Alternative Awaiting and Broadcast for Two-Way Relay Fading Channels

Abstract

We investigate a two-way relay (TWR) fading channel based on
store-and-forward (SF), where two source nodes wish to exchange information
with the help of a relay node. A new upper bound on the ergodic sum-capacity
for the TWR fading system is derived when delay tends to infinity.We further
propose two alternative awaiting and broadcast (AAB) schemes: pure partial
decoding (PPD) with SF-I and combinatorial decoding (CBD) with SF-II, which
approach the new upper bound at high SNR with unbounded and bounded delay
respectively. Numerical results show that the proposed AAB schemes
significantly outperform the traditional physical layer network coding (PLNC)
methods without delay. Compared to the traditional TWR schemes without delay,
the proposed CBD with SF-II method significantly improves the maximum sum-rate
with an average delay of only some dozen seconds in the relay buffer.

Full text

Page 1

arXiv:1109.1949v1 [cs.IT] 9 Sep 2011
Alternative Awaiting and Broadcast for Two-Way
Relay Fading Channels
Jianquan Liu†, Youyun Xu†∗, Meixia Tao†,
†Dept. of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China
∗PLA University of Science and Technology, Nanjing 210007, China
Emails: {jianquanliu, xuyouyun, mxtao}@sjtu.edu.cn
Abstract—We investigate a two-way relay (TWR) fading chan-
nel based on store-and-forward (SF), where two source nodes
wish to exchange information with the help of a relay node. A new
upper bound on the ergodic sum-capacity for the TWR fading
system is derived when delay tends to infinity. We further propose
two alternative awaiting and broadcast (AAB) schemes: pure
partial decoding (PPD) with SF-I and combinatorial decoding
(CBD) with SF-II, which approach the new upper bound at
high SNR with unbounded and bounded delay respectively.
Numerical results show that the proposed AAB schemes signifi-
cantly outperform the traditional physical layer network coding
(PLNC) methods without delay. Compared to the traditional
TWR schemes without delay, the proposed CBD with SF-II
method significantly improves the maximum sum-rate with an
average delay of only some dozen seconds in the relay buffer.
Index Terms—Two-way relaying, physical layer network cod-
ing, store-and-forward, partial decoding
I. INTRODUCTION
We consider the fading version of a classic three-node two-
way relay (TWR) problem studied in [1], [2]. As shown
in Fig. 1, two source nodes, denoted as 0 and 2, wish to
exchange information with the help of a relay node, denoted
as 1. We consider a two-dimensional network layout, where
the three nodes can be located arbitrarily. The channel on
each communication link is assumed to be corrupted with
small-scale fading, shadowing, path loss and additive white
Gaussian noise (AWGN). The instantaneous signal-to-noise
ratio (SNR) from node i to node j in certain time t is denoted
as γij[t] =
σ2
j
channel gain hij[t] from node i to node j, average transmit
power Piat the node i and AWGN power σ2
Note that | · | and E{·} stand for the magnitude of a complex
scalar and the expectation operator, respectively. The ergodic
capacity¯Cij in bit/s/Hz is determined by the SNR on the
link as¯Cij= E{Cij} = E{C(γij)} = E{log2(1 + γij)}. For
simplicity, we also assume the channel gains are reciprocal
and unchanged during one round of information exchange
between two source nodes, which is defined as one time
unit or two time slots. Then, we have hij[t] = hji[t] and
γij[t] = γji[t], for any t ∈ {2k + 1,2k + 2},k ∈ N. In
this paper, we focus on two-phase TWR system with equal
time slot, which can be divided into a multiple access (MAC)
phase and a broadcast (BC) phase. We assume that all the
nodes operate in the half-duplex mode and the direct link
between two source nodes can not be used. In this paper, we let
Pi|hij[t]|2
, for i,j ∈ {0,1,2}. It counts the tth
jat the node j.
Relay
Node 1
Source
Node 0
Source
Node 2
MAC Phase
Relay
Node 1
Source
Node 0
Source
Node 2
BC Phase
Fig. 1: System model of two-way relay fading channels.
P0= P1= P2= P,σ2
letters to denote vectors and lower letters to denote elements.
For simplicity, we introduce an ergodic sum-rate to describe
the performance for TWR fading channels. An ergodic sum-
rate of¯Rs is considered as the average sum-rate over all
channel distributions, such as¯Rs= E{Rs}, where Rsdenotes
an instantaneous sum-rate. The instantaneous sum-capacity Cs
and ergodic sum-capacity¯Cs are then the supremums of Rs
and¯Rs, respectively.
Related Work. Two-way relaying has recently obtained a
lot of research interests [1]–[8]. Rankov et al. analyzed the
ergodic sum-rates of the AF, DF for the TWR half-duplex
fading channels with no direct link [1]. Due to that the authors
in [1] used the full decoding in the MAC phase and superpo-
sition in the BC phase, both the achievable ergodic sum-rates
in [1, eq. (24)] and upper bound in [1, eq. (69)] are relatively
poor. For the two-phase TWR Gaussian channels, Oechtering,
et al. obtained the capacity region of the BC phase in terms
of jointly random coding [9]. Kim, et al. further broadened
the frontier of the achievable rate region by allowing time
sharing between different transmission phases [3]. We restrict
our attention to the case of reciprocal channel with equal time
allocation, then the upper bound on sum-rate in [3] for the two-
phase TWR system is equivalent to min{C01,C21}. Similar
results are gained in [4], [5], [8]. Note that both the sum-rates
achieved in [4], [5] by lattice codes can approach the upper
bound Cu
intuitively extend these results from TWR Gaussian channels
to reciprocal fading channels. However, none of the works in
0= σ2
1= σ2
2= σ2. We use bold upper
mat high SNR for each considered rate case. We can

