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中国科技论文在线
Performance of Cooperative Spectrum Sensing Over Fading
Channels With Low SNR#**
WANG Qian1, YUE Dianwu1, LIU Francis2
(1. College of Information Science and Technology, Dalian Maritime University, 5
LiaoNing DaLian 116026;
2. Department of Electronic and Information Engineering, The Hong Kong Polytechnic
University)
Foundations: Research Fund for the Doctoral Program of Higher Education of China Ministry of Education under
Grant 20092125110006
Brief author introduction:王谦，(1983-)，男，大连海事大学博士研究生，主要研究方向为认知无线电技术
Correspondance author: 岳殿武，(1965-)，男，大连海事大学教授，博士生导师，主要研究方向为MIMO
技术、协作通信技术、认知无线电技术、信道编码、信息安全
Abstract: In this paper, we investigate the performance of cooperative spectrum sensing over fading
channels. We derive the upper-and lower-bounds of the overall false alarm probability and overall 10
detection probability as a function of the bit error probabilities of the cognitive radio-fusion center
(CR-FC) links. Given an overall false alarm probability or overall detection probability, we derive the
relationships of error probabilities of the CR-FC links that should be satisfied. We also investigate the
relationship between the overall detection probability and the overall false alarm probability when the
SNR of the primary user signal at each CR node is small. 15
Keywords: Wireless communications; cognitive radio; cooperative spectrum sensing; energy detection;
fading channel
0 Introduction
In recent years, driven by an increasing demand for extra frequency bands, cognitive radio 20
(CR) technology has been proposed and studied for improving the spectrum efficiency of existing
frequency bands. To ensure that the primary services in the considered frequency bands are not
affected by CR transmissions, CR nodes are required to possess the ability of detecting the
presence of an active primary user (PU) [1]. Among various kinds of sensing schemes, energy
detection attracts more attention because of its simple implementation. However, when there 25
exists fading in the PU-CR link, the sensing task becomes more difficult and consequently the
sensing performance may degrade significantly. Subsequently, cooperative spectrum sensing has
been proposed to enhance the sensing performance [2]-[4]. Cooperative spectrum sensing can be
regarded as an advanced version of the single-user sensing and has been extensively studied. In
cooperative spectrum sensing, each CR node senses the channel independently and then sends the 30
information to the fusion centre (FC) via the channel between the CR node and the FC (CR-FC
link). Based on all the received messages, the FC makes a final decision on the presence or
absence of a PU. We consider decision fusion in which each CR node sends its local decision to
the FC, and the FC makes a final decision based on the decoded signals and a fusion rule. The
counting rule, which is also known as the K -out-of- N rule, is a common fusion rule for 35
decision fusion. In the study of cooperative spectrum sensing, the CR-FC links are usually
assumed to be error-free. Such an assumption does not hold in general because of the presence of
Gaussian noise at the receiver of the FC. If the CR-FC links are further subject to common fading
appearing in a wireless environment, more errors will occur at the FC receiver. Consequently,
incorrect sensing decisions will be made by the FC. 40
In this paper, we will take into account the errors in the CR-FC links as we investigate
cooperative spectrum sensing with decision fusion at the FC. We will study the effect of the errors
in the CR-FC links on the overall sensing performance. In particular, the upper-and lower-bounds
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中国科技论文在线
of the overall false alarm probability and overall detection probability will be derived as a function
of the bit error probabilities of the CR-FC links. Given an overall false alarm probability or overall 45
detection probability, we derive the region of error probabilities of the CR-FC links that can be
allowed. We also investigate the relationship between the overall detection probability and the
overall false alarm probability when the SNR of the primary user signal at the CR node is small.
