A New Hybrid Half-Duplex/Full-Duplex
Relaying System with Antenna Diversity
Cheng Li, Bin Xia, Zhiyong Chen

arXiv:1802.07801v1 [] 21 Feb 2018

Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai,
China
Emails: {lichengg, bxia, zhiyongchen}@sjtu.edu.cn

Abstract
The hybrid half-duplex/full-duplex (HD/FD) relaying scheme is an effective paradigm to overcome
the negative effects of the self-interference incurred by the full-duplex (FD) mode. However, traditional
hybrid HD/FD scheme does not consider the diversity gain incurred by the multiple antennas of the
FD node when the system works in the HD mode, leading to the waste of the system resources. In
this paper, we propose a new hybrid HD/FD relaying scheme, which utilizes both the antennas of the
FD relay node for reception and transmission when the system works in the HD mode. With multiple
antennas, the maximum ratio combining/maximum ratio transmission is adopted to process the signals
at the relay node. Based on this scheme, we derive the exact closed-form system outage probability and
conduct various numerical simulations. The results show that the proposed scheme remarkably improves
the system outage performance over the traditional scheme, and demonstrate that the proposed scheme
can more effectively alleviate the adverse effects of the residual self-interference.

I. I NTRODUCTION
The full-duplex (FD) communications are receiving more and more interest from both industry
and academia due to the ultimate utilization of radio resources [1], [2]. Compared to the halfduplex (HD) mode, the FD mode bears the capability of reception and transmission on the
same time and frequency resource [3]. The FD mode is usually achieved with two antennas,
one for reception and the other for transmission [4]. The signals leakage, which is called selfinterference, from the transmit antenna to the local receive antenna would severely limit the
This work was supported in part by the National Key Science and Technology Specific Program under Grant 2016ZX03001015,
in part by the National Nature Science Foundation of China under Grant 61531009, and in part by the National High Technology
Research and Development Program of China under 863 5G Grant 2014AA01A704.

February 23, 2018

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1

system performance. Although the self-interference can be greatly cancelled with various methods
[5], the performance gain of the FD mode over the HD mode is limited by the residual selfinterference (RSI).
On the other side, the relay stations are usually deployed in the remote area to extend the
coverage of the cellular networks. Integrating the FD mode into the relay communication is an
effective way to improve the rate of the cell edge users [6]. The performance of the FD relay
system have been investigated in [7]–[12]. Specifically, in [7]–[9], the authors have analyzed the
achievable rate and system outage probabilities of the FD relay networks. The diversity gain of
the FD relay system with direct link is investigated by the authors in [10]. In addition to the
performance analysis of the FD relay systems, the research has also shown that the FD system
could achieve better performance compared to their HD counterparts when the RSI is below a
certain threshold [11], [12].
To alleviate the adverse effects of the RSI, the hybrid half-duplex/full-duplex (HD/FD) relaying
scheme was proposed [13]. With the hybrid HD/FD relaying scheme, the system can dynamically
change between the HD mode and FD mode. When the RSI is high, the system converts to the
HD mode, thus, the RSI can be inherently eliminated. However, in the previous works, although
the authors considered the hybrid HD/FD relaying scheme, the system resources have not been
fully utilized. Specifically, the authors only used one antenna to receive and transmit signals when
the system works in the HD mode even if the FD relay node is equipped with two antennas
[13]. This leads to certain diversity loss, resources waste and performance degradation.
In this paper, we propose a new hybrid HD/FD relaying scheme. In this scheme, the MRC/MRT,
which utilizes the antenna diversity to combat the channel fading, is adopted to process signals
at the FD relay node when the system works in the HD mode. For performance evaluation,
we derive the exact closed-form system outage probability, which is based on the FD outage
probability and the conditional HD outage probability of the proposed HD/FD relaying scheme.
In addition, we compare the proposed scheme with the traditional scheme proposed in [13]
in terms of the outage performance. The results show that the proposed scheme improves the
system outage performance over the traditional scheme and alleviate the adverse effects of the
RSI. Finally, numerical simulations corroborates the theoretical results.

