456 ieee transactions on communications, vol. 60, no. 2, … · 2019-12-18 · 456 ieee...

456 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 2, FEBRUARY 2012

Aggregate Interference Modeling in CognitiveRadio Networks with Power and Contention ControlZengmao Chen, Cheng-Xiang Wang, Senior Member, IEEE, Xuemin Hong, John S. Thompson, Member, IEEE,

Sergiy A. Vorobyov, Senior Member, IEEE, Xiaohu Ge, Senior Member, IEEE, Hailin Xiao, and Feng Zhao

Abstract—In this paper, we present interference models forcognitive radio (CR) networks employing various interferencemanagement mechanisms including power control, contentioncontrol or hybrid power/contention control schemes. For thefirst case, a power control scheme is proposed to govern thetransmission power of a CR node. For the second one, a con-tention control scheme at the media access control (MAC) layer,based on carrier sense multiple access with collision avoidance(CSMA/CA), is proposed to coordinate the operation of CR nodeswith transmission requests. The probability density functions(PDFs) of the interference received at a primary receiver froma CR network are first derived numerically for these two cases.For the hybrid case, where power and contention controls arejointly adopted by a CR node to govern its transmission, theinterference is analyzed and compared with that of the firsttwo schemes by simulations. Then, the interference PDFs underthe first two control schemes are fitted by log-normal PDFs toreduce computation complexity. Moreover, the effect of a hiddenprimary receiver on the interference experienced at the receiver

Paper approved by O. Oyman, the Editor for Cooperative and Heteroge-neous Networks of the IEEE Communications Society. Manuscript receivedJuly 20, 2010; revised November 22, 2010, April 1, 2011 and August 30,2011.

Z. Chen and C.-X. Wang are with the Joint Research Institute for Signaland Image Processing, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK (e-mail: {zengmao.chen, cheng-xiang.wang}@hw.ac.uk). C.-X. Wang is the corresponding author.

X. Hong is with the School of Information Science and Tech-nology, Xiamen University, Xiamen 361005, Fujian, China (e-mail:[email protected]).

J. Thompson is with the Institute for Digital Communications, Universityof Edinburgh, Edinburgh, EH9 3JL, UK (e-mail: [email protected]).

S. A. Vorobyov is with the Department of Electrical and ComputerEngineering, University of Alberta, Edmonton, AB, T6G 2V4, Canada (e-mail: [email protected]).

X. Ge is with the Department of Electronics and Information Engineering,Huazhong University of Science and Technology, Wuhan 430074, Hubei,China (e-mail: [email protected]).

H. Xiao and F. Zhao are with Guilin University of ElectronicTechnology, Guilin 541004, China (e-mail: [email protected],[email protected]).

The authors acknowledge the support from the RCUK for the UK-ChinaScience Bridges Project: R&D on (B)4G Wireless Mobile Communications.Z. Chen, C.-X. Wang, and J. Thompson acknowledge the support from theScottish Funding Council for the Joint Research Institute in Signal andImage Processing between the University of Edinburgh and Heriot-WattUniversity, as part of the Edinburgh Research Partnership in Engineering andMathematics (ERPem). S. A. Vorobyov acknowledges the support in part fromthe Natural Sciences and Engineering Research Council (NSERC) of Canada.X. Ge acknowledges the support from National Natural Science Foundationof China (NSFC) (Grant No.: 60872007), National 863 High TechnologyProgram of China (Grant No.: 2009AA01Z239), Hubei Provincial Scienceand Technology Department (Grant No.: 2011BFA004), and the Ministryof Science and Technology (MOST) of China, International Science andTechnology Collaboration Program (Grant No.: 0903). F. Zhao acknowledgesthe support from the NSFC (Grant No.: 61172055). C.-X. Wang and F.Zhao acknowledge the support of the Key Laboratory of Cognitive Radioand Information Processing (Guilin University of Electronic Technology),Ministry of Education, China (Grant No.: 2011KF01).

Digital Object Identifier 10.1109/TCOMM.2011.012012.100426

is investigated. It is demonstrated that both power and contentioncontrols are effective approaches to alleviate the interferencecaused by CR networks. Some in-depth analysis of the impactof key parameters on the interference of CR networks is givenas well.

Index Terms—Cognitive radio, interference modeling, powercontrol, contention control, hidden primary receiver.

I. INTRODUCTION

W ITH the requirement to improve spectrum utilization,the emerging cognitive radio (CR) technology [1]–[4]

has attracted increasing attention. A CR network is envisionedto be capable of reusing the unused or underutilized spectra ofincumbent systems (also known as primary networks) by sens-ing its surrounding environment and adapting its operationalparameters autonomously. A CR system may coexist with aprimary network on either an interference-free or interference-tolerant basis [5], [6]. For the former case, the CR system onlyexploits the unused spectra of the primary network, whichconsequently guarantees no interference to primary users. Forthe latter case, the CR system is allowed to share the spectraassigned to the primary network, under the condition thatthe CR network must not impose detrimental interference onthe primary network. Therefore, modeling and analyzing theinterference caused by CR networks is of great importanceto reveal how the service of a primary network is deterioratedand how CR networks may be deployed to protect the primarynetwork against detrimental interference.

In the literature, the existing research on interference model-ing for CR networks mainly falls into three categories: spatial,frequency-domain and accumulated interference modeling. Forspatial interference modeling, the fraction of white spacesavailable for CR networks was investigated in [7]–[9]. In [10],the region of interference for CR receivers and region ofcommunication for CR transmitters were studied for thecase where a CR network coexists with a cellular network.The interference from CR devices to wireless microphonesoperating in TV bands was analyzed in [11], where the lossof reliable communication area of a wireless microphone dueto the existence of CR devices was examined. CR interferencein the frequency domain was also researched in the literature,e.g., the interference due to out-of-band emission of a wirelessregional area network (WRAN) was analyzed in [12].

As for accumulated interference modeling, in [13], theaggregate interference power from a sea of CR transmitterssurrounding a primary receiver was derived. The performanceof a primary system was evaluated in [14] in terms of outage

0090-6778/12$31.00 c⃝ 2012 IEEE

CHEN et al.: AGGREGATE INTERFERENCE MODELING IN COGNITIVE RADIO NETWORKS WITH POWER AND CONTENTION CONTROL 457

probability caused by the interference from CR networks. Theoutage probability was derived for both underlay and overlayspectrum sharing cases. In [15], the aggregate interferencefrom multiple CR transmitters following a Poisson pointprocess was approximated by a Gamma distribution and theprobability of interference at a primary receiver was alsogiven. It is worth noting that only pathloss was assumed forthe interfering channel in [13]–[15]. Their work was extendedby taking both shadowing and fading into account in [16]and [17]. Moreover, the probability density function (PDF)for accumulated interference and outage probability due tothe aggregate interference from CR nodes were also derivedin [16] and [17], respectively.

However, in all the previous works [7]–[17], the CR trans-mitters were assumed to transmit at a fixed power level. More-over, the CR nodes were all assumed to communicate witheach other simultaneously. Thus, no contention control schemewas employed at the cognitive media access control (MAC)layer. In [18], some preliminary interference modeling was re-ported by considering power and contention control schemes.In this paper, we present more realistic and comprehensiveinterference models by significantly extending the work of[18]. Firstly, a more realistic power control scheme than that in[18] is proposed, and a new interference management scheme- hybrid power/contention control scheme is introduced. Sec-ondly, the PDFs of interference perceived at a primary networkfrom a CR network are derived numerically for the cases ofpower or contention control. The interference distribution ofthe hybrid control scheme is also analyzed and comparedwith that of the pure power control and pure contentioncontrol schemes by simulations. Furthermore, to reduce thecomplexity of the numerical interference PDF’s computationused in [18], cumulant-based approximations are applied tofitting the interference distributions. Finally, the impact of ahidden primary receiver on the aggregate interference is alsoinvestigated. The interference modeling presented in this workconsiders several basic interference management schemes,which forms a fundamental basis for the development of otheradvanced interference models for CR networks. Moreover, itgives insights into CR deployment by figuring out how tooptimize CR operation parameters. Finally, the interferencemodeling lays a foundation for performance evaluation ofprimary networks, e.g., outage capacity of primary systemscan be derived based on the interference PDFs.

The remainder of this paper is organized as follows. Thesystem model is elaborated in Section II. The detailed inter-ference modeling is presented in Section III. In Section IV,the interference distributions are approximated by log-normaldistributions. We incorporate the hidden primary receiverproblem in Section V. The impact of several key parameterson the interference is analyzed via numerical studies in Sec-tion VI. Finally, Section VII concludes the paper.

