > 1), existing algorithms for low-
signal s(t) exhibits a sparse property in frequencydomain as rank MC can be implemented to complete the matrix. It has
a large percentage of spectrum is normally underutilized in been proved that the exact signal or matrix can be recovered
practice.Thissparsepropertymakesthesub-Nyquistsampling if the number of available measurements are no less than the
at SUs by adopting CS techniques. After CS technique is minimumbound.Withenoughnumberofmeasurements,exact
implemented at SUs, the compressed measurements collected transmitted signals can be obtained at the FC. Then energy
at SU can be expressed as detection is adopted to determine the spectrum occupancies,
j
in which the decision is made by comparing the energy of
xj =Φjrj(t)=ΦjFj−1rj(f)=Θj(s(f)+nj(f)), (8) recovered signal with a threshold defined as [21]
whereΦ CΛ n(Λ<n)isthemeasurementmatrixutilized
atondconlljec(jtft∈)herecfoe×mrptorestsheedtmraenassmuriettmedenstisgxnjal∈s aCnΛd×t1h,eanAdWsG(fN) λ= σs2+σ2 1+ Q−1n(cid:0)/P¯2d(cid:1)!. (10)
(cid:0) (cid:1)
receivedatinfrequencydomain.ThecompressionratioatSUs Once the sensing decisions are deteprmined at the FC, they
is definedasκ= Λn,(0≤κ≤1). In addition,Θj =ΦjFj−1, would be sent back to the participating SUs in the CSS
where j−1 is inverse discrete Fourier transform (IDFT) networkbythereportingchanneltostartthedatatransmission
F
matrix. period.
Afterthe compressedmeasurementsare collected,SUs will
send them to a remote FC by a error-free reporting channel. III. THROUGHPUT ANALYSIS OF ASINGLE SECONDARY
In both the single node and multiple nodes scenarios, FC is USER
proposed to perform signal recovery efficiently. By adopting In this section, the closed form expressions are derived for
such a powerful FC, energy constrained SUs can get rid of the power outage probability of spectrum sensing and data
signal recovery process and continue harvesting energy from transmission, respectively. With CS technique implemented,
PSs. At the FC, the compressed measurements X can be the analysis with the targetof maximizingthroughputof each
expressed as SU locally is provided in this section.
X =ΘRf =Θ(Sf +Nf), (9) A. Power outage probability analysis
where R = [r (f),r (f),...,r (f)] which is in size of We assume there exists a threshold transmit power, below
f 1 2 J
n J, and S and N refer to the matrix constructed by whichthespectrumsensinginthesecondslotorthedatatrans-
f f
×
5
mission in the fourth slot cannot be scheduled. We introduce B. Compressive spectrum sensing
power outage probability, i.e., probability that the harvested
In the scenario of CS based single SU with WPT, the
energy is not sufficient to perform spectrum sensing or carry
original signals can be obtained by solving the l norm
1
outthedatatransmissionatacertaindesiredquality-of-service
optimization problem as follows:
(QoS)level.Inpracticalscenarios,weexpectaconstantpower
forthedatatransmission.Therefore,wealsodenotethepower minksj(f)k1,
athsrethsheotlrdanPssmiatsptohweesrenosfinthgepSoUw,erreisnpethcetivseelcyo.nTdheslofotlalonwdiPngt s.t.kΘj ·sj(f)−xjk22 ≤εj, (17)
theorem provides the exact analysis for the power outage where εj is the error bound related to the noise level. This
probability at the single SU scenario. optimization problem can be solved by many existing algo-
rithms for CS, such as by adopting many existing algorithms
Theorem1. Thepoweroutageprobabilityofspectrumsensing
such as CoSaMP [23], SpaRCS [24] etc.