February 23, 2018

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2

II. S YSTEM M ODEL
In this section, we elaborate on the channel mode, the proposed hybrid HD/FD relaying scheme
and the specific signaling process.

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Fig. 1. The hybrid HD/FD relay system with FD mode.





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(a) Sub-time slot 1.

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(b) Sub-time slot 2.
Fig. 2. The hybrid HD/FD relay system with HD mode.

A. Channel Model
In this paper, we consider the one-way decode-and-forward FD relay system, which consists of
a source node S, a destination node D and a relay node R. The source node S and the destination

February 23, 2018

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3

node D are both equipped with one antenna and can only work in the HD mode. However, the
relay node R is equipped with two antennas Ant-1 and Ant-2, and each antenna is equipped
with a Rx module and a Tx module. Thus, the relay node has the capability to work in either
the FD mode or the HD mode. We assume that the direct link between the source node S and
the destination node D does not exist due to the deep fading and strong shadowing effects [11].
The channel coefficients of h1,1 , h1,2 and h2,1 , h2,2 are shown in the Fig. 1. Since the antennas
Ant-1 and Ant-2 can naturally achieve 40 dB isolation, thus h1,1 and h1,2 , h2,1 and h2,2 are
assumed to be independent with each other [13]. We model the links from the source S to
the relay R and the relay R to the destination D as complex Gaussian channels. Then, we
can easily prove that the envelops of the channels are subject to the Rayleigh distribution. Let
E{|hi,j |2 } = Ωi,j , i, j ∈ {1, 2} to denote the channel parameters. In this work, we assume that
all the links are block fading wireless channels.
B. The Proposed Hybrid HD/FD Relaying Scheme
The proposed hybrid HD/FD scheme is elaborated on in the following.
The Full-Duplex Mode: The source node S transmits signals to the relay node, and the relay
node R forwards the previously decoded signals to the destination node D simultaneously as
depicted in Fig. 1. In the FD mode, we assume that the antenna Ant-1 is connected to the Rx
module to receive signals from the source node S and the antenna Ant-2 is connected to the
Tx module to forward the signals to the destination node D. Hence, the simultaneous reception
and transmission is achieved. It is worth noting that in the FD mode, although several selfinterference cancellation techniques can be adopted to cancel the self-interference, the system is
harassed by the RSI [13].
The Half-Duplex Mode: The source node S transmits signals to the relay node, and the relay
node R forwards the decoded signals to the destination node D in the next time slot as depicted
in Fig. 2. In the HD mode, the system is divided into two phases: reception phase and relaying
phase. During the reception phase, both the two antennas Ant-1 and Ant-2 of the relay node
are connected to the Rx modules to receive signals from the source node S. Within the relaying
phase, both the two antennas Ant-1 and Ant-2 are connected to the Tx modules to forward the
recoded signals to the destination node D. In order to reap the diversity gain, we adopt the
maximum ratio combining (MRC) to combine the signals received at the two antennas and the
maximum ratio transmission (MRT) to forward the signals to the node D.

February 23, 2018

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4

The Hybrid Scheme: In our proposed hybrid HD/FD scheme, the system works either in the
FD mode or the HD mode determining on the instantaneous channel capacity. If the instantaneous
capacity of the FD mode surpasses that of the HD mode, i.e., Cf d > Chd , the system chooses
to work in the FD mode. The benefits of the FD mode are two-fold: i) The FD mode could
inherently achieve higher spectrum efficiency; ii) The latency at the relay node could be greatly
reduced. On the other side, if the instantaneous capacity of the HD mode is larger than that of
the FD mode, i.e., Chd > Cf d , the system converts to the HD mode. The benefits of the HD
mode are also two-fold: i) The system can inherently eliminate the self-interference perfectly;
ii) The MRC/MRT can be adopted to combat the channel fading.
C. The Specific Signaling Process
The FD mode: In this mode, the signals received at the relay node and the destination node
can be expressed as
yr = h1,1 xs + hsi xr + nr ,

