modeling time-varying aggregate interference in cr system and application to primary exclusive zone...

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 1, JANUARY 2014 429

Modeling Time-Varying Aggregate Interference inCognitive Radio Systems, and Application to

Primary Exclusive Zone DesignMohd. Shabbir Ali and Neelesh B. Mehta, Senior Member, IEEE

AbstractAccurately characterizing the time-varying interfer-ence caused to the primary users is essential in ensuring asuccessful deployment of cognitive radios (CR). We show thatthe aggregate interference at the primary receiver (PU-Rx) frommultiple, randomly located cognitive users (CUs) is well modeledas a shifted lognormal random process, which is more accuratethan the lognormal and the Gaussian process models consideredin the literature, even for a relatively dense deployment of CUs. Italso compares favorably with the asymptotically exact stable andsymmetric truncated stable distribution models, except at highCU densities. Our model accounts for the effect of imperfectspectrum sensing, which depends on path-loss, shadowing, andsmall-scale fading of the link from the primary transmitter to theCU; the interweave and underlay modes of CR operation, whichdetermine the transmit powers of the CUs; and time-correlatedshadowing and fading of the links from the CUs to the PU-Rx. Itleads to expressions for the probability distribution function, levelcrossing rate, and average exceedance duration. The impact ofcooperative spectrum sensing is also characterized. We validatethe model by applying it to redesign the primary exclusive zoneto account for the time-varying nature of interference.

Index TermsCognitive radio, interference, spectrum sensing,underlay, interweave, shadowing, fading, time-variations, ran-dom process, lognormal, primary exclusive zone.

I. INTRODUCTION

COGNITIVE radio (CR) offers a promising solution tothe problem of under utilization of the spectrum. Acommon paradigm of CR classifies users into two categories,namely, primary users (PUs), which have unfettered accessto the spectrum, and cognitive users (CUs), which can usethe spectrum but under tight constraints on the aggregateinterference their transmissions cause to the PUs [1][6]. Asuccessful design and deployment of CR, therefore, requiresas a first step an accurate model for the aggregate interferencecaused to the PUs by transmissions from one or many CUs.This characterization feeds into the design and evaluationof transmission policies for the CUs and techniques to helpmitigate their interference.

Several factors together affect the aggregate interference,and must be accounted for in order to arrive at an accurate

Manuscript received April 30, 2013; revised August 6 and October 8, 2013;accepted October 12, 2013. The associate editor coordinating the review ofthis paper and approving it for publication was R. Zhang.

The authors are with the Dept. of Electrical Communication Eng.,Indian Institute of Science (IISc), Bangalore, India (e-mail: [email protected], [email protected]).

This work was partially supported by a research grant from ANRC.A part of the work has been accepted for presentation in the IEEE Global

Conf. on Communications (Globecom), 2013.Digital Object Identifier 10.1109/TWC.2013.113013.130762

model for it. It is affected by propagation characteristics ofthe channels between the CUs and PUs, such as path-loss,shadowing, and fading. Furthermore, the number of CUs thattransmit and their locations affect the interference. Imperfectspectrum sensing also directly affects it as it determines theCUs transmit powers and whether or not they transmit. Theuse of cooperation, in which the CUs cooperate with eachother and fuse their decisions, affects the accuracy of spectrumsensing and, thus, the aggregate interference.

Given the importance of interference modeling in CR,one of the approaches pursued in the literature is based onmeasurements from test deployments [7][9]. However, thesedeployments are typically small because of the difficulty insetting up an experiment with many interferers that capturesthe many sources of randomness. Furthermore, the modelsdeduced are location-specific. Therefore, a second approachhas focused on developing statistical models for the aggregateinterference [1][5]. However, no closed form exists for itsprobability distribution function (PDF). Therefore, severalapproximate analytical models have been investigated.

Interference in Underlay CR Mode: In the underlay mode,a CU can transmit even when it senses that the PU transmitter(PU-Tx) is transmitting [10]. However, it does so with a muchlower power in order to avoid excessive interference to thePU receiver (PU-Rx). In [5], the aggregate interference at thePU-Rx from a fixed number of CUs, which are distributeduniformly over a region, is modeled as a lognormal randomvariable (RV). However, only large-scale shadow fading istaken into account. A spatial Poisson point process (SPPP)model is instead assumed to model the randomness in the CUnumber and locations in [4], and the aggregate interferenceis modeled as a lognormal RV. Power control as well ascontention control, in which CUs too close to each other donot transmit simultaneously, are accounted for. Furthermore,lognormal shadowing and Nakagami-m fading are modeled.However, imperfect spectrum sensing and the time-varyingnature of aggregate interference are not studied in [4], [5].

Interference in Interweave CR Mode: In the interweavemode, a CU transmits only when it senses the PU-Tx to beoff [10]. The transmission can be in the same band as thePU-Tx or in a different band. In [2], the amplitude of theaggregate interference is modeled as a symmetric truncated-stable (STS) RV. Unlike the stable distribution model [1],the STS model ensures finite second and higher moments.An SPPP model determines the CU locations and an energydetector (ED) is used for spectrum sensing. A detect-and-

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430 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 1, JANUARY 2014

avoid strategy in which the transmit power level of the CUis adapted based on the received signal strength is considered.Cooperative spectrum sensing is accounted for in [3], whichmodels the aggregate interference as a shifted lognormal(SLN) RV assuming an SPPP model for the CUs. However,time variations are not modeled in [1][3].

A. ContributionsWe develop a comprehensive model for the aggregate inter-

ference that captures its snapshot statistics, i.e., PDF, as well asits time-variation statistics, which is measured in terms of theauto-correlation function, average level crossing rate (LCR),and average exceedance duration (AED) [11]. We show thatthe aggregate interference is well modeled as a wide sensestationary (WSS) SLN random process (RP), except when itis small. Our model allows the CUs to operate in both theinterweave and underlay modes, in which they transmit in thesame band as the PU-Tx with a high power when they sensethe PU-Tx to be off and with a low power when they sense thePU-Tx to be on. It accounts for imperfect spectrum sensing,which depends on the location of the CU relative to the PU-Tx.Also, it accounts for the combined effect of time-correlatedshadowing and Rayleigh fading on the various links. We alsoextensively benchmark the proposed model with several othermodels proposed in the literature. The randomness in the CUlocations and number is also captured using the SPPP model.

