Statistical modeling of cognitive network interference
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Rabbachin, Alberto, Tony Q. S. Quek, and Moe Z. Win.
“Statistical Modeling of Cognitive Network Interference.”
GLOBECOM 2010, 2010 IEEE Global Telecommunications
Conference 2010. 1-6. © 2011 IEEE.
As Published
http://dx.doi.org/10.1109/GLOCOM.2010.5683500
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Institute of Electrical and Electronics Engineers
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings.
Statistical Modeling of
Cognitive Network Interference
Alberto Rabbachin∗, Tony Q.S. Quek†, and Moe Z. Win‡
† Institute
∗ Joint
Research Centre European Commission 21027 Ispra, Italy,
for Infocomm Research, A∗ STAR, 1 Fusionopolis Way, #21-01 Connexis, Singapore 138632,
‡ Massachusetts Institute of Technology Cambridge, MA 02139, USA.
Abstract—In this paper, we propose a new statistical interference model for cognitive network based on the amplitude
aggregate interference, which accounts for the parameters related
to the sensing procedure, spatial reuse protocol employed by secondary users, and environment dependent conditions like channel
fading and shadowing. We derive the characteristic function
and the nth cumulant of the cognitive network interference
on the primary user. By using the theory of truncated-stable
distribution, we show how we can approximate the cognitive
network interference analytically. We further show how to
apply our model to derive system performance measure such
as bit error probability in the presence of cognitive network
interference. Moreover, this work can serve to bring additional
understanding of cognitive network interference for successful
deployment of cognitive networks in the future.
I. I NTRODUCTION
Opportunistic spectrum access allows the opening of underutilized portions of the licensed spectrum for secondary reuse,
provided that the transmissions of secondary radios do not
cause harmful interference to primary network. To enable
secondary users to accurately detect and to access the idle
spectrum, cognitive radio has been proposed as the enabling
technology.However, spectrum sensing is challenged by the
uncertainty in the aggregate interference in the network. Such
uncertainty can be resulted from the unknown number of
interferers, the unknown location of the interferers, the effect
of channel fading, shadowing, and other uncertain environment
dependent conditions Therefore, it is crucial to study how such
uncertainty can be incorporated in the statistical aggregate
interference model and the effect of the cognitive network
interference on the primary network system performance.
Throughout this paper, we refer to the aggregate interference
generated by secondary users sharing the same spectrum of
the primary user as cognitive network interference [1]. In
[2], the authors study the coexistence between wideband and
narrowband communication systems, where they have applied
stable distribution to model the aggregate interference. In
[3], the authors assume the presence of a control channel to
indicate the presence of primary receivers and the aggregate
interference power is modeled as the sum of all interferers’
power. In [4], the authors compare bounded and unbounded
path-loss models, showing that the unbounded path-loss model
might results in significant deviations from more realistic
performance. In [5], an expression for the moment of the
aggregate interference distribution generated by nodes located
in an arbitrary area is derived. Although the above models
allow one to analyze cognitive network interference, they
cannot be easily extended to perform error probability analysis
of primary network in the presence of cognitive network interference. Moreover, these models usually assume unbounded
path-loss model and are unable to capture the instantaneous
cognitive interference conditions.
In this paper, we propose a new statistical interference
model for cognitive network based on the amplitude aggregate interference, which accounts for the parameters related
to the sensing procedure, spatial reuse protocol employed
by secondary users, and environment dependent conditions
like channel fading and shadowing. Moreover, our statistical
model allows us to model the cognitive network interference
generated by secondary users confined in a limited region. By
using the theory of truncated-stable distribution, we show how
we can approximate the cognitive network interference analytically. We further show how to derive system performance
measure such as bit error probability (BEP) in the presence
of cognitive network interference. The paper is organized as
follows. Section II presents the system model. Section III
derives the exact and approximate instantaneous interference
distribution. Section IV demonstrates how these results can be
applied in BEP analysis. Section V provides numerical results
to illustrate the coexistence between primary and secondary
networks depends on various system parameters. Section VI
gives the conclusion.