(1)

yd = h2,2 xr + nd ,

(2)

respectively, where xs and xr are the signals transmitted by the source S and the relay R
with power PS = E{|xs |2 } and PR = E{|xr |2 }, respectively. nr and nd denote the thermal
noise over the relay R and the destination D, respectively, with zero mean and variance σ 2 ,
i.e., nr , nd ∽ CN (0, σ 2 ). The relay R receives the signals from the source S as well as the
self-interference signals from the antenna Ant-2 to Ant-1. hsi denotes the self-interference
channel coefficient. In this paper, we assume that the relay node could apply the self-interference
cancellation techniques in [5] to cancel hsi xr . Hence, the system is harassed by the RSI after selfinterference cancellation. We denote the RSI as e
hsi x
er . Whatever the specific distributions of e
hsi

and x
er are, due to the imperfect estimation of the self-interference channel and distortion of the
self-interference signals during the cancellation process, we assume that the effects of e
hsi x
er are
characterized by the Gaussian distribution [13], i.e., e
hsi x
er ∽ CN (0, σ 2 ), where σ 2 = Kr PR .
RSI

RSI

Kr indicates the self-interference capability of the relay node. Thus, the received signals at the

relay node after self-interference cancellation can be expressed as
yer = h1,1 xs + e
hsi x
er + nr ,

February 23, 2018

(3)

DRAFT

5

Hence, the signal-to-interference-plus-noise ratio (SINR) at the relay node and the destination
node can be expressed as
|h2,2 |2 PR
|h1,1 |2 PS
,
γ
=
,
(4)
f,d
k r PR + σ 2
σ2
denote the SINRs at the relay node R and the destination node D under the
γf,r =

where γf,r and γf,d

FD mode, respectively.
The HD Mode: In this mode, the time slot is divided into two sub-time slots. In the first
sub-time slot, the relay node R receives signals from the source node S
yr = H1 xs + nr

(5)

where vector yr = [y1,r , y2,r ]T denotes the received signals at the antennas Ant-1 and Ant-2.
H1 = [h1,1 , h1,2 ]T denotes the estimated channel vector between the source S and the relay R.
nr = [n1,r , n2,r ]T denotes the Gaussian noises over the antennas Ant-1 and Ant-2 and we assume
n1,r , n2,r ∼ CN (0, σ 2). In order to maximize the SINR at the relay R, we adopt the MRC to
combine the received signals at the antennas Ant-1 and Ant-2. The combined signals can be
expressed as
y′ r = W1 H H1 xs + W1 H nr
where (·)H denotes the conjugate transpose, W1 =

H1
||H1 ||F

(6)

denotes the processing matrix of the

MRC and || · ||F denotes the Frobenius norm.
In the second sub-time slot, the relay node R uses the MRT technique to pre-process the
signals and then forwards the signals to the destination node D. The received signals at the
destination node can be expressed as
yd = W2 H H2 xr + nd ,
where W2 =

H2
||H2 ||F

(7)

is the processing matrix of the MRT and H2 = [h2,1 , h2,2 ]T is the estimated

channel vector from the antennas at the relay node R to the destination node D.
Based on the MRC/MRT, the SINR at the relay node R and the destination node D can be
expressed as

|h1,1 |2 + |h1,2 |2
PS ,
(8)
σ2
|h2,1 |2 + |h2,2 |2
γh,d =
PR
(9)
σ2
denote the SINRs at the relay node and the destination node under the HD
γh,r =

where γh,r and γh,d
mode, respectively.

February 23, 2018

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6

III. O UTAGE P ERFORMANCE A NALYSIS
In this section, we analyze the system outage performance of the considered one-way decodeand-forward FD relay system under the proposed hybrid HD/FD relaying scheme.
The system outage probability under the FD mode and HD mode can be defined as
fd
Pout
= P r{Cf d = log2 (1 + min{γf,r , γf,d}) < R0 },
hd
Pout
= P r{Chd =

1
log2 (1 + min{γh,r , γh,d}) < R0 },
2

(10)
(11)

respectively, where Cf d and Chd denote the system capacity of the FD mode and the HD
mode, respectively. P r{x} denotes the probability of the event x. We can note that due to the
simultaneous transmission, the pre-factor 1/2 disappears under the FD mode, which indicates
that the FD mode could effectively recover the spectrum efficiency.
With the proposed hybrid HD/FD scheme, the system outage probability can be calculated as
sys
Pout
= P r{Cf d < R0 , Chd < R0 }