Pout inthesecondtimeslotandthepoweroutageprobability
s The performance metric of spectrum sensing at single SU
of data transmission Pout in the fourth time slot can be
t can be measured by the probability of detection Pd and
expressed in closed-form as follows:
probability of false alarm P . For a target probability of
f
Pout =exp πλpδ M−1Γ m+δ,µζdξ0 , (11) tdheetecthtiroensh,oP¯ldd,cwanithbwehdiecthertmheinPedUsbyare(1s0u)ffiacciceonrtdlyingplryoteifcttehde,
ζ − µδ (cid:16) m! (cid:17)
ζ m=0 number of samples n is fixed. As a result, Pf at a single SU
X
wδh=er2e/ξζ,∈and(sΓ,t(),,µ)sis=theηPuβppPApsαer1,inµctom=plePtte(1Gη−Paαpm1A−m(βαa−1+αfuα2n)2+c)tPisoβn,. can be dPeriv=edQasQfoll1owP¯s: σs2+σ2 + nσs2 , (18)
Proof: Base·d· on (2), the power outage probability of f − d r σ2 r2σ2!
spectrum sensing can be expressed as (cid:0) (cid:1)
where σ2 and σ2 refer to the noise power and signal power,
Psout =Pr{PH1 ≤Ps} respectively. s
Assumingtheenergycostpersamplee inspectrumsensing
s
=E Pr h 2 dξµ is fixed, the energy consumption of spectrum sensing is
Φp k pk ≤ p s 
proportionalto the number of collected samples as given by:
p∈Φp,Ykdpk≥d0 n o
βTP
=EΦp Fkhpk2 dξpµs ,  (12) n= es s. (19)
p∈Φp,Ykdpk≥d0 (cid:0) (cid:1) In fact, the number of collected samples n is determined
where F h 2 is the cumulative density function (CDF) of by the sensing time slot. In this case, the energy consumed
h 2 ankdpiks given by  by reporting collected measurements and receiving decision
p
k k
results between SUs to the FC is ignored. Substituting (19)
F h 2(x)=1 e−x M−1xm . (13) into (18), we obtain
k pk − m!!
Applying the moment generating funcmXti=on0, we rewrite (12) as Pf = 1−Q Q−1 P¯d (cid:18)1+ σσs22(cid:19)+rβ2TePssσσs22!!.
(cid:0) (cid:1) (20)
Psout =exp−λp 1−Fkhpk2 dξpµs ddp. (14)
RZ2 (cid:16) (cid:0) (cid:1)(cid:17) C. Throughput analysis
Then changingtopolarcoordinatesandsubstituting(13) into
Byconsideringthepoweroutageprobability,thethroughput
(14), we obtain
at the single SU in the CR networks can be expressed as:
Psout =exp"−2πλpM−1µms Rd∞0 dpmmξ+!1e−µsdξpddp#. τ = 1−Psout × 1−Ptout ×(1−Pf)
m=0 (1 α β α )τ , (21)
X (15) (cid:0)× − 1(cid:1)− (cid:0)− 2 t (cid:1)
Applying [22, Eq. (3.381.9)] to calculate the integral, we where τ = log 1+ Pt is the throughput for the data
t 2 N0
obtain (11). transmissionslot,a(cid:16)ndN0re(cid:17)ferstotheAWGNlevelinthedata
Similarly,basedon(4),thepoweroutageprobabilityofdata transmission channel. Here we simplify the data transmission
transmission Pout can be expressed as follows: process and do not consider the fading, which means the
t
Pout =Pr P P throughput τt is only determined by the transmit signal-to-
t { T2 ≤ t} noise-radio (SNR).
=Pr max hp 2dp−ξ µt . (16) When implementing CS technique to achieve sub-Nyquist
(cid:26)p∈Φp,kdpk≥d0nk k o≤ (cid:27) sampling rate at an SU, It has been proven that the exact
Following the similar procedure as (15) and applying µt signal recovery can be guaranteed if the number of collected
→
µs, we obtain Ptout in (11). measurements satisfies Λ C Klog(n/K), where C is
The proof is completed. someconstantdependingon≥each·instance[25].Ifthesignalis
6
recovered successfully by Λ samples, the achieved P would on number of tuples Z generated for calculation. This
f
be the same as that no CS technique is implemented with n method is efficient for solving (23) as it does not rely
samples.Therefore,the necessarysensingtimeslotto achieve on advancedoptimizationtechniquesandthe methodof
the same P can be reduced to κβ with CS implemented. generating (α ,β,α ,P ) is efficient.