We then develop a corresponding model for the aggregateinterference for cooperative spectrum sensing, in which CUsthat are close to each other cooperate and arrive at a commondecision. We show that the SLN RP again models the aggre-gate interference well, except when it is small, and brings outthe reduction in the aggregate interference due to cooperation.

We then demonstrate the usefulness and tractability of theproposed model by refining the design of the primary exclusivezone (PEZ) [4], [12], [13]. The PEZ is defined as the regionaround the PU-Rx within which no CU is allowed to transmit,and helps protect the PU-Rx from excessive interference. Itsarea affects the aggregate interference experienced by the PU-Rx. We propose a novel criterion for determining the PEZradius that incorporates the impact of the time-varying natureof the aggregate interference.

B. Comparisons with LiteratureThe SLN model has been considered before in the litera-

ture [3], [14]. Further, our channel model and the SPPP modelfor CUs is similar to that in [3]. However, the generalizationfrom an SLN RV to an SLN RP, and the demonstration of itsaccuracy in modeling the time-varying nature of the aggregateinterference when all the aforementioned physical layer effectsare accounted for is novel and is a contribution of this paper.The following is a list of our specific contributions and themany ways in which our approach and results differ from thosein [3] and other related works [1], [2], [12][17]:

Overall goal: While [3] focuses on characterizing thesnapshot statistics and models the aggregate interferenceas an SLN RV, we characterize the time-varying behaviorof the aggregate interference as it also affects the PU-Rx.For example, in [15], it has been argued that long dips

in the signal-to-noise-plus-interference-ratio (SINR) aredetrimental to the PU-Rx. Our model is also more generalthan the LCR analysis in [16], which considers a systemwith one CU and one PU, only models shadowing, andassumes perfect spectrum sensing.While the stable model is provably accurate in the asymp-totic regime of a large number of CUs [1], generalizingit or the STS model to incorporate time-variations is anopen problem. These two models also require accurate,numerically stable techniques to compute the cumula-tive distribution function (CDF) from the characteristicfunction (CF) that they characterize. Furthermore, as weshall see, the asymptotic exactness manifests itself onlyat higher CU densities. In [17], the aggregate interferenceis instead modeled as a gamma RP. However, shadowingand imperfect spectrum sensing are not modeled and thenumber of CUs is assumed to be fixed. In [14], the SLNRV has instead been used for approximating the decisionstatistics of a multichannel energy detector. Thus, itsmodel and results are very different from ours.

Analytical results: While moments of the aggregate in-terference are derived in [3], we derive new expressionsfor the moments and the autocorrelation of the aggregateinterference. These then help determine all the parametersthat are required to specify the SLN RP, and beget newanalytical results for the LCR and AED of the aggregateinterference. Our moment expressions also turn out to bedifferent due to differences in our CU transmission andspectrum sensing models, which are summarized below.

CU transmission model: While [3] focuses on the inter-weave mode, we analyze a hybrid mode of operation thatcombines the interweave and underlay modes.

Spectrum sensing model: In [3], an out-of-band spectrumsensing model is assumed in which each CU senses thesignal it receives from the full-duplex PU-Rx on a sep-arate control channel. In our model, however, spectrumsensing is based on the signal received from the PU-Txand is in-band [2], [18]. This also avoids the need forthe CUs to simultaneously sense an out-of-band beacon.Another difference, which also affects spectrum sensing,is that we incorporate the PEZ in our model.

Extensive benchmarking and application: Our paper alsodemonstrates the utility of the proposed RP model byapplying it to the design of the PEZ, and showing that it isreasonably accurate over a wide range of parameters. Thisis unlike [3]. Further, the benchmarking of the proposedmodel is more extensive in our paper.

New PEZ design criteria: The incorporation of the time-varying nature of the interference in the PEZ design isa contribution of the paper, and has not been consideredin [12], [13]. Further [12], did not consider imperfectspectrum sensing, shadowing, and small-scale fading,while imperfect spectrum sensing was not modeledin [13]. Our quantification of how cooperative spectrumsensing shrinks the PEZ is also novel.

The paper is organized as follows. Sec. II describes thesystem model. The aggregate interference process model isdeveloped in Sec. III. Cooperative spectrum sensing is con-

ALI and MEHTA: MODELING TIME-VARYING AGGREGATE INTERFERENCE IN COGNITIVE RADIO SYSTEMS, AND APPLICATION TO PRIMARY . . . 431

p PUTx

Correctdetection

R

RPEZ

PURx

Misseddetection

Area A

CUs

Fig. 1. System model showing the spectrum sensing by the CUs that arescattered over a region of area A.

sidered in Sec. IV. Simulation results and PEZ redesign arepresented in Sec. V. Our conclusions follow in Sec. VI.

II. SYSTEM MODELWe shall use the following notation. Expectation is denoted

by E [.], and EX [.] denotes expectation conditioned over theRV X . The notation X(t) N (X , 2X , CX()) shall meanthat X(t) is a WSS Gaussian RP with mean X , variance 2X ,and covariance function CX(). Similarly, X N (X , 2X)denotes a Gaussian RV, and X exp() denotes an expo-nential RV with mean .

A. System LayoutFigure 1 shows the system layout. The number and locations

of the CUs is modeled using a homogeneous SPPP, which ischaracterized by a density parameter . Therefore, the numberof CUs NCR that occur in a region of area A is a PoissonRV with mean A. The PU-Rx is located at the center ofthe region and the PU-Tx is located at a distance p from thePU-Rx on the x-axis. The region of radius RPEZ around thePU-Rx is the PEZ, within which no CU transmits [4], [12],[13]. The PEZ can be constructed using an in-band beaconsignal from the PU-Rx or using geo-location [4].1 Note thatthis beacon aided model is different from that in [3], in whichthe CU transmissions are governed by an out-of-band beaconsent by the full-duplex PU-Rx anytime the PU-Tx transmits.