= P r{Cf d < R0 }P r{Chd < R0 | Cf d < R0 },

(12)

According to the Total Probability Theorem, the system outage probability of the FD mode
can be divided into three mutual exclusive events A, B and C as follows
fd
Pout
= P r{Cfsrd < R0 , Cfrdd > R0 }
{z
}
|
A

+

P r{Cfsrd

+

P r{Cfsrd

|
|

> R0 , Cfrdd < R0 }
{z
}
B

< R0 , Cfrdd < R0 },
{z
}

(13)

C

where Cfsrd = log2 (1 + γf,r ) and Cfrdd = log2 (1 + γf,d ) denote the channel capacities of the links
from the source S to the relay R and from the relay R to the destination D, respectively. Event
A denotes that the link from the source S to the relay R is in the outage state but the link from
the relay R to the destination D is not. Event B denotes that the link from the source S to the
relay R is not in the outage state whereas the link from the relay R to the destination D is.
Event C denotes that both the links from the source S to the relay R and from the relay R to
the destination D are in the outage state.

February 23, 2018

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7

Applying the Total Probability Theorem again, the system outage probability can be expressed
as
sys
Pout
= P r{A}P r{Chd < R0 |A}

+ P r{B}P r{Chd < R0 |B}
+ P r{C}P r{Chd < R0 |C}.

(14)

Next, we will derive the system outage probabilities under the conditions of events A, B and
C, respectively.
A. System Outage Probability Under the Event A
Recall that the complex Gaussian channels have the parameter E{|hi,j |2 } = Ωi,j , i, j ∈ {1, 2},
we can easily prove that the random variable |hi,j |2 is subject to the exponential distribution.
The probability density function (pdf) of |hi,j |2 is expressed as
f|hi,j |2 (x) = λi,j e(−λi,j x) ,
where λi,j =

1
,
Ωi,j

(15)

i, j ∈ {1, 2}.

Thus, we have
P r{A} = P r{Cfsrd < R0 , Cfrdd > R0 }
= P r{log2 (1 + γf,r ) < R0 , log2 (1 + γf,d) > R0 }
|h1,1 |2 PS
|h2,2 |2 PR
<
t
,
> t1 }
1
KR P R + σ 2
σ2
Z t1 (KR PR +σ2 )Z +∞
PS
=
f|h1,1 |2 (x)f|h2,2 |2 (y)dxdy
2
= P r{

t1 σ
PR

0

(−λ1,1 t1

= (1 − e

KR PR +σ 2
)
PS

(−λ2,2

)e

t1 σ 2
)
PR

,

(16)

where t1 = 2R0 − 1. Under the condition of the event A, the outage probability of the HD mode
can be calculated as
sr
P r{Chd < R0 |A} = P r{Chd
< R0 |A}
rd
+ P r{Chd
< R0 |A}
sr
rd
− P r{Chd
< R0 , Chd
< R0 |A},

February 23, 2018

(17)

DRAFT

8

sr
rd
where Chd
and Chd
denote the channel capacity of the links from the source S to the relay R

and from the relay R to the destination D, respectively.
sr
Then, we first calculate the P r{Chd
< R0 |A}
sr
P r{Chd
< R0 |A}

1
|h1,1 |2 + |h1,2 |2
=P r{ log2 (1 +
PS ) < R0 |A}
2
σ2
t1 (KR PR + σ 2 )
(|h1,1 |2 + |h1,2 |2 )PS
< t2 |h1,1 |2 <
}
=P r{
2
σ
PS
Z min{ t1 (KRPPR +σ2 ) , t2Pσ2Z} tP2 σ2 −x
S
S
S
=
f(|h1,1 |2 |A) (x)f|h1,2 |2 (y)dxdy
0