f 1 2 t
Replacing Pout, Pout and P in (21) by (11) and (20)
s t f
respectively, the full expression of the throughput with CS IV. THROUGHPUT ANALYSIS OF MULTIPLESECONDARY
implemented can be expressed as follows: USERS
In this section, the closed-form expressions for the power
M 1Γ m+δ,µcsdξ
πλpδ − ζ 0 outage probability of data transmission are deprived for both
τ = 1 exp
cs  − − δ (cid:16) m! (cid:17) theactiveandinactiveSUs,respectively.Afterconsideringthe
ζ=Y{s,t}  µcζs mX=0  throughputoptimization of a single SU locally in Section III,
  (cid:16) (cid:17) σ2 βTP σ2  weoptimizethethroughputoftheCSSnetworkswithmultiple
1 Q Q 1 P¯ 1+ s + s s
× − − d (cid:18) σ2(cid:19) r 2es σ2!! SMUCstiencchlundiqinuge.the activeandinactiveoneswith implementing
(cid:0) (cid:1)
P
(1 α κβ α ) log 1+ t . (22)
× − 1− − 2 × 2 N
(cid:18) 0(cid:19) A. Power outage probability analysis
If there is no CS technique implemented at SUs, the time In this subsection, we provide power outage probability
slot fraction for spectrum sensing follows the condition that analysis for spectrum sensing and data transmission for both
0 β 1. When the CS technique is implemented at SUs, active and inactive SUs. In the considered CSS networks, the
≤ ≤
the time slot for spectrum sensing follows the condition that number of active participating SUs is J and the number of
1
CKlog(n/K) κβTPs n.Bycombiningthesetwocondi- inactive SUs is J .
≤ es ≤ 2
tions, the constraint for β becomes es(CKlog(n/K)) β 1. 1) Power Outage Probability of Spectrum Sensing: Note
κTPs ≤ ≤
With the implementation of CS, the β in (11) is replaced by that the power outage probability of sensing is always zero.
κβ. This behaviorcan be explainedas follows:1) For active SUs,
Furthermore, the throughput can be maximized by solving itis assumedthattheharvestedenergyfromthe firsttime slot
the following problem: isenoughforthemtotakemeasurementsforspectrumsensing.
Therefore, the power outage probability of spectrum sensing
(P0): max τcs, (23) foractiveSUsarezero;and2) FortheinactiveSUs, theywill
α1,β,α2,Pt
not perform spectrum sensing and only transmit data. As a
s.t.C :0 α 1,
1 1
≤ ≤ consequence, there is no spectrum sensing outage.
e (CKlog(n/K))
C : s β 1, 2) Power Outage Probability of Data Transmission: For
2
κTP ≤ ≤
s those active SUs, the power outage probability of data trans-
C :α α 1,
3 2,min 2 mission is same that of the single SU scenario. For those
≤ ≤
C :0 1 α κβ α 1, inactive SUs, as all the time slots before data transmission
4 1 2
≤ − − − ≤
C :P P P , areusedforenergyharvesting,thepoweroutageprobabilityis
5 t,min t t,max
≤ ≤ differentasthatofactiveSUs.Thefollowingtheoremprovides
where α2,min refers to the minimum time slot fraction for the exact analysis for the power outage probability of data
signal recovery at the FC and data transmission between the transmissionforbothactiveandinactiveSUsinCSSnetworks.
SU and FC. P and P refer to the allowed minimum
t,min t,max
Theorem 2. The power outage probability of data transmis-
and maximum power levels for data transmission period. It
sion at the active and inactive SUs in the fourth time slot can
is noticed that (23) is a constrained nonlinear optimization
be expressed in closed-form as follows:
problemandtheobjectivefunctionisverycomplex.However,
the constraints are linear, which motives us to solve the M 1Γ m+δ,µ dξ
optimization problem by the following three methods: Pout =exp πλpδ − ψ 0 (24)
ψ − µδ (cid:16) m! (cid:17)
1) Grid search: The grid search algorithm for solving (23) ψ m=0
X
can be described in Algorithm 1. The grid search al-  
gorithm can find out the global optimal value if the where ψ ∈ (a,i), µa = Pt(1η−Pαp1A−(βα−1+αα2)2+)Psβ, and
step size ∆i(i=1,2,3,4) for α1, β, α2, Pt are small µi = ηPPt(p1A−(αα11−+ββ−+αα22)).
enough. However, the computational complexity would Proof:Basedon(6),thepoweroutageprobabilityofdata
greatly increase if the step is set to be small enough. transmission at the active SUs is the same as Pout as given
t
2) fmincon: fmincon is a toolbox in MATLAB which is in (11) by replacing µa µζ.
