We propose a new timedivision halfduplex estimateandforward (EF) relaying strategy suitable for a relay with mobility. We reconfigure EF relaying to guarantee a strong relaydestination link which is required to achieve a high rate using EF relaying. Based on the reconfigured model, we optimize the relaying strategy to attain a high rate irrespective of the relay position with preserving the total transmit bandwidth and energy. The proposed relaying strategy achieves high communication reliability for any relay position, which differs from conventional EF and decodeandforward (DF) relaying schemes.
1. Introduction
D
ecodeandforward (DF) and estimateandforward (EF) relaying protocols are two prominent relaying strategies for the three terminal relay channel composed of source, relay and destination nodes
[1]
[2]
. The performance of DF relaying is highly dependent on the relay position between the source and the destination
[3]
. The performance of EF relaying shows less dependency on the relay position under the condition of a strong relaydestination link
[3]
[4]
[5]
[6]
. In wireless communications, however, a strong relaydestination link is not feasible when the relay is not in close proximity to the destination. Thus, the performance of EF relaying is also highly dependent on the relay position. If the relay has mobility, as in the mobile relay and the nomadic relay systems
[7]
[8]
, the relay can change its position randomly. Recently, a mobile relay for a LTEAdvanced system was investigated in
[9]
. A mobile relay equipped with amplifyandforward (AF) protocol which utilizes minimum power was proposed in
[10]
. In fact, we need a new relaying strategy with sufficiently high resultant communication reliability, irrespective of the relay position. The new relaying strategy should not require bandwidth expansion and an energy increase for operation.
As a solution, we propose a new timedivision halfduplex EF relaying strategy, where halfduplex relaying is preferred in practical cooperative communication systems due to the absence of a selfinterference (echo) signal
[11]
, and the timedivision relaying strategy can be easily extended to frequencydivision relaying. First, we reconfigure EF relaying such that the actual relaydestination channel is included in the newly defined relay unit in order to ensure that the resultant relaydestination link is always strong. Based on the reconfigured model, we manipulate equations associated with the achievable rate of relay channels, and propose a systematic way to optimize the relaying strategy to maximize the achievable rate. The proposed relaying strategy includes quantization at the relay as well as power allocation between the source and the relay. When optimizing the relaying strategy, we consider the actual relaydestination channel condition as well as the modulation order at the source node and the relay node with constraints on the transmit bandwidth and energy. With the same transmit bandwidth and energy, the proposed relaying strategy achieves higher communication reliability than direct communication and conventional EF relaying. The performance of the proposed relaying strategy is much less sensitive to the relay position than conventional EF and DF relaying. These observations show that the proposed relaying strategy is suitable for a relay with mobility.
2. System Model
We consider a timedivision halfduplex relay channel composed of a source node (S), a relay node (R) and a destination node (D). Channels from S to R, from R to D and from S to D are called SR, RD and SD channels, respectively, each of which has the channel gain
h_{sr}
,
h_{rd}
and
h_{sd}
, respectively.
The relay channel operates in a broadcast (BC) mode during the first time fraction
t
and in a multipleaccess (MAC) mode during the next time fraction 1 −
t
, where 0 <
t
< 1, as depicted in
Fig. 1
. Suppose that S does not transmit a signal (silent) in MAC mode.
Halfduplex relay channel operating in BC mode and MAC mode.
In BC mode, S transmits a symbol
X
, which is received as
V
at R after passing through the SR channel and is received as
Z
_{1}
at D after the SD channel. In MAC mode, R transmits a symbol
W
which is received as
Z
_{2}
at D after the RD channel. Note that
W
is formed at R by quantizing
V
as
W
=
Q_{t}
(
V
), where
Q_{t}
denotes an
L_{q}
bin quantization function. We consider a scalar quantization
Q_{t}
with low complexity because it does not result in remarkable information loss compared with an optimal vector quantization requiring a great deal of computation
[12]
[13]
. No additional channel coding is applied to
W
to avoid the resultant bandwidth expansion of the RD channel. Then, the relay channel is modeled as
where
N_{R}
,
N
_{D1}
and
N
_{D2}
are zeromean additive circular symmetric complex white Gaussian noises with the singlesided power spectral density of
N
_{0}
. At D, loglikelihood ratios (LLR's) of bits mapped to
X
are computed from
Z
_{1}
in BC mode and from
Z
_{2}
in MAC mode, which are combined to use in the detection and decoding processes.
For simple analysis, we suppose that all nodes are aligned on a straight line with R located between S and D as shown in
Fig. 2
. The distance of R from S and from D is
d
and 1 −
d
, respectively, with 0 <
d
< 1. Then, 
h_{sd}

