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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 1,
JANUARY 2014 429
Modeling Time-Varying Aggregate Interference inCognitive Radio
Systems, and Application to
Primary Exclusive Zone DesignMohd. Shabbir Ali and Neelesh B.
Mehta, Senior Member, IEEE
AbstractAccurately characterizing the time-varying interfer-ence
caused to the primary users is essential in ensuring asuccessful
deployment of cognitive radios (CR). We show thatthe aggregate
interference at the primary receiver (PU-Rx) frommultiple, randomly
located cognitive users (CUs) is well modeledas a shifted lognormal
random process, which is more accuratethan the lognormal and the
Gaussian process models consideredin the literature, even for a
relatively dense deployment of CUs. Italso compares favorably with
the asymptotically exact stable andsymmetric truncated stable
distribution models, except at highCU densities. Our model accounts
for the effect of imperfectspectrum sensing, which depends on
path-loss, shadowing, andsmall-scale fading of the link from the
primary transmitter to theCU; the interweave and underlay modes of
CR operation, whichdetermine the transmit powers of the CUs; and
time-correlatedshadowing and fading of the links from the CUs to
the PU-Rx. Itleads to expressions for the probability distribution
function, levelcrossing rate, and average exceedance duration. The
impact ofcooperative spectrum sensing is also characterized. We
validatethe model by applying it to redesign the primary exclusive
zoneto account for the time-varying nature of interference.
Index TermsCognitive radio, interference, spectrum
sensing,underlay, interweave, shadowing, fading, time-variations,
ran-dom process, lognormal, primary exclusive zone.
I. INTRODUCTION
COGNITIVE radio (CR) offers a promising solution tothe problem
of under utilization of the spectrum. Acommon paradigm of CR
classifies users into two categories,namely, primary users (PUs),
which have unfettered accessto the spectrum, and cognitive users
(CUs), which can usethe spectrum but under tight constraints on the
aggregateinterference their transmissions cause to the PUs [1][6].
Asuccessful design and deployment of CR, therefore, requiresas a
first step an accurate model for the aggregate interferencecaused
to the PUs by transmissions from one or many CUs.This
characterization feeds into the design and evaluationof
transmission policies for the CUs and techniques to helpmitigate
their interference.
Several factors together affect the aggregate interference,and
must be accounted for in order to arrive at an accurate
Manuscript received April 30, 2013; revised August 6 and October
8, 2013;accepted October 12, 2013. The associate editor
coordinating the review ofthis paper and approving it for
publication was R. Zhang.
The authors are with the Dept. of Electrical Communication
Eng.,Indian Institute of Science (IISc), Bangalore, India (e-mail:
[email protected], [email protected]).
This work was partially supported by a research grant from
ANRC.A part of the work has been accepted for presentation in the
IEEE Global
Conf. on Communications (Globecom), 2013.Digital Object
Identifier 10.1109/TWC.2013.113013.130762
model for it. It is affected by propagation characteristics
ofthe channels between the CUs and PUs, such as
path-loss,shadowing, and fading. Furthermore, the number of CUs
thattransmit and their locations affect the interference.
Imperfectspectrum sensing also directly affects it as it determines
theCUs transmit powers and whether or not they transmit. Theuse of
cooperation, in which the CUs cooperate with eachother and fuse
their decisions, affects the accuracy of spectrumsensing and, thus,
the aggregate interference.
Given the importance of interference modeling in CR,one of the
approaches pursued in the literature is based onmeasurements from
test deployments [7][9]. However, thesedeployments are typically
small because of the difficulty insetting up an experiment with
many interferers that capturesthe many sources of randomness.
Furthermore, the modelsdeduced are location-specific. Therefore, a
second approachhas focused on developing statistical models for the
aggregateinterference [1][5]. However, no closed form exists for
itsprobability distribution function (PDF). Therefore,
severalapproximate analytical models have been investigated.
Interference in Underlay CR Mode: In the underlay mode,a CU can
transmit even when it senses that the PU transmitter(PU-Tx) is
transmitting [10]. However, it does so with a muchlower power in
order to avoid excessive interference to thePU receiver (PU-Rx). In
[5], the aggregate interference at thePU-Rx from a fixed number of
CUs, which are distributeduniformly over a region, is modeled as a
lognormal randomvariable (RV). However, only large-scale shadow
fading istaken into account. A spatial Poisson point process
(SPPP)model is instead assumed to model the randomness in the
CUnumber and locations in [4], and the aggregate interferenceis
modeled as a lognormal RV. Power control as well ascontention
control, in which CUs too close to each other donot transmit
simultaneously, are accounted for. Furthermore,lognormal shadowing
and Nakagami-m fading are modeled.However, imperfect spectrum
sensing and the time-varyingnature of aggregate interference are
not studied in [4], [5].
Interference in Interweave CR Mode: In the interweavemode, a CU
transmits only when it senses the PU-Tx to beoff [10]. The
transmission can be in the same band as thePU-Tx or in a different
band. In [2], the amplitude of theaggregate interference is modeled
as a symmetric truncated-stable (STS) RV. Unlike the stable
distribution model [1],the STS model ensures finite second and
higher moments.An SPPP model determines the CU locations and an
energydetector (ED) is used for spectrum sensing. A detect-and-
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430 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO.
1, JANUARY 2014
avoid strategy in which the transmit power level of the CUis
adapted based on the received signal strength is
considered.Cooperative spectrum sensing is accounted for in [3],
whichmodels the aggregate interference as a shifted lognormal(SLN)
RV assuming an SPPP model for the CUs. However,time variations are
not modeled in [1][3].
A. ContributionsWe develop a comprehensive model for the
aggregate inter-
ference that captures its snapshot statistics, i.e., PDF, as
well asits time-variation statistics, which is measured in terms of
theauto-correlation function, average level crossing rate (LCR),and
average exceedance duration (AED) [11]. We show thatthe aggregate
interference is well modeled as a wide sensestationary (WSS) SLN
random process (RP), except when itis small. Our model allows the
CUs to operate in both theinterweave and underlay modes, in which
they transmit in thesame band as the PU-Tx with a high power when
they sensethe PU-Tx to be off and with a low power when they sense
thePU-Tx to be on. It accounts for imperfect spectrum sensing,which
depends on the location of the CU relative to the PU-Tx.Also, it
accounts for the combined effect of time-correlatedshadowing and
Rayleigh fading on the various links. We alsoextensively benchmark
the proposed model with several othermodels proposed in the
literature. The randomness in the CUlocations and number is also
captured using the SPPP model.
