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### MS. THESIS

### Secrecy Performance of Full-Duplex Relay System With Randomly Located

### Eavesdroppers

## 무작위로 위치한 다수 도청자가 존재하는 전이중 중계 시스템의 보안 성능

### BY

### Donghyun Jung

### AUGUST 2017

### DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

### COLLEGE OF ENGINEERING

**Abstract**

### Secrecy Performance of Full-Duplex Relay System With Randomly Located

### Eavesdroppers

### Donghyun Jung Department of Electrical and Computer Engineering The Graduate School Seoul National University

Full-duplex (FD) operation improves the spectral eﬃciency of a relay system where the relay simultaneously transmits and receives signals on the same channel. The performance of FD relay systems is limited by self-interference, i.e., signal leakage from the relay’s transmit antenna to its receive antenna.

However, in the FD relay system with malicious eavesdroppers, simultaneous transmission from the source and relay confuses the eavesdroppers.

In this thesis, we investigate a FD relay system where a source commu- nicates with a destination via a decode-and-forward relay in the presence of randomly located eavesdroppers. We derive analytical expressions for the secrecy outage probability and the average secrecy rate of the system. Sim- ulation results show that the secrecy performance of the system is improved as the density of eavesdroppers decreases. It is also shown that the FD re- lay system has higher secrecy performance than the half-duplex relay system when a large portion of self-interference is cancelled.

**Keywords:** Physical layer security, stochastic geometry, full-duplex relay
system, randomly located eavesdroppers, secrecy outage probability, average
secrecy rate.

**Student Number:** 2015-20987.

**Contents**

**Abstract** **i**

**Contents** **iii**

**List of Figures** **iv**

**Chapter 1** **Introduction** **1**

**Chapter 2** **System Model** **5**

**Chapter 3** **Secrecy Performance Analysis** **11**

3.1 SINR Distribution . . . 12 3.2 Secrecy Outage Probability . . . 17 3.3 Average Secrecy Rate . . . 20

**Chapter 4** **Simulation Results** **22**

**Chapter 5** **Conclusion** **38**

**List of Figures**

Figure 4.1 Secrecy outage probability versus density of eavesdrop-
pers*λ*_{e}with various values of self-interference cancellation fac-
tor *ρ*. . . 27
Figure 4.2 Secrecy outage probability versus self-interference can-

cellation factor *ρ* with various values of density of eavesdrop-
pers *λ**e*. . . 28
Figure 4.3 Secrecy outage probability versus transmit SNR *P/N*_{0}

with various values of density of eavesdroppers *λ*_{e}. . . 30
Figure 4.4 Secrecy outage probability versus transmit SNR *P/N*0

with various values of self-interference cancellation factor *ρ*. . . 31
Figure 4.5 Average secrecy rate versus density of eavesdroppers

*λ**e* with various values of self-interference cancellation factor *ρ*. 33

Figure 4.6 Average secrecy rate versus self-interference cancella-
tion factor *ρ* with various values of density of eavesdroppers
*λ*_{e}. . . 34
Figure 4.7 Average secrecy rate versus transmit SNR *P/N*_{0} with

various values of density of eavesdroppers *λ*_{e}. . . 36
Figure 4.8 Average secrecy rate versus transmit SNR *P/N*_{0} with

various values of self-interference cancellation factor *ρ*. . . 37

**Chapter 1**

**Introduction**

Full-duplex (FD) operation improves the spectral eﬃciency of a relay system where the relay simultaneously transmits and receives signals on the same channel [1]. However, the main limitation of the FD relay system is that simultaneous transmission and reception at the relay causes a high level of self-interference, i.e., signal leakage from the relay’s transmit antenna to its receive antenna. Several techniques to mitigate the eﬀect of the self- interference at FD nodes have been investigated, such as passive cancellation, analog cancellation, and digital cancellation [2]-[5].

Communication is secure when a legitimate receiver successfully decodes its received signal from a transmitter while an eavesdropper cannot overhear the signal. Physical layer security guarantees the secure communication by using the characteristics of wireless channels without encryption methods [6], [7]. The eﬀects of malicious eavesdroppers who attempt to overhear the communication between the legitimate transmitter and receiver have been considered.

Randomly located eavesdroppers have been recently investigated where eavesdroppers’ positions are modeled by using stochastic geometry [8]-[13].

In [8], secrecy performance of a downlink cellular system was analyzed where a multi-antenna base station communicates with multiple malicious users in the presence of external eavesdroppers. In [9], an artificial noise-aided multi- antenna transmission scheme was studied where optimal power allocation between an information signal and an artificial noise which minimizes the secrecy outage probability was obtained. In [10], secrecy performance of a downlink cellular system in the presence of randomly located eavesdroppers was analyzed where a multi-antenna base station employs transmit antenna

selection to enhance the secrecy performance. In [11], a secure transmission scheme in a half-duplex (HD) relay system was investigated where the source and relay transmit artificial noises along with information signals to confuse eavesdroppers.