1 System Model
We consider a cooperative CR network with N CR nodes and a FC [5]. The cooperative 50
spectrum sensing is operated in two successive stages. In the first stage, the i th CR node
( 1, , )i N= … performs energy detection based on its received signal. Due to the fading in the
PU-CR link, the received SNR at the i th CR node, denoted by ( )1
iγ , is a random variable. Let
( )
1
( )
1( )i
ifγ γ and ( )1 iγ be the probability density function (PDF) and expected value of ( )1 iγ ,
respectively. In addition, we denote the channel link between PU and the i th CR node by ( )1
ih 55
with second moment ( )1
iΩ . We assume that the i th CR nodes apply the energy threshold ( )iλ
and the number of samples ( )2 iu in a sensing duration. Denote the false alarm probability and the
detection probability of the i th CR node by ( )ifP and
( )i
dP , respectively. Then, we have [6]
( )
( )
( ) ( )
( )
( )
, / 2i i
i
f i
u
P
u
λΓ= Γ (1)
( )( ) ( )
1
( ) ( ) ( ) ( ) ( )
1 1 10
2 , ( )di ii i i i id uP Q fγγ λ γ γ
∞= ∫ (2) 60
where ( )Γ ⋅ and ( ),Γ ⋅ ⋅ denote the gamma function [7] and the incomplete gamma function
[7], respectively. Moreover, ( ),mQ ⋅ ⋅ represents the generalized Marcum Q -function of order m
which is given by () in [8], .,
( ) ( )2 2 111, exp 2mm mm y
t xQ x y t I xt dt
x
∞
−−
⎛ ⎞+= −⎜ ⎟⎝ ⎠∫ (3)
with 1mI − being the modified Bessel function of order 1m − . 65
After the energy detection in each CR node, the CR network proceeds to the second stage in
which the CR nodes send their local decisions to the FC via the CR-FC links. For the i th CR-FC
link, we denote the received SNR by ( )2
iγ ; the expected SNR by ( )2 iγ ; and the PDF of ( )2iγ
by ( )( )
2
( )
2i
ifγ γ . The FC decodes the binary local decisions from the received signals. Suppose binary
phase-shift-keying (BPSK) is used to send the binary data in the CR-FC links. The bit error 70
probability for the i th CR-FC link, denoted by ( )ibP , can be expressed as [8]
( ) ( )( )
2
( ) ( ) ( ) ( )
2 2 20
2 dii i i ibP Q fγγ γ γ
∞= ∫ . When ( )2iγ follows a Rayleigh distribution, ( )ibP can be further
simplified to ( )( )( ) ( ) ( )2 21 / 1 / 2i i ibP γ γ= − + . For the i th PU-CR-FC link, the false alarm
probability ( )ifP% and the detection probability ( )idP% can be expressed, respectively, as
( ) ( )( ) ( ) ( ) ( ) ( )1 1 i i i i if f b f bP P P P P= − + −% (4) 75
( ) ( )( ) ( ) ( ) ( ) ( )1 1 . i i i i id d b d bP P P P P= − + −% (5)
Finally, the FC decodes the binary decisions sent by the CR nodes and makes its final
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中国科技论文在线
decision, based on the K -out-of- N rule, on whether a PU is active or not. We define D as the
peripheral set of the index of PU-CR-FC link, . {1, 2, , }N=D L and use 1D and 0D to
record the index of PU-CR-FC link whose output is decoded as presence and absence, respectively. 80
So we have 1 0 =∪D D D . Furthermore, we denote ( )jD as the set of all the possible sets 1D
whose cardinality is j . So the cardinality of ( )jD is
N
j
⎛ ⎞⎜ ⎟⎝ ⎠
. Consequently, the overall false alarm
probability fQ and the overall detection probability dQ can be expressed, respectively, as
( )( ) ( )( )
( )1 1 1
1
N
i i
f f f
j K j i i
Q P P
= ∈ ∈ ∉
= −∑ ∑ ∏ ∏
D D D D
% % (6)
( )( ) ( )( )
( )1 1 1
1
N
i i
d d
j K j i i
dQ P P
= ∈ ∈ ∉
= −∑ ∑ ∏ ∏
D D D D
% % (7) 85
2 Overall Sensing Bounds With Errors in The CR-FC Links
When the average received SNR at the FC (., ( )2
iγ , 1, ,i N= L ) is small, the bit error
probability ( )ibP of the CR-FC link increases. In this section, we investigate the impact of the bit
error probabilities on the overall false alarm probability fQ and overall detection probability dQ .
