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(1)

HAL Id: tel-01807651

https://tel.archives-ouvertes.fr/tel-01807651

Submitted on 5 Jun 2018

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Spectrum sensing for half and full-duplex interweave cognitive radio systems

Abbass Nasser

To cite this version:

Abbass Nasser. Spectrum sensing for half and full-duplex interweave cognitive radio systems. Physics

[physics]. Université de Bretagne occidentale - Brest, 2017. English. �NNT : 2017BRES0006�. �tel-

01807651�

(2)

–

THÈSE / UNIVERSITÉ DE BRETAGNE OCCIDENTALE

sous le sceau de l’Université Bretagne Loire pour obtenir le titre de DOCTEUR DE L’UNIVERSITÉ DE BRETAGNE OCCIDENTALE Mention : Sciences et Technologies de l’Information et de la

Communication École Doctorale SICMA

présentée par

Abbass NASSER

Préparée au Lab-STICC UMR CNRS 6285 à l'UBO & Ensta-Bretagne

Spectrum Sensing for Half and Full-Duplex Interweave Cognitive Radio Systems

Thèse soutenue le 17 janvier 2017 devant le jury composé de :

M. Christian Jutten

Professeur, Université Joseph Fourier, Examinateur

M. Gilles Burel

Professeur, Université de Bretagne Occidentale, Examinateur

M. Karim Abed Meraim

Professor, Université d'Orléans, Rapporteur

M. Yannick Deville

Professeur, Université Paul Sabatier Toulouse 3, Rapporteur

M. Koffi-Clément Yao,

Maitre de Conférence, Université de Bretagne Occidentale, Encadrant

M. Ali Mansour,

Professeur, Ensta-Bretagne, Directeur de thèse M. H. Charara,

Maitre de Conférence, Université Libanaise, Membre invité

(3)

(4)

&

(5)

etc

(6)

(7)

(8)

(9)

(10)

T_p T_p T_p T_or

Tav

(11)

T_p H₀ Tp H1

(12)

f(T S|H₀) f(T S|H1)

(p_{f a};p_d) = (0.1; 09) (p_{f a};p_d) = (0.1; 09) ρ

p_{f a} = 0.1 p_md= 1−p_d

p_{f a}= 0.1

p_md= 1−p_d p_{f a} = 0.05

N = 10000 N_s

N_s = 3

(13)

N_s −12dB N = 1500

N = 1000

p_{f a}= 0.1 N_s

N_s= 4 8

=−5 (p_{f a};p_d) = (0.1; 0.9)

p_d p_{f a} = 0.1

H₀ H₁

p_d = 0.9 p_{f a} = 0.1

γ_d γs δ

(p^H_{f a} ; p^H_d) = (0.1 ; 0.9)

i.e.

T_s T_t

p_d N_t

(14)

p^p_d

p_{f a} N = 1500 N_s= 4 p^p_d

p_{f a} −10 N_s= 4

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

⇠

etc.

%

etc

(23)

(24)

(25)

Spectrum Spectrum Spectrum PU

SU

(a): Undelay Access (b): Overlay Access (c): Interweave Access

(26)

(27)

(28)

(29)

Radio Environment

Spectrum Sensing

Spectrum Decision Spectrum Mobility

Spectrum Sharing

etc

(30)

(31)

etc

(32)

(33)

(34)

S

S&T

T S T S T

S&T S&T

SU is idle SU is idle SU is idle

(a): HD-CR

(b): FD-CR

SU is active SU is active SU is active

If PU is absent If PU is absent

H

0

H

1

y(n)

(35)

8 <

:

H

0

: y(n) = w(n)

H

₁

: y(n) = hs(n) + w(n)

h s(n) w(n) = w

_p

(n) +

iw

q

(n) i.i.d

i.e. E[w(n)] = 0 = E[w

²

(n)] E[.]

w

_p

(n) w

_q

(n) w(n)

E[w

²_p

(n)] = E[w

²_q

(n)] = σ

_w²

2 σ

_w²

= E[ | w(n) |

²

] s(n)

γ γ = | h |

²

σ

_w²

8 <

:

H

0

: y(n) = x

g

(n) + w(n)

H

₁

: y(n) = hs(n) + x

_g

(n) + w(n)

x

_g

(n) x(n) R

_X

x(n) T

X

x

g

(n)

