This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE DySPAN 2010 proceedings
Primary User Detection in Distributed Cognitive
Radio Networks under Timing Inaccuracy
Jari Nieminen, Riku Jäntti
Lijun Qian
Department of Communications and Networking
Aalto University, Finland
{jari.nieminen, riku.jantti}@tkk.fi
Department of Electrical and Computer Engineering
Prairie View A&M University, USA
[email protected]
Abstract—In this paper we study energy detection of primary
users signals’ in the case of distributed Cognitive Radio (CR)
networks. We concentrate on the effects of interference caused
by other CR users due to timing misalignments and propose a
novel mathematical model for calculating the effects of interfering
nodes on energy detection and derive closed-form solutions for
the probabilities of detection and false alarm. We verify our
model by simulations and show that the impact of interference
is severe in the presence of timing errors in distributed CR
networks. To the best of authors’ knowledge, the problem of
inaccurate timing has not been investigated before. However, as
we show in this paper, without interference cancellation or precise
synchronization, timing errors will degrade sensing performance
heavily in decentralized secondary networks and the throughput
of CR networks will be significantly lower as well. The results
obtained in this work can be used to guide the design and
performance analysis of energy-based detection in distributed CR
networks when accurate time synchronization is not available.
I. I NTRODUCTION
Due to the continuous increase in demand for high-speed
wireless communication applications, the problem of inefficient exploitation of the scarce spectrum resources has arised.
Several recent studies show that a considerable number of
licensed parts of the spectrum are rarely occupied [1], [2].
It has been foreseen that Cognitive Radios (CR) [3] could be
the solution that would enable flexible and efficient spectrum
usage. The purpose of CRs is to exploit momentarily unused
parts of the spectrum opportunistically in order to improve
spectrum utilization. The general operation of cognitive radios
includes spectrum sensing, spectrum management, spectrum
mobility and dynamic spectrum sharing [4].
One of the most critical issues in CR networks is the ability
to measure the environment so that secondary users can avoid
interfering with incumbent systems. Hence, without successful
Primary User (PU) detection, CR technology cannot prosper.
In order to sense the presence of the incumbent efficiently,
all secondary users in a CR network should have a common
quiet period. For this purpose, CR users should be perfectly
time synchronized, which is usually not the case in practice.
As a result, CR users will cause interference among each other
during spectrum sensing.
Radio communication networks can be synchronized by
using in-band or out-of-band solutions. Global Positioning
System (GPS) is the most common out-of-band synchronization method and can provide precise timing. However, GPS
may suffer from availability problems due to failure, blockage
or jamming. In addition, GPS does not work in situations
such as indoors. Furthermore, cost and power consumption
of GPS receivers makes it an infeasible solution for certain
applications, such as wireless sensor networks [5].
Current in-band time synchronization schemes cannot provide as accurate time synchronization as GPS. Furthermore,
in-band time synchronization algorithms will periodically
consume resources of the network. Therefore, it would be
desirable to run in-band synchronization algorithms as rarely
as possible. On the other hand, clock drifts grow as a function
of time. The measurement results presented in [6] show that
the deviation of clock drift of a basic clock used in WLAN
devices is approximately 10−6 . Hence, the tradeoff between
resource consumption, quality of clocks and precision of timing needs to be addressed. The problem of inaccurate timing is
especially significant in distributed multi-hop networks since
the maximum of the time synchronization error grows as a
function of hop count [7].
Spectrum sensing can be based on energy, matched filter or
cyclostationary detection [8]. Energy detectors simply compare the power of the received signal to the threshold and
therefore, are not able to differentiate between PU’s signal
and noise. Thus, energy detectors may cause false alarms
or even miss PU’s transmissions. On the other hand, energy
detectors are simple to implement, fast and do not require any
knowledge of the primary user’s signal.
Energy detection was first discussed in [9] and the author derived closed-form solutions for detection and false
alarm probabilities when energy detection is exploited in the
presence of unknown deterministic signals. The analysis was
extended in [10] for a signal with a random amplitude. Furthermore, the impact of different fading channels on energy-based
detection was studied in [11].
