Download Cooperative Spectrum Sensing with an Improved Energy

Cooperative Spectrum Sensing with an Improved
Energy Detector in Cognitive Radio Network
Ajay Singh, Manav R. Bhatnagar, and Ranjan K. Mallik
Department of Electrical Engineering
Indian Institute of Technology Delhi
Hauz Khas, IN-110016 New Delhi, India
Email: ajaysingh8226@gmail.com,{manav,rkmallik}@ee.iitd.ac.in
Abstract— In this paper, performance of cooperative spectrum
sensing with an improved energy detector is discussed. The
Cognitive radios (CRs) utilize an improved energy detector for
taking a decision of the presence of the primary user (PU). The
improved energy detector uses an arbitrary positive power 𝑝
of the amplitude of the received samples of the primary user’s
signals. The decision of each CR is orthogonally forwarded to
a fusion center (FC), which takes final decision of the presence
of the PU. We minimize sum of the probability of false alarm
and missed detection in cooperative spectrum sensing to obtain
an optimized number of CRs for detecting a spectrum hole. An
optimized value of 𝑝 and sensing threshold of each CR is also
obtained by minimizing the total probability of error.
I. I NTRODUCTION
Detecting an unused spectrum and sharing it without interfering the primary users (PUs) is a key requirement of the
cognitive radio network. For detection of a spectrum hole, the
cognitive communication network relies upon the cognitive
radios (CRs) or secondary users [1]. Spectrum sensing is
a procedure in which a CR captures the information of a
band available for transmission and then shares the frequency
band without interfering the primary users. Cooperation among
the CRs provides improvement in detection of the primary
signals [2]. Due to the random nature of the wireless channel,
the signals sensed by the CRs are also random in nature.
Therefore, sometimes a CR can falsely decide the presence
or absence of the PU [3]. In [3], a cooperation based spectrum sensing scheme is proposed to detect the PU with an
optimal linear combination of the received energies from the
cooperating CRs in a fusion center (FC). The cooperative
spectrum sensing scheme provides better immunity to fading,
noise uncertainty, and shadowing as compared to a cognitive
network that does not utilize cooperation among the CRs [4].
Probability of miss 𝑃𝑚 and probability of false alarm 𝑃𝑓 play
important role in describing a reliable communication. We
can fix either of the probabilities at a particular value and
then optimize the other one for improving the probability of
detection of the primary signals [5]. In [6], optimization of
the total probability of error is performed by minimizing sum
of the probability of false alarm and the probability of missed
detection of an cooperative spectrum sensing scheme. It is
explored in [6] that how many CRs are needed for optimum
detection of the primary user in a cooperative spectrum sensing
based scheme, such that the total error probability does not
exceed a fixed value.
For the detection of unknown deterministic signals corrupted by the additive white Gaussian noise (AWGN), an
energy detector is derived in [7], which is also called conventional energy detector. The conventional energy detector [7]
utilizes square of the magnitude of the received data sample,
which is equal to the energy of the received sample. It is
shown in [8] that the performance of a cognitive radio network
can be improved by utilizing an improved energy detector in
the CRs, where the conventional energy detector is modified
by replacing the squaring operation of the received signal
amplitude with an arbitrary positive power 𝑝.
In this paper, we consider optimization of the cooperative
spectrum sensing scheme with an improved energy detector to
minimize the sum of the probability of false alarm and missed
detection. We derive a closed form analytical expression to find
an optimal number of the CRs required for cooperation while
utilizing the improved energy detector in each CR. Moreover,
we also perform optimization of the threshold and the energy
detector to minimize the total error rate.
II. S YSTEM M ODEL
We consider a system consisting of 𝑁 number of CRs, one
primary user, and a fusion center. There are two hypothesis
𝐻0 and 𝐻1 in the 𝑖-th CR for the detection of the spectrum
hole
𝐻0 : 𝑦𝑖 (𝑡) = 𝑣𝑖 (𝑡),
if PU is absent,
𝐻1 : 𝑦𝑖 (𝑡) = 𝑠(𝑡) + 𝑣𝑖 (𝑡),
if PU is present,
𝒩 (0, 𝜎𝑠2 ),
(1)
(2)
𝜎𝑠2
where
is the
where 𝑖 = 1, 2....𝑁 , 𝑠(𝑡) ∼
average transmitted power of the PU, denotes a zero mean
Gaussian signal transmitted by the PU, and 𝑣𝑖 (𝑡) ∼ 𝒩 (0, 𝜎𝑛2 )
is additive white Gaussian noise (AWGN) with zero mean
and 𝜎𝑛2 variance. The variance of the signal received at each
secondary user under 𝐻1 will be 𝜎𝑠2 + 𝜎𝑛2 . It is assumed that
each CR contains an improved energy detector [8]. The 𝑖-th
CR utilizes the following statistic for deciding of the presence
of the PU [8]
𝑊 =∣ 𝑦𝑖 ∣𝑝 , 𝑝 > 0.
