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Performance Analysis of Mixed
Amplify-and-Forward and Decode-and-Forward
Protocol in Underlay Cognitive Networks
Tran Trung Duy, Hyung Yun Kong
Abstract
In this paper, we propose and evaluate outage performance of a mixed amplify-and-forward
(AF) and decode-and-forward (DF) relaying protocol in underlay cognitive radio. Different
from the conventional AF and DF protocols, in the proposed protocol, a secondary source
attempts to transmit its signal to a secondary destination with help of two secondary relays.
One secondary relay always operates in AF mode, while the remaining one always operates in
DF mode. Moreover, we also propose a relay selection method, which relies on the decoding
status at the DF relay. For performance evaluation and comparison, we derive the exact and
approximate closed-form expressions of the outage probability for the proposed protocol over
Rayleigh fading channel. Finally, we run Monte Carlo simulations to verify the derivations.
Results presented that the proposed protocol obtains a diversity order of three and the outage
performance of our scheme is between that of the conventional underlay DF protocol and that
of the conventional underlay AF protocol.
Keywords: Underlay cognitive network, cooperative communication, outage probability,
mixed AF and DF protocols, Rayleigh fading channel.
1. Introduction
Cooperative communication protocols [1] can be classified into two groups:
decode-and-forward (DF) and amplify-and-forward (AF). In the DF protocol [2-3], relays
attempt to decode the source signal and forward the decoded signal to the destination. In the
AF protocol [2], [4], relays amplify the signal received from the source and forward the
amplified signal to the destination. As a result, the AF protocol offers a simpler hardware
circuitry than the DF protocol because it avoids the hard decoding at relays. To combine the
merits of both AF and DF schemes, the hybrid decode-amplify-forward protocols (HDAF)
[5-8], in which the relay will amplify the received signal and forward the amplified signal to
the destination if it cannot decode the received signal correctly, have been proposed and
analyzed. Although the HDAF protocol performs better than the AF protocol and DF protocol,
its implementation which requires relays to operate in both DF mode and AF mode is a very
complex work. Recently, the authors in [9-10] proposed a framework to design the mixed AF
and DF relay system (MADF). Different from the HDAF, the MADF includes an AF
sub-system and a DF sub-system. In AF (DF) sub-system, relays only use AF (DF) technique
to forward the source signal to the destination.
To overcome the spectrum scarcity issue in wireless communication network, cognitive radio
(CR) [11-12] has been proposed to enhance the efficient spectrum utilization. In CR, licensed
users (or primary users) are allowed to access the licensed bands at any time, while un-licensed
users (or secondary users) can only access these bands if they are not occupied by licensed
users. Traditionally, the secondary users must detect the presence (or absence) of the primary
users [13-14]. Recently, underlay cognitive radio schemes, in which the secondary users can
use the licensed bands at the same time with primary users, have been proposed and
investigated. In these schemes, secondary users must adjust their transmit power so that their
operations are not harmful to the quality of service of primary users. In [15-20], the authors
studied the performance of DF cooperative cognitive relaying protocols with different relay
selection methods and combining techniques. In [21-25], underlay AF cognitive protocols
were proposed and analyzed. In [7], the authors considered the outage performance of the
HDAF protocol with partial relay selection in underlay cognitive networks.
To the best of our knowledge, there have been no publications related to the MADF scheme in
underlay cognitive radio network. Hence, in this paper, we propose and evaluate the
performance of such a protocol. In particular, we consider an underlay scheme in which two
secondary relays are ready to help the secondary source forward the signal to the secondary
destination. Of the two relays, one relay always operates in AF mode, while the remaining
relay always uses DF mode to relay the source signal. At the first time slot, the secondary
source broadcasts its signal to the secondary destination and the secondary relays. If the
destination can decode the signal correctly, it feeds back an ACK message to inform the
decoding status and, so the two relays do nothing. Otherwise, it sends a NACK message to
request a retransmission. Then, one of the two relays is chosen to forward the signal to the
destination. For the performance evaluation, we derive exact and approximate closed-form
expressions of the outage probability for the proposed protocol over Rayleigh fading channel.
Our derivations are verified through Monte Carlo simulations. The results show that the
proposed protocol obtains a diversity order of three and the outage performance of the
proposed protocol is between that of the corresponding DF protocol and that of the
corresponding AF protocol.
The rest of the paper is organized as follows. The system model and the proposed scheme are
described in Section 2. In Section 3, the performance evaluation of the protocol is described.
