480 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL … · 480 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 2, FEBRUARY 2011 Cognitive Network Interference

480 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 2, FEBRUARY 2011

Cognitive Network InterferenceAlberto Rabbachin, Member, IEEE, Tony Q.S. Quek, Member, IEEE, Hyundong Shin, Member, IEEE, and

Moe Z. Win, Fellow, IEEE

Abstract—Opportunistic spectrum access creates the openingof under-utilized portions of the licensed spectrum for reuse,provided that the transmissions of secondary radios do not causeharmful interference to primary users. Such a system wouldrequire secondary users to be cognitive—they must accuratelydetect and rapidly react to varying spectrum usage. Therefore,it is important to characterize the effect of cognitive networkinterference due to such secondary spectrum reuse. In this paper,we propose a new statistical model for aggregate interference ofa cognitive network, which accounts for the sensing procedure,secondary spatial reuse protocol, and environment-dependentconditions such as path loss, shadowing, and channel fading.We first derive the characteristic function and cumulants ofthe cognitive network interference at a primary user. Usingthe theory of truncated-stable distributions, we then developthe statistical model for the cognitive network interference. Wefurther extend this model to include the effect of power controland demonstrate the use of our model in evaluating the systemperformance of cognitive networks. Numerical results show theeffectiveness of our model for capturing the statistical behavior ofthe cognitive network interference. This work provides essentialunderstanding of interference for successful deployment of futurecognitive networks.

Index Terms—Opportunistic spectrum access, cognitive ra-dio, cognitive network interference, detection-and-avoidance,truncated-stable distribution.

I. INTRODUCTION

W ITH the emergence of new wireless applications anddevices, there is a dramatic increase in the demand for

radio spectrum. Due to the scarcity of radio spectrum and theunder-utilization of assigned spectrum, government regulatorybodies such as the U.S. Federal Communications Commission(FCC) have started to review their spectrum allocation policies[1], [2]. Conventional rigid spectrum allocation forbids flexiblespectrum usage that severely hinders efficient utilization ofscarce spectrum since bandwidth demands vary along time and

Manuscript received 1 December 2009; revised 30 May 2010. This researchwas supported, in part, by the MIT Institute for Soldier Nanotechnologies, theOffice of Naval Research Presidential Early Career Award for Scientists andEngineers (PECASE) N00014-09-1-0435, and the National Science Founda-tion under grant ECCS-0901034, and the National Research Foundation ofKorea (KRF) grant funded by the Korea government (MEST) (No. 2010-0014773).A. Rabbachin is with the Institute for the Protection and Security of the

Citizen of the Joint Research Center, European Commission, 21027 Ispra,Italy (e-mail: [email protected]).T. Q. S. Quek is with the Institute for Infocomm Research, A∗STAR, 1

Fusionopolis Way, #21-01 Connexis South Tower, Singapore 138632 (e-mail:[email protected]).H. Shin is with the Department of Electronics and Radio Engineering,

Kyung Hee University, 1 Seocheon-dong, Giheung-gu, Yongin-si, Gyeonggi-do, 446-701 Korea (e-mail: [email protected]).M. Z. Win is with the Laboratory for Information & Decision Systems

(LIDS), Massachusetts Institute of Technology, 77 Massachusetts Avenue,Cambridge, MA 02139 (e-mail: [email protected]).Digital Object Identifier 10.1109/JSAC.2011.110219.

space dimensions. Therefore, opportunistic spectrum accesstogether with a cognitive radio (CR) technology has becomea promising solution to resolve this problem [3]–[7].Opportunistic spectrum access creates the opening of under-

utilized portions of the licensed spectrum for reuse, providedthat the transmissions of secondary radios do not causeharmful interference to primary users. For secondary users toaccurately detect and access the idle spectrum, CR has beenproposed as an enabling technology [3], [4], [7]. For example,if a communication channel is active between the primary andsecondary networks, the busy channel assessment can be basedon the detection of a preamble shared between the primary andsecondary networks or on the energy sensing of the primarynetwork radio signals [8]–[10]. Moreover, the CR network canimplement a detect-and-avoid protocol where the transmissionpower levels of the CR devices are based on the sensed powerof the primary network signals.Spectrum sharing is however challenging due to the un-

certainty associated with the aggregate interference in thenetwork. Such uncertainty can be resulted from the unknownnumber of interferers and unknown locations of the interferersas well as channel fading, shadowing, and other uncertainenvironment-dependent conditions [11], [12]. Therefore, itis crucial to incorporate such uncertainty in the statisticalinterference model in order to quantify the effect of thecognitive network interference on the primary network systemperformance.1 A unifying framework for characterizing thenetwork interference was proposed to investigate a varietyof issues involving aggregate interference generated asyn-chronously in a wireless environment subject to path loss,shadowing, and multipath fading [13], [14]. The originalmotivation for this work was to quantify the aggregate networkemission of randomly located ultra-wide bandwidth (UWB)radios [15]–[17] in terms of their spatial density [18]–[20].This framework has also been used to study the coexistenceissues in heterogeneous wireless networks [21]–[25]. A com-mon theme of all these work is the use of a Poisson pointprocess [26] for positions of the emitting nodes. The Poissonpoint process has been widely used in diverse fields such asastronomy [27], [28], positron emission tomography [29], cellbiology [30], optical communications [31]–[34], and wirelesscommunications [28], [35]–[40]. More recently, the Poissonmodel has been applied for spatial node distributions in avariety of wireless networks such as random access, ad hoc,relay, cognitive radio, or femtocell networks [41]–[52].To address the coexistence problem arisen by secondary

1Throughout this paper, we refer to the aggregate interference generated bysecondary users sharing the same spectrum with the primary user as cognitivenetwork interference.

0733-8716/11/$25.00 c© 2011 IEEE

RABBACHIN et al.: COGNITIVE NETWORK INTERFERENCE 481

cognitive networks, it is of great importance to accuratelymodel the aggregate interference generated by multiple ac-tive secondary users in the network. In [48], the momentexpression for the aggregate interference generated by Poissonnodes in an arbitrary area was derived assuming the typicalunbounded path-loss model. However, the unbounded path-loss model results in significant deviations from a realisticperformance [49]. For cognitive radio networks, the log-normal distribution was proposed to model the sum of allinterferers’ powers [45]. This log-normal approximation wasalso used for the aggregate interference at primary userswithout accounting for the channel uncertainty due to fading[46]. The optimal power control strategies for secondary userswere determined in [47] based on the Poisson model of theprimary network.In this paper, we propose a new statistical model for per-

dimension (real or imaginary part) aggregate interference of acognitive network, accounting for the sensing procedure, sec-ondary spatial reuse protocol, spatial density of the secondaryusers and environment-dependent conditions such as path loss,shadowing, and channel fading. Moreover, our framework al-lows us to model the cognitive network interference generatedby secondary users in a limited or finite region, taking intoaccount the shape of the region and the position of the primaryuser. As an example, we consider two types of secondaryspatial reuse protocols, namely, single-threshold and multiple-threshold protocols. For each protocol, we first express thecharacteristic function (CF) of the cognitive network inter-ference, from which we derive its cumulants. Using thesecumulants, we then model the cognitive network interferenceas truncated-stable random variables. We further extend thismodel to include the effect of power control and demonstratethe use of our model in evaluating a system performance suchas the bit error probability (BEP) in the presence of cognitivenetwork interference. Numerical results verify the validity ofour model in capturing the effect of the cognitive networkinterference in different scenarios.The paper is organized as follows. Section II presents the

system model. Section III derives the instantaneous inter-ference distribution and its truncated-stable model for eachsecondary spatial reuse protocol. Section IV demonstratesapplications of our statistical model for cognitive networkinterference. Section V provides numerical results to illustratethe effectiveness of our framework for characterizing thecoexistence between primary and secondary networks in termsof various system parameters. Section VI gives the conclusion.We relegate the glossary of statistical symbols used throughoutthe paper to Appendix A and the derivations of cumulants toAppendix B.

