A Cooperative Spectrum Detection Technique in Non-Gaussian ...

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A Cooperative Spectrum Detection Technique in Non-Gaussian ...

2012 International Conference on Biological and Biomedical Sciences

Advances in Biomedical Engineering, Vol.9

A Cooperative Spectrum Detection Technique in Non-Gaussian Noise ∗

Xiaomei Zhu, Weiping Zhu

Institute of Signal Processing and Transmission, Nanjing University of Posts and

Telecommunications, Nanjing, P. R. of China

njiczxm@163.com, zwp@njupt.edu.cn

Keywords: Cognitive radio, Rao detector, Non-Gaussian noise

Abstract. This paper addresses the problem of primary user detection in non-Gaussian noise for

cognitive radio systems. A Rao detector-based cooperative detection scheme is presented, in which

the mixed Gaussian noise model is used to fit non-Gaussian noise and the Rao detector is empleyed

to detect the primary user signal. It can be shown that the proposed detector is asymptotically

optimal. Analytical expressions of the probability of detection for Rao detection and multi-users

cooperative detection are derived. Simulation results indicate that, to achieve the same detection

probability, the mixed-Gaussian Rao detector needs about 11dB less SNR in non-Gaussian noise

than in Gaussian noise, and the proposed cooperative Rao detection can improve the overall

probability of detection by 5% compared to non-cooperative detection.

1. Introduction

Cognitive radio is to improve the spectrum utilization efficiency and realize a dynamic management

of the use of spectrum. It is becoming the key technology for reusing spectrum resources for the next

generation wireless networks. One of the important challenges in cognitive radio systems is to

identify the primary users over a wide range of spectrum as quickly as possible. This process is very

difficult as we need to identify various primary users’ signals embedded in various interferences

generated by other secondary users and thermal noises. Some methods such as transmitter detection

and cooperative detection have been proposed with Gaussian noise assumption [1] .However, the

detection problem in practical applications is more complicated due to the fact that the noise is most

likely non-Gaussian because of the presence of CR user’s interference [2] . Furthermore, the

performances of the spectrum detector optimized against Gaussian noise may degrade drastically

when non-Gaussian noise or interference is present [3] [4] . In view of these problems it is advisable to

seek useful solutions to the spectrum detection in practical non-Gaussian noises and to evaluate the

detection performance.

In this paper, the detection problem in non-Gaussian noise condition is studied. The non-Gaussian

noise model and the Rao detector are presented in Section II. Section III presents multi-user

cooperative detection model based on Rao detection, and investigates the detection performance

expression. In Section IV, some simulation results with analysis are provided. Section VI concludes

the paper.

2. Rao Detection in Non-Gaussian Noise

2.1 Non-Gaussian Noise Model and Parameter Estimation

∗ This work was supported by the Startup Fund of Nanjing University of Posts and Telecommunications under grant number

NY207031.

978-1-61275-027-9/10/$25.00 ©2012 IERI ICBBS 2012

283


Assuming that x (n)

,n=0,1,…,N-1, is the second-order mixed-Gaussian noise sequence with zero

mean in cognitive radio, namely,the sequence can be seen as the sum of samples from the Gaussian

2

2

distribution μ (0, σ 1

) with probability 1−

ε and samples from Gaussian distribution μ(0,

σ 2

) with

probabilityε , the probability density function (PDF) for x (n)

is then expressed as

2

2

1−ε

x ( n)

ε x ( n)

p( x(

n);

ε)

= exp( − ) + exp( − ) (1)

2

2

2

2 2σ

2

πσ

1 2πσ


2

where ε is a mixture parameter between 0 and 1. By changing the parameterε , the Gaussian mixture

model can almost fit any non-Gaussian distribution [5]. Hereσ 2

1

andσ

2 2

are known, andε is unknown,

which may vary with the different locations of the cognitive users and communication environments.

Usually, ε can be estimated through matrix estimation [5] . Noting that the mean of x(n)

is zero, the

variance of the overall PDF is

1 N n 0

Using ∑ − x

N =

1

2

E(

x

= ( 1− ε ) σ + εσ

2

1

( n)

instead of E ( x

2 ( n))

, namely,

2

1

∫ ∞ −∞

2

2

( n))

= x ( n)

p(

x(

n);

ε ) dx(

n)

(2)

2

2

1

N

N 1

∑ −

n=

0

2

x ( n)

= (1 − ε ) σ + εσ

2

1

2

2

(3)

The estimate of ε is given by


ε =

1

N

N 1

2

∑ − x

n=

0

2

σ

2

( n)