II. SYSTEM MODEL

The system model is illustrated in Fig. 1. It consists of a CRnetwork coexisting with a primary transmitter-receiver pair.The interference region (IR) is adopted by the primary networkto protect itself against interference from the CR network.No CR transmission is allowed within the IR. There exist

−1.5 −1.0 −0.5 0 0.5 1.0 1.5(km)−1.5

−1.0

−0.5

0

0.5

1.0

1.5(km)

Primary receiverInterference regionCR transmitter

Fig. 1. System model for CR networks coexisting with a primarynetwork (𝑅 = 250 m).

two main types of techniques to identify the IR for a primarynetwork: geo-location technique and spectrum sensing [19].For geo-location-based CR networks, the global positioningsystem (GPS) can be incorporated within the CR network. Itenables CR transmitters to determine whether they are locatedfar enough outside the protected service contour of the primarysystem. The geo-location technology usually leads to a circularIR around the primary system. As for spectrum-sensing-basedCR, the IR is usually not circular but more irregular than thatof the geo-location-based CR due to fading and/or imperfectsensing. In this paper, we focus on the former type and thus,a circular IR with a radius 𝑅 is considered.

The underlying interference channels from CR transmittersto the primary receiver experience pathloss, shadowing andfading. The pathloss function 𝑔(𝑟𝑗) is

𝑔(𝑟𝑗) = 𝑟−𝛽𝑗 (1)

where 𝑟𝑗 is the distance between the 𝑗th (𝑗 = 1, 2, ⋅ ⋅ ⋅ ) activeCR transmitter and the primary receiver and 𝛽 is the pathlossexponent. The composite model for shadowing and fadingcan be expressed as the product of the long term shadowingand the short term multipath fading. In this paper, log-normalshadowing and Nakagami fading are considered. Let ℎ𝑗 denotethe channel gain for the composite shadowing and fading ofthe interference channel from the 𝑗th active CR transmitterto the primary receiver. The PDF of the composite channelgain ℎ𝑗 can be approximated by the following log-normaldistribution [20] (Page 102)

𝑓h(𝑥) ≈ 1√2𝜋𝜎𝑥

exp

{− (ln(𝑥) − 𝜇)2

2𝜎2

}(2)

where the mean 𝜇 and variance 𝜎2 can be expressed as

𝜇 =

(𝑚−1∑𝑘=1

1

𝑘− ln(𝑚) − 0.5772

)+ 𝜇Ω (3)

𝜎2 =

∞∑𝑘=0

1

(𝑚 + 𝑘)2+ 𝜎2

Ω (4)


with 𝑚 standing for the Nakagami shape factor and 𝜇Ω and 𝜎2Ω

denoting the mean and variance of the log-normal distribution,respectively.

Let 𝑝𝑗 denote the transmission power of the 𝑗th activeCR transmitter. The accumulated power of the instantaneousinterference received at the primary receiver can be expressedas

𝑌 =

∞∑𝑗=1

𝑝𝑗𝑔(𝑟𝑗)ℎ𝑗 . (5)

In this paper, we investigate the characteristics of the ag-gregate interference from all CR transmitters employing thefollowing three different interference management schemes:(i) power control, (ii) contention control, and (iii) hybridpower/contention control.

A. Power Control

In this scenario, the distribution of active CR transmittersfollows a Poisson point process with a density parameter 𝜆for the density of CR transmitters on the plane.

The transmission power of a CR transmitter is governed bythe following power control law

𝑝pwc(𝑟cc𝑗 ) =

{ (𝑟cc𝑗𝑟pwc

)𝛼𝑃max, 0 < 𝑟cc𝑗 ≤ 𝑟pwc

𝑃max, 𝑟cc𝑗 > 𝑟pwc

(6)

where 𝑟cc𝑗 is the distance from the 𝑗th active CR transmitterto its nearest neighboring active CR transmitter, 𝛼 is the powercontrol exponent, 𝑃max is the maximum transmission powerfor CR transmitters, and 𝑟pwc is the power control range,which determines the minimum 𝑟cc𝑗 leading to maximum CRtransmission power 𝑃max. Compared to the power control lawin [18], a new parameter 𝑟pwc is introduced here to adjust therange of the power control. The interference caused by the 𝑗thactive CR transmitter to its nearest active CR transmitter dueto pathloss is 𝑝pwc(𝑟cc𝑗 )𝑔(𝑟cc𝑗 ). The above proposed powercontrol scheme is designed in such a manner that when 𝛼 = 𝛽within the power control range 𝑟pwc, this interference is equalto a constant 𝑃max/𝑟

𝛼pwc. Beyond the power control range,

the interference is smaller than that constant. This meansthat at any CR transmitter, the interference from the nearestneighboring CR transmitter is capped and independent of thenearest neighbor distance within the power control range.

A CR network can be deployed as either an infrastructureor an ad hoc network [2]. For a CR infrastructure network, theabove power control scheme is applicable to CR base stations(BSs), whose transmission powers are usually determinedby their coverages. Moreover, the CR BSs locations areusually fixed, which minimizes the network planning load todetermine the transmission power of CR BSs. When CR BSsfollow a Poisson point distribution, the PDF of 𝑟cc𝑗 can begiven as [22]

𝑓cc(𝑥) = 2𝜋𝜆𝑥𝑒−𝜆𝜋𝑥2

. (7)

B. Contention Control

Unlike the above mentioned power control scheme, for thecase of contention control, every active CR transmitter hasfixed transmission power 𝑝, but their transmission is governedby contention control to determine which CR transmitters can

transmit at a given time. We assume that the multiple accessprotocol carrier sense multiple access with collision avoidance(CSMA/CA) is employed, like in IEEE 802.11 networks.Every CR transmitter senses the medium before transmission.If the medium is busy, namely, the CR transmitter detectstransmission from other CR transmitters within its contentionregion, it defers its transmission. Otherwise, the CR transmitterstarts its transmission. As a result of the contention control, allthe active CR transmitters are separated from each other by atleast the contention distance, which is the minimum distance𝑑min between two concurrent CR transmitters. The contentioncontrol scheme can be applied to either a CR ad hoc networkor distributed multiple-access users of a CR infrastructurenetwork.

The distribution of active CR transmitters under the con-tention control scheme can be modeled as a Matern-hardcore(MH) point process [21]. The MH point process Φmh can beobtained by thinning a Poisson point process Φ, which can beexpressed as follows [22]

Φmh = {𝑥 ∈ Φ ∣ 𝑚(𝑥) < 𝑚(𝑦) for all y in Φ ∩ 𝐶(𝑥, 𝑑min)}.(8)

Each point 𝑥 in the original Poisson point process Φ is markedwith a random variable 𝑚(𝑥) uniformly distributed in (0, 1).A point 𝑥 is retained from the thinning process only if itsmark is smaller than that of all the other points within thedisk 𝐶(𝑥, 𝑑min) centered at point 𝑥 with the radius 𝑑min.Otherwise, the point 𝑥 is removed. The retaining probability𝑞mh for the MH point process, which is the probability of apoint from a Poisson point process with a density 𝜆 survivingthe thinning process, is given by [22]

𝑞mh =1 − 𝑒−𝜆𝜋𝑑

2min

𝜆𝜋𝑑2min

. (9)

C. Hybrid Power/Contention Control

A natural extension of the above two interference man-agement schemes is to implement both schemes in the samesystem. This is termed hybrid power/contention control and itworks in the following manner. The contention control schemeis first applied, resulting in a set of active CR transmittersfollowing an MH point process. Then, a power control schemesimilar to (6) is employed to adjust the transmission power ofeach active CR transmitter according to the distance to thenearest neighboring active transmitter. The following powercontrol law is adopted in the hybrid control scheme

𝑝hyb(𝑟) =

⎧⎨⎩(𝑟𝑐𝑐𝑑min

)𝛼𝑝, 𝑑min ≤ 𝑟𝑐𝑐 ≤ 𝑟hyb(

𝑟hyb𝑑min

)𝛼𝑝, 𝑟𝑐𝑐 > 𝑟hyb

(10)

where 𝑟𝑐𝑐 is the distance from an active CR transmitter toits nearest neighboring active CR transmitter, 𝛼 is the powercontrol exponent as in (6), and 𝑟hyb is the power controlrange similar to 𝑟pwc in (6) except that it also determines

the maximum transmission power, i.e.,(𝑟hyb𝑑min

)𝛼𝑝. The above

power control law (10) guarantees that when a pathloss chan-nel is considered for each active CR transmitter, the perceivedinterference caused by its nearest neighboring CR transmitteris 𝑝hyb(𝑟𝑐𝑐)𝑔(𝑟𝑐𝑐), which is (i) a constant 𝑝/𝑑𝛼min (if 𝛼 = 𝛽)


within the power control range 𝑟hyb and (ii) less than theconstant when 𝑟 is larger than the power control range.