B. Channel ModelThe power PRi(t) received at the PU-Rx from the ith CU,

which is located at a distance ri(t) from it, is given by

PRi(t) = PK

(d0ri(t)

)eXi(t)hi(t), (1)

where P is the transmit power of the CU. The path-losscomponent is K

(d0

ri(t)

), where K =

(

4d0

)2, is the

carrier wavelength, d0 is the break-point distance, and isthe path-loss exponent [11]. The Rayleigh fading component ishi(t) exp (1). Note that our model can also handle any other

1The CUs within the PEZ are all assumed to know that they are within it.This is justified because several measurements of the PU-Rx beacon collectedover a sufficiently long duration of time can be used to ensure this.

mean for hi(t). The normalized covariance function Chi()is given as per the Jakes fading model [11]:

Chi() = J20 (2fm) , (2)

where fm is the maximum Doppler spread and J0 is theBessel function of first kind of order zero [19]. The shad-owing component is eXi(t), where = log 1010 and Xi(t) N (0, 2sh, CXi()). The covariance function CXi() is givenby the modified Gudmundsons model as [15]

CXi() = 2sh exp

(v

22

2D2

), (3)

where v is the speed of the CU and D is the decorrelationdistance. The shadowing and fading seen by different CUs ontheir links from the PU-Tx and to the PU-Rx are independentand identically distributed (IID).

C. PU-Tx Based Spectrum Sensing (SS)The accuracy of SS by the CU depends on the strength of

the signal it receives from the PU-Tx and the SS algorithmused. We consider the popular ED-based SS algorithm [20]assuming that all the CUs periodically spectrum sense togetherfor a short duration.2 In [20], the SS algorithm declares thePU-Tx to be on if the energy received by the CU from it overa time T and bandwidth B exceeds a threshold .

For this detector, the false alarm probability PFA at the ith

CU is Q(N0TBN20TB

), where N0 is the noise power and Q

is the Gaussian Q function. The correct detection probabilitydepends on the signal-to-noise-ratio (SNR) (i, Yi, gi) ofthe PU-Tx signal at the CU, which in turn depends on thedistance i of the CU from the PU-Tx, shadowing eYi , andRayleigh fading gi during the time of sensing. Here, Yi N (0, 2sh) and gi exp (1). The expression for the correctdetection probability, denoted by PD(i, Yi, gi), in terms ofPFA is [20]

PD(i, Yi, gi) = Q

(Q1(PFA) (i, Yi, gi)

TB

1 + 2 (i, Yi, gi)

).

(4)Note that our model can include other SS algorithms as

well, e.g., [22]; the expressions for PFA and PD will differ.

D. CU Transmission and Interference ModelIf a CU detects the PU-Tx to be on, which we refer to

as hypothesis H1, then it operates in the underlay mode andtransmits with a lower power Pu. Else, if the CU detects thePU-Tx to be off, which we refer to as hypothesis H0, then itoperates in the interweave mode and transmits with a higherpower Po. In both cases, it transmits in the same band as thePU-Tx. Therefore, from (1), the interference power Ii(t) atthe PU-Rx from the ith CU, which is ri(t) distance away, is

Ii(t)=

PuK

(d0

ri(t)

)eXi(t)hi(t), if H1 is detected,

PoK(

d0ri(t)

)eXi(t)hi(t), if H0 is detected.

(5)2If the CUs sense the spectrum asynchronously then the aggregate inter-

ference from the concurrent transmissions by other CUs has to be taken intoaccount along the lines of [21].


Note that this two-level transmit power model may not bethe most suitable one in the presence of interference fromother systems. One possible way to improve it is to use themore advanced detect-and-avoid strategy [2]. Further, energydetection might no longer be the right SS method to use sinceit cannot differentiate between the sources of interference.

III. INTERFERENCE MODELING: NON-COOPERATIVE SSThe aggregate interference I(t) at the PU-Rx from the

CUs is

I(t) =

NCRi=1

Ii(t). (6)

Our goal is to accurately model the time evolution of I(t)in between the SS durations when the PU-Tx is on. We nowdevelop the moment-matching-based WSS SLN RP model, asper which I(t) is modeled as

I(t) eZ(t) + s, (7)where Z (t) N (Z , 2Z , CZ()) and s is called the shiftparameter. Our goal is to determine the constants Z , Z , ands, and CZ() by matching them with the corresponding termsof I(t). To do so, we first express these parameters in termsof the cumulants of I(t). The cumulants are then written interms of E [Ii(t)m], for m = 1, 2, 3, and E [Ii(t)Ii(t+ )],which are derived in Results 1 and 2. It is here that the maincontribution of this section lies.

In terms of the cumulants of I(t), the parameters Z , Z ,and s are given by [14]

2Z = log

(1

4

23 + 4

23 1

), (8)

Z =1

2log

(I(2)

e2Z 1

) 1

22Z , (9)

s = I(1) exp(Z +

1

22Z

), (10)

where = 4I(3) + 44 + (I(3))

2 and I(m) is themth cumulant of I(t), which is defined as [19, (26.1.12)]

I(m) =1

jmdm log

(E[ejI(t)

])dm

=0

,

=1

jm

dm log(ENCR

[(E[ejIi(t)

])NCR])dm

=0

,

(11)where j =

1. The second equality is obtained by using (6),conditioning on NCR, and using the fact that all Ii(t), for1 i NCR, are IID for the SPPP that governs the locationsof the CUs. Upon taking the expectation over NCR and thenevaluating the derivative, we get

I(m) = (R2 R2PEZ

)E [Ii(t)

m] . (12)The SLN RP model also requires CZ(). As shown in

Appendix A, it is equal to

CZ() = log((R2 R2PEZ

)E [Ii(t)Ii(t+ )]

+ e2Z+2Z

) (2Z + 2Z) . (13)

A. With Path-loss and ShadowingWe first analyze the case when small-scale fading is aver-

aged out. This is of interest when the PU-Rx can average overthe fast variations of the small-scale fading [11, Chap. 3].

