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IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 5, MAY 2015 783
Outage Probability of Full-Duplex AF Relaying With
Processing Delay and Residual Self-Interference
Qiang Wang, Member, IEEE, Yue Dong, Xiaodong Xu, Member, IEEE, and Xiaofeng Tao, Senior Member, IEEE
Abstract—This letter investigates the outage performance in
a full-duplex relay (FDR) channel that adopts an amplify-
and-forward (AF) protocol. We provide a new closed-form ex-
pression for the outage probability that captures the joint effect
of residual self-interference (RSI) and a direct link. In addition,
when the processing delay of the relay is larger than 1, which is
often the case in practice, the proposed closed-form outage perfor-
mance expression still applies. Finally, all the theoretical results
are validated through numerical simulations. The results indicate
that AF-FDR can outperform selective decode-and-forward FDR
(SDF-FDR) in the low-to-intermediate-rate regions because of less
processing delay.
Index Terms—Full-duplex relay, amplify-and-forward, pro-
cessing delay, self-interference, cooperative diversity, outage
probability.
I. INTRODUCTION
RECENTLY, full-duplex relaying (FDR) has attracted con-
siderable attention as a means to improve the spectral
efficiency of relay networks [1]–[3]. However, the main lim-
itation of full-duplex operation is loopback self-interference
due to signal leakage between the transmission and reception
at the relay. In practice, self-interference cannot be completely
mitigated, even with multiple stages of cancellation [4]–[6].
Several residual self-interference (RSI) models have been
adopted for theoretical studies. References [7]–[11] assumed
that the RSI is caused by the self-interfering channel estimation
error and that the variance in the RSI is proportional to the av-
erage transmitted power. However, recent experimental results
in [12] suggested that the RSI is additive and noise-like and that
the variance of the RSI is proportional to the b-th power of the
transmitted power, where bis often less than one. The authors
in [13] investigated the performance of an amplify-and-forward
(AF) full-duplex relay using this RSI model with no direct link.
When a non-negligible direct link is present, coopera-
tive diversity is available via combining [10], [11]. In [10],
Manuscript received October 1, 2014; revised February 25, 2015; accepted
March 3, 2015. Date of publication March 9, 2015; date of current version
May 7, 2015. This work was supported in part by the National Natural Science
Foundation of China under Grant (61302082, 61325006) and in part by the
National High-tech Research and Development Program of China under Grant
2014AA01A701. The associate editor coordinating the review of this paper and
approving it for publication was D. Michalopoulos.
Q. Wang, X. Xu, and X. Tao are with the National Engineering Laboratory
for Mobile Network Security, School of Information and Communication Engi-
neering, Beijing University of Posts and Telecommunications, Beijing 100876,
China (e-mail: wangq@bupt.edu.cn; xuxiaodong@bupt.edu.cn; taoxf@bupt.
edu.cn).
Y. Dong is with the State Key Laboratory of Networking and Switching
Technology, School of Information and Communication Engineering, Beijing
University of Posts and Telecommunications, Beijing 100876, China (e-mail:
dongyue@bupt.edu.cn).
Digital Object Identifier 10.1109/LCOMM.2015.2411596
by adopting a selective decode-and-forward (SDF) relay, the
authors developed an approximate, yet accurate, closed-form
expression for the end-to-end outage probability that captures
the joint effects of RSI and a direct link. In SDF, the relay node
assists only when it is able to decode the source message. The
work in [11] focused on the same AF relaying that we target,
and the closed-form outage performance was analyzed only
for two extreme cases: either the RSI or the direct link was
considered. However, to the best of our knowledge, the joint
effect of non-ideal cancellation and the presence of a direct link
on AF relaying performance under the more realistic RSI model
remains unclear.
In this work, we investigate the outage performance of AF-
FDR under a more realistic RSI model, and we focus on the
scenarios in which a non-negligible direct link is present and
in which the source-relay link is fading-free (to be elaborated
in subsequent sections). Due to the difference in the processing
capabilities of the relays, the processing delay may not always be
the same. Taking the processing delay of the relay into account,
we consider the communication over a block of symbols; thus,
the received vector at the destination over one block is a super-
position of shifted versions of the source vector arriving through
different paths. In particular, we derive a novel expression
for the end-to-end equivalent signal-to-interference and noise
ratio (SINR) in which the processing delay of the relay can
be larger than 1. Based on this, an approximate, yet accurate,
closed-form expression for the outage probability is presented.