Bounds of Overall Detection Probability 90
Consider the scenario of counting rule. Given 1 ( )k∈D D causing a present decision and
'
2 ( )k∈D D , 'k k< , if 2D contains 1D , then the set 2D will also cause a present decision. So,
the counting rule is a monotonic rule based on which we have [9]
( ) 0, 1, , .
d
i
d
Q
i N
P
∂ > =∂ L% (8)
Then we differentiate %the probability of false alarm ( )idP% in each PU-CR-FC link (., (4)) 95
. ( )idP and we have
( )
( )
( ) 1 2 0, 1, ,
i
id
bi
d
dP
P i N
dP
= − > =% L (9)
where the last inequality holds because the bit error probability ( )ibP is always smaller than
1/2. So, we conclude that ( )idQ is monotonically increasing in
( )i
dP because
( )
( )( ) 0
i
d d d
ii
d dd
Q Q P
P PP
∂ ∂ ∂= >∂ ∂∂
%
% . Consequently, for fixed N , K and
( )i
bP , the upper-bound and 100
lower-bound of dQ can be found, respectively, by applying
( ) 1, 1, ,idP i N= = L and
( ) 0, 1, ,idP i N= = L into (6). Denoting the upper-bound and lower-bound of dQ under such
conditions as upperdQ and
lower
dQ , respectively, we can write
( )( ) ( )( )
( )1 1 1
upper 1
N
i i
d b b
j K j i i
Q P P
= ∈ ∈ ∉
= −∑ ∑ ∏ ∏
D D D D
(10)
( )( ) ( )( )
( )1 1 1
lower 1
N
i i
d b b
j K j i i
Q P P
= ∈ ∈ ∉
= −∑ ∑ ∏ ∏
D D D D
(11) 105
Although these two bounds are not reachable because ( )iλ can never be set so that ( )idP
draws value at 1 or 0 in practice, they can be regarded as very tight bounds for analysis purposes
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中国科技论文在线
when ( ) , 1, ,idP i N= L are set very close to 0 or 1.
By considering that the lowerdQ in (11) has the same formation with (7), it can be readily
shown that lower ( )/ 0id bQ P∂ ∂ > . Comparing the expressions for upperdQ and lowerdQ , it can be 110
observed that they become equivalent after replacing ( )ibP with
( )1 ibP− in one of the expressions.
The results mean that the lower-bound of the overall detection probability lowerdQ increases
monotonically with ( )ibP while the upper-bound
upper
dQ decreases monotonically with
( )i
bP . We
then consider the scenario when K and N are fixed. If the average received SNR for the
CR-FC links ( )2 , 1, ,
i i Nγ = L decreases, the bit error probability ( ) , 1, ,ibP i N= L will increase. 115
Consequently, the upper-bound of the total detection probability upperdQ decreases and the
lower-bound of the total detection probability lowerdQ increases. At a certain critical bit error
probability, the upper bound upperdQ may coincide with the lower bound
lower
dQ , implying that a
valid overall detection probability dQ cannot be achieved. (Equivalently, a valid
( ) , 1, ,idP i N= L
cannot be found.) In other words, a preset overall detection probability givendQ , can only be 120
achieved if and only if valid bit error probabilities ( ) , 1, ,ibP i N= L are found such that
upper given lower
d d dQ Q Q> > , .,
( )( ) ( )( )
( )1 1 1
given lower1
N
i i
d b b d
j K j i i
Q P P Q
= ∈ ∈ ∉
> − =∑ ∑ ∏ ∏
D D D D
(12)
( )( ) ( )( )
( )1 1 1
given upper 1 .
N
i i
d b b d
j K j i i
Q P P Q
= ∈ ∈ ∉
< − =∑ ∑ ∏ ∏
D D D D
(13)
Consider a special case when (1) ( )Nb b bP P P= = =L , equation (10) and (11) are simplified to 125
upper (1 )
N
j N j
d b b
j K
N
Q P P
j
−
=
⎛ ⎞= −⎜ ⎟⎝ ⎠∑ (14)
lower (1 )
N
j N j
d b b
j K
N
Q P P
j
−
=
⎛ ⎞= −⎜ ⎟⎝ ⎠∑ (15)
and the bit error probability bP should satisfy the relationship given in (12) and (13), .
given lower (1 )
N
j N j
d b b d
j K
N
Q P P Q
j
−
=
⎛ ⎞> − =⎜ ⎟⎝ ⎠∑ (16)
given upper (1 ) .