II. SYSTEM MODEL

For cognitive networks, the secondary users need to sensechannels before transmission in order not to cause harmfulinterference to a primary network. In this paper, we considerthe primary network in frequency division duplex mode.Therefore, to detect the presence of active primary users,the secondary user senses the primary users’ uplink channel.Furthermore, we consider the secondary network as a simple

ad-hoc network where secondary users join or exit the net-work, and sense or access the channel independently withoutcoordinating with other secondary users [53]–[55]. As such,there exists the possibility that secondary users can transmitat the same time regardless of their distances between eachother.2

A. Cognitive Network Activity Model

The activity of each secondary user depends on the strengthof the received uplink signal transmitted by the primaryuser. In the following, we consider two types of secondaryspatial reuse protocols, namely: single-threshold and multiple-threshold protocols.1) Single-Threshold Protocol: In this case, the ith sec-

ondary user is active if

KPpYi

R2bi

≤ β, (1)

or equivalently,

R−2bi Yi ≤ ζ, (2)

where β is the activating threshold; ζ � βKPp

is the normalizedthreshold; Pp is the transmitted power of the primary user; Yiis the squared fading path gain of the channel from the primaryuser to the ith secondary user; K is the gain accounting forthe loss in the near-field; Ri is the distance between theprimary and the ith secondary user; and b is the amplitudepass-loss exponent.3 We assume that Yi’s are independentand identically distributed (IID) with the common cumulativedistribution function (CDF) FY (·). Therefore, the activityof the secondary network users can be represented by theBernoulli random variable:

1[0,ζ]

(R−2bi Yi

) ∼ Bern (FYi (R2bi ζ)), (3)

with the indicator function defined as

1[p,q] (x) ={

1, if p ≤ x ≤ q,0, otherwise, (4)

where the value one of the Bernoulli variable denotes that thesecondary user is active.2) Multiple-Threshold Protocol: For this case, the transmis-

sion power of the secondary network users is set accordingto the detected power level of the primary network uplinksignal [56]. We consider N − 1 normalized threshold valuesζ1, ζ2, . . . , ζN−1 in increasing order to identify N differentclasses (or sets) of active secondary users, denoted by Ak,k = 1, 2, . . . , N . Let ζ0 = 0 and ζN = ∞. Then, the kthactive class Ak obeys the following activation rule:4

1[ζk−1,ζk]

(R−2bi Yi

) ∼ Bern(μ(pt)Yi

(0,R2b

i ζk−1,R2bi ζk)).

(5)

2When a more intelligent medium access control protocol that involvesa form of coordination or local information exchange is feasible for thesecondary network, our results can still serve as a worst-case scenario analysis.3For brevity, we assume that the noise effect is negligible on the primary

detection procedure as in [45].4The zeroth-order partial moment μ(pt)

X (0, l, u) of the random variable Xcan be written in terms of its CDF as

μ(pt)X (0, l, u) = FX (u) − FX (l) .


Note that the power of the received primary user’s signal at theactive secondary users in the class Ak is between KPpζk−1

and KPpζk.

B. Interference Model

The interference signal at the primary receiver generated bythe ith cognitive interferer can be written as

Ii =√PIR

−bi Xi, (6)

where PI is the interference signal power at the limit ofthe near-far region;5 Ri is the distance between the ithcognitive interferer and the primary receiver; and Xi is theper-dimension fading channel path gain of the channel fromthe ith cognitive interferer to the primary receiver.6 In thefollowing, we assume that Xi’s are IID with the commonprobability density function (PDF) fX (·), which are mutuallyindependent of Yi’s.We consider that the secondary users are spatially scattered

according to an homogeneous Poisson point process in atwo-dimensional plane R

2, where the victim primary user isassumed to be located at the center of the region. Let S ⊂ Z

+

be the index set of secondary users in a region R ⊂ R2. Then

the probability that k secondary users lie inside R dependsonly on the total area AR of the region, and is given by [26]

P {|S| = k} =(λAR)k

k!e−λAR , k = 0, 1, 2, . . . (7)

where λ is the spatial density (in nodes per unit area).Furthermore, we assume that the region R is constrained inthe annulus prescribed by two radius dmin and dmax, which areminimum and maximum distances from the primary receiver,respectively.7 This allows us to consider a scenario where thesecondary users are located within a limited region.

III. INSTANTANEOUS INTERFERENCE DISTRIBUTION

To characterize the cognitive network interference, we firstderive the cumulants for the cases of full network activity(all secondary users are active) and regulated activity (eachsecondary user is regulated by a spatial reuse protocol) inSection III-A and III-B, respectively. Using these cumulantexpressions, we develop the symmetric truncated-stable modelfor the cognitive network interference in Section III-C.

A. Full Activity

In this case, the cognitive network interference is generatedby all the secondary users present in the region R and can bewritten as

Ifa =√PI

∑i∈SR−bi Xi︸︷︷︸

Zfa(R)

. (8)

5We consider the near-far region limit at 1 meter.6Note that Xi = �{Hi}, where Hi is the complex path gain of the channel

from the ith cognitive interferer to the primary receiver.7Note that Ri in (6) can be smaller than 1. Therefore, the received

interference power can be larger than PI but it is finite since dmin > 0.

By using [14, Theorem 3.1], the CF of Zfa (R) can beexpressed as

ψZfa(R) (jω) = exp

(−2πλ

∫X

∫ dmax

dmin

[1 − exp

(jωxr−b

)]×fX (x) rdrdx

), (9)

where j =√−1. Using (9), we can then calculate the nth

cumulant of the interference Zfa (R) as follows:

κZfa(R) (n) =1jndn lnψZfa(R) (jω)

dωn

∣∣∣∣∣ω=0

= 2πλ∫X

∫ dmax

dmin

xnr1−nbfX (x) drdx

=2πλnb− 2

(d2−nbmin − d2−nb

max

)μX (n) . (10)

Using the cumulant of Zfa (R), we can obtain the nth cu-mulant of the cognitive network interference Ifa for the fullactivity case as follows:

κIfa (n) = Pn/2I κZfa(R) (n) . (11)

B. Regulated Activity

1) Single-Threshold Protocol: In this spatial reuse protocol,the activity of the secondary users in the region R is regu-lated by the single normalized threshold ζ according to (3).Therefore, the cognitive network interference for the single-threshold protocol can be written as

Ist =√PI

∑i∈Ast

R−bi Xi︸︷︷︸

Zst(ζ;R)

, (12)

where Ast defines the index set of active secondary users inthe region R:

Ast ={i ∈ S : 1[0,ζ]

(R−2bi Yi

)= 1}. (13)

Similar to (9), the CF of Zst (ζ;R) can be expressed as

ψZst(ζ;R) (jω)

= exp

(−2πλ

∫X

∫Y

∫ dmax

dmin

[1 − exp

(jωxr−b

)]

×1[0,ζ]

(r−2by

)fX (x) fY (y) rdrdydx

), (14)

from which the cumulant κZst(ζ;R) (n) is derived in Ap-pendix B-A. Using the cumulant of Zst (ζ;R), we can obtainthe nth cumulant of the cognitive network interference Ist forthe single-threshold protocol as follows:

κIst (n) = Pn/2I κZst(ζ;R) (n) . (15)

Remark 1: As ζ → ∞, the second and third terms in (38)vanish and hence,

limζ→∞

κIst (n) = κIfa (n) , (16)

as expected. Therefore, the full activity can be viewed as anextreme case of the single-threshold spatial reuse such thatζ → ∞.


2) Multiple-Threshold Protocol: Using (5), the per-classcognitive interference generated by the secondary users in Ak

can be written as

Imt,k =√PI,k

∑i∈Ak

R−bi Xi︸︷︷︸

Zk(R)

, (17)

where PI,k is the transmitted power of the secondary users inthe kth active class Ak and

Ak ={i ∈ S : 1[ζk−1,ζk]

(R−2bi Yi

)= 1}. (18)

The N power levels PI,1, PI,2, . . . , PI,N are set in decreasingorder such that users active in classes characterized by higherdetected power level of the primary signal transmit with lowerpower. Similar to (14), the CF of Zk (R) can be expressed as

ψZk(R) (jω) = exp

(−2πλ

∫X

∫Y

∫ dmax

dmin

[1 − exp

(jωxr−b

)]

× 1[ζk−1,ζk]

(r−2by


).

(19)

The cognitive network interference generated by the sec-ondary users in all the N classes is then given by

Imt =N∑k=1

√PI,kZk (R) . (20)

Since all the Zk (R)’s are statistically independent,8 weobtain the nth cumulant of the cognitive network interferenceImt for the multiple-threshold protocol as

κImt (n) =N∑k=1

Pn/2I,k κZk(R) (n) , (21)

where κZk(R) (n) are given by (40), (44), and (46) in Ap-pendix B-B for k = 1, k = 2, 3, . . . , N − 1, and k = N ,respectively.Remark 2: Using the cumulant expressions (10), (15), and

(21), we can characterize statistical properties (e.g., mean,variance, and other higher order statistics) of the cognitivenetwork interference for each secondary spatial reuse protocol.For example, the second-order cumulant can be used tomeasure the power of the cognitive network interference.

