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

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)

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