Nevertheless, as far as we know, none of the previous papers
studied the impact of interfering nodes on the performance of
energy detection, which is a critical issue in CR networks. The
fact that energy detection will be exploited in CR networks,
such as in IEEE 802.22 networks [12], motivated us to study its
performance under timing inaccuracy. According to [12], IEEE
802.22 CR networks will exploit two different quiet periods for
incumbent sensing, called fast sensing (1ms) and fine sensing
(20ms). Consequently the effects of interference and detection
978-1-4244-5188-3/10/$26.00 ©2010 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE DySPAN 2010 proceedings
models can be divided into two different categories depending
on the relationship between the length of the sensing slot and
the standard deviation of the timing errors. The contributions
of this paper are as follows:
• We demonstrate the impact of timing errors on sensing in
distributed CR networks and show that the interference
caused by other secondary users is a significant problem.
• We propose a novel detection model which takes into
account the impact of interference from other CR users
on energy-based spectrum sensing and derive closed-form
solutions for correct detection and false alarm probabilities.
• We also deduce a simplified model for cases where the
standard deviation of timing errors is large compared to
the length of sensing slots. This result would be very
useful when there are many channels to be sensed and
the time for sensing each channel is very short.
• Correctness of the proposed models is verified by simulations and by matching the classical detection models in
specific cases.
In general, the results obtained in this paper can serve as a
guideline in designing energy-based detection in distributed
CR networks when accurate time synchronization is not available.
The rest of the paper is organized as follows. The system
model is given in Section II. The impact of interference caused
by other secondary users due to timing misalignments will
be demonstrated in Section III. Furthermore, in Section IV
we propose a novel model for calculating the probabilities
of detection and false alarm in decentralized CR networks
and deduce a simplified model for certain situations. Finally,
Section V concludes the paper.
II. S YSTEM M ODEL
We assume that the network in question consists of primary
users and additional cognitive users that exploit the spectrum
while primary users are not transmitting. Moreover, there is no
fixed infrastructure among the CR users, i.e. the CR network
is a distributed multi-hop ad hoc network. For simplicity, we
also assume that the transmissions of primary users start in
the beginning of a sensing slot and finish at the end so that
primary users’ signals do not disappear or reappear during a
sensing slot. All signals are assumed to be complex.
The detection problem is usually presented with two hypotheses. However, the basic modeling of the detection problem does not include the interference coming from other
CR users. If the cognitive radio network consists of multiple
CR users, other users’ activities will have an impact on the
received signal of a particular CR user (unless there is perfect
time synchronization among the CR nodes which is not a
practical assumption). The interference on each sample n can
be denoted as
z(n) =
M
m=1
hm (n) · sm (n)
(1)
where m is the interfering node, n is the index of the discrete
sample, hm (n) is the complex channel response between the
transmitter and the receiver and sm (n) is the transmitted signal
from node m and the total number of interfering nodes is M .
We assume that all channels experience fast fading and thus,
hm (n) can be defined as Gaussian.
If the primary user is not present, the hypothesis H0 for
each sample n can be presented as follows
H0 : r(n) = w(n) + z(n),
(2)
where r(n) is the received signal and w(n) denotes Additive
White Gaussian Noise (AWGN). The second hypothesis H1 ,
for cases in which the primary user’s signal is present, can be
formulated as
H1 : r(n) = h0 · s(n) + w(n) + z(n),
(3)
where s(n) is the transmitted primary user’s signal and h0
stands for the complex channel from the primary user. Furthermore, the false alarm probability, i.e. decision metric is larger
than the threshold λL in case of no primary user’s presence,
is defined as
Pf = P r(L > λL |H0 ),
(4)
where L is the decision metric. Respectively, the correct
detection probability, i.e. decision metric is larger than the
threshold in case of primary user’s presence, is
Pd = P r(L > λL |H1 ).
(5)
By utilizing the probability of correct detection, the miss
detection probability can be defined which demonstrates a
situation in which a primary user is active but the decision
metric is under the threshold
Pm = P r(L ≤ λL |H1 ) = 1 − Pd .