In a FD relay system with an eavesdropper, simultaneous transmission from the source and relay confuses the eavesdropper [14]-[17]. In [14], a FD jamming relay system was proposed where the relay receives an information signal from the source and simultaneously transmits a jamming signal to the eavesdropper. In [15], a power allocation algorithm which maximizes the secrecy rate of the FD relay system was presented. In [16], the secrecy outage probability of a two-way FD relay system with optimal relay seletion was analyzed. In [17], an Almouti-based beamforming and artificial noise design was proposed to maximize the secrecy rate of the two-way FD relay system.

However, most previous works focused on eavesdropper(s) at deterministic position(s) in the FD relay system.

In this thesis, we consider a FD relay system with randomly located eaves- droppers. We derive analytical expressions for the secrecy outage probability

and the average secrecy rate of the system. Computer simulations verify the validity of our analysis. Also, the performance of a HD relay system with randomly located eavesdroppers is compared with the FD relay system.

The rest of this thesis is organized as follows. In chapter 2, the system model is described. In chapter 3, the secrecy outage probability and the aver- age secrecy rate are analyzed. In chapter 4, simulation results are provided.

Finally, conclusions are drawn in chapter 5.

**Chapter 2**

**System Model**

Consider a FD relay system where a source *s* communicates with a desti-
nation *d* via a decode-and-forward (DF) relay *r* in the presence of multiple
eavesdroppers. Assume that the source, the relay, and the destination are
located at fixed positions and there is no direct link between the source and
the destination. Assume that the eavesdroppers are located according to
a homogeneous Poisson point process (PPP) Φ_{e} with density *λ*_{e}. Suppose
that the FD relay has two antennas, one for transmission and one for recep-
tion while the source, the destination, and the eavesdroppers have a single

antenna.

Suppose that transmission takes place in blocks. In each block, the source transmits a signal to the relay and, at the same time, the relay transmits a signal to the destination. The relay receives not only the signal transmitted from the source but also the signal transmitted from itself which becomes interference. The relay decodes and re-encodes the received signal, and in the next block, transmits the re-encoded signal to the destination.

The eavesdroppers attempt to overhear the signals transmitted from the source and the relay. Suppose that the source and the relay use diﬀerent codebooks, so that the eavesdroppers can only individually decode the re- ceived signal from the first hop and that from the second hop [11]. Assume that the eavesdroppers are non-colluding, i.e., each eavesdropper does not cooperate with other eavesdroppers [9].

Assume that all channels remain constant over a block and vary inde-
pendently from one block to another. Assume that the coeﬃcient of the
channel from node *i* to node *j* in block *k*, *h*_{ij}[*k*], *i, j* *∈ {s, r, d, e}*, *e* *∈*Φ_{e}, is
an independent complex Gaussian random variable with zero mean and unit

variance. Assume that the channels have an additive white Gaussian noise
(AWGN) with zero mean and variance *N*_{0}.

In block *k*, the source transmits a signal *x*_{s}[*k*] with power *P*_{s} to the
relay and the relay transmits the re-encoded signal of *x*_{s}[*k* *−*1], denoted
by ˆ*x*_{s}[*k* *−*1], with power *P*_{r} to the destination. Let *ρ*, *ρ* *∈* [0*,*1], denote
the self-interference cancellation factor which represents a residual portion
of self-interference power at the relay after cancellation [5]. The received
signal at the relay is given by

*y*_{r}[*k*] =√

*d*^{−}_{sr}^{α}*h*_{sr}[*k*]*x*_{s}[*k*] +*√*

*ρh*_{rr}[*k*]ˆ*x*_{s}[*k−*1] +*n*_{r}[*k*] (2.1)

where *d**sr* is the distance from the source to the relay, *α* is the path-loss
exponent, and*n*_{r}[*k*] is an AWGN. The received signal at an eavesdropper (at
position)*e*, *e∈*Φ_{e}, is given by

*y*_{e}[*k*] =√

*d*^{−}_{se}^{α}*h*_{se}[*k*]*x*_{s}[*k*] +√

*d*^{−}_{re}^{α}*h*_{re}[*k*]ˆ*x*_{s}[*k−*1] +*n*_{e}[*k*] (2.2)

where*d**se*and *d**re* are the distance from the eavesdropper*e* to the source and

that to the relay, respectively, and *n*_{e}[*k*] is an AWGN.

