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Aggregate Interference and Capacity-Outage
Analysis in a Cognitive Radio Network
Mahsa Derakhshani, Student Member, IEEE, Tho Le-Ngoc, Fellow, IEEE
Abstract –This paper presents a study on interference caused
by Secondary Users (SUs) due to miss-detection and its effects on
the capacity-outage performance of the Primary User (PU) in a
cognitive radio network assuming Rayleigh and Nakagami
fading channels. The effect of beacon transmitter placement on
aggregate interference distribution and capacity-outage
performance is studied considering two scenarios of beacon
transmitter placement: beacon transmitter located at PU
transmitter or at PU receiver. Based on the developed statistical
models for the interference distribution, closed-form expressions
for the capacity-outage probability of the PU are derived to
examine the effects of various system parameters on the
performance of the PU in presence of interference from SUs. It is
shown that beacon transmitter at PU receiver imposes less
interference and hence better capacity-outage probability to the
PU than beacon transmitter at PU transmitter. Furthermore, the
model is extended to investigate the cooperative sensing effect on
aggregate interference statistical model and capacity-outage
performance considering OR (i.e., logical OR operation) and
Maximum Likelihood (ML) cooperative detection techniques.
Simulation results are also provided to verify the developed
analytical models.
Index Terms—Cognitive radio, Aggregate interference,
Capacity-outage probability, Beacon, Cooperative sensing.
I. INTRODUCTION
In cognitive communications [1]-[4], beacon signaling can
be used by the Primary User (PU) in order to facilitate
Secondary Users (SUs) in detection of spectrum holes. Upon
detecting the beacon, SUs will keep silent to avoid
interference to the PU. Although beacon is designed to
improve the PU detection performance at each SU, there is a
non-zero probability of beacon miss-detection due to noise
and channel fading, and in such a case, SU transmission will
cause interference to the PU [5]-[7]. How this interference
Manuscript received July 10, 2010; revised March 28, 2011, June 20, 2011
and September 19, 2011; accepted October 2, 2011. The editor coordinating
the review of this paper and approving it for publication was Prof. Y. Gong.
This work was supported in part by the Natural Sciences and Engineering
Research Council (NSERC) Strategic Project Grant, and in part by the
NSERC Discovery Grant.
Copyright (c) 2011 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
M. Derakhshani and T. Le-Ngoc are with the Department of Electrical
Engineering, Department of Electrical and Computer Engineering, McGill
University, Montreal, QC, Canada H3A 2A7 (e-mail:
mahsa.derakhshani@mail.mcgill.ca; tho.le-ngoc@mcgill.ca).
caused by SUs affects the performance of the PU and how it
relates to design parameters are of interest.
In a cognitive network with beacon, different locations are
considered for the beacon transmitter to study aggregate
interference. The beacon transmitter is located at PU receiver
[7]-[8] or at PU transmitter [9]-[10]. Then, it would be
interesting to analyze the effect of beacon transmitter location
on aggregate interference caused by SUs and performance of
the PU dealing with aggregate interference.
In this paper, the probabilistic properties of interference
caused by SUs are investigated in a cognitive wireless
network by taking into account random variations in location
of SUs as well as propagation characteristics. In the case of
locating beacon transmitter at PU receiver, analysis shows
that the aggregate interference has a Gamma distribution in
Rayleigh fading channels. For the case of beacon transmitter
located at PU transmitter, it is shown that shifted-Gamma
distribution is a better fit to characterize the aggregate
interference in Rayleigh fading channels. Furthermore,
interference distribution analysis is extended to Nakagami
fading channels by proposing Gamma distribution as a close
approximation for interference distribution. In this case,
beacon transmitter is assumed to be located at PU receiver
since it causes lower interference than the case with beacon
transmitter at PU transmitter. In addition, the capacity-outage
probabilities of the PU are developed to examine the effects
of various system parameters on the performance of the PU in
the presence of interference from SUs for two scenarios in
Rayleigh fading channels. Then, the effect of beacon
transmitter location is analyzed by comparing capacity-outage
probability in two different scenarios
1
.
This paper also presents a study on the effect of applying
cooperative sensing [12]-[15] in a cognitive radio network on
mitigating interference considering OR and ML cooperative
techniques by looking into interference distribution and
capacity-outage probability of PU assuming that the PU
beacon transmitter located at PU receiver since locating
beacon transmitter at PU receiver imposes lower interference.