(3)
It can be seen from (3) that for 𝑝 = 2, 𝑊 reduces to statistics
corresponding to the conventional energy detector [7]. In a
cooperative sensing scheme, multiple CRs exists in a cognitive
radio network such that each CR makes independent decision
regarding the presence or absence of PU. We consider a
cooperative scheme in which each secondary user sends its
binary decision 𝑑𝑖 (0 or 1) to the fusion center by using an
improved energy detector over error free orthogonal channels.
The fusion center combines these binary decisions to find the
presence or absence of the PU as follows :
𝐷=
𝑁
∑
𝑑𝑖 ,
(4)
𝑖=1
where D is the sum of the all 1-bit decisions from the CRs. Let
𝑛, 𝑛 ≤ 𝑁 corresponds to a number of cooperating CRs out of
𝑁 CRs. The FC uses a majority rule for deciding the presence
or absence of the PU. As per the majority decision rule if 𝐷
is greater than 𝑛 then hypothesis ℋ1 holds and if 𝐷 is smaller
than 𝑛 then hypothesis ℋ0 will be true. The hypothesis ℋ0
and ℋ1 can be written as
ℋ0 : 𝐷 < 𝑛, if PU is absent,
(5)
ℋ1 : 𝐷 ≥ 𝑛, if PU is present.
(6)
III. P ROBABILITY OF FALSE A LARM AND MISSED
DETECTION IN THE CR S
The cumulative distribution function (c.d.f.) of the improved
energy detector can be written as
1
Pr(∣𝑦𝑖 ∣𝑝 ≤ 𝑦) =Pr(∣𝑦𝑖 ∣ ≤ 𝑦 𝑝 )
1
1
=Pr(𝑦𝑖 ≤ 𝑦 𝑝 ) − Pr(𝑦𝑖 < −𝑦 𝑝 ),
(7)
where Pr(⋅) denotes the probability. For finding the probability
density function (p.d.f.) of the decision statistics 𝑊, we need
to differentiate (8) with respect to (w.r.t.) 𝑦 to get
1
1
1 1−𝑝
1 1−𝑝
(8)
𝑓𝑊 (𝑦) = 𝑦 𝑝 𝑓𝑦𝑖 (𝑦 𝑝 ) + 𝑦 𝑝 𝑓𝑦𝑖 (−𝑦 𝑝 ),
𝑝
𝑝
where 𝑓𝑦𝑖 (⋅) is the p.d.f. of received signal at the FC under
hypothesis 𝐻0 or 𝐻1 depending upon the presence or absence
of the primary signal. Let 𝑓𝑦𝑖∣𝐻0 (⋅) and 𝑓𝑦𝑖∣𝐻0 (⋅) be the
p.d.f.s of the received signal under hypothesis 𝐻0 and 𝐻1 ,
respectively. From (5) and (6), it can be deduced that
(
)
1
𝑥2
exp − 2 ,
(9)
𝑓𝑦𝑖∣𝐻0 (𝑥) = √
2𝜎𝑛
2𝜋𝜎𝑛2
1
𝑓𝑦𝑖∣𝐻1 (𝑥) = √
2𝜋(𝜎𝑠2 + 𝜎𝑛2 )
(
exp −
𝑥2
+ 𝜎𝑛2 )
2(𝜎𝑠2
𝑍1
where 𝑍1 is the decision region corresponding to hypothesis
𝐻1 . Let us assume that the decision threshold in each CR is
given as 𝜆. The probability of false alarm 𝑃𝑓 in each CR is
derived in Appendix I as
(
)
2
3 𝜆𝑝
1
,
,
(14)
𝑃𝑓 = √ 𝛤
2 2𝜎𝑛2
𝜋
where 𝛤 (𝑎) is the gamma function [9, Eq. (6.1.1)]
and 𝛤 (𝑎,
∫ ∞𝑥) is the incomplete gamma function
𝛤 (𝑎, 𝑥)= 𝑥 𝑡𝑎−1 exp(−𝑡)𝑑𝑡 [9, Eq. (6.5.3)]. The probability
of miss 𝑃𝑚 is defined by [10, Eq. (41), Chapter 2]
∫
𝑃𝑚 ≜
𝑓𝑊 ∣𝐻1 (𝑦)𝑑𝑦,
(15)
𝑍0
where 𝑍0 is the decision region corresponding to the hypothesis 𝐻0 . In Appendix I, it is shown that the probability of miss
𝑃𝑚 in each CR will be
(
)
2
𝜆𝑝
3
1
,
𝑃𝑚 = √ 𝛾
,
(16)
2 2(𝜎𝑠2 + 𝜎𝑛2 )
𝜋
where ∫𝛾(𝑎, 𝑥) is the incomplete gamma
𝑥
𝛾(𝑎, 𝑥)= 0 𝑡𝑎−1 exp(−𝑡)𝑑𝑡 [9, Eq. (6.5.2)].