N
j N j
d b b d
j K
N
Q P P Q
j
−
=
⎛ ⎞< − =⎜ ⎟⎝ ⎠∑ (17) 130
Denote the overall detection probability that satisfies lower givend dQ Q= by bˆP . Since the
lower-bound of the total detection probability lowerdQ increases monotonically with bP , we can
conclude that the inequality in (16) is satisfied ˆ[0, )b bP P∀ ∈ . Comparing the expressions for
upper
fQ and
lower
fQ in (16) and (17), it can be observed that they become equivalent after replacing
bP with 1 bP− in one of the expressions. Because of this feature, it can be readily proved that the 135
inequality in (17) is satisfied ˆ[0,1 )b bP P∀ ∈ − . Consequently, the two inequalities in (16) and (17)
are satisfied simultaneously ˆ ˆ[0, min( ,1 ))b b bP P P∀ ∈ − . If the transmit power in the CR nodes are
controlled such that the bit error probability bP satisfies that ˆ ˆmin( ,1 )b b bP P P< − , the preset
overall detection probability will be achievable by setting the thresholds of energy detectors at the
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中国科技论文在线
CR nodes. 140
Bounds of Overall Detection Probability
Comparing the expressions for the overall false alarm probability fQ (6) and the overall
detection probability dQ (7), we can easily observe that they become identical if
( ) ( )i i
f dP P=% % for
all i s. Comparing the expressions for the detection probability ( )idP% (5) and the false alarm
probability ( )ifP% (4) further reveals that ( ) ( ) , 1, ,i id fP P i N= =% % L if ( ) ( ) , 1, ,i if dP P i N= = L . 145
Consequently, the effect of ( )ifP and
( )i
bP on fQ is the same as the effect of
( )i
dP and
( )i
bP
on dQ , which has been elaborated in the previous section.
3 Relationship Between The overall detection probability and the
overall false alarm probability When Average SNR at a CR Node
is Small 150
In this section, we study the relationship between the overall detection probability dQ and
the overall false alarm probability fQ when the average received SNRs at each CR node are
small. Substituting [7]
2
0
1( )
! ( 1) 2
m k
m
k
zI z
k m k
+∞
=
⎛ ⎞= ⎜ ⎟Γ + + ⎝ ⎠∑ into (3) and exchanging the order of
integration and summation, we can re-write the Marcum Q -function as
( ) ( )2 2 2/2
0
1, , / 2 .
2! ( )
kk
m x
k
xQ x y m k y
e k m k
∞
=
⎛ ⎞== Γ +⎜ ⎟⎝ ⎠Γ +∑ (18) 155
Here, we further define ( ) ( ) ( ) ( ) ( )1 1 1
i i i i iγ γ β γ β= = Ω where ( )iβ is a random variable with unit
mean and is drawn according to the PDF of ( )1
iγ , and γ denotes the transmitting SNR of PU.
We also assume that all moments of ( )iβ exist. This assumption can be easily justified, especially
for Rayleigh, Rician and Nakagami-$m$ fading channels [8]. Replacing x , y and m in (18)
with ( ) ( )12
i iγ β , ( )iλ and ( )iu , respectively, the probability that the PU signal is detected for a 160
given ( )1
iγ can be written as
( ) ( ) ( )( ) ( )1( ) ( ) ( )1( ) ( ) ( ) ( ) ( )1 ( )
0
2 , , / 2 .
! ( )
i i
i
ki i
i i i i i
iu
k
e
Q u k
k u k
γ β γ βγ β λ λ
−∞
=
= Γ +Γ +∑ (19)
Then ( )idP can be derived by integrating (19) over the whole range of
( )iβ , .,
( )
1
( )
( ) ( )
( ) ( )
1( ) 0
0
( , / 2) ( ) ( )d
! ( )
i
i
i i
i i k
d i
k
u kP e f
k u k
γ β
β
λ γ β β β∞ ∞ −
=
Γ += Γ +∑ ∫ (20)
where ( ) ( )ifβ β denotes the PDF of ( )iβ . We substitute ( ) ( )
( )
1( )
1
0
exp
!