In Nakagami fading channels, the aggregate interference with
cooperative sensing is approximated as a Gamma random
variable. The analysis of capacity-outage probability of the
PU shows that cooperation can be used to maintain the
capacity-outage probability of PU at a desired level by
offering lower interference instead of enhancing the
sensitivity of the individual detectors. Comparing the
capacity-outage probabilities and interference means, it can be
1
Parts of this paper have been presented in [11]
concluded that ML cooperative sensing imposes less
interference in comparison with OR cooperative sensing.
Recently, the characteristics of aggregate interference in
cognitive radio networks are investigated in several works.
Past studies mostly investigate the aggregate interference
analysis in underlay cognitive radio networks [4] without
taking into account sensing errors [16]-[21]. In this paper, we
focus on interweave cognitive radio networks in which SUs
identify and utilize spectrum holes. In interweave cognitive
radio networks, interference characteristics have been
explored in [8]-[9], [22]-[24] considering sensing errors.
In [8], the effect of cooperative sensing is discussed on the
aggregate interference and distribution of aggregate
interference is approximated by a Log-normal distribution. In
comparison with [8], we provide a more precise
approximation for interference distribution assuming
Nakagami fading channels by proposing Gamma distribution.
In addition, we study the effect of interference on the
performance of the PU by deriving closed-form expressions
for the capacity-outage probability of PU. Also, the beacon
placement effect on the aggregate interference is studied in
this paper.
In [9], the upper bounds for mean and variance of aggregate
interference in Rayleigh fading channels are derived.
Furthermore, the aggregate interference is approximately
modeled with a Gaussian random variable based on the
central limit theorem for a large number of SUs. However, in
this paper, it is shown that the aggregate interference is in fact
a positively skewed and right-tailed random variable.
The effect of sensing performance and power control on the
aggregate interference are discussed in [22]. Although, the
aggregate interference is modeled as a deterministic value
since the effect of random SU locations are not taken into
account in that work. Moreover, a statistical model for the
aggregate interference caused by SUs is developed in [23],
although only path loss is assumed for the propagation
channels. In [24], the characteristic function and the
cumulants of aggregate interference are numerically derived
considering fading channels.
The remainder of this paper is organized as follows. After a
brief overview of the system configuration and modeling in
Section II, Section III provides the probability density
function (pdf) of the aggregate interference over Nakagami
fading channels and the closed-form expression for the
capacity-outage probability when beacon transmitter is at PU
receiver. Section IV presents interference distribution and the
closed-form expression for the capacity-outage probability for
the case in which beacon transmitter is at PU transmitter and
performance comparison to study the beacon transmitter
placement effect. The interference and capacity-outage
analysis of a network with cooperative sensing are studied in
Section V. Finally, Section VI provides the conclusions.
II. S
YSTEM CONFIGURATION AND MODELING
Consider a PU communications link of distance
surrounded by secondary (cognitive) users. The distance
between the
SU transmitter and the PU receiver,
, is in
the range from to , i.e.,
where
. In other
i
Fig. 1. Network model.
words, is the minimum allowable distance between a
transmitter and a receiver, and SUs are assumed to be
uniformly located with a density of SUs per unit area in a
ring centered at the PU receiver with inner radius and outer
radius of . Under these assumptions,
is a random variable
with pdf
2
, where
, and the
angle
which the
SU transmitter makes to the line
connecting the PU transmitter and receiver, and is uniformly
distributed between 0 and 2 (see Figure 1).
The wireless channel model includes path loss and small-
scale fading, i.e., the channel response can be expressed as
/
, where is the distance between the
transmitter and the receiver under consideration, 2 is the
path-loss exponent, is a constant dependent on the
frequency and transmitter/receiver antenna gain,
represents
the small-scale fading component. Since such model is valid
for 0, we assume that transmitters cannot be located
closer than an arbitrarily small distance 0 (i.e., minimum
allowable distance between a transmitter and a receiver). For
simplicity, without loss of generality, we normalize 1 in
the following discussion. The channel response of the PU
transmitter-receiver link is denoted as
while that of the
interference link from the
SU transmitter to the PU
receiver is denoted as
.
Each SU has an active factor of , which is the probability
that the user is actively transmitting. To indicate the presence
of PU transmission, a beacon signal is transmitted on an out-
of-band control channel. If a SU correctly detects the beacon,
it will be silent for the whole PU transmission period. In the
case it miss-detects the beacon, the SU transmits concurrently
with the PU with a probability that is its activity factor , and,
as a result, it may introduce interference to the PU. Assume
an energy detection scheme in which the SU declares the
beacon presence if its received power from the beacon is
larger than a threshold. The received beacon signal at the
SU transmitter can be presented as
/
where
is the transmitted beacon signal with power
,
is the distance between the beacon transmitter and the
SU
transmitter,
represents the small-scale fading over the link
between the beacon transmitter and the
SU transmitter,
and
~0,
is AWGN. Hence, the received signal-to-
noise ratio (SNR) of the beacon signal at SU transmitter is
|
|
.