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the literatures has sufficiently exploited the potential benefits
of asymmetric channel gains. All the aforementioned schemes
are restricted by the poor channel.
Contribution of This Work. We first show that a new upper
bound on the ergodic sum-capacity for the TWR fading system
is given by¯Cu
the instantaneous sum-capacity, given as Cu
the MAC channel with partial decoding [3], [4], the new upper
bound¯Cu
forward (SF) protocol with infinite delay. Since for asymmetric
and symmetric rate cases the upper bound Cu
by the lattice codes at high SNR in [4] and [5] respectively,
we further propose two alternative awaiting and broadcast
(AAB) schemes for the asymmetric rate case. The achievable
ergodic sum-rates¯Rs, for both two AAB schemes: pure partial
decoding (PPD) with SF-I and combinatorial decoding (CBD)
with SF-II, approach the new upper bound¯Cu
with unbounded and bounded delay respectively.
s=1
2(¯C01+¯C21). Based on an upper bound on
m= C01+C21, for
sis derived in Section II by introducing store-and-
mare approached
sat high SNR
II. ALTERNATIVE AWAITING AND BROADCAST (AAB)
For the TWR reciprocal fading channels the side of higher
upload rate suffers from the poor channel because of imme-
diate forwarding at the relay node. We introduce a protocol:
store-and-forward (SF), which has the potential to improve the
exchange rate by trading the delay.
A. Upper Bound with Delay
Let Ru
instantaneous capacity of one side i → j during the tth
exchanging — from the (2k+1)thtime slot to the (2k+2l)th
time slot. Therein, l denotes delay and l ∈ Z+. According to
the results derived in [3]–[5], we obtain
ijd[t],i,j ∈ {0,2}, denote the upper bound on
Ru
02d[t]=
1
2min
1
2min
?
?
C01[2k + 1],C12[2k + 2l]
?
?
,
(1)
Ru
20d[t]=
C21[2k + 1],C10[2k + 2l].
(2)
Obviously, the Eqs. (1)-(2) can avoid the performance loss
compared with the transmission rates of two independent
point-to-point links when C01[2k + 1] = C12[2k + 2l] and
C21[2k+1] = C10[2k+2l] even if C01[2k+1] ?= C21[2k+1].
Intuitively, we attain a new upper bound on the ergodic sum-
capacity as following
¯Cu
s=1
2
?¯C01+¯C21
?
.
(3)
Note that ¯Cu
and E{|h10|2} ≥ E{|h21|2} along with an average delay
L = E{l} → +∞ in view of the upper bound on instantaneous
sum-capacity Cu
scan be achieved if E{|h12|2} ≥ E{|h01|2}
min the MAC phase is achievable.
B. Pure partial Decoding (PPD) with SF-I
In this subsection, we propose an AAB scheme which
is named as PPD with SF-I. The partial decoding and two
distinct lattice codes are used in the MAC phase and a new
transmission protocol SF-I is proposed for broadcast.
1) Descriptions of Process Flow: For TWR Gaussian chan-
nels with lattice codes, more details can be seen in [4].
Considering the fading channels, we treat variations of the
channel gains |hij[2k + 1]|2as fluctuations of the power
Pij[2k + 1]. Different from [4], we use a SF protocol. The
received information flow at the relay is transmitted during
certain time slot after the current time slot in SF-I. The order
of the broadcasted information flows is determined by the
matched channel gain pairs between the MAC phase and the
BC phase, no matter which one is received firstly.
2) Derivations of achievable rate: For an asymmetrical
Gaussian TWR channels, Nam, et al. have achieved a rate
pair (R02,R20) in [4] given as
R02≤1
2min
??
log2(
P0
P0+ P2
P2
P0+ P2
+P0
σ2
1
)
?+
?+
,log2(1 +P1
σ2
2
)
?
,
(4)
R20≤1
2min
??
log2(+P2
σ2
1
),log2(1 +P1
σ2
0
)
?
. (5)
Similar to analysis of the new upper bound¯Cu
is also introduced. Let P0 = P|h01|2, P2 = P|h21|2,
P|h12|2
σ2
,
σ2
σ2
and σ2
from the Gaussian case to fading channels
s, the delay l
P1
σ2
2=
P1
0=
P|h10|2
1= σ2, we extend Eqs.(4)-(5)
R02d[t]
≤
1
2min
log2(1 + γ12[2k + 2l])
1
2min
log2(1 + γ10[2k + 2l])
??
log2(Ψ[2k + 1] + γ01[2k + 1])
?+
,
?
,
(6)
R20d[t]
≤
??
log2(1 − Ψ[2k + 1] + γ21[2k + 1])
?+
,
?
,
(7)
where Ψ[2k + 1] =
{i,j} ∈ {0,1,2}, t ∈ [2k + 1,2k + 2l],k ∈ N.
In general, we achieve an ergodic sum-rate¯RPPD
|h01[2k+1]|2
|h01[2k+1]|2+|h21[2k+1]|2, γij =
P|hij|2
σ2
,
s
given by
¯RPPD
s
≤1
2E
??
log2(Ψ + γ01)
?+
+
?
log2(1 − Ψ + γ21)
?+?
.
(8)
However, the average delay L = E{l} still tends to infinity.
C. Combinatorial Decoding (CBD) with SF-II
As depicted in Fig. 2, we present another AAB scheme
which is named as CBD with SF-II. Therein, both the succes-
sive decoding and the partial decoding are considered and a
novel transmission protocol SF-II is also proposed.
1) Descriptions of Process Flow: Firstly, we consider the
(2k + 1)thtime slot — a MAC phase. Without loss of
generality, we assume that |h01[2k + 1]|2≥ |h21[2k + 1]|2,
namely R01[2k + 1] ≥ R21[2k + 1]. The source node 0
splits the message S0[2k + 1] into two parts: S1
S2
0[2k + 1]. Therein, the length of one part, e.g. S1
is equal to that of the message S2[2k +1] which is wanted to
transmit from the source node 2. S1
are then encoded to C1
code L and a Gaussian code G respectively. After operating
X1
0[2k + 1] = (C1
modulating C2
0[2k + 1] and
0[2k + 1],
0[2k + 1] and S2
0[2k + 1] by a lattice
0[2k + 1]
0[2k + 1] and C2
0[2k + 1] + D0[2k + 1]) mod?nand
0[2k+1] to X2
0[2k+1], the source node 0 forms