^{2}
= 1, 
h_{sr}

^{2}
= 1 /
d^{α}
and 
h_{rd}

^{2}
= 1 / (1 −
d
)
^{α}
, where
α
is a channel attenuation factor.
Relay channel with all nodes aligned on a straight line.
As a reference system, we consider a direct communication system in which the
M_{o}
ary symbol is transmitted from S to D with the energy per symbol
E_{T}
. The relay system is constrained to utilize the same transmit energy and overall transmission bandwidth as those of a reference system. The energy constraint is expressed by
where
E_{S}
and
E_{R}
denote the energy per symbol at S and at R, respectively. To satisfy the bandwidth constraint, the relay system must transmit the same number of symbols as the reference system. For this purpose, alphabet sizes of
X
and
W
must be
M
=
and
L
=
, respectively, where
X
∈ {
x
_{0}
,…,
x
_{M1}
} and
W
∈ {
w
_{0}
,…,
w
_{L−1}
}. We consider
M
and
L
be powers of two. It follows that
t
=
and
L_{q}
= 2
^{b(a/log2Mo−1)}
with positive integers
a
and
b
.
3. Relaying Strategy
 3.1 Conventional EF Relaying
Let us consider a conventional configuration of the timedivision halfduplex EF relaying as depicted in
Fig. 3
, where the relay unit (R) is represented by a dotted box. The achievable rate by this configuration is given by
[11]
[12]
subject to
where
R_{rd}
is the achievable rate over the RD channel in MAC mode, and
p
(
x
) and
p
(
w
) denote the distribution of
X
and
W
, respectively. This is obtained from
[11]
[12]
by assuming that S is silent in MAC mode. Suppose that
V
is quantized as
W
without exploiting the side information
Z
_{1}
, which results in
I
(
V
;
W

Z
_{1}
) =
I
(
V
;
W
) . Note that
I
(
V
;
W
) =
H
(
W
) −
H
(
W

V
) =
H
(
W
) because
W
is a deterministic function of
V
, i.e.,
W
=
Q_{t}
(
V
). Thus, the constraint (4) is simplified as
Note that
R_{rd}
≤ log
_{2}
(1+
h_{rd}

^{2}
). Suppose
t
= 0.5,
d
= 0.2,
α
= 2,
L_{q}
≥ 4 and
p
(
w_{i}
) = 1 /
L
which is identical for all
i
= 0,…,
L
 1. Then, we have 2 ≤
H
(
W
) ≤
and it follows that
(1 
d
)
^{α}
≤
. Since
≥
E_{R}
by (2), we obtain
≥ (1 
t
)
(1 
d
)
^{α}
= 0.1773 dB. This means that
≥ 0.1773 dB is required to satisfy constraint (4). If channel coded bits with the rate
R_{c}
= 1/2 are to be transmitted by EF relaying. then
≥ 2.833 dB is required to satisfy (4) because
=
R_{c}
. In case of direct communication, reliable communication is available at
E_{b}
/
N
_{0}
much lower than 2.833dB by using the rate 1/2 turbo codes
[14]
. This analysis tells us that conventional EF relaying may result in poorer performance than direct communication with some operating parameters. To avoid this problem, R is placed near D when EF relaying is used. Consequently, a conventional EF relaying strategy is not suitable for a relay with mobility. To resolve this problem, we propose a new EF relaying strategy under a new framework working well for various
d
values, even at low SNR.
The models of conventional and reconfigured EF relaying.
 3.2 Proposed Relaying Strategy
The proposed relaying strategy operates in the same manner as introduced in Sec. 2. S broadcasts
M
ary symbol
X
∈ {
x
_{0}
,…,
x
_{M−1}
} in BC mode during the time fraction
t
, where
X
is received as
V
at R and as
Z
_{1}
at D. At R,
V
is quantized as
L
ary symbol
W
∈ {
w
_{0}
,…,
w
_{L−1}
} by
W
=
Q_{t}
(
V
). At D, LLR’s of bits mapped to
X
are computed from
Z
_{1}
. In MAC mode during 1 −
t
, R transmits
W
which is received as
Z
_{2}
at D. Then, LLR’s of bits mapped to
X
are computed from
Z
_{2}
and combined with LLR’s obtained in BC mode. The SR channel can be represented by the transition probability
p
(
w_{k}