=

Z

0

min{m1 ,m2 }

Z

t2 σ2
PS

−x

0

0

f|h1,1 |2 (x)
f|h |2 (y)dxdy
P|h1,1 |2 (m1 ) 1,2



=

1
(1 − e(−λ1,1 min{m1 ,m2 }) )
A
P|h
(m
)
2
1
1,1 |

−

λ1,1 e(−λ1,1 m2 ) ((λ1,2 −λ1,1 ) min{m1 ,m2 })
(e
− 1) ,
λ1,2 − λ1,1

where t2 = 4R0 − 1, m1 =

t1 (KR PR +σ2 )
PS

and m2 =

t2 σ2
.
PS

(18)

P|hA1,1 |2 (m1 ) = 1 − e−λ1,1 m1 denotes the

probability of |h1,1 |2 < m1 under event A.
rd
By the similar way, we can derive the probability of P r{Chd
< R0 |A} as follows
rd
P r{Chd
< R0 |A}
Z m3 Z m3 −y
=
f(|h2,2 |2 |A) (y)f(|h2,1 |2 ) (x)dxdy
0

=
−
where m3 =

t1 σ2
.
PR

0

1
P|hA2,2 |2 (m3 )

 (−λ2,2 m3 )
(e
− e(−λ2,2 m2 ))

λ2,2 e(−λ2,1 m2 ) ((λ2,1 −λ2,2 )m2 )
(e
− e((λ2,2 −λ2,1 )m3 ) ) ,
λ2,1 − λ2,2

(19)

P|hA2,2 |2 (m3 ) = e−λ2,2 m3 denotes the probability of |h2,2 |2 > m3 under event A.

Next, we derive the probability of both the links from the source S to the relay R and from
the relay R to the destination D locate in the outage region under the HD mode, i.e.,
sr
rd
P r{Chd
< R0 , Chd
< R0 |A}
Z mmin
Z m2 −yZ1 mZ2 m2 −y2
1
f (x1 )f (y1 )f (x2 )f (y2 )dx1 dy1 dx2 dy2
=
P (A) 0
0
m3 0
(a)

sr
rd
=P r{Chd
< R0 |A} × P r{Chd
< R0 |A},

(20)

where mmin = min{m1 , m2 }. P (A) = P|hA1,1 |2 (m1 ) × P|hA2,2 |2 (m2 ) denotes the probability of
the whole event A. (a) is achieved by the independence of the links from the source S to the

February 23, 2018

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9

relay R and from the relay R to the destination D. f (x1 ) = f|h1,1 |2 (x), f (y1) = f|h1,2 |2 (y1 ),
f (x2 ) = f|h2,1 |2 (x2 ) and f (y2) = f|h2,2 |2 (y2 ) are the pdfs of the corresponding channels.
Substituting (18), (19) and (20) into (17), We can get the system outage probability under the
condition of event A. Similar to the event A, the system outage probability under the event B
and event C can also be derived in the similar methods. Due to the length limit of this paper, the
details of the specific derivations procedures are omitted here. Then according to the (14), we
can obtain the whole system outage probability of the proposed hybrid HD/FD relaying scheme.
IV. S IMULATION R ESULTS
In this section, we investigate the proposed hybrid HD/FD relaying scheme by the numerical
simulations.
0

10

System outage probability Pout

−1

10

−0.0805

10

−0.0815

10
−2

5.99

6

6.01

10

Tra_Theory
Pro_Hybrid_Theory
Tra_M−C_Sim
Pro_Hybrid_M−C_Sim

−3

10

−4

10

0

5

10

15
20
25
Transmit power PR (dB)

30

35

40

Fig. 3. The comparison with regard to the transmit power PR .

In Fig. 3, we compare the traditional HD/FD relaying scheme in [13] with the proposed
HD/FD relaying scheme with the variation of the transmit power PR . The transmit power of the
source node PS is set to 30 dBm (dB in the following). The variances of RSI and the Gaussian
noises are set to 1. The minimum data rate threshold is set to 3 bps/Hz. The results show that
within the whole regime of the transmit power PR , the proposed hybrid HD/FD relaying scheme
outperforms the traditional hybrid HD/FD relaying scheme. Especially, in the high transmit
power regime of PR , the proposed hybrid HD/FD relaying scheme dramatically improves the
system outage performance compared to the traditional HD/FD relaying scheme. The theoretical