ji
i
j j
γ βγ β ∞
=
−− = ∑ into 165
(20) and exchange the order of integration and summation, yielding
( )( ) ( ) ( )1
0 0
( 1) ( , / 2) ( )
! ! ( )
j j ki i i
d j k
k j
u kP M
k j u k
λ β γ∞ ∞ ++
= =
− Γ += Γ +∑∑ (21)
where ( )( )ijM β represents the j th moment of the random variable ( )iβ . As ( ) ( )ifβ β
represents the PDF of ( )iβ and the first moment of ( )iβ has been normalized to 1, we have
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中国科技论文在线
( )( )( ) 00 0( ) d 1iiM fββ β β β∞= =∫ (22) 170
( )( )( )1 0( ) d fββ β β β∞= =∫ (23)
Inserting (22) and (23) into (21) gives
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
1( ) ( ) ( )
( , / 2) ( 1, / 2) ( , / 2)
( ) ( 1) ( )
i i i i i i
i i
d i i i
u u uP
u u u
λ λ λ γ⎛ ⎞Γ Γ + Γ= + −⎜ ⎟Γ Γ + Γ⎝ ⎠
( )( ) ( ) ( ) ( )1( )
1 1
( 1) ( , / 2) ( ) .
! ! ( )
j i i j ki i
j ki
k j
u k M
k j u k
λ β γ∞ ∞ ++
= =
− Γ ++ Γ +∑∑ (24)
Note that ( ) ( )1 1
i iγ γ= Ω . When the average received SNR ( )1 iγ is small, say much less than 1, 175
(24) will be dominated by the first two terms. Consequently, we can approximate the detection
probability by
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
1( ) ( ) ( )
( , / 2) ( 1, / 2) ( , / 2) .
( ) ( 1) ( )
i i i i i i
i i
d i i i
u u uP
u u u
λ λ λ γ⎛ ⎞Γ Γ + Γ≈ + − Ω⎜ ⎟Γ Γ + Γ⎝ ⎠ (25)
Comparing (25) with (1) shows that the first term in (25) equals the false alarm probability
( )i
fP , ., 180
( ) ( ) ( ) ( )
( ) ( ) ( )
1( ) ( )
( 1, / 2) ( , / 2)
( 1) ( )
i i i i
i i i
d f i i
u uP P
u u
λ λ γ⎛ ⎞Γ + Γ≈ + − Ω⎜ ⎟Γ + Γ⎝ ⎠ (26)
( ) ( ) ( ) ( )
( ) ( ) ( )
1( ) ( )
( 1, / 2) ( , / 2) .
( 1
) ( )
i i i i
i i i
d f i i
u uP P
u u
λ λ γ⎛ ⎞Γ + Γ⇒ − ≈ − Ω⎜ ⎟Γ + Γ⎝ ⎠ (27)
We can further conclude that the detection probability ( )idP is close to the false alarm
probability ( )ifP when the average received SNR
( )
1
iγ is very small, ., ( )1 0iγ → . As ( )1 iγ
increases, the detection probability ( )idP increases. Since the second term in \eqref{eq:pd4} is 185
independent of the PDF of ( )1
iγ , we conclude that the speed at which ( )idP improves is
independent of the fading type of the PU-CR link.
We consider the case in which cooperative spectrum sensing with the K -out-of- N rule is
used when the average SNR ( )1
iγ for each PU-CR link is small, say much less than 1. We apply
the approximation in (25) into (5) and obtain 190
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1( ) ( ) ( )
( , / 2) ( 1, / 2) ( , / 2) (1 2 )
( ) ( 1) ( )
i i i i i i
i i i i
d b bi i i
u u uP P P
u u u
λ λ λ γ⎡ ⎤⎛ ⎞Γ Γ + Γ≈ + + − Ω −⎢ ⎥⎜ ⎟Γ Γ + Γ⎝ ⎠⎣ ⎦
%
( ) ( )ifP a i γ= +% (28)
where the last equation holds because of (4) and
( ) ( ) ( ) ( )
( ) ( )
1( ) ( )
( 1, / 2) ( , / 2)( ) (1 2 )
( 1) ( )
i i i i
i i
b i i
u ua i P
u u
λ λ⎛ ⎞Γ + Γ= − − Ω⎜ ⎟Γ + Γ⎝ ⎠ . dQ can be derived by substituting (28)
into (7), giving 195
( ) ( )( ) ( ) ( )( )