II. S YSTEM M ODEL
For cognitive networks, the secondary users need to sense
channels before transmission in order not to cause harmful interference to a primary network. In this paper, we
consider the primary network in frequency division duplex
mode. Therefore, to detect the presence of active primary
users, the secondary user senses the uplink channel while the
transmission occupies the downlink channel of the primary
system. Furthermore, we consider the secondary network as
a simple ad-hoc network where secondary users join/exit the
network and sense/access the channel independently without
coordinating with other secondary users As such, there exists
the possibility that secondary users can transmit at the same
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time regardless of their distances between each other.1
B. Interference Model
A. Cognitive Network Activity Model
The interference signal at the primary receiver generated by
the ith cognitive interferer can be written as
Ii = PI ri−ν Xi ,
(6)
The activity of each secondary user depends on its received
signal strength of the uplink signal transmitted by the primary
user. In the following, we consider two types of secondary
spatial reuse protocols, namely: single-threshold and multiplethreshold protocols.
1) Single-Threshold Protocol: In this case, a secondary user
is said to be active if
KPp Y
≤ β,
(1)
r2ν
or equivalently,
r−2ν Y ≤ ζ,
(2)
β
is the normalized
where β is the activating threshold; ζ KP
p
threshold; Pp is the transmitted power of the primary user; Y
is the squared fading path strength from the primary user to
the secondary user; K is the gain accounting for the loss in the
near-field; r is the distance between the primary and secondary
users; and ν is the amplitude pass-loss exponent.2 Therefore,
the activity of the secondary network users can be represented
by the Bernoulli random variable:
1[0,ζ] r−2ν Y ∼ Bern FY r2ν ζ ,
(3)
with the indicator function defined as
1, if a ≤ x ≤ b
1[a,b] (x) =
0, otherwise,
(4)
where the value one of the Bernoulli variable denotes that the
secondary user is active.
2) Multiple-Threshold Protocol: For this case, the transmission power of the secondary network users is set according
to the detected power level of the primary network uplink
signal [6]. We consider N − 1 normalized threshold values
ζ1 , ζ2 , . . . , ζN −1 in increasing order to identify N different
zones (or sets) of active secondary users, denoted by Ak ,
k = 1, 2, . . . , N . Let ζ0 = 0 and ζN = ∞. Then, the kth
active zone Ak obeys the following activating rule:3
(pt) 0, r2ν ζk−1 , r2ν ζk . (5)
1[ζk−1 ,ζk ] r−2ν Y ∼ Bern μY
Note that the received primary network signal power at the
active secondary user in the zone Ak is between KPp ζk−1
and KPp ζk .
1 When a more intelligent medium access control protocol that involves
some form of coordination or local information exchange is feasible for the
secondary network, our results can still serve as a worst-case scenario analysis.
2 For brevity, we assume that the noise effect is negligible on the primary
detection procedure as in [3]. This assumption is intuitively due to the fact
that the harmful interference is especially caused by nodes in the vicinity of
the victim receiver. In this case, the power of the primary user signal is much
higher than the noise power.
3 The zeroth-order partial moment μ(pt) (0, l, u) of the random variable X
X
can be written in terms of its cumulative distribution function (CDF) as
(pt)
μX
(0, l, u) = FX (u) − FX (l) .
where PI is the interference signal power at the limit of the
near-far region;4 ri is the distance between the ith interferer
and the primary receiver; and Xi is the per-dimension fading
channel gain, i.e., the real or imaginary part of the channel gain
Hi from the ith cognitive interferer to the primary receiver. In
the following, we assume that Y in (1) and Xi in (6) are
statistically independent.
In our model, we consider that the secondary users are
spatially scattered according to an homogeneous Poisson point
process in a two-dimensional plane R2 , where the victim
primary user is assumed to be located at the center of the
region. The probability that k secondary users lie inside a
region R ⊂ R2 depends only on the total area AR of the
region, and is given by [7]
(λAR )k −λAR
e
,
k = 0, 1, 2, . . . (7)
k!
where λ is the spatial density (in nodes per unit area).
Furthermore, we assume that the region R is limited by
minimum and maximum distances from the primary receiver,
denoted by Rs and Rf , respectively.5 In this way, this model
will enable us to consider a scenario where the secondary users
are only located within a limited region.