C. Truncated-Stable Distribution Model

The truncated-stable distributions are a relatively new classof distributions that follow from the class of stable distribu-tions [57]. The attractivenesses of using stable distributions tomodel interference in wireless networks are: 1) the ability tocapture the spatial distribution of the interfering nodes; and 2)the ability to accommodate heavy tail behavior exhibiting thedominant contribution of a few interferers in the vicinity of theprimary user [58]. However, as shown in [14], the aggregate

8The cumulants have the linear property for independent random variables,i.e., if X and Y are independent, then

κX+Y (n) = κX (n) + κY (n) .

interference converges to a stable distribution only if theinterferers are scattered in the entire plane. Stable distributionshave unbounded (infinite) second-order moment due to thesingularity at r = 0 and thus, care must be taken when usingthis model. The truncated-stable distributions have smoothedtails and finite moments, offering an alternative statistical toolto model the aggregate interference in more realistic scenarioswithout this singularity.The CF of a symmetric truncated-stable random variableT ∼ St (γ′, α, g) is given by [59]

ψT (jω)

= exp(γ′Γ (−α)

[(g − jω)α

2+

(g + jω)α

2− gα

]),

(22)

where Γ (·) is the Euler’s gamma function; and γ′, α andg are the parameters associated with the truncated-stabledistribution. The parameters γ′ and α are akin to the dispersionand the characteristic exponent of the stable distribution, re-spectively. The parameter g is the argument of the exponentialfunction used to smooth the tail of the stable distribution.The nth cumulant of the truncated-stable distribution can beobtained using (22) as

κT (n) =

{γ′Γ (−α) gα−n

∏n−1i=0 (α− i) , for even n

0, for odd n.(23)

For given α, using (23), the parameters γ′ and g can beexpressed in terms of the first two nonzero cumulants, namely,the second- and fourth-order cumulants.To model the cognitive network interference using the

truncated-stable distribution, we first fix the characteristicexponent to α = 2/b. This choice is motivated by the factthat as dmin → 0 and dmax → ∞, the cognitive networkinterference follows a stable distribution with the characteristicexponent α = 2/b. Let IA be the cognitive network interfer-ence corresponding to the activity model A ∈ {fa, st,mt},i.e., full activity, regulated activity with the single-thresholdprotocol, or the multiple-threshold protocol. Then, we canmodel the cognitive network interference IA as the symmetrictruncated-stable random variable, i.e.,

IA ∼ St (γ′A, α = 2/b, gA) , (24)

where the dispersion and smoothing parameters γ′A and gAare given in terms of the second and fourth cumulants of IAas

γ′A =κIA (2)

Γ (−α)α (α− 1)[κIA(2)(α−2)(α−3)

κIA(4)

]α−22

, (25)

gA =

√κIA (2) (α− 2) (α− 3)

κIA (4). (26)

To validate our statistical model, we consider an annulusregion defined by dmin = 1 meter, dmax = 60 meters,and λ = 0.1 users/m2. Both primary and secondary signalsexperience Rayleigh fading, i.e.,

√Yi ∼ Rayleigh (1/2) and

|Hi| ∼ Rayleigh (1/2). We consider the multiple-threshold


−60 −40 −20 0 20 40 60−60

−40

−20

0

20

40

60

meters

meters

Fig. 1. Node displacements of a CR network with the multiple-thresholdprotocol (not only a single realization snapshot). dmin = 1 meter, dmax = 60meters, λ = 0.1 users/m2, N = 3, ζ1 = −42 dBm, and ζ2 = −20 dBm.The green (asterisk), blue (circle), and red (diamond) colors (markers)represent the classes A1, A2, and A3, respectively.

protocol implemented using two thresholds (i.e., N = 3)with the following parameters: the secondary network userstransmit with power PI,1 = 0 dBm if the signal powercoming from the primary user is lower than ζ1 = −42 dBm,with power PI,2 = −23.7 dBm if the signal power comingfrom the primary user is between ζ1 and ζ2 = −20 dBm,and with power PI,3 = −38.7 dBm if the signal powercoming from the primary user is higher than ζ2. Fig. 1 showsrealization snapshots of active secondary users regulated bythis multiple-threshold protocol, while Figs. 2 and 3 showthe PDF and complementary CDF (CCDF) of the cognitivenetwork interference Imt. We can observe from Figs. 2 and 3that the simulation results match well with the truncated-stablestatistical model.With the symmetric truncated-stable model, we can also

account for shadowing in the characterization of the cogni-tive network interference. For example, consider the single-threshold protocol shadowing environment with obstacles suchthat the whole region R can be divided into different subre-gions R0, and R1,R2, . . . ,RL corresponding to the positionsof the obstacles. Due to shadowing, these L subregions expe-rience additional attenuation behind those obstacles. Then, thecognitive network interference can be written as

Ist =L∑�=0

√PI,�

∑i∈Ast,�

R−bi Xi

︸︷︷︸Zst(ζβ�;R�)

, (27)

where

Ast,� ={i ∈ S ∩R� : 1[0,ζβ�]

(R−2bi Yi

)= 1}. (28)

For � = 1, 2, . . . , L, PI,� and β� account for an additionalattenuation for the subregion R� behind the obstacle, andPI,0 = PI and β0 = 1. The CF of Zst

(ζβ�;R�) can be

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

0

0.05

0.1

0.15

simulationtruncated−stable

f Im

t(x

)

x

Fig. 2. PDF of the cognitive network interference Imt for the multiple-threshold protocol with the same parameters as in Fig. 1. PI,1 = 0 dBm forA1, PI,2 = −23.7 dBm for A2, and PI,3 = −38.7 dBm for A3.

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.210

−4

10−3

10−2

10−1

100

FI m

t(x

)

x

simulationtruncated-stable

Fig. 3. CCDF of the cognitive network interference Imt for the multiple-threshold protocol with the same parameters as in Fig. 1. PI,1 = 0 dBm forA1, PI,2 = −23.7 dBm for A2, and PI,3 = −38.7 dBm for A3.

expressed as

ψZst(ζβ�;R�) (jω)

= exp

(−θ�λ

∫X

∫Y

∫ b�

a�

[1 − exp

(jωxr−b

)]× 1[0,ζβ�]

(r−2by


).

(29)

where a� and b� are the limits of the subregion R�; and θ�is the angle covered by R�. If the obstacle is present, theangle θ� corresponds to the angle covered by the obstacle.For a single obstacle placed at distance d from the origin,we have two subregions in front and behind the obstacle:


−60 −40 −20 0 20 40 60−60

−40

−20

0

20

40

60meters

meters

Fig. 4. Node displacements of a CR network with the single-thresholdprotocol in the presence of shadowing (not only a single realization snapshot).dmin = 1 meter, dmax = 60 meters, λ = 0.01 users/m2 , ζ = −40 dBm,θ1 = θ2 = π/2, and β1 = β2 = 20 dB. The shadowing is characterizedby two obstacles present at 10 and 25 meters from the primary receiver,covering the angle of π/2, and causing additional attenuation of 20 dB. Theblue (circle) and green (asterisks) colors (markers) represent inactive andactive nodes, respectively.

(a1, b1) = (dmin, d) and (a2, b2) = (d, dmax), respectively.The nth cumulant of the cognitive network interference forthe single-threshold protocol in the presence of shadowing canbe written as

κIst (n) =L∑�=0

Pn/2I,� κZst(ζβ�;R�) (n) , (30)

where the cumulant κZst(ζβ�;R�) (n) is obtained from

κZst(ζ;R) (n) in (38) by replacing ζ, 2π, dmin, and dmax

with ζβ�, θ�, a�, and b�, respectively. Fig. 4 shows real-ization snapshots of active secondary users regulated by thesingle-threshold protocol with ζ = −40 dBm in the regionprescribed by dmin = 1 meter and dmax = 60 meters forλ = 0.01 users/m2. The shadowing is characterized bytwo obstacles present at 10 and 25 meters from the primaryreceiver, covering the angle of π/2 and causing additionalattenuation of 20 dB. Accordingly, we set θ1 = θ2 = π/2, andβ1 = β2 = 20 dB. Figs. 5 and 6 show the PDF and CCDF ofthe cognitive network interference Ist in this situation. Fromthese figures, we can observe again that the truncated-stablemodel captures a remarkably accurate statistical behavior ofthe cognitive network interference.

IV. APPLICATIONS

A. Effect of the Primary Network Power Control

Power control is often used in cellular systems to overcomethe near-far problem. If the primary network uses powercontrol, the transmitting power of the primary user variesdepending on the distance Rp and channel gain Hp betweenthe base station and primary receiver. Therefore, the transmitpower Pp is random and it is important to understand the effectof power control on the cognitive network interference. Under

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

0.01

0.02

0.03

0.04

0.05

0.06

f Ist

(x)

x


Fig. 5. PDF of the cognitive network interference Ist for the single-thresholdprotocol in the presence of shadowing with the same parameters as in Fig. 4.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.510

−4

10−3

10−2

10−1

100

FI s

t(x

)

x


Fig. 6. CCDF of the cognitive network interference Ist for the single-threshold protocol in the presence of shadowing with the same parameters asin Fig. 4.

perfect power control, Pp is set such that Pp|Hp|2/(R2bp ) ≥

P , where P is the minimum required power level. Fordiscrete power control, the set of possible power levels arefinite. Assuming that there are L possible transmit powerlevels P1, P2, . . . , PL, we have the following probability massfunction (PMF) for Pp at these power levels:

P {Pp = P�}

=

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

P

{P�R2b

p|Hp|2 ≤ P1

}, for � = 1,

P

{P�−1 <

P�R2bp

|Hp|2 ≤ P�

}, for � = 2, 3, . . . , L− 1,

P

{P�R2b

p|Hp|2 > PL

}, for � = L,

(31)


Fig. 7. Circular-section approximation of the non-circular region. The greensquare represents the primary user. Different colored sections correspond todifferent secondary user densities.

which can be determined empirically. In this case, the nthcumulant of the cognitive network interference for the single-threshold protocol can be written as

κIst (n) = E

{Pn/2I κ

Zst

“β

KPp;R

” (n)}

= Pn/2I

L∑�=1

P {Pp = P�} κZst

“β

KP�;R

” (n) . (32)

B. Effect of Secondary Interference Avoidance

Instead of allowing all the active secondary users in thesame class to transmit at the same power, we can also employsecondary power control, which will be effective in reducinginterference and improving power efficiency [60], [61]. Inaddition, we can effectively design a more power-efficientsecondary network if the knowledge of the secondary users’positions is available. For example, each secondary user avoidstransmitting using on-off power control if the average receivedsignal-to-noise ratio at its desired receiver is very low. Hence,with the location-awareness, we can regulate each secondaryuser to transmit only if its desired secondary receiver is withina certain maximum transmission range R, which correspondsto the maximum distance beyond which reliable transmissionis not possible. Let Ps and Rs be the random variablesthat represent the secondary transmit power and the distancefrom the intended receiver, respectively. Then, for the single-threshold spatial reuse protocol with power control, the nthcumulant of the cognitive network interference becomes:9

κIst (n) = μ√Ps

(n) κZst(ζ;R) (n) . (33)

9The nth moment μ√Ps

(n) depends on the power control and intendedreceiver selection strategies of the secondary network. For example, we havePs ∼ PI Bern (FRs (R�)) for the on-off secondary power control with themaximum transmission range R� . Hence,

μ√Ps

(n) = Pn/2I FRs (R�) ,

and μ√Ps

(n) → Pn/2I as R� → ∞ (no power control).