We then develop a corresponding model for the
aggregateinterference for cooperative spectrum sensing, in which
CUsthat are close to each other cooperate and arrive at a
commondecision. We show that the SLN RP again models the aggre-gate
interference well, except when it is small, and brings outthe
reduction in the aggregate interference due to cooperation.
We then demonstrate the usefulness and tractability of
theproposed model by refining the design of the primary
exclusivezone (PEZ) [4], [12], [13]. The PEZ is defined as the
regionaround the PU-Rx within which no CU is allowed to
transmit,and helps protect the PU-Rx from excessive interference.
Itsarea affects the aggregate interference experienced by the
PU-Rx. We propose a novel criterion for determining the PEZradius
that incorporates the impact of the time-varying natureof the
aggregate interference.
B. Comparisons with LiteratureThe SLN model has been considered
before in the litera-
ture [3], [14]. Further, our channel model and the SPPP modelfor
CUs is similar to that in [3]. However, the generalizationfrom an
SLN RV to an SLN RP, and the demonstration of itsaccuracy in
modeling the time-varying nature of the aggregateinterference when
all the aforementioned physical layer effectsare accounted for is
novel and is a contribution of this paper.The following is a list
of our specific contributions and themany ways in which our
approach and results differ from thosein [3] and other related
works [1], [2], [12][17]:
Overall goal: While [3] focuses on characterizing thesnapshot
statistics and models the aggregate interferenceas an SLN RV, we
characterize the time-varying behaviorof the aggregate interference
as it also affects the PU-Rx.For example, in [15], it has been
argued that long dips
in the signal-to-noise-plus-interference-ratio (SINR)
aredetrimental to the PU-Rx. Our model is also more generalthan the
LCR analysis in [16], which considers a systemwith one CU and one
PU, only models shadowing, andassumes perfect spectrum
sensing.While the stable model is provably accurate in the
asymp-totic regime of a large number of CUs [1], generalizingit or
the STS model to incorporate time-variations is anopen problem.
These two models also require accurate,numerically stable
techniques to compute the cumula-tive distribution function (CDF)
from the characteristicfunction (CF) that they characterize.
Furthermore, as weshall see, the asymptotic exactness manifests
itself onlyat higher CU densities. In [17], the aggregate
interferenceis instead modeled as a gamma RP. However, shadowingand
imperfect spectrum sensing are not modeled and thenumber of CUs is
assumed to be fixed. In [14], the SLNRV has instead been used for
approximating the decisionstatistics of a multichannel energy
detector. Thus, itsmodel and results are very different from
ours.
Analytical results: While moments of the aggregate in-terference
are derived in [3], we derive new expressionsfor the moments and
the autocorrelation of the aggregateinterference. These then help
determine all the parametersthat are required to specify the SLN
RP, and beget newanalytical results for the LCR and AED of the
aggregateinterference. Our moment expressions also turn out to
bedifferent due to differences in our CU transmission andspectrum
sensing models, which are summarized below.
CU transmission model: While [3] focuses on the inter-weave
mode, we analyze a hybrid mode of operation thatcombines the
interweave and underlay modes.
Spectrum sensing model: In [3], an out-of-band spectrumsensing
model is assumed in which each CU senses thesignal it receives from
the full-duplex PU-Rx on a sep-arate control channel. In our model,
however, spectrumsensing is based on the signal received from the
PU-Txand is in-band [2], [18]. This also avoids the need forthe CUs
to simultaneously sense an out-of-band beacon.Another difference,
which also affects spectrum sensing,is that we incorporate the PEZ
in our model.
Extensive benchmarking and application: Our paper
alsodemonstrates the utility of the proposed RP model byapplying it
to the design of the PEZ, and showing that it isreasonably accurate
over a wide range of parameters. Thisis unlike [3]. Further, the
benchmarking of the proposedmodel is more extensive in our
paper.
New PEZ design criteria: The incorporation of the time-varying
nature of the interference in the PEZ design isa contribution of
the paper, and has not been consideredin [12], [13]. Further [12],
did not consider imperfectspectrum sensing, shadowing, and
small-scale fading,while imperfect spectrum sensing was not
modeledin [13]. Our quantification of how cooperative
spectrumsensing shrinks the PEZ is also novel.
The paper is organized as follows. Sec. II describes thesystem
model. The aggregate interference process model isdeveloped in Sec.
III. Cooperative spectrum sensing is con-
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ALI and MEHTA: MODELING TIME-VARYING AGGREGATE INTERFERENCE IN
COGNITIVE RADIO SYSTEMS, AND APPLICATION TO PRIMARY . . . 431
p PUTx
Correctdetection
R
RPEZ
PURx
Misseddetection
Area A
CUs
Fig. 1. System model showing the spectrum sensing by the CUs
that arescattered over a region of area A.
sidered in Sec. IV. Simulation results and PEZ redesign
arepresented in Sec. V. Our conclusions follow in Sec. VI.
II. SYSTEM MODELWe shall use the following notation. Expectation
is denoted
by E [.], and EX [.] denotes expectation conditioned over theRV
X . The notation X(t) N (X , 2X , CX()) shall meanthat X(t) is a
WSS Gaussian RP with mean X , variance 2X ,and covariance function
CX(). Similarly, X N (X , 2X)denotes a Gaussian RV, and X exp()
denotes an expo-nential RV with mean .
A. System LayoutFigure 1 shows the system layout. The number and
locations
of the CUs is modeled using a homogeneous SPPP, which
ischaracterized by a density parameter . Therefore, the numberof
CUs NCR that occur in a region of area A is a PoissonRV with mean
A. The PU-Rx is located at the center ofthe region and the PU-Tx is
located at a distance p from thePU-Rx on the x-axis. The region of
radius RPEZ around thePU-Rx is the PEZ, within which no CU
transmits [4], [12],[13]. The PEZ can be constructed using an
in-band beaconsignal from the PU-Rx or using geo-location [4].1
Note thatthis beacon aided model is different from that in [3], in
whichthe CU transmissions are governed by an out-of-band beaconsent
by the full-duplex PU-Rx anytime the PU-Tx transmits.