T

_X

R

_X

R

_X

ˆ

x

_g

(n) x

_g

(n) y(n) y(n) ˆ

ˆ y(n)

ˆ

y(n) = y(n) − x ˆ

_g

(n)

i.e. x ˆ

_g

(n) = x

_g

(n) y(n) = ˆ y(n)

(36)

OX ADC Evaluation of TS

LNA Comparison with a pre-

determined threshold

Decision

� � � �

y(t)

y(n)

T

_X

R

_X

(37)

PU CR

� _�

PU signal

Self-Interference

PU

transmitting antenna

�

� � >>d

etc

(38)

λ 8

<

:

H

₀

: T S < λ : P U is idle

H

₁

: T S ≥ λ : P U is transmitting

p

_re

p

_re

= P r

✓

T S < λ | H

₀

◆ p

_{f a}

p

_{f a}

= P r

✓

T S ≥ λ | H

₀

◆ p

_d

p

_d

= P r

✓

T S ≥ λ | H

₁

◆

(39)

p

_md

p

_md

= P r

✓

T S < λ | H

₁

◆ p

_re

+ p

_{f a}

= 1 p

_d

+ p

_md

= 1

H

0

H

1

H

₀

p

_{f a}

= 0.1 p

_d

= 0.9

p

_{f a}

� �� )

� �� ) Reject

Missed Detection

False Alarm

Detection

�

f(T S|H⁰) f(T S|H1)

(40)

H

₀

H

₁

p

_d

p

_{f a}

T

_ED

N λ

T

_ED

= 1 N

X

N n=1

| y(n) |

²

λ p

_{f a}

H

0

p

_{f a}

T

_ED

H

₀

χ

²

2N T

_ED

T

ED H0

⇠ N (µ

^ED₀

, V

₀^ED

)

Hi

⇠ H

i

, i 2 { 0, 1 } N (µ, V )

µ V

(41)

H

₁

T

_ED

s(n) w(n) s(n) T

_ED

H

1

T

_ED ^H

⇠ N

¹

(µ

^ED₁

, V

₁^ED

)

N

p

^ED_{f a}

= Q λ − σ

_w²

p1 N

σ

_w²

!

p

^ED_d

= Q λ − (σ

_w²

+ σ

_s²

)

p1

N

(σ

_w²

+ σ

²_w

)

!

Q(x)

Q(x) = 1 p 2π

Z

+1 x

e

⁻^u²

du λ

λ = 1

p N Q

⁻¹

(p

_{f a}

)σ

_w²

+ σ

_w²

λ

(42)

s(n) = X

k

b

_k

g(n − kN

_s

)

b

_k

g(n) N

_s

≥ 1

r

ss

(m) s(n) m

r

ss

(m) = E[s(n)s

^⇤

(n − m)] 6 = 0

etc r

ss

(m)

r

_ss

(m) = 8

> >

<

> >

:

σ

_s²

1 − | m | N

_s

!

; | m |  N

s

0; | m | > N

_s

σ

_s²

= E[ | s(n) |

²

]

r

yy

(m) y(n)

r

_yy

(m) = E[y(n)y

^⇤

(n − m)]

= E

"

(hs(n) + w(n))(hs(n − m) + w(n − m))

^⇤

#

= E

"

| h |

²

s(n)s

^⇤

(n − m)

# + E

"

w(n)w

^⇤

(n − m)

#

+ E

"

h

^⇤

w(n)s

^⇤

(n − m)

# + E

"

hw

^⇤

(n − m)s(n)

#

w(n)

E[w(n)w

^⇤

(n − m)] = σ

_w²

δ(m)

δ(m) = 8

<

:

1 if m = 0

0 elsewhere

(43)

s(n) w(n) E[h

^⇤

w(n)s

^⇤

(n − m)] + E[hw

^⇤

(n − m)s(n)] = 0

r

_yy

(m) = E

"

| h |

²

s(n)s

^⇤

(n − m)

#

+ E [w(n)w

^⇤

(n − m)]

= | h |

²

r

_ss

(m) + σ

²_w

δ(m)

r

yy

(m) m 6 = 0

1  m  N

s

1  m  N

s

r

_yy

(m) = 8

<

:

0 under H

₀

6 = 0 under H

₁

{ r

_yy

(m) } m 2 [1; N

_s

− 1] T

_ACD

T

_ACD

= 1 ˆ r

_yy

(0)

Ns−1

X

m=1

c

m

Re { r ˆ

yy

(m) }

Re { . } ˆ r

_yy

(m) =

_N¹

P

N

n=1

y(n)y

^⇤

(n − m)

r

_yy

(m) c

_m

r ˆ

_yy

(0)

ˆ r

yy

(0)

H

₀

p

^ACD_{f a}

= Q λ s N

λ

²

+ ϕ

!

p

^ACD_d

= Q λ −

_γ+1^βγ

p V

_ACD

!