P {k in R} =
III. I NSTANTANEOUS I NTERFERENCE D ISTRIBUTION
To introduce our statistical model for the cognitive network
interference, we first derive the cumulants of the cognitive
network interference for random activity (the activity of each
secondary user is regulated by a spatial reuse protocol) in
Section III-A. Next, in Section III-B we develop the symmetric truncated-stable approximation for the cognitive network
interference using its cumulant expressions.
A. Random Activity
1) Single-Threshold Protocol: In this spatial reuse protocol,
the secondary users in the region R are activated by the single
normalized threshold ζ using (3). Therefore, the cognitive
network interference for the single-threshold protocol can be
written as
−ν
Ist = PI
ri Xi ,
(8)
i∈Ast
Zst (ζ;R)
where Ast defines the set of secondary users activated by the
single normalized threshold ζ in the region R:
Ast = i ∈ R : 1[0,ζ] ri−2ν Y = 1 .
(9)
consider the near-far region limit at 1 meter.
that ri in (6) can be smaller than 1. Therefore, the received
interference power can be larger than PI but it is finite since Rs > 0.
4 We
5 Note
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The characteristic function (CF) of Zst (ζ; R) can be expressed
as
Rf
exp jsxr−ν − 1
ψZst (ζ;R) (js) = exp 2πλ
X
×1[0,ζ] r
−2ν
Y
Rs
y fX (x) fY (y) rdrdydx .
(10)
Using (10), we can then calculate the nth cumulant of the
interference as
1 dn ln ψZst (js) Zst (ζ; R) = n
j
dsn
s=0
2πλ μX (n)
Rs2−nν − Rf2−nν FY Rs2ν ζ
=
nν − 2
nν−2
2
−
nν
(pt)
2ν
2ν
, Rs ζ, Rf ζ
+ ζ 2ν μY
2ν
2−nν (pt)
2ν
2ν
− Rf
0, Rs ζ, Rf ζ .
μY
(11)
Using the cumulant of Zst (ζ; R), we can obtain the nth
cumulant of the cognitive network interference Ist for the
single-threshold protocol as follows:
n/2
κIst (n) = PI
κZst (ζ;R) (n) .
(12)
2) Multiple-Threshold Protocol: Using (5), the per-zone
cognitive interference generated by the secondary users in the
kth active zone Ak can be written as
ri−ν Xi ,
(13)
Imt,k = PI,k
i∈Ak
Zk (R)
where PI,k is the transmitted power of the secondary users in
the kth active zone Ak and
(14)
Ak = i ∈ R : 1[ζk−1 ,ζk ] ri−2ν Y = 1 .
Similar to (10), the CF of Zk (R) can be expressed as
Rf
exp jsxr−ν − 1
ψZk (R) (js) = exp 2πλ
×1[ζk−1 ,ζk ] r
X
−2ν
Y
Rs
y fX (x) fY (y) rdrdydx .
(15)
The cognitive network interference generated by the secondary users in all the N active zones is then given by
Imt
N
=
PI,k Zk (R) .
(16)
k=1
Since all the Zk (R)’s are statistically independent,6 we obtain
the nth cumulant of the cognitive network interference Imt for
6 The 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) .
the multiple-threshold protocol as
κImt (n) =
N
k=1
n/2
PI,k κZk (R) (n) ,
(17)
where κZk (R) (n) are given by (11), (18), and (20) for k = 1,
k = 2, 3, . . . , N − 1, and k = N , respectively.
B. Truncated-Stable Distribution Approximation
The truncated-stable distributions are a relatively new class
of distributions that follow from the class of stable distributions [8]. The attractiveness of using stable distributions to
model interference in wireless networks are: 1) the ability to
capture the spatial distribution of the interfering nodes; and
2) the ability to accommodate heavy tails that exhibit the
dominant contribution of a few interferers in the vicinity of
the primary user [9]. However, as shown in [1], the aggregate
interference converges to a stable distribution only if the
number of interferers is unbounded and if ri ∈ [0, ∞]. The
main drawback of this model is that, with small but nonzero
probability, the distance r can be zero and hence, the interference power can be infinite. This singularity at r = 0 is taken
account of in the stable distributions with unbounded (infinite)
second or higher order moments. However, the truncatedstable distributions with smoothed tails and finite moments
can offer an alternative statistical tool to model the aggregate
interference in more realistic scenarios where this singularity
can be avoided.