B. Channel ModelThe power PRi(t) received at the PU-Rx from the
ith CU,
which is located at a distance ri(t) from it, is given by
PRi(t) = PK
(d0ri(t)
)eXi(t)hi(t), (1)
where P is the transmit power of the CU. The path-losscomponent
is K
(d0
ri(t)
), where K =
(
4d0
)2, is the
carrier wavelength, d0 is the break-point distance, and isthe
path-loss exponent [11]. The Rayleigh fading component ishi(t) exp
(1). Note that our model can also handle any other
1The CUs within the PEZ are all assumed to know that they are
within it.This is justified because several measurements of the
PU-Rx beacon collectedover a sufficiently long duration of time can
be used to ensure this.
mean for hi(t). The normalized covariance function Chi()is given
as per the Jakes fading model [11]:
Chi() = J20 (2fm) , (2)
where fm is the maximum Doppler spread and J0 is theBessel
function of first kind of order zero [19]. The shad-owing component
is eXi(t), where = log 1010 and Xi(t) N (0, 2sh, CXi()). The
covariance function CXi() is givenby the modified Gudmundsons model
as [15]
CXi() = 2sh exp
(v
22
2D2
), (3)
where v is the speed of the CU and D is the
decorrelationdistance. The shadowing and fading seen by different
CUs ontheir links from the PU-Tx and to the PU-Rx are
independentand identically distributed (IID).
C. PU-Tx Based Spectrum Sensing (SS)The accuracy of SS by the CU
depends on the strength of
the signal it receives from the PU-Tx and the SS algorithmused.
We consider the popular ED-based SS algorithm [20]assuming that all
the CUs periodically spectrum sense togetherfor a short duration.2
In [20], the SS algorithm declares thePU-Tx to be on if the energy
received by the CU from it overa time T and bandwidth B exceeds a
threshold .
For this detector, the false alarm probability PFA at the
ith
CU is Q(N0TBN20TB
), where N0 is the noise power and Q
is the Gaussian Q function. The correct detection
probabilitydepends on the signal-to-noise-ratio (SNR) (i, Yi, gi)
ofthe PU-Tx signal at the CU, which in turn depends on thedistance
i of the CU from the PU-Tx, shadowing eYi , andRayleigh fading gi
during the time of sensing. Here, Yi N (0, 2sh) and gi exp (1). The
expression for the correctdetection probability, denoted by PD(i,
Yi, gi), in terms ofPFA is [20]
PD(i, Yi, gi) = Q
(Q1(PFA) (i, Yi, gi)
TB
1 + 2 (i, Yi, gi)
).
(4)Note that our model can include other SS algorithms as
well, e.g., [22]; the expressions for PFA and PD will
differ.
D. CU Transmission and Interference ModelIf a CU detects the
PU-Tx to be on, which we refer to
as hypothesis H1, then it operates in the underlay mode
andtransmits with a lower power Pu. Else, if the CU detects
thePU-Tx to be off, which we refer to as hypothesis H0, then
itoperates in the interweave mode and transmits with a higherpower
Po. In both cases, it transmits in the same band as thePU-Tx.
Therefore, from (1), the interference power Ii(t) atthe PU-Rx from
the ith CU, which is ri(t) distance away, is
Ii(t)=
PuK
(d0
ri(t)
)eXi(t)hi(t), if H1 is detected,
PoK(
d0ri(t)
)eXi(t)hi(t), if H0 is detected.
(5)2If the CUs sense the spectrum asynchronously then the
aggregate inter-
ference from the concurrent transmissions by other CUs has to be
taken intoaccount along the lines of [21].
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432 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO.
1, JANUARY 2014
Note that this two-level transmit power model may not bethe most
suitable one in the presence of interference fromother systems. One
possible way to improve it is to use themore advanced
detect-and-avoid strategy [2]. Further, energydetection might no
longer be the right SS method to use sinceit cannot differentiate
between the sources of interference.
III. INTERFERENCE MODELING: NON-COOPERATIVE SSThe aggregate
interference I(t) at the PU-Rx from the
CUs is
I(t) =
NCRi=1
Ii(t). (6)
Our goal is to accurately model the time evolution of I(t)in
between the SS durations when the PU-Tx is on. We nowdevelop the
moment-matching-based WSS SLN RP model, asper which I(t) is modeled
as
I(t) eZ(t) + s, (7)where Z (t) N (Z , 2Z , CZ()) and s is called
the shiftparameter. Our goal is to determine the constants Z , Z ,
ands, and CZ() by matching them with the corresponding termsof
I(t). To do so, we first express these parameters in termsof the
cumulants of I(t). The cumulants are then written interms of E
[Ii(t)m], for m = 1, 2, 3, and E [Ii(t)Ii(t+ )],which are derived
in Results 1 and 2. It is here that the maincontribution of this
section lies.
In terms of the cumulants of I(t), the parameters Z , Z ,and s
are given by [14]
2Z = log
(1
4
23 + 4
23 1
), (8)
Z =1
2log
(I(2)
e2Z 1
) 1
22Z , (9)
s = I(1) exp(Z +
1
22Z
), (10)
where = 4I(3) + 44 + (I(3))
2 and I(m) is themth cumulant of I(t), which is defined as [19,
(26.1.12)]
I(m) =1
jmdm log
(E[ejI(t)
])dm
=0
,
=1
jm
dm log(ENCR
[(E[ejIi(t)
])NCR])dm
=0
,
(11)where j =
1. The second equality is obtained by using (6),conditioning on
NCR, and using the fact that all Ii(t), for1 i NCR, are IID for the
SPPP that governs the locationsof the CUs. Upon taking the
expectation over NCR and thenevaluating the derivative, we get
I(m) = (R2 R2PEZ
)E [Ii(t)
m] . (12)The SLN RP model also requires CZ(). As shown in
Appendix A, it is equal to
CZ() = log((R2 R2PEZ
)E [Ii(t)Ii(t+ )]
+ e2Z+2Z
) (2Z + 2Z) . (13)
A. With Path-loss and ShadowingWe first analyze the case when
small-scale fading is aver-
aged out. This is of interest when the PU-Rx can average overthe
fast variations of the small-scale fading [11, Chap. 3].
The distance q (ri(t), p, cos i) between the PU-Tx and theith CU
is equal to
q (ri(t), p, cos i) =ri(t)2 + p2 2ri(t)p cos i, (14)
where i is the azimuth angle of the CU. The PDF pi of iand the
PDF pri of ri(t), conditioned on the CU not lying inthe PEZ, are
given by
pi(x) =1
2, 0 x < 2,
pri(x) =2x
R2 R2PEZ, RPEZ x < R. (15)
Result 1: The mth moment of the interference from anarbitrary CU
i is then given by
E [Ii(t)m]
2Pmo Kmdm0
(R2mPEZ R2m
)(m 2) (R2 R2PEZ)
e12m
222sh
2 (Pmo Pmu )Km
Wc (R2 R2PEZ)e
12m
222sh
Whn1=1
wh (n1)
Wln2=1
wl (n2)
Wcn3=1
RRPEZ
dm0rm1i
PD(q (ri, p, ac (n3)) ,
2shah (n1) , al (n2)
)dri. (16)
Proof: The proof is given in Appendix B.Here, wh (n) and ah (n),
for n = 1, . . . ,Wh , de-
note the weights and the abscissas, respectively, of
Gauss-Hermite quadrature, ac (n), for n = 1, . . . ,Wc , denote
theabscissas of Gauss-Chebyshev quadrature, and wl (n) andal (n),
for n = 1, . . . ,Wl , denote the weights and theabscissas,
respectively, of Gauss-Laguerre quadrature
[19,(25.4.38),(25.4.45),(25.4.46)].
