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

2012 International Conference on Biological and Biomedical Sciences

Advances **in** Biomedical Eng**in**eer**in**g, Vol.9

A **Cooperative** **Spectrum** **Detection** **Technique** **in** **Non**-**Gaussian** Noise ∗

Xiaomei Zhu, Weip**in**g Zhu

Institute of Signal Process**in**g and Transmission, Nanj**in**g University of Posts and

Telecommunications, Nanj**in**g, P. R. of Ch**in**a

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 **in**dicate 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 becom**in**g the key technology for reus**in**g 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 **in**terferences

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 **in**terference [2] . Furthermore, the

performances of the spectrum detector optimized aga**in**st **Gaussian** noise may degrade drastically

when non-**Gaussian** noise or **in**terference 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 **in**vestigates 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 Nanj**in**g University of Posts and Telecommunications under grant number

NY207031.

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

283

Assum**in**g 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

2σ

2

where ε is a mixture parameter between 0 and 1. By chang**in**g 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] . Not**in**g that the mean of x(n)

is zero, the

variance of the overall PDF is

1 N n 0

Us**in**g ∑ − x

N =

1

2

E(

x

= ( 1− ε ) σ + εσ

2

1

( n)

**in**stead 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 test**in**g 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. Suppos**in**g the primary user signal is a weak

s**in**usoidal 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. Accord**in**g 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 determ**in**ed 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, us**in**g (1) **in**to (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 **in**formation 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

k**in**d. 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

2π

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 suffer**in**g different shadow**in**g **in**fluence will receive weak

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

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

**in**terference to the PU system **in**evitably. CU4 and CU3 could receive the transmitter signal with**in**

the radius scope of primary transmitter's emissive power.

1

Figure1. **Cooperative** spectrum sens**in**g model

Based on the above detection scenario, the reliability that some s**in**gle 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 f**in**al 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 **in**dividual 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)

F**in**ally, the cooperative detection probability based on the OR rule is

2

where λ

i

= SNR

iσ

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 correspond**in**g 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 us**in**g 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 **in**creas**in**g 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** **in**terference backgrounds, the mixed-**Gaussian** Rao detection has ga**in**ed 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 result**in**g ROC curves are shown **in** Fig.4. Obviously, under

the same probability of false alarm conditions, the probability of detection is **in**creased as the SNR

**in**creases.

Figure 4. **Non**-cooperative detection and traditional OR cooperation

Figure 5. Proposed cooperation based on Rao detection

Fig.4 also **in**dicates 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 us**in**g

288

the fusion method of traditional OR rule. However, us**in**g the improved fusion method, the FC

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

SNR is higher than the average SNR, and then **in**tegrates 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 **in**creased by 5% as compared to the probability of detection of CU4 alone when P

f

is 0.1.

5. Conclud**in**g 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

select**in**g users whose SNRs are higher than the average SNR. The results also illustrate that the

multi-user cooperation br**in**gs significant performance ga**in** for cognitive wireless networks.

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