III. INTERFERENCE MODELING

We intend to model the aggregate interference from CRtransmitters employing the three different interference man-agement schemes introduced in Section II by finding theircorresponding PDFs. We apply the characteristic-function-based method used, for example, in [16] and [23] to derivethe PDFs. First, the characteristic functions of the interferenceunder different system models are derived. Then, the PDFsof the aggregate interference are obtained by performing aninverse Fourier transform of characteristic functions.

A. Power Control

When all the CR transmitters follow a Poisson point processdistribution and employ the power control scheme proposedin (6), we can adopt the characteristic function-based methodas in [16], [23]-[26] and obtain the following characteristicfunction 𝜙Y(𝜔) of the aggregate interference 𝑌 at a primaryreceiver from all CR transmitters

𝜙Y(𝜔) = exp

(𝜆𝜋

∫𝐻

𝑓h(ℎ)

∫𝑃

𝑓p(𝑝)𝑇 (𝜔𝑝ℎ)𝑑𝑝 𝑑ℎ

)(11)

where 𝑓p(⋅) is the PDF of the CR transmission power𝑝pwc(𝑟cc𝑗 ) defined in (6) and

𝑇 (𝜔𝑝ℎ) = 𝑅2(1 − 𝑒𝑖𝜔𝑔(𝑅)𝑝ℎ) + 𝑖𝜔𝑝ℎ

∫ 𝑔(𝑅)

0

[𝑔−1(𝑡)]2𝑒𝑖𝜔𝑡𝑝ℎ𝑑𝑡.

(12)

In (12), 𝑔−1(⋅) denotes the inverse function of 𝑔(⋅) in (1).For the derivation of (11), the following fact is used: thedistance from a CR transmitter to the primary receiver 𝑟 hasindependent and identical uniform distributions for a givennumber of CR transmitters [23]. The corresponding PDFs havethe form [23]

𝑓r(𝑟) =

{2𝑟/(𝑙2 −𝑅2), 𝑅 ≤ 𝑟 ≤ 𝑙0, otherwise

(13)

when CR transmitters are distributed within an annular ringwith inner radius 𝑅 and outer radius 𝑙. In (11), 𝑝 is a functionof 𝑟cc as shown in (6), so the expectation of 𝑇 (𝜔𝑝ℎ) over 𝑝equals that of 𝑇 (𝜔𝑝pwc(𝑟cc)ℎ) over 𝑟cc. Using the PDF of 𝑟ccgiven in (7), (11) can be rewritten as

𝜙Y(𝜔) = exp

(𝜆𝜋

∫𝐻

𝑓h(ℎ)

∫𝑟cc

𝑓cc(𝑥)𝑇 (𝜔𝑝pwc(𝑟cc)ℎ)𝑑𝑥𝑑ℎ

).

(14)

Moreover, (14) can be written as (see Appendix A for thedetailed derivation procedure)

𝜙Y(𝜔)=exp

{𝜆𝜋

∫𝐻

𝑓h(ℎ)

∫ 𝑟pwc

0

𝑓cc(𝑥)

[𝑅2

(1−𝑒

𝑖𝜔𝑥𝛼𝑃max𝑔(𝑅)ℎ𝑟pwc𝛼

)

+𝑖𝜔𝑥𝛼𝑃maxℎ

𝑟pwc𝛼

∫ 𝑔(𝑅)

0

𝑡−2𝛽 𝑒

𝑖𝜔𝑡𝑥𝛼𝑃maxℎ𝑟pwc𝛼 𝑑𝑡

]𝑑𝑥𝑑ℎ

+ 𝜆𝜋

∫𝐻

𝑓h(ℎ)

∫ ∞

𝑟pwc

𝑓cc(𝑥)[𝑅2(

1 − 𝑒𝑖𝜔𝑔(𝑅)𝑃maxℎ)

+ 𝑖𝜔𝑃maxℎ

∫ 𝑔(𝑅)

0

𝑡−2𝛽 𝑒𝑖𝜔𝑡𝑃maxℎ𝑑𝑡

]𝑑𝑥𝑑ℎ

}.

(15)

Fig. 2. Intersection of the IR and contention region (dotted regionA ), and the hidden primary receiver (denoted in the bracket) whichis hidden from all CR transmitters distributed in the shaded region.

Finally, we obtain the PDF of the interference by performingthe inverse Fourier transform of 𝜙Y(𝜔) as

𝑓Y(𝑦) =1

2𝜋

∫ +∞

−∞𝜙Y(𝜔)𝑒−2𝜋𝑖𝜔𝑦𝑑𝜔. (16)


As mentioned in Section II.B, the distribution of CR trans-mitters can be modeled as an MH point process when thecontention control is adopted. However, the thinning processdepicted in (8) does not consider the existence of the IR.For CR transmitters right outside of the IR, their retainingprobability and also the density of the MH process areaffected by the IR. More specifically, for a CR transmitterwith 𝑅 < 𝑟 < 𝑅 + 𝑑min whose contention region intersectswith the IR as shown in Fig. 2, its retaining probability andhence the density of retained CR transmitters is determinedby the dotted region A shown in Fig. 2. After some algebraicmanipulations, we obtain the area 𝐴 of the dotted region asfollows

𝐴 = 𝑅2(𝛾 − sin𝛾 cos𝛾) + 𝑑2min(𝜂 − sin𝜂 cos𝜂) (17)

where

𝛾 = arccos𝑅2 + 𝑟2 − 𝑑2min

2𝑅𝑟(18)

𝜂 = arccos𝑟2 + 𝑑2min −𝑅2

2𝑟𝑑min. (19)

The density of active CR transmitters after adopting thecontention control scheme can be expressed as

𝜆cnt(𝑟) =

⎧⎨⎩

𝜆mh = 𝜆𝑞mh = 1−𝑒−𝜆𝜋𝑑2min

𝜋𝑑2min, 𝑟 ≥ 𝑅 + 𝑑min

𝜆roIR(𝑟) = 1−𝑒−𝜆(𝜋𝑑2min−𝐴)

𝜋𝑑2min−𝐴, 𝑅 < 𝑟 < 𝑅 + 𝑑min

0, otherwise(20)


where 𝜆roIR(𝑟) denotes the density of active CR transmittersright outside of the IR with 𝑅 < 𝑟 < 𝑅 + 𝑑min. It can beproved that 𝜆roIR(𝑟) is a monotonically decreasing functionof 𝑟 with 𝜆roIR(𝑅 + 𝑑min) = 𝜆mh.

For active CR transmitters located in the region 𝑟 ≥𝑅 + 𝑑min, the distribution of their aggregate interference isequivalent to that of a Poisson point process with density𝜆mh. This is due to the fact that the thinning process (8)is independent of 𝑟 in this region. Therefore, the active CRtransmitters in this region have a uniformly distributed valueof 𝑟 similar to (13) given a total number of active CRtransmitters. While, for active CR transmitters located rightoutside of the IR with 𝑅 < 𝑟 < 𝑅 + 𝑑min, the characteristicfunction of the aggregate interference analogous to (11) isdifficult to obtain due to the fact that 𝜆roIR(𝑟) is not fixedlike 𝜆mh, but changes over 𝑟. To model the distribution ofthe aggregate CR interference under the contention controlscheme, we can approximate the thinning process by ignoringthe impact of the IR. The approximated density of active CRtransmitters can be written as follows

𝜆apprx =

{𝜆mh, 𝑟 > 𝑅0, otherwise.

(21)

The aggregate CR interference distribution under contentioncontrol can be approximated by that of a Poisson pointprocess with density 𝜆mh and fixed transmission power 𝑝. Thecharacteristic function of the accumulated interference can befound as

𝜙Y(𝜔) = exp

(𝜆mh𝜋

∫𝐻

𝑓h(ℎ)𝑇 (𝜔𝑝ℎ)𝑑ℎ

). (22)

The detailed derivation of (22) is presented in Appendix B.By comparing (20) and (21), one can find out that the

approximated thinning process underestimates the aggregateCR interference due to the monotonically decreasing nature of𝜆roIR(𝑟). Some active CR transmitters located in the region𝑅 < 𝑟 < 𝑅+𝑑min are ignored during the approximation, i.e.,the approximated distribution (22) serves as a lower bound forthe CR interference distribution under the contention controlscheme. The accuracy of the above approximation depends onthe difference between 𝜆roIR(𝑟) and 𝜆mh. It can be seen from(17)–(20) that 𝜆roIR(𝑟) is determined by 𝜆, 𝑑min and 𝑅. InFig. 3, we evaluate the impact of these parameters on the ratio𝜆roIR(𝑅)/𝜆mh, which relates to the accuracy of the approxi-mation. It is easy to see that the approximation becomes moreaccurate as the ratio 𝜆roIR(𝑅)/𝜆mh decreases and approaches1. By observing Fig. 3, the following points may be observed:(i) the approximation tends to be less accurate as the density 𝜆and/or IR radius𝑅 increase; (ii) for small 𝜆, the approximationaccuracy is inversely proportional to 𝑑min; while, for large 𝜆,the accuracy is proportional to 𝑑min; (iii) the density 𝜆 has lessinfluence on the approximation accuracy as it increases, sincethe ratio saturates. The approximation accuracy is evaluatednumerically in Section IV.B as well.