In [25], the exact detection probability of an energy detector
as a function of
is given by
√
where
is the false alarm probability and is the product of
the energy detector’s integration time and channel bandwidth.
In [8], it is shown that for small
(e.g.,
0.01), the
detection probability can be approximated as
0
1
where
√
⁄
. Hence, the average
beacon miss-detection probability of the
SU can be
approximated as
.
(1)
Assuming
, then
|
|
|
|
.
(2)
Let
and
be the transmitted signals from the PU and the
SU with power of
and , respectively. The received
signal at the PU receiver can be written as
(3)
where
~0,
is AWGN and
which indicates the
coincident transmission of the
SU with the PU
transmission, is a Bernoulli random variable as
1
01
.
(4)
Since
in (3) are independent and zero-mean signals with
power , according to (3) and (4), the aggregate interference
caused by SUs becomes
,
(5)
where
denotes the level of interference that
the
SU causes to the PU receiver if it miss-detects the
beacon signal. According to (5), the contribution level of the
SU to the aggregate interference depends on the product
.
III. B
EACON TRANSMITTER AT PU RECEIVER
A. Interference model
In this section, the probabilistic properties of aggregate
interference are studied in small-scale fading channels when
beacon transmitter is located at PU receiver.
1) Rayleigh fading: In a Rayleigh fading channel,
|
|
has an exponential distribution with parameter 1. Therefore,
according to (2), the beacon miss-detection probability is
given by
1
.
(6)
Fig. 2: CDF of aggregate interference in a Rayleigh fading channel for
beacon transmitter at PU receiver.
The beacon miss-detection probability must be kept low,
typically lower than 3% for good operation, and in this range,
it can be further approximated as
1
1
1
.
(7)
When the beacon transmitter is located at the primary
receiver,
, and, from (5) and (7),
. Since
has an exponential distribution with parameter 1,
’s are
i.i.d. random variables and have the exponential distribution
with parameter 1
⁄
. As a result,
∑
has a Gamma
distribution with shape parameter , scale parameter and
.
;,
⁄
Γ
,
0
⁄
(8)
Figure 2 illustrates the plot of the cumulative distribution
function (CDF) of aggregate interference. The plot confirms
that Gamma distribution with calculated parameters has an
accurate fit for the interference distribution
2
.
2) Nakagami fading: In a Nakagami fading channel
parameterized by average received power
and the fading
parameter , channel power (i.e.,
and
|
|
) has a
Gamma distribution with and
assuming
1. Considering CDF of the Gamma distribution,
according to (2), the miss-detection probability is
|
|
,
⁄
,
(9)
where
is the Gamma function and
,
is the lower incomplete Gamma
function. For small 1, ,
⁄
.
Therefore, according to (5), the aggregate interference is
,
Γ
.
(10)
2
0.01,0.01,10
,1,1,
5 and 50 are used
for the numerical results shown in Figures 2-15 unless specified.
4 5 6 7 8 9 10
x 10
-7
10
-4
10
-3
10
-2
10
-1
10
0
Rayleigh fading
I
0
CDF
Simulation
Gamma distribution
In order to present a statistical model for the interference,
the mean and variance of the aggregate interference are
calculated. Since
2
, where
,
then for ,
. Therefore, the mean
of aggregate interference becomes
Γ
2
1
2
.
(11)
Since
,
then
(12)
By using mean squared-error curve-fitting for different
numbers of SUs, Gamma distribution is found to have a good
agreement with the simulation results. For a given set of
{
,
’s are Gamma random variables, and as a sum of
Gamma random variables,
(conditioned on
) is a
Gamma random variable too with
|
∑
and
|
∑
. For a large number of SUs
independently and identically distributed uniformly in the
circular area with
,
(unconditioned) can be
approximated as a Gamma random variable with
and
in (11) and (12).
According to (11) and (12), the shape parameter and the
scale parameter of Gamma distribution can be derived as
4
1
1
1
2
2
1
Γ
21
2
1
1
.
(13)
Figure 3 compares the simulations results with Gamma
distribution with calculated parameters for different . It
shows that Gamma approximation matches closely the
simulation results.