The distance q (ri(t), p, cos i) between the PU-Tx and theith CU is equal to

q (ri(t), p, cos i) =ri(t)2 + p2 2ri(t)p cos i, (14)

where i is the azimuth angle of the CU. The PDF pi of iand the PDF pri of ri(t), conditioned on the CU not lying inthe PEZ, are given by

pi(x) =1

2, 0 x < 2,

pri(x) =2x

R2 R2PEZ, RPEZ x < R. (15)

Result 1: The mth moment of the interference from anarbitrary CU i is then given by

E [Ii(t)m]

2Pmo Kmdm0

(R2mPEZ R2m

)(m 2) (R2 R2PEZ)

e12m

222sh

2 (Pmo Pmu )Km

Wc (R2 R2PEZ)e

12m

222sh

Whn1=1

wh (n1)

Wln2=1

wl (n2)

Wcn3=1

RRPEZ

dm0rm1i

PD(q (ri, p, ac (n3)) ,

2shah (n1) , al (n2)

)dri. (16)

Proof: The proof is given in Appendix B.Here, wh (n) and ah (n), for n = 1, . . . ,Wh , de-

note the weights and the abscissas, respectively, of Gauss-Hermite quadrature, ac (n), for n = 1, . . . ,Wc , denote theabscissas of Gauss-Chebyshev quadrature, and wl (n) andal (n), for n = 1, . . . ,Wl , denote the weights and theabscissas, respectively, of Gauss-Laguerre quadrature [19,(25.4.38),(25.4.45),(25.4.46)].

Result 2: The auto-correlation of the interference from aCU is given as follows:

E [Ii(t)Ii(t+ )] P 2oK

2d20

(R22PEZ R22

)( 1) (R2 R2PEZ)

e22sh(1+exp

( v22

2D2

)) 2

(P 2o P 2u

)K2

Wc (R2 R2PEZ)

e22sh(1+exp

( v22

2D2

)) Whn1=1

wh (n1)

Wln2=1

wl (n2)

Wcn3=1 R

RPEZ

d20r21i

PD

(q (ri, p, ac (n3)) ,


)dri.

(17)Proof: The proof is given in Appendix C.

The final expressions in (16) and (17) are in the form ofsingle integrals. These cannot be simplified further becauseof the presence of the PD term inside the integrand, whichdepends on the SS algorithm. It is typically quite involved,as can be seen from (4). The integrals are easily evaluatednumerically. In general, as the number of the Gauss-quadrature


terms increases, the error between the integral and the approx-imating sum decreases [19, (25.4.38),(25.4.45),(25.4.46)]. Wehave found that for our problem Wh = Wc = Wl = 6 issufficient to ensure accurate complementary CDF (CCDF) andLCR curves for sh 12.

Note that the analysis can be extended to the case wherePU-Rx is not at the center of the region. In this case, thedistance between the ith CU and the PU-Rx, which arisesin (5), changes. However, the expressions of the moments andthe auto-correlation of Ii(t) can still be obtained in a singleintegral form. The case where the distance p between the PU-Rx and PU-Tx is an RV can also be incorporated. In this case,an additional expectation over p will appear in (16) and (17).

B. With Path-loss, Shadowing, and FadingIf the PU-Rx cannot average over the small-scale fading,

then this should be taken into account in the aggregate in-terference model. When small-scale fading is also consideredalong with path-loss and shadowing, the expression for themth moment E [Ii(t)m] in (32) will get multiplied by a factorE [hi(t)

m]. Using E [hi(t)m] = m!, and simplifying furtheryields the expressions for moments that contain an additionalfactor of m! compared to (16). Similarly, the auto-correlationE [Ii(t)Ii(t+ )] in (35) will get multiplied by a factorE [hi(t)hi(t+ )]. Consequently, from (2), the autocorrelationin (17) gets scaled by a factor (J20 (2fm) + 1).C. CCDF, LCR, and AED of I(t) Based on SLN RP Model

For a threshold Ith, the CCDF of I(t) is given by [3, (23)]:Pr (I(t) Ith) = Q

(log (Iths)Z

Z

). The LCR LI(t)(Ith)

of I(t) can be obtained from the level crossing theory ofGaussian processes [23], the key steps for which are shownbelow. From [23, Lemma 10.2], the LCR can be written interms of Z(t) as

LI(t)(Ith) =

limn

[ni=1

Pr(Z

(i 1n

)< log (Ith s) < Z

(i

n

))].

(18)Using [23, Theorem 10.1], we then get

LI(t)(Ith) =

Z

2Zexp

( (log (Ith s) Z)

2

22Z

), Ith > s,

(19)where Z = d2d2CZ()

=0

is computed from (13) and (17).Finally, the AED I(t)(Ith) of I(t) is the ratio of the CCDFand the LCR:

I(t)(Ith) =Pr (I(t) Ith)

LI(t)(Ith). (20)

IV. INTERFERENCE MODELING: COOPERATIVE SSWe now incorporate cooperative SS into our model, and

characterize how it affects the aggregate interference RP. Forthis purpose, we use the OR fusion rule because it is morepreferable than the AND and majority rules from the point ofview of protecting the PU-Rx from excessive interference [3].

In it, a CU that has detected the PU-Tx to be on will broadcastits decision to the CUs that lie within a cooperation region ofradius RC around it. All these CUs will take a logical OR withtheir decision, which means that they will also then assumethat the PU-Tx is on. This is a one shot process the CUsdo not broadcast their decisions yet again. We assume thatthe interference caused by the short transmissions of the CUsfor enabling cooperative SS is negligible. Furthermore, thecommunication is assumed to be error-free. This is justifiablesince only one bit of information needs to be communicated,and can be sent with sufficient protection.

Note that the interference from the CUs will be correlatedbecause their decisions are correlated due to cooperation. Tomake the analysis tractable, we use a decoupling approxima-tion, in which the interferences from the CUs are assumed tobe uncorrelated, but the effect of correlation in their decisionsis captured in the probability of correct decision. Such adecoupling approximation has been used to good effect inanalyzing wireless local area networks [24]. With this, theexpressions for the parameters of the SLN RP model of theaggregate interference with cooperative SS are given by (8),(9), (10), and (13).

As before, to characterize the aggregate interference withcooperative SS, we need the expressions for E [Ii(t)m] andE [Ii(t)Ii(t+ )]. These are given below when path-loss andshadowing are considered. The distance between the PU-Txand the j th CU that is located within the cooperation rangeof the ith CU is q (rij(t), q (ri(t), p, cos i) , cos ij), which isgiven by (14), where rij(t) is the distance between the ith andj th CUs, and ij is the angle subtended by the lines from thej th CU and the PU-Tx to the ith CU.