C. Hybrid Power/Contention Control

To model the aggregate interference under the hybridcontrol scheme, the nearest neighboring distance distributionfunction analogous to (7) for active CR transmitters is in-dispensable to evaluate the transmission power designated

0 50 100 150 200 250 3001

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

λ ( x10−4 user/m2)

λ roIR

(R)/λ m

h

dmin

= 10 m, R=100 m

dmin

= 20 m, R=100 m

dmin

= 40 m, R=100 m

dmin

= 80 m, R=100 m

dmin

= 10 m, R=1000 m

dmin

= 40 m, R=1000 m

Fig. 3. Impact of system parameters on the density 𝜆roIR of activeCR transmitters right outside the IR.

in (10). Unfortunately, there is no closed-form expressionfor the nearest neighbor distance distribution function foran MH point process [27]. Thus, we approach this problemnumerically.

The PDF for the aggregate interference under the hybridcontrol scheme is simulated in Fig. 4, where the interferencePDFs for power and contention control are given as well forthe purpose of comparison. It can be seen from this figurethat with the setup of 𝑑min = 𝑟pwc, 𝑝 = 𝑃max and 𝛼 = 𝛽both the mean and variance of the aggregate interferenceincrease for the hybrid control scheme compared to eitherpower or contention control schemes. However, the boostedinterference is paid off by the increased CR communicationarea (coverage) for the hybrid control scheme. We define thecoverage of each CR transmitter as a circular disk centered ata CR transmitter with radii being min(𝑟/2, 𝑟pwc/2), 𝑑min/2and min(𝑟/2, 𝑟hyb/2) for power control, contention controland hybrid power/contention control schemes, respectively.Then, the received signal power at cell edge of a CRtransmitter due to pathloss is 2𝛽𝑃max/𝑟pwc

𝛽 , 2𝛽𝑝/𝑑𝛽min and2𝛽𝑝/𝑑𝛽min, which is same for all the three schemes under theaforementioned setup. With this setup, the overall coverageratio obtained numerically among the power, contention andhybrid power/contention control is 1.0093 : 1 : 2.0229. Twointeresting facts are unveiled from this experiment. Firstly,with the setup of 𝑑min = 𝑟pwc, 𝑝 = 𝑃max and 𝛼 = 𝛽, thepower and contention control schemes lead to almost the sameamount of interference and coverage, but the latter is lesscomplex to implement. Secondly, introducing power controlinto the contention control scheme can enlarge the coveragecompared to the pure contention control scheme at the cost ofcausing higher interference to the primary system.

IV. ANALYTICAL APPROXIMATION

In the previous section, in order to derive the PDFs for ag-gregate interference, the characteristic function-based methodhas been used which consists of two steps. Namely, charac-teristic function computation and Fourier transformation. This


0 1 2 3 4 5

x 10−7

0

0.5

1

1.5

2

2.5

x 107

Aggregate interference power (W)

PD

F

Power controlContention controlHybrid power/contention control

Fig. 4. Comparison of interference distributions for power, contentionand hybrid power/contention control schemes (𝑅 =100 m, 𝜆 =3user/104m2, 𝛽 =4, 𝑟pwc = 20 m, 𝛼 = 4, 𝑃max = 1 W, 𝑝 = 1 W,𝑑min = 20 m and 𝑟hyb = 30 m).

interference modeling approach is extremely computation-intensive, since generally closed-form expressions are notadmitted for either step and the computations in both stepshave to be performed numerically. It is desirable to modelthe aggregate interference with less complexity. An alternativeapproach to model the interference, which greatly reducescomplexity, is to approximate interference PDFs as certainknown distributions. Observations from Fig. 4 suggest that theinterference distribution for either power or contention controlis positively skewed and heavy-tailed, which suggests a log-normal distribution. Thus, in this section, we fit the aggregateinterference under power and contention control schemes tolog-normal distributions. The theory behind the log-normal fit-ting is based on the following two facts. It has been shown thatthe sum of interference from uniformly distributed interferersin a circular area is asymptotically log-normal [17], [28]. Thisensures that the aggregate interference in these two schemescan be approximated as log-normal distributed. Meanwhile,the sum of randomly weighted log-normal variables can bemodeled as a log-normal distribution as well [29], whichguarantees that the aggregate interference is still log-normaldistributed even if the effect of shadow fading (2) is taken intoaccount. In what follows, the log-normal fitting is performedusing a cumulant-matching approach [30], where the first twoorder cumulants of the aggregate interference 𝑌 in (5) areused to estimate the mean and variance of the log-normaldistribution function. Therefore, the exact PDFs of interferencecan be obtained. Fortunately, these cumulants have closed-form expressions for both control schemes. Consequently,it significantly reduces the complexity compared to the in-terference modeling carried out in Section III. Moreover,compared to the characteristic function-based method, therelationship between CR system parameters and the resultinginterference becomes much clearer for the cumulant-basedPDFs approximation.

For a log-normal random variable, its mean 𝜇 and variance𝜎2 can be estimated using the first two order cumulants 𝑘1and 𝑘2 as follows [31]:

𝜇 = ln𝑘1√𝑘2𝑘21

+ 1(23)

𝜎2 = ln

(𝑘2𝑘21

+ 1

). (24)

In the context of interference distribution fitting, the 𝑛th cumu-lant 𝑘𝑛 of the aggregate interference 𝑌 can be obtained fromits characteristic function 𝜙Y(𝜔) via the following equation

𝑘𝑛 =1

𝑖𝑛

[∂𝑛ln𝜙Y(𝜔)

∂𝜔𝑛

]𝜔=0

. (25)

A. Power Control

From (15) and (25), the cumulants for aggregate interfer-ence under the power control scheme can be derived as (seeAppendix C for detailed derivation)

𝑘𝑛 =2𝜆𝜋𝑃𝑛max𝑒

𝑛𝜇+𝑛2𝜎2

2

(𝑛𝛽 − 2)𝑅𝑛𝛽−2

[𝑛𝛼(𝑛𝛼− 2) ⋅ ⋅ ⋅ 2𝑟pwc

𝑛𝛼(2𝜋𝜆)𝑛𝛼2

(1−𝑒−𝜆𝜋𝑟pwc

2)

−𝑛𝛼2 −1∑𝑖=1

𝑛𝛼(𝑛𝛼− 2) ⋅ ⋅ ⋅ (𝑛𝛼− 2𝑖 + 2)

(2𝜋𝜆𝑟pwc2)𝑖

𝑟pwc𝑛𝛼−2𝑖𝑒−𝜆𝜋𝑟pwc

2

⎤⎦.

(26)

It can be seen from (26) that 𝑘𝑛 is proportional to 𝑃𝑛max

and 1/𝑅𝑛𝛽−2, and all cumulants are most sensitive to the IRradius 𝑅 since it has the highest exponent compared to otherparameters. The power control range 𝑟pwc and the density 𝜆have similar impact on all cumulants, but the impact of theformer is larger than that of the latter, since the former has alarger exponent.

To evaluate the accuracy of the approximation for thepower control case, comparisons are performed in Fig. 5(a),where cumulative distribution functions (CDFs) are used toimprove readability. It can be seen from Fig. 5(a) that there isfairly good agreement among the interference CDFs derivedin Section III, the approximated counterparts and the MonteCarlo simulation. This approximation approach can be appliedto both the pathloss-only and shadow fading channels.