→
efficient but it may return a local optimal value. Similarly,basedon(6),thepoweroutageprobabilityofdata
3) Random sampling: A set S = v ,v , ,v that transmission at the inactive SUs can be expressed as follows:
1 2 Z
{ ··· }
satisfies the conditions in (23) is generated randomly,
Pout =Pr P <P (25)
wherev =(α ,β,α ,P )(i 1,2, ,Z ) isatuple i { H3 t}
i 1 2 t
∈{ ··· }
of generated random samples, and Z is the number of =Pr max hp 2dp−ξ <µi .
generated tuples. The accuracy of this method depends (cid:26)p∈Φp,kdpk≥d0nk k o (cid:27)
7
Algorithm 1 Grid search tosimplifythecase,wechooseh =1.Consequently,Q can
j f
1: Initialization: ∆1, ∆2, ∆3, ∆4, and τtemp = . be expressed as
∅
2: forallα1 ∈(0:∆1 :1),β ∈ es(CKκlToPg(sn/K)) :∆2 :1 , σ2 nJ σ2
3: α2w∈hi(lαe20,min1:∆3α:11),κPβt ∈α((cid:16)P2t,mi1nd:o∆4 :Pt,max) do(cid:17) Qf =Q Q−1 P¯d r1+ σs2 +r 2 σs2!. (29)
4: τtemp≤=[τ−temp,−τcs] w−here ≤τcs is expressed as (22). (cid:0) (cid:1)
5: end while
6: end for C. Throughput analysis
7: Return τmax =τtemp. By applying the MC technique at the FC, β in (24) should
be replaced by κβ. In addition, the throughput of considered
CSS networks with multiple SUs can be expressed as
Following the similar procedure as (15) and applying µ
i
→ J
µ , we can obtain Pout in (24).
s i τ = 1 Pout(j) (1 Q )
The proof is completed. mc − ψ × − f
j=1
X(cid:0) (cid:1)
(1 α κβ α )τ , (30)
1 2 t
× − − −
B. Matrix completion based cooperative spectrum sensing
where κ is defined as the compression ratio at active SUs.
As the spectrum is normally underutilized in practice, the
ψ = a refers to the J number of active SUs and ψ = i
1
transmitted signals exhibit a sparse property in frequency
refers to the rest of inactive SUs in the considered networks.
domain, which can be transformed into a low-rank property
If no MC technique is implementedat the FC and each SU is
of the matrix X at the FC. When only the J active SUs
1 supposed to take samples at Nyquist rate, all the participating
sendcompressedsamplesto the FC, the matrixwithcollected
SUs will send samples to the FC. Then the throughputof the
samples is incomplete at the FC. The exact matrix can be considerednetworkscanbegivenbyreplacingPout(j)in(30)
ψ
obtained by solving the following low-rank MC problem: with Pout(j) and κ = 1. The constraints for the multiple
a
nodescasearethesameasthatforthesinglenodecaseexcept
min rank(S(f))
s.t. vec(Θ S(f)) vec(X) 2 ε , (26) the condition for β. In the multiple nodes case, the condition
k · − k2 ≤ for β follows Cµ2νKlog6ν κβTPsJ1 nJ. Combining
≤ es ≤
where rank() is the rank function of a matrix whose value the condition for β in case of no MC implemented, β should
is equal to t·he number of nonzero singular values of the satisfy es(Cµ2νKlog6ν) β 1.
matrix, and ε = [ε1,...,εJ]. However, the problem in (26) ThethrouκgThPpsuJt1can b≤e ma≤ximizedbysolvingthe following
is NP-hard due to the combinational nature of the function problem:
rank(). It has been proved that the nuclear norm is the best
·
convex approximation of the rank function over the unit ball (P1): max τmc (31)
of matrixes with norm no less than one [26]. Therefore, (26) α1,β,α2,Pt
s.t. C ,C ,C andC ,
can be replaced by the following convex formulation: 1 3 4 5
e Cµ2νklog6ν
s
min S(f) C6 : β 1,
s.t. kvec(Θk∗ S(f)) vec(X) 2 ε . (27) (cid:0) ρTPsJ1 (cid:1) ≤ ≤
k · − k2 ≤
where Cµ2νklog6ν is the lower bound of the number of
The problem in (27) such as singular value decomposi-
observedmeasurementsattheFCtoguaranteetheexactmatrix
tion [27], nuclear norm minimization [12] and so on.