Finally, the simulation results demonstrate that the AF-FDR can
provide superior outage performance compared to SDF-FDR if
the decreased processing delay in AF relaying is considered.
Note that different forwarding strategies at the relay will cause
a different processing delay, and AF is the simplest scheme and
clearly has a considerably smaller processing delay than DF.
II. SYSTEM MODEL
We consider an infrastructure relay system that transfers
information from a base station to a randomly located mobile
station with the help of a fixed-location relay station, as shown
in Fig. 1. The relay employs an AF protocol and works in
full-duplex mode. At time t, source Scontinuously transmits
its information symbol x[t]to relay Rand destination D.
Simultaneously, due to the processing delay τ,Ramplifies and
forwards the signal received in the previous time (t−τ)using
an amplification coefficient G.
Furthermore, we assume that an imperfect interference can-
cellation scheme is used at the relay. In this study, we assume
that the RSI is Gaussian and equal to v[t]∼CN(0,V)to be
specified later [12], which can also be considered the worst-
case scenario [14].
1558-2558 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
784 IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 5, MAY 2015
Fig. 1. System model.
We can now write the signal received at Rand Dat time tas
yr[t]=Pshsrx[t]+v[t]+nr[t],(1)
yd[t]=Pshsdx[t]+Prhrdxr[t]+nd[t],(2)
where hij is the channel gain between node i∈{s, r}and node
j∈{r, d},nj[t]is the additive white Gaussian noise (AWGN)
at node j∈{r, d}denoted by nj[t]∼CN(0,N
j),x[t]is the
signal transmitted by S, and xr[t]is the signal transmitted by R
at time t, which can be written as
xr[t]=Gyr[t−τ].(3)
Let E{|x[t]|2}=q1;Sthen expends an average power of q1Ps.
To maintain a power constraint of q2Prat R,E{|xr[t]|2}=
E{|Gyr[t−τ]|2}=q2. Therefore, the amplification coeffi-
cient Gcan be written as
G=q2/[q1Ps|hsr|2+V+Nr](4)
where the variance of RSI Vis modeled from [12] as V=
a(q2Pr)b, where aand b(0 ≤b≤1) are constants that depend
on the cancellation technique. The values of aand bcan be
found in [12], e.g., based on the experimental results, b=0.21
for analog cancellation. By antenna separation between the
transmitting antenna and the receiving antenna on the relay, we
can get smaller values of aand b.
By substituting (1) and (3) into (2), the received signal at the
destination can be rewritten as
yd[t]= Pshsdx[t]+PrPshrdhsrGx[t−τ]
+PrhrdG(v[t−τ]+nr[t−τ]) + nd[t].(5)
All channel gains remain constant over a block duration
of L+τtime slots corresponding to Lsuccessive codewords
transmitted from the source, in addition to the τtime slots delay
due to relay processing. Hence, we rewrite (5) in vector form to
jointly account for the L+τreceived signals as
yd=Hx +Q(nr+v)+nd,(6)
where yd=(yd[1],...,y
d[L+τ])T,x=(x[1],...,x[L])T,
nr=(nr[1],...,n
r[L])T,v=(v[1],...,v[L])T,nd=(nd[1],
...,n
d[L+τ])T, and
H=PshsdIL
0τ×L+PrPshrdhsr G0τ×L
IL,(7)
Q=PrhrdG0τ×L
IL.(8)
To calculate the outage probabilities, we adopt the same
channel models presented in [7], in which we can reasonably
approximate that S−Ris non-fading. For this reason, Sand
Rare fixed nodes in this model. Furthermore, S−Dand
R−Dlinks exhibit Rayleigh block fading. Thus, hsd and hrd
remain constant over one block and vary independently from
one block to another following a circularly symmetric complex
Gaussian distribution with a zero mean and variance πsd and
πrd, respectively.