(44)

ϕ = 1 2

Ns−1

X

l=1

c

²_l

β =

Ns−1

X

l=1

c

_l

r

ss

(l)

r

_ss

(0) V

_ACD

= (1 + γ)λ

²

− 4βγλ + Σ + ηγ N (γ + 1)

²

η =

Ns−1

X

l=1 Ns−1

X

k=1;l6=k

c

_l

c

_k

r

_ss

(l − k) + r

_ss

(l + k)

r

_ss

(0) γ

etc

s(n) T

_α

r

_ss

(n, m) = E[s(n)s

^⇤

(n − m)] = r

_ss

(n + T

_α

, m)

r

ss

(n, m) n

r

ss

(n, m) = X

1 p=−1

R

ss

p T

_α

, m

!

exp j2π p T

_α

n

!

R

_ss

(

_T^p

α

, m) {

_T^p_α

}

R

_ss

(α, m)

R

_ss

(α, m) = lim

N−!+1

1 N

N

X

2

n=−^N₂

s(n)s

^⇤

(n − m)) exp ( − j2παn)

R

_ss

(α, m) 6 = 0 α 2 {

_T^p_α

} α α = 0 R(0, m)

m = 0 α = 0 R

_ss

(0, 0)

s(n)

(45)

R

_yy

(α, m)

R

_yy

(α, m) = lim

N−!1

N

X

2

n=−^N₂

y(n)y

^⇤

(n − m) exp ( − j2παn)

= lim

N−!1

N

X

2

n=−^N₂

hs(n) + w(n)

!

h

^⇤

s

^⇤

(n − m) + w

^⇤

(n − m)

!

exp ( − j2παn)

= lim

N−!1

Ni

X

2

n=−^N₂

| h |

²

s(n)s

^⇤

(n − m) + w(n)w

^⇤

(n − m) + s(n)w

^⇤

(n − m)

+ w(n)s

^⇤

(n − m)

!

exp ( − j2παn)

= R

_ss

(α, m) + R

_ww

(α, m) + R

_sw

(α, m) + R

_ws

(α, m)

w(n) R

_ww

(α, m) = 0 8 α 6 = 0

R

_ws

(α, m) = R

_sw

(α, m) = 0 s(n)

w(n)

R

_yy

(α, m) = R

_ss

(α, m)

R

_yy

(α, m) R ˆ

_yy

(α, m)

R ˆ

_yy

(α, m) ' 1 N

N 2−1

X

n=−^N₂

y(n)y

^⇤

(n − m) exp ( − j2παn)

T

CAF

R ˆ

_yy

(α, m)

T

_CAF

= | R ˆ

_yy

(α, m) |

²

T

_CAF

(46)

T

_CSD

= N r ˆ

_y

r ˆ

_y^T

T r ˆ

_y

ˆ r

_y

=

"

Re { R ˆ

_yy

(α, m

₁

) } , Re { R ˆ

_yy

(α, m

₂

) } ..., Re { R ˆ

_yy

(α, m

_M

) } , Im { R ˆ

_yy

(α, m

₁

) } , Im { R ˆ

_yy

(α, m

₂

) } ..., Im { R ˆ

_yy

(α, m

_M

) }

#

Re { . } Im { . } M

= 1 2

"

Re {P + U} Im {U − P}

Im {P + U} Re {P − U}

#

−1

P = P

_pq

!

U = U

_pq

!

P

_p,q

=

L−1

X

2

l=¹⁻₂^L

f (l) ˆ R

_yy

α + 2πl N , m

_p

!

R ˆ

^⇤_yy

α + 2πl N , m

_q

!

U

_p,q

=

L−1

X

2

l=¹⁻₂^L

f (l) ˆ R

_yy

α + 2πl N , m

_p

!

R ˆ

_yy

α − 2πm N , m

_q

!