In block *k*+ 1, the received signal at the destination is given by

*y*_{d}[*k*+ 1] =

√

*d*^{−}_{rd}^{α}*h*_{rd}[*k*+ 1]ˆ*x*_{s}[*k*] +*n*_{d}[*k*+ 1] (2.3)

where *d**rd* is the distance from the relay to the destination and *n**d*[*k*+ 1] is
an AWGN. The received signal at the eavesdropper *e* is given by

*y*_{e}[*k*+ 1] =√

*d*^{−}_{re}^{α}*h*_{re}[*k*+ 1]ˆ*x*_{s}[*k*]

+√

*d*^{−}_{se}^{α}*h*_{se}[*k*+ 1]*x*_{s}[*k*+ 1] +*n*_{e}[*k*+ 1]*.* (2.4)

The signal-to-interference-plus-noise ratios (SINR) at the relay is given by

*γ*_{sr} = *P**s**d*^{−}_{sr}^{α}*|h**sr*[*k*]*|*^{2}

*ρP*_{r}*|h*_{rr}[*k*]*|*^{2}+*N*_{0}*.* (2.5)
The signal-to-noise ratio (SNR) at the destination is given by

*γ*_{rd} = *P**r**d*^{−}_{rd}^{α}*|h**rd*[*k*+ 1]*|*^{2}

*N*_{0} *.* (2.6)

For DF relaying, the end-to-end SINR from the source to the destination is given by

*γ*_{d}= min*{γ*_{sr}*, γ*_{rd}*}.* (2.7)

The SINR at the eavesdropper*e*for the signal transmitted from the source
is given by

*γ*_{se}= *P*_{s}*d*^{−}_{se}^{α}*|h*_{se}[*k*]*|*^{2}

*P*_{r}*d*^{−}_{re}^{α}*|h*_{re}[*k*]*|*^{2}+*N*_{0}*.* (2.8)
The SINR at the eavesdropper*e* for the signal transmitted from the relay is
given by

*γ*_{re} = *P*_{r}*d*^{−}_{re}^{α}*|h*_{re}[*k*+ 1]*|*^{2}
*P**s**d*^{−}_{se}^{α}*|h**se*[*k*+ 1]*|*^{2}+*N*0

*.* (2.9)

Since the source and relay use diﬀerent codebooks, the SINR at the eaves-
dropper*e* is given by [11]

*γ**e*= max*{γ**se**, γ**re**}.* (2.10)

For non-colluding eavesdroppers, the secrecy rate of the system is determined by the most detrimental eavesdropper, i.e., an eavesdropper with the highest

SINR which is given by

*γ*_{e}*∗* = max

*e**∈*Φ*e*

*γ*_{e}*.* (2.11)

**Chapter 3**

**Secrecy Performance Analysis**

In this chapter, we first analyze the cumulative distribution functions (CDFs)
of *γ*_{d} and *γ*_{e}*∗*. Then, we derive analytical expressions for the secrecy outage
probability and the average secrecy rate of the FD relay system.

**3.1** **SINR Distribution**

The CDF of the end-to-end SINR from the source to the destination is given by

*F*_{γ}_{d}(*x*) = Pr[*γ*_{d}*< x*]

=1*−*Pr[min*{γ*_{sr}*, γ*_{rd}*} ≥x*]

=1*−*Pr[*γ*_{sr}*≥x, γ*_{rd} *≥x*]

=1*−*(1*−F*_{γ}_{sr}(*x*)) (1*−F*_{γ}_{rd}(*x*)) (3.1)

where *F*_{γ}_{sr}(*x*) and *F*_{γ}_{rd}(*x*) are the CDFs of *γ*_{sr} and *γ*_{rd}, respectively. The
CDF of the SINR at the relay is given by

*F**γ**sr*(*x*) = Pr[*γ**sr* *< x*]

= Pr

[ *P*_{s}*d*^{−}_{sr}^{α}*U*
*ρP*_{r}*V* +*N*_{0} *< x*

]

= Pr [

*U <* *ρP**r**V* +*N*0

*P*_{s}*d*^{−}_{sr}^{α} *x*
]

=

∫ _{∞}

0

∫ ^{ρPr v+N}^{0}

*Psd**−**α*
*sr*

*x*
0

*f*_{U,V}(*u, v*)*dudv* (3.2)

where*U* =*|h*_{sr}[*k*]*|*^{2},*V* =*|h*_{rr}[*k*]*|*^{2}, and*f*_{U,V}(*u, v*) is the joint probability den-
sity function (PDF) of*U* and*V*. Since *U* and*V* are independent exponential
random variables with unit mean, we have

*F**γ**sr*(*x*) =

∫ _{∞}

0

∫ ^{ρPr v+N}^{0}

*Psd**−α*
*sr*

*x*
0

*f**U*(*u*)*f**V*(*v*)*dudv*

=

∫ _{∞}

0

exp(*−v*)