The simulation results are presented in Section 4. Finally, the paper is concluded in Section 5.
2. System Model
S
D
DF
R
AF
R
PU
Fig.1. Cooperative relaying scheme in underlay cognitive radio network.
In Fig. 1, we present system model of the mixed AF and DF protocol, named MADF, in
underlay cognitive radio network. In this figure, the secondary source (S) communicates with
the secondary destination (D) with help from two secondary relays. The DF relay,
DF
R
, relays
the source's signal by using the DF technique and the AF relay,
AF
R
, uses the AF technique to
forward the received signal to the destination. In underlay cognitive radio network, the
secondary transmitters such as the source and the relays must adapt their transmit power to
satisfy a maximum interference threshold, i.e., Q, at the primary user (PU). In addition, their
transmit power is also below a maximum threshold level, i.e., P. Let us denote
0
h
,
AF
1
h
,
DF
1
h
,
AF
2
h
,
DF
2
h
,
3
h
,
AF
4
h
and
DF
4
h
as the channel coefficients of the
SD→
,
AF
SR→
,
DF
SR→
,
AF
RD→
,
DF
RD→
,
S PU→
,
AF
R PU→
and
DF
R PU→
links, respectively. Assume that
all of the channels follow a Rayleigh fading distribution. Hence, the channel gains
2
00
||h
γ
=
,
AF AF 2
11
||h
γ
=
,
DF DF 2
11
||h
γ
=
,
AF AF 2
22
||h
γ
=
,
DF DF 2
22
||h
γ
=
,
2
33
||h
γ
=
,
AF AF 2
44
||h
γ
=
and
DF DF 2
44
||h
γ
=
are exponential random variables (RVs) with parameters
0
λ
,
1A
λ
,
1D
λ
,
2A
λ
,
2D
λ
,
3
λ
,
4A
λ
and
4D
λ
, respectively.
In this paper, we assume that all of the nodes are placed in a two-dimensional plane. The
co-ordinates of the source S, the relay
AF
R
, the relay
DF
R
, the destination D, and the primary
user are (0,0),
( )
AF
R,0x
,
( )
DF
R,0x
, (1,0) and
( )
PU PU
,xy
, respectively. Therefore, the distances
between the
SD
→
,
AF
SR
→
,
DF
SR→
,
AF
RD→
,
DF
RD→
,
S PU→
,
AF
R PU→
and
DF
R PU
→
links can be calculated respectively as
S-D
1
d=
,
AF AF
S-R R
dx=
,
DF DF
S-R R
dx=
,
AF AF
R -D R
1dx= −
,
DF DF
R -D R
1dx= −
,
( ) ( )
22
S-PU PU PU
d xy= +
,
( )
( )
AF AF
22
R -PU PU R PU
d xx y=−+
and
( )
( )
DF DF
22
R -PU PU R PU
d xx y=−+
. Similar to [2], to
take path-loss into account, we can model the parameters as follows:
0 S-D
d
β
λ
=
,
AF
1A S-R
d
β
λ
=
,
DF
1D S-R
d
β
λ
=
,
AF
2A R -D
d
β
λ
=
,
DF
2D R -D
d
β
λ
=
,
3 S-PU
d
β
λ
=
,
AF
4A R -PU
d
β
λ
=
and
DF
4D R -PU
d
β
λ
=
.
Now, we describe the operation of the proposed protocol. At the first time slot, the source S
broadcasts its signal
S
x
with transmit power
S
P
, where
( )
S3
min , /P PQ
γ
=
. The signals
received at the destination and the relays in this time slot can be given, respectively, as
S D S 0S 1
,y Phx n
→
= +
(1)
AF
AF
S R S1 S 2
,y Ph x n
→
= +
(2)
DF
DF
S R S1 S 3
,y Ph x n
→
= +
(3)
where
1
n
,
2
n
and
3
n
are zero-mean and
0
N
-variance Gaussian noises at the destination and
at the relays
AF
R
and
DF
R
, respectively.
From (1)-(3) , the corresponding instantaneous signal-to-noise ratios (SNR) can be given as
( )
2
0 S0 0 0 0 3
| | / min , / ,
th th
Ph N P Q
γ γγ
Ψ= =
(4)
( )
AF AF 2 AF AF
1 S1 0 1 1 3
| | / min , / ,
th th
Ph N P Q
γ γγ
Ψ= =
(5)
( )
DF DF 2 DF DF
1 S1 0 1 1 3
| | / min , / ,
th th
Ph N P Q
γ γγ
Ψ= =
(6)
where
0
/
th
P PN=
and
0
/
th
Q QN=
.