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t
SF-II
01
h
21
h
+
2 S
2
C
2
X
1Y
L
L
0
D
2
D
()th
k12
+
Time Slot
1
3
5
2 +
k
2 +
k
2 +
k
12 −
k
32 −
k
52 −
k
2
4
6
2 +
k
2 +
k
2 +
k
k2
22 −
k
42 −
k
()th
k22
+
Time Slot
0
S
1
0
S
2
0 S
1
0
C
2
0
C
1
0
X
2
0
X
0
X
θ
θ
−
1
G
12
h
10
h
1
X
L
0
D
0 Y
2 Y
1
0 S
1
0
C
L
2
D
G
2
C
2 S
() l2kS12
2
2
−+
() l2kC12
2
2
−+
Fig. 2: CBD with SF-II.
the transmitted signal X0[2k+1] =?θ[2k + 1]X1
?1 − θ[2k + 1]X2
cation coefficient θ[2k+1]. At the same time, the source node
2 generates the transmitted signal X2[2k+1] through mapping
the message S2[2k + 1] to C2[2k + 1] by an identical lattice
code L and operating X2[2k+1] = (C2[2k+1]+D2[2k+1])
mod
?n. Note that the random dither vectors D0[2k+1] and
D2[2k + 1] are mutually independent of each other and are
also known at both the relay node and two source nodes.
At the relay, the received superimposed signal is given as
0[2k+1]+
0[2k+1] by superposition and a power allo-
Y1[2k + 1] = h01[2k + 1]X0[2k + 1]
+h21[2k + 1]X2[2k + 1] + Z1[2k + 1]
??
+
?
+h21[2k + 1]X2[2k + 1] + Z1[2k + 1]
= T[2k + 1] +
?
+Z1[2k + 1].
(9)
= h01[2k + 1]
θ[2k + 1]X1
0[2k + 1]
1 − θ[2k + 1]X2
0[2k + 1]
?
(10)
1 − θ[2k + 1]h01[2k + 1]X2
0[2k + 1]
(11)
The relay first decodes S2
a function T[2k + 1] as noise1, subtracts X2
received signals, and then decodes T[2k + 1]. Obviously, we
should set θ[2k+1] =|h21[2k+1]|2
to satisfy that two lattice coded signals have the same received
SNR, e.g. γ1
D1[2k + 1]) mod
?nto form X1
dither vector D1[2k +1]. Due to the reciprocity between two
relay channels, we have |h12[2k + 2]|2≤ |h10[2k + 2]|2,
namely R12[2k+2] ≤ R10[2k+2]. Therefore, the relay stores
S2
0[2k + 1] and awaits a special condition of channel gain
|h12[2k+2]|2≥ |h10[2k+2]|2for broadcast. At the same time,
a fractional message, e.g. S2[2k +1−2l], for l ∈ Z+, which
has been received and stored in the former (2k + 1 − 2l)th
time slot at the relay, may be picked up and encoded to form
C2[2k+1] by a Gaussian code. C2[2k+1] will be modulated
to X2
the transmitted signal X1[2k+1] =
0[2k +1] (X2
0[2k +1]) by treating
0[2k + 1] off its
|h01[2k+1]|2, θ[2k+1] ∈ [0,1], in order
01= γ21. Then, the relay operates (T[2k + 1] +
1[2k + 1] by a random
1[2k+1] and superimposed with X1
1[2k+1] for generating
?η[2k + 2]X1
1[2k+1]+
1Ong, et al. have discussed another two encoding/decoding schemes of