1
=Pr(−𝑦 𝑝 ≤ 𝑦𝑖 ≤ 𝑦 𝑝 )
1
The probability of false alarm 𝑃𝑓 is defined as [10, Eq. (41),
Chapter 2]
∫
𝑃𝑓 ≜
𝑓𝑊 ∣𝐻0 (𝑦)𝑑𝑦,
(13)
)
.
(10)
The p.d.f. of 𝑊 under hypothesis 𝐻0 is given from (8) and (9)
as follows:
2
√ 1−𝑝
𝑦𝑝
2𝑦 𝑝 exp(− 2𝜎
)
𝑛
√
.
(11)
𝑓𝑊 ∣𝐻0 (𝑦) =
𝑝 𝜋𝜎𝑛
From (8) and (10), the p.d.f. of 𝑊 under hypothesis 𝐻1 can
be obtained as
2
√ 1−𝑝
𝑝
2𝑦 𝑝 exp(− 2(𝜎𝑦2 +𝜎2 ) )
𝑛
𝑠
.
(12)
𝑓𝑊 ∣𝐻1 (𝑦) =
√ √
𝑝 𝜋 (𝜎𝑠2 + 𝜎𝑛2 )
function
IV. O PTIMIZATION OF COOPERATIVE SPECTRUM SENSING
SCHEME
The probability of false alarm 𝑃𝐹 of the FC for cooperative
sensing will be
𝑃𝐹 = Pr(ℋ1 ∣ℋ0 ),
(17)
From (5), (6), and (17), the probability of false alarm of the
FC can be written as
)
𝑁 (
∑
𝑁
𝑃𝐹 =
(18)
𝑃𝑓𝑙 (1 − 𝑃𝑓 )𝑁 −𝑙 .
𝑙
𝑙=𝑛
In the FC, the probability of missed detection 𝑃𝑀 will be
𝑃𝑀 = Pr(ℋ0 ∣ℋ1 ).
(19)
The probability of missed detection of the FC can be obtained
from (5), (6), and (19) as follows:
)
𝑁 (
∑
𝑁
𝑃𝑀 = 1 −
(20)
(1 − 𝑃𝑚 )𝑙 (𝑃𝑚 )𝑁 −𝑙 .
𝑙
𝑙=𝑛
Let us define 𝑍, which denotes the total error rate in the
cooperative scheme as
𝑍 ≜𝑃𝐹 + 𝑃𝑀
)
𝑁 (
∑
𝑁
=1 +
𝑃𝑓𝑙 (1 − 𝑃𝑓 )𝑁 −𝑙
𝑙
𝑙=𝑛
𝑁
∑( 𝑁 )
−
(1 − 𝑃𝑚 )𝑙 (𝑃𝑚 )𝑁 −𝑙 ,
𝑙
𝑙=𝑛
(21)
0
5
10
4.5
4
−1
10
Optimal number of CRs
Total error probability
3.5
−2
10
n=1
n=2
n=3
n=4
n=5
n=6
−3
10
0
1
2
3
4
5
p
6
7
8
2.5
λ=10
λ=20
λ=50
2
1.5
1
0.5
−4
10
3
9
10
0
0
1
2
3
4
5
p
6
7
8
9
10
Fig. 1. Total probability of error versus 𝑝 for different number of CRs for
𝜆 = 10 and SNR=10 dB.