∫ ^{ρPr v+N}^{0}

*Psd**−**α*
*sr*

*x*
0

exp(*−u*)*dudv*

=

∫ _{∞}

0

exp(*−v*)
(

1*−*exp
(

*−ρP*_{r}*v*+*N*_{0}
*P*_{s}*d*^{−}_{sr}^{α} *x*

))
*dv*

=1*−*exp
(

*−N*_{0}*d*^{α}_{sr}
*P*_{s} *x*

) ∫ _{∞}

0

exp (

*−*
(

1 + *ρP*_{r}*d*^{α}_{sr}
*P*_{s} *x*

)
*v*

)
*dv*

=1*−*exp
(

*−N*_{0}*d*^{α}_{sr}
*P**s*

*x*
) (

1 + *ρP*_{r}*d*^{α}_{sr}
*P**s*

*x*
)_{−}1

(3.3)

where*f**U*(*u*) and *f**V*(*v*) are the PDFs of *U* and *V*, respectively. The CDF of
the SNR at the destination is given by

*F*_{γ}_{rd}(*x*) = Pr[*γ*_{rd}*< x*]

= Pr [

*|h*_{rd}[*k*+ 1]*|*^{2} *<* *N*0*d*^{α}_{rd}
*P*_{r} *x*

]

=1*−*exp
(

*−N*_{0}*d*^{α}_{rd}
*P*_{r} *x*

)

(3.4)

where *|h*_{rd}[*k* + 1]*|*^{2} has an exponential distribution with unit mean. From
(3.1), (3.3), and (3.4), the CDF of the end-to-end SINR from the source to
the destination is obtained as

*F*_{γ}_{d}(*x*) = 1*−*exp
(

*−N*_{0}
(*d*^{α}_{sr}

*P**s*

+*d*^{α}_{rd}
*P**r*

)
*x*

)(

1 + *ρP*_{r}*d*^{α}_{sr}
*P**s*

*x*
)_{−}1

*.* (3.5)

The CDF of the SINR at the most detrimental eavesdropper is given by

*F**γ**e∗*(*x*) = Pr[*γ**e*^{∗} *< x*]

=EΦ*e*

[∏

*e**∈*Φ*e*

Pr[max*{γ*_{se}*, γ*_{re}*}< x*]

]

=EΦ*e*

[∏

*∈*

Pr[*γ**se**< x*] Pr[*γ**re* *< x*]

]

=EΦ*e*

[∏

*e**∈*Φ*e*

*F*_{γ}_{se}(*x*)*F*_{γ}_{re}(*x*)
]

(3.6)

where*F*_{γ}_{se}(*x*) and*F*_{γ}_{re}(*x*) are the CDFs of*γ*_{se}and*γ*_{re}, respectively. Similarly
to the derivation of (3.3), the CDF of the SINR at the eavesdropper *e* for
the signal transmitted from the source is given by

*F*_{γ}_{se}(*x*) = 1*−*exp
(

*−N*_{0}*d*^{α}_{se}
*P**s*

*x*
) (

1 + *P*_{r}
*P**s*

(*d*_{se}
*d**re*

)*α*

*x*
)_{−}1

(3.7)

and the CDF of the SINR at the eavesdropper *e* for the signal transmitted
from the relay is given by

*F*_{γ}_{re}(*x*) = 1*−*exp
(

*−N*_{0}*d*^{α}_{re}
*P*_{r} *x*

) (
1 + *P*_{s}

*P*_{r}
(*d*_{re}

*d*_{se}
)*α*

*x*
)_{−}1

*.* (3.8)

From (3.6), (3.7), and (3.8), the CDF of the SINR at the most detrimental eavesdropper is obtained as

*F*_{γ}_{e}(*x*) =*E*_{Φ}_{e}
[∏

*e**∈K**e*

(

1*−*exp
(

*−N*_{0}*d*^{α}_{se}
*P*_{s} *x*

) (
1 + *P*_{r}

*P*_{s}
(*d*_{se}

*d*_{re}
)*α*

*x*
)_{−}1)

*×*
(

1*−*exp
(

*−N*0*d*^{α}_{re}
*P*_{r} *x*

) (
1 + *P**s*

*P*_{r}
(*d**re*

*d*_{se}
)*α*

*x*

)_{−}1)]