(6)
Since the noise signal w(n) is AWGN, it can be modeled
as zero-mean Gaussian distributed with standard deviation
σw . Furthermore, we assume that the primary user’s signal is
Orthogonal Frequency Division Modulation (OFDM) -based
and since OFDM signals can be seen as a combination
of many independent random signals in time-domain, the
primary user’s signal s(n) can be modeled with a zero-mean
Gaussian distribution with standard deviation σs . Respectively,
a secondary user’s signal sm (n) can be modeled as zero-mean
Gaussian distributed and thus, the sum of interference signals
z(n) will be Gaussian distributed with standard deviation σz
as well. Furthermore, the effect of shadow fading assures that
the assumption of normally distributed signal powers is correct
[13].
Synchronization error depends on the used synchronization
scheme and, for example, for two-way synchronization the
probability of negative and positive errors is the same [7].
In consequence, we can model synchronization error with a
zero-mean Gaussian distribution. We assume that two-way
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE DySPAN 2010 proceedings
Received Interference on Average (dBm)
−60
−62
−64
−66
−68
−70
−72
−74
−76
φ = −10 ms
φ = 0 ms
φ = +10 ms
−78
−80
0
2
4
6
8
10
12
14
16
18
20
Sensing time (ms)
Figure 1.
Impact of timing error on CR frames
Figure 2.
synchronization is executed periodically and therefore, timing
error φ at time t can be modeled by using synchronization
error and clock drift c at a certain time t in the following
way
φ(t) = + c(t),
(7)
so the total timing error φ at time t will follow a zero-mean
Gaussian distribution as well. It is supposed that all the clocks
of secondary users are similar and so the behaviour of clocks
will be the same in the long run.
Finally, we also assume that the used Medium Access
Control (MAC) protocol enables sensing in the middle of
data transmissions so that the presence of a primary user can
be detected quickly and resources can be reserved for nonpredetermined periods if necessary. One example of such a
MAC protocol is C-MAC [14] by Cordeiro et al. Naturally,
if there is a contention period before or after sensing, the
experienced interference will be slightly different due to
random access and backoff procedures.
III. I NTERFERENCE DUE TO T IMING E RRORS
In this section we show that because of timing errors other
secondary users will create a significant amount of interference
during a sensing period in decentralized CR networks, which
will lead to the degradation of sensing performance. Figure 1
demonstrates the general impact of timing errors on CR frames
and sensing. In the figure, User 1 has the correct time reference
and since the clock of User 2 is ahead, User 2 will start
transmitting too early and as a result, create interference
in the end of the sensing slot. Respectively, User 3 cannot
stop transmitting in time in the beginning of the sensing slot
since its clock is behind the reference clock. The amount of
timing error will determine how much interference is caused
and it will be different for different nodes with different
timing errors. Naturally, network topology, channel conditions
and traffic patterns will have an impact on the amount of
interference as well.
For each sensing periodic the impact of timing error, and
thus the amount of interference, will be different for various
secondary users depending on the timing offset of each individual node. Now φi denotes the discrete timing error of
an individual CR user i with respect to global time and we
Received interference for users with different timing errors
model timing error as normally distributed with zero-mean,
i.e. N (0, σφ ). Moreover, φij stands for relative clock drift
conditioned on φi .
We mark the maximum amount of timing error in samples
as N and thus, the amount of interference each node i
experiences on each sample n during a sensing interval is
χφ >n · χaj =1 ,
n ≤ N/2,
zi (n) = j=i ij
χ
·
χ
,
n
> N/2,
aj =1
j=i φij <n−N
(8)
where
1,
aj =
0,
if node j transmits,
otherwise,
(9)
and χE is an indication function of an event E giving value
1 when E occurs and zero otherwise. In other words, the
interference on each sample is the sum of received signals
from all secondary users that transmit at that time and depends
on the relative clock drift between users i and j. Furthermore,
the expected value of each interfered sample for a given timing
error is
n ≤ N/2,
P r{φi − φj > n|φi } · Pj ,
P r{φi − φj < n − N |φi } · Pj , n > N/2,
(10)
where Pj is the probability of transmission of user j, i.e.
P r{aj } = Pj .
To show an example of the interference caused by other CR
users due to timing misalignments, we simulated a scenario
in which 21 CR users where randomly positioned on a
rectangular area of 1 km2 and each node had a fixed timing
error between −φmax and φmax . The channel model by Erceg
et al. [15] for suburban areas, which is based on empirical data,
was used. Since the model is not in fact designed for terminalto-terminal communications, we added extra attenuation to
compensate that.