Fig. 2.
Optimal number of CRs versus 𝑝 for 𝜆 = 10, 20, 50, and
SNR=10 dB.
It can be seen from (21) that 𝑍 denotes the total error rate in
the case when 𝑃𝐹 and 𝑃𝑀 are equiprobable.
B. Optimization of the Improved Energy Detector
A. Optimization of Number of Cooperating CRs
In a cognitive network with very large number of cooperative CRs, large delays are incurred in deciding the presence
of the spectrum hole. Therefore, it is expedient to find an
optimized number of CRs which significantly contribute in
deciding the presence of the PU.
Theorem 1: Optimal number of CRs (n*) for a fixed value
of the sensing threshold 𝜆, the positive power 𝑝 of the
amplitude of received sample of PU, and the signal-to-noise
ratio (SNR) between the PU-CR link will be
( ⌈
⌉)
𝑁
,
(22)
𝑛∗ = min 𝑁,
1+𝛼
In order to get an optimized value of 𝑝 we need to
differentiate (21) w.r.t. 𝑝 keeping 𝑛, 𝜆, and SNR fixed and
set the result to zero. After some algebra we get
𝑁
∑
∂𝑍
=−
∂𝑝
𝑙=𝑛
√1 𝛤
𝜋
(
ln
𝛼=
1− √1𝜋 𝛾
(
√1 𝛾
𝜋
ln
2
3 𝜆𝑝
2 , 2𝜎 2
𝑛
2
3
𝜆𝑝
2 , 2(𝜎 2 +𝜎 2 )
𝑠
𝑛
2
3
𝜆𝑝
2 , 2(𝜎 2 +𝜎 2 )
𝑠
𝑛
1− √1𝜋 𝛤
(
2
3 𝜆𝑝
2 , 2𝜎 2
𝑛
)
)
)
(𝑁 − 𝑙)𝑃𝑓 𝑙 (1 − 𝑃𝑓 )𝑁 −𝑙−1
∂𝑃𝑓
∂𝑝
and
∂𝑃𝑚
∂𝑝
are given as
3
,
(23)
∂𝑃𝑓
∂𝑝
𝑙=𝑛
where
)
𝑁
𝑙
)
𝑁 (
∑
∂𝑃𝑓
𝑁
𝑙𝑃𝑓 𝑙 (1 − 𝑃𝑓 )𝑁 −𝑙
+
𝑙
∂𝑝
𝑙=𝑛
(
)
𝑁
∑
∂𝑃𝑚
𝑁
(𝑁 − 𝑙)𝑃𝑚 𝑁 −𝑙−1 (1 − 𝑃𝑚 )𝑙
−
𝑙
∂𝑝
𝑙=𝑛
)
𝑁 (
∑
∂𝑃𝑚
𝑁
,
(25)
𝑙𝑃𝑚 𝑁 −𝑙 (1 − 𝑃𝑚 )𝑙−1
+
𝑙
∂𝑝
where
(
(
2
𝑝
𝜆
𝜆 𝑝 𝑙𝑜𝑔𝜆 exp(− 2𝜎
2 )
∂𝑃𝑓
𝑛
√
=
,
∂𝑝
2𝜋𝑝2 𝜎𝑛3
(26)
)
3
2
𝜆𝑝
𝜆 𝑝 𝑙𝑜𝑔𝜆 exp(− 2(𝜎2 +𝜎2 ) )
∂𝑃𝑚
𝑛
𝑠
=− √
.
3
∂𝑝
2𝜋𝑝2 (𝜎𝑛2 + 𝜎𝑠2 ) 2
(27)
and ⌈.⌉ denotes the ceiling function.
Proof: In order to obtain an optimized number of the CRs,
we need to differentiate (21) w.r.t. 𝑛 and set the result equal
to zero, to get
(
)
]
[ 𝑁 −𝑛
𝑁
(1 − 𝑃𝑚 )𝑛−𝑃𝑓 𝑛 (1 − 𝑃𝑓 )𝑁 −𝑛 = 0. (24)
𝑃𝑚
𝑙
It is difficult to obtain a closed form solution of 𝑝, however, we
can obtain an optimized value of 𝑝 numerically by using (25).
After some algebra and with the help of results given in [6]
we get (22) from (24).