x_{j}
),
j
= 0,…,
M
 1 and
k
= 0,…,
L
 1 and defined by
where
U_{k}
denotes the quantization bin of
V
mapped to
w_{k}
. The quantization at R can also be conducted by using the matched filtered signal
, by which (6) can be written in another form as
where the bin bounded by
is mapped to
w_{k}
, and ℜ{⋅} and ℑ{⋅} denote the real part and the imaginary part, respectively. The RD channel can be represented by the transition probability
f
(
z
_{2}

w_{k}
),
k
= 0,…,
L
 1, defined by
Let
b_{ℓ}
be the
ℓ
th bit in the bit stream mapped to
X
, then the LLR of
b_{ℓ}
is computed in BC mode as
where
and
denote the set of symbols
X
whose corresponding
ℓ
th bit is 0 and 1, respectively. In MAC mode, the LLR of
b_{ℓ}
is computed as
The LLR’s of each bit
b_{ℓ}
computed in BC mode and MAC mode are combined as
L
(
b_{ℓ}
) =
L
(
b_{ℓ}

z
_{1}
) +
L
(
b_{ℓ}

z
_{2}
) to be used in the detection and decoding processes.
Although the proposed relaying strategy and the conventional EF relaying scheme
[12]
operate in a similar manner, they differ in determining optimal parameters of relaying schemes. In the proposed relaying strategy, the modulation scheme of
W
and the actual RD channel condition are taken into account to optimize operating parameters. On the other hand, in the conventional EF relaying scheme, the RD channel is assumed errorfree. For finding optimal parameters of the relaying strategy, let us reconfigure the timedivision halfduplex EF relaying as depicted in
Fig. 3
, where the new relay unit is represented by a solid box with the label R
^{new}
. Note that R
^{new}
includes the actual RD channel as well as R. The newly defined RD channel is free of additive noise and attenuation so that
R_{rd}
→ ∞ and the constraint (4) is not needed. Note that
W
is the output of R in the conventional configuration. Considering that
Z
_{2}
is the output of R
^{new}
, the optimization problem (3) with constraint (4) in the conventional configuration can be modified as
without a constraint in the new configuration. At low SNR,
Z
_{1}
and
Z
_{2}
are almost independent so
I
(
X
;
Z
_{1}
,
Z
_{2}
) ≈
I
(
X
;
Z
_{1}
) +
I
(
X
;
Z
_{2}
) . In addition,
I
(
X
;
Z
_{1}
) and
I
(
X
;
Z
_{2}
) are maximized when
X
is uniformly distributed. Thus, at low SNR, (11) is approximated by
At low SNR,
I
(
X
;
Z
_{1}
) is upperbounded by the capacity of
M
ary input additive white Gaussian noise (AWGN) channel, denoted by
C_{M}
(
E_{S}
/
N
_{0}
), because
X
is modulated as an
M
ary signal. For given
d
,
E_{T}
and
N
_{0}
,
p
(
w
) is determined by
E_{S}
and
Q_{t}
. Consequently, for given
d
and
E_{T}
, (12) can be written by
With
p
(
x
) =
, we have
where
and
In the second equality of (16),
p
(
x_{j}