10 20 30 40 50 6010

15

20

25

30

−80

−60

−40

−20

meters

meters

Aggregate interference power [dBm]

Fig. 8. Aggregate interference power (dBm) generated by FBSs placed withdensity λ = 0.01 FBSs/m2 in the first and fourth apartments of the firstrow and in the third and fifth apartments of the second row for PI = 0 dBm,walls absorbing 20 dB of the radio signal, and |Hi| ∼ Nakagami (2, 1).

If the intended receiver is the nearest neighbor, then Rs ∼Rayleigh (1/ (2πλr)) follows from the properties of Poissonpoint processes, where λr is the density of secondary receivers.

C. Non-circular Regions

When the primary and secondary users are confined in alimited or finite region, the position of the primary user andthe shape of the region affect the distribution of the distancebetween the primary and secondary users and, therefore, alsothat of the aggregate interference. In the framework devel-oped in Sections II and III, we implicitly consider the polarcoordinate system and place the primary user at the centerof the region. This coordinate system is natural for analyzingthe interferers scattered in a circular section. To extend thisframework to a non-circular region, we can first divide the areaof interest into infinitesimal circular sections (see for example,Fig. 7) and use (30) to approximate the nth cumulant of thecognitive network interference. Using this approach, we canalso consider any position of the primary user, shadowing withmultiple obstacles, and areas with different densities within theregion of interest.Remark 3 (Femtocells): We can apply the approach for

non-circular regions to model the aggregate interference gen-erated by femtocell base stations (FBSs) in the macrocellnetworks [62]. Since the FBSs are randomly deployed withoutany coordination with the macrocell network, they can causeharmful interference to the macrocell users. For example,using (30) with the cumulants for the full network activity(10) instead of κZst(ζβ�;R�) (n), we can characterize thestatistics of the aggregate interference generated by the FBSsin any environment. In Fig. 8, the aggregate interferenceis calculated in one of the reference environments chosenin the femtocells standardization process. Each large squarerepresents a (10 × 10)-meter square apartment. Each smallsquare represents a point where the aggregate interferencepower is measured, which corresponds to the interferenceaffecting a macrocell user.

D. BEP Analysis

Consider a binary phase-shift keying (BPSK) narrowbandsystem in the presence of interference generated by the cogni-tive network confined within the regionR, where transmission


0 5 10 15 20

10−4

10−3

10−2

10−1

100

analisys

BEP

SIR = −8 dB

SIR = −12 dB

SIR = −16 dB

Eb/N0 [dB]

simulation

Fig. 9. BEP of BPSK versus Eb/N0 in the presence of the cognitive networkinterference Ist for the single-threshold protocol when SIR = −16, −12, and−8 dB. λ = 0.1 users/m2 and ζ = −40 dBm. For comparison, the BEP inthe absence of interference is also plotted (dashed line).

activities of the nodes are regulated according to (3). Thedecision variable of the primary received symbol after thecorrelation receiver can be written as

V = GU√Eb + Ist +W , (34)

where G is the channel fading affecting the victim signal;U ∈ {1,−1} is the information data; Eb is the energy perbit; Ist is the congnitive network interference; and W is thezero-mean additive white Gaussian noise with variance N0/2.Conditioned on G, Ist, and U = +1, the CF of the decisionvariable V can be written as

ψV(jω∣∣G, Ist,U = +1

)= exp

{jω(G√Eb + Ist

)− N0ω

2

4

}. (35)

Assuming that G and Ist are statistically independent, the CFof the decision variable conditioned on U = +1 is given by

ψV(jω∣∣U = +1

)= ψG

(jω√Eb

)ψIst (jω) exp

(−N0ω

2

4

).

(36)

For the cognitive network interference Ist, we use thesymmetric truncated-stable model Ist ∼ St (γ′st, α = 2/b, gst),where the parameters γ′st and gst are determined by using(25) and (26), respectively. Since Ist is approximated as asymmetric random variable, the average BEP is equal to theBEP conditioned on U = +1, which can be expressed, usingthe inversion theorem [63], as

Pe = P{V < 0

∣∣U = +1}

=12

+12π

∫ ∞

0

ψV(−jω∣∣U = +1

)−ψV (+jω∣∣U = +1)

jωdω.

(37)

10−4

10−3

10−2

10−1

10−6

10−5

10−4

10−3

10−2

10−1

100

λ = 0.1λ = 0.01λ = 0.001

Normalized activating threshold ζ

BEP

Fig. 10. BEP of BPSK as a function of the normalized activating threshold ζfor the single-threshold protocol when λ = 0.1, 0.01, and 0.001 users/m2.Eb/N0 = 10 dB and SIR = −10 dB. For comparison, the BEP in theabsence of interference is also plotted (dashed line).

V. NUMERICAL RESULTS

In this section, we illustrate the use of cognitive networkinterference model to provide insight into the coexistencebetween primary and secondary networks. In numerical ex-amples, we consider dmin = 1 meter, dmax = 60 meters, b= 1.5, and Rayleigh fading for both primary and secondarysignals unless differently specified. We first investigate theeffect of the cognitive network interference on the BEPperformance of the primary user. In Fig. 9, the BEP of BPSKversus Eb/N0 is depicted at the signal-to-interference ratioSIR � Eb/PI = −16, −12, and −8 dB when the secondarynetwork having density λ = 0.1 users/m2 employs the single-threshold protocol with ζ = −40 dBm. We can observe fromFig. 9 that the simulation agrees well with the analyticalresults, which confirms the BEP analysis in Section IV-D andagain validates the truncated-stable interference model.To ascertain the effect of the activating threshold and spatial

density of secondary users on the primary BEP performance,Fig. 10 shows the BEP of BPSK as a function of the normal-ized activity threshold ζ for the single-threshold protocol atEb/N0 = 10 dB and SIR = −10 dB when λ = 0.1, 0.01,and 0.001 users/m2. As expected, we can observe that theprimary BEP degrades severely as the node density λ and/orthe threshold ζ increase. For a given secondary density, ouranalytical framework enables us to design an activity thresholdthat guarantees a target BEP at the primary user.To demonstrate the effect of fading on the cognitive network

interference, we next consider Nakagami-m fading for bothprimary and secondary signals, i.e.,

√Yi ∼ Nakagami (m, 1)

and |Hi| ∼ Nakagami (m, 1). Fig. 11 shows the variance(or equivalently, average power) of the cognitive networkinterference Ist as a function of the maximum distance dmax

from the primary user for Nakagami fading parametersm = 1,3, and 5. The secondary network has the user density λ = 0.01users/m2, each transmits with PI = 0 dBm according tothe single-threshold protocol with ζ = −30 dBm. This


0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

−3

dmax (meters)

Var

{I st}

m = 1m = 3m = 5

Fig. 11. Variance of the cognitive network interference Ist for the single-threshold protocol as a function of the maximum distance dmax of theregion R when the Nakagami fading parameters m = 1, 3, and 5.λ = 0.01 users/m2, ζ = −30 dBm, PI = 0 dBm, and Nakagami-m fading for primary and secondary links

√Yi ∼ Nakagami (m, 1) and

|Hi| ∼ Nakagami (m, 1).

example reveals that for a fixed threshold ζ, as the fadingparameter m increases (less severe fading), the cognitivenetwork interference vanishes at the primary user due to raresecondary activity. We can also see that milder fading (i.e.,larger m) reduces the cognitive network interference powerfor all the values of dmax. This is due to the fact that milderfading decreases the activity of the secondary users in theproximity of the primary user, leading consequently to a lowercognitive interference power. Moreover, we observe that thecognitive network interference power tends to saturate as dmax

increases since secondary users located far from the primaryuser contribute marginally to aggregate interference.The effect of power control on the cognitive network

interference is illustrated in Fig. 12, where the variance ofthe cognitive network interference Ist for the single-thresholdprotocol as a function of the activating threshold β is depictedin the presence of primary power control. In this example,K = 0 dBm and the density and transmit power of thesecondary users are λ = 0.1 users/m2 and PI = 0 dBm,respectively. The primary user is distributed in a circular areadefined by minimum and maximum distances dminp = 1 meterand dmaxp = 1000 meters from the base station, respectively,and its communication link experiences Rayleigh fading, i.e.,|Hp| ∼ Rayleigh (1/2). For the primary power control policy,we set four power levels −5, −15, −25, −35 in dBm and theminimum required power level to P = −95 dBm. We can seefrom the figure that if the primary network uses power control,the variance of the cognitive network interference increasesfor all the values of β. This is due to the fact that when theprimary user is close to the base station, its transmission powerdecreases. As a consequence, the secondary users will increasetheir activity, leading to a larger number of active secondaryusers.In Fig. 13, the variance of the cognitive network interferenceIfa as a function of the maximum transmission range R