If the destination D can decode the source's signal successfully, it sends an ACK message to all
of the nodes to confirm the status. In this case, the transmission is successful, so the relays do
nothing. Otherwise, the destination D sends a NACK message to request a retransmission
from one of the two relays.
If the relay
AF
R
is chosen to retransmit the source's signal to the destination, it amplifies the
received signal
AF
SR
y
→
and forward the amplified signal to the destination at the second time
slot. In this case, the received signal at the destination is
AF
AF AF
RD SR 2 4
,y Gy h n
→→
= +
(7)
where
4
n
is the zero-mean and
0
N
-variance Gaussian noise at the destination, G is the
amplification factor given as
( )
AF
AF
R S1 0
/GP P N
γ
= +
, where
AF
R
P
is transmit power of the
relay
AF
R
given as
( )
AF
AF
R4
min , /P PQ
γ
=
.
From (7), the received instantaneous SNR at the destination can be formulated as
AF AF
AF 12
DAF AF
12
,
1
ΨΨ
Ψ=
Ψ+Ψ+
(8)
where
( )
AF AF AF AF
2 2 24
min , /
th th
PQ
γ γγ
Ψ=
.
Next, we consider the case in which the relay
DF
R
is selected to forward the signal
S
x
. In this
case, the signal received at the destination in the second time slot is expressed as
DF DF
DF DF
R D R 2S 5
,y P hx n
→= +
(9)
where
5
n
is the zero-mean and
0
N
-variance Gaussian noise at the destination in this time slot,
and the transmit power
( )
DF
DF
R4
min , /P PQ
γ
=
.
From (9), the received instantaneous SNR at the destination is expressed as
DF
DF 2 DF
R2
DF DF 2
D2
DF
04
||
min , .
th th
Ph PQ
N
γ
γγ
Ψ= =
(10)
Next, we introduce the relay selection strategy of the proposed protocol. First, if the relay
DF
R
cannot decode the signal
S
x
correctly, it sends a NACK message to inform the decoding
status. In this case, the relay
AF
R
will forward the received signal to the destination by using
AF technique. Second, for the case that the relay
DF
R
successfully decodes the signal, it also
informs the status by generating an ACK message. In this case, the best relay selected to
forward the signal
S
x
in next time slot is realized by using the following strategy:
( )
AF DF
best D D
R : arg max , .ΨΨ
(11)
The equation above implies that the relay providing the higher instantaneous SNR at the
destination is chosen for the cooperation. Next, the destination again decodes the received
signal, and if it fails to decode it in this time, the signal is dropped.
3. Performance Evaluation
If the received instantaneous SNR at the destination D (the relays
AF
R
and
DF
R
) is higher
than a predetermined threshold
th
γ
, we can assume that the destination D and the
DF
R
decodes the signal successfully. Otherwise, we assume that the signal
S
x
is not decoded
correctly. Hence, the outage probability of the proposed protocol can be formulated as
( ) ( )
( )
out out
12
MADF DF AF DF AF DF
Exact 0 1 D 0 1 D D
PP
P Pr , , Pr , ,max , .
th th th th th th
γγγ γγ γ
=Ψ≤Ψ≤Ψ≤ +Ψ≤Ψ> ΨΨ≤
(12)
In (12), the first term
out
1
P
presents the probability that the DF relay cannot decode the signal
received from the source successfully
( )
DF
1th
γ
Ψ≤
and the destination cannot also decode the
signal received from the source and the AF relay at both time slots correctly
( )
AF
0D
,
th th
γγ
Ψ≤ Ψ ≤
. The second term
out
2
P
in (12) indicates the probability that the
destination cannot receive the source signal at the first time slot successfully
( )
0th
γ
Ψ≤
but the
DF relay can
( )
DF
1th
γ
Ψ>
, and the transmisison from the best relay to the destination are not
correct
( )
( )
AF DF
DD
max ,
th
γ
ΨΨ ≤
.