the TWR Gaussian model with no delay [10]. However, both two schemes
decrease the transmission rate of the side of inferior channel gain comparing
with the scheme of identical transmission rate for two source nodes.
?1 − η[2k + 2]X2
allocation coefficient with η[2k + 2] ∈ [0,1].
In SF-II, the received Gaussian coded message at the relay
is transmitted during certain time slot after the current time
slot while the lattice coded message T[2k + 1] is transmitted
immediately in the next (2k + 2)thtime slot. Note that
the storage and extraction of each received Gaussian coded
message obey the rule of First-In First-Out (FIFO).
In the (2k+2)thtime slot — the immediate BC phase, the
superimposed signal X1[2k +1] is broadcasted to two source
nodes by the relay node. At two source nodes, the received
signals are given as
1[2k + 1]. Here, η[2k + 2] is also a power
Yi[2k + 2] = h1i[2k + 2]X1[2k + 1] + Zi[2k + 2], (12)
??
+
?
where i ∈ {0,2}. Here we have |h12[2k+2]|2≤ |h10[2k+2]|2
when we set |h10[2k + 2]|2= |h01[2k + 1]|2and |h12[2k +
2]|2= |h21[2k+1]|2. As knowing S2[2k+1−2l], the source
node 2 first subtracts X2
then decodes the lattice coded message S1
a lattice code book
?
source node 0 first decodes2S2[2k +1−2l] (X2
treating X1
received signal, and then decodes the lattice coded message
S2[2k + 1] by using a lattice code book
2) Derivations of achievable rate: For a symmetrical Gaus-
sian TWR channels, Wilson, et al. have achieved an identical
transmission rate for two source nodes in [5] given as
= h1i[2k + 2]
η[2k + 2]X1
1[2k + 1]
1 − η[2k + 2]X2
1[2k + 1]
?
+ Zi[2k + 2],
(13)
1[2k + 1] off its received signal and
0[2k + 1] by using
T,C1
. At the same time, the
0∈?n?
1[2k +1]) by
1[2k + 1] off its
1[2k + 1] as noise. Subtracting X2
?
T,C2∈?n?
.
R02
≤
1
2min
1
2min
??
log2(1
2+P0
2+P2
σ2
1
)
?+
?+
,log2(1 +P1
σ2
2
)
?
,(14)
R20
≤
??
log2(1
σ2
1
)
,log2(1 +P1
σ2
0
)
?
,(15)
where P0= P2= P1= P,σ2
P|h01|2, P2 = P|h21|2,
0= σ2
=
2= σ2
P|h12|2
σ2
1= σ2. Let P0=
P1
σ2
0
=
P1
σ2
2
,
P|h10|2
σ2
and
2It is also possible to decode the lattice codeword first by treating the
Gaussian codeword as noise. However, this scheme doesn’t adequately utilize
a performance improvement of1
phase and that achieved in the BC phase for the latticed coded TWR channels.
2bit between the rate achieved in the MAC