(*a*)= exp
(

*−λ*_{e}

∫ _{∞}

0

∫ _{2π}

0

*d*_{se}
{

exp (

*−N*_{0}*d*^{α}_{se}
*P*_{s} *x*

) (
1 + *P*_{r}

*P*_{s}
(*d*_{se}

*d*_{re}
)*α*

*x*
)_{−}1

+ exp (

*−N*_{0}*d*^{α}_{re}
*P*_{r} *x*

) (
1 + *P*_{s}

*P*_{r}
(*d*_{re}

*d*_{se}
)*α*

*x*
)_{−}1

*−*exp
(

*−N*_{0}
(*d*^{α}_{se}

*P*_{s} + *d*^{α}_{re}
*P*_{r}

)
*x*

) (
1 + *P*_{r}

*P*_{s}
(*d*_{se}

*d*_{re}
)*α*

*x*
)_{−}1

*×*
(

1 + *P*_{s}
*P*_{r}

(*d*_{re}
*d*_{se}

)*α*

*x*
)_{−}1}

*dθdd*_{se}

(*b*)= exp
(

*−λ**e*

∫ _{∞}

0

∫ 2*π*
0

*d**se*

{ exp

(

*−N*_{0}*d*^{α}_{se}
*P*_{s} *x*

)

*×*
(

1 + *P*_{r}
*P*_{s}

(

*d*_{se}

√*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d*_{sr}*d*_{se}cos*θ*
)*α*

*x*
)_{−}1

+ exp (

*−N*_{0}√

*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d*_{sr}*d*_{se}cos*θ*^{α}

*P*_{r} *x*

)

*×*
(

1 + *P*_{s}
*P**r*

(√*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d**sr**d**se*cos*θ*
*d**se*

)*α*

*x*
)_{−}1

*−*exp
(

*−N*_{0}
{

*d*^{α}_{se}
*P*_{s} +

√*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d*_{sr}*d*_{se}cos*θ*^{α}
*P*_{r}

}
*x*

)

*×*
(

1 + *P*_{r}
*P*_{s}

(

*d*_{se}

√*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d*_{sr}*d*_{se}cos*θ*
)*α*

*x*
)_{−}1

*×*
(

1 + *P**s*

*P*_{r}

(√*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d*_{sr}*d*_{se}cos*θ*
*d*_{se}

)*α*

*x*
)_{−}1

*dθdd*_{se}

(3.9)

where (*a*) follows by using the probability generating functional of the PPP
Φ_{e} which is given by [18]

EΦ*e*

[∏

*x**∈*Φ*e*

*f*(*x*)
]

= exp (

*−λ*_{e}

∫

R^{2}

(1*−f*(*x*))*dx*
)

(3.10)

and by changing to the polar coordinates, and (*b*) follows from the law of
cosines, i.e.,*d*_{re} =√

*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d*_{sr}*d*_{se}cos*θ*.

**3.2** **Secrecy Outage Probability**

The secrecy rate of the system is given by

*R* = [log(1 +*γ*_{d})*−*log(1 +*γ*_{e}*∗*)]^{+} (3.11)

where [*x*]^{+ ∆}= max*{*0*, x}*. The secrecy outage probability of the system is
given by [12]

*P*_{out} = Pr[*R≤*0]

= Pr[log(1 +*γ*_{d})*−*log(1 +*γ*_{e}*∗*)*≤*0]

= Pr[*γ*_{d}*≤γ*_{e}*∗*]

=

∫ _{∞}

0

∫ _{∞}

*x*

*f*_{γ}_{d}(*x*)*f*_{γ}_{e∗}(*y*)*dydx*

=1*−*

∫ _{∞}

0

*f*_{γ}_{d}(*x*)*F*_{γ}_{e∗}(*x*)*dx* (3.12)

where*f**γ*_{d}(*x*) and*f**γ**e∗*(*y*) are the PDFs of*γ**d* and *γ**e*^{∗}, respectively. The PDF
of the end-to-end SINR from the source to the destination is given by

*f*_{γ}_{d}(*x*) =*dF*_{γ}_{d}(*x*)
*dx*

=*−* *d*
*dx*

{ exp

(

*−N*_{0}
(*d*^{α}_{sr}

*P*_{s} +*d*^{α}_{rd}
*P*_{r}

)
*x*

) (

1 + *ρP*_{r}*d*^{α}_{sr}
*P*_{s} *x*

)_{−}1}

= exp (

*−N*_{0}
(*d*^{α}_{sr}

*P*_{s} +*d*^{α}_{rd}
*P*_{r}

)
*x*

) (

1 + *ρP*_{r}*d*^{α}_{sr}
*P*_{s} *x*

)_{−}1

*×*
(

*N*0

(*d*^{α}_{sr}
*P*_{s} +*d*^{α}_{rd}

*P*_{r}
)

+
( *P*_{s}

*ρP*_{r}*d*^{α}_{sr} +*x*
)_{−}1)

*.* (3.13)

From (3.9), (3.12), and (3.13), the secrecy outage probability is obtained as

*P*_{out} =1*−*

∫ _{∞}

0

(

1 + *ρP*_{r}*d*^{α}_{sr}
*P*_{s} *x*

)_{−}1(
*N*_{0}

(*d*^{α}_{sr}
*P*_{s} +*d*^{α}_{rd}

*P*_{r}
)