recovery [28]. Here, ν =max(n,J) and µ =O √logν . It
Once the exact matrix is recovered, energy detection can
isnoticedthatthestructureof(P1)issimilaras(P0).Thuswe
be used to determine the spectrum occupancy. In the CSS (cid:0) (cid:1)
can use the similar methods to obtain the optimal throughput
networks, as all the participating SUs send the collected
for the CSS network.
samples to the FC, the data fusion is considered. Suppose
the channel coefficients from the PUs to each participating
SUs are known. When the channel coefficients are unknown,
V. NUMERICAL RESULTS
the weighting factor associated with the jth SU is set to be
g = 1 . By using the maximal ratio combining, probability In the simulation, we set the frame period to be T = 1 s,
j √J
of false alarm of the CSS networks Qf becomes thetransmitpowerofPBs to bePp =43dBm, thenumberof
antennasofPBtobeM =32,thecarrierfrequencyforpower
σ2 n σ2 transfer to be 900 MHz, and the energy conversion efficiency
Q =Q Q 1 P¯ 1+ s H+ sH . (28)
f − d Jσ2 2J σ2 of WPT to be η = 0.8. In addition, it is assumed that the
(cid:0) (cid:1)r r ! target probability of detection P¯d is 90%. In this paper, we
J let d 1 m to make sure the path loss of WPT is equal or
where H = j=1|hj|2 refers to the channel coefficients for all great0er≥than one. In the following part, SNR = σσs22 refers to
theparticipatPingSUsintheconsideredCSSnetworks.Inorder the SNR in sensing channels.
8
A. Numerical results on optimizing throughput of single sec-
100
ondary user
In this subsection, simulation results of the optimized y
throughput of single SU are demonstrated after the derived bilit
power outage probability in (11) and P in (20) are verified ba10-1
f o
by Monte Carlo simulations. pr P =10 dBm
e s
Fig. 2 plots the power outage probability of spectrum ag 5 dBm
sensing versus density of PBs with different power threshold out 0 dBm
P . The black solid and dash curves are used to represent er 10-2
s w Analytical d =1.0 m
the analytical results with d0 = 1 m and d0 = 1.5 m, Po 0
respectively, which are both obtained from (11). Monte Carlo Analytical d =1.5 m
0
simulations are marked as “ ” to verify our derivation. The Simulation
• 10-3
figure shows the precise agreement between the simulation 10-3 10-2 10-1
and analytical curves. One can be observed is that as density
Density of PBs
of PBs increases, the power outage probability dramatically
Fig. 2: Power outage probability of spectrum sensing versus
decreases. This is because the multiuser diversity gain is
density of PBs with M = 32, P = 43 dBm, α = 0.25,
improvedwith increasing numberof PBs when chargingwith p 1
α =0.2, and β =0.25.
WPT. The figure also demonstrates that the outage occurs 2
more frequently as the power threshold P and the radius of
s
protection zone d increase.
0
1
Fig. 3 plots the probability of false alarm Pf versus SNR Analytical
in sensing channels with different compression ratio κ. It is κ=100%
observed that Pf decreases with higher SNR, which would m0.8 κ=50%
improve the throughput of secondary network. The black ar κ=25%
al
solid curve is used to represent the analytical result which e
s0.6
is obtained from (20) with enough protection to PUs being al
provided. Monte Carlo simulations with compression ratio of f
y
κ = 100% are marked as “◦” to verify our derivation, which bilit0.4
represents the scenario without CS technique implemented. a
b
o
The figure shows precise agreement between the simulation Pr0.2
andanalyticalcurves.Whenκisreducedto50%,itisnoticed
that P is still well matched with the analytical result, which
f
0
means the performance of spectrum sensing would not be
-20 -15 -10 -5
degraded when only 50% of the samples are collected at SNR (dB)
an SU. When κ is further reduced to 25%, P is increased,
f Fig. 3: Probability of false alarm versus SNR in sensing
which means the signal recovery is not exact any more. As a
channels with different compression ratio κ, P¯ = 90%, and
result,throughputofthesecondarynetworkwouldbedegraded d
sparsity level =12.5%.