Result 2: The auto-correlation of the interference from aCU is
given as follows:
E [Ii(t)Ii(t+ )] P 2oK
2d20
(R22PEZ R22
)( 1) (R2 R2PEZ)
e22sh(1+exp
( v22
2D2
)) 2
(P 2o P 2u
)K2
Wc (R2 R2PEZ)
e22sh(1+exp
( v22
2D2
)) Whn1=1
wh (n1)
Wln2=1
wl (n2)
Wcn3=1 R
RPEZ
d20r21i
PD
(q (ri, p, ac (n3)) ,
2shah (n1) , al (n2)
)dri.
(17)Proof: The proof is given in Appendix C.
The final expressions in (16) and (17) are in the form ofsingle
integrals. These cannot be simplified further becauseof the
presence of the PD term inside the integrand, whichdepends on the
SS algorithm. It is typically quite involved,as can be seen from
(4). The integrals are easily evaluatednumerically. In general, as
the number of the Gauss-quadrature
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ALI and MEHTA: MODELING TIME-VARYING AGGREGATE INTERFERENCE IN
COGNITIVE RADIO SYSTEMS, AND APPLICATION TO PRIMARY . . . 433
terms increases, the error between the integral and the
approx-imating sum decreases [19, (25.4.38),(25.4.45),(25.4.46)].
Wehave found that for our problem Wh = Wc = Wl = 6 issufficient to
ensure accurate complementary CDF (CCDF) andLCR curves for sh
12.
Note that the analysis can be extended to the case wherePU-Rx is
not at the center of the region. In this case, thedistance between
the ith CU and the PU-Rx, which arisesin (5), changes. However, the
expressions of the moments andthe auto-correlation of Ii(t) can
still be obtained in a singleintegral form. The case where the
distance p between the PU-Rx and PU-Tx is an RV can also be
incorporated. In this case,an additional expectation over p will
appear in (16) and (17).
B. With Path-loss, Shadowing, and FadingIf the PU-Rx cannot
average over the small-scale fading,
then this should be taken into account in the aggregate
in-terference model. When small-scale fading is also
consideredalong with path-loss and shadowing, the expression for
themth moment E [Ii(t)m] in (32) will get multiplied by a factorE
[hi(t)
m]. Using E [hi(t)m] = m!, and simplifying furtheryields the
expressions for moments that contain an additionalfactor of m!
compared to (16). Similarly, the auto-correlationE [Ii(t)Ii(t+ )]
in (35) will get multiplied by a factorE [hi(t)hi(t+ )].
Consequently, from (2), the autocorrelationin (17) gets scaled by a
factor (J20 (2fm) + 1).C. CCDF, LCR, and AED of I(t) Based on SLN
RP Model
For a threshold Ith, the CCDF of I(t) is given by [3, (23)]:Pr
(I(t) Ith) = Q
(log (Iths)Z
Z
). The LCR LI(t)(Ith)
of I(t) can be obtained from the level crossing theory
ofGaussian processes [23], the key steps for which are shownbelow.
From [23, Lemma 10.2], the LCR can be written interms of Z(t)
as
LI(t)(Ith) =
limn
[ni=1
Pr(Z
(i 1n
)< log (Ith s) < Z
(i
n
))].
(18)Using [23, Theorem 10.1], we then get
LI(t)(Ith) =
Z
2Zexp
( (log (Ith s) Z)
2
22Z
), Ith > s,
(19)where Z = d2d2CZ()
=0
is computed from (13) and (17).Finally, the AED I(t)(Ith) of
I(t) is the ratio of the CCDFand the LCR:
I(t)(Ith) =Pr (I(t) Ith)
LI(t)(Ith). (20)
IV. INTERFERENCE MODELING: COOPERATIVE SSWe now incorporate
cooperative SS into our model, and
characterize how it affects the aggregate interference RP.
Forthis purpose, we use the OR fusion rule because it is
morepreferable than the AND and majority rules from the point
ofview of protecting the PU-Rx from excessive interference [3].
In it, a CU that has detected the PU-Tx to be on will
broadcastits decision to the CUs that lie within a cooperation
region ofradius RC around it. All these CUs will take a logical OR
withtheir decision, which means that they will also then assumethat
the PU-Tx is on. This is a one shot process the CUsdo not broadcast
their decisions yet again. We assume thatthe interference caused by
the short transmissions of the CUsfor enabling cooperative SS is
negligible. Furthermore, thecommunication is assumed to be
error-free. This is justifiablesince only one bit of information
needs to be communicated,and can be sent with sufficient
protection.
Note that the interference from the CUs will be
correlatedbecause their decisions are correlated due to
cooperation. Tomake the analysis tractable, we use a decoupling
approxima-tion, in which the interferences from the CUs are assumed
tobe uncorrelated, but the effect of correlation in their
decisionsis captured in the probability of correct decision. Such
adecoupling approximation has been used to good effect inanalyzing
wireless local area networks [24]. With this, theexpressions for
the parameters of the SLN RP model of theaggregate interference
with cooperative SS are given by (8),(9), (10), and (13).
As before, to characterize the aggregate interference
withcooperative SS, we need the expressions for E [Ii(t)m] andE
[Ii(t)Ii(t+ )]. These are given below when path-loss andshadowing
are considered. The distance between the PU-Txand the j th CU that
is located within the cooperation rangeof the ith CU is q (rij(t),
q (ri(t), p, cos i) , cos ij), which isgiven by (14), where rij(t)
is the distance between the ith andj th CUs, and ij is the angle
subtended by the lines from thej th CU and the PU-Tx to the ith
CU.