Since finding an exact closed-form expression for
out
1
P
and
out
2
P
is a difficult work, we can
attempt to find the approximate closed-form expressions by using the well-known
approximation (see [6, eq. (7)], [7, eq. (17)], [23, eq. (2)]):
( )
AF AF AF
D 12
min ,Ψ≈ ΨΨ
. Hence,
out
1
P
and
out
2
P
can be approximated as
( )
1
2
1 DF AF AF
out 3 0 1 1 2
W
DF AF AF
01 1
32
33 3
W
P Pr , , ,min ,
Pr , , ,min , ,
th th th th th th th
th
th th th th
th th th
th
QPP P
P
QQ Q Q
P
γ γγ γ γ γ γ
γγ γ
γ γγ γ
γγ γ
≈ ≤ ≤ ≤ Ψ≤
+ > ≤ ≤ Ψ≤
(13)
( )
( )
3
4
2 DF AF AF DF
out 3 0 1 1 2 D
W
DF AF AF DF
01 1
3 2D
33 3
W
P Pr , , ,max min , ,
Pr , , ,max min , ,
th th th th th th th
th
th th th th
th th th
th
QPP P
P
QQ Q Q
P
γ γγ γ γ γ γ
γγ γ
γ γγ γ
γγ γ
≈ ≤ ≤ > Ψ Ψ≤
+ > ≤ > Ψ Ψ≤
.
(14)
Now, we calculate the terms
1
W
,
2
W
,
3
W
and
4
W
in (13) and (14). At first, by using [16,
Eq. (8)], we can rewrite the CDF of
AF
2
Ψ
as follows:
()( )
AF
2
2A A
2A
A
1 exp exp ,
th
th th th
yQ
y
Fy y
P yQ P
λκ
λ
κ
Ψ
+
=−− + −
+
(15)
where
A 2A 4A
/
κλλ
=
.
Using (15), we respectively obtain the exact closed-form expressions of
1
W
and
3
W
as
( ) ( ) ( ) ( ) ( )
( )
()
DF AF AF
30
1 12
1
3 0 1D
1D 2A
W / / / 11 / 1
1 exp 1 exp 1 exp
1 exp exp
th th th th th th th th th
th th th
th th th
th th
th
th th t
FQ PF PF P F P F
Q
PPP
PP
γγ
γγ
γγ γ γ
λ λγ λ γ
λγ γ
λγγ
Ψ
= −− −
=−− −− −−
×− − − −
( )
2A A
A
exp ,
th th
h th th
Q
QP
λγ κ
κ
+
−
+
(16)
( ) ( ) ( )
( )
( ) ( ) ( )
( )
()
DF
30 1
DF AF AF
D1 2
3
3 0 1D
2D
2D
D
W / /1 /
11 / 1
1 exp 1 exp exp
1 exp exp
th th th th th th
th th th th
th th th
th th th
th
th th
th th th
FQ PF P F P
F F PF
Q
P PP
PQ
γγ γ
γ
γγ
γγ γ
λ λγ λ γ
λγ
λγ γ
γκ
ΨΨ
= −
× −− −
=−− −− −
+
×− − + −
+
( )
( )
D
2A A
1D 2A
A
1 exp exp exp .
th
th
th th
th th
th
th th th th th
Q
P
Q
PP Q P
κ
λγ κ
λγ γ
λγγκ
+
×− − − − −
+
(17)
where
( )
DF
D
.F
Ψ
is the CDF of
DF
D
Ψ
which can be obtained from (15) by replacing
2A
λ
and
A
κ
with
2D
λ
and
D
κ
, respectively, with
D 2D 4D
/
κλλ
=
.
Considering the term
2
W
in (13), similar to [22], it can be formulated as
( ) ( )
2 3 32
/
W exp W
th th
QP
x x dx
λλ
+∞
= −
∫
, (18)
where,
()( )
( )
DF AF AF
01 12
2
W 11 1 .
th th th th
th th th
x F xF x F x F
QQ Q
γγγ
γγ γ γ
Ψ
= −− −
(19)
Combining (15), (18) and (19), and after some carefully manipulation, we have
( )
( )
3 0 3 1D
33
23
3 0 3 1D
3 0 1D
3
3 0 1D
2A A
2A
A
exp exp
W exp
exp
exp exp
th th th th
th th
th th
th
th th th th th
th th
th
th th th
th
th
th
th th th
QQ
QQ
PP
Q
PQ Q
Q
Q
QP
PQ
λλγ λλγ
λλ
λλλγ λλγ
λ λλγ
λ
λ λλγ
λγ κ
γ
λγγκ
++
−−
=−− −
++
++
+−
++
+
−− − −
+
()
()
( )
( )
()
( )
3 0 1A
3 1A 3
3
3 1D 3 0 1D
3 1A 1D 3 0 1A 1D
33
3 1A 1D
exp
exp
exp exp
th
th
th th
th th th
th
th
th
th th th th
th th th th
th th
th th
th th
Q
P
Q
QQ
QP
P
QQ
QQ
QQ
PP
Q
λ λλγ
λ λγ λ
λ
λ λγ λ λ λ γ
λ λ λγ λ λλ λγ
λλ
λ λ λγ
++
+−
−
−
+ ++
×++ +++
−−
−+
++
()
3 0 1A 1D
.