B. Capacity-outage probability
In the presence of interference from SUs,
, the
instantaneous capacity of the PU is
1
|
|
⁄
. Given a required PU threshold rate
,
the capacity-outage probability can be calculated as
|
(14)
where
|
and
2
1
. Assuming Rayleigh fading
channel,
has the exponential distribution with parameter
1 and
has the Gamma distribution represented by its PDF
;,
given by (8). As a result,
1
(15)
(a) 2
(b) 3
Fig. 3: CDF of aggregate interference in Nakagami fading channels with non-
cooperative sensing and beacon transmitter at PU receiver.
and
1
;,
1
1
1
1
.
(16)
For low outage probability, e.g., 3% or less, the above
expression can be approximated as
11
1
2
1
.
(17)
In the above expression,
is the average power of the
received PU signal, while represents the average total
transmitted power from SUs. can be interpreted as the
effective interference, and
/
represents the
average signal-to-SU-interference-and-noise ratio (SINR) at
the PU receiver. In other words, the above expression
indicates that the capacity-outage probability is approximately
1 1.5 2 2.5 3 3.5 4
x 10
-9
10
-4
10
-3
10
-2
10
-1
10
0
Nakagami fading,m=2
I
0
CDF
Simulation
Gamma distribution
0.5 1 1.5 2
x 10
-11
10
-4
10
-3
10
-2
10
-1
10
0
Nakagami fading, m=3
I
0
CDF
Simulation
Gamma distribution
proportional to the inverse of the SINR, and exponentially
increases with the required PU threshold rate
. The
expression is also applicable for the case of no SU by setting
,, or to 0. Figure 4 illustrates the plots of the capacity-
outage probability versus the PU threshold rate
(in b/s/Hz)
for different SINR values. The plot confirms the precision of
the analytical derivation in (16) as it closely matches the
simulation results. It also supports the result that the capacity-
outage probability is approximately proportional to the
inverse of the SINR by comparing different plots for a given
.
IV. B
EACON TRANSMITTER AT PU TRANSMITTER
A. Interference Model
1) Rayleigh fading: When the beacon transmitter is
located at the PU transmitter, the distance between the beacon
transmitter and the
SU transmitter is
2
and, from (5) and (7), the
instantaneous interference from the
SU transmitter to the
PU receiver is
1
2
/
(18)
where
is a random variable with pdf
2
, where
, and the angle
which the
SU
transmitter makes to the line connecting the PU transmitter
and receiver, is uniformly distributed between 0 and 2 as
previously discussed. The aggregate interference from SUs is
1
2
/
.
(19)
In order to present a statistical model for the interference, the
aggregate interference mean is calculated. Since
1,
1
2
/
. By approximating
with four terms, the mean of aggregate interference
becomes
1
2
⁄
1
2
2
⁄
1
2
⁄
2
4
.
(20)
Since
0,
0.5 and
,
10.5
.
(21)
Based on the probabilistic properties of
,
,
, sample
values of
can be generated by simulation to obtain the
histogram of its distribution as shown in Figure 5. The
aggregate interference from SUs can be approximated as
.
(22)
Fig. 4. PU capacity-outage probability versus required threshold rate for
various SINR in a Rayleigh fading channel with beacon transmitter at
PU receiver.
Fig. 5. Histogram and pdf of interference I
0
, in a Rayleigh fading channel and
path-loss exponent
=2.1.
In other words, in a Rayleigh channel, the aggregate
interference from SUs can be approximated as a shifted-
Gamma distributed random variable with the PDF
;,,
Γ
,
,
(23)
where n, and
are, respectively, the shape parameter,
the scale parameter and the shift parameter. According to (21)
and (22),
0.5
. The shift
parameter empowers us to match the skewness of the
distribution, in addition to the scale and shape parameters. If
,
and are considered as mean, variance and skewness of
aggregate interference
, then 4
, 2
⁄
and
2
.
The CDF of aggregate interference plotted in Figure 6
shows that the shifted-Gamma approximation closely follows
simulation results and is more accurate than the Gamma
approximation.
10
-1
10
0
10
1
10
-3
10
-2
10
-1
10
0
Rayleigh fading, Beacon transmitter at PU receiver
PU threshold rate C
0
Outage-capacity probability
Analytical
Simulation
SINR=-10dB
SINR=10dB
SINR= 0dB
0 0.5 1 1.5 2 2.5 3
x 10
-6
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
6
I
0
Rayleigh fading
f(I
0
)
Simulation
Shifted gamma distribution
Fig. 6. CDF of aggregate interference in a Rayleigh fading channel for beacon
transmitter at PU transmitter.
B. Capacity-Outage Probability
A closed-form expression for the outage probability of the
PU related to the mean and variance of the interference can be
derived to quantify the effect of SUs by using (15) where
has the shifted-Gamma distribution represented by its pdf
;,,
given by (23). As a result,
1
;,,
1
1
.