Following the similar steps as in Appendix C and giventhe characteristic function (22) for the aggregate interferenceunder contention control and also using (25), we can find the𝑛th cumulant 𝑘𝑛 of aggregate interference as

𝑘𝑛 =𝜆𝜋𝑞mh

𝑖𝑛

∫𝐻

𝑓h(ℎ)[−𝑅2 (𝑖𝑝𝑔(𝑅)ℎ)

𝑛

+𝑛 (𝑖𝑝ℎ)𝑛∫ 𝑔(𝑅)

0

𝑡𝑛−1− 2𝛽 𝑑𝑡

]𝑑ℎ

= 𝜆𝜋𝑞mh

(𝑛

𝑛− 2𝛽

𝑔𝑛−2𝛽 (𝑅) −𝑅2𝑔𝑛(𝑅)

)𝑝𝑛∫𝐻

𝑓h(ℎ)ℎ𝑛𝑑ℎ

=2𝑝𝑛

(1 − 𝑒−𝜆𝜋𝑑

2min

)𝑒𝑛𝜇+

𝑛2𝜎2

2

(𝑛𝛽 − 2)𝑑2min𝑅𝑛𝛽−2

. (27)


0 1 2 3 4 5 6

x 10−7

0

0.2

0.4

0.6

0.8

1


CD

F

(a)

Monte Carlo simu.Derived CDFApproximated CDF

0 1 2 3 4 5 6

x 10−7

0

0.2

0.4

0.6

0.8

1


CD

F

(b)

Monte Carlo simu.Derived CDFApproximated CDF

Pathloss onlyλ=3x10−4 user/m2

Shadow fadingλ=3x10−4 user/m2



Shadow fadingλ=3x10−4 user/m2


Fig. 5. Log-normal approximation for interference distribution under(a) power control (𝑅 =100 m, 𝛽 =4, 𝑟pwc = 20 m, 𝛼 = 4, 𝑃max = 1W, 𝜇 = 0 and 𝜎 = 4 dB) or (b) contention control (𝑅 =100 m,𝛽 =4, 𝑑min = 20 m, 𝑝 = 1 W, 𝜇 = 0 and 𝜎 = 4 dB).

As we can see from (27), 𝑘𝑛 is proportional to 𝑝𝑛 and1/𝑅𝑛𝛽−2, which suggests that IR radius 𝑅 is the mosteffective parameter to control the aggregate interference dueto its highest exponent. The cumulants are not sensitive to theCR density 𝜆 for large 𝜆. The contention range 𝑑min has littleimpact on the cumulants when 𝑑min is small. Itcanalsobeseenfrom (26) and (27) that shadow fading has the same impact oncumulants for the power and contention control schemes.

The accuracy evaluation of approximations under the con-tention control scheme is also performed and shown inFig. 5(b). It can be seen from this figure that the log-normalapproximation is fairly accurate compared to the derivedinterference CDFs for either pathloss-only or shadow fadingchannels. Moreover, it can be observed that the approximatedthinning process tends to be less accurate as the CR density𝜆 increases, which agrees with the analysis in Section III.B.

V. IMPERFECT PRIMARY SYSTEM KNOWLEDGE

In practice, some information about the primary systemmay not be perfectly known. One prominent example is thelocation of the primary receivers, which is usually requiredby CR networks in order to protect primary receivers frominterfering CR transmitters. However, this information is notalways available, especially in the case of passive primaryreceivers, i.e., when the primary receivers are hidden from CRnetworks. It is widely accepted that passive receiver detectiontechniques can be used or developed in the context of CRnetworks. For example, one of such primary receiver detectiontechniques is reported in [32]. Nevertheless, its applicability isstill not convincingly viable since it requires deploying sensornodes close to primary receivers and much coordination isinvolved between these sensors and CR networks as well.The most commonly used and also the simplest approach toprotect the primary receiver is to regulate the transmissionof the CR network based on primary transmitter sensing,assuming that primary receivers are in close proximity to theprimary transmitter. In this section, we evaluate the effectof a hidden primary receiver on the resulting interference toprimary receivers.

Consider a primary and CR coexisting systems depicted inFig. 2, where an IR with radius 𝑅 centered at the primarytransmitter is introduced. All CR transmitters are distributed inthe shaded concentric ring with inner radius𝑅 and outer radius𝑙. Let 𝜃 be the angle between the line joining the primarytransmitter (in brackets) and a CR transmitter and the linejoining the primary transmitter-receiver pair (in brackets). Thedistance from the CR transmitter to the primary transmitter is𝑟 and the distance between the primary transmitter-receiverpair is 𝑟p. Then, the distance between the CR transmitter andthe primary receiver 𝑟cp can be expressed as

𝑟cp(𝑟, 𝜃) =[𝑟2 + 𝑟2p − 2𝑟𝑟pcos𝜃

] 12 , 𝑟 ∈ [𝑅, 𝑙]; 𝜃 ∈ [0, 2𝜋]

(28)where 𝑟 is distributed as in (13) and 𝜃 is uniformly distributedin [0, 2𝜋] if a Poisson point process is assumed for the CRtransmitter distribution.

A. Power Control

Under the power control scheme proposed in Section II.Aand the system model given in Fig. 2, the characteristicfunction of aggregate interference 𝜙Y(𝜔) can be written asfollows (see Appendix D for the detailed derivation):

𝜙Y(𝜔)= lim𝑙→∞

exp

{𝜆

∫𝐻

𝑓h(ℎ)

∫ 𝑟pwc

0

𝑓cc(𝑥)

∫ 2𝜋

0

∫ 𝑙

𝑅

𝑒𝑖𝜔

(𝑟

𝑟pwc

)𝛼𝑃max(𝑥)𝑔(𝑟cp(𝑟,𝜃))ℎ𝑟− 𝑟 𝑑𝑟 𝑑𝜃 𝑑𝑥 𝑑ℎ

+ 𝜆

∫𝐻

𝑓h(ℎ)

∫ ∞

𝑟pwc

𝑓cc(𝑥)

∫ 2𝜋

0

∫ 𝑙

𝑅

𝑒𝑖𝜔𝑃max(𝑥)𝑔(𝑟cp(𝑟,𝜃))ℎ𝑟 − 𝑟 𝑑𝑟 𝑑𝜃 𝑑𝑥 𝑑ℎ}.

(29)

Applying the log-normal approximation method used in


Section IV, we obtain the 𝑘th cumulant of interference as

𝑘𝑛 = lim𝑙→∞

𝜆

{∫𝐻

𝑓h(ℎ)

∫ 𝑟pwc

0

𝑓cc(𝑥)

∫ 2𝜋

0

∫ 𝑙

𝑅

(𝑟𝛼𝑃max(𝑥)𝑔(𝑟cp(𝑟, 𝜃))ℎ)𝑛

𝑟pwc𝑛𝛼

𝑟𝑑𝑟 𝑑𝜃 𝑑𝑥 𝑑ℎ

+

∫𝐻

𝑓h(ℎ)

∫ ∞

𝑟pwc

𝑓cc(𝑥)

∫ 2𝜋

0

∫ 𝑙

𝑅

[𝑃max(𝑥)𝑔(𝑟cp(𝑟, 𝜃))ℎ]𝑛𝑟𝑑𝑟 𝑑𝜃 𝑑𝑥 𝑑ℎ} .

(30)

As can be seen from (30), unlike (26), the 𝑘th cumulant doesnot have a closed-form expression. However, the complexity ofobtaining the exact interference PDF from (30) is still smallerthan that of the numerical method in Section III.

An experiment is done in Fig. 6(a) to examine the effect ofhidden primary receiver on the resulting interference comparedto the interference for the case of perfect knowledge ofprimary receiver location. It can be seen from the figure thatthe hidden primary receiver problem boosts the interference interms of increased interference mean and variance. This figurealso shows that the log-normal approximation stillfitswellwithboth the derivedCDF and Monte Carlo simulations.


Under the contention control scheme proposed in Sec-tion II.B and the system model given in Fig. 2, the characteris-tic function of aggregate interference 𝜙𝑌 (𝜔) can be expressedas

𝜙𝑌 (𝜔) = lim𝑙→∞

exp{𝑞mh𝜆𝜋𝐷𝑙

(𝐸(𝑒𝑖𝜔𝑝𝑔(𝑉 )ℎ

)− 1)}

= lim𝑙→∞

exp {𝑞mh𝜆𝜋𝐷𝑙

×(∫𝐻

𝑓h(ℎ)

∫ 2𝜋

0

∫ 𝑙

𝑅

𝑟𝑒𝑖𝜔𝑝𝑔(𝑟cp(𝑟,𝜃))ℎ

𝜋𝐷𝑙𝑑𝑟 𝑑𝜃 𝑑ℎ− 1

)}

= lim𝑙→∞

exp

{𝑞mh𝜆

∫𝐻

𝑓h(ℎ)

∫ 2𝜋

0

∫ 𝑙

𝑅

𝑟𝑒𝑖𝜔𝑝𝑔(𝑟cp(𝑟,𝜃))ℎ − 𝑟 𝑑𝑟 𝑑𝜃 𝑑ℎ}

(31)

with 𝐷𝑙 = 𝑙2 −𝑅2.Using the same log-normal approximation method as in

Section IV, the 𝑘th cumulant of interference can be writtenas

𝑘𝑛= lim𝑙→∞

𝑞mh𝜆

∫𝐻

𝑓h(ℎ)

∫ 2𝜋

0

∫ 𝑙

𝑅

[𝑝𝑔(𝑟cp(𝑟, 𝜃))ℎ]𝑛𝑟− 𝑟𝑑𝑟𝑑𝜃𝑑ℎ.