An optimal value of 𝜆 for given 𝑛, 𝑝, and SNR is obtained
by differentiating (21) w.r.t. 𝜆 and setting the derivative equal
C. Optimization of Sensing Threshold
−1
10
−0.001
10
−0.002
10
−2
Total error rate
Probability of detection
10
−0.003
10
−0.004
10
−3
10
p=1.75
p=2.00
p=2.25
p=2.50
−0.005
p=1.75
p=2.00
p=2.25
p=2.50
10
−0.006
10
−4
10
0
1
2
3
4
5
SNR (dB)
6
7
8
9
10
Fig. 3. Total error rate versus SNR for 𝑝 = 1.75, 2.00, 2.25, 2.50, 𝑛 = 1,
and sensing threshold 𝜆 = 50.
to zero. After some algebra we get
𝑁 (
∑
∂𝑍
𝑁
=−
𝑙
∂𝜆
𝑙=𝑛
𝑁 (
∑
𝑁
+
𝑙
𝑙=𝑛
(
𝑁
∑
𝑁
−
𝑙
𝑙=𝑛
𝑁 (
∑
𝑁
+
𝑙
)
)
)
)
∂𝑃𝑓
∂𝜆
and
∂𝑃𝑚
∂𝜆
(𝑁 − 𝑙)𝑃𝑓 𝑙 (1 − 𝑃𝑓 )𝑁 −𝑙−1
𝑙𝑃𝑓 𝑙 (1 − 𝑃𝑓 )𝑁 −𝑙
∂𝑃𝑓
∂𝜆
∂𝑃𝑓
∂𝜆
(𝑁 − 𝑙)𝑃𝑚 𝑁 −𝑙−1 (1 − 𝑃𝑚 )𝑙
𝑙𝑃𝑚 𝑁 −𝑙 (1 − 𝑃𝑚 )𝑙−1
∂𝑃𝑚
,
∂𝜆
∂𝑃𝑚
∂𝜆
(28)
can be obtained as follow:
3
2
𝑝
𝜆
𝜆 𝑝 −1 exp(− 2𝜎
2 )
∂𝑃𝑓
𝑛
√
=−
,
2
3
∂𝜆
2𝜋𝑝 𝜎𝑛
3
2
4
6
8
10
SNR (dB)
12
14
16
18
20
Fig. 4. Probability of detection versus SNR for 𝑝 = 1.75, 2.00, 2.25, 2.50,
𝑛 = 1, and sensing threshold 𝜆 = 50.
V. N UMERICAL R ESULTS
𝑙=𝑛
where
0
(29)
2
𝑝
𝜆 𝑝 −1 exp(− 2(𝜎𝜆2 +𝜎2 ) )
∂𝑃𝑚
𝑛
𝑠
=− √
.
3
∂𝜆
2𝜋𝑝(𝜎𝑛2 + 𝜎𝑠2 ) 2
(30)
The optimized value of 𝜆 can be found numerically from (28).
D. Probability of Detection of Spectrum Hole
Probability of detection of spectrum hole 𝑃𝐷 in the FC can
be obtain from (20) as follows:
𝑃𝐷 =1 − 𝑃𝑀
(
)]𝑙
)[
2
𝑁 (
∑
𝜆𝑝
3
1
𝑁
,
=
1− √ 𝛾
𝑙
2 2(𝜎𝑠2 + 𝜎𝑛2 )
𝜋
𝑙=𝑛
)]𝑁 −𝑙
(
[
2
𝜆𝑝
3
1
,
× √ 𝛾
.
(31)
2 2(𝜎𝑠2 + 𝜎𝑛2 )
𝜋
In Fig. 1, we have plotted the total error rate versus 𝑝 plots
for different number of cooperative CRs 𝑛 = 1, 2, 3, 4, 5, 6,
𝜆 = 10, and SNR=10 dB. It can be seen from Fig. 1 that the
total error rate is a convex function of 𝑝 for a fixed value of
𝑛, 𝜆, and SNR. It can be seen from Fig. 1 that for 𝜆 = 10
and SNR=10 dB, the total error rate is minimum for 𝑛 = 2.