th th
Q
λ λλ λγ
+++
(20)
Similarly, we can obtain
() ( )
()
( )
( )
( )
( )
( )
( )
4 3 34
/
2D D
2D
D
3 31
3 1D
3 3 01
3 01
2A
W exp W
1 exp exp
exp /
exp /
exp
th th
QP
th th
th th
th th th th
th th D th th
th th
th th D th th
th D th
th t
th
x x dx
Q
PQ P
QQ P
Q
QQ P
Q
P
λλ
λγ κ
λγ γ
γκ
λ λ λγ
λ λγ
λ λ λλγ
λ λλγ
λγ γ
+∞
= −
+
=−− + −
+
−+
+
− ++
−++
−− −
×
∫
( )
()
()
()
()
( )
()
2A A
A
3 1A 1
3
3 1A 1
3 0 1A 1
3
3 0 1A 1
exp
exp
exp
th th
h
th th th
th D th
th
th D th th
th D th
th
th D th th
Q
QP
Q
Q
QP
Q
Q
QP
λγ κ
γκ
λ λ λγ
λ
λ λ λγ
λ λλ λγ
λ
λ λλ λγ
+
−
+
++
−
++
×
+++
−−
+++
,
(21)
where
( ) ( ) ( )
( )
DF DF AF AF
01D 1 2
4
W 1 11 1 .
th th th
th th
th th th
x F x F xF F x F
QQ Q
γγγ
γγ γ
γγ
ΨΨ
= − −− −
From (12)-(14), (16)-(17), (20) and (21), an approximate closed-form expression of the outage
probability for the proposed protocol can be expressed as
MADF
Exact 1 2 3 4
P W W W W.≈+++
(22)
It should be noted that the exact outage probability
MADF
Exact
P
reaches
1234
WWWW+++
at
small values of
0
N
. Therefore, instead of finding an asymptotic expression for
MADF
Exact
P
, we can
find an asymptotic expression for
1234
WWWW+++
. In order to find a such expression, we
can use Maclaurin expansion for
1234
WWWW+++
at
0
0N=
.
First, we easily obtain
( )
0
1234
0
lim W W W W 0
N→
+++ =
,
( )
0
1234
00
WWWW
lim 0
NN
→
∂ +++ =
∂
and
( )
0
21234
2
00
WWWW
lim 0
N
N
→
∂ +++ =
∂
.
Moreover, we can obtain the following equation:
( ) ( )
( )
0
31 2 3 4 2A 4A 3
3 0 1D 1D 2A
32
00 4A
3 4A 3
0 1D 2A 2 23 2
4A 3 3
31A 1A 2A 1
0 1D 33 2 2
33
W W W W exp /
lim 6 1 exp
12 2
6 exp
6 62λ3
6 exp
th
N
th
QP
Q
N PP P Q
Q
P P Q Q PQ
Q
P Q PQ
λλ
λ λλ λλ γ
λ
λλ
λλ λ γ
λ λλ
λλλ λ
λλ λλ
→
∂ +++ −
+
=−− +
∂
+
+ − ++
+
+− + +
( ) ( )
( )
3
A 2A 1A 2A
23
3
2D 4D 2A 4 3
02D 1D 2A
3
4D 4A
1A 22 2
33
4D
0 2D 3
4D 3
2A
2λ
exp / exp /
6 1 exp
2 21
exp /
1
6 exp
.
th
th
PQ P
QP QP
Q
P PP Q P Q
Q PQ P
QP
Q
PP Q PQ
λλγ
λ
λλ λλ
λλ λλ
λγ
λλ
λλλ
λ
λλ λ λλ
λ
++
+
− −
+
+− − + +
++
−
+ −+
+
+
( )
3
4A
3 4A
.
exp /
1
th
QP
PQ P Q
γ
λ
λλ
−
+
(23)
From results above, the asymptotic expression of
1234
WWWW
+++
(
MADF
Exact
P
) can be given
as
( )
0
0
33
12340
1234 3
0
00
WWWW
W W W W lim .
3!