− σ

− σ

2

1

2

1

(4)

2.2 Rao detection Algorithm and Performance Analysis

Under hypothesis H

1

and H , the hypothesis testing is described as

0

⎧ x(

n)

= w(

n)


⎩x(

n)

= Hθ

+ w(

n)

H

H

0

1

Or

⎧θ

= 0


⎩θ

≠ 0

H

H

0

1

(5)

where x(n)

is the N × 1observation vector, H is the known N × p observation matrix with rank p

and N > p ,θ is the p × 1 unknown parameter vector under the hypotheses H

1, w (n)

is the secondorder

mixed-Gaussian noise sequence with zero mean. Supposing the primary user signal is a weak

sinusoidal signal with unknown amplitude, the detection problem can be described as

⎧ x(

n)

= w(

n)


⎩x(

n)

= Acos(2πf

+

where A is the unknown signal amplitude. According to (6), we have

+

0

n ϕ)

w(

n)

H

H

0

1

(6)

284



θ = A


⎩H

= [cos( ϕ),cos(2π

f + ϕ),

,cos(2π

( N −1)

f0

+

T

0

ϕ)]

(7)

For = θ 0

= 0

θ , the Rao test statistic is determined by [6]

∂ ln p(

x;

θ ) T

−1

∂ ln p(

x;

θ ) T

( x)

=

I ( θ

0

)

∂θ

θ = θ

0

∂θ

θ = θ

0

T R

1

T T − T

H( H H)

H

= y y iA ( )


[

N −1


n=

0

=

N −1

i(

A)

y(

n)cos(2πf

n=

0

[cos(2πf

0

0

+ ϕ)]

+ ϕ)]

2

2

(8)

where y = [ g[0] g[1] g[ N −1]] T , and g (⋅)

is a Gaussian-based filter,

and i (A)

is the essentially mathematical expectation of sample

Hence, using (1) into (9), we can get y (n)

. If the test statistic

of the Rao detector, then Rao test decision is H

1.

dp(

x(

n);

ε ) / dx

g(

x(

n))

= −

(9)

p(

x(

n);

ε )

2

2

[ g ( x(

n))]

, as

⎛ dp(

w)


⎜ ⎟


∞ ⎝ d(

w)

1

=


2 1 N

2

i ( A)

∫ dw = E{ [ g(

x(

n))]

} ≈

−∞

∑[

y(

n)]

(10)

p(

w)

N n=

0

T R

'

( x)

> γ , where γ '

is the threshold

In large sample conditions, the theoretic performance of Rao detector is consistent with the

asymptotic performance of the generalized likelihood ratio test [7] , so T R

(x)

is distributed as follows

T R

2

⎧χ

(1,0)

( x)

~ ⎨ 2

⎩ χ (1, λ )

H

0

H

1

(11)

Here χ 2 (1,0 ) represents a chi-square distribution with 1 degree of freedom and χ

2 (1, λ ) represents a

noncentral chi-square distribution with 1 degree of freedom and non-centrality parameter

T T

2

λ=

i( A)

θ1

H Hθ

1

≤SNR⋅

σ I

[8] , where I

f

f

is the Fisher information array element, θ 1

is the true value of

θ under the hypotheses H . The PDF of

1

T R

(x)

is given by

1/ 2

⎧(1/

2)

⎪ x

f =

Γ(1/

2)

T R

( x)

⎨ 1−2

( )

⎪ 1 x

4

( ) e

⎪⎩

2 λ

e

(1/ 2−1)

−x/ 2

λ+

x


2

I

(

1/ 2−1

λx)

H

H

0

1

(12)

285


where Γ(.)

is the gamma function and I

v

(.) is the vth-order modified Bessel function of the first

kind. From (12) the false alarm probability and the detection probability of Rao detection are given

by

where Q a ∫ ∞ 1 1 2

( ) = exp( − t ) dt

a


2

P

'

'

= P(

T > γ | H ) 2Q(

γ )

(13)

FA R

0

=

3. Primary User Detection based on Multiuser Cooperation

'

'

'

PD = P(

TR

> γ | H ) = Q(

γ − λ)

+ Q(

γ + λ)

(14)

A simulated model of CR networks can be described as Fig.1, where CUs (cognitive user) are

random distribution, CU1, CU2 and CU3 suffering different shadowing influence will receive weak

PU signal, which may give a wrong decision. If these CUs transmit their own data on the certain

spectrum paragraph, they likely affect the PU (primary user) receiver’s results, and create

interference to the PU system inevitably. CU4 and CU3 could receive the transmitter signal within

the radius scope of primary transmitter's emissive power.