(24)
For low outage probability, e.g., 3% or less, the above
expression can be approximated as
11
1
2
1
.
(25)
The above expression is very much similar to (17) with the
same average power of the received PU signal, represented by
while the effective interference becomes
,
which is increased by an additional term
, as compared to
that in the case of beacon transmitter at PU receiver.
Figure 7 illustrates the plots of the capacity-outage
probability versus the PU threshold rate
(in b/s/Hz) for
different values of SINR
⁄
. The plots
confirm the precision of the analytical derivation in (24) as
they closely match the simulation results. Furthermore, as
compared to results in Figure 4, for the same SINR and PU
threshold rate
, beacon transmitter at PU transmitter yields a
higher capacity-outage probability than beacon transmitter at
PU receiver.
C. Beacon Transmitter Placement Comparison
In this section, with focus on the general network model in
Fig. 1, the effect of beacon transmitter placement on the
performance of the network is studied by comparing the
aggregate interference mean and the capacity-outage
probability of PU for two different scenarios, the beacon
transmitter at the PU transmitter or at the PU receiver.
Fig. 7. PU capacity-outage probability versus required threshold rate for
various SINR in a Rayleigh fading channel with beacon transmitter at
PU transmitter.
As shown in (5), the contribution level of the
SU to the
aggregate interference depends on the product
. The
probability of miss detection
(also the probability of
causing interference) depends on the distance from the
SU
transmitter to the beacon transmitter,
, while the
interference level
depends on the distance from the
SU
transmitter to the PU receiver,
. This fact makes the location
of beacon transmitter (at PU transmitter or at PU receiver)
influencing the effective aggregate interference differently. In
locations of high interference level
(i.e., when SU
transmitter is close to PU receiver), locating beacon
transmitter at PU transmitter increases the probability of
causing interference
while locating beacon transmitter at
PU receiver decreases the probability of causing interference.
As a result, locating beacon transmitter at PU transmitter
causes higher
and hence higher effective aggregate
interference in comparison with locating beacon transmitter at
PU receiver.
As shown in (25), as compared to the case with the beacon
transmitter at the PU receiver, the use of beacon transmitter at
the PU transmitter introduces extra interference represented
by
, and, hence, increases the PU capacity-outage
probability. In other words, considering the network model in
Fig. 1, setting the beacon transmitter at PU receiver is more
beneficial to avoid interference increase although it is more
practical to put the beacon transmitter at the PU transmitter.
Figure 8 illustrates the plots of aggregate interference
versus
for two different scenarios based on analytical
results and simulation results. It is shown that the analytical
results according to
for the case with beacon
transmitter at primary receiver and (21) match closely the
simulation results. It is apparent that the mean value (i.e.,
) is independent of
when beacon transmitter
located at PU receiver. However, when beacon transmitter is
located at PU transmitter, the mean value according to (21) is
an increasing function of
. The plots show that the
interference increase caused by locating the beacon
transmitter at the PU transmitter is larger when the PU
transmitter-receiver link is longer.
4 5 6 7 8 9 10
x 10
-7
10
-4
10
-3
10
-2
10
-1
10
0
Rayleigh fading
I
0
CDF
Simulation
Shifted Gamma distribution
Gamma distribution
10
-1
10
0
10
1
10
-3
10
-2
10
-1
10
0
Rayleigh fading, Beacon transmitter at PU transmitter
PU threshold rate C
0
Outage-capacity probability
Analytical
Simulation
SINR=-10dB
SINR=0dB
SINR=10dB
Fig. 8. Interference mean
vs. PU transmitter-receiver distance
.
Fig. 9. PU capacity-outage probability versus required threshold rate.
Figure 9 compares the capacity-outage probability of two
different beacon transmitter placements, and shows that
setting beacon transmitter at PU receiver improves the
capacity-outage performance of PU by reducing the effective
interference. Furthermore, in accordance with the results in
Fig. 8, it confirms that this PU capacity-outage performance
improvement increases with larger PU transmitter-receiver
distance
.
V. C
OOPERATIVE SENSING
This section provides a study on the effect of cooperative
sensing (i.e., OR sensing and ML sensing) on alleviating the
aggregate interference in Nakagami fading channels assuming
beacon transmitter at PU receiver (i.e.,
and
).
Since our objective is to highlight the impact of SU
cooperation on the aggregate interference rather than
designing advanced cooperation schemes, we simply consider
the following cooperation protocol. Upon detecting a beacon,
a SU broadcasts a signal which can be the preliminary result
as one bit in OR cooperation or the received beacon signal
power in ML cooperation to its neighbors within its
cooperation range. In this simple protocol, SUs broadcast
their messages in different time-slots of a control frame on a
dedicated control channel.