L f (l) P

L

l=1

f(l) = 1

T

CSD

H

0

H

1

χ

²

2M

H

₀

χ

²

H

₁

N r ˆ

_y

r ˆ

_y^T

8

>>

<

>>

: TCSD

H1

∼ χ²_2M TCSD

H1

∼ χ²_2M Nrˆx rˆx^T

!

(47)

p

_{f a}

p

_d

H

₀

H

₁

p

^CSD_{f a}

= Γ(λ/2, M ) p

^CSD_d

= Q

M

p λ, q

N r ˆ

x

r ˆ

^T_x

!

Γ(a, b) Q

_M

(a, b)

Γ(a, b) = Z

b

0

t

^a⁻¹

exp ( − t)dt Q

_M

(a, b) = 1

a

^M⁻¹

Z

₊₁

b

t

^M

exp

✓

− t

²

+ a

²

2 !

I

_M₋₁

(at)dt

I

_M

(t) M

etc

(48)

{ β

_i

}

i=1,...,m

; (m  N

_s

) m ⇥ m C

_yy

C

yy

= 2 6 6 6 6 6 4

E[y(n)y

^⇤

(n)] E[y(n)y

^⇤

(n − 1)] . . . E[y(n)y

^⇤

(n − m)]

E[y(n − 1)y

^⇤

(n)] E[y(n − 1)y

^⇤

(n − 1)] . . . E[y(n − 1)y

^⇤

(n)]

. . .

E[y(n − m)y

^⇤

(n)] . . . E[y(n − )y

^⇤

(n − 1)] . . . E[y(n − m)y

^⇤

(n − m)]

3 7 7 7 7 7 5

C

yy

H

0

C

_yy

β

₁

= β

₂

= ... = β

_m

= σ

_w²

H

1

s(n) w(n) C

yy

C

_ss

C

_ww

s(n) w(n)

β

₁

= β

₁^s

+ σ

_w²

β

₂

= β

₂^s

+ σ

_w²

. . .

β

_m

= β

_m^s

+ σ

²_w

β

₁^s

≥ β

₂^s

... ≥ β

^s_m

β

_max

β

_min

C

_yy

β

_max

/β

_min

T

_ED

/β

_min

β

min

(49)

C

_yy

w(n) || w(n) ||

²

χ

²

χ

²

χ

²

T

_GoF

T

_GoF

= − X

N n=1

✓ ln(F

₀

(y(n)))

N − n + 1/2 + ln(1 − F

₀

(y(n))) n − 1/2

◆ F

0

χ

²

T

_GoF

(50)

p

_d

p

_{f a}

p

_d

p

_{f a}

p

_d

p

_{f a}

p

_d

p

_{f a}

N

s

= 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

pfa

p d

SNR=−12 dB, N=1000 samples

ED CSD ACD

p

_d

p

_{f a}

= 0.1 N

_s

= 8 N = 1000

p

_d

p

_{f a}

N

_s

(51)

−18 −16 −14 −12 −10 −8 −6 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SNR (dB)

p d

pfa=0.1; N=1000 samples

ED CSD ACD

i.e. p

_d

, p

_{f a}

N

(p

_{f a}

= 0.1, p

_d

= 0.9)

N

_s

= 3 N

(52)

10² 10³ 10⁴ 10⁵

−25

−20

−15

−10

−5 0

SNR (dB)

Number of samples (pfa ; p

d) = (0.1 ; 0.9); N s=3 sps

ED CSD ACD

(pf a;pd) = (0.1; 09)

C

ED

N | y(n) |

²

N − 1

| y(n) |

²

2N − 1

C

_ED

= 2N − 1

C

CSD

= (N

s

− 1)N (L + 1) + 4(N

s

− 1)L

²

+ 8(N

s

− 1)

³

+ 6(N

s

− 1)

²

+ 2(N

s

− 1))

c

m

= 1 2N − 1

ˆ

r

_yy

(m)

(53)

N

_s

N

_s

− 1 m = 1, ..., N

_s

− 1

C

_ACD

= (N

_s

− 1)(2N − 1) + N

_s

− 2

= 2(N

_s

− 1)N − 1

(p

_{f a}

= 0.1; p

_d

= 0.9)

-18 -16 -14 -12 -10 -8 -6

SNR (dB) 10²

10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸

Number of required operations

(pfa ; p

d) = (0.1 ; 0.9); N s=3 sps

ED CSD ACD

(pf a;pd) = (0.1; 09)

etc

(54)