+ (

*x*+ *P*_{s}
*ρP*_{r}*d*_{sr}

)_{−}1)

*×*exp
(

*−N*
(*d*^{α}_{sr}

*P*_{s} +*d*^{α}_{rd}
*P*_{r}

)

*x−λ*_{e}

∫ _{∞}

0

∫ 2*π*
0

*d*_{se}
{

exp (

*−N*0*d*^{α}_{se}
*P*_{s} *x*

)

*×*
(

1 + *P*_{r}
*P*_{s}

(

*d*_{se}

√*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d*_{sr}*d*_{se}cos*θ*
)*α*

*x*
)_{−1}

+ exp (

*−N*0

√*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d**sr**d**se*cos*θ*^{α}
*P**r*

*x*
)

*×*
(

1 + *P*_{s}
*P*_{r}

(√*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d*_{sr}*d*_{se}cos*θ*
*d*_{se}

)*α*

*x*
)_{−}1

*−*exp
(

*−N*_{0}
(

*d*^{α}_{se}
*P*_{s} +

√*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d*_{sr}*d*_{se}cos*θ*^{α}
*P*_{r}

)
*x*

)

*×*
(

1 + *P*_{r}
*P**s*

(

*d*_{se}

√*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d*_{sr}*d*_{se}cos*θ*
)*α*

*x*
)_{−}1

*×*
(

1 + *P*_{s}
*P*_{r}

(√*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d*_{sr}*d*_{se}cos*θ*
*d*_{se}

)*α*

*x*
)_{−}1

*dθdd**se*

*dx.*

(3.14)

**3.3** **Average Secrecy Rate**

The average secrecy rate of the system is given by [19]

*R*_{avg} =E[*R*]

= 1 ln 2

∫ _{∞}

0

*F**γ**e∗*(*x*)

1 +*x* (1*−F*_{γ}_{d}(*x*))*dx.* (3.15)

From (3.5), (3.9), and (3.15), the average secrecy rate is obtained as

*R*_{avg} = 1
ln 2

∫ _{∞}

0

(1 +*x*)^{−}^{1}
(

1 + *ρP*_{r}*d*_{sr}
*P**s*

*x*
)_{−}1

*×*exp
(

*−N*0

(*d*^{α}_{sr}
*P*_{s} +*d*^{α}_{rd}

*P*_{r}
)

*x−λ**e*

∫ _{∞}

0

∫ 2*π*
0

*d**se*

{ exp

(

*−N*0*d*^{α}_{se}
*P*_{s} *x*

)

*×*
(

1 + *P*_{r}
*P*_{s}

(

*d*_{se}

√*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d*_{sr}*d*_{se}cos*θ*
)*α*

*x*
)_{−}1

+ exp (

*−N*_{0}√

*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d*_{sr}*d*_{se}cos*θ*^{α}

*P*_{r} *x*

)

*×*
(

1 + *P*_{s}
*P**r*

(√*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d**sr**d**se*cos*θ*
*d**se*

)*α*

*x*
)_{−}1

*−*exp
(

*−N*_{0}
(

*d*^{α}_{se}
*P*_{s} +

√*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d*_{sr}*d*_{se}cos*θ*^{α}
*P*_{r}

)
*x*

)

*×*
(

1 + *P*_{r}
*P*

(

*d*_{se}

√*d*^{2} +*d*^{2} *−*2*d* *d* cos*θ*
)*α*

*x*
)_{−}1

*×*
(

1 + *P*_{s}
*P*_{r}

(√*d*^{2}_{sr}+*d*^{2}_{se}*−*2*d*_{sr}*d*_{se}cos*θ*
*d*_{se}

)*α*

*x*
)_{−}1

*dθdd**se*

*dx.*

(3.16)

Closed-form expressions of (3.14) and (3.16) are diﬃcult to obtain, but they can be computed by using Gaussian quadrature method or software such as Mathematica [11]-[13].

**Chapter 4**

**Simulation Results**

Consider a FD relay system which consists of a source, a DF relay, a des-
tination, and multiple eavesdroppers. Suppose that the path-loss exponent
*α* = 4, the distances from the source to the relay and from the relay to the
destination*d*_{sr}=*d*_{rd} = 10 m, and the transmit SNRs of the source and relay
*P*_{s}*/N*_{0} =*P*_{r}*/N*_{0} =*P/N*_{0}. In figures 4.1, 4.2, 4.5, and 4.6, the performance of
a HD relay system is also depicted for comparison [20].