Result 3: The mth moment of interference E [Ii(t)m] fromthe ith CU with cooperative SS is then given by

E [Ii(t)m]

2Pmo Kmdm0

(R2mPEZ R2m

)(m 2) (R2 R2PEZ)

e12m

222sh

2 (Pmo Pmu )Km

Wc (R2 R2PEZ)e

12m

222sh

Wcn4=1

RRPEZ

dm0rm1i

(1 1 f1 (ri, ac (n4))

1 f2 (ri, ac (n4))eR2Cf2(ri,ac(n4))

)dri, (21)

where

f1 (ri, ac (n4)) =

Whn1=1

wh (n1)

Wln2=1

wl (n2)

PD(q (ri, p, ac (n4)) ,


), (22)

f2 (ri, ac (n4)) =2

WcR2C

Whn1=1

wh (n1)

Wln2=1

wl (n2)

Wc

n3=1

RC0

PD

(q (rij, q (ri, p, ac (n4)) , ac (n3)) ,


)rijdrij. (23)


TABLE ISIMULATION PARAMETERS

Parameter Variable ValueTransmit power of PU-Tx PTx 10 dBmTransmit power of CU in interweave mode Po 2 dBmTransmit power of CU in underlay mode Pu -6 dBmNoise power N0 -100 dBmDensity of CUs 100 CUs/km2System bandwidth B 1 MHzCarrier frequency fc 900 MHzRadius of region considered R 1000 mRadius of PEZ RPEZ 200 mDistance between PU-Tx and PU-Rx p 500 mPath-loss exponent 4Standard deviation of shadow fading sh 6Break-point distance d0 10 mSpeed of CUs v 5 ms1False alarm probability PFA 10%Spectrum sensing duration T 50 sec

The auto-correlation E [Ii(t)Ii(t+ )] is given by

E [Ii(t)Ii(t+)] P 2oK

2d20

(R22PEZ R22

)( 1) (R2 R2PEZ)

e22sh(1+exp

( v22

2D2

)) 2

(P 2o P 2u

)K2

Wc (R2 R2PEZ)

e22sh(1+exp

( v22

2D2

)) Wcn4=1

RRPEZ

d20r21i

(1 1 f1 (ri, ac (n4))

1 f2 (ri, ac (n4))eR2Cf2(ri,ac(n4))

)dri. (24)

Proof: The proof is given in Appendix D.Compared to the expressions for the mth moment and auto-correlation in Sec. III-A, the PD term is different and themoments now depend on the cooperation range RC .

As before, when Rayleigh fading is also accounted for, theexpression for E [Ii(t)m] in (21) will get scaled by a factor m!.The expression for E [Ii(t)Ii(t+ )] in (24) will get scaled bya factor

(J20 (2fm) + 1

).

V. NUMERICAL RESULTSWe now verify our analysis using Monte Carlo simulations.

The parameters used are listed in Table I. The simulationsmake measurements over up to 105 drops. In each drop, arandom number of CUs and their locations are generated as perthe SPPP. Each CU moves with a fixed speed v in a randomlychosen direction. Every CU performs SS as per Sec. II-C, incase of non-cooperative SS, and as per Sec. IV, in case ofcooperative SS. The aggregate interference I(t) from all theCUs at the PU-Rx is then measured.

We first present results for non-cooperative SS and thenfor cooperative SS. In each case, we show results for thesnapshot statistics and then for the time-varying behavior. Forsnapshot statistics, we compare the CDF and the CCDF ofthe various models, as has been done in [2][5], [25], [26].The CDF evaluates the accuracy in matching smaller aggregateinterference values, while the CCDF evaluates the accuracy inmatching larger aggregate interference values, where the CDFsaturates to unity. To compare the accuracy in modeling the

100 95 90 85 80 75104

103

102

101

100

I(t) in dBm

CDF

and

CCDF

SimulationGaussianLognormalStableSTSSLN

CDF

CCDF

Fig. 2. Non-cooperative SS: Comparison of CDF and CCDF of I(t) fromvarious models with path-loss and shadowing.

time-variation of the aggregate interference, we study the LCRof the various models, as has been done in [16], [17], [27].

A. BenchmarkingWe compare the SLN model with the following: Gaussian model: This model is motivated by the central

limit theorem. In it, I(t) is modeled as a Gaussian RP. Lognormal model: In this model, log (I(t)) is modeled

as a Gaussian RP, along lines similar to [27]. STS model: In this model, the amplitude of I(t) is mod-

eled as an STS RV [2]. The parameters that determine itsCF are obtained by matching the first, second, and fourthcumulants of the STS RV with the corresponding cumu-lants of the amplitude of I(t). The CF is numericallyintegrated to get the CDF, as closed-form expressions forthe latter are not known. Finally, the CDF of I(t) isdetermined using a variable transformation.

Stable distribution model: In this model, I(t) is modeledas stable RV [1]. The parameters that determine the CFare obtained from simulations using the method presentedin [28]. The CDF of I(t) is then obtained by numericallyintegrating the CF, as a closed-form expression for it isnot known.

B. Non-cooperative SS: With Path-loss and Shadowing1) Snapshot Statistics Comparisons: Figure 2 compares the

CDF and the CCDF of the various models when small-scalefading is averaged out. For lower values of aggregate interfer-ence (I(t) < 90 dBm), the SLN model underestimates theCDF. However, none of the models proposed in the literatureare accurate in this regime.3 One reason behind this inaccuracyis the use of the moment-matching method, which penalizesless the approximation errors for lower values of interferencethan for higher values of the interference [26]. Another reasonis that the probability that I(t) is less than s is zero in theSLN model. For moderate to high values of the aggregateinterference (I(t) > 90 dBm), the SLN model matches the

3Since the lognormal model overestimates the CDF while the SLN modelunderestimates it, better accuracy can be achieved by a mixture model whoseCDF is the arithmetic mean of the CDFs of these two models [29].


100 95 90 85 80104

103

102

101

100

Ith in dBm

LCR

SimulationGaussianLognormalSLN

Fig. 3. Non-cooperative SS: Comparison of LCR of I(t) from variousmodels with path-loss and shadowing.

CCDF well and is more accurate than all the other models.It captures the skewness of the interference distribution betterthan the other models [3]. Note that the 90 dBm value abovearises due to the particular choice of simulation parameters. Ingeneral, it increases as the mean of the aggregate interferenceincreases.