Page 4

σ2
fading channels
1= σ2, we extend Eqs.(14)-(15) from the Gaussian case to
R02[t]
≤
1
2min
log2(1 +P|h12[2k + 2]|2
??
log2(1
2+P|h01[2k + 1]|2
σ2
?
)
?+
,
σ2
2+P|h21[2k + 1]|2
),
(16)
R20[t]
≤
1
2min
log2(1 +P|h10[2k + 2]|2
??
log2(1
σ2
?
)
?+
,
σ2
),
(17)
where |h01[2k + 1]|2= |h21[2k + 1]|2= |h01[2k + 1]|2=
|h21[2k+1]|2. However, h01[2k+1] is not equal to h21[2k+1]
generally. According to the descriptions in Subsection II-C-
1), we obtain an instantaneous rate pair (R01[t],R21[t]) in the
MAC phase given by
R01[t] ≤1
2
??
log2(1
2+P|h21[2k + 1]|2
σ2
)
?+
+log2(1 +P(|h01[2k + 1]|2− |h21[2k + 1]|2)
σ2+ 2P|h21[2k + 1]|2
R21[t] ≤1
2
and rate pair (R10[t],R12[t]) in the BC phase as
)
?
, (18)
?
log2(1
2+P|h21[2k + 1]|2
σ2
)
?+
,
(19)
R12[t] ≤1
2log2(1 +η[2k+2]P|h12[2k+2]|2
η[2k+2]P|h10[2k+2]|2
σ2+(1−η[2k+2])P|h10[2k+2]|2)
+log2(1 +(1−η[2k+2])P|h10[2k+2]|2
σ2
),
(20)
R10[t] ≤1
2
?
log2(1 +
σ2
)
?
.
(21)
In order to satisfy that we should not decrease the transmis-
sion rate of the side of inferior channel gain, we need
1
2log2
≥1
2log2
1
2log2
≥1
2log2
?
1 +η[2k + 2]P|h12[2k + 2]|2
σ2
?
?1
2+P|h21[2k + 1]|2
η[2k + 2]P|h10[2k + 2]|2
σ2+ (1 − η[2k + 2])P|h10[2k + 2]|2
?1
σ2
Combining the former two inequations, we obtain
σ2
?
,
(22)
?
1 +
?
2+P|h21[2k + 1]|2
?
.
(23)
η[2k + 2] ≥





