10−6

10−5

10−4

10−3

10−2

10−4

10−3

10−2

10−1

100

Var

{I st}

Activating threshold β

power control onpower control off

Fig. 12. Variance of the cognitive network interference Ist for the single-threshold protocol in the presence of primary power control as a function ofthe activating threshold β. λ = 0.1 users/m2 , K = 0 dBm, PI = 0 dBm,|Hp| ∼ Rayleigh (1/2), dminp = 1 meter, and dmaxp = 1000 meters. Thepower levels of the primary user are −5, −15, −25, and −35 dBm with theminimum required power level P � = −95 dBm.

of the secondary users for the case of full activity (i.e.,ζ → ∞) is depicted in the presence of the on-off secondarypower control for various values of λ. In this example,√Yi ∼ Nakagami (2, 1) and |Hi| ∼ Nakagami (2, 1). For

the secondary power control policy, we set PI = 0 dBm,Ps ∼ Bern (FRs (R)), Rs ∼ Rayleigh (1/ (2πλr)), andλr = λ. Hence, μ√

Ps(n) in (33) becomes 1−e−πλrR

�2, which

reveals that the interference power increases and approachesexponentially to one (i.e., PI = 0 dBm without powercontrol) as the transmission range R increases. We can seefrom Fig. 13 that the cognitive interference power reduces,especially at low values of λ, as the range R decreases.

Fig. 14 shows the PDFs of the cognitive network interfer-ence Ifa at the primary user in a (200 × 200)-meter square (seeFig. 7) for the case of full activity (ζ → ∞) and PI = 0 dBm.The primary and secondary links have Nakagami-m fading,i.e.,

√Y ∼ Nakagami (2, 1) and |Hi| ∼ Nakagami (2, 1); and

the square region has two different secondary spatial densities:λ = 0.01 in the red sections and λ = 0 (i.e., no secondaryusers) in the yellow sections. The PDFs fIfa (x) are plottedfor three cases of the primary user location: i) at the centerof the large square, ii) at the center of the low (zero) densityregion, and iii) at the top-right corner of the large square.We can observe from Fig. 14 that the cognitive networkinterference becomes less severe as the primary user movesto the corner. This is due to the fact that the distance betweenthe primary and secondary users increases when the primaryuser is located at the corner. Moreover, using this framework,we can also consider a nonuniform spatial distribution ofthe secondary users in the region of interest. Therefore, ourstatistical interference model enables us to characterize theposition where the primary user is less vulnerable to the effectof cognitive network interference.


20 40 60 80 100 120 14010

−10

10−8

10−6

10−4

10−2

λ = 0.01λ = 0.001λ = 0.0001λ = 0.00001

Maximum transmission range R (meters)

Var

{I fa}

Fig. 13. Variance of the cognitive network interference Ifa for full activity(ζ → ∞) in the presence of the on-off secondary power control as afunction of the maximum transmission range R� of the secondary users whenλ = 0.01, 0.001, 0.0001, and 0.00001 users/m2. PI = 0 dBm, Ps ∼Bern (FRs (R�)), Rs ∼ Rayleigh (1/ (2πλr)), λr = λ, and Nakagami-m fading for primary and secondary links

√Yi ∼ Nakagami (2, 1) and

|Hi| ∼ Nakagami (2, 1).

VI. CONCLUSIONS

In this paper, we proposed a new statistical model foraggregate interference of cognitive networks, which accountsfor the sensing procedure, the spatial distribution of nodes,secondary spatial reuse protocol, and environment-dependentconditions such as path loss, shadowing, and channel fading.We considered two types of secondary spatial reuse protocols,namely, single-threshold and multiple-threshold protocols. Foreach protocol, we derived the characteristic function and thecumulant of the cognitive network interference at the primaryuser. By using the truncated-stable distributions, we obtainedthe statistical model for the cognitive network interference.We further extended this model to include the effect ofpower control and shadowing, and derived the BEP in thepresence of cognitive network interference. Numerical resultsdemonstrated the effectiveness of our model for capturing thestatistical behavior of the cognitive network interference in avariety of scenarios. The framework developed in the paperenables us to characterize cognitive network interference forsuccessful deployment of future cognitive networks. Further-more, this framework can also be applied in the study of theeffect of inter-tier interference caused by randomly deployedclosed-access femtocells on the macrocell users in multi-tiernetworks.

APPENDIX AGLOSSARY OF STATISTICAL SYMBOLS

We adopt the convention of using upper-case letters withoutserifs for random variables and the corresponding lower-caseletters with serifs for their realizations and dummy arguments.

−1.5 −1 −0.5 0 0.5 1 1.50

0.05

0.1

0.15

0.2

0.25

0.3

−0.1−0.05 0 0.05 0.10.04

0.05

0.06

0.07

f Ifa

(x)

x

PU at the center of the squarePU at the center of the low density regionPU at the top-right corner of the square

Fig. 14. PDF of the cognitive network interference Ifa at the primary user(PU) in a (200 × 200)-meter square (see Fig. 7) for full activity (ζ → ∞).PI = 0 dBm and Nakagami-m fading for primary and secondary links

√Yi ∼

Nakagami (2, 1) and |Hi| ∼ Nakagami (2, 1). The secondary spatial densityis equal to λ = 0.01 users/m2 in the red sections, whereas λ = 0 users/m2

(i.e., no secondary users) in the yellow sections.

E {·} Expectation operatorP {·} Probability measurefX (x) Probability density function of XFX (x) Cumulative distribution function of XFX (x) Complementary cumulative

distribution function of X:FX (x) = 1 − FX (x)

ψX (jω) Characteristic function of X:ψX (jω) � E

{ejωX

}where j =

√−1μX (n) nth moment of X: μX (n) � E {Xn}μ

(pt)X (n, l, u) nth partial moment of X calculated

within the interval [l, u]:μ

(pt)X (n, l, u) �

∫ ulxnfX (x) dx

κX (n) nth cumulant of X:

κX (n) � 1jn

dn lnψX(jω)dωn

∣∣∣ω=0

Bern (p) Bernoulli distribution with mean p:if X ∼ Bern (p), then P {X = 1} = pand P {X = 0} = 1 − p

St (γ′, α, g) Symmetric truncated-stabledistribution with the dispersion γ′,characteristic exponent α,and smoothing parameter g

Rayleigh(σ2)

Rayleigh distribution with theparameter σ2:

fX (x) = xσ2 exp

(− x2

2σ2

), x ≥ 0

Nakagami (m,Ω) Nakagami distribution withthe fading severity parameter mand power parameter Ω:fX (x) = 2mmx2m−1

ΩmΓ(m) exp(−mx2

Ω

),

x ≥ 0


APPENDIX BDERIVATIONS OF THE CUMULANTS

A. Cumulant of Zst (ζ;R) for the Single-Threshold Protocol

We start by deriving the nth cumulant of Zst (ζ;R) in (12)for the single-threshold protocol. Using (14), we obtain

κZst(ζ;R) (n) = 2πλ∫X

∫Y

∫ dmax

dmin

xnr1−nb1[0,ζ]

(r−2by

)× fX (x) fY (y) drdydx

= 2πλμX (n)∫ d2b

maxζ

0

∫ dmax

maxndmin,(y/ζ)

12b

o r1−nbfY (y) drdy


minζ

0

∫ dmax

dmin

r1−nbfY (y) drdy

+ 2πλμX (n)∫ d2b

maxζ

d2bminζ

∫ dmax

(y/ζ)12b

r1−nbfY (y) drdy

=2πλμX (n)nb− 2

∫ d2bminζ

0

(d2−nbmin − d2−nb

max

)fY (y) dy

+2πλμX (n)nb− 2

∫ d2bmaxζ

d2bminζ

[(y/ζ)

2−nb2b − d2−nb

max

]fY (y) dy


[(d2−nbmin − d2−nb

max

)FY(d2bminζ

)+ ζ

nb−22b μ

(pt)Y

(2 − nb

2b, d2b

minζ, d2bmaxζ

)

− d2−nbmax μ

(pt)Y

(0, d2b

minζ, d2bmaxζ

)].