Result 3: The mth moment of interference E [Ii(t)m] fromthe ith
CU with cooperative SS is then given by
E [Ii(t)m]
2Pmo Kmdm0
(R2mPEZ R2m
)(m 2) (R2 R2PEZ)
e12m
222sh
2 (Pmo Pmu )Km
Wc (R2 R2PEZ)e
12m
222sh
Wcn4=1
RRPEZ
dm0rm1i
(1 1 f1 (ri, ac (n4))
1 f2 (ri, ac (n4))eR2Cf2(ri,ac(n4))
)dri, (21)
where
f1 (ri, ac (n4)) =
Whn1=1
wh (n1)
Wln2=1
wl (n2)
PD(q (ri, p, ac (n4)) ,
2shah (n1) , al (n2)
), (22)
f2 (ri, ac (n4)) =2
WcR2C
Whn1=1
wh (n1)
Wln2=1
wl (n2)
Wc
n3=1
RC0
PD
(q (rij, q (ri, p, ac (n4)) , ac (n3)) ,
2shah (n1) , al (n2)
)rijdrij. (23)
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434 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO.
1, JANUARY 2014
TABLE ISIMULATION PARAMETERS
Parameter Variable ValueTransmit power of PU-Tx PTx 10
dBmTransmit power of CU in interweave mode Po 2 dBmTransmit power
of CU in underlay mode Pu -6 dBmNoise power N0 -100 dBmDensity of
CUs 100 CUs/km2System bandwidth B 1 MHzCarrier frequency fc 900
MHzRadius of region considered R 1000 mRadius of PEZ RPEZ 200
mDistance between PU-Tx and PU-Rx p 500 mPath-loss exponent
4Standard deviation of shadow fading sh 6Break-point distance d0 10
mSpeed of CUs v 5 ms1False alarm probability PFA 10%Spectrum
sensing duration T 50 sec
The auto-correlation E [Ii(t)Ii(t+ )] is given by
E [Ii(t)Ii(t+)] P 2oK
2d20
(R22PEZ R22
)( 1) (R2 R2PEZ)
e22sh(1+exp
( v22
2D2
)) 2
(P 2o P 2u
)K2
Wc (R2 R2PEZ)
e22sh(1+exp
( v22
2D2
)) Wcn4=1
RRPEZ
d20r21i
(1 1 f1 (ri, ac (n4))
1 f2 (ri, ac (n4))eR2Cf2(ri,ac(n4))
)dri. (24)
Proof: The proof is given in Appendix D.Compared to the
expressions for the mth moment and auto-correlation in Sec. III-A,
the PD term is different and themoments now depend on the
cooperation range RC .
As before, when Rayleigh fading is also accounted for,
theexpression for E [Ii(t)m] in (21) will get scaled by a factor
m!.The expression for E [Ii(t)Ii(t+ )] in (24) will get scaled bya
factor
(J20 (2fm) + 1
).
V. NUMERICAL RESULTSWe now verify our analysis using Monte Carlo
simulations.
The parameters used are listed in Table I. The simulationsmake
measurements over up to 105 drops. In each drop, arandom number of
CUs and their locations are generated as perthe SPPP. Each CU moves
with a fixed speed v in a randomlychosen direction. Every CU
performs SS as per Sec. II-C, incase of non-cooperative SS, and as
per Sec. IV, in case ofcooperative SS. The aggregate interference
I(t) from all theCUs at the PU-Rx is then measured.
We first present results for non-cooperative SS and thenfor
cooperative SS. In each case, we show results for thesnapshot
statistics and then for the time-varying behavior. Forsnapshot
statistics, we compare the CDF and the CCDF ofthe various models,
as has been done in [2][5], [25], [26].The CDF evaluates the
accuracy in matching smaller aggregateinterference values, while
the CCDF evaluates the accuracy inmatching larger aggregate
interference values, where the CDFsaturates to unity. To compare
the accuracy in modeling the
100 95 90 85 80 75104
103
102
101
100
I(t) in dBm
CDF
and
CCDF
SimulationGaussianLognormalStableSTSSLN
CDF
CCDF
Fig. 2. Non-cooperative SS: Comparison of CDF and CCDF of I(t)
fromvarious models with path-loss and shadowing.
time-variation of the aggregate interference, we study the LCRof
the various models, as has been done in [16], [17], [27].
A. BenchmarkingWe compare the SLN model with the following:
Gaussian model: This model is motivated by the central
limit theorem. In it, I(t) is modeled as a Gaussian RP.
Lognormal model: In this model, log (I(t)) is modeled
as a Gaussian RP, along lines similar to [27]. STS model: In
this model, the amplitude of I(t) is mod-
eled as an STS RV [2]. The parameters that determine itsCF are
obtained by matching the first, second, and fourthcumulants of the
STS RV with the corresponding cumu-lants of the amplitude of I(t).
The CF is numericallyintegrated to get the CDF, as closed-form
expressions forthe latter are not known. Finally, the CDF of I(t)
isdetermined using a variable transformation.
Stable distribution model: In this model, I(t) is modeledas
stable RV [1]. The parameters that determine the CFare obtained
from simulations using the method presentedin [28]. The CDF of I(t)
is then obtained by numericallyintegrating the CF, as a closed-form
expression for it isnot known.
B. Non-cooperative SS: With Path-loss and Shadowing1) Snapshot
Statistics Comparisons: Figure 2 compares the
CDF and the CCDF of the various models when small-scalefading is
averaged out. For lower values of aggregate interfer-ence (I(t)
< 90 dBm), the SLN model underestimates theCDF. However, none of
the models proposed in the literatureare accurate in this regime.3
One reason behind this inaccuracyis the use of the moment-matching
method, which penalizesless the approximation errors for lower
values of interferencethan for higher values of the interference
[26]. Another reasonis that the probability that I(t) is less than
s is zero in theSLN model. For moderate to high values of the
aggregateinterference (I(t) > 90 dBm), the SLN model matches
the
3Since the lognormal model overestimates the CDF while the SLN
modelunderestimates it, better accuracy can be achieved by a
mixture model whoseCDF is the arithmetic mean of the CDFs of these
two models [29].
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ALI and MEHTA: MODELING TIME-VARYING AGGREGATE INTERFERENCE IN
COGNITIVE RADIO SYSTEMS, AND APPLICATION TO PRIMARY . . . 435
100 95 90 85 80104
103
102
101
100
Ith in dBm
LCR
SimulationGaussianLognormalSLN
Fig. 3. Non-cooperative SS: Comparison of LCR of I(t) from
variousmodels with path-loss and shadowing.
CCDF well and is more accurate than all the other models.It
captures the skewness of the interference distribution betterthan
the other models [3]. Note that the 90 dBm value abovearises due to
the particular choice of simulation parameters. Ingeneral, it
increases as the mean of the aggregate interferenceincreases.