correspondingly. Actually, the lowest compression ratio for
successful signal recovery is dependent on the sparsity levels
of received signals and the available prior information. This
is out of the scope of this paper and would not be discussed assignedto thethirdslotα alwaysequalsto thelowerbound
2
further. α . This givesa sign thatthe throughputcan be improved
2,min
Fig. 4 plots the achievable throughputτ versus the lower if the time slot for the signal recovery at the FC and data
cs
bound of time allocated to the third slot α . Here α is transmission between SUs and the FC is reduced. In other
2 2
reserved for signal recovery at the FC and data transmission words, the energy harvesting should be done mainly in the
betweentheSUandtheFC.Inthisfigure,severalobservations first time slot α to reduce the power outage probability in
1
aredrawnasfollows:1)Themaximalthroughputachievedby the followingspectrum sensing slot if the signal recoveryand
gridsearchmethodisslighthigherthanthatoffminconmethod data transmission between the SU and FC can be promised.
as fmincon relies on the initial input and may return a local Fig. 5 plots the achieved optimal throughput τ versus
cs
optimalvalue.However,theaccuracyofgridsearchmethodis lower bound α and compression ratio κ when solving the
2
dependenton the step sizes; 2) The random sampling method problemin (23). It shows that the achieved maximalthrough-
achieveslowerthroughputthangridsearchandfminconmeth- put increases with decreasing κ and increasing α . This
2,min
ods, which demonstrates the benefits of the presented grid behavior can be explained as follows: as κ decreases, the
searchandfminconmethods.Whenthegeneratedsetsincrease number of samples to be collected for spectrum sensing at
for random sampling, the achieved maximal throughput get an SU is reduced as the signal in original size of N 1 can
×
closer to the optimal but the computational complexity is berecoveredfromlessnumberofΛmeasurementsbyutilizing
much increase; 3) It is seen that the optimal value of time CS technique. When the time slot assigned for spectrum
9
0.22
fmincon
0.6 0.5
0.2 grid search Hz)
random sampling s/ 0.5 0.45
s/s/Hz)0.18 ut (bit/ 0.4 0.4
bit0.16 hp 0.35
oughput (0.14 Throug 00..32 00..235
hr0.12
T 0.1 0.2
0.1 0
0.15
0.5
0.08 κ 0.4 0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 0 0.2α
α (T) 2,min
2,min
Fig. 4: Throughput of single SU τ versus lower bound of Fig. 5: Optimized throughput of single SU τ versus lower
cs cs
the third time slot α , SNR= 10 dB, and κ=1. bound of the third time slot α and compression ratio κ,
2,min 2,min
−
and SNR= 10 dB in sensing channels.
−
sensing is reduced, the energy consumption for spectrum
sensingisreduced.Asaresult,thetimewhichcanbeassigned Fig.6plotsthepoweroutageprobabilityP versusdensity
out
fordatatransmissionisincreased.Therefore,thethroughputof of PBs with different power threshold P . In this case, the
s
the secondary network is improved. By optimizing the trans- power outage probability here is for the whole system, which
mission power P , the energy harvested in the current frame can be calculated as P = 1 (1 Ps ) (1 Pt ).
t out − − out × − out
period would be fully utilized and the maximal throughput Both the single SU scenario and multiple SUs scenario are
can be achievedaccordingly.It shouldbe noted thatwhen the illustrated in the figure. It can be observed that as density
compression ratio κ is set to 100% and the lower bound of of PBs increases, the power outage probability dramatically
thethirdtimeslotα iszero,theachievedthroughputcan decreases,whichiscausedbythatthemultiuserdiversitygain
2,min
be regarded as that of the traditional frame structure design is improved with increasing number of PBs when charging
without considering sub-Nyquist sampling. The case when with WPT. We can also observe that the P of multiple
out
κ is set to 100% can be regarded as a benchmark for the SUs scenario is lower than that of single SU scenario. This is
performance metric. because the power outage probability of spectrum sensing is
alwayszeroin multipleSUs scenario,whichin turnlowerthe
poweroutageprobabilityofthewholesystem.Forthemultiple
B. Numerical results on optimizing throughput of multiple
SUs scenario, the P of the active SUs, inactive SUs and
out
secondary users
the averageP of the CSS networksare all presentedin the
out
In the case of optimizing the throughput of the CSS net- figure. It is notedthe averagedPout falls between the Pout of
works, the total number of participating SUs is set be to J = active SUs and inactive SUs, which as we expected.