Figure 2 shows the impact of timing misalignments for
different users in terms of received interference for the length
of a sensing slot. Interfering signals were summed coherently
and thus, the simulation results show the average of the
E{zi (n)|φi } =
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE DySPAN 2010 proceedings
0.4
Noise plus Interference
Noise
Standard Deviation of Received Interference (dB)
Probability of False Alarm
0.3
0.25
0.2
0.15
0.1
0.05
2
4
6
8
10
12
14
16
18
maximum interferences for users with different timing errors.
As we can see, the amount of intra-system interference an
individual node experiences depends on the timing error of a
particular node. The node with a perfect timing will experience
symmetrical interference in both ends of the sensing slot
while users with maximum errors will experience interference
only in the other end of the sensing slot. Furthermore, since
the amount of interference is different for different samples
and users, the classical detection models are not suitable for
modeling sensing in decentralized cognitive radio networks
and thus, new models are required. The simulation results
correspond to the model presented in Equation (8).
In practice, system design is generally based on the false
alarm probability due to the lack of knowledge about PUs’ signals. This means that the detection threshold will be calculated
by fixing the target false alarm probability as follows [11]
Pf =
w
Γ(N )
σ = 5 samples
φ
50
σ = 20 samples
φ
40
30
20
10
0
2
4
6
8
10
12
14
16
18
20
Sensing time (samples)
Probability of false alarm with and without interference
λL
Γ(N, 2σ
2 )
φ
σφ = 2 samples
0
20
Standard Deviation of Timing Error
Figure 3.
σ = 1 sample
60
0.35
,
(11)
where Γ is the gamma function and Γ(x, y) is the upper
incomplete gamma function [16].
Increased interference levels mean that received signal levels are higher, respectively, and as a consequence of this
detection probabilities will grow. On the other hand, false
alarm probabilities will grow as well and thus, the performance
of energy detectors will be different than planned. Figure 3
shows the effects of different timing misalignments on the
probability of false alarm. The figure also shows the 95%
confidence intervals for the results. In this simulation the
timing error of each user was randomly selected according
to the distributions of timing errors and the target probability
of false alarm was set as 10%.
A. Impact on the Throughput of CR Networks
As we can see, the probability of false alarm increases
when the standard deviation of timing error grows. In practice this means that the throughput of a secondary network
would degrade due to unnecessary stops in data transmissions.
Figure 4. Standard deviation of interference caused by a single user due to
different timing errors as a function of samples
However, since the probability of detection increases as well,
the presence of incumbents will not be missed as often and
thus, from the incumbent’s point of view timing errors in
secondary networks are beneficial. Nevertheless, timing errors
will deteriorate the throughput of CR networks since data
transmissions may be discontinued for nothing.
Moreover, the confidence intervals indicate that interfering
nodes will also add more uncertainty to the detection process.
As a result, the performance of energy detectors under interference caused by other CR users will be more unstable. From
the figure it is also evident that by using the basic formulas for
estimating detection parameters, the performance of detection
will not be close to the analytical results in practice.
B. Impact on the Length of Sensing Slots
Since timing errors are zero-mean Gaussian distributed, the
effect of other interfering secondary users on primary user
detection will be similar for all the CR users in a network on
average. Figure 4 illustrates the standard deviation of intranetwork interference. Only one interfering node was randomly
positioned around the sensing node in order to see the impact
of an individual interfering node with different timing offsets.
As we can see from the figure, received interference levels
may be very different for different samples depending on the
standard deviation of timing errors and the length of sensing
slots.
We conclude that if the standard deviation of timing error is
at least as large as the length of the sensing slot, the impact of
interference will be the same for all the samples on average. In
fact, more important than the actual value of timing errors is
the relation between timing errors and sensing slot lengths
(σφ /N ). For instance, if the length of the sensing slot is
1ms and the standard deviation of timing error is 1ms as
well, the experienced interference will have a constant impact
over all the samples on average. Therefore, in addition to the
following general model that takes into account intra-network
interference due to timing inaccuracies, we will also deduce a
simplified model that can be applied if the standard deviation
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE DySPAN 2010 proceedings
of timing error is sufficiently large compared to the length of
the sensing slot. From the design point of view, an “optimal”
length of the sensing slot may be determined when (σφ /N )
can be estimated.