( )1 1 1
1
K
i i
d f t f t
j k j i i
Q P P i P P iγ γ
= ∈ ∈ ∉
≈ + − −∑ ∑ ∏ ∏
D D D D
( ) ( ) ( )
( )
2
1 11 1
1 2
2 1
1
K
i i
f f t
j k j ii i
i i
P P i P γ
= ∈ ∈∈ ∈≠
⎧⎡ ⎤⎪⎢ ⎥= + +⎨⎢ ⎥⎪⎢ ⎥⎣ ⎦⎩
∑ ∑ ∑∏ ∏
D D DD D
L
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中国科技论文在线
( )( ) ( ) ( )( )2
11 1
1 2
2 1
11 1
i i
f f t
ii i
i i
P P i P γ
∉∉ ∉≠
⎫⎡ ⎤⎪⎢ ⎥× − − − + ⎬⎢ ⎥⎪⎢ ⎥⎣ ⎦⎭
∑∏ ∏
DD D
L
( ) ( )( )
( ) ( )
( ) ( ) ( )( )2
1 1 11 1 1 1
1 2
2 1
11 { 1
K K
i i i i
f f f f
j k j kj j ii i i i
i i
P P P i P P
= =∈ ∈ ∈∈ ∉ ∈ ∉
≠
= − + −∑ ∑ ∑ ∑ ∑∏ ∏ ∏ ∏
D D D D DD D D D
( ) ( ) ( )( )
1 1 1
1 2
2 1
1 1 }
i i
f f t
i i i
i i
P i P P γ
∉ ∈ ∉≠
− − +∑ ∏ ∏
D D D
L 200
( ) ( ) ( )( )
( )
2
1 1 1 1
1 2
2 1
1{ 1
K
i i
f f f
j k j i i i
i i
Q P i P P
= ∈ ∈ ∈ ∉
≠
= + −∑ ∑ ∑ ∏ ∏
D D D D D
( ) ( ) ( )( )
1 1 1
1 2
2 1
1 1 }
i i
f f t
i i i
i i
P i P P γ
∉ ∈ ∉≠
− − +∑ ∏ ∏
D D D
L (29)
where the last equation holds due to the relationship shown in (6). When the average received
SNR 1γ is much smaller than 1, dQ can be approximated by the first two terms in (29), .,
( ){ 21 1 1 1
1 2 2 1
( ) ( )
1( ) ,
( ) 1
N
i i
d f f fj i i i i i
j K
Q Q a i P P∈ ∈ ∈ ≠ ∈/=
≈ + −∑ ∑ ∏ ∏∑ D D D D D% % 205
( ) ( )}21 1 1
1 2 2 1
( )( )
1 ,
( ) 1 iif fi i i i ia i P P γ∈ ∈ ∈ ≠/ /− −∑ ∏ ∏D D D% % (30)
( ){ 21 1 1 1
1 2 2 1
( ) ( )
1( ) ,
( ) 1
N
i i
d f f fj i i i i i
j K
Q Q a i P P∈ ∈ ∈ ≠ ∈/=
⇒ − ≈ −∑ ∑ ∏ ∏∑ D D D D D% %
( ) ( )}21 1 1
1 2 2 1
( )( )
1 ,
( ) 1 .iif fi i i i ia i P P γ∈ ∈ ∈ ≠/ /− −∑ ∏ ∏D D D% % (31)
Consider a special case with assumptions )a all the received SNR in PU-CR links are
identical, . (1) ( )1 1 1
NΩ = = Ω = ΩL or (1) ( )1 1 1Nγ γ γ= = =L ; )b all the CR nodes are with identical 210
samples and thresholds, . (1) ( )Nu u u= = =L and (1) ( )··· Nλ λ λ= = = ; )c all the bit error
probabilities in CR-FC links are identical, . (1) ( )Nb b bP P P= = =L . Then, we have
(1) ( )N
f f fP P P= = =% % %L and (1) ( )a a N a= = =L . The assumptions )a and )b are common in the
CR literatures. Consequently, the right hand side (RHS) of (31) can be simplified as
( ) ( ) 11 1 ( ) 1N N j N jj jd f f f f f
j K
N
Q Q jP P N j P P a
j
γ− − −−
=
⎛ ⎞ ⎡ ⎤− ≈ − − − −⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠∑ % % % % 215
( )1 1 .N KKf f fNQ KP P aK γ−−⎛ ⎞= + −⎜ ⎟⎝ ⎠ % % (32)
As shown in (31) and (32), the overall detection probability dQ will be close to the overall
false alarm probability fQ when the average SNR
( )
1
iγ for each PU-CR link is small. The same
equation also indicates that dQ improves with an increasing
( )
1
iγ . Unlike in the high SNR region
where the choice of the fusion rule influences significantly the overall detection probability dQ 220
[11], the choice of the fusion rule here (in the low SNR region) only affects the rate that the
overall detection probability improves as the average SNR ( )1
iγ increases.