(38)

B. Cumulants of Zk (R) (k = 1, 2, . . . , N ) for the Multiple-Threshold Protocol

We now derive the nth cumulant of Zk (R) in (17) for Ak

of the multiple-threshold protocol. Using (19), we obtain

κZk(R) (n) = 2πλ∫X

∫Y

∫ dmax

dmin

xnr1−nb1[ζk−1,ζk]

(r−2by

)× fX (x) fY (y)drdydx


maxζk

d2bminζk−1

∫ minndmax,(y/ζk−1)

12b

o

maxndmin,(y/ζk)

12b

o r1−nb

× fY (y) drdy. (39)

1) k = 1: It is obvious from (39) that

κZ1(R) (n) = κZst(ζ1;R) (n) . (40)

2) k = 2, 3, . . . , N − 1: We can evaluate the integral in(39) by dividing the integration interval of y into three disjointones, namely:

d2bminζk−1 ≤ y < min

{d2bmaxζk−1, d

2bminζk

},

min{d2bmaxζk−1, d

2bminζk

} ≤ y < max{d2bmaxζk−1, d

2bminζk

},

max{d2bmaxζk−1, d

2bminζk

} ≤ y ≤ d2bmaxζk,

involving two different cases d2bminζk ≥ d2b

maxζk−1 andd2bminζk < d2b

maxζk−1.

i) Case d2bminζk ≥ d2b

maxζk−1: In this case, we have

κZk(R) (n)

=

[2πλμX (n)

∫ d2bmaxζk−1

d2bminζk−1

∫ (y/ζk−1)12b

dmin

r1−nbfY (y) drdy

+∫ d2b

minζk

d2bmaxζk−1

∫ dmax

dmin

r1−nbfY (y) drdy

+∫ d2b

maxζk

d2bminζk

∫ dmax

(y/ζk)12b

r1−nbfY (y)drdy

](41)


[d2−nbmin μ

(pt)Y

(0, d2b

minζk−1, d2bmaxζk−1

)− ζ

nb−22b

k−1 μ(pt)Y

(2 − nb

2b, d2b


)+(d2−nbmin − d2−nb

max

)μ

(pt)Y

(0, d2b

maxζk−1, d2bminζk

)+ ζ

nb−22b

k μ(pt)Y

(2 − nb

2b, d2b

minζk, d2bmaxζk

)

− d2−nbmax μ

(pt)Y

(0, d2b

minζk, d2bmaxζk

)]. (42)

ii) Case d2bminζk < d2b

maxζk−1: Similarly, we have

κZk(R) (n)

=2πλμX (n)

[∫ d2bminζk

d2bminζk−1

∫ (y/ζk−1)12b

dmin

r1−nbfY (y) drdy

+∫ d2b

maxζk−1

d2bminζk

∫ (y/ζk−1)12b

(y/ζk)12b

r1−nbfY (y) drdy

+∫ d2b

maxζk

d2bmaxζk−1

∫ dmax

(y/ζk)12b

r1−nbfY (y)drdy

]


[d2−nbmin μ

(pt)Y

(0, d2b

minζk−1, d2bminζk

)− ζ

nb−22b

k−1 μ(pt)Y

(2 − nb

2b, d2b

minζk−1, d2bminζk

)+(ζ

nb−22b

k − ζnb−2

2b

k−1

)× μ

(pt)Y

(2 − nb

2b, d2b

minζk, d2bmaxζk−1

)+ ζ

nb−22b

k μ(pt)Y

(2 − nb

2b, d2b

maxζk−1, d2bmaxζk

)

− d2−nbmax μ

(pt)Y

(0, d2b

maxζk−1, d2bmaxζk

)].

(43)


Now, combining (42) and (43), we obtain the nth cumulantof Zk (R) for k = 2, 3, . . . , N − 1 as follows:

κZk(R) (n) =2πλμX (n)nb− 2

[d2−nbmin μ

(pt)Y

(0, d2b

minζk−1,Δmin

)− ζ

nb−22b

k−1 μ(pt)Y

(2 − nb

2b, d2b

minζk−1,Δmin

)+ c1 μ

(pt)Y (c2,Δmin,Δmax)

+ ζnb−2

2b

k μ(pt)Y

(2 − nb

2b,Δmax, d

2bmaxζk

)

− d2−nbmax μ

(pt)Y

(0,Δmax, d

2bmaxζk

)], (44)

where Δmin = min{d2bmaxζk−1, d

2bminζk

}, Δmax =

max{d2bmaxζk−1, d

2bminζk

}, and

(c1, c2)

=

⎧⎨⎩(d2−nbmin − d2−nb

max , 0), if d2b

minζk ≥ d2bmaxζk−1(

ζnb−2

2b

k − ζnb−2

2b

k−1 , 2−nb2b

), if d2b

minζk < d2bmaxζk−1.

(45)

3) k = N : Since ζN = ∞, it is obvious that d2bminζk ≥

d2bmaxζk−1 and the third term in (41) vanishes for k =N . Hence, it follows immediately from (42) along withμ

(pt)Y (0, a,∞) = FY (a) that

κZN (R) (n)=2πλμX (n)nb− 2

[d2−nbmin μ

(pt)Y

(0, d2b


)− ζ

nb−22b

k−1 μ(pt)Y

(2 − nb

2b, d2b


)

+(d2−nbmin − d2−nb

max

)FY(d2bmaxζk−1

)]. (46)

REFERENCES

[1] FCC, “Policy task force report (et docket-135),” 2002.[2] M. A. McHenry, “NSF Spectrum Occupancy Measurements Project

Summary,” Shared Spectrum Company, 2005.[3] FCC, “Notice of proposed rule making, in the matter of facilitating

opportunities for flexible, efficient and reliable spectrum use employingcognitive radio technologies (et docket no. 03-108) and authorizationand use of software defined radios (et docket no. 00-47), FCC 03-322,”Dec. 2003.

[4] S. Haykin, “Cognitive radio: Brain-empowered wireless communica-tions,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb.2005.

[5] I. F. Akyildiz, W. Lee, M. C. Vuran, and S. Mohanty, “Next gen-eration/dynamic spectrum access/cognitive radio wireless networks: Asurvey,” Computer Networks, vol. 50, no. 13, pp. 2127–2159, Sep. 2006.

[6] Q. Zhao and B. M. Sadler, “A survey of dynamic spectrum access,”IEEE Signal Process. Mag., vol. 24, no. 3, pp. 79–89, May 2007.

[7] A. Goldsmith, S. A. Jafar, I. Maric, and S. Srinivasa, “Breaking spectrumgridlock with cognitive radios: An information theoretic perspective,”Proc. IEEE, vol. 97, no. 5, pp. 894–914, May 2009.

[8] A. Ghasemi, , and E. S. Sousa, “Spectrum sensing in cognitive radionetworks: Requirements, challenges and design trade-offs,” IEEE Com-mun. Mag., vol. 46, no. 4, pp. 32–39, Apr. 2008.

[9] D. Cabric, S. M. Mishra, and R. W. Brodersen, “Implementation issuesin spectrum sensing for cognitive radios,” in Proc. IEEE Asilomar Conf.on Signals, Systems, and Computers, Pacific Grove, CA, Nov. 2004, pp.772–776.

[10] A. Sonnenschein and P. M. Fishman, “Radiometric detection of spread-spectrum signals in noise,” IEEE Trans. Aerosp. Electron. Syst., vol. 28,no. 3, pp. 654–660, Jul. 1992.

[11] R. Tandra and A. Sahai, “SNR walls for signal detection,” IEEE J. Sel.Topics Signal Process., vol. 2, no. 1, pp. 4–17, Feb. 2008.

[12] R. Tandra, S. M. Mishra, and A. Sahai, “What is a spectrum hole andwhat does it take to recognize one: Extended version,” Proc. IEEE,vol. 97, no. 5, pp. 824–848, May 2009.

[13] M. Z. Win, “A mathematical model for network interference,” IEEECommun. Theory Workshop, Sedona, AZ, May 2007.

[14] M. Z. Win, P. C. Pinto, and L. A. Shepp, “A mathematical theory ofnetwork interference and its applications,” Proc. IEEE, vol. 97, no. 2,pp. 205–230, Feb. 2009.

[15] M. Z. Win and R. A. Scholtz, “Impulse radio: How it works,” IEEECommun. Lett., vol. 2, no. 2, pp. 36–38, Feb. 1998.

[16] ——, “Ultra-wide bandwidth time-hopping spread-spectrum impulseradio for wireless multiple-access communications,” IEEE Trans. Com-mun., vol. 48, no. 4, pp. 679–691, Apr. 2000.

[17] M. Z. Win, “A unified spectral analysis of generalized time-hoppingspread-spectrum signals in the presence of timing jitter,” IEEE J. Sel.Areas Commun., vol. 20, no. 9, pp. 1664–1676, Dec. 2002.

[18] R. A. Scholtz, “Private conversation,” University of Southern California,Sep. 1997, Los Angeles, CA.

[19] J. H. Winters, “Private conversation,” AT&T Labs-Research, Mar. 2001,Middletown, NJ.

[20] L. A. Shepp, “Private conversation,” AT&T Labs-Research, Mar. 2001,Middletown, NJ.

[21] P. C. Pinto and M. Z. Win, “Communication in a Poisson field ofinterferers – Part I: Interference distribution and error probability,” IEEETrans. Wireless Commun., vol. 9, no. 7, pp. 2176–2186, Jul. 2010.