(32)

The effect of hidden primary receiver under contentioncontrol is evaluated in Fig. 6(b), where a pathloss-only channelis assumed. As we can see from this figure, the uncertaintyabout the primary receiver location leads to interference withlarger mean and variance as compared to that in the case withperfect knowledge of primary receiver location. Moreover, itcan be seen from this figure that the log-normal fitting forthe interference is fairly accurate compared to the MonteCarlo simulations. Thus, the approximation approach is stillapplicable in this scenario.

2 4 6 8 10 12

x 10−8

0

0.2

0.4

0.6

0.8

1


CD

F

(a)

Perfect primary Rx knowledgeMonte Carlo simulationHidden primary Rx (approx.)Hidden primary Rx (derived)

2 4 6 8 10 12

x 10−8

0

0.2

0.4

0.6

0.8

1


CD

F

(b)

Perfect primary Rx knowledgeMonte Carlo simulationHidden primary Rx (approx.)Hidden primary Rx (derived)

Fig. 6. Log-normal approximation for interference distribution witha hidden primary receiver under (a) power control (𝑅 =200 m, 𝜆 =3user/104m2, 𝛽 =4, 𝑟pwc = 20 m, 𝛼 = 4, 𝑃max = 1 W and 𝑟𝑝 =0.5𝑅) or (b) contention control (𝑅 =200 m, 𝜆 =3 user/104m2, 𝛽 =4,𝑑min = 20 m, 𝑝 = 1 W and 𝑟𝑝 = 0.5𝑅).

VI. NUMERICAL RESULTS AND ANALYSIS

The aggregate interference power from CR transmittersemploying power control or contention control is investigatednumerically in this section. For the power control scheme,Fig. 7(a) shows the effect of different power control param-eters on their resulting aggregate interference. The detailedsetup for the initial power control scheme is as follows: themaximum transmission power for each CR transmitter 𝑃max =1 W, the density of CR transmitter 𝜆 = 3 user/104m2, the IRradius 𝑅 = 100 m, the power control range 𝑟pwc = 20 m,the pathloss exponent 𝛽 = 4 and the power control exponent𝛼 = 4. From the two rightmost PDFs in this figure, it canseen that introducing power control scheme actually shifts theinterference distribution leftwards compared to the distributionwithout power control. It means that the power control scheme


can reduce the interference experienced at the primary receiverin terms of reducing its mean and slightly decreasing itsvariance. When deploying a CR network under the powercontrol scheme, its resulting interference can be controlledby manipulating several parameters including 𝑃max, 𝑟pwc, 𝜆,and 𝑅. It can be seen in Fig. 7(a) that the interference can bereduced by either decreasing the maximum transmission powerand/or CR density, or increasing the power control rangeand/or IR radius. Interestingly, it also suggests that adjustingthe IR radius is an effective way to control the interference,since the interference is more sensitive to the IR radius than toany other parameter as demonstrated in Fig. 7(a). Meanwhile,the interference is least sensitive to the CR user density in thesense that halving 𝜆 leads to higher interference compared todoubling 𝑟pwc, halving 𝑃max or doubling 𝑅.

For the contention control scheme, the impact of contentioncontrol parameters on the resulting interference is depictedin Fig. 7(b), whose initial setup is the same as that ofFig. 7(a) except that the transmission power for each CRtransmitter is 𝑝 = 1 W and the contention control range is𝑑min = 20 m. It can be seen from the two rightmost PDFsin Fig. 7(b) that the contention control scheme results in aninterference distribution with reduced mean like the powercontrol scheme in Fig. 7(a). Meanwhile, the interference canbe reduced by decreasing 𝑝, 𝜆, and/or increasing 𝑅 or 𝑑min.It can be observed by comparing Fig. 7(b) with Fig. 7(a)that (i) increasing the IR radius is an effective approach toreduce the interference for both the power and contentioncontrol schemes. However, the power control scheme is moresensitive to the IR radius than the contention control one;(ii) reducing the transmission power and/or CR transmitterdensity affects the interference in the very similar manner forthese two control schemes.

Finally, the impact of shadow fading on the aggregate in-terference is investigated for different values of the Nakagamishaping factor 𝑚 under power and contention control schemes,respectively, in Figs. 8(a) and 8(b). The initial setup in thisexample is the same as the one used for Figs. 7(a) and7(b), except that the standard variance is 𝜎Ω = 4 dB. When𝑚 = 1 the interfering channel becomes a Rayleigh channel,which is dominated by the log-normal shadowing. Whereas,when 𝑚 = 100 the fluctuations of the channel are reducedsignificantly compared to the Rayleigh fading channel. Onefact observed in Fig. 8 is that the interference distributionshave larger variance and heavier tails when shadow fadingis incorporated for both control schemes. Interestingly, fadingtends to make the interference distribution more heavy-tailedthan shadowing, i.e., the interference under shadowing hasbetter outage property than that under fading. Moreover, theshadow fading has the similar effect for both control schemes,which agrees with the analysis in Section IV.

VII. CONCLUSIONS

Interference at a primary receiver caused by CR trans-mitters with different interference management mechanismsincluding power control, contention control, and hybridpower/contention control schemes has been characterized.The PDFs of interference for the first two mechanisms havebeen evaluated analytically while, the interference distribution

0 0.5 1 1.5 2

x 10−7

0

0.5

1

1.5

2

x 108


PD

F

(a)

Original power controlPDF without power controlHalved transmission powerHalved densityDoubled power control rangeDoubled IR radius

0 0.5 1 1.5 2

x 10−7

0

0.5

1

1.5

2

x 108


PD

F

(b)

Original contention controlPDF without contention controlHalved tansmission powerHalved densityDoubled contention rangeDoubled IR radius

Fig. 7. Impact of various CR deployment parameters on the aggregateinterference for CR networks with (a) power control (𝑅 =100 m,𝜆 =3 user/104m2, 𝛽 =4, 𝑟pwc = 20 m, 𝛼 = 4 and 𝑃max = 1W) or (b) contention control (𝑅 =100 m, 𝜆 =3 user/104m2, 𝛽 =4,𝑑min =20 m, and 𝑝 = 1 W).

under the hybrid power/contention control has been studiednumerically. It has been found that the proposed powercontrol and contention control schemes are two effectiveapproaches to alleviate interference caused by CR transmitters.The hybrid control scheme causes higher interference to aprimary receiver, but leads to larger CR coverage as comparedto either power or contention control schemes. Then, theinterference distributions for power and contention controlschemes have been approximated by log-normal distributionswith greatly reduced complexity using the cumulant-basedapproach. Furthermore, the effect of a hidden primary receiveron the perceived interference has also been investigated forthe primary receiver. Numerical studies have demonstrated the


0 0.5 1 1.5 2 2.5 3 3.5

x 10−7

0

0.5

1

1.5

2

2.5

3x 10

7


PD

F

(a)

Pathloss (PL) onlyPL + shadow fading m=1PL + shadow fading m=100

0 0.5 1 1.5 2 2.5 3 3.5

x 10−7

0

0.5

1

1.5

2

2.5

3x 10

7


PD

F

(b)

Pathloss (PL) onlyPL + shadow fading m=1PL + shadow fading m=100

Fig. 8. Impact of shadow fading on the aggregate interference forCR networks (𝑅 =100 m, 𝜆 =3 user/104m2, 𝛽 =4) with (a) powercontrol (𝑟pwc = 20 m, 𝛼 = 4 and 𝑃max = 1 W) or (b) contentioncontrol (𝑑min =20 m and 𝑝 = 1 W).

impact of some CR deployment parameters on the resultingaggregate interference under power and contention controlschemes. It has been shown that increasing the IR radius is aneffective way to reduce the interference. Moreover, the powercontrol scheme is more sensitive to the IR radius than thecontention control counterpart. Finally, the impact of shadowfading on the aggregate interference has been analyzed as well.