1

Figure1. Cooperative spectrum sensing model

Based on the above detection scenario, the reliability that some single cognitive user, such as

CU1,CU2,CU3, detect available spectrum will be low, and thus some of these CUs are selected to

take part in spectrum detection at the same time, which can improve the detection performance.

Figure2. Block diagram of cooperative spectrum detection based on Rao

Fig.2 is the block diagram of the proposed cooperative spectrum detection based on the Rao

detector, where M cognitive users take part in cooperation. Cognitive users transmit local detection

results u = [ u1, u2... u M

] and SNR = [ SNR1, SNR2... SNR M

] to the fusion center (FC) after the estimation of

ε and the Rao detection. The FC chooses the users whose SNR are larger than the average SNR

286


value of all the cooperative cognitive users. Then, the selected users cooperatively detect the

primary users signal based on OR rule. The final decision declares whether the primary user is

present or not. The cooperative detection probability and the cooperative false alarm probability

when L cognitive users take participate in the cooperation are given by [9] :

If the cooperative probability of false alarm

probability of false alarm, , is given by

P

fa , i

L

−∏ (1 −

P fa

= 1 P , P d

= 1 − ∏ (1 − P

(15)

, c

fa,

i

)

1

L

, c

d , i

)

1

P

fa ,

is to be fixed, from (15) the individual CU’s

c

P L

fa, i

1 − (1 − Pfa,

c

)

= (16)

From (13) and (16), the threshold of each user for the Rao detector is

' −1

1

2

γ

i

= ( Q ( P

fa , i

))

2

i = 0,1, L

(17)

Finally, the cooperative detection probability based on the OR rule is

2

where λ

i

= SNR


i I

f , i

.

P

L


= 1 − (1 −P

)

dc , di ,

1

L

'

'

1 ∏(1 ( Q( ri λi) Q( ri λi)))

1

= − − − + +

(18)

4. Simulation Results

4.1 Mixed-Gaussian Rao and Gaussian Rao detection

Figure 3. ROC for Mixed-Gaussian Rao and Gaussian Rao detection

287


2

2

Simulations are carried out with σ

1

= 1, σ

2

= 81 , and ε = 0. 3 corresponding to

2 '

σ = 25 and I

f

≈ 0.56 . When given false alarm probability P

f a

is 0.01 and 0.03, the threshold γ is

6.635 and 4.709, respectively. By using the threshold γ '

and non-centrality parameter λ , the

detection probability can be derived, as shown in Fig.3. We can see that the detection probability is

a monotonically increasing function of λ (or SNR). For the mixed-Gaussian Rao

2

detection λ = SNR(

σ I

f

) , while for the Gaussian Rao detection, λ = SNR . With the same signal and

non-Gaussian interference backgrounds, the mixed-Gaussian Rao detection has gained an

additional SNR improvement about 11dB, which is in agreement with the theoretical value:

Δ= I = dB

(19)

2

10lg( σ

f

) 11.4

4.2 Cooperative Detection and Non-Cooperative Detection

Suppose three users CU1, CU2 and CU4 in Fig.1 are chosen to cooperate, and their SNRs are set to

-3dB, -5dB, -10Db, respectively. The resulting ROC curves are shown in Fig.4. Obviously, under

the same probability of false alarm conditions, the probability of detection is increased as the SNR

increases.

Figure 4. Non-cooperative detection and traditional OR cooperation

Figure 5. Proposed cooperation based on Rao detection

Fig.4 also indicates that when multiple users cooperate and the SNR of CU2 is very low, the

probability of cooperative detection is less than the probability of detection by CU4 alone if using

288


the fusion method of traditional OR rule. However, using the improved fusion method, the FC

compares the SNRs of individual cognitive users in the fusion center, and chooses the users whose

SNR is higher than the average SNR, and then integrates detection results via the OR rule. In Fig.5,

we have selected CU4, CU1 with SNR -3dB,-5dB respectively, and observed that the detection

performance is increased by 5% as compared to the probability of detection of CU4 alone when P

f

is 0.1.

5. Concluding Remarks

In this paper, we have studied the use of Rao detector for the detection of primary user under non-

Gaussian noise in cognitive radio networks. Our simulation results show that the mixed-Gaussian

Rao detector can improve the detection performance by about 11dB. The proposed cooperative

detection based on the Rao detection can also improve the overall detection performance further by

selecting users whose SNRs are higher than the average SNR. The results also illustrate that the

multi-user cooperation brings significant performance gain for cognitive wireless networks.

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