We see that the Gaussian and lognormal models fail toprovide a
good fit. For small values of the interference, theCDF of the
Gaussian model saturates at Q
(GG
), where G
is the mean and 2G is the variance. Intuitively, this failureis
because the rate of convergence of the sum of lognormalRVs is very
slow, which is due to the skewed nature of thelognormal PDF [30].
The stable distribution model is better inmatching the CDF.
However, it gives a poor CCDF match forhigher values of the
aggregate interference. The STS modelmatches the CCDF better but
its CDF saturates for smallvalues, as was the case with the
Gaussian model.
2) Time-varying Behavior Comparisons: The LCR of theSLN RP model
and the benchmark models is shown in Fig. 3.We see that the LCR
curve is not symmetric about the meanof the aggregate interference,
which is because the PDF of theaggregate interference is asymmetric
about its mean [3]. Forlow threshold values, the LCR is small
because the aggregateinterference mostly stays above it and seldom
crosses thethreshold. As the threshold increases, the LCR increases
andreaches a maximum value, which depends on the speed of theCUs.
However, as the threshold increases further, the LCRagain starts
decreasing because the interference is less likelyto be high enough
to cross it. We again see that the GaussianRP model is quite
inaccurate for both small and large valuesof interference. For Ith
< 90 dBm, the lognormal RP modeloverestimates the LCR while the
SLN RP model underesti-mates it, which is in line with Fig. 2. For
Ith > 90 dBm, theSLN RP model matches the LCR accurately, and is
the mostaccurate model. As mentioned earlier, corresponding
resultsfor the STS model and the stable distribution model are
notshown because a time-varying model for them is not known.In both
figures, for a CU density of = 100 CUs/km2, theaverage number of
CUs in the entire annular region is 302.As the CU density increases
and exceeds 500 CUs/km2, theproposed model does become more
inaccurate.
100 95 90 85 80 75104
103
102
101
100
I(t) in dBm
CDF
and
CCDF
SimulationLognormalStableSTSSLN
CDF
CCDF
Fig. 4. Non-cooperative SS: Comparison of CDF and CCDF of I(t)
fromvarious models with path-loss, shadowing, and Rayleigh
fading.
100 95 90 85 80 75 70103
102
101
100
101
102
Ith in dBm
LCR
SimulationGaussianLognormalSLN
Fig. 5. Non-cooperative SS: Comparison of LCR of I(t) from
variousmodels with path-loss, shadowing, and Rayleigh fading.
C. Non-cooperative SS: Path-loss, Shadowing, and Fading1)
Snapshot Statistics: Figure 4 compares the CDF and the
CCDF obtained using the various models. To avoid clutter,we do
not show the Gaussian model curves as they are quiteinaccurate. The
observations for the stable and the STS modelsare qualitatively
similar to those in Fig. 2. However, comparedto Fig. 2, the CCDF
curve shifts to the right due to theadditional fluctuations induced
by fading. Again, the SLNmodel matches the CCDF well and is more
accurate than theother models.
2) Time-varying Behavior: The LCR with both shadowingand
Rayleigh fading is shown in Fig. 5. The trends are similarto Fig.
3. However, one important difference is that the max-imum value of
LCR is 100 times higher than in Fig. 3. Thisis because of the
faster fluctuations due to small-scale fading.For Ith < 90 dBm,
the lognormal RP model overestimatesthe LCR whereas the SLN RP
model underestimates it, whichis in line with Fig. 4. For Ith >
90 dBm, the SLN RP modelmatches the LCR accurately, and is the most
accurate model.
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436 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO.
1, JANUARY 2014
90 85 80 75104
103
102
101
100
I(t) in dBm
CCDF
SimulationStableSLN
Without cooperationWith cooperation
Fig. 6. Comparison of CCDF of I(t) with non-cooperative and
co-operative SS from various models with shadowing and Rayleigh
fading( = 200 CUs/km2 and RC = 100 m).
100 95 90 85 80 75 70104
103
102
101
100
101
Ith in dBm
LCR
SimulationGaussianLognormalSLN
Fig. 7. Cooperative SS: Comparison of LCR of I(t) from various
modelswith shadowing and Rayleigh fading ( = 200 CUs/km2 and RC =
100 m).
D. Cooperative SS: With path-loss, Shadowing, and Fading
1) Snapshot Statistics: Figure 6 compares the CCDF ofI(t) using
the stable and SLN models with cooperativeand non-cooperative SS. A
CU density of 200 CUs/km2and a cooperation range of RC = 100 m are
considered.The Gaussian, lognormal, and STS models are not shownto
avoid clutter. The stable models CCDF deviates from thesimulations
for higher values of the interference. We again seethat the SLN
model provides an accurate match.
We also see in Fig. 6 that the CCDF of the aggregateinterference
with cooperation is 2 dB to the left of the CCDFwithout
cooperation. This is because the aggregate interferencehas
decreased. In general, as RC increases, more CUs willcooperate,
detect the PU-Tx to be on, and transmit with lowpower. This reduces
the aggregate interference.
2) Time-varying Behavior: The LCR of the aggregate in-terference
with cooperative SS is shown in Fig. 7 for theGaussian, lognormal,
and SLN RP models. The SLN modelmatches the LCR for higher values
of the interference betterthan the other models. The Gaussian model
is again theleast accurate. For Ith < 90 dBm, the lognormal
modeloverestimates the LCR, and for higher values of
interference,it underestimates the LCR.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1300
350
400
450
500
550
600
650
700
R PEZ
in
met
ers
From simulations without cooperationFrom simulations with
cooperationFrom proposed SLN RP model
Transition point
=
= 2 sec
Fig. 8. Zoomed-in view of PEZ radius as a function of using the
SLN modelwith path-loss and shadowing. Results for non-cooperative
and cooperative SSare compared ( = 100 CUs/km2, RC = 100 m, and Ith
= 95 dBm).The transition point shows the value of below which the
AED constraint isactive.
E. Application to PEZ DesignWe now apply the analytical model
for aggregate inter-
ference to redesign the PEZ based on two constraints. Thefirst
constraint is the classical outage probability constraint,which
mandates that the probability that the aggregate inter-ference I(t)
is greater than a threshold Ith should not exceed(1 ) [3]. The
second constraint is the outage durationconstraint, which is new.
It is motivated by the minimumoutage duration concept [15]. It
mandates that the averagetime duration for which I(t) remains above
Ith should notexceed . An alternate approach is to cast the above
constraintsin terms of the SINR of the PU, as has been done in
[5].However, the analysis is more involved.