February 23, 2018

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10

0

10

−1

System outage probability Pout

10

−0.0523

10
−2

10

−0.0533

10

9.495

−3

10

9.5

9.505

−4

10

Tra_Hybrid_Theory
Pro_Hybrid_Theory
Tra_Hybrid_M−C_Sim
Pro_Hybrid_M−C_Sim

−5

10

−6

10

1

2

3

4

5

6

7

8

9

10

Minimum data threshold R0 (bps/Hz)

Fig. 4. The comparison with regard to the minimum data rate R0 .

evaluations of both the traditional and the proposed HD/FD scheme are validated by the Monte
Carlo (M-C in the figure) simulations.
In Fig. 4, we compared the system outage probability of the traditional HD/FD relaying
scheme proposed in [13] with that of the proposed HD/FD relaying scheme with the minimum
data rate threshold R0 . The transmit power of PS and PR are set to 30 dB. The simulation results
demonstrate the superiority of the proposed hybrid HD/FD relaying scheme over the traditional
HD/FD relaying scheme. Especially in the low data threshold R0 regime, the system outage
performance is improved by two or three orders of magnitude. These insights have revealed that
when all the antennas of the FD node is utilized within the HD mode, the system performance
can be greatly improved by the incurred diversity gain.
In Fig. 5, we plot the system outage probabilities of the proposed hybrid HD/FD relaying
scheme, the considered FD mode and the HD mode with the the transmit power PS . From
the simulation results we can observe that in the low transmit power regime of PS , the FD
mode outperforms the HD mode. However, with the utilizing of the two antennas, the outage
probabilities of the HD mode decrease faster than that of the FD mode with the increase of
the transmit power PS . In the high transmit power regime, the HD mode outperforms the FD
mode. Nevertheless, the proposed hybrid HD/FD relaying scheme can always achieve better
performance than the FD mode and the HD mode in the low transmit power and high transmit
power regimes of PS .

February 23, 2018

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11

0

10

System outage probability Pout

10

0

−1

10

−0.003

10

0

2

4
−0.00164

10
−2

10

−0.00166

10

3.998 4 4.002
−3

10

−4

10

0

FD_Theory
HD_Theory
Pro_Hybrid_Theory
FD_M−C_Sim
HD_M−C_Sim
Pro_Hybrid_M−C_Sim

5

10

15
20
25
Transmit power Ps (dB)

30

35

40

Fig. 5. The system outage probability of the proposed scheme vs. PS .

0

System outage probability Pout

10

FD_Theory
HD_Theory
Pro_Theory
FD_M−C_Sim
HD_M−C_Sim
Pro_M−C_Sim

−0.606

10
−1

10

−0.614

10

18 19 20
−20

−15

−10
−5
0
5
10
Residual self−interference level σ2RSI (dB)

15

20

2
Fig. 6. The system outage probability of the proposed scheme vs. σRSI
.

In Fig. 6, we plot the outage probabilities of the FD mode, the HD mode and the proposed
hybrid HD/FD relaying scheme with the RSI. The results show that the RSI would severely limit
the system outage performance of the FD mode whereas the outage probabilities of the HD mode
are irrelevant to the RSI. In addition, it is noted that when the self-interference can be suppressed
to a certain threshold, the FD mode is superior to the HD mode. However, the proposed hybrid
HD/FD relaying scheme always achieves a better performance than the FD mode and HD mode
within the whole RSI regime. This indicates that the proposed hybrid HD/FD relaying scheme
can restrict the severe adverse effects of the RSI.

February 23, 2018

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12

V. C ONCLUSION
In this paper, we proposed a new hybrid HD/FD relaying scheme, in which the two antennas
of the FD relay node could be fully utilized when the system worked in the HD mode. In
addition, we adopted the MRC/MRT technique to combine and transmit the signals to reap the
diversity gain. The simulation results showed that the proposed hybrid HD/FD relaying scheme
could dramatically improve the system outage performance compared to the traditional hybrid
HD/FD relaying scheme. Moreover, the proposed hybrid HD/FD relaying scheme could more
efficiently alleviate the adverse effects of the RSI incurred by the FD mode.
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