1 −
σ2
2P|h21[2k+1]|2,
2,P|h21[2k+1]|2
if
0 <
(2P|h21[2k+1]|2−σ2)(P|h10[2k+2]|2+σ2)
P|h10[2k+2]|2(σ2+2P|h21[2k+1]|2)
1
2<|h21[2k+1]|2
0,
if
0 ≤P|h21[2k+1]|2
|h21[2k+1]|2
|h10[2k+2]|2 ≤1
σ2
≥1
2,
,
if
|h10[2k+2]|2 ≤ 1,P|h21[2k+1]|2
σ2
≥1
2,
σ2
<1
2.
(24)
For maximizing the additional broadcast rate from the relay
to source node 0, we should select the minimum of η[2k +2]
for three considered conditions.
Without loss of generality, we assume that R01[t] − R21[t]
can be successfully broadcasted during 2l1[2k + 1] time slots
— from the (2k + 2)thto the (2k + 2l1[2k + 1])thtime slot.
Let Γ[2k + 1] = 1 +P(|h01[2k+1]|2−|h21[2k+1]|2)
σ2+2P|h21[2k+1]|2
?
σ2
and ∆
?
2k +
2l(k)= 1 +(1−η[2k+2l])P|h12[2k+2l]|2
, we have
log2
?
Γ[2k + 1]
?
≤
l1[2k+1]
?
l(k)=l0[2k+1]+1
φ
?
l(k)
?
log2
?
∆
?
2k + 2l(k)
??
+
l1[2k0+1]
?
l(k0)=l0[2k0+1]+1
?
Γ[2k0+ 1]
φ
?
l(k0)
?
log2
?
∆
?
2k0+ 2l(k0)
??
−log2
?
,
(25)
where 2k0+1 denotes the time slot in which adjacent former
information R01[t0]−R21[t0], namely log2
been generated. Moreover, l0[2k+1] = l1[2k0+1]−(k−k0)
denotes the extra time units should be used from the current
time slot 2k+1 in order to successfully broadcast the adjacent
former information log2
Γ[2k0 + 1]
satisfies φ(l(k)) = 1 only if |h01[2k+2l]|2≤ |h21[2k +2l]|2,
for l(k) ∈
?
l0[2k + 1],l1[2k + 1]
Then we obtain an instantaneous delay l1[2k + 1] given as
?
Γ[2k0+1]
?
, has
??
. Note that φ(l(k))
?
, else φ(l(k)) = 0.
l1[2k + 1]
=min
l1[2k+1]∈Z+
?
l1[2k+1]
?
l(k)=l0[2k+1]+1
?
∆
?
2k + 2l(k)
??φ(l(k))
×
l1[2k0+1]
?
l(k0)=l0[2k0+1]+1
?
∆
?
2k0+ 2l(k0)
??φ(l(k0))
≥ Γ[2k + 1] × Γ[2k0+ 1]
?
.
(26)
Note that the unit of delay studied in this paper is time unit
— one time unit equals two time slots. Then, we obtain the
corresponding achievable rate pair (R02d[t],R20d[t]) given by
R02d[t] ≤ max
?
?
R01[t],R12[t]
?
?
,
(27)
R20d[t] ≤ min
R21[t],R10[t],
(28)
with an average delay L = E{l} = E{l1}.
In general, we achieve an ergodic sum-rate¯RCBD
s
given by
¯RCBD
s
≤ E
??
log2
?1
P||h01|2− |h21|2|
σ2+ 2P min{|h01|2,|h21|2}
III. NUMERICAL RESULTS
In this section, we present some numerical results to illus-
trate performance. We assume that the distance between two
source nodes 0 and 2 is normalized to 1 and the location of
the relay is determined using the projections x and y. The
source nodes 0 and 2 are located at the coordinates (-0.5,0)
and (0.5,0), respectively. We set {x,y} ∼ U[−0.5,0.5], where
2+P min{|h01|2,|h21|2}
σ2
??+
+1
2log2
?
1 +
??
.
(29)

Page 5

−10−7.5−5−2.502.557.510 12.515
0
1
2
3
4
5
6
7
8
P/σ2(dB)
Maximum Sum−Rate(b/s/Hz)
{x,y} ∼ U[−0.5,0.5]