Figure 8 plots RPEZ as a function of for different valuesof with
only path-loss and shadowing. Figure 9 plots thecorresponding
results with path-loss, shadowing, and fading.Results obtained by
using the formulae developed for the SLNmodel are compared with
those obtained from an extensiveMonte Carlo simulation-based
search. In both figures, resultsare shown for both non-cooperative
and cooperative SS. The = case corresponds to only the outage
constraint beingactive. Observe the good agreement between the
simulationresults and the results obtained using the SLN model. As
increases, the outage constraint becomes tighter and RPEZ
in-creases. Furthermore, cooperation shrinks RPEZ, which meansthat
more CUs can transmit closer to the PU-Rx withoutexcessively
interfering with it. For = 2 sec in Fig. 8 and = 30 msec in Fig. 9,
the AED constraint is active for < 0.95 and < 0.86,
respectively. Thus, the AED constraintis active for a large range
of values of with and withoutcooperative SS.
VI. CONCLUSIONS
We characterized the aggregate interference caused by CUs,which
transmit with different powers depending on whetherthey sense the
PU-Tx to be on or off. Our model accounted forthe dependence of the
aggregate interference on the randomlocations and number of the
CUs, their imperfect spectrumsensing, and the time-varying nature
of large-scale shadowing
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ALI and MEHTA: MODELING TIME-VARYING AGGREGATE INTERFERENCE IN
COGNITIVE RADIO SYSTEMS, AND APPLICATION TO PRIMARY . . . 437
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9300
350
400
450
500
550
600
650
700
R PEZ
in
met
ers
From simulations without cooperationFrom simulations with
cooperationFrom proposed SLN RP model
Transition point
=
= 30 msec
Fig. 9. Zoomed-in view of PEZ radius as a function of using the
SLN modelwith path-loss, shadowing, and Rayleigh fading for
non-cooperative andcooperative SS ( = 200 CUs/km2, RC = 100 m, and
Ith = 95 dBm).The transition point shows the value of below which
the AED constraint isactive.
and small-scale fading of the various links between the PU-Tx,
PU-Rx, and CUs. We saw that the aggregate interferenceprocess is
well characterized by SLN RP for moderate to highvalues of the
interference, and we developed expressions forits moments and
autocorrelation. These led to expressions forits CCDF, LCR, and
AED. We also saw how cooperativeSS helps reduce the aggregate
interference. Upon applyingthe model to PEZ design, we saw that a
new constraint onthe AED is often active, and should be accounted
for. Oneinteresting problem for future work is to characterize
theaggregate interference in a more general scenario in
whichmultiple PU-Txs communicate with multiple PU-Rxs.
APPENDIXA. Derivation of Covariance Functions CZ() and CI()
To obtain CZ(), we match the covariance functions ofI(t) and
eZ(t) + s. It can be easily shown that
CeZ(t)+s() = E[eZ(t)+Z(t+)
](E
[eZ(t)
])2. (25)
From the expression for the moment generating function(MGF) of
the jointly Gaussian RVs Z(t) and Z(t + ), weget
CeZ(t)+s() = e2Z+
2Z+CZ() e2Z+2Z . (26)
The covariance function CI() of I(t) is given by
CI() = E
[NCRi=1
Ii(t)
NCRk=1
Ik(t+ )
](E
[NCRi=1
Ii(t)
])2.
(27)Conditioning over NCR and evaluating the expectation
overIi(t) and Ii(t+ ), which are independent of NCR, we get
CI() = ENCR
NCR
i=1
NCRk=1,k =i
E [Ii(t)]E [Ik(t+ )]
+
NCRi=1
E [Ii(t)Ii(t+ )]
](ENCR
[NCRi=1
E [Ii(t)]
])2.
(28)
Averaging over NCR, we get
CI() = E [NCR (NCR 1)]E [Ii(t)]E [Ik(t+ )]+ E [NCR]E [Ii(t)Ii(t+
)] (E [NCR]E [Ii(t+ )])2 .
(29)Substituting E [NCR] = A and E
[N2CR
]= A + (A)2,
where A = (R2 R2PEZ
), we get
CI() = (R2 R2PEZ
)E [Ii(t)Ii(t+ )] . (30)
Upon equating (26) and (30), we get (13).
B. Non-cooperative SS: Derivation of E [Ii(t)m] with Path-loss
and Shadowing
Recall that the ith CU transmits with power Pu if itdetects the
PU-Tx to be on, which happens with probabilityPD(q (ri(t), p, cos
i) , Yi, gi). Else, it transmits with powerPo. From the law of
total probability and (5), we get
E [Ii(t)m] = E
[(PoK
(d0ri(t)
)eXi(t)
)m (1 PD(q (ri(t), p, cos i) , Yi, gi))
+
(PuK
(d0ri(t)
)eXi(t)
)mPD(q (ri(t), p, cos i) , Yi, gi)
].
(31)Since Xi(t) is independent of the RVs ri(t), i, gi, and
Yi,rearranging terms results in
E [Ii(t)m] = Pmo K
mE
[(d0ri(t)
)m]E
[emXi(t)
] (Pmo Pmu )KmE
[emXi(t)
] E
[(d0ri(t)
)mPD(q (ri(t), p, cos i) , Yi, gi)
]. (32)
Substituting the MGF of Xi(t) and the PDFs of Yi, gi, andi (from
(15)) in (32), we get
E [Ii(t)m] = Pmo K
me12m
222shE
[(d0ri(t)
)m]
(Pmo Pmu )Kme12m
222sh
e y
2i
22sh
sh2
0
egi 20
1
2
E[(
d0ri(t)
)mPD(q (ri(t), p, cos i) , yi, gi)
]dyidgidi.
(33)Using Gauss-Hermite, Gauss-Laguerre, and Gauss-
Chebyshev quadratures [19] to evaluate the integrals over yi,gi,
and i, respectively, yields
E [Ii(t)m] Pmo Kme
12m
222shE
[(d0ri(t)
)m]
1Wc
(Pmo Pmu )Kme12m
222sh
Whn1=1
wh (n1)
Wln2
wl (n2)
Wcn3=1
E
[(d0ri(t)
)m
PD(q (ri(t), p, ac (n3)) ,
2shah (n1) , al (n2)
)]. (34)
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438 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO.
1, JANUARY 2014
Given the total number of CUs, the location of the ith CU
isuniformly distributed over the region. Substituting the PDF
ofri(t) (from (15)) and simplifying yields (16).
C. Non-cooperative SS: Derivation of E [Ii(t)Ii(t+ )]
withPath-loss and Shadowing
Along lines similar to Appendix B, the autocorrelation ofIi(t)
can be written as
E [Ii(t)Ii(t+ )] = P2oK
2E
[e(Xi(t)+Xi(t+))
] E
[(d20
ri(t)ri(t+ )
)]
(P 2o P 2u)K2E [e(Xi(t)+Xi(t+))] E
[(d20
ri(t)ri(t+ )
)PD(q (ri(t), p, cos i) , Yi, gi)
].