ˆ σ

²_w

ˆ σ

_w²

2  1

κ σ

²_w

; κσ

_w²

7 σ

_w²

κ κ ≥ 1

ˆ σ

_w²

f

_σ_ˆ2

w

( ˆ σ

²_w

)

fεˆ(ˆε) = 8

><

>: 1

2ρ, ε−ρ≤εˆ≤ε+ρ 0, elsewhere

ε = 10 log

₁₀

(σ

_w²

) ˆ ε = 10 log

₁₀

(ˆ σ

²_w

) ρ = 10 log

₁₀

(κ)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

pfa 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

pd

SNR=-10 dB, N=1200 samples

ED: ρ = - 0 dB CSD: ρ = - 0 dB ACD: ρ = - 0dB ED: ρ = - 0.75 dB CSD: ρ = - 0.75 dB ACD: ρ = - 0.75 dB ED: ρ = - 2 dB CSD: ρ = - 2 dB ACD: ρ = - 2dB

ρ

p

_{f a}

(p

f a

< 0.5; p

d

> 0.5) (κ − 1/κ)

N

_s

= 2 ρ = 0

N

s

(55)

2N − 1

(2(N

_s

− 1)N )

(N

_s

− 1)N (L + 1) + 4(N

_s

− 1)L

²

+ 8(N

_s

− 1)

³

+ 6(N

_s

− 1)

²

+ 2(N

_s

− 1))

!

etc

(56)

N = 1000

N

_a

= 5

(57)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p

p d

N=1000 samples, SNR=−10 dB, Rayleigh Channel

Local SCS OR AND

Majority: 3 out of 5

N

sig

N

a

N

sig

 N

a

(58)

(59)

(60)

N

_a

H

_η

: y

_i

(n) = ηh

_i

s(n) + w

_i

(n)

η 2 { 0; 1 } H

₀

H

₁

y

i

(n) 1 ⇥ N ith

N s(n) w

_i

(n)

ith

σ

_w²_i

h

_i

ith

(61)

(n) (n)

=

⁻¹ ⁻¹

= Cov[ (n), (n)] C ˆ

ˆ = 1

N X

N n=1

(n)

^H

(n)

H

(n) (n)

(n) = [z

₁

(n), z

₂

(n), ..., z

_N_a₁

(n)]

^T

(n) = [y

1

(n), y

2

(n), ..., y

_N_a₂

(n)]

^T

y

i

(n) z

i

(n) ith

N

_a₁

N

_a₂

N

_a

N

_a₁

N

_a₂

N

_a

(n) (n)

(n, m, α) = (n − m) exp (j2παn) α

s(n) m P

_N

n=1

s(n)s

^⇤

(n − m)e

⁻^j2παn

α s

^⇤

(n) s(n)

(62)

α

ˆ

ˆ = 1

N X

N n=1

(n)

^H

(n, m, α) 1

N X

N n=1

(n)

^H

(n − m) exp ( − j2παn)

ˆ ˆ

^H

ˆ

α ˆ N

ˆ = 1

N X

N n=1

(n − m) exp (j2παn)

^H

(n − m) exp ( − j2παn)

= 1 N

X

N n=1

(n − m)

^H

(n − m)

ˆ

ˆ = ˆ

⁻¹

ˆ ˆ

⁻¹

ˆ

^H

N

_a

= 1 ˆ =

_N¹

P

N

n=1

y

₁

(n)y

₁^⇤

(n − m) exp ( − j2παn) = ˆ R

_y₁_y₁

(α, m) y

₁

(n)

ˆ =

_N¹

P

_N

n=1

| y

₁

(n − m) |

²

' R ˆ

_y₁_y₁

(0, 0) N R ˆ

_y₁_y₁

(0, 0) =

1 N

P

N

n=1

| y

₁

(n) |

²

y

₁

(n) z

₁

(n) = y

₁

(n −

m) exp (j2παn)

ˆ = ˆ R

_y⁻₁¹_y₁

(0, 0) ˆ R

y1y1

(α, m) ˆ R

⁻_y₁¹_y₁

(0, 0) ˆ R

^⇤_y₁_y₁

(α, m))

= 1

| R ˆ

_y₁_y₁

(0, 0) |

²

| R ˆ

_y₁_y₁

(α, m) |

²

N

_a

> 1 ˆ

ˆ =

2 6 6 6 6 6 4

R ˆ

_y₁_y₁

(α, m) R ˆ

_y₁_y₂

(α, m) . . . R ˆ

_y₁_y_Na

(α, m) R ˆ

_y₂_y₁

(α, m) R ˆ

_y₂_y₂

(α, m) . . . R ˆ

_y₂_y_Na

(α, m)

. . .