The CF of a symmetric truncated-stable random variable
T ∼ St (γ , α, g) is given by [11]
α
α
(g + js)
(g − js)
α
+
−g
,
ψT (js)=exp γ Γ (−α)
2
2
(21)
where Γ (·) is the Euler’s gamma function; and γ , α and g are
the parameters associated with the truncated-stable distribution. The parameters γ and α are akin to the dispersion and the
characteristic exponent of the stable distribution, respectively.
The parameter g is the argument of the exponential function
used to smooth the tail of the stable distribution. The nth
cumulant of the truncated-stable distribution can be obtained
using (21) as
n−1
1 dn ln ψT (js) α−n
=
γ
Γ
(−α)
g
(α − i) . (22)
jn
dsn
i=0
s=0
To approximate the cognitive network interference to the
truncated-stable distribution, we first fix the characteristic
exponent to α = 2/ν. This choice is motivated by the fact that
as Rs → 0 and Rf → ∞, the cognitive network interference
follows a stable distribution with the characteristic exponent
α = 2/ν. Let IA be the cognitive network interference
corresponding to the activity model A ∈ {fa, st, mt}, i.e.,
full activity, the single-threshold protocol, or the multiplethreshold protocol. Then, we can approximate the cognitive
network interference IA to the symmetric truncated-stable
random variable as IA ∼ St (γA
, α = 2/ν, gA), where the
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nν−2
2πλ μX (n) 2−nν (pt) 2 − nν 2ν
(pt)
2ν
Rs
, Rs ζk−1 , Δmin
0, Rs2ν ζk−1 , Δmin − ζk−1
κZk (R) (n) =
μY
μY
nν − 2
2ν
nν−2
2 − nν
(pt)
(pt)
+ c1 μY (c2 , Δmin , Δmax ) + ζk 2ν μY
, Δmax , Rf2ν ζk
2ν
(pt) − Rf2−nν μY
0, Δmax , Rf2ν ζk ,
where Δmin = min Rf2ν ζk−1 , Rs2ν ζk , Δmax = max Rf2ν ζk−1 , Rs2ν ζk , and
2−nν
− Rf2−nν , 0 ,
if Rs2ν ζk ≥ Rf2ν ζk−1
Rs
nν−2
nν−2
(c1 , c2 ) =
2ν
ζk 2ν − ζk−1
, if Rs2ν ζk < Rf2ν ζk−1 .
, 2−nν
2ν
(19)
nν−2
2πλ μX (n) 2−nν (pt) 2 − nν 2ν
(pt)
2ν
Rs
, Rs ζk−1 , Rf2ν ζk−1
κZN (R) (n) =
0, Rs2ν ζk−1 , Rf2ν ζk−1 − ζk−1
μY
μY
nν − 2
2ν
+ Rs2−nν − Rf2−nν F̄Y Rf2ν ζk−1 .
simulation
truncated−stable distribution
−1
CCDF
10
−2
10
−3
10
Ist =
−4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Received signal strength
dispersion and smoothing parameters γA
and gA are given
in terms of the second and fourth cumulants of IA , using the
relationship between the cumulants and parameters in (22), as
follows [10]:
gA
L
PI,
ri−ν Xi ,
=1
Fig. 1. CCDF of the cognitive network interference Imt for the multiplethreshold protocol with λ = 0.1, PI,1 = 0 dBm for A1 , PI,2 = −23.7 dBm
for A2 , and PI,3 = −38.7 dBm for √
A3 . Both primary and secondary
signals experience Rayleigh fading, i.e., Y ∼ Rayleigh (1/2) and |Hi | ∼
Rayleigh (1/2).
γA
κIA (2)
=
,
α−2
2
κIA (2)(α−2)(α−3)
Γ (−α) α (α − 1)
κIA (4)
κIA (2) (α − 2) (α − 3)
.
=
κIA (4)
(20)
two thresholds (i.e., N = 3). Fig. 1 shows the complementary
CDF (CCDF) of the cognitive network interference Imt . We
can observe from Fig 1 that the simulation results match well
with the truncated-stable statistical approximation.