w_{k}
,
z
_{2}
) =
p
(
x_{j}

w_{k}
) is used because
X
,
W
and
Z
_{2}
form a Markov chain in this order as well as in the reverse order, and in the last equality,
is used. A closed form of
I
(
X
;
Z
_{2}
) is obtained by plugging (15)(17) into (14) with the aid of (6)(8). Then, by using the closed form of
I
(
X
;
Z
_{2}
), we can find numerically optimal values of
t
and
E_{S}
as well as the optimal quantizer
Q_{t}
that maximize
t
(
C_{M}
(
E_{S}
/
N
_{0}
) +
I
(
X
;
Z
_{2}
)) and we can evaluate the resultant achievable rate
R_{EF}
for given
d
,
E_{T}
and
N
_{0}
.
4. Numerical Results
Consider BPSK and QPSK modulated direct communication schemes (
M_{o}
= 2 and
M_{o}
= 4) as reference systems. We suppose that
d
is known to all nodes, and let
α
= 2 and
N
_{0}
= 2. Optimal values of
t
and
E_{S}
as well as the optimal quantizer
Q_{t}
in the proposed relaying strategy for given
d
and
E_{T}
are obtained as follows. We first find the candidate values of
t
satisfying
t
=
with an arbitrary positive integer
a
, as introduced in Sec. 2, and fix the value of
t
as one of cadidates. Then, by varying the value of
E_{S}
, we do the following. For each
E_{S}
, we find threshold values of
Q_{t}
that maximize the value of
t
(
C_{M}
(
E_{S}
/
N
_{0}
) +
I
(
X
;
Z
_{2}
)) by using the closed form obtained in Sec. 3. The quantization at R is conducted for the matched filtered signal
. Each phase component of
, i.e.,
_{I}
and
_{Q}
, is quantized independently based on the set of perphase thresolds
, by which
L_{q}
level quantization is obtained. Then, we find the value of
E_{S}
and its corresponding
which results in the maximum value of
t
(
C_{M}
(
E_{S}
/
N
_{0}
) +
I
(
X
;
Z
_{2}
)) for a given
t
. By repeating the same procedure for all candidate values of
t
, we find the optimal combination of
t
,
E_{S}
and
, and evaluate the achievable rate
R_{EF}
. The number of quantization bins
L_{q}
is suboptimally chosen as the value above which the performance is not improved significantly by increasing
L_{q}
compared with the resultant growth of complexity. The values of perphase thresholds are set as symmetric, i.e.,
θ_{k}
=
,
k
= 1,…,
 1.
First, consider the case that BPSKmodulated direct communication (
M_{o}
= 2) is a reference system. Through a numerical search, we determine
t
= 0.5 resulting in
M
= 4, and
L_{q}
=
L
= 16 as optimal values. The set of perphase thresholds is in the form of {−
θ
_{1}
,0,
θ
_{1}
}. We choose QPSK and 16QAM as the modulation schemes for
X
and
W
, respectively. Let us consider
x
_{0}
= 
a

ja
with
a
=
, which is a point in the signal constellation for the QPSK symbol
X
. The transition probability
p
(
w
_{0}

x
_{0}
) is obtained by using (7) as
where
Q
(
x
) =
. All other transition probabilities
p
(
w_{k}

x_{j}
) are obtained in a similar manner. Optimal parameters for some
d
and perbit SNR,
E_{b}
/
N
_{0}
, with
t
= 0.5 and
L_{q}
= 16 are listed in
Table 1
as samples.
Optimal parameters of the proposed relaying strategy and resultantREFfor somedandEb/N0witht= 0.5,Lq= 16,α= 2 andN0= 2, where the reference system is BPSKmodulated direct communication. In numerical search, increments ofESandθ1are 0.001.
Optimal parameters of the proposed relaying strategy and resultant R_{EF} for some d and E_{b} / N_{0} with t = 0.5, L_{q} = 16, α = 2 and N_{0} = 2, where the reference system is BPSKmodulated direct communication. In numerical search, increments of E_{S} and θ_{1} are 0.001.
Next, consider the case that QPSKmodulated direct communication (
M_{o}
= 4) is a reference system. We determine
t
= 0.5 resulting in
M
= 16, and
L_{q}
=
L
= 256 as optimal values. The set of perphase thresholds is in the form of {−
θ
_{7}
,…,−
θ
_{1}
,0,
θ
_{1}
,…,
θ
_{7}
}. We choose 16QAM and 256QAM as the modulation schemes for
X
and
W
, respectively. Transition probabilities
p
(
w_{k}