R ˆ

_y_Na_y₁

(α, m) . . . R ˆ

_y_Na_y₂

(α, m) . . . R ˆ

_y_Na_y_Na

(α, m)

3 7 7

7 7

7 5

(63)

ˆ

T

CCST

= − N log

Na

Y

i=1

(1 − β

i

)

!

{ β

_i

} , 1  i  N

_a

ˆ 1 ≥ β

₁

≥ β

₂

≥ ...β

_N_a

≥ 0 i.e.β

i

 1

ˆ

⁻¹

ˆ

⁻¹

ˆ

H

0

N

a

ˆ ' β

₁

= β

₂

, ..., β

_N_a

= 0 T

_CCST

H

₁

ˆ s(n)

T

_CCST

T

_CCST

ˆ T

CCST

(n)

H

₀

H

₁

H

₀

H

₁

8 <

:

H

0

: y

1

(n) = w

1

(n)

H

₁

: y

₁

(n) = h

₁

s(n) + w

₁

(n)

(64)

V

V = [m

₁

, m

₂

, ..., m

_P

] P

P

N

n=1

s(n − m

_p

)s

^⇤

(n − m

_k

)e

⁻^j2παn

6 = 0 8 m

_p

, m

_k

2 V

1

(n, [m

1

; m

_k

])

1

(n, [m

1

; m

k1

]) = h

y

1

(n − m

1

), y

1

(n − m

2

), ... y

1

(n − m

_k₁

) i

T

1 < k

1

 P

1

(n, [m

1

; m

_k₁

])

1

(n, [m

₁

; m

_k₂

], α)

1

(n, [m

₁

; m

_k₂

], α) =

₁

(n, [m

₁

; m

_k₂

]) exp (j2παn), 8 k

₁

, k

₂

2 [1; P ]

min (k

₁

; k

₂

) > l l

k

₁

≥ k

₂

> 1

1

(n, [m

₁

; m

_k₁

])

₁

(n, [m

₁

; m

_k₂

], α) y

1

(n)

y(n)

R ˆ

SAS

ˆ

_SAS

= ˆ

⁻_{1 1}¹

ˆ

1 1

ˆ

⁻¹

1 1

ˆ

1 1

ˆ

1 1

ˆ

1 1

ˆ

1 1

ˆ

1 1

ˆ

1 1

ˆ

1 1

H

₀

ˆ

⁰

1 1

ˆ

⁰

1 1

= 2 6 6 6 6 6 4

R ˆ

ww

(α, [m

1

; m

1

]) R ˆ

ww

(α, [m

1

; m

2

]) . . . R ˆ

ww

(α, [m

1

; m

_k₂

]) R ˆ

_ww

(α, [m

₂

; m

₁

]) R ˆ

_ww

(α, [m

₂

; m

₂

]) . . . R ˆ

_ww

(α, [m

₂

; m

_k2

])

. . .

R ˆ

_ww

(α, [m

_k₁

, m

₁

]) . . . R ˆ

_ww

(α, [m

_k₁

; m

₂

]) . . . R ˆ

_ww

(α, [m

_k₁

; m

_k₂

])

3 7 7

7 7

7 5

(65)

R ˆ

_ww

(α, [m

_i

; m

_j

])

R ˆ

_ww

(α, [m

_i

; m

_j

]) = 1 N

X

N n=1

w

₁

(n − m

_i

)w

₁^⇤

(n − m

_j

)e

⁻^j2παn

w

₁

(n) α 6 = 0 ˆ

⁰

1 1

'

1

(n, [m

₁

; m

_k₂

])

₁

(n, [m

₁

; m

_k₂

], α)

H

1

ˆ

¹

1 1

ˆ

¹

1 1

(α) = ˆ R

_ss

(α) + ˆ R

_sw

(α) + ˆ R

_ws

(α) + ˆ R

⁰_rq

(α) R ˆ

_ws

(α) R ˆ

_sw

(α)

R ˆ

_ss

(α)

R ˆ

ss

(α) = | h |

²

2 6 6 6 6 6 4

R ˆ

_ss

(α, [m

₁

; m

₁

]) R ˆ

_ss

(α, [, [m

₁

; m

₂

]) . . . R ˆ

_ss

(α, [m

₁

; m

_k₂

]) R ˆ

ss

(α, [m

2

; m

1

]) R ˆ

ss

(α, [m

2

; m

2

]) . . . R ˆ

ss

(α, [m

2

; m

_k2

])

. . .