With the symmetric truncated-stable approximation, we can
also account for shadowing in the statistical characterization
of the cognitive network interference. For example, consider
the single-threshold protocol and divide the whole region R
into different subregions R1 , R2 , . . . , RL corresponding to
the positions of the obstacles and the additional attenuation
for users behind those obstacles. Then, the cognitive network
interference can be written as
0
10
10
(18)
(23)
To validate our statistical model, we consider a circular
region defined by Rs = 1 meter, Rf = 60 meters. We
consider the multiple-threshold protocol implemented using
Zst (ζ β̆ ;R )
(25)
where PI, and β̆ account for an additional attenuation for the
subregion R behind the obstacle and
Ast, = i ∈ R : 1[0,ζ β̆ ] ri−2ν Y = 1 .
(26)
The CF of Zst ζ β̆ ; R ) can be expressed as
ψZst (ζ β̆ ;R ) (js)
= exp θ λ
(24)
i∈Ast,
×1[0,ζ β̆ ] r
X
−2ν
Y
b
a
exp jsxr−ν − 1
y fX (x) fY (y) rdrdydx .
(27)
where a and b are the limits of the subregion R ; and θ is
the angle covered by R . If the obstacle is present, the angle
θ corresponds to the angle covered by the obstacle. For a
single obstacle placed at distance d from the origin, we have
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0
60
10
analysis
simulation
40
SIR = −16 dB
−1
10
SIR = −12 dB
20
−2
BEP
meters
10
0
−3
10
SIR = −8 dB
−20
−4
10
−40
−60
−60
−40
−20
0
20
40
60
0
5
Fig. 2. Node displacements of a cognitive radio network with the singlethreshold protocol in the presence of shadowing (not only a single realization
snapshot). Rs = 1 meter, Rf = 60 meters, λ = 0.01, ζ = −40 dBm,
θ1 = θ2 = π/2, and β̆1 = β̆2 = 20 dB. The shadowing is characterized by
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. The blue and
green colors represent inactive and active nodes, respectively.
two subregions in front and behind the obstacle: (a1 , b1 ) =
(Rs , d) and (a2 , b2 ) = (d, Rf ), respectively. The nth cumulant
of the cognitive network interference for the single-threshold
protocol in the presence of shadowing can be written as
κIst (n) =
L
=1
n/2
10
15
20
25
Eb /N0 [dB]
meters
PI, κZst (ζ β̆ ;R ) (n)
(28)
where the cumulant κZst (ζ β̆ ;R ) (n) is obtained from
κZst (ζ;R) (n) in (11) by setting ζ, 2π, Rs , and Rf to ζ β̆ ,
θ , a , and b , respectively. Fig. 2 shows realization snapshots
of secondary users activated by the single-threshold protocol
in the presence of shadowing.
C. BEP Analysis
Consider a binary phase-shift keying (BPSK) narrowband
system in the presence of interference generated by the cognitive network confined within the region R. The nodes of the
cognitive network are active according to (3). The decision
variable of the primary received symbol after the correlation
receiver can be written as
V = GU Eb + Ist + W ,
(29)
Fig. 3. BEP of BPSK versus Eb /N0 in the presence of the cognitive network
interference Ist for the single-threshold protocol when SIR = −16, −12, and
−8 dB. λ = 0.1 and ζ = −40 dBm, and Rayleigh fading for primary and
secondary links. For comparison, the BEP in the absence of interference is
also plotted (black line)
Assuming that G and Ist are statistically independent, the CF
of the decision variable conditioned on U = +1 is given by
N 0 s2
.
ψV jsU = +1 = ψG js Eb ψIst (js) exp −
4
(31)
For the cognitive network interference Ist , we
use
the
symmetric
truncated-stable
approximation
, α = 2/ν, gst ), where the parameters γst
Ist ∼ St (γst
and gst are determined by using (23) and (24), respectively.
Since Ist is approximated as a symmetric random variable, the
average BEP Pe is equal to the BEP conditioned on U = +1,
which can be expressed, using the inversion theorem [12], as
Pe = P V < 0U = +1
∞
ψV jsU = +1 + ψV −jsU = +1
1
1
ds.