It implies that for 𝜆 = 10 and SNR=10 dB only two CRs
are required for deciding the presence of the PU by using an
improved energy detector. We have shown plots of the optimal
number of CRs required for cooperation versus 𝑝 for 𝜆 =
10, 20, 50, and SNR=10 dB in Fig. 2. It can be seen from
Fig. 2 that the optimal number of the cooperative CRs varies
with the value of 𝑝 and 𝜆 at SNR=10 dB. For example, for
𝑝 = 3 the optimal number of the CRs is 2, 3, and 4 for
𝜆 = 50, 20, and 10, respectively. Fig. 3 shows that the total
error rate versus SNR plot with 𝑝 = 1.75, 2, 2.25, 2.50, 𝑛 = 1,
and 𝜆 = 50. It can be seen from Fig. 3 that the total error rate
decreases by increasing the SNR and the value of 𝑝. Further, it
can be seen from Fig. 3 that the improved energy detector with
𝑝 = 2.50 significantly outperforms the conventional energy
detector, i.e., when 𝑝 = 2. For 𝑝 = 1.75, 2, 2.25, 2.50, 𝑛 = 1,
and 𝜆 = 50, the probability of detection of a spectrum hole
versus SNR curve is plotted in Fig. 4. It is shown in Fig. 4
that the probability of detection of a spectrum hole increases
with increase in the value of 𝑝.
Fig. 5 illustrates that for a fixed range of 𝜆 and 𝑝 there
exists an optimal number of CRs which minimizes the total
error rate. In Fig. 6, we have plotted the total error rate versus
𝑝 and 𝜆 plots for different number of cooperative CRs at 10 dB
SNR. It can be seen from Fig. 6 that there exists an optimal
value of 𝑝 and 𝜆 for which the total error rate is minimized for
a fixed number of CRs. The plots of Figs. 5 and 6 suggest that
in order to optimize the overall performance of the cooperative
cognitive system utilizing the improved energy detector, we
need to jointly optimize the number of cooperative CRs, the
energy detector, and the sensing threshold in the CR.
Fig. 5.
Optimal number of CRs versus 𝜆 and 𝑝 at SNR=10 dB.
VI. C ONCLUSIONS
We have discussed how to choose an optimal number of
cooperating CRs in order to minimize the total error rate of
a cooperative cognitive communication system by utilizing an
improved energy detector. It is shown that the performance of
the cooperative spectrum sensing scheme depends upon the
choice of the power of the absolute value of the received data
sample and the sensing threshold of the cooperating CRs as
well. We have obtained an analytical expression of the optimal
number of CRs required for taking the decision of the presence
of the primary user when the CRs utilize the improved energy
detector. It is shown by simulations that an optimal value of the
total error rate can be obtained by utilizing a joint optimization
of the number of cooperating CRs, threshold in each CR, and
the energy detector.
A PPENDIX I
P ROOF OF (14) AND (16)
From (11) and (13), the probability of false alarm 𝑃𝑓 in
each CR can be obtained as
∫ ∞
𝑓𝑊/𝐻0 (𝑦)𝑑𝑦
𝑃𝑓 =
∫
𝜆
√
1−𝑝
𝑝
2
𝑝
𝑦
exp(− 2𝜎
)
𝑛
√
𝑑𝑦.
=
𝑝 𝜋𝜎𝑛
𝜆
After applying some algebra in (32) we get
∫ ∞
√
1
𝑡 exp(−𝑡)𝑑𝑡.
𝑃𝑓 = √
2
𝑝
𝜋 𝜆2
∞
2𝑦
(32)
(33)
2𝜎𝑛
By comparing (33) with the expression of the incomplete
gamma function [9, Eq. (6.5.3)] we get (14).
From (12) and (15), the probability of miss 𝑃𝑚 in each CR
will be
∫ ∞
𝑓𝑊/𝐻1 (𝑦)𝑑𝑦
𝑃𝑚 =
𝜆
∫
=
∞
𝜆
2
√ 1−𝑝
𝑝
2𝑦 𝑝 exp(− 2(𝜎𝑦2 +𝜎2 ) )
𝑛
𝑠
𝑑𝑦.
√ √
𝑝 𝜋 (𝜎𝑠2 + 𝜎𝑛2 )
(34)
Fig. 6. Total error rate versus 𝜆 and 𝑝 for different number cooperating users
𝑛 = 1, 2, 3, 4, 5, 6 at SNR=10 dB.
After some algebra we get
𝑃𝑚
1
=√
𝜋
2
𝜆𝑝
2 +𝜎 2 )
2(𝜎𝑠
𝑛
∫
0
√
𝑟 exp(−𝑟)𝑑𝑟.
(35)
From (35) and [9, Eq. (6.5.2)] we get (16).
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