R ˆ

_ss

(α, [m

_k₁

, m

₁

]) . . . R ˆ

_ss

(α, [m

_k₁

; m

₂

]) . . . R ˆ

_ss

(α, [m

_k₁

; m

_k₂

]) 3 7 7 7 7 7 5

R ˆ

_ss

(α, [m

_i

; m

_j

]) s(n) m

_i

m

_j

R

_ss

(α)

s(n) α

α

T

_SAS

T

_SAS

= − N log

k1

Y

i=1

(1 − β

_i^s

) { β

_i^s

} , i = 1, 2, ..., N

_a

1 ≥ β

^s₁

≥ β

₂^s

≥ ... ≥ β

_N^s_a

l l

(66)

l = 1 T

_SAS

T

SAS

= − N log(1 − β

₁^s

)

T

_SAS

λ

T

_SAS

H1

R

H0

λ

T

_SAS

R ˆ

_SAS

R ˆ

_SAS

T

_SAS

T

_SAS

λ

(n, m) (n, p, α)

(n, [m

₁

; m

_k₁

]) = [

₁

(n, [m

₁

; m

_k₁

]),

₂

(n, [m

₁

; m

_k₁

]), ...,

_N_a

(n, [m

₁

; m

_k₁

])]

^T

(n, [m

₁

; m

_k₂

], α) = [

₁

(n, [m

₁

; m

_k₂

], α),

₂

(n, [m

₁

; m

_k₂

], α), ...,

_N_a

(n, [m

₁

; m

_k₂

], α)]

^T

i

(n, [m

₁

; m

_k₂

]), 1  i  N

_a

ith

_i

(n, [m

₁

; m

_k₂

], α) =

i

(n, [m

₁

; m

_k₂

])e

^j2παn

1  i  N

_a

α (n, [m

₁

; m

_k₁

]) (n, [m

₁

; m

_k₂

], α)

ˆ

_{M AS}

= ˆ

⁻¹

ˆ ˆ

⁻¹

ˆ

(67)

T

M AS

= − N log(1 − β

₁^m

)

β

₁^m

ˆ

_{M AS}

(n, [0; 0]) (n, [m

1

; m

1

], α)

1µs F

s

i.e. F

s

= 8B B

ith

V

_sim

= [0, T

_s

, 2T

_s

, 3T

_s

, 4T

_s

, 5T

_s

, 6T

_s

, 7T

_s

] T

_s

=

_F¹

s

1

(n, [m

₁

; m

_k₁

])

k

₁

= 8 k

₂ ₁

(n, [m

₁

; m

_k₂

], α)

i.e. V

sim

p

_d

p

_{f a}

k

₂

(68)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pf

p d

SNR=−10 dB, N=2000 samples

CCST−S:p₂=1 CCST−S: p₂=5 CCST−S: p₂=8 GLRT

-8-8-8

pfa

p

_{f a}

V

sim 1

(n, [m

1

; m

_k₁

])

1

(n, [m

₁

; m

_k₂

], α) i.e. k

₁

= k

₂

= p p

_{f a}

= 0.1

p

_d

(n, [m

₁

; m

_k₁

]) (n, [m

₁

; m

_k₂

], α)

V

_sim

(69)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 pfa

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p d

SNR=-8 dB, N=1500 samples

CCST-S, NU=0 dB ED, NU=0 dB CCST-S, NU=0.5 dB ED, NU=0.5 dB CCST-S, NU=1.5 dB ED, NU=1.5 dB

−10 −8 −6 −4 −2 0 2 4

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SNR (dB)

p d

pfa=0.1, N=1000 samples

p=2 p=4 p=6 p=8

pf a= 0.1

(70)