We see that the Gaussian and lognormal models fail toprovide a good fit. For small values of the interference, theCDF of the Gaussian model saturates at Q

(GG

), where G

is the mean and 2G is the variance. Intuitively, this failureis because the rate of convergence of the sum of lognormalRVs is very slow, which is due to the skewed nature of thelognormal PDF [30]. The stable distribution model is better inmatching the CDF. However, it gives a poor CCDF match forhigher values of the aggregate interference. The STS modelmatches the CCDF better but its CDF saturates for smallvalues, as was the case with the Gaussian model.

2) Time-varying Behavior Comparisons: The LCR of theSLN RP model and the benchmark models is shown in Fig. 3.We see that the LCR curve is not symmetric about the meanof the aggregate interference, which is because the PDF of theaggregate interference is asymmetric about its mean [3]. Forlow threshold values, the LCR is small because the aggregateinterference mostly stays above it and seldom crosses thethreshold. As the threshold increases, the LCR increases andreaches a maximum value, which depends on the speed of theCUs. However, as the threshold increases further, the LCRagain starts decreasing because the interference is less likelyto be high enough to cross it. We again see that the GaussianRP model is quite inaccurate for both small and large valuesof interference. For Ith < 90 dBm, the lognormal RP modeloverestimates the LCR while the SLN RP model underesti-mates it, which is in line with Fig. 2. For Ith > 90 dBm, theSLN RP model matches the LCR accurately, and is the mostaccurate model. As mentioned earlier, corresponding resultsfor the STS model and the stable distribution model are notshown because a time-varying model for them is not known.In both figures, for a CU density of = 100 CUs/km2, theaverage number of CUs in the entire annular region is 302.As the CU density increases and exceeds 500 CUs/km2, theproposed model does become more inaccurate.

100 95 90 85 80 75104

103

102

101

100

I(t) in dBm

CDF

and

CCDF

SimulationLognormalStableSTSSLN

CDF

CCDF

Fig. 4. Non-cooperative SS: Comparison of CDF and CCDF of I(t) fromvarious models with path-loss, shadowing, and Rayleigh fading.

100 95 90 85 80 75 70103

102

101

100

101

102

Ith in dBm

LCR


Fig. 5. Non-cooperative SS: Comparison of LCR of I(t) from variousmodels with path-loss, shadowing, and Rayleigh fading.

C. Non-cooperative SS: Path-loss, Shadowing, and Fading1) Snapshot Statistics: Figure 4 compares the CDF and the

CCDF obtained using the various models. To avoid clutter,we do not show the Gaussian model curves as they are quiteinaccurate. The observations for the stable and the STS modelsare qualitatively similar to those in Fig. 2. However, comparedto Fig. 2, the CCDF curve shifts to the right due to theadditional fluctuations induced by fading. Again, the SLNmodel matches the CCDF well and is more accurate than theother models.

2) Time-varying Behavior: The LCR with both shadowingand Rayleigh fading is shown in Fig. 5. The trends are similarto Fig. 3. However, one important difference is that the max-imum value of LCR is 100 times higher than in Fig. 3. Thisis because of the faster fluctuations due to small-scale fading.For Ith < 90 dBm, the lognormal RP model overestimatesthe LCR whereas the SLN RP model underestimates it, whichis in line with Fig. 4. For Ith > 90 dBm, the SLN RP modelmatches the LCR accurately, and is the most accurate model.


90 85 80 75104

103

102

101

100

I(t) in dBm

CCDF

SimulationStableSLN

Without cooperationWith cooperation

Fig. 6. Comparison of CCDF of I(t) with non-cooperative and co-operative SS from various models with shadowing and Rayleigh fading( = 200 CUs/km2 and RC = 100 m).

100 95 90 85 80 75 70104

103

102

101

100

101

Ith in dBm

LCR


Fig. 7. Cooperative SS: Comparison of LCR of I(t) from various modelswith shadowing and Rayleigh fading ( = 200 CUs/km2 and RC = 100 m).

D. Cooperative SS: With path-loss, Shadowing, and Fading

1) Snapshot Statistics: Figure 6 compares the CCDF ofI(t) using the stable and SLN models with cooperativeand non-cooperative SS. A CU density of 200 CUs/km2and a cooperation range of RC = 100 m are considered.The Gaussian, lognormal, and STS models are not shownto avoid clutter. The stable models CCDF deviates from thesimulations for higher values of the interference. We again seethat the SLN model provides an accurate match.

We also see in Fig. 6 that the CCDF of the aggregateinterference with cooperation is 2 dB to the left of the CCDFwithout cooperation. This is because the aggregate interferencehas decreased. In general, as RC increases, more CUs willcooperate, detect the PU-Tx to be on, and transmit with lowpower. This reduces the aggregate interference.

2) Time-varying Behavior: The LCR of the aggregate in-terference with cooperative SS is shown in Fig. 7 for theGaussian, lognormal, and SLN RP models. The SLN modelmatches the LCR for higher values of the interference betterthan the other models. The Gaussian model is again theleast accurate. For Ith < 90 dBm, the lognormal modeloverestimates the LCR, and for higher values of interference,it underestimates the LCR.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1300

350

400

450

500

550

600

650

700

R PEZ

in

met

ers

From simulations without cooperationFrom simulations with cooperationFrom proposed SLN RP model

Transition point

=

= 2 sec

Fig. 8. Zoomed-in view of PEZ radius as a function of using the SLN modelwith path-loss and shadowing. Results for non-cooperative and cooperative SSare compared ( = 100 CUs/km2, RC = 100 m, and Ith = 95 dBm).The transition point shows the value of below which the AED constraint isactive.

E. Application to PEZ DesignWe now apply the analytical model for aggregate inter-

ference to redesign the PEZ based on two constraints. Thefirst constraint is the classical outage probability constraint,which mandates that the probability that the aggregate inter-ference I(t) is greater than a threshold Ith should not exceed(1 ) [3]. The second constraint is the outage durationconstraint, which is new. It is motivated by the minimumoutage duration concept [15]. It mandates that the averagetime duration for which I(t) remains above Ith should notexceed . An alternate approach is to cast the above constraintsin terms of the SINR of the PU, as has been done in [5].However, the analysis is more involved.