The probabilistic properties of aggregate interference are
investigated for OR and ML cooperative sensing in Nakagami
fading channels. The closed-form expressions for capacity-
outage probability are derived in Rayleigh fading channels.
Comparing the mean of interference and capacity-outage
probability, it can be concluded that increasing cooperation
range (i.e., increasing the number of cooperating SUs) offers
lower interference and hence better capacity-outage
probabilities. In addition, it is shown that employing ML
cooperative sensing offers lower interference at the cost of
requiring more signaling.
A. OR Detector
1) Interference model: In OR cooperative detection, it is
assumed that each SU sends its preliminary detection decision
as one bit via the cooperation channel. The SUs within a
certain range (cooperation
) can receive correctly other
SUs’ preliminary decisions. Each SU will use the OR rule on
the preliminary decisions to decide finally if the PU exists.
Then, the new beacon miss-detection probability of each SU
with OR sensing is the product of preliminary miss-detection
probability of that SU and the probability that all of the SUs
in its cooperation range miss-detect the beacon.
According to (9), the miss-detection probability can be
written as
∏
,
where
1
is the
number of cooperating SUs. Considering
, where is
the outer radius of the network, all
in the cooperation range
of the
SU are assumed equal to
to simplify the problem.
Therefore, the new beacon miss-detection probability is
,
. For small 1, it can be
approximated as
mΓ
.
(26)
Since
, according to (5) and (26),
,
Γ
.
(27)
Similar to (11) and (12), the mean and variance of aggregate
interference can be derived as
(28)
.
(29)
Similar to the non-cooperative case, by using mean squared-
error curve-fitting for different number of SUs, Gamma
distribution matches closely the simulation results. For a
given set of {
,
’s are Gamma random variables, and as a
sum of Gamma random variables,
(conditioned on
) is
also a Gamma random variable with
|
10
0
10
1
8
8.5
9
9.5
10
10.5
11
x 10
-7
Rayleigh fading
R
0
E[I
0
]
Beacon at PU transmitter, analytical
Beacon at PU transmitter, simulation
Beacon at PU receiver, analytical
Beacon at PU receiver, simulation
10
-1
10
0
10
1
10
-2
10
-1
10
0
Rayleigh fading, SINR=0 dB
PU threshold rate C
0
Outage-capacity probability
Beacon at PU transmitter, R
0
=10
Beacon at PU transmitter, R
0
=5
Beacon at PU receiver, R
0
=10
Beacon at PU receiver, R
0
=5
∑
and
|
∑
. For a large
number of SUs independently and identically distributed
uniformly in the circular area with
,
(unconditioned) can be approximated as a Gamma random
variable with
and
in (28) and (29). According to (28)
and (29), the shape parameter and the scale parameter of
Gamma distribution are derived as
.
(30)
Figures 10 shows the CDF of aggregate interference for
10 in Nakagami fading channel with =1 and =2. It
shows that Gamma distribution provides a close fit for the
interference.
2) Capacity-outage probability: In this section, the
capacity-outage probability of PU is derived in the Rayleigh
fading channel. Since aggregate interference is approximated
as a Gamma random variable with calculated parameter in
(30), similar to (16), the capacity-outage probability becomes
1
1
1
1
.
(31)
For low outage probability, e.g., 3% or less, the above
expression can be approximated as
11
1
2
1
.
(32)
In the above expression,
represents the
average total transmitted power from SU’s applying OR
cooperative sensing, which can be interpreted as the effective
interference, and
represents the average
signal-to-SU-interference-and-noise (SINR) at the PU
receiver. In other words, the above expression indicates that
the capacity-outage probability is approximately proportional
to the inverse of the SINR, and exponentially increases with
the required PU threshold rate
similar to (17) and (25). The
result is also applicable for the non-cooperative case sensing
when
1 because SINR turns to
.
Figures 11 illustrates the plots of the capacity-outage
capacity probabilities versus the PU threshold rate
(in
b/s/Hz) for
10 for different values of
⁄
.
(a) 1
(b) 2
Fig. 10: CDF of aggregate interference in Nakagami fading channels with OR
sensing,
10.