N
N
N
N
→
→
∂ +++
+++ →
∂
(24)
From (24), it is obvious that the diversity gain of the proposed protocol equals 3, i.e.,
( )
( )
()
( )
0
00
33
12340
3
MADF 00
Exact
1/ 1/
00
WWWW
log lim 3!
log P
diversity=- lim lim 3
log 1/N log 1/N
N
NN
N
N
→
→+∞ →+∞
∂ +++
∂
=−=
.
Next, we evaluate the outage performance of the considered scheme when
QP<<
and
QP>>
. First, when
QP<<
or
/0QP→
, the outage probability
MADF
Exact
P
can be rewritten as
( )( )
( ) ( )
( )( )
( ) ( )
out
1
AF AF AF
DF 13 24
MADF 01
Exact AF AF AF
33 13 24
P
AF AF AF
DF DF 13 24
012
DF AF AF AF
33 4 1 3 24
//
P Pr , , / /1
//
Pr , ,max , / /1
th th
th
th th
th th
th
th th
QQ
QQ
QQ
QQQ
γγ γγ
γγ
ρρ γ
γγ γγ γγ
γγ γγ
γγγ
ρρ
γγ γ γγ γγ
=≤≤ ≤
++
+≤> ++
out
2
P
,
th
γ
≤
(25)
where
0
//
th th th
QN Q
ργ γ
= =
.
We can observe from (25) that the outage probability
MADF
Exact
P
in this case only depends on the
interference constraint Q, which corresponds with the case that the transmit power constraint
is relaxed [15],[17].
Considering the term
out
1
P
in (25), similarly, it can be formulated by the following equation:
( ) ( )
out out
1 3 31
0
P exp P .x x dx
λλ
+∞
= −
∫
(26)
In (26), the conditioned probability
( )
out
1
Px
can be calculated as
( ) ( ) ( )
DF
01
out
1
P Pr ,
1
YZ
x F xF x YZ
γγ
ρρ ρ
= ≤
++
(27)
where
AF
1
/
th
YQ x
γ
=
and
AF AF
24
/
th
ZQ
γγ
=
.
Now, our objective here is to find
Pr 1
YZ t
YZ
≤
++
. With the same manner as in [22] and
with the CDF
( )
1A
1 exp
Y
th
xy
Fy Q
λ
=−−
and the CDF
()
A
Z
th
z
Fz zQ
κ
=+
(see [15, Eq. (3)]),
we can given an exact expression of
Pr 1
YZ t
YZ
≤
++
as
( )
()
A1
A
2
1A A A
1A 1A
1
2AA
A
Pr 1 exp
1
1exp . . ,
th
th th
th th
th th th th
th th
Qt
YZ tx
YZ t Q Q
tt Q tQ t tt
x xE x
QtQ QtQ
Qt Q
κλ
κ
λκ κ
λλ
κκ
κ
≤=− −
++ +
+
−+
+−
++
+
(28)
where
( )
1
.E
is exponential integral [24].
Plugging (26), (27) and (28) together, using [24, 6.228] with
2
ν
=
and
( ) ( )
1i
E x Ex
ββ
−=−
for the corresponding integral, and after some careful manipulation, we can obtain
out
1
P
as
follow
( ) ( )
( )
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
out 0 03
1
30 3 1D 3 0 1D
03 03
A
A3 1A 3 0 1A 3 1D 1A 3 0 1D 1A
2
1A A
3 1A
AA
2
1A A
3 0 1A
A
1A 3 A
2
A
P
1/ /
,
1/ ,
1/
th th
th
th
QQ
Q
Q
λρ λλρ
λ λρ λλρλ λλρ
λλρ λλρ
κ
ρκ λλρλ λλρ λ λ λρλ λλ λρ
λκ ρ ρρ
λλ
ρκ ρκ
λκ ρ ρ
λ λρ λ
ρκ
λ λρ ρ κ
ρκ
= −
++ ++
−−
++ ++ + + +++
−
+
Ξ+
++
−
−Ξ + + +
+
++
( )
( ) ( )
A
2
1A A
3 1D 1A
AA
2
1A A
3 0 1D 1A
AA
/
,
1/ /
,
1/ /
,
th
th th
th th
Q
QQ
QQ
ρ
ρκ
λκ ρ ρρ
λ λρ λ
ρκ ρκ
λκ ρ ρρ
λ λλρ λ
ρκ ρκ
+
+
−
+
−Ξ + +
++
−
+
+Ξ + + +
++
(29)
where,
( ) ( )
2
11
, log .