Figure 8 plots RPEZ as a function of for different valuesof with only path-loss and shadowing. Figure 9 plots thecorresponding results with path-loss, shadowing, and fading.Results obtained by using the formulae developed for the SLNmodel are compared with those obtained from an extensiveMonte Carlo simulation-based search. In both figures, resultsare shown for both non-cooperative and cooperative SS. The = case corresponds to only the outage constraint beingactive. Observe the good agreement between the simulationresults and the results obtained using the SLN model. As increases, the outage constraint becomes tighter and RPEZ in-creases. Furthermore, cooperation shrinks RPEZ, which meansthat more CUs can transmit closer to the PU-Rx withoutexcessively interfering with it. For = 2 sec in Fig. 8 and = 30 msec in Fig. 9, the AED constraint is active for < 0.95 and < 0.86, respectively. Thus, the AED constraintis active for a large range of values of with and withoutcooperative SS.

VI. CONCLUSIONS

We characterized the aggregate interference caused by CUs,which transmit with different powers depending on whetherthey sense the PU-Tx to be on or off. Our model accounted forthe dependence of the aggregate interference on the randomlocations and number of the CUs, their imperfect spectrumsensing, and the time-varying nature of large-scale shadowing


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9300

350

400

450

500

550

600

650

700

R PEZ

in

met

ers

From simulations without cooperationFrom simulations with cooperationFrom proposed SLN RP model

Transition point

=

= 30 msec

Fig. 9. Zoomed-in view of PEZ radius as a function of using the SLN modelwith path-loss, shadowing, and Rayleigh fading for non-cooperative andcooperative SS ( = 200 CUs/km2, RC = 100 m, and Ith = 95 dBm).The transition point shows the value of below which the AED constraint isactive.

and small-scale fading of the various links between the PU-Tx, PU-Rx, and CUs. We saw that the aggregate interferenceprocess is well characterized by SLN RP for moderate to highvalues of the interference, and we developed expressions forits moments and autocorrelation. These led to expressions forits CCDF, LCR, and AED. We also saw how cooperativeSS helps reduce the aggregate interference. Upon applyingthe model to PEZ design, we saw that a new constraint onthe AED is often active, and should be accounted for. Oneinteresting problem for future work is to characterize theaggregate interference in a more general scenario in whichmultiple PU-Txs communicate with multiple PU-Rxs.

APPENDIXA. Derivation of Covariance Functions CZ() and CI()

To obtain CZ(), we match the covariance functions ofI(t) and eZ(t) + s. It can be easily shown that

CeZ(t)+s() = E[eZ(t)+Z(t+)

](E

[eZ(t)

])2. (25)

From the expression for the moment generating function(MGF) of the jointly Gaussian RVs Z(t) and Z(t + ), weget

CeZ(t)+s() = e2Z+

2Z+CZ() e2Z+2Z . (26)

The covariance function CI() of I(t) is given by

CI() = E

[NCRi=1

Ii(t)

NCRk=1

Ik(t+ )

](E

[NCRi=1

Ii(t)

])2.

(27)Conditioning over NCR and evaluating the expectation overIi(t) and Ii(t+ ), which are independent of NCR, we get

CI() = ENCR

NCR

i=1

NCRk=1,k =i

E [Ii(t)]E [Ik(t+ )]

+

NCRi=1

E [Ii(t)Ii(t+ )]

](ENCR

[NCRi=1

E [Ii(t)]

])2.

(28)

Averaging over NCR, we get

CI() = E [NCR (NCR 1)]E [Ii(t)]E [Ik(t+ )]+ E [NCR]E [Ii(t)Ii(t+ )] (E [NCR]E [Ii(t+ )])2 .

(29)Substituting E [NCR] = A and E

[N2CR

]= A + (A)2,

where A = (R2 R2PEZ

), we get

CI() = (R2 R2PEZ

)E [Ii(t)Ii(t+ )] . (30)

Upon equating (26) and (30), we get (13).

B. Non-cooperative SS: Derivation of E [Ii(t)m] with Path-loss and Shadowing

Recall that the ith CU transmits with power Pu if itdetects the PU-Tx to be on, which happens with probabilityPD(q (ri(t), p, cos i) , Yi, gi). Else, it transmits with powerPo. From the law of total probability and (5), we get

E [Ii(t)m] = E

[(PoK

(d0ri(t)

)eXi(t)

)m (1 PD(q (ri(t), p, cos i) , Yi, gi))

+

(PuK

(d0ri(t)

)eXi(t)

)mPD(q (ri(t), p, cos i) , Yi, gi)

].

(31)Since Xi(t) is independent of the RVs ri(t), i, gi, and Yi,rearranging terms results in

E [Ii(t)m] = Pmo K

mE

[(d0ri(t)

)m]E

[emXi(t)

] (Pmo Pmu )KmE

[emXi(t)

] E

[(d0ri(t)

)mPD(q (ri(t), p, cos i) , Yi, gi)

]. (32)

Substituting the MGF of Xi(t) and the PDFs of Yi, gi, andi (from (15)) in (32), we get

E [Ii(t)m] = Pmo K

me12m

222shE

[(d0ri(t)

)m]

(Pmo Pmu )Kme12m

222sh

e y

2i

22sh

sh2

0

egi 20

1

2

E[(

d0ri(t)

)mPD(q (ri(t), p, cos i) , yi, gi)

]dyidgidi.

(33)Using Gauss-Hermite, Gauss-Laguerre, and Gauss-

Chebyshev quadratures [19] to evaluate the integrals over yi,gi, and i, respectively, yields

E [Ii(t)m] Pmo Kme

12m

222shE

[(d0ri(t)

)m]

1Wc

(Pmo Pmu )Kme12m

222sh

Whn1=1

wh (n1)

Wln2

wl (n2)

Wcn3=1

E

[(d0ri(t)

)m

PD(q (ri(t), p, ac (n3)) ,


)]. (34)


Given the total number of CUs, the location of the ith CU isuniformly distributed over the region. Substituting the PDF ofri(t) (from (15)) and simplifying yields (16).

C. Non-cooperative SS: Derivation of E [Ii(t)Ii(t+ )] withPath-loss and Shadowing

Along lines similar to Appendix B, the autocorrelation ofIi(t) can be written as

E [Ii(t)Ii(t+ )] = P2oK

2E

[e(Xi(t)+Xi(t+))

] E

[(d20

ri(t)ri(t+ )

)]

(P 2o P 2u)K2E [e(Xi(t)+Xi(t+))] E

[(d20

ri(t)ri(t+ )

)PD(q (ri(t), p, cos i) , Yi, gi)

].