IV. D ETECTION IN THE P RESENCE OF T IMING E RRORS
In this section we introduce a novel model for estimating
the impact of interference created by other CR users on energy
detection and derive closed-form solutions for the probabilities
of detection and false alarm. First, we will derive the general
model which can be used in all situations. Secondly, we deduce
a simplified model which can be exploited in cases where the
standard deviation of timing errors is at least as large as the
length of sensing slots. Moreover, we show that our model
converges to the classical notations as well.
Energy detectors simply compare the received signal power
to the threshold. So the decision metric for an energy detector
is
L(r) =
N
1 |r(n)|2
N n=1
(13)
and respectively with primary user presence, hypothesis H1
(3), the decision metric will be
N
1 L=
|s(n) + w(n) + z(n)|2 .
N n=1
(14)
P r(L ≤ λL |H0 ) = 1 −
(15)
K
Pk · τK exp(TH0 ,k λL )1
(16)
k=1
where 1 is a column vector of ones, H0,k is the matrix that
includes the signal’s properties experienced by this particular
user for each transmission combination k, i.e.
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
TH ,k = ⎢
⎢
0
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
− 21
ξ (1)
k
1
ξ2 (1)
k
0
0
− 21
ξ (2)
k
1
ξ2 (2)
k
···
.
.
.
.
0
.
0
0
− 21
ξ (3)
k
.
.
.
0
.
.
.
0
.
0
···
0
0
0
···
.
.
.
.
.
⎤
0
.
0
.
.
.
1
ξ2 (n−1)
k
− 21
ξ (n)
k
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
and the probability vector τK = 1/K 0 0 · · · 0
[17], since each transmission combination is associated with
exactly one Markov process and the probability of starting in
state 1 is 1, scaled by the total number of combinations K.
Similarly
P r(L ≤ λL |H1 ) = 1 −
K
Pk · τK exp(TH1 ,k λL )1
(17)
k=1
A. General Model
The following model enhances the classical energy detection
models by taking into account the impact of other CR users
that will interfere sensing due to timing errors. Each user i
transmits with a probability Pi and has a timing error of φi
and thus, interference experienced by each node is different for
each sensing interval even though the experienced interference
will be same for all nodes in the long run since the mean of
the timing error is zero.
During a sensing interval, a node i experiences interference
because of transmissions of some neighbors. However, it is
clear that all its neighbors cannot be transmitting simultaneously on the same frequency channel and thus, a transmission
matrix T can be derived which contains all possible link
combinations, i.e. transmitter-receiver pairs. The total amount
of possible transmission combinations is marked with K and
the probability of a certain link combination Pk is factorial
of all the transmission pairs that belong to the particular
combination.
Since the variables s(n), w(n) and z(n) are complex
2
and σz2 , respectively, the signal
Gaussian with variances σs2 , σw
x(n) = |s(n) + w(n) + z(n)|2 follows exponential distribution
2
+ σz2 and therefore
with mean σs2 + σw
X
).
2 + σ 2 (n)
σs2 + σw
z
We define the noise plus interference power as ξ 2 (n) =
2
+ σz2 (n). Now, sum of N exponential distributed random
σw
variables can be expressed as a mixture of hypoexponential
distributions as follows
(12)
and thus, when the primary user’s signal is not present,
hypothesis H0 (2), the decision metric will be
N
1 L=
|w(n) + z(n)|2
N n=1
P r(x(n) ≤ X) = 1 − exp(−
where
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
TH ,k = ⎢
⎢
1
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
−
1
σ 2 +ξ2 (1)
s,k
k
0
0
1
− 2
σ
+ξ2 (2)
s,k
k
1
σ 2 +ξ2 (2)
s,k
k
0
0
.
.
.
0
.
.
.
0
.
0
···
0
0
0
···
1
σ 2 +ξ2 (1)
s,k
k
−
1
σ 2 +ξ2 (3)
s,k
k
.
···
.
.
.
.
.
.
.
0
.
0
.
0
.
.
.
.