4 Results and Discussions
In this section, we present some numerical results. We set the number of samples for sensing
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中国科技论文在线
at each CR node to be 10, ., (1) ( )2 2 10Nu u= = =L and hence ( ) 5, 1, ,iu i N= = L . Moreover, 225
we set all the thresholds of energy detectors in the CR nodes are identical, . (1) ( )Nλ λ= =L .
-30 -25 -20 -15 -10 -5 0 5 10 15 20
0
1
� �K%�
B
ou
nd
s
of
Q
d
K=1
K=5
K=8
Plot of the upper-bounds (solid lines) and lower-bounds (dashed lines) of the overall detection probability
dQ versus the average received SNR 2γ for each CR-FC link. ( ) 5, 1, ,iu i N= = L , 1,5,8K = and 10N = .
230
First, we set 1,5,8K = and 10N = in the K -out-of- N decision rule. We assume all
received SNR in the FC are identical, . (1) ( )2 2 2
Nγ γ γ= = =L , which makes the corresponding bit
error probabilities hold the relationship that (1) ( )Nb bP P= =L . Figure 1 plots the upper- and
lower-bounds of the overall detection probability ((14) and (15)) versus the average received
SNR 2γ for each CR-FC link. The results show that for a fixed K , the corresponding 235
lower-bound decreases while the upper-bound increases with 2γ . For a particular value of 2γ ,
the range of achievable overall detection probability dQ is determined by the corresponding
upper-bound and lower-bound values. For example, when 5K = and 2 15γ = − dB, the
upper-bound and lower-bound of dQ are given by and , respectively. As a result, the
overall detection probability dQ beyond the range [ ], is not achievable under such 240
conditions. We can also observe that for 1K = , the range of achievable dQ is very small when
2γ falls below 5− dB. As implied in Sect. , the same set of curves can be used to represent the
bounds of the overall false alarm probability fQ . In consequence, similar conclusions can be
made on the overall false alarm probability.
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中国科技论文在线
-20 -15 -10 -5 0 5 10
10
-5
10
-4
10
-3
10
-2
10
-1
��K%�
P
d-
P
f
m=1
m=2
m=3
Approximate Results
245
Fig. 2 Plot of difference between detection probability ( )idP and false alarm probability
( )i
fP (.,
( ) ( )i i
d fP P− )
versus the average received SNR ( )1
iγ at i th CR node. The channels between the PU and CR nodes are
Nakagami- m fading channels with 1,2,3m = . ( ) 5iu = and ( ) = .
Next, we compare the exact and approximate detection probabilities ( )idP at each CR node 250
when the average received SNR ( )1
iγ at each CR node (., each PU-CR link) is small. The exact
value is given by (2) and the approximate value is evaluated using (26) or (27). We assume that
the channels between the PU and CR nodes are Nakagami- m fading channels with m taking the
values 1, 2,3 . We also set the false alarm probability ( )ifP for each CR node at . Instead of
the plotting the detection probability ( )idP , we plot the difference between the detection 255
probability ( )idP and the false alarm probability
( )i
fP , .,
( ) ( ) ( ) i id f dP P P− = − . The reason for
doing so will become obvious when we describe the results below. Figure. 2 plots the exact
difference between ( )idP and
( )i
fP for 1, 2,3m = and also the approximate difference (27) as the
average received SNR 1γ at each CR node changes from 20− dB to 10 dB. Note that there is
only one curve for the approximate difference (27) because of its independency on the channel 260
type. Considering the curves for the exact ( ) ( )i id fP P− ( ( ) = − ), we observe that they are
almost identical for 1, 2,3m = , indicating that the value of $m$ has little impact on detection
probability ( )idP when
( )
1
iγ is small. Moreover, we find that the exact values for ( ) ( )i id fP P− and
the approximate ones (27) are almost the same, implying that the exact detection probabilities
( )i
dP and the approximate ones (26) are very close. Since the expression for the approximate 265
detection probability (26) is not dependent on the type of channel, we can conclude that the type
of fading has little impact on the sensing performance (detection probability) of each CR node
when the average received SNR 1γ at each CR node is small. In this log-log plot, we observe that
( ) ( ) 55 10i id fP P
−− ≈ × when ( )1 20iγ = − dB. It indicates that the detection probability ( )idP is almost
the same as the false alarm probability ( )ifP when
( )
1
iγ is small, as implied in (27). Moreover, in 270
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中国科技论文在线
this log-log plot, ( ) ( )i id fP P− increases linearly with a unit slope as ( )1 iγ increases, indicating
that ( ) ( )i id fP P− also increases linearly with ( )1 iγ in a linear plot (as suggested in (27)).