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 10⁻³

10⁻² 10⁻¹ 10⁰

Number of SU receiving antennas: M

p md

pfa=0.1; N=2000 samples, SNR=−10 dB

CCST−M CCST−D

pmd= 1−pd

pf a = 0.1

p

_md

N

_a

N = 1000 − 10 p

_{f a}

= 0.1

N

a

p

_md

N

_a

N

a

= 5 p

md

' 0.2 p

md

' 0.06 N

a

= 7

p

_md

0.1 p

_md

= 0.004

E[w

i

(n)w

^⇤_j

(n)] = 8

<

:

σ

_w²

i = j σ

_w²

γ

^|ⁱ⁻^j^|

i 6 = j

; 1  i, j  M ;

γ 0  γ  1

(71)

−16 −14 −12 −10 −8 −6 −4 10⁻³

10⁻² 10⁻¹ 10⁰

pfa=0.05; M=5; N=1000 samples

SNR (dB)

p md

CCST−M CCST−D

pmd= 1−pd

pf a = 0.05

N

_a

= 5 N = 1000

p

_md

= 0.5 = − 14dB = − 12dB

N

a

− 12 N

_a

= 6 N = 2000

p

_md

= 0.1 p

_{f a}

= 0.03

p

_md

p

_{f a}

= 0.5

(72)

10⁻³ 10⁻² 10⁻¹ 10⁰ 10⁻⁴

10⁻³ 10⁻² 10⁻¹ 10⁰

p md

pfa

SNR=−12 dB, M=6; N=2000 samples

CCST−D CCST−M

pmd= 1−pd

pf a = 0.05

(73)

(74)

i.e. N

s

≥ 2

N

s

= 2

N

_s

= 2

(75)

s(n) B

s(n) s

_p

(n) s

_q

(n)

s(n) s(n) = X

k

b

_k

g(n − k + N

s

) = s

p

(n) + js

q

(n)

b

_k

g(n) N

_s

N

_s

=

^F_B^s

≥ 2 F

_s

s

_p

(n) s

_q

(n) s(n)

H

₀

H

₁

8 <

:

H

₀

: y(n) = w(n)

H

₁

: y(n) = hs(n) + w(n)

h w(n)

σ

_w²

N (0, σ

_w²

) w(n) = w

_p

(n) + jw

_q

(n) i.i.d i.e. E[w

²

(n)] = 0 w

_p

(n) w

_q

(n) w(n)

E[w

²_p

(n)] = E[w

²_q

(n)] = σ

_w²

2 σ

_w²

= E[ | w(n) |

²

] E[.]

s(n) γ

γ = | h |

²

σ

_w²

(76)

P

_x

(k) x(n) r

_xx

(m)

r

_xx

(m) = E[x(n)x

^⇤

(n − m)]

P

_x

(k) = lim

N!+1

N

X

2

m=^N₂−1

r

_xx

(m) exp ( − j2πk m N )

w(n) w(n)

r

_ww

(m) = E[w(n)w

^⇤

(n − m)] = σ

_w²

δ(m)

P

_w

(k) = σ

²_w

σ

_w²

s(n) P

_s

(k)

(y(n))

x(n)

P ˆ

x

(k) = 1

N | X(k) |

²

X(k) x(n) N

X(k) =

N

X

2

n=^N₂−1

x(n) exp ⇣

− j2πk n N

⌘

(77)

Lemma 1 X(k) x(n) = x

_p

(n) + jx

_q

(n)

X(k) E[X

²

(k)] = 0

X(k) x(n)

E[X

²

(k)] = E 2 6 4

X

N

m,n=−^N₂−1

x(n) exp ⇣

− j2πk n N

⌘

x(m) exp ⇣

− j2πk m N

⌘ 3 7 5

=

X

N m=n=−^N₂−1

E[x

²

(n)] exp

✓

− j2πk 2n N

◆ | {z }

=0;using the circularity property of x(n)

+ X

N m6=n

E [x(n)x(m)] exp

✓

− j2πk n + m N

◆ | {z }

= 0as x(n)is i.i.d. and zero mean

= 0

Lemma 1 W (k) w(n)

CP

y

(k) y(n)

CP

_y

(k) = X

k u=v

P ˆ

_y

(u), , 8 k 2 [v; l],

P ˆ

_y

(u) y(n)

Ψ(k) y(n)

Ψ(k) = CP

_y

(k)

(l − v + 1)σ

_w²