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

⎧
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)
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