βµ
µβ µ β µβ µ
+
Ξ= −
+
(30)
Similarly, the outage probability
2
out
P
in (25) can be expressed under an exact closed-form
expression as
( ) ( )
( )
( ) ( )
( )
( )
( )
( )
( )
( )
( ) ( )
2
out
D
03 03A
3 1D 3 0 1D A 3 1A 1D 3 0 1A 1D
2
1A A
3 1D 1A
AA
1A 3 A
22
A 1A A
3 0 1D 1A
AA
P
1/ /
,
1/
1/ /
,
th th
th
th th
QQ
Q
QQ
ρ
ρκ
λλρ λλκ ρ
λλρλ λλρ ρκ λ λ λρλ λλ λρ
λκ ρ ρρ
λ λρ λ
ρκ ρκ
λ λρ ρ κ
ρκ λ κ ρ ρρ
λ λλρ λ
ρκ ρκ
=+
−
+ ++ + + + +++
−
+
Ξ+ +
×
++
+
+
+ −
+
−Ξ + + +
++
.
(31)
Substituting (29) and (31) into (25), we obtain an exact expression for
MADF
Exact
P
when
QP<<
.
Similarly, by using the third-order derivative, we can obtain an asymptotic closed-form
expression of
MADF
Exact
P
as
QP<<
by
( ) ( )
0
3
3 MADF 3
MADF Exact 0 1D 0 2D 0
2A 1A 2A 1A
Exact 0
32
03 4A 3 3 4D 4A 3
0
P2
32
1
P.
6!
th
N
N
NQ
N
λλ λλ γ
λλ λλ
λ λ λ λλ λ λ
→
∂
→ = ++ +
∂
(32)
Now, we consider the case in which the interference Q is relaxed, i.e.,
QP>>
or
/QP→ +∞
. Also, from (12)-(14), we can rewrite the exact outage probability
MADF
Exact
P
in this case as
( )( )
( ) ( )
()( )
( ) ( )
AF AF
12
MADF DF
Exact 0 1 AF AF
12
AF AF
12
DF DF
01 2 AF AF
12
P Pr , , 1
Pr , ,max , ,
1
th
th
th
th
th th
th
PP
PP
PP
PPP
γγ
γ θγ θ γ
γγ
γγ
γ θγ θ γ γ
γγ
=≤≤ ≤
++
+≤≤ ≤
++
(33)
where
0
//
th th th
PN P
θγ γ
= =
.
By using the result presented in [25, Eq. (8)], we can obtain
( )( )
( ) ( )
AF AF
12
AF AF
12
Pr 1
th
th
th
PP
PP
γγ γ
γγ
≤
++
,
and then,
MADF
Exact
P
can be expressed by an exact closed-form expression as
( )
( )
( )
( )
( )
( )
( )
( )
MADF
Exact 0 1D 2D
22
1A 2A 1A 2A 1 1A 2A
P 1 exp 1 exp
22
1 exp ,
th
th th th th
th th th
K
P PP
λθ λ λ θ
γ
λλ γ γ λ λ λλ γ γ
=− − − −+
×− + − + +
(34)
where
( )
1.K
is the first order modified Bessel function of the second kind [24].
Finally, by using
( )
0
1 exp
x
xx
→
− −→
,
( )
0
exp 1
x
xx
→
− →−
and
()
101/
x
Kx x
→
→
for (34), we finally
obtain an asymptotic expression for (34) as follow
( )( )
0
3
MADF 0
Exact 0 1D 2D 1A 2A
0
P.
th
N
N
P
γ
λλ λ λ λ
→
→+ +
(35)
For the performance comparision, we introduce the operation of the corresponding
decode-and-forward (DF) and amplify-and-forward (AF) protocols in underlay cognitive
radio. In the DF (AF) protocol, two secondary relays operate in DF (AF) mode. Similar to the
proposed protocol, at the first time slot, if the destination cannot decode the source's signal
successfully, it requests a retransmission from one of the two relays. In the DF protocol, if only
a relay can decode the signal successfully, it forwards the decoded signal to the destination at
the second time slot. If both relays can correctly receive the source's signal, the relay having
the higher instantaneous SNR of the link between itself and the destination is chosen to
forward the decoded signal. In the AF protocol, the relays providing higher instantaneous SNR
of the source-relay-destination link is chosen for the cooperation.
4. Simulation Results
In this section, Monte-Carlo simulations are provided to verify the mathematical derivations.
In all of the simulations, we assume that the path-loss exponent
β
equals 3 and the threshold
th
γ
equals 1.