1
2
σ
+ξ2 (n−1)
s,k
k
1
− 2
σ
+ξ2 (n)
s,k
k
Equations (16) and (17) model the impact of other secondary users in some definite situation. In order to calculate
the probabilities of false alarm and detection, the model
requires calculations over all the possible link combinations
and thus, a large amount of sensing intervals has to be
considered in practice. The probabilities are calculated over all
the interference situations and hence, the model gives average
results over multiple sensing slots in a certain situation.
However, if the aim is to calculate the probabilities of false
alarm and detection for a single sensing slot, it is enough
to consider only the received interference powers for each
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥.
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE DySPAN 2010 proceedings
Pf,1 = P r{L > λL |H0 } = τ exp(TH0 λL )1
(18)
where
⎡
TH0
⎢
⎢
⎢
⎢
⎢
⎢
⎢
=⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
− ξ21(1)
1
ξ2 (1)
0
0
− ξ21(2)
1
ξ2 (2)
0
0
− ξ21(3)
.
.
.
0
.
.
.
0
..
0
0
0
···
and now τ = 1 0 0 · · · 0
probability of correct detection is
..
.
0
..
.
0
..
.
0
.
···
0
···
.
.
.
1
ξ2 (n−1)
− ξ21(n)
•
•
•
Case
Case
Case
Case
1:
2:
3:
4:
σz (n)
σz (n)
σz (n)
σz (n)
=
=
=
=
0.8
0.7
Case 1: Analysis
Case 1: Simulation
Case 2: Analysis
Case 2: Simulation
Case 3: Analysis
Case 3: Simulation
Case 4: Analysis
Case 4: Simulation
0.6
0.5
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
0.4
−4
−3
−2
−1
0
1
2
3
4
5
SNR
Figure 5.
(19)
Therefore, if the interference power can be estimated at the
receiver, by using this model the expected values of false alarm
and correct decision probabilities can be determined.
For demonstration, we consider a simplified example scenario in which the power of an interfering signal is assumed
to be the same as the power of the primary user’s signal.
Now, since all the signals are Gaussian distributed, the sum
of interfering signals will be the PU’s signal multiplied by the
amount of interfering users for each sample.
As we can see from Figure 2, the amount of interference
experienced by each user will be different for users with
different timing errors and hence, we carried out simulations
by using different example scenarios in order to see the impact
of different timing errors on the sensing performance. The
amount of samples, i.e. the length of the sensing slot, was
set as 20. Signal-to-Noise Ratio (SNR) is specified as σσws and
noise power was fixed as 1 dB.
We simulated the probability of detection for different
CR users that have different timing errors, see Figure 2 for
demonstration, by using the following interference cases:
•
0.9
⎤
and respectively, the
Pd,1 = P r{L > λL |H1 } = τ exp(TH1 λL )1.
1
Probability of Detection
sample. Now Pf,1 and Pd,1 are the probabilities of false alarm
and detection conditioned on interference being equal to the
k = 1 case. In case of only one sensing slot, the probability
of false alarm can be described in terms of hypoexponential
distribution, which is a special case of phase-type distribution,
as
[0, 0, 0, 0, · · · , 0, 0, 0, 0]
[2σs , σs , 0, · · · , 0, σs , 2σs ]
[4σs , 3σs , 2σs , σs , 0, · · · , 0]
[3σs , 2σs , σs , 0, · · · , 0, σs , 2σs , 3σs ]
In the first case, the probability of correct detection was
simulated without any interference caused by other secondary
users, which is the theoretical upper bound of performance.
Case 2 represents a situation where the timing error is 0 and
the third case shows the situation for nodes with timing error
of −φmax or φmax . Case 4 illustrates the impact of increased
interference compared to the second case.
Figure 5 illustrates both analytical and simulation results
as a function of SNR and as we can see, our model models
Analysed and simulated results for example cases
the behaviour of energy detection correctly in all the cases.
Furthermore, the effect of interference is significant since the
probability of correct detection is much higher for the first case
than for the other cases and thus, intra-network interference
cannot be neglected when designing CR systems. We can also
conclude that the probability of detection will be lower for
the nodes that have larger timing misalignments since in the
third case the performance is worse than in the second case.