APPENDIX

A. Derivation of (15)Substituting (6) and (7) into (14), we have

𝜙Y(𝜔)=exp

{𝜆𝜋

∫𝐻

𝑓h(ℎ)

∫𝑟cc

𝑓cc(𝑥)[𝑅2(

1−𝑒𝑖𝜔𝑔(𝑅)𝑝pwc(𝑥)ℎ)

+𝑖𝜔𝑝pwc(𝑥)ℎ

∫ 𝑔(𝑅)

0

(𝑔−1(𝑡))2𝑒𝑖𝜔𝑡𝑝pwc(𝑥)ℎ𝑑𝑡

]𝑑𝑥 𝑑ℎ

}

= exp

{𝜆𝜋

∫𝐻

𝑓h(ℎ)

∫ 𝑟pwc

0

𝑓cc(𝑥)[𝑅2(

1−𝑒𝑖𝜔( 𝑥𝑟pwc

)𝛼𝑃max𝑔(𝑅)ℎ)

+𝑖𝜔𝑥𝛼𝑃maxℎ

𝑟𝛼pwc

∫ 𝑔(𝑅)

0

(𝑔−1(𝑡))2𝑒𝑖𝜔𝑡( 𝑥

𝑟pwc)𝛼𝑃maxℎ𝑑𝑡

]𝑑𝑥𝑑ℎ

+𝜆𝜋

∫𝐻

𝑓h(ℎ)

∫ ∞

𝑟pwc

𝑓cc(𝑥)[𝑅2(

1 − 𝑒𝑖𝜔𝑔(𝑅)𝑃maxℎ)

+𝑖𝜔𝑃maxℎ

∫ 𝑔(𝑅)

0

(𝑔−1(𝑡))2𝑒𝑖𝜔𝑡𝑃maxℎ𝑑𝑡

]𝑑𝑥 𝑑ℎ

}.

(33)

Using (1) and (33), the characteristic function (15) is obtained.B. Derivation of (22)

Following similar steps as in [16], the characteristic functionof the aggregate interference can be expressed as

𝜙Y(𝜔) = lim𝑙→∞

𝑒𝜆𝜋(𝑙2−𝑅2)(𝑄−1) (34)

where

𝑄 = 𝐸(𝑒𝑖𝜔𝑃𝑔(𝑉 )𝐻

)=

∫𝐻

𝑓h(ℎ)

∫ 𝑙

𝑅

𝐸[𝑒𝑖𝜔𝑃𝑔(𝑟)ℎ

] 2𝑟

𝑙2 −𝑅2𝑑𝑟 𝑑ℎ

=

∫𝐻

𝑓h(ℎ)

∫ 𝑙

𝑅

[(1 − 𝑞mh) + 𝑞mh𝑒

𝑖𝜔𝑝𝑔(𝑟)ℎ] 2𝑟

𝑙2 −𝑅2𝑑𝑟 𝑑ℎ

= 1 − 𝑞mh + 𝑞mh

∫𝐻

𝑓h(ℎ)

∫ 𝑙

𝑅

𝑒𝑖𝜔𝑝𝑔(𝑟)ℎ2𝑟

𝑙2 −𝑅2𝑑𝑟 𝑑ℎ.

(35)

When 𝑙 → ∞, the integral in the last equality of (35) can bewritten as

lim𝑙→∞

∫𝐻

𝑓h(ℎ)

∫ 𝑙

𝑅

𝑒𝑖𝜔𝑝𝑔(𝑟)ℎ2𝑟

𝑙2 − 𝑅2𝑑𝑟 𝑑ℎ

= 1 +1

𝑙2 −𝑅2

∫𝐻

𝑓h(ℎ)𝑇 (𝜔𝑝ℎ)𝑑ℎ (36)

where 𝑇 (𝜔𝑝ℎ) is given in (12). Substituting (35) and (36) into(34), we obtain (22).

C. Derivation of (26)

From (15) and (25), we have

𝑘𝑛 =𝜆𝜋

𝑖𝑛

∫𝐻

𝑓h(ℎ)

∫ 𝑟pwc

0

𝑓cc(𝑥)

[−𝑅2

(𝑖𝑥𝛼𝑃max𝑔(𝑅)ℎ

𝑟pwc𝛼

)𝑛

+𝑛 (𝑖𝑥𝛼𝑃maxℎ)

𝑛

𝑟pwc𝑛𝛼

∫ 𝑔(𝑅)

0


]𝑑𝑥 𝑑ℎ

+𝜆𝜋

𝑖𝑛

∫𝐻

𝑓h(ℎ)

∫ ∞

𝑟pwc

𝑓cc(𝑥)[−𝑅2 (𝑖𝑃max𝑔(𝑅)ℎ)

𝑛

+𝑛 (𝑖𝑃maxℎ)𝑛∫ 𝑔(𝑅)

0


]𝑑𝑥 𝑑ℎ

= 𝜆𝜋

∫𝐻


(𝑛

𝑛− 2𝛽

𝑔𝑛−2𝛽 (𝑅)−𝑅2𝑔𝑛(𝑅)

)

×[∫ 𝑟pwc

0

𝑓cc(𝑥)(𝑥𝛼𝑃max)𝑛

𝑟pwc𝑛𝛼

𝑑𝑥+

∫ ∞

𝑟pwc

𝑓cc(𝑥)𝑃𝑛max𝑑𝑥

]


=2𝜆𝜋𝑃𝑛max

(𝑛𝛽 − 2)𝑅𝑛𝛽−2

∫𝐻


×(∫ 𝑟pwc

0

𝑓cc(𝑥)𝑥𝑛𝛼

𝑟pwc𝑛𝛼

𝑑𝑟 +

∫ ∞

𝑟pwc

𝑓cc(𝑥)𝑑𝑥

).

(37)

The first equality of (37) is obtained based on the followingfact [

∂𝑛

∂𝜔𝑛

]𝜔=0

𝑒𝑎𝜔 =

[∂𝑛

∂𝜔𝑛

]𝜔=0

∞∑𝑖=0

(𝑎𝜔)𝑛

𝑛!= 𝑎𝑛. (38)

In the last equality of (37), the first integral can be expressedas ∫

𝐻

𝑓h(ℎ)ℎ𝑛𝑑ℎ = 𝑒𝑛𝜇+𝑛2𝜎2

2 (39)

with 𝜇 and 𝜎2 given in (3) and (4), respectively. Also, the sumof the last two integrals in (37) can be simplified as∫ 𝑟pwc

0

𝑓cc(𝑥)𝑥𝑛𝛼

𝑟pwc𝑛𝛼

𝑑𝑥 +

∫ ∞

𝑟pwc

𝑓cc(𝑥)𝑑𝑥

=𝑛𝛼(𝑛𝛼− 2) ⋅ ⋅ ⋅ 2𝑟pwc

𝑛𝛼(2𝜋𝜆)𝑛𝛼2

(1 − 𝑒−𝜆𝜋𝑟pwc

2)

−𝑛𝛼2 −1∑𝑖=1

𝑛𝛼(𝑛𝛼− 2) ⋅ ⋅ ⋅ (𝑛𝛼− 2𝑖+ 2)

(2𝜋𝜆𝑟pwc2)𝑖

𝑟pwc𝑛𝛼−2𝑖𝑒−𝜆𝜋𝑟pwc

2

.

(40)

Substituting (39) and (40) into (37) yields (26).D. Derivation of (29)

𝜙Y(𝜔) = lim𝑙→∞

exp{𝜆𝜋𝐷𝑙

(𝐸(𝑒𝑖𝜔𝑝pwc𝑔(𝑉 )ℎ

)−1)}

= lim𝑙→∞

exp

{𝜆𝜋𝐷𝑙

[∫𝐻

𝑓h(ℎ)

∫ ∞

0

𝑓cc(𝑥)

∫ 2𝜋

0

∫ 𝑙

𝑅

𝑟𝑒𝑖𝜔𝑝pwc(𝑥)𝑔(𝑟cp(𝑟,𝜃))ℎ

𝜋𝐷𝑙𝑑𝑟 𝑑𝜃 𝑑𝑥 𝑑ℎ−1

]}

= lim𝑙→∞

exp

{𝜆

∫𝐻

𝑓h(ℎ)

∫ ∞

0

𝑓cc(𝑥)

∫ 2𝜋

0

∫ 𝑙

𝑅

𝑟𝑒𝑖𝜔𝑝pwc(𝑥)𝑔(𝑟cp(𝑟,𝜃))ℎ − 𝑟 𝑑𝑟 𝑑𝜃 𝑑𝑥 𝑑ℎ}

(41)

with 𝐷𝑙 = 𝑙2 − 𝑅2. The first equality in (41) is obtained inthe same way as (34) and (35). Equation (29) can be obtainedimmediately from (41).

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Zengmao Chen received his B.Sc. degree fromNanjing University of Posts & Telecommunications(NUPT), China, and his M.Eng. degree from Bei-jing University of Posts and Telecommunications(BUPT), China, in 2003 and 2006, respectively, andhis joint Ph.D. degree from Heiot-Watt Universityand the University of Edinburgh, UK, in 2011.

Dr Chen is now a Post-doc Research Associatewith the Joint Research Institute for Signal and Im-age Processing, Heriot-Watt University, UK. From2006 to 2007, he worked as a Research & Design

Engineer in Freescale Semiconductor (China) Ltd. His research interests in-clude: cognitive radio networks, MIMO communication systems, interferencemodeling, and interference cancellation.