Figure 4.1 shows the secrecy outage probability of the relay systems versus

the density of eavesdroppers*λ*_{e} with various values of self-interference cancel-
lation factor*ρ* for the transmit SNR *P/N*_{0} = 50*,*60 dB. It is shown that the
analytical results perfectly match the simulation results. It is shown that as
*λ*_{e} increases, the secrecy outage probability of the relay systems increases. It
is also shown that the FD relay system has lower secrecy outage probability
than the HD relay system for *ρ* = *−*80*,−*60*,−*50 dB when *P/N*_{0} = 50 dB
and for *ρ*=*−*80*,−*60 dB when *P/N*_{0} = 60 dB.

Figure 4.2 shows the secrecy outage probability of the relay systems versus
the self-interference cancellation factor *ρ* with various values of the density
of eavesdroppers*λ*_{e} for the transmit SNR*P/N*_{0} = 60 dB. It is shown that as
*ρ*increases, the secrecy outage probability of the FD relay system increases.

It is shown that the FD relay system has lower secrecy outage probability
than the HD relay system for small values of*ρ*.

Figure 4.3 shows the secrecy outage probability of the FD relay system
versus the transmit SNR *P/N*_{0} with various values of the density of eaves-
droppers *λ*_{e} for the self-interference cancellation factor *ρ*=*−*80*,−*60 dB. It
is shown that as *P/N*_{0} increases, the secrecy outage probability of the FD

relay system decreases until it reaches a minimum, and then increases. It is
shown that as *λ*_{e} decreases, the optimal value of*P/N*_{0} which minimizes the
secrecy outage probability of the FD relay system decreases.

Figure 4.4 shows the secrecy outage probability of the FD relay system
versus the transmit SNR *P/N*_{0} with various values of the self-interference
cancellation factor *ρ* for the density of eavesdroppers *λ*_{e} = 10^{−}^{4} m^{−}^{2}. It is
shown that as *ρ* decreases, the optimal value of *P/N*_{0} which minimizes the
secrecy outage probability of the FD relay system increases.

Figure 4.5 shows the average secrecy rate of the relay systems versus the
density of eavesdroppers*λ*_{e} with various values of self-interference cancella-
tion factor *ρ* for the transmit SNR *P/N*_{0} = 50*,*60 dB. It is shown that the
analytical results perfectly match the simulation results. It is shown that as
*λ*_{e} increases, the average secrecy rate of the relay systems decreases. It is
shown that the FD relay system has higher average secrecy rate than the HD
relay system for *ρ*=*−*80*,−*60*,−*50 dB.

Figure 4.6 shows the average secrecy rate of the relay systems versus
the self-interference cancellation factor *ρ* with various values of density of

eavesdroppers *λ*_{e} for the transmit SNR *P/N*_{0} = 60 dB. It is shown that as
*ρ* increases, the average secrecy rate of the FD relay system decreases. It is
shown that the FD relay system has higher average secrecy rate than the HD
relay system for small values of *ρ*.

Figure 4.7 shows the average secrecy rate of the FD relay system versus
the transmit SNR *P/N*_{0} with various values of density of eavesdroppers *λ*_{e}
for the self-interference cancellation factor*ρ*=*−*80*,−*60 dB. It is shown that
as*P/N*_{0} increases, the average secrecy rate of the FD relay system increases
until it reaches a maximum, and then decreases.

Figure 4.8 shows the average secrecy rate of the FD relay system versus
the transmit SNR *P/N*_{0} with various values of the self-interference cancel-
lation factor *ρ* for the density of eavesdroppers *λ*_{e} = 10^{−}^{4} m^{−}^{2}. It is shown
that as*ρ* decreases, the optimal value of *P/N*_{0} which minimizes the average
secrecy rate of the FD relay system increases.

10 -4

10 -3

10 -2 10

-1 10

0

Density of eavesdroppers, e

(m -2

)

= 40 dB, Anal.

= 40 dB, Sim.

= 50 dB, Anal.

= 50 dB, Sim.

= 60 dB, Anal.

= 60 dB, Sim.

= 80 dB, Anal.

= 80 dB, Sim.

HD relay system Secrecyoutageprobability,P out

(a)*P/N*_{0}= 50 dB

10 -4

10 -3

10 -2 10

-1 10

0

Density of eavesdroppers, e

(m -2

)

= 40 dB, Anal.

= 40 dB, Sim.

= 50 dB, Anal.

= 50 dB, Sim.

= 60 dB, Anal.

= 60 dB, Sim.

= 80 dB, Anal.

= 80 dB, Sim.

HD relay system Secrecyoutageprobability,P out

(b)*P/N*0= 60 dB

Figure 4.1. Secrecy outage probability versus density of eavesdroppers *λ**e*

with various values of self-interference cancellation factor*ρ*.