Moreover, the detection performances of cases 3 and 4 are
closely-spaced.
The impact of timing errors is negligible if only a small
amount of samples experience interference. However, when
the amount of affected samples increases, the performance
degradation becomes more severe as well. This indicates that
small timing errors can be tolerated but in case of larger timing
errors, the sensing performance will deteriorate. As a result,
high quality oscillators or precise time synchronization will
be required for successful implementation of distributed CR
networks.
The previously presented model is suitable for all cases
regardless of the distribution of interference. However, the
model converges to a simplified solution if the variance of
interference is constant during the sensing slot such as in
case of large timing errors or small sensing slots, as will be
discussed next.
B. Simplified Model
In large distributed CR networks in which the standard
deviation of the timing errors among the nodes are large
compared to the length of the sensing interval, each of the
samples experiences the same interfering signal on average.
Moreover, if multiple channels need to be sensed during one
sensing period, the time available for sensing each channel has
to be reduced and thus, the standard deviation of timing errors
may be larger than the length of the sensing slot per channel as
well. Hence, the following model is also suitable for evaluating
the impact of interference in case of sensing of multiple
channels. Now, the standard deviation of the interfering signal
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE DySPAN 2010 proceedings
probabilities will be higher as well which leads to inefficient
operation of secondary networks.
0.8
0.7
V. C ONCLUSIONS
Probability of Miss
0.6
0.5
σ = 0 dB
z
σ = 2 dB
0.4
z
σz = 4 dB
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Probability of False Alarm
Figure 6.
ROC curves with different distributions of interference
can be denoted simply as σz .
False alarm and correct detection probabilities can be once
again calculated by using equations (18) and (19). However,
since now we have several identical phases in sequence, the
matrices H0 and H1 follow Erlang distribution and thus, the
probability of false alarm can be presented in a simplified form
by using the CDF of Erlang distribution as follows
Pf = 1 −
L
γ(N, 2(σ2λ+σ
2) )
w
(N − 1)!
z
(20)
where γ(x, y) is the lower incomplete gamma function
x
γ(x, y) =
e−t ty−1 dt.
(21)
0
Moreover, in this case the probability of detection Pd is
Pd = 1 −
λL
γ(N, 2(σ2 +σ
2 +σ 2 ) )
w
z
(N − 1)!
s
.
In this paper, we studied the effects of timing errors
on energy-based incumbent detection in decentralized CR
networks and proposed a novel detection model that takes
into account the interference caused by other CR users. We
also derived closed-form solutions for correct detection and
false alarm probabilities and demonstrated that the interference
caused by other secondary users is a significant problem. We
also deduced a simplified model for special cases and verified
the correctness of the models by simulations and by matching
the classical detection models in specific cases.
Since the impact of timing errors will cause degradation of
sensing performance in distributed CR networks, our future
research includes designing of a compensation algorithm that
minimizes the effects of interfering users. On the other hand,
by developing efficient time synchronization protocols for CR
networks the effect of interfering users can be minimized. In
this work, the received signals are assumed to be Gaussian,
which represents the popular and important cases in practice.
Studying the detection model and the impacts of timing errors
when the received signals are not Gaussian is one of our future
efforts.
ACKNOWLEDGEMENT
This research work is supported in part by TEKES (Finnish
Funding Agency for Technology and Innovation) as part of
the Wireless Sensor and Actuator Networks for Measurement
and Control (WiSA-II) program.
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Therefore, our results converge to the results presented in [11]
for signals over AWGN channels, see (11), even though the
notation is slightly different.
The performance of detectors can be analyzed by using Receiver Operating Characteristic (ROC) which can be obtained
by plotting the probability of detection (or the probability
of miss) as a function of false alarm probability. Figure 6
demonstrates the impact of various interference variances on
ROC. Again, SNR was set as 1 dBs and the amount of samples
N was 20. As we can see from the figure, the interference due
to large timing errors will also have a significant impact on
the sensing performance.
Moreover, if the impact of interference is taken into account
when setting the detection threshold, the probability of miss
will increase depending on the amount of interference as
Figures 5 and 6 show. However, since the system design
is usually based on the probability of false alarm without
any knowledge of interference, interfering CR users increase
correct detection probabilities but at the same time, false alarm
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