Upper Bound with Delay
PPD with SF−I
CBD with SF−II
Upper Bound without Delay
Lattice with DNF
Fig. 3: Ergodic sum-rates versus P/σ2.
−10 −7.5−5−2.502.55 7.5 1012.51517.520
10
20
30
40
50
60
70
80
P/σ2(dB)
Average Delay(s/Hz)
{x,y} ∼ U[−0.5,0.5]


|h01|2 ≥ |h21|2
|h01|2 ≤ |h21|2
Fig. 4: Average delay versus P/σ2for
proposed CBD with SF-II.
0.010.0350.060.0850.110.135
ρ (b/s/Hz)
0.160.1850.210.235 0.26
10
1
10
2
10
3
10
4
P/σ2=10dB,{x,y}∼U[−0.5,0.5]
Delay of Bit Service(s/Hz)


Proposed CBD with SF−II, R02
Proposed CBD with SF−II, R20
CBD with SF−II, Relay Buffer, R02
CBD with SF−II, Relay Buffer, R20
Upper Bound without Delay, R02
Upper Bound without Delay, R20
Lattice with DNF, R02
Lattice with DNF, R20
45s/Hz
Fig. 5: System service delay versus packet
arrival rate.
U denotes Uniform distribution. Suppose that the channel gain
hij, for {i,j} ∈ {0,1,2}, is modeled by hij = αij· d−β/2
where β is the path loss exponent and fixed at 3, αijand dij
denote the Rayleigh fading with i.i.d αij∼ CN(0,1) and the
distance between node i and j, respectively. For comparison,
the performance of the upper bound without delay and lattice
with DNF derived in [3]–[5] are also shown in the figures.
Fig. 3 shows the ergodic maximum sum-Rates (MSRs) of
different TWR strategies, when P/σ2increases from −10dB
to 15dB. No matter what P/σ2is, the proposed upper bound
with delay, PPD with SF-I and CBD with SF-II always obtain
the larger MSRs than the upper bound without delay and
the lattice with DNF methods. Moreover, both the lattice
with DNF and PPD with SF-I approach the corresponding
upper bounds very well, especially in the high P/σ2regime.
Although the CBD with SF-II yet doesn’t approach the upper
bound with delay even if P/σ2is 15dB, the gaps between
them are decreased slowly along with increasing P/σ2. At
the same time, all the gaps between the considered methods
with delay and methods without delay are enlarged because
of the influence of inferior channel gains.
Fig. 4 illustrates the variations of the average delay L versus
that of P/σ2. Interestingly, it can be seen that L is increased
slowly along with increasing P/σ2. For all considered P/σ2,
the average delay L are less than 75 s/Hz even if P/σ2is
set as 20dB. This figure confirms that the proposed CBD with
SF-II is effective and practical. Note that if P/σ2= 10dB, we
have L ≈ 45 s/Hz which will be used in the next subsection.
Fig. 5 shows that the delay of bit service with the length
of each packet is fixed as Ω = 10 bits. Suppose that both two
source nodes have buffers for all considered TWR strategies
while the relay has buffer only for the proposed CBD with
SF-II. We assume that the packet arrival rate at two source
nodes follows Poisson distribution with mean ρ. The proposed
CBD with SF-II always outperforms the other two TWR
methods significantly. Both the bit delay for two transmission
directions at the relay are less than the bit delay at two source
nodes for considered CBD with SF-II and stop increasing
and maintain at approximately 45 s/Hz when ρ ≥ 0.235.
However, in the same regime of ρ, the bit delay of two source
nodes for proposed CBD with SF-II is exponentially increased.
This figure confirms again that the average delay analyzed in
ij
,
Section V is correct and negligible.
IV. CONCLUSION
In this research, we considered joint network coding and
SF for TWR fading channels. Under the model of the MAC
and BC phases using equal time allocation and reciprocal
channel gains, we presented a new upper bound on the ergodic
sum-capacity which is no longer limited by the poor channel
compared to the traditional upper bound without delay. We
further proposed two AAB schemes: PPD with SF-I and CBD
with SF-II, which approach the new upper bound at high SNR
with unbounded and bounded delay respectively. Particularly,
we derived an average delay of the proposed CBD with SF-II
method given by some dozen seconds. It is very small and
negligible. Numerical results confirm that the proposed AAB
schemes outperform the traditional PLNC methods without
delay significantly and approach the new upper bound with
delay at asymptotically large SNRs for TWR fading channels.
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