-100 -80 -60 -40 -20 0 10

-3 10

-2 10

-1 10

0

FD, e

=10 -3

m -2

FD, e

=10 -4

m -2

FD, e

=10 -5

m -2

HD, e

=10 -3

m -2

HD, e

=10 -4

m -2

HD, e

=10 -5

m -2

Self-interference canncellation factor, (dB) Secrecyoutageprobability,P out

Figure 4.2. Secrecy outage probability versus self-interference cancellation
factor *ρ* with various values of density of eavesdroppers *λ*_{e}.

40 60 80 100 10

-4 10

-3 10

-2 10

-1 10

0

e

=10 -3

m -2

e

=10 -4

m -2

e

=10 -5

m -2 Secrecyoutageprobability,P out

Transmit SNR, P/N 0

(dB)

(a)*ρ*=*−*80 dB

40 60 80 100 10

-3 10

-2 10

-1 10

0

e

=10 -3

m -2

e

=10 -4

m -2

e

=10 -5

m -2 Secrecyoutageprobability,P out

Transmit SNR, P/N 0

(dB)

(b)*ρ*=*−*60 dB

Figure 4.3. Secrecy outage probability versus transmit SNR *P/N*0 with var-
ious values of density of eavesdroppers*λ*_{e}.

40 60 80 100 10

-3 10

-2 10

-1 10

0

Transmit SNR, P/N 0

(dB)

= 40 dB

= 50 dB

= 60 dB

= 80 dB

SecrecyOutageProbability

Figure 4.4. Secrecy outage probability versus transmit SNR *P/N*_{0} with var-
ious values of self-interference cancellation factor*ρ*.

10 -4

10 -3

10 -2 0.0

0.5 1.0 1.5 2.0

= 40 dB, Anal.

= 40 dB, Sim.

= 50 dB, Anal.

= 50 dB, Sim.

= 60 dB, Anal.

= 60 dB, Sim.

= 80 dB, Anal.

= 80 dB, Sim.

HD relay system

Averagesecrecyrate,R avg

(bps/Hz)

Density of eavesdroppers, e

(m -2

)

(a)*P/N*0= 50 dB

10 -4

10 -3

10 -2 0

1 2 3 4

= 40 dB, Anal.

= 40 dB, Sim.

= 50 dB, Anal.

= 50 dB, Sim.

= 60 dB, Anal.

= 60 dB, Sim.

= 80 dB, Anal.

= 80 dB, Sim.

HD relay system

Averagesecrecyrate,R avg

(bps/Hz)

Density of eavesdroppers,

e (m

-2

)

(b)*P/N*0= 60 dB

Figure 4.5. Average secrecy rate versus density of eavesdroppers *λ**e* with
various values of self-interference cancellation factor*ρ*.

-100 -80 -60 -40 -20 0 0

1 2 3 4 5 6

FD, e

=10 -3

m -2

FD, e

=10 -4

m -2

FD, e

=10 -5

m -2

HD, e

=10 -3

m -2

HD, e

=10 -4

m -2

HD, e

=10 -5

m -2

Averagesecrecyrate,R avg

(bps/Hz)

Self-interference cancellation factor, (dB)

Figure 4.6. Average secrecy rate versus self-interference cancellation factor
*ρ*with various values of density of eavesdroppers *λ**e*.

40 60 80 100 0

2 4 6 8 10 12

Transmit SNR, P/N 0

(dB) e

=10 -3

m -2

e

=10 -4

m -2

e

=10 -5

m -2

Averagesecrecyrate,R avg

(bps/Hz)

(a)*ρ*=*−*80 dB

40 60 80 100 0

1 2 3 4 5 6 7 8

e

=10 -3

m -2

e

=10 -4

m -2

e

=10 -5

m -2

Transmit SNR, P/N 0

(dB) Averagesecrecyrate,R avg

(bps/Hz)

(b)*ρ*=*−*60 dB

Figure 4.7. Average secrecy rate versus transmit SNR *P/N*_{0} with various
values of density of eavesdroppers *λ*_{e}.

40 60 80 100 0

2 4 6 8 10

= 40 dB

= 50 dB

= 60 dB

= 80 dB

Transmit SNR, P/N 0

(dB) Averagesecrecyrate,R avg

(bps/Hz)

Figure 4.8. Average secrecy rate versus transmit SNR *P/N*0 with various
values of self-interference cancellation factor*ρ*.

**Chapter 5**

**Conclusion**

In this thesis, we consider a full-duplex (FD) relay system in the presence of randomly located eavesdroppers. We analyze the cumulative density func- tions of the end-to-end signal-to-interference-plus-noise ratio (SINR) from the source to the destination and the SINR at the most detrimental eaves- dropper. We derive analytical expressions for the secrecy outage probability and the average secrecy rate of the system. Simulation results perfectly match the analytical results of the secrecy outage probability and the aver- age secrecy rate. It is shown that the secrecy performance of the system is