[22] ——, “Communication in a Poisson field of interferers – Part II: Channelcapacity and interference spectrum,” IEEE Trans. Wireless Commun.,vol. 9, no. 7, pp. 2187–2195, Jul. 2010.

[23] A. Rabbachin, T. Q. S. Quek, P. C. Pinto, I. Oppermann, and M. Z.Win, “Non-coherent UWB communication in the presence of multiplenarrowband interferers,” IEEE Trans. Wireless Commun., vol. 9, no. 11,pp. 3365-3379, Nov. 2010.

[24] P. C. Pinto, A. Giorgetti, M. Z. Win, and M. Chiani, “A stochasticgeometry approach to coexistence in heterogeneous wireless networks,”IEEE J. Sel. Areas Commun., vol. 27, no. 7, pp. 1268–1282, Sep. 2009.

[25] P. C. Pinto and M. Z. Win, “Spectral characterization of wirelessnetworks,” IEEE Wireless Commun. Mag., vol. 14, no. 6, pp. 27–31,Dec. 2007.

[26] J. F. Kingman, Poisson Processes. Oxford University Press, 1993.[27] S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev.

Modern Phys., vol. 15, no. 1, pp. 1–89, Jan. 1943.[28] S. E. Heath, “Applications of the Poisson Model to Wireless Telephony

and to Cosmology,” Ph.D. dissertation, Department of Statistics, Rut-gers University, Piscataway, NJ, Mar. 2004, thesis advisor: ProfessorLawrence A. Shepp.

[29] M. Y. Vardi, L. Shepp, and L. Kaufman, “A statistical model for positronemission tomography,” Journal of the American Statistical Association,vol. 80, no. 389, pp. 8–20, Mar. 1985.

[30] M. Beil, F. Fleischer, S. Paschke, and V. Schmidt, “Statistical analysis of3D centromeric heterochromatin structure in interphase nuclei,” Journalof Microscopy, vol. 217, pp. 60–68, 2005.

[31] D. L. Snyder, “Filtering and detection for doubly stochastic Poissonprocesses,” IEEE Trans. Inf. Theory, vol. 18, no. 1, pp. 91–102, Jan.1972.

[32] J. R. Pierce, “Optical channels: practical limits with photon-counting,”IEEE Trans. Inf. Theory, vol. 26, no. 12, pp. 1819–1821, Dec. 1978.

[33] J. R. Pierce, E. C. Posner, and E. R. Rodemich, “The capacity of thephoton counting channel,” IEEE Trans. Inf. Theory, vol. 27, no. 1, pp.61–77, Jan. 1981.

[34] J. L. Massey, “Capacity cutoff rate, and coding for direct detectionoptical channel,” IEEE Trans. Commun., vol. 29, no. 11, pp. 1615–1621,Nov. 1981.

[35] E. S. Sousa, “Performance of a spread spectrum packet radio networklink in a Poisson field of interferers,” IEEE Trans. Inf. Theory, vol. 38,no. 6, pp. 1743–1754, Dec. 1992.

[36] J. Ilow, D. Hatzinakos, and A. N. Venetsanopoulos, “Performance of FHSS radio networks with interference modeled as a mixture of Gaussianand alpha-stable noise,” IEEE Trans. Commun., vol. 46, no. 4, pp. 509–520, Apr. 1998.

[37] C. C. Chan and S. V. Hanly, “Calculating the outage probability in aCDMA network with spatial Poisson traffic,” IEEE Trans. Veh. Technol.,vol. 50, no. 1, pp. 183–204, Jan. 2001.

[38] F. Baccelli, B. Błaszczyszyn, and F. Tournois, “Spatial averages ofcoverage characteristics in large CDMA networks,” Wireless Networks,vol. 8, no. 6, pp. 569–586, Nov. 2002.


[39] X. Yang and A. P. Petropulu, “Co-channel interference modeling andanalysis in a Poisson field of interferers in wireless communications,”IEEE Trans. Signal Process., vol. 51, no. 1, pp. 64–76, 2003.

[40] J. Orriss and S. K. Barton, “Probability distributions for the numberof radio transceivers which can communicate with one another,” IEEETrans. Commun., vol. 51, no. 4, pp. 676–681, Apr. 2003.

[41] S. P. Weber, X. Yang, J. G. Andrews, and G. de Veciana, “Transmissioncapacity of wireless ad hoc networks with outage constraints,” IEEETrans. Inf. Theory, vol. 51, no. 12, pp. 4091–4102, Dec. 2005.

[42] O. Dousse, F. Baccelli, and P. Thiran, “Impact of interferences onconnectivity in ad hoc networks,” IEEE/ACM Trans. Netw., vol. 13,no. 2, pp. 425–436, Apr. 2005.

[43] O. Dousse, M. Franceschetti, and P. Thiran, “On the throughput scalingof wireless relay networks,” IEEE Trans. Inf. Theory, vol. 52, no. 6, pp.2756–2761, Jun. 2006.

[44] L. Song and D. Hatzinakos, “Cooperative transmission in Poissondistributed wireless sensor networks: Protocol and outage probability,”IEEE Trans. Wireless Commun., vol. 5, no. 10, pp. 2834–2843, Oct.2006.

[45] A. Ghasemi and E. S. Sousa, “Interference aggregation in spectrum-sensing cognitive wireless networks,” IEEE J. Sel. Topics Signal Pro-cess., vol. 2, no. 1, pp. 41–56, Feb. 2008.

[46] R. Menon, R. M. Buehrer, and J. H. Reed, “On the impact of dynamicspectrum sharing techniques on legacy radio systems,” IEEE Trans.Wireless Commun., vol. 7, no. 11, pp. 4198–4207, Nov. 2008.

[47] W. Ren, Q. Zhao, and A. Swami, “Power control in spectrum overlaynetworks: How to cross a multi-lane highway,” IEEE J. Sel. AreasCommun., vol. 27, no. 7, pp. 1283–1296, Sep. 2009.

[48] E. Salbaroli and A. Zanella, “Interference analysis in a Poisson fieldof nodes of finite area,” IEEE Trans. Veh. Technol., vol. 58, no. 4, pp.1776–1783, May 2009.

[49] H. Inaltekin, M. Chiang, H. V. Poor, and S. B. Wicker, “The behavior ofunbounded path-loss models and the effect of singularity on computednetwork characteristics,” IEEE J. Sel. Areas Commun., vol. 27, no. 7,pp. 1078–1092, Sep. 2009.

[50] R. K. Ganti and M. Haenggi, “Interference and outage in clusteredwireless ad hoc networks,” IEEE Trans. Inf. Theory, vol. 55, no. 9,pp. 4067–4086, Sep. 2009.

[51] F. Baccelli, B. Błaszczyszyn, and P. Muhlethaler, “Stochastic analysis ofspatial and opportunistic Aloha,” IEEE J. Sel. Areas Commun., vol. 27,no. 7, pp. 1105–1119, Sep. 2009.

[52] V. Chandrasekhar and J. G. Andrews, “Uplink capacity and interfer-ence avoidance for two-tier femtocell networks,” IEEE Trans. WirelessCommun., vol. 8, no. 7, pp. 3498–3509, Jul. 2009.

[53] Q. Zhao, L. Tong, A. Swami, and Y. Chen, “Decentralized cognitiveMAC for opportunistic spectrum access in ad hoc networks: A POMDPframework,” IEEE J. Sel. Areas Commun., vol. 25, no. 3, pp. 589–600,Apr. 2007.

[54] O. Simeone, I. Stanojev, S. Savazzi, Y. Bar-Ness, U. Spagnolini,and R. Pickholtz, “Spectrum leasing to cooperating secondary ad hocnetworks,” IEEE J. Sel. Areas Commun., vol. 26, no. 1, pp. 203–213,Jan. 2008.

[55] L. C. Wang and A. Chen, “Effects of location awareness on concurrenttransmissions for cognitive ad hoc networks overlaying infrastructure-based systems,” IEEE Trans. Mobile Comput., vol. 8, no. 5, pp. 577–589,May 2009.

[56] ECC, “Technical requirements for UWB DAA (detect and avoid) devicesto ensure the protection of radiolocation services in the bands 3.1 - 3.4GHz and 8.5 - 9 GHz and BWA terminals in the band 3.4 - 4.2 GHz,”2008.

[57] G. Samoradnitsky and M. Taqqu, Stable Non-Gaussian Random Pro-cesses. Chapman and Hall, 1994.

[58] R. K. Ganti and M. Haenggi, “Interference in ad hoc networks withgeneral motion-invariant node distributions,” in Proc. IEEE Int. Symp.on Inf. Theory, Toronto, CANADA, Jul. 2008, pp. 1–5.

[59] P. Carr, H. Geman, D. B. Madan, and M. Yor, “The fine structure of assetreturns: An empirical investigation,” The Journal of Business, vol. 75,no. 2, pp. 305–332, Apr. 2002.

[60] N. Bambos and S. Kandukuri, “Power controlled multiple access(PCMA) schemes for next-generation wireless packet networks,” IEEETrans. Wireless Commun., vol. 9, no. 3, pp. 58–64, Mar. 2002.