50,includingboththeactiveandinactiveSUs.Comparingthe Fig. 7 plots the throughput averaged on per SU τmc with
formatof(P1)and(P0),wecanseethatbothofthemarelinear different number of active SUs J and different compression
1
constrained.Therefore,similaras(P0),thegridsearchmethod ratios κ. In this case, it is assumed that the exact matrix
can be appliedto obtainthe optimalthroughputbut with non- completioncanbeguaranteedwhenthenumberofactiveSUs
negligible complexity, especially for the case of optimizing J isintherangeof10to50withcompressionratioκchanges
1
throughput of the whole cooperative network. The fmincon from 20% to 100%. It shows that the average throughput
method can be adopted to obtain the sub-optimal throughput achievesthebestperformancewhenthenumberofactiveSUs
efficiently. In the following simulations, the fmincon method J issettobetheminimalnumberincomparisonwiththatof
1
isutilized tosolvethe optimizationproblem(P1).Inaddition, caseJ =J.Thisbenefitsfromthenon-activeSUs,whichcan
1
as aforementioned in the introduction part, many algorithms saveenergyforspectrumsensingandharvestmoreenergyfor
have been proposed for the low-rank MC based cooperative data transmission. With compression ratio κ decreasing from
spectrumsensing. With Pˆ =90%,the detectionperformance 75% to 20%, the achieved optimal throughputis increased as
d
with different compression ratios is presented in the authors’ the necessary time slot for spectrum sensing is reduced.
previouswork in [14], whichwould notbe demonstratedhere Fig.8 plotstheachievedmaximumthroughputaveragedon
again to reduce redundancy. In the following simulations, per SU versus differentcompression ratio κ and lower bound
how the achieved throughput is influenced by parameters, for the third time slot α . In this case, the number of
2,min
such as the number of active SUs J , compression ratio κ active SUs is J = 30. The achieved throughput shown in
1 1
and the lower bound of the third time slot α , would be Fig.8isbasedontheconditionthattheexactmatrixrecovery
2,min
demonstrated. canbeguaranteedwiththegivencompressionratio.Asshown
10
100 z)
H 1 0.9
s/
ability10-1 ut (bits/0.5 00..78
b p
o h
pr ug 0.6
ge 10-2 P single SU hro
a out T 0 0.5
ut P active SU
o out
er Average Pout whole network 0.8 0.4
ow10-3 Pout inactive SU 0.6 0.3
P
0.4
0.2
10-4 α2,min 0.2 0.2 0.4 0.6 0.8 1
10-3 10-2 10-1
Density of PBs κ
Fig. 6: Power outage probability comparison for single SU Fig. 8: Optimized throughputaveraged on per SU of multiple
and multiple SUs with versus density of PBs, P = 0 dBm, SUs τ versus lower boundα and compression ratio κ,
s mc 2,min
α =0.25, α =0.20, and β =0.25. and SNR= 10 dB in sensing channels.
1 2
−
1
scheme. In addition, sub-Nyquist sampling was performed
at SUs to reduce the energy consumption during spectrum
0.99
sensing. The compressive sensing and matrix completion
z)
H0.98 techniques were adopted at a remote fusion center to per-
s/ κ=0.20
ut (bits/0.97 κκ==00..5705 foaocsrcmiunpgtalhenecsyes.icgBonnyadloarpryeticmuosviezerirnyagnfdtohrethdfeeotuwerchtitooilmneecmosalookptiesn,rgatthiovrnoeusngpehetpwcutorturkomsf
p0.96
gh were maximized, respectively. Simulation results showed that
u
hro0.95 the throughput can be improved in the proposed new frame
T structure design. We conclude that by carefully tuning the
0.94 parameters for different time slots and transmit power, WPT
can be used along with compressive sensing and matrix com-
0.93
10 20 30 40 50 pletiontoprovideahighqualityofthroughputperformancefor
J cognitiveradionetwork,withsignificantlyenergycomputation
1
reduction at power-limited SUs.
Fig. 7: Optimized throughputaveraged on per SU of multiple
SUs τ versus number of active SUs J and compression
mc 1
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