III. OUTAGE PERFORMANCE OF AF-FDR
A. Instantaneous End-to-End Capacity
The mutual information per block between xand ydin (6),
which is achieved using i.i.d. complex Gaussian inputs, can be
expressed as
I(x;yd)=logdetIL+τ+q1HHHNdIL+τ
+(Nr+V)QQH−1.(9)
Noting that τis considerably smaller than L, we assume that
the destination can also observe the noise transmitted from the
relay in the L+τtime for mathematical tractability. Therefore,
Qis replaced by √PrhrdGIL+τ, and we substitute it into (9) as
I(x;yd)≈log det IL+τ+q1HHHNdIL+τ
+(Nr+V)Pr|hrd |2G2IL+τ−1
=logdetIL+1
NHHH,(10)
where N=Nd+Pr|hrd|2G2(Nr+V)
q1and HHH=αIL+
βBτ
L+β∗Fτ
L, with α=Ps|hsd|2+PsPr|hsr |2|hrd|2G2and
β=Ps√PrGhsdh∗
srh∗
rd.BL(FL)denotes a square backward
(forward) shift matrix of size L, with values of ones only on
the first subdiagonal (superdiagonal) and zeros elsewhere.
By exploiting [10, Eqs. 13, 14, and 15] and modifying the
signal models thereof, the average mutual information for AF
transmission can be approximated as
IAF =I(x;yd)
L+τ≈L
L+τlog 1+ α
N
=L
L+τlog(1 + γ),(11)
for the case where L=mτ, m ∈Z+. Here, γ=α
Ndenotes
the instantaneous end-to-end equivalent SINR at D.By
substituting (4) into αand N,γcan be further expressed as
γ=γsrγrd +γsd γsr +γsd
1+γsr +γrd
,(12)
where γsd =q1Ps
Nd|hsd|2,γrd =q2Pr
Nd|hrd|2and γsr =
q1Ps
Nr+V|hsr|2denote the instantaneous SINRs for the S−D,
R−Dand S−Rlinks, respectively. Note that γsr is a
constant because the S−Rlink is assumed to be fading-free
in this work.
WANG et al.: OUTAGE PROBABILITY OF FULL-DUPLEX AF RELAYING WITH PROCESSING DELAY AND RSI 785
B. Outage Analysis
The outage probability of the AF-FDR scheme can be
written as
Pout =PL
L+τlog(1 + γ)<r
=Fγ(T),(13)
where ris the target rate, TΔ
=2
r(1+ τ
L)−1, and Fγ(·)is the
cumulative distribution function (cdf) of γ. The cdf of γcan be
obtained as
Fγ(x)=
∞
0
Pγsry+γsr γsd +γsd
1+γsr +y<x
fγrd (y)dy
=
∞
0
Fγsd x+(x−γsr)y
1+γsr fγrd (y)dy, (14)
where fγrd (·)is the probability density function (pdf) of γrd and
Fγsd (·)is the cdf of γsd. Because hsd and hrd are zero-mean
complex Gaussian variables, fγrd(·)and Fγsd (·)are written as
fγrd (y)=λrde−λrdy,(15)
Fγsd (x)=1−e−λsd x,x≥0
0,x<0,(16)
where λsd
Δ
=1/(q1Ps
Ndπsd)and λrd
Δ
=1/(q2Pr
Ndπrd).
By substituting (15) and (16) into (14), the outage probability
is obtained as
Pout =Fγ(T)=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
1−λsd (T−γsr)
1+γsr
λrd+λsd (T−γsr )
1+γsr
e−λrd (1+γsr)T
γsr−T
−λrd
λrd+λsd (T−γsr )
1+γsr
e−λsdT,T<γ
sr
1−λrd
λrd+λsd (T−γsr )
1+γsr
e−λsdT,Tγsr ,
(17)
where γsr =q1Ps
Nr+V|hsr|2and V=a(q2Pr)b. Clearly, aand b
can influence the value of the parameter γsr, and thus, they have
an effect on the outage probability.