= +
2 2π 0
js
(32)
IV. N UMERICAL RESULTS
In this section, we illustrate how our interference model can
provide insight into the coexistence of primary and secondary
networks. In numerical examples, we consider Rs = 1 meter,
Rf = 60 meters, ν = 1. We first investigate the effect of the
cognitive network interference on the BEP performance of the
primary user. In Fig. 3, the BEP of BPSK versus Eb /N0 is
depicted at the signal-to-interference ratio SIR Eb /PI in the
presence of Rayleigh fading for both primary and secondary
where G is the channel fading affecting the victim signal;
signals. We can observe from Fig. 3 that the simulation and
U ∈ {1, −1} is the information data; Eb is the energy per bit;
analytical results match very well, which verify both our BEP
and W is the zero-mean additive white Gaussian noise with
analysis and the truncated-stable interference approximation.
variance N0 /2. Conditioned on G, Ist , and U = +1, the CF
To demonstrate the effect of fading on the cognitive network
of the decision variable V can be written as
interference,
we next consider Nakagami-m fading for both
N s2 !
primary
and
secondary signals. Fig. 4 shows the variance
0
.
ψV jsG, Ist , U = +1 = exp js G Eb + Ist −
(or
equivalently,
average power) of the cognitive network
4
(30) interference Ist for the single-threshold protocol.as a function
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−3
2
x 10
−1
10
m=1
m=3
m=5
1.8
1.6
Rf
Rf
Rf
Rf
−2
10
1.4
= 30m, ζ = −40dB
= 60m, ζ = −40dB
= 60m, ζ = −60dB
= 120m, ζ = −60dB
−3
Var {Ist }
Var {Ist }
10
1.2
1
0.8
−4
10
−5
10
0.6
0.4
−6
10
0.2
0
0
10
20
30
40
50
60
Maximum distance Rf (meters)
−7
10
1
1.5
2
2.5
3
3.5
4
4.5
5
ν
Fig. 4. Variance of the cognitive network interference Ist for the singlethreshold protocol as a function of the maximum distance Rf of the region
R when the Nakagami fading parameters m = 1, 3, and 5. λ = 0.01,
ζ = −30 dBm,√PI = 0 dBm, and Nakagami-m fading for primary and
secondary links Y ∼ Nakagami (m, 1) and |Hi | ∼ Nakagami (m, 1).
of the maximum distance Rf from the primary user.This
example reveals that for fixed threshold ζ, as the the fading
parameter m increases (less severe fading), the cognitive
interference vanishes in the vicinity of the primary user due to
rare secondary activity. We can also see that less severe fading
(i.e., larger m) reduces the cognitive interference power for
all the values of Rf . This is due to the fact that more severe
fading increases the secondary activity in the proximity of
the primary user, leading consequently to a higher cognitive
interference power. This finding is in contrast to common
folklore that more severe fading on the interference signals
would be beneficial. Moreover, we observe that the aggregate
average interference power tends to saturate since secondary
users located far from the primary user contribute marginally
to cognitive interference.
An important insight is given in Fig. 5 where the interference power is plotted versus the amplitude path-loss
coefficient. As shown in this figure, the cognitive network
interference exhibits a non-monotonic behavior with an increasing path-loss coefficient. Similarly to the channel fading, a variation of the path-loss coefficient affects both the
interference signal power and the primary signal detection.
Under certain condition of m, Rf , and ζ, an higher path-loss
coefficient might lead to an higher activity of the nodes that
is not compensated by a reduction of the interference signal
power.
V. C ONCLUSIONS
In this paper, we have proposed a new statistical interference
model for cognitive network based on the amplitude aggregate
interference. Based on two spatial reuse protocols, we have
derived the characteristic function and the nth cumulant of the
cognitive network interference at the primary user. By using
the theory of truncated-stable distribution, we have shown
how we can approximate the cognitive network interference
Fig. 5. Variance of the cognitive network interference Ist for the singlethreshold protocol as a function of the path-loss coefficient ν of the region
R for λ = 0.1, √
ζ = −30 dBm, PI = 0 dBm, and fading for primary and
secondary links Y ∼ Nakagami (2, 1) and |Hi | ∼ Nakagami (2, 1).
analytically. We further extendeded our model to include the
effect of shadowing and showed how to derive the bit error
probability in the presence of cognitive network interference.
In summary, the proposed framework can help to understand
cognitive interference for successful deployment of future
cognitive networks.
ACKNOWLEDGMENT
The authors gratefully acknowledge helpful discussions with
H. Shin.
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