-30 -25 -20 -15 -10 -5 0
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
��K%�
Q
d-
Q
f
Exact Results, K=1
Exact Results, K=5
Exact Results, K=10
Approximate Results
Fig. 3 Plot of difference between overall detection probability dQ and overall false alarm probability fQ
(., d fQ Q− ) versus the average received SNR at each CR node. All PU-CR links and CR-FC links are subject to 275
Rayleigh fading. All the average received SNRs at CR nodes are
identical, . (1) ( )1 1 1
Nγ γ γ= = =L . All the average received SNRs at the FC equals 20 dB.
( ) 5, 1, ,iu i N= = L , = , 1,5,10K = and 10N = .
Finally, we compare the exact and approximate overall detection probabilities dQ when all 280
the average received SNRs ( )1 , 1, ,
i i Nγ = L at each CR node (., each PU-CR link) are small.
The exact value is given in (7) with (1) ( )Nd dP P= =% %L and the approximate value is evaluated using
(32). We set 1,5,10K = and 10N = in the K -out-of- N decision rule. We assume an overall
false alarm probability of , ., = . We also adjust the transmit SNRs at the CR nodes
such that the average received SNR 2γ at the FC equals 20 dB. Consequently, based on the 285
results in Fig. 1, we can ensure that = is achievable. Similar to the previous case, we plot
the difference between the overall detection probability dQ and the overall false alarm
probability fQ , ., f dQ Q Q− = − . Assuming that all PU-CR links and CR-FC links are
subject to Rayleigh fading, Fig. 3 plots the exact d fQ Q− and the approximate one given by (31)
as the average received SNR 1γ at each CR node changes from -30 dB to 0 dB. We can observe 290
that the values for the exact d fQ Q− and the approximate ones are almost the same when
1 10γ < dB. In this log-log plot, we observe that 410d fQ Q −− ≈ when 1 30γ = − dB and
1,5,10K = . It indicates that the overall detection probability dQ is almost the same as the overall
false alarm probability fQ when 1γ is small, as implied in (31). Furthermore, the value of K is
found to have a minimal effect on the overall detection probability when 1γ is small. 295
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中国科技论文在线
5 Conclusion
We have analyzed the performance of cooperative spectrum sensing. We find that there is a
limited range of overall false alarm probability and overall detection probability that can be
achieved when errors exist in the CR-FC links. Moreover, the range diminishes as the error rate
increases. When the received SNR at each CR node is small, we further conclude the overall 300
detection probability is almost the same as the overall false alarm probability.
References
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[2] GHASEMI A, SOUSA E S. Collaborative Spectrum Sensing for Opportunistic Access in Fading
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[5] ZHANG W, LETAIEF K B. Cooperative spectrum sensing with transmit and relay diversity in cognitive radio
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[6] DIGHAM F F, ALOUINI M S, SIMON M K. On the energy detection of unknown signals over fading
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[9] THOMOPOULOS S C A, VISWANATHAN R, BOUGOULIAS D K. Optimal Distributed Decision Fusion[J].
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[10] WANG Z, GIANNAKIS G B. A simple and general parameterization quantifying performance in fading
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325
低信噪比下衰落信道协作频谱感知性能
王谦1，岳殿武1，刘重明2
（1. 大连海事大学信息科学技术学院，辽宁 大连 116026；
2. 香港理工大学电子与信息工程学院）
摘要：本文研究衰落信道上的协作频谱感知性能，给出了虚警概率和检测概率的取值上下界。330
在给定的虚警概率或检测概率下，推导出了控制信道上错误概率的取值范围。最后研究了检
测概率和虚警概率在低信噪比下的近似关系。
关键词：无线通信；认知无线电；协作频谱感知；能量检测；衰落信道
中图分类号：
335