C. Power Allocation With a Sum Constraint
From a system perspective, imposing a constraint on the
total transmit power of the system may be reasonable, as is
occasionally assumed in the related literature. In the global
constraint scenario with Ps=Pr=Ptand q1+q2=1,the
optimization problem is formulated as
Pout =argmin
q1
Fγ(T)s.t. q1+q2=1,0q11.(18)
We conjecture that the objective function is quasi-convex in
0q11based on the observed quasi-convex shape of the
function when evaluated numerically for numerous combina-
tions of system parameters. Therefore, the bisection method is
used to identify the optimal power allocation.
IV. N UMERICAL RESULTS
In this section, we provide numerical examples for the outage
probability performance of the AF-FDR scheme. We compare
the AF-FDR scheme with two existing schemes in the literature:
1) two-hop AF, in which the direct source transmissions are
Fig. 2. Outage probability vs. πsd (L=20,τAF =τSDF =2,r=1 bit/s/Hz,
Pt=20dB, b=0.1,a=1).
Fig. 3. Outage probability vs. Ptfor different values of band a(L=20,
τAF =τSDF =2,r=1bit/s/Hz, πsd =−5dB).
treated as interference at the destination [7], and 2) SDF-FDR,
which was proposed in [10] and in which the relay node assists
only when it is able to decode the source message. Without any
loss of generality, it is assumed that Nd=Nr=1,πsr =1and
that |hsr|2=1for all simulations.
Figs. 2 and 3 compare the outage probabilities obtained from
numerical simulations and theoretical analyses for different
values of direct link variance and total power. The results from
the theoretical analysis correspond well with the simulation
results, thereby validating the accuracy of our derivations. The
disparity primarily arises from the approximation in (10).
First, we plot the outage performance over a range of direct
link variances. As shown in Fig. 2, the AF-FDR and SDF-FDR
schemes yield approximately the same performance over a wide
range of πsd, and the performance gap grows between two-
hop and AF-FDR with increasing πsd. The basic reason for
this trend is that as the direct link strength increases, the direct
signal becomes non-negligible at the destination; therefore,
simply treating it as interference is no longer applicable.
786 IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 5, MAY 2015
Fig. 4. Outage probability vs. rfor different values of band τSDF (L=30,
πsd =−5dB, Pt=30dB, a=1).
Next, we investigate the outage performance against Ptfor
afixedπsd =−5dB and different values of aand b.As
shown in Fig. 3, the outage probability of AF-FDR decreases
with the total power more rapidly than with two-hop due to
the additional cooperative diversity. Alternatively, the AF-FDR
scheme is more sensitive to the values of aand bcompared
with the SDF-FDR scheme. This result occurs because in the
AF-FDR scheme, the relay amplifies and forwards not only
the received signal but also the RSI. Notably, only a small
performance gap exists between AF-FDR and SDF-FDR in
low-RSI power regions (aand bare small).
Finally, we plot the outage probability over a range of target
rates and investigate the effects of different processing delays
caused by the AF and DF protocols. Because AF is the most
practical and simplest scheme, the processing delay is hypoth-
esized to be smaller than that under DF. As shown in Fig. 4,
the SDF-FDR scheme with τSDF =10and b=0.7provides
the worst outage performance, indicating that AF-FDR can
outperform SDF-FDR when the processing delay caused by
SDF is considerably greater than that under AF. However,
the performance is similar in the high-RSI power region (b=
0.7) if only a small processing delay gap exists between AF-
FDR and SDF-FDR (τSDF =3 and τAF =1). In the low-
to-intermediate-rate regions, AF-FDR provides better outage
performance than SDF-FDR with τSDF =3. In addition, the
benefit provided by a smaller processing delay will gradually
become negligible as the rate increases (when b=0.7).
V. C ONCLUSION
We analyzed the performance of an infrastructure relay sys-
tem with a non-negligible direct link. Using a more realistic RSI
model based on recent experimental results, we presented an
approximate outage performance expression for AF full-duplex
relaying that captures the joint effects of the RSI and coopera-
tive diversity. In addition, the proposed closed-form expression
for the outage probability also applies to the cases in which the
relay’s processing delay τis larger than 1. The numerical results
demonstrated that AF-FDR and SDF-FDR yield approximately
the same performance in lower-RSI power regions. In partic-
ular, AF-FDR can outperform SDF-FDR if we consider the
different processing delays caused by the AF and DF because
AF has a considerably smaller processing delay than DF.
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