x_{j}
),
j
= 0,…,15,
k
= 0,…,255, are obtained in a similar manner to the case of BPSKmodulated reference system as introduced above. Optimal parameters for some
d
and perbit SNR,
E_{b}
/
N
_{0}
, with
t
= 0.5 and
L_{q}
= 256 are listed in
Table 2
as samples.
Optimal parameters of the proposed relaying strategy and resultantREFfor somedandEb/N0witht= 0.5,Lq= 256,α= 2 andN0= 2, where the reference system is QPSKmodulated direct communication. In numerical search, increments ofESandθkare 0.001.
Optimal parameters of the proposed relaying strategy and resultant R_{EF} for some d and E_{b} / N_{0} with t = 0.5, L_{q} = 256, α = 2 and N_{0} = 2, where the reference system is QPSKmodulated direct communication. In numerical search, increments of E_{S} and θ_{k} are 0.001.
The achievable rates evaluated by (13) with optimally chosen parameters for
d
= 0.95 are plotted in
Fig. 4
for the cases that the reference system is BPSK and QPSK modulated direct communications. The capacity of the reference system (direct link) and the upper bound on the achievable rate of relay channel are also plotted. For fair comparison, the same transmit bandwidth and energy are considered for all schemes. The upper bound on the achievable rate of relay channel corresponds to the capacity of 1 × 2 SIMO (singleinput multipleoutput) AWGN channel under the assumption that the RD channel is perfect and S remains silent in MAC mode. It is observed from
Fig. 4
that the achievable rate of the proposed scheme approaches the upper bound of the relay channel at low SNR especially when the reference system is BPSKmodulated direct communication. Achievable rates of the proposed relaying strategy for various relay positions are plotted in
Fig. 5
and
Fig. 6
. Achievable rates of the conventional EF relaying are also plotted for various relay positions in
Fig. 7
and
Fig. 8
. It is observed that the achievable rate of the proposed relaying strategy is insensitive to the relay position. On the other hand, the conventional EF relaying scheme shows high dependency on the relay position in terms of the achievable rate, and may not provide a reliable communication link when
d
is small, i.e., R is far from D. It is also observed that the proposed relaying strategy results in a higher rate than the conventional EF relaying scheme when
d
is not close to 1.
Achievable rate of the proposed relaying strategy, the direct link capacity and the upper bound on the achievable rate of a relay channel, where the relay position d = 0.95.
Achievable rates of the proposed relaying strategy for different relay positions d, where the reference system is BPSKmodulated direct communication.
Achievable rates of the proposed relaying strategy for different relay positions d, where the reference system is QPSKmodulated direct communication.
Achievable rates of the conventional EF relaying strategy for different relay positions d in case that the reference system is BPSKmodulated direct communication, where the achievable rate of the proposed relaying scheme for d = 0.9 is also plotted.
Achievable rates of the conventional EF relaying strategy for different relay positions d in case that the reference system is QPSKmodulated direct communication, where the achievable rate of the proposed relaying scheme for d = 0.9 is also plotted.
We plot the simulated bit error rate (BER) performances of the proposed relaying, conventional EF relaying and DF relaying strategies in
Fig. 9

Fig. 12
. In simulations, 16200bit irregular lowdensity parity check (LDPC) codes
[15]
[16]
with 50 iterations of message passing decoding are used. For simulations associated with BPSKmodulated reference system, we use rate 1/3 irregular LDPC codes whose degree distribution polynomials are defined by
λ
(
x
) = 0.4
x
+ 0.2
x
^{2}
+ 0.4
x
^{11}
and
ρ
(
x
) =
x
^{4}
. For simulations associated with QPSKmodulated reference system, we use rate 1/5 irregular LDPC codes whose degree distribution polynomials are defined by
λ
(
x
) = 0.5333
x
+ 0.1111
x
^{2}
+ 0.3556
x
^{11}
and
ρ
(
x
) = 0.2
x
^{2}
+ 0.8
x
^{3}
. Degree distribution polynomials of irregular LDPC codes are chosen to show sufficiently good BER performances over a direct link. In conventional EF relaying
[12]
, no data loss over the RD channel is assumed, i.e.,
z
_{2}
=
w_{k}
. Since there is, in fact, signal attenuation and additive noise over the RD channel, we first demodulate
z
_{2}
as
,
k
= 0,…,
L
 1, and compute
L
(
b_{ℓ}