(35)To simplify further, we assume that the distance ri(t) betweenthe CU and the PU-Rx is larger than the distance traveled bythe CU in a time duration : ri(t+ ) ri(t). Else, the jointPDF of ri(t) and ri(t+ ) needs to be taken into account.However, the time-variation of shadowing is accounted for.Rearranging terms and substituting the joint MGF of Xi(t)and Xi(t+ ) (by using (3)), we get

E [Ii(t)Ii(t+ )] P 2oK2e22sh

(1+exp

(v222D2

))

E[

d20ri(t)2

] (P 2o P 2u)K2e22sh

(1+exp

( v22

2D2

))

E[(

d0ri(t)

)2PD(q (ri(t), p, cos i) , Yi, gi)

]. (36)

Averaging over the RVs Yi, gi, and i, as in Appendix B,we get

E [Ii(t)Ii(t+ )] P 2oK2e22sh

(1+exp

( v22

2D2

))

E[

d20ri(t)2

](P 2o P 2u

)K2

Wce22sh

(1+exp

( v22

2D2

))

Whn1=1

wh (n1)

Wln2=1

wl (n2)

Wcn3=1

E

[(d0ri(t)

)2

PD(q (ri(t), p, ac (n3)) ,


)]. (37)

Substituting the PDF of ri(t), which is given in (15), andsimplifying yields (17).

D. Cooperative SS: With Path-loss and Shadowing1) Derivation of E [Ii(t)m]: The probability of detection

by a CU with cooperative SS depends on the SNRs of allits cooperating CUs, which are within a radius RC fromit. Let M be the number of cooperating CUs and let =[1, 2, . . . , M ] denote their SNRs. Then, the probability of

detection P (i)D,OR() of the ith CU using the OR fusion rule isgiven by [18]

P(i)D,OR() = 1 (1 PDi)

Mk=1,k =i

(1 PDj) , (38)

where

PDi = PD(q (ri(t), p, cos i) , Yi, gi) , (39)PDj = PD(q (rij(t), q (ri(t), p, cos i) , cos ij) , Yj , gj) , k = i,

(40)

And, neglecting boundary effects, M is a Poisson RV withmean R2C .

Proceeding along lines similar to Appendix B, the mthmoment of Ii(t) is given by

E [Ii(t)m] Pmo Kme

12m

222shE

[(d0ri(t)

)m] (Pmo Pmu )Kme

12m

222sh

E( d0

ri(t)

)m1 (1 PDi) Mk=1,k =i

(1 PDj) .(41)

Conditioning on the position of the ith CU and M , the secondexpectation term above becomes

E

( d0

ri(t)

)m1 (1 PDi) Mk=1,k =i

(1 PDj)

= Eri(t),i

[(d0ri(t)

)m(1 (1 E [PDi])EM

[(1 E [PDj])M1

])], (42)

where

E [PDi] f1 (ri(t), cos i) =

e y

2i

22sh

sh2

0

egiPD(q (ri, p, i) , yi, gi) dgidyi, (43)

E [PDj] f2 (ri(t), cos i) =

e y

2i

22sh

sh2

0

egi

20

RC0

rijR2C

PD(q (rij, q (ri(t), p, cos i) , cos ij) , yi, gj)

drijdijdgjdyi. (44)

As in Appendix B, after using Gauss quadrature to evaluatethe above integrals, the expressions for f1 and f2 simplifyto (22) and (23), respectively. Substituting (42) in (41) andaveraging over i, ri(t), and M yields (21).

2) Derivation of E [Ii(t)Ii(t+ )]: Along lines similar toAppendices B and C, the autocorrelation of Ii(t) can be


written as

E [Ii(t)Ii(t+ )] = P2oK

2e22sh

(1+exp

(v222D2

))

E[(

d0ri(t)

)2] (P 2o P 2u)K2e22sh

(1+exp

(v222D2

))

E[(

d0ri(t)

)2P

(i)D,OR()

]. (45)

Simplifying further, we get (24).

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Mohd. Shabbir Ali received his Bachelor ofEngineering degree in Electrical and ElectronicsEng. from the MuffakhamJah College of Eng. andTechnoglogy (MJCET), Hyderabad, India in 2010.He is an M.Sc. (Eng.) student in the Dept. ofElectrical Communication at the Indian Institute ofScience (IISc), Bangalore. His research interestsinclude cognitive radio networks, homogeneous andheterogeneous networks, and interference modelingand management.

Neelesh B. Mehta (S98-M01-SM06) received hisBachelor of Technology degree in Electronics andCommunications Eng. from the Indian Institute ofTechnology (IIT), Madras in 1996, and his M.S. andPh.D. degrees in Electrical Eng. from the CaliforniaInstitute of Technology, Pasadena, USA in 1997 and2001, respectively. He is now an Associate Professorin the Dept. of Electrical Communication Eng.,Indian Institute of Science (IISc), Bangalore. Beforejoining IISc in 2007, he was a research scientist inUSA from 2001 to 2007 in AT&T Laboratories, NJ,

Broadcom Corp., NJ, and Mitsubishi Electric Research Laboratories (MERL),MA.

His research includes work on link adaptation, multiple access protocols,cellular systems, MIMO and antenna selection, cooperative communications,energy harvesting networks, and cognitive radio. He was also actively involvedin the Radio Access Network (RAN1) standardization activities in 3GPPfrom 2003 to 2007. He has co-authored 45+ IEEE transactions papers,70 conference papers, and 3 book chapters, and is a co-inventor in 20issued US patents. He has served as a TPC co-chair for tracks/symposia inCOMSNETS 2014, Globecom 2013, ICC 2013, WISARD 2010 and 2011,NCC 2011, VTC 2009 (Fall), and Chinacom 2008. He is an Editor ofIEEE WIRELESS COMMUNICATIONS LETTERS, IEEE TRANSACTIONS ONCOMMUNICATIONS, and the Journal of Communications and Networks. Hecurrently serves as the Director of Conference Publications on the Board ofGovernors of IEEE ComSoc.

modeling time-varying aggregate interference in cr system and application to primary exclusive zone...

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