In Figs. 2-3, we present the outage probability as a function of
( )
0
1/N
in dB. In these figures,
we assume that the coordinates of
AF
R
,
DF
R
and PU are
( )
0.5,0
,
( )
0.5,0
and
( )
0.5,0.5
,
respectively. We can observe from Figs. 2-3 that the outage probability is lower when one of
two constaints P and Q is relaxed. It is also seen that the simulation results match very well
with the theoretical results at high
0
1/N
values. It is noted that there exists a small gap
between the theoretical results and simulation results because we used the approximation
( )
AF AF AF
D 12
min ,Ψ≈ ΨΨ
to derive the outage probability. In addition, we can see that the slopes
of the outage probability curves at high
0
1/N
region are three, so the diversity order of the
proposed protocol equals three.
Fig. 2. Outage probability as a function of
0
1/N
in dB when
AF DF
RR
0.5xx= =
, and
PU PU
0.5xy= =
.
Fig. 3. Outage probability as a function of
0
1/N
in dB when
AF DF
RR
0.5xx= =
, and
PU PU
0.5xy= =
.
In Figs. 4-5, we present the outage performance of the proposed protocol as a function of P and
Q, respectively. In these simulations, we assign the value of
AF
R
x
,
DF
R
x
,
PU
x
,
PU
y
and
0
1/N
by 0.4, 0.4, 0.3, 0.3 and 10 dB, respectively. In Fig. 4, we can see that the outage probability
decreases with the increasing of P. In addition, at high P value, the outage probability does not
depend on P. It is because that when the value of P is high, the ratio Q/P goes to zero and, so
the outage performance only depends on the Q value as presented in (25). Similarly, as
observed in Fig. 5, the outage probability does not depend on the value of Q when the ratio
Q/P is quite high.
Fig. 4. Outage probability as a function of the maximum transmit power
P
when
AF DF
RR
0.4xx= =
,
PU PU 0.3xy= =
, and
0
1/ 10dBN=
.
Fig. 5. Outage probability as a function of the maximum interference
Q
when
AF DF
RR
0.4
xx
= =
,
PU PU
0.3xy= =
, and
0
1/ 10N dB=
.
Fig. 6. Outage probability as a function of
AF
R
x
when
1PQ= =
,
PU PU 0.35xy= =
, and
0
1/ 15N dB=
.
Fig. 7. Outage probability as a function of
DF
R
x
when
1PQ= =
,
AF
R0.5x=
, and
0
1/ 15N dB=
.
In Figs. 6-7, we observe the effect of the relays' postions on the outage performance. In Fig. 6,
we fix the position of the primary user at
( )
0.35,0.35
while change the co-ordinate
AF
R
x
of
the relay
AF
R
from 0.05 to 0.95. We can see that the outage probability change with the
change of
AF
R
x
and
DF
R
x
. In this figure, the performance of the proposed protocol is better
when the relay
DF
R
is near the source (
DF
R0.1x=
). It is due to the fact that when it is closed to
the source, the probability that relay
DF
R
can successfully decode the source's signal is higher,
which hence decreses the outage probability of the proposed protocol. In Fig. 7, we change the
position of the relay
DF
R
while fixing the position of the relay
AF
R
at
( )
0,0.5
. In this figure,
it can be seen that the outage performance is better when
PU PU
0.4xy= =
. It is due to the fact
that when the primary user is further from the secondary network region, the transmit power of
the source and relays is higher, which causes an enhancement of the outage performance.
Fig. 8. Outage probability as a function of
0
1/N
in dB when
AF DF
RR
0.5xx= =
,
PU PU
0.4xy= =
,
P=Q=0.5.
In Fig. 8, we compare the outage performance of the proposed protocol with that of the DF and
AF protocols introduced above. We can see that the performance of the prosed protocol is
between that of the DF protocol and that of the AF protocol. However, the implementation of
our scheme is easier that of the DF protocol, because the process operation of a DF relay is
more complex than that of a AF relay. In addition, it should noted in this figure that, the
theoretial results of the MADF protocol is same with the simulation results of the DF protocol.
5. Conclusion
In this paper, we propose a mixed amplify-and-forward and decode-and-forward relaying
scheme for underlay cognitive radio network. The main contribution of this paper is to derive
the exact and approximate closed-form expressions of the outage probability over Rayleigh
fading channel. In addition, we find the asymptotic expressions to prove that the diversity gain
of the proposed protocol equals three. Next, Monte Carlo simulations are presented to verify
our derivations and to compare the performance of our scheme with the corresponding AF and
DF protocols.
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