Fig. 11: PU capacity-outage probability versus required threshold rate for
various SINR with OR sensing in a Rayleigh fading channel.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10
-14
10
-4
10
-3
10
-2
10
-1
10
0
R
c
=10, Nakagami fading, m=1, OR sensing
I
0
CDF
Simulation
Gamma distribution
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10
-24
10
-4
10
-3
10
-2
10
-1
10
0
R
c
=10, Nakagami fading, m=2, OR sensing
I
0
CDF
Simulation
Gamma distribution
10
-1
10
0
10
1
10
-11
10
-10
10
-9
10
-8
R
c
=10, Nakagami fading, m=1, OR sensing
PU threshold rate C
0
Capacityoutage probability
Analytical
Simulation
SINR=-10dB
SINR=0dB
SINR=10dB
B. ML Detector
1) Interference model: Considering ML cooperative
detection, each SU sends the received beacon signal power to
the other CRs in its cooperation range. Thus, at each SU, final
detection decision will be made based on the total sum of
received beacon powers including directly from the beacon
transmitter and the nearby SUs. Then, according to (1), the
new beacon miss-detection probability of each SU with ML
sensing will be
where
denotes total received beacon SNR at the
SU.
can
be presented as
∑
́
where
is the beacon
SNR which is received at the
SU directly from beacon
transmitter and ́
is the received beacon SNR relayed from
the
SU. Due to double AWGN at relay and at destination
SU, ́
where
represents the directly received
beacon SNR from beacon transmitter at the
SU. Hence,
∑
∑
,
.
Considering
, all
in the cooperation range of the
SU are considered equal to
. Then,
∑
,
. Hence, the miss-detection
probability becomes
∑
,
∑
2
.
(33)
In order to find a closed-form expression, the miss-detection
probability of ML cooperative sensing is approximated with a
close upper-bound for
1. Considering Nakagami fading
with and
1,
∑
has a Gamma distribution with
and
. According to CDF of Gamma
distribution, the miss-detection is derived as
Γ,2
⁄
Γ
Γ
,2
Γ
.
(34)
For small , the lower incomplete gamma function can be
approximated as Γ
,
. Therefore,
according to (5) and (34), the aggregate interference becomes
,
2
Γ
.
(35)
Similar to (11) and (12), the mean and variance of aggregate
interference are computed as
(36)
.
(37)
Similar to non-cooperative sensing and OR sensing, Gamma
distribution also matches closely the simulation results in ML
sensing with the following scale and shape parameters
.
(38)
(a) 1.
(b) 2
Fig. 12: CDF of aggregate interference in Nakagami fading channels with ML
sensing,
10.
Fig. 13: Interference mean
vs. cooperation range
in Nakagami fading
channels.
0.5 1 1.5 2 2.5
x 10
-16
10
-4
10
-3
10
-2
10
-1
10
0
R
c
=10, Nakagami fading, m=1, ML sensing
I
0
CDF
Simulation
Gamma distribution
0 0.5 1 1.5
x 10
-27
10
-4
10
-3
10
-2
10
-1
10
0
R
c
=10, Nakagami fading, m=2, ML sensing
I
0
CDF
Simulation
Gamma distribution
0 5 10 15 20 25 30 35 40 45 50
10
-200
10
-150
10
-100
10
-50
10
0
Nakagami fading
R
c
E[I
0
]
OR, simulation
OR, analytical
ML, simulation
ML, analytical
m=2
m=1
Figures 12 shows the CDF of aggregate interference for
10 in Nakagami fading channels with =1 and =2. It
shows that Gamma distribution provides a close estimate of
the interference CDF.
In Figures 13, the mean of interference is plotted as a
function of cooperation range based on simulation results and
analytical results according to (28) and (36). Fig. 13 confirms
the precision of the analytical derivations in (28) and (36)
since they closely match the simulation results. It is also
shown that the mean of interference is a decreasing function
of
since increasing the number of cooperating users (i.e.,
increasing
) is beneficial to reduce miss-detection
probability, and hence, decrease the mean of interference.
When
is equal to (e.g.
50), it represents the
scenario where all SUs cooperate for spectrum sensing. It
illustrates that cooperation of all SUs will be beneficial to
cause the least interference mean. In addition, they show that
ML sensing offers lower interference due to further detection
improvement based on more signaling information as
compared to the OR detector.
2) Capacity-outage probability: In this section, the
capacity-outage probability of PU for ML cooperative sensing
is studied. According to (16) and (38), the outage probability
is
1
1
1
1
.
(39)
For low outage probability, e.g., 3% or less, the above
expression can be approximated as
11
1
2
1
,
1.
(40)
Comparing to the capacity-outage probability in OR sensing,
it is obvious that the outage probability is smaller with
cooperative ML sensing because of lower effective
interference
for
1. Figures 14
illustrates the plots of the capacity-outage capacity
probabilities versus the PU threshold rate
(in b/s/Hz) for
10 for different values of
⁄
.