z
_{2}
) by modifying (10) as
for MAC mode simulations of conventional EF relaying, where
is the demodulated symbol of
z
_{2}
. When optimizing the conventional EF relaying scheme given by (3) and (4), a
L
ary input AWGN channel capacity is used as
R_{rd}
. In DF relaying,
V
is decoded, reencoded and mapped to
M
ary symbols
W
at R, where the same channel codes are used at S and R, and the transmit power is allocated between S and R to achieve the maximum rate
[1]
[3]
.
BER performances of the proposed relaying and conventional EF relaying strategies, where the reference system is BPSKmodulated direct communication and the code rate is 1/3.
BER performances of the proposed relaying and conventional EF relaying strategies, where the reference system is QPSKmodulated direct communication and the code rate is 1/5.
BER performances of the proposed relaying and DF relaying strategies, where the reference system is BPSKmodulated direct communication and the code rate is 1/3.
BER performances of the proposed relaying and DF relaying strategies, where the reference system is QPSKmodulated direct communication and the code rate is 1/5.
It is observed that the proposed relaying strategy achieves a BER performance gain over the reference system for all
d
values with satisfying bandwidth and energy constraints. The proposed relaying scheme shows good BER performances, irrespective of
d
, which is different from conventional EF relaying
[12]
. The performance of conventional EF relaying gets poorer as
d
decreases. It is also observed that DF relaying shows poorer BER performance than the proposed scheme for large and small
d
values, where the performance of DF relaying is highly dependent on
d
.
In
Fig. 13
and
Fig. 14
, we plot the minimum values of
E_{b}
/
N
_{0}
required by the proposed relaying, conventional EF relaying and DF relaying strategies to achieve the target rate with various relay positions, where minimum
E_{b}
/
N
_{0}
required by direct communication is also plotted for comparison. For cases whose reference systems are BPSK and QPSKmodulated direct communications, the target rates are set as 1/3 and 1/5, respectively. Although the minimum
E_{b}
/
N
_{0}
is achieved by an infinitelength optimal coding scheme, this performance index provides enough insight on the BER performances of coded relaying schemes under comparison. The minimum
E_{b}
/
N
_{0}
requirement of the proposed relaying strategy is insensitive to the relay position, which differs from conventional EF relaying and DF relaying schemes. In case of conventional EF relaying, a higher value of minimum
E_{b}
/
N
_{0}
is required as
d
gets smaller. For great or small
d
, the proposed relaying strategy requires lower minimum
E_{b}
/
N
_{0}
than DF relaying.
Minimum values of E_{b} / N_{0} required by the proposed relaying, conventional EF relaying and DF relaying strategies to achieve the rate 1/3 with various relay positions, where the reference system is BPSKmodulated direct communication.
Minimum values of E_{b} / N_{0} required by the proposed relaying, conventional EF relaying and DF relaying strategies to achieve the rate 1/5 with various relay positions, where the reference system is QPSKmodulated direct communication.
Through various analyses, it is observed that the performance of the proposed relaying strategy is insensitive to the relay position. Consequently, the proposed relaying strategy is suitable for the relay system with mobility in which a relay position varies randomly.
5. Conclusion
We proposed a new timedivision halfduplex EF relaying strategy resulting in reliable communication for all relay positions. We reconfigured EF relaying such that the channel between the relay and the destination is perfect, which is the basic requirement for EF relaying to achieve a high rate. From the reconfigured model, we manipulated equations associated with an achievable rate and found a systematic way to optimize an EF relaying strategy to maximize the achievable rate. The proposed EF relaying strategy enables reliable communication for all relay positions without bandwidth expansion and energy increase for transmission. Consequently, the proposed relaying scheme is suitable for a relay system with mobility.
BIO
Inho Hwang received B.S. and M.S. degrees in electrical engineering from ChungAng University, Seoul, Korea, in 2008 and 2010, respectively. He is currently pursuing Ph.D. degree at ChungAng University, Seoul, Korea. His research interests include cooperative communications and coding theory
Jeong Woo Lee is an associate professor in the School of Electrical and Electronics Engineering, ChungAng University, Seoul, Korea. He received B.S. and M.S. degrees from Seoul National University, Seoul, Korea in 1994 and 1996, respectively, and Ph.D. degree from the University of Illinois at UrbanaChampaign, IL, USA in 2003, all in electrical engineering. His current research interests include cooperative communications, mobile relay system and coding theory.
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