[61] G. J. Foschini and Z. Miljanic, “A simple distributed autonomous powercontrol algorithm and its convergence,” IEEE Trans. Veh. Technol.,vol. 42, no. 4, pp. 641–646, Nov. 1993.

[62] V. Chandrasekhar, J. G. Andrews, and A. Gatherer, “Femtocell networks:A survey,” IEEE Commun. Mag., vol. 46, no. 9, pp. 59–67, Sep. 2008.

[63] J. Gil-Pelaez, “Note on the inversion theorem,” Biometrika, vol. 38, no.3/4, pp. 481–482, Dec. 1951.

Alberto Rabbachin (S’03–M’07) received the M.S.degree from the University of Bologna (Italy) in2001 and the Ph.D. degree from the University ofOulu (Finland) in 2008.Since 2008 he is a Postdoctoral researcher with

the Institute for the Protection and Security of theCitizen of the European Commission Joint ResearchCenter. He has done research on ultrawideband(UWB) impulse-radio techniques, with emphasis onreceiver architectures, synchronization, and rangingalgorithms, as well as on low-complexity UWB

transceiver design. He is the author of several book chapters, internationaljournal papers, conference proceedings, and international standard contribu-tions. His current research interests include aggregate interference statisticalmodeling, cognitive radio, and wireless body area networks. Dr. Rabbachinreceived the Nokia Fellowship for year 2005 and 2006, the IEEE Globecom2010 Best Paper Award, the IEEE Globecom 2010 GOLD Best Paper Award,and the European Commission JRC Best Young Scientist Award 2010. Hehas served on the Technical Program Committees of various internationalconferences.

Tony Q.S. Quek (S’98–M’08) received the B.E.and M.E. degrees in Electrical and ElectronicsEngineering from Tokyo Institute of Technology,Tokyo, Japan, in 1998 and 2000, respectively. AtMassachusetts Institute of Technology (MIT), Cam-bridge, MA, he earned the Ph.D. in Electrical Engi-neering and Computer Science in Feb. 2008.Since 2008, he has been with the Institute for In-

focomm Research, A∗STAR, where he is currently aPrincipal Investigator and Senior Research Engineer.He is also an Adjunct Assistant Professor with the

Division of Communication Engineering, Nanyang Technological University.His main research interests are the application of mathematical, optimization,and statistical theories to communication, detection, information theoreticand resource allocation problems. Specific current research topics includecooperative networks, interference networks, heterogeneous networks, greencommunications, wireless security, and cognitive radio.Dr. Quek has been actively involved in organizing and chairing sessions,

and has served as a member of the Technical Program Committee (TPC) ina number of international conferences. He served as the Technical ProgramChair for the Services & Applications Track for the IEEE Wireless Com-munications and Networking Conference (WCNC) in 2009, the CognitiveRadio & Cooperative Communications Track for the IEEE Vehicular Tech-nology Conference (VTC) in Spring 2011, and the Wireless CommunicationsSymposium for the IEEE Globecom in 2011; as Technical Program Vice-Chair for the IEEE Conference on Ultra Wideband in 2011; and as theWorkshop Chair for the IEEE Globecom 2010 Workshop on FemtocellNetworks and the IEEE ICC 2011 Workshop on Heterogeneous Networks.Dr. Quek is currently an Editor for WILEY JOURNAL ON SECURITY AND

COMMUNICATION NETWORKS. He was Guest Editor for the JOURNALOF COMMUNICATIONS AND NETWORKS (Special Issue on HeterogeneousNetworks) in 2011.Dr. Quek received the Singapore Government Scholarship in 1993, Tokyu

Foundation Fellowship in 1998, and the A∗STAR National Science Scholar-ship in 2002. He was honored with the 2008 Philip Yeo Prize for OutstandingAchievement in Research and the IEEE Globecom 2010 Best Paper Award.


Hyundong Shin (S’01–M’04) received the B.S.degree in Electronics Engineering from Kyung HeeUniversity, Korea, in 1999, and the M.S. and Ph.D.degrees in Electrical Engineering from Seoul Na-tional University, Seoul, Korea, in 2001 and 2004,respectively.From September 2004 to February 2006, Dr.

Shin was a Postdoctoral Associate at the Labora-tory for Information and Decision Systems (LIDS),Massachusetts Institute of Technology (MIT), Cam-bridge, MA, USA. In March 2006, he joined the

faculty of the School of Electronics and Information, Kyung Hee Univer-sity, Korea, where he is now an Assistant Professor at the Department ofElectronics and Radio Engineering. His research interests include wirelesscommunications, information and coding theory, cooperative/ collaborativecommunications, and multiple-antenna wireless communication systems andnetworks.Professor Shin served as a member of the Technical Program Committee

in the IEEE International Conference on Communications (2006, 2009), theIEEE International Conference on Ultra Wideband (2006), the IEEE GlobalCommunications Conference (2009, 2010), the IEEE Vehicular TechnologyConference (2009 Fall, 2010 Spring, 2010 Fall), the IEEE InternationalSymposium on Personal, Indoor and Mobile Communications (2009, 2010),and the IEEE Wireless Communications and Networking Conference (2010).He served as a Technical Program co-chair for the IEEE Wireless Communica-tions and Networking Conference PHY Track (2009). Dr. Shin is currently anEditor for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS andthe KSII Transactions on Internet and Information Systems. He was a GuestEditor for the 2008 EURASIP Journal on Advances in Signal Processing(Special Issue on Wireless Cooperative Networks).Professor Shin received the IEEE Communications Society’s Guglielmo

Marconi Best Paper Award (2008) and the IEEE Vehicular TechnologyConference Best Paper Award (2008 Spring).

Moe Z. Win (S’85-M’87-SM’97-F’04) receivedboth the Ph.D. in Electrical Engineering and M.S. inApplied Mathematics as a Presidential Fellow at theUniversity of Southern California (USC) in 1998.He received an M.S. in Electrical Engineering fromUSC in 1989, and a B.S. (magna cum laude) inElectrical Engineering from Texas A&M Universityin 1987.Dr. Win is an Associate Professor at the Mas-

sachusetts Institute of Technology (MIT). Prior tojoining MIT, he was at AT&T Research Laboratories

for five years and at the Jet Propulsion Laboratory for seven years. Hisresearch encompasses developing fundamental theories, designing algorithms,and conducting experimentation for a broad range of real-world problems. Hiscurrent research topics include location-aware networks, intrinsically securewireless networks, aggregate interference in heterogeneous networks, ultra-wide bandwidth systems, multiple antenna systems, time-varying channels,optical transmission systems, and space communications systems.Professor Win is an IEEE Distinguished Lecturer and elected Fellow of

the IEEE, cited for “contributions to wideband wireless transmission.” Hewas honored with two IEEE technical field awards: the IEEE Kiyo TomiyasuAward (2011) for “fundamental contributions to high-speed reliable commu-nications over optical and wireless channels” and the IEEE Eric E. SumnerAward (2006, jointly with R.A. Scholtz) for “pioneering contributions to ultra-wide band communications science and technology.” Together with studentsand colleagues, his papers have received several awards including the IEEECommunications Society’s Guglielmo Marconi Best Paper Award (2008)and the IEEE Antennas and Propagation Society’s Sergei A. SchelkunoffTransactions Prize Paper Award (2003). His other recognitions include theOutstanding Service Award of the IEEE ComSoc Radio CommunicationsCommittee (2010), the Laurea Honoris Causa from the University of Ferrara,Italy (2008), the Technical Recognition Award of the IEEE ComSoc RadioCommunications Committee (2008), Wireless Educator of the Year Award(2007), the Fulbright Foundation Senior Scholar Lecturing and ResearchFellowship (2004), the U.S. Presidential Early Career Award for Scientists andEngineers (2004), the AIAA Young Aerospace Engineer of the Year (2004),and the Office of Naval Research Young Investigator Award (2003).Professor Win has been actively involved in organizing and chairing a

number of international conferences. He served as the Technical ProgramChair for the IEEE Wireless Communications and Networking Conferencein 2009, the IEEE Conference on Ultra Wideband in 2006, the IEEECommunication Theory Symposia of ICC-2004 and Globecom-2000, and theIEEE Conference on Ultra Wideband Systems and Technologies in 2002;Technical Program Vice-Chair for the IEEE International Conference onCommunications in 2002; and the Tutorial Chair for ICC-2009 and theIEEE Semiannual International Vehicular Technology Conference in Fall2001. He is an elected Member-at-Large on the IEEE CommunicationsSociety Board of Governors (2011-2013). He was the chair (2004-2006)and secretary (2002-2004) for the Radio Communications Committee of theIEEE Communications Society. Dr. Win is currently an Editor for IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS. He served as AreaEditor for Modulation and Signal Design (2003-2006), Editor for WidebandWireless and Diversity (2003-2006), and Editor for Equalization and Diversity(1998-2003), all for the IEEE TRANSACTIONS ON COMMUNICATIONS. Hewas Guest-Editor for the PROCEEDINGS OF THE IEEE (Special Issue onUWB Technology & Emerging Applications) in 2009 and IEEE JOURNAL ONSELECTED AREAS IN COMMUNICATIONS (Special Issue on Ultra -WidebandRadio in Multiaccess Wireless Communications) in 2002.

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