Figures 15 compares the capacity-outage probabilities for
ML and OR sensing in a certain SINR for small and large
.
It is shown that the larger number of cooperating SUs offers
larger performance gains for ML sensing in comparison with
OR sensing. Moreover, it is shown how cooperative sensing
with larger number of cooperating SUs improves the capacity-
outage probability by offering lower interference. For
example, increasing the cooperation range from
5 to 10
approximately reduces capacity-outage probability of PU by 6
orders of magnitude for OR cooperation and 7 orders of
magnitude with ML cooperation.
Fig. 14: PU capacity-outage probability versus required threshold rate for
various SINR with ML sensing in a Rayleigh fading channel.
Fig. 15: PU capacity-outage probability versus required threshold rate for ML
and OR sensing in a Rayleigh fading channel.
VI. CONCLUSION
In this paper, aggregate interference model and distribution
are discussed and its effect on the performance of a network
with beacon consisting multiple SUs and single PU is
investigated in terms of capacity-outage probability of PU.
Furthermore, the effect of propagation channel models,
beacon transmitter placement and cooperation sensing are
studied on aggregate interference and capacity-outage
probability of PU.
First, closed-form expressions representing the aggregate
interference from SUs to PU receivers and PU capacity-
outage probability are derived for both cases: beacon
transmitter at PU transmitter and at PU receiver. They can be
used to establish cognitive radio network parameters and/or to
estimate its performance. Simulation and numerical results are
provided to verify the derived closed-form equations. It is
shown that locating beacon transmitter at PU receiver could
be beneficial to enhance the performance by comparing mean
10
-1
10
0
10
1
10
-11
10
-10
10
-9
10
-8
R
c
=10, Nakagami fading, m=1, ML sensing
PU threshold rate C
0
Capacity-outage probability
Analytical
Simulation
SINR=0dB
SINR=10dB
SINR=-10dB
10
-1
10
0
10
1
10
-11
10
-10
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
Nakagami fading, m=1, SINR=0 dB
PU threshold rate C
0
Capacity-outage probability
OR sensing, R
c
=5
ML sensing, R
c
=5
OR sensing, R
c
=10
ML sensing, R
c
=10
value of aggregate interference and the capacity-outage
probability of PU.
Then, closed-form expressions for interference and
capacity-outage probability are derived for OR and ML
cooperative sensing. Simulation and analytical results indicate
that cooperation can be used to maintain the capacity-outage
probability of PU at the desired level when it is not practical
to enhance the sensitivity of the individual detectors.
Comparing the capacity-outage probabilities and interference
means, it is shown that ML cooperative sensing offers lower
interference in comparison with OR cooperative sensing.
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Mahsa Derakhshani received her B.Sc. degree and
M.Sc. degree in Electrical Engineering from Sharif
University of Technology, Tehran, Iran, in 2006 and
2008, respectively. Since September 2008, she has
been with the Department of Electrical and
Computer Engineering, McGill University, Canada,
where she is currently working toward her Ph.D.
degree. She received the John Bonsall Porter Prize
and McGill Engineering Doctoral Award (MEDA)
at McGill University.
Her research interests lie in the area of wireless
communications and include spectrum sensing techniques and resource
allocation in cognitive radio networks.
Tho Le-Ngoc obtained his B.Eng. (with Distinction)
in Electrical Engineering in 1976, his M.Eng. in
1978 from McGill University, Montreal, and his
Ph.D. in Digital Communications in 1983 from the
University of Ottawa, Canada. During 1977-1982, he
was with Spar Aerospace Limited and involved in
the development and design of satellite
communications systems. During 1982- 1985, he
was an Engineering Manager of the Radio Group in
the Department of Development Engineering of
SRTelecom Inc., where he developed the new point-
to-multipoint DA-TDMA/TDM Subscriber Radio System SR500. During
1985-2000, he was a Professor at the Department of Electrical and Computer
Engineering of Concordia University. Since 2000, he has been with the
Department of Electrical and Computer Engineering of McGill University.
His research interest is in the area of broadband digital communications. He
is a senior member of the Ordre des ingénieurs du Québec and a fellow of the
Institute of Electrical and Electronics Engineers (IEEE), the Engineering
Institute of Canada (EIC), the Canadian Academy of Engineering (CAE) and
the Royal Society of Canada (RSC). He is the recipient of the 2004 Canadian
Award in Telecommunications Research, and recipient of the IEEE Canada
Fessenden Award 2005. He holds a Canada Research Chair (Tier I) on
Broadband Access Communications, and a Bell Canada/NSERC Industrial
Research Chair on Performance & Resource Management in Broadband
xDSL Access Networks