Cheng-Xiang Wang (S’01-M’05-SM’08) receivedthe BSc and MEng degrees in Communicationand Information Systems from Shandong University,China, in 1997 and 2000, respectively, and the Ph.D.degree in Wireless Communications from AalborgUniversity, Denmark, in 2004. He has been withHeriot-Watt University, Edinburgh, UK, since 2005,first as a Lecturer, then as a Reader in 2009, andthen was promoted to a Professor in Wireless Com-munications since 2011. He is also an HonoraryFellow of the University of Edinburgh, UK, a Chair

Professor of Shandong University, a Guest Professor of Huazhong Universityof Science and Technology, an Adjunct Professor of Guilin University ofElectronic Technology, and a Guest Researcher of Xidian University, China.He was a Research Fellow at the University of Agder, Grimstad, Norway, from2001-2005, a Visiting Researcher at Siemens AG-Mobile Phones, Munich,Germany, in 2004, and a Research Assistant at Technical University ofHamburg-Harburg, Hamburg, Germany, from 2000-2001. His current researchinterests include wireless channel modeling and simulation, green communi-cations, cognitive radio networks, vehicular communication networks, mobilefemtocell networks, cooperative (relay) MIMO communications, and (beyond)4G wireless communications. He has edited 1 book and published one bookchapter and over 150 papers in refereed journals and conference proceedings.He is leading several projects funded by EPSRC, Mobile VCE, and industries,including the RCUK funded UK-China Science Bridges: R&D on (B)4GWireless Mobile Communications. Dr Wang is currently serving as anAssociate Editor for IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY,an Editor for Wireless Communications and Mobile Computing Journal (JohnWiley & Sons) and Security and Communication Networks Journal (JohnWiley & Sons), and a Scientific Board Member for InTech (Open Access Pub-lisher). He also served as an Editor for IEEE TRANSACTIONS ON WIRELESSCOMMUNICATIONS (2007-2009) and Hindawi Journal of Computer Systems,Networks, and Communications (2007-2011). He was the leading Guest Editorfor IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, SpecialIssue on Vehicular Communications and Networks. He served or is serving asa TPC member, TPC Chair, and General Chair for more than 60 internationalconferences. He received the IEEE Globecom’10 Best Paper Award in 2010and the IEEE ICCT’11 Best Paper Awards in 2011. Dr Wang was listed inthe Dictionary of International Biography 2008 and 2009, Who’s Who in theWorld 2008 and 2009, Great Minds of the 21st Century 2009, and 2009 Manof the Year. He is a Senior Member of the IEEE, a member of the IET, aFellow of the HEA, and a member of EPSRC Peer Review College.

Xuemin Hong received his BSc degree in Com-munication Engineering from Beijing InformationScience and Technology University, China, in 2004and his Ph.D. degree in Wireless Communicationsfrom Heriot-Watt University, UK, in 2008. SinceJuly 2011, he has been an Associate Professor withthe School of Information Science and Technology,Xiamen University, China. From August 2009 toJuly 2011, he was a Post-doc Research Associatewith the Joint Research Institute for Signal and Im-age Processing, Heriot-Watt University, UK. From

January 2009 to July 2009, he was a Post-doc Research Fellow with theDepartment of Electrical and Computer Engineering, University of Waterloo,Canada. From 2004 to 2005, he was affiliated with the Centre for Telecom-munication Research, King’s College London, UK.

Dr Hong’s research interests include MIMO and cooperative systems,wireless radio channel modeling, cognitive radio networks, and (beyond) 4Gwireless communications. He has published more than 20 technical papers inmajor international journals and conferences and 1 book chapter in the areaof wireless communications.

John S. Thompson received his BEng and Ph.D.degrees from the University of Edinburgh in 1992and 1996, respectively. From July 1995 to August1999, he worked as a postdoctoral researcher atEdinburgh, funded by the UK Engineering andPhysical Sciences Research Council (EPSRC) andNortel Networks. He was appointed as a lecturerat what is now the School of Engineering at theUniversity of Edinburgh in 1999. He was recentlypromoted to a personal chair in Signal Processingand Communications.

Dr John S. Thompson’s research interests currently include energy efficientcommunications systems, antenna array techniques and multi-hop wirelesscommunications. He has published over 200 papers to date including anumber of invited papers, book chapters and tutorial talks, as well as co-authoring an undergraduate textbook on digital signal processing. He is overallproject leader for the £1.2M EPSRC Islay project involving four universitieswhich investigates efficient hardware implementation of complex algorithms.He is also the academic deputy leader for the £2M EPSRC/Mobile VCEGreen Radio project, which involves a number of international communicationcompanies. He is currently editor-in-chief of the IET Signal Processing journaland was a technical programme co-chair for the Globecom conference inMiami in December 2010.

Sergiy A. Vorobyov (M’02-SM’05) received theM.Sc. and Ph.D. degrees in systems and controlfrom Kharkiv National University of Radio Elec-tronics, Ukraine, in 1994 and 1997, respectively.

Since 2006, he has been with the Department ofElectrical and Computer Engineering, University ofAlberta, Edmonton, AB, Canada where he becomean Associate Professor in 2010 and Full Professor in2012. Since his graduation, he also occupied variousresearch and faculty positions in Kharkiv NationalUniversity of Radio Electronics, Ukraine; Institute

of Physical and Chemical Research (RIKEN), Japan; McMaster University,Canada Duisburg-Essen University and Darmstadt University of Technology,Germany; and Joint Research Institute between the Heriot-Watt and EdinburghUniversities, U.K. His research interests include statistical and array signalprocessing, applications of linear algebra, optimization, and game theorymethods in signal processing and communications, estimation, detection, andsampling theories, and cognitive systems.

He is a recipient of the 2004 IEEE Signal Processing Society Best PaperAward, 2007 Alberta Ingenuity New Faculty Award, 2011 Carl Zeiss Award(Germany), and other research awards. He was an Associate Editor for theIEEE TRANSACTIONS ON SIGNAL PROCESSING (2006–2010) and for theIEEE SIGNAL PROCESSING LETTERS (2007–2009). He is a member ofSensor Array and Multi-Channel Signal Processing and Signal Processingfor Communications and Networking Technical Committees of IEEE SignalProcessing Society. He has served as a Track Chair in Asilomar’11 and as aTechnical Co-chair in IEEE CAMSAP’11.


Xiaohu Ge (M’09-SM’11) is currently a Professorwith the Department of Electronics and InformationEngineering at Huazhong University of Science andTechnology (HUST), China. He received his PhDdegree in Communication and Information Engi-neering from HUST in 2003. He has worked atHUST since Nov. 2005. Prior to that, he worked asan assistant researcher at Ajou University (Korea)and Politecnico Di Torino (Italy) from Jan. 2004 toOct. 2005. He was a visiting researcher at Heriot-Watt University, Edinburgh, UK from June to Au-

gust 2010. His research interests are in the area of mobile communications,traffic modeling in wireless networks, green communications, and interferencemodeling in wireless communications. He has published about 50 papers inrefereed journals and conference proceedings and has been granted about10 patents in China. He is leading several projects funded by NSFC,China MOST, and industries. He is taking part in several international jointprojects, such as the RCUK funded UK-China Science Bridges: R&D on(B)4G Wireless Mobile Communications and the EU FP7 funded project:Security, Services, Networking and Performance of Next Generation IP-basedMultimedia Wireless Networks.

Dr Ge is currently serving as an Associate Editor for International Journalof Communication Systems (John Wiley & Sons) and KSII Transactions onInternet and Information Systems. Since 2005, he has been actively involved inthe organisation of more than 10 international conferences, such as PublicityChair of IEEE Europecomm 2011 and Co-Chair of workshop of GreenCommunication of Cellular Networks at IEEE GreenCom 2010. He is a SeniorMember of the IEEE, a Senior member of the Chinese Institue of Electronics,a Senior member of the China Institute of Communications, and a memberof the NSFC and China MOST Peer Review College.

Hailin Xiao was born in 1976. He received theB.S. and M.S. degrees from Wuhan University andGuangxi Normal University in 1998 and 2004 re-spectively. He received his Ph.D. degree from theUniversity of Electronic Science and Technology ofChina (UESTC) in 2007. He was Professor in theSchool of Information and Communications, GuilinUniversity of Electronic Technology. Now he is aResearch Fellow with Joint Research Institute forSignal and Image Processing, School of Engineering& Physical Sciences, Heriot-Watt University. His

research interests include MIMO wireless communications, cooperative com-munications, and smart antenna techniques.

Feng Zhao was born in 1974. He received the Ph.D.degree in Communication and Information Systemfrom Shandong University, China, in 2007. Now heis a Research Associate in the School of Informationand Communications of Guilin University of Elec-tronic Technology. His research interests includewireless communication systems, information pro-cessing, and information security. He has publishedmore than 30 papers in journals and internationalconferences.

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