(35)To simplify further, we assume that the distance ri(t)
betweenthe CU and the PU-Rx is larger than the distance traveled
bythe CU in a time duration : ri(t+ ) ri(t). Else, the jointPDF of
ri(t) and ri(t+ ) needs to be taken into account.However, the
time-variation of shadowing is accounted for.Rearranging terms and
substituting the joint MGF of Xi(t)and Xi(t+ ) (by using (3)), we
get
E [Ii(t)Ii(t+ )] P 2oK2e22sh
(1+exp
(v222D2
))
E[
d20ri(t)2
] (P 2o P 2u)K2e22sh
(1+exp
( v22
2D2
))
E[(
d0ri(t)
)2PD(q (ri(t), p, cos i) , Yi, gi)
]. (36)
Averaging over the RVs Yi, gi, and i, as in Appendix B,we
get
E [Ii(t)Ii(t+ )] P 2oK2e22sh
(1+exp
( v22
2D2
))
E[
d20ri(t)2
](P 2o P 2u
)K2
Wce22sh
(1+exp
( v22
2D2
))
Whn1=1
wh (n1)
Wln2=1
wl (n2)
Wcn3=1
E
[(d0ri(t)
)2
PD(q (ri(t), p, ac (n3)) ,
2shah (n1) , al (n2)
)]. (37)
Substituting the PDF of ri(t), which is given in (15),
andsimplifying yields (17).
D. Cooperative SS: With Path-loss and Shadowing1) Derivation of
E [Ii(t)m]: The probability of detection
by a CU with cooperative SS depends on the SNRs of allits
cooperating CUs, which are within a radius RC fromit. Let M be the
number of cooperating CUs and let =[1, 2, . . . , M ] denote their
SNRs. Then, the probability of
detection P (i)D,OR() of the ith CU using the OR fusion rule
isgiven by [18]
P(i)D,OR() = 1 (1 PDi)
Mk=1,k =i
(1 PDj) , (38)
where
PDi = PD(q (ri(t), p, cos i) , Yi, gi) , (39)PDj = PD(q (rij(t),
q (ri(t), p, cos i) , cos ij) , Yj , gj) , k = i,
(40)
And, neglecting boundary effects, M is a Poisson RV withmean R2C
.
Proceeding along lines similar to Appendix B, the mthmoment of
Ii(t) is given by
E [Ii(t)m] Pmo Kme
12m
222shE
[(d0ri(t)
)m] (Pmo Pmu )Kme
12m
222sh
E( d0
ri(t)
)m1 (1 PDi) Mk=1,k =i
(1 PDj) .(41)
Conditioning on the position of the ith CU and M , the
secondexpectation term above becomes
E
( d0
ri(t)
)m1 (1 PDi) Mk=1,k =i
(1 PDj)
= Eri(t),i
[(d0ri(t)
)m(1 (1 E [PDi])EM
[(1 E [PDj])M1
])], (42)
where
E [PDi] f1 (ri(t), cos i) =
e y
2i
22sh
sh2
0
egiPD(q (ri, p, i) , yi, gi) dgidyi, (43)
E [PDj] f2 (ri(t), cos i) =
e y
2i
22sh
sh2
0
egi
20
RC0
rijR2C
PD(q (rij, q (ri(t), p, cos i) , cos ij) , yi, gj)
drijdijdgjdyi. (44)
As in Appendix B, after using Gauss quadrature to evaluatethe
above integrals, the expressions for f1 and f2 simplifyto (22) and
(23), respectively. Substituting (42) in (41) andaveraging over i,
ri(t), and M yields (21).
2) Derivation of E [Ii(t)Ii(t+ )]: Along lines similar
toAppendices B and C, the autocorrelation of Ii(t) can be
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ALI and MEHTA: MODELING TIME-VARYING AGGREGATE INTERFERENCE IN
COGNITIVE RADIO SYSTEMS, AND APPLICATION TO PRIMARY . . . 439
written as
E [Ii(t)Ii(t+ )] = P2oK
2e22sh
(1+exp
(v222D2
))
E[(
d0ri(t)
)2] (P 2o P 2u)K2e22sh
(1+exp
(v222D2
))
E[(
d0ri(t)
)2P
(i)D,OR()
]. (45)
Simplifying further, we get (24).
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Mohd. Shabbir Ali received his Bachelor ofEngineering degree in
Electrical and ElectronicsEng. from the MuffakhamJah College of
Eng. andTechnoglogy (MJCET), Hyderabad, India in 2010.He is an
M.Sc. (Eng.) student in the Dept. ofElectrical Communication at the
Indian Institute ofScience (IISc), Bangalore. His research
interestsinclude cognitive radio networks, homogeneous
andheterogeneous networks, and interference modelingand
management.
Neelesh B. Mehta (S98-M01-SM06) received hisBachelor of
Technology degree in Electronics andCommunications Eng. from the
Indian Institute ofTechnology (IIT), Madras in 1996, and his M.S.
andPh.D. degrees in Electrical Eng. from the CaliforniaInstitute of
Technology, Pasadena, USA in 1997 and2001, respectively. He is now
an Associate Professorin the Dept. of Electrical Communication
Eng.,Indian Institute of Science (IISc), Bangalore. Beforejoining
IISc in 2007, he was a research scientist inUSA from 2001 to 2007
in AT&T Laboratories, NJ,
Broadcom Corp., NJ, and Mitsubishi Electric Research
Laboratories (MERL),MA.
His research includes work on link adaptation, multiple access
protocols,cellular systems, MIMO and antenna selection, cooperative
communications,energy harvesting networks, and cognitive radio. He
was also actively involvedin the Radio Access Network (RAN1)
standardization activities in 3GPPfrom 2003 to 2007. He has
co-authored 45+ IEEE transactions papers,70 conference papers, and
3 book chapters, and is a co-inventor in 20issued US patents. He
has served as a TPC co-chair for tracks/symposia inCOMSNETS 2014,
Globecom 2013, ICC 2013, WISARD 2010 and 2011,NCC 2011, VTC 2009
(Fall), and Chinacom 2008. He is an Editor ofIEEE WIRELESS
COMMUNICATIONS LETTERS, IEEE TRANSACTIONS ONCOMMUNICATIONS, and the
Journal of Communications and Networks. Hecurrently serves as the
Director of Conference Publications on the Board ofGovernors of
IEEE ComSoc.