Int. J. Communications, Network and System Sciences, 2011, 4, 568-577
doi:10.4236/ijcns.2011.49068 Published Online September 2011 (http://www.SciRP.org/journal/ijcns)
A Simple Symbol Estimation for Soft Information
Relaying in Cooperative Relay Channels
Azlan Abd Aziz, Yasunori Iwanami
Department of C om put e r Sci ence and Engin eer i ng, Graduate School of Engineering,
Nagoya Institute o f Technology, Nagoya, Japan
Received July 12, 2011; revised August 11, 2011; accepted August 22, 2011
Abstract
This paper proposes Symbol-based Soft Forwarding (SSF) protocol for coded transmissions which is based
on a simple proposed soft symbol estimation at relay nodes. We present a simple strategy of forwarding soft
information based on a simple linear summation of likelihood functions of each symbol. Specifically, with
SSF, we demonstrate that exclusion of decoding at the relays costs no significant performance loss. To vali-
date our claims, we examine bit error rate (BER) performance for the proposed scheme against the baseline
SF scheme through computer simulations. We find that the proposed scheme can obtain considerable per-
formance gains compared to the conventional relaying protocol.
Keywords: Symbol Estimation, Soft Information Relaying, Cooperative Relay, Symbol LLR
1. Introduction
Soft Forwarding (SF) [1,2] has been widely studied as an
effective relaying protocol which combines the features
of Amplify-and-Forward (AF) and Decode-and-Forward
(DF) [3]. AF preserves the reliability information but
ignores the channel coding. DF, however, enjoys the
coding gain but suffers from the error propagation.
Therefore, SF is introduced to reap the benefits of both
previous strategies with applications to various systems.
For instance, SF can be represented by the bit log likeli-
hood ratios (LLR) values generated by a channel decoder.
Soft values are re-transmitted by relays in different ways
e.g., based on the power constraint at relays (AF strategy)
as in [1], or transmitting the expectation values in terms
of the mean squared error (MSE) namely Estimate-
and-Forward (ENF) [2,4], Decode-Estimate-Forward sch-
eme (DENF) [5], or Soft-Decode-and-Forward (SDF) [6].
If these works assume Gaussian distribution in bit LLR
computations for simplicity, [6,7] improve LLR compu-
tations based on more accurate distribution. Moreover,
these methods require high search operations in bit LLR
computations, especially if higher modulations are ap-
plied.
In conventional SF schemes, channel between source-
relay link has to be highly reliable enough to ensure er-
ror-free re-transmission at the relay. In [1,2], to have
reliable re-transmissions from the relays, the authors
employ error correcting codes and cyclic redundancy
checks (CRC) to assist the detection at the destination.
Generally, early coded cooperative relay schemes used
convolutional or Turbo codes [5-8], but recently apply-
ing Low Density Parity Check (LDPC) codes have
gained more attention [9]. A drawback for single-input
single output (SISO) coded scheme under block fading
channel is the lack of coding gain due to the constant
fading coefficient in each block of codeword [10]. This
becomes a challenging task for LDPC coded schemes to
cope with the erroneous and unreliable decoded signals
at the relays. Another strategy to solve error propagations
from the relays has been proposed in [11,12] based on
Maximum Likelihood (ML) criterion bit LLR combining
at the destination. However, this strategy increases the
computational complexity at the destination due to the
requirement of perfect S-R channel knowledge for opti-
mal detection.
Unlike [1,2,5], which require bit-wise detection and
decoding at relays, we propose a simple relaying strategy
which is known as Symbol-based Soft Forwarding (SSF).
This method provides a novel soft symbol estimation and
a simple forwarding strategy by transmitting the expecta-
tion values from a linear combination of posteriori prob-
abilities of each symbol in the constellation set. In par-
ticular, this strategy avoids severe impact of decoding

A. ABD AZIZ ET AL.
569
errors and hence, expected values can be accurately
computed. This idea is mooted from the fact that a relay
behaves like a repeater. Thus, detail processing and
bit-wise analysis of receive signals are not necessary at
this stage. To achieve these goals, we simplify the relay
architecture by developing a soft symbol estimation
technique, as opposed to [2,5] which is based hyperbolic
tangent functions. More importantly, it will be shown
that the proposed schemes improve bit error rate (BER)
remarkably in various simulation setups. For comparison,
we re-name the SF as bit-based Soft Forwarding (BSF).
The organization of this paper is as follows: Section 2
is system description followed by the proposed scheme
in Section 3. We present simulation results in Section 4,
and in Section 5 the paper is summarized. Finally the
derivation of symbol LLR algebra for quadrature phase-
shift keying (QPSK) is described in the Appendix.
2. System Description
2.1. Cooperative Relay System
We consider a hybrid relay network as in Figure 1 with a
source, destination and parallel/serial relays. It is as-
sumed that there is also a direct transmission from the
source (S) to the destination (D). The total number of
serial relays in one link is denoted as
s
L and the total
number of parallel relays as
p
L. We denote ,
p
s as a
relay node with the s-th hop an d p-th bran ch. We assume
that all receiving nodes have perfect channel state infor-
mation (CSI). All nodes have only one antenna working
in a half-duplex mode. In this system, S broadcasts to D
and to all branch relays of the 1st hop1,1 ,1
p
L.
Then, these relays forward the received signal to its re-
spective relay nodes in the p-th branch as shown in Fig-
ure 1 above and th e final s-th hop forwards the sign al to
the destination. It is assumed that each link does not dis-
turb each other which can be implemented by using or-
thogonal chan nels.
R
(,,R)
}
R
At the source, information bits
of length with are encoded by an LDP C
encoder of rate
()
{,1,2,,
c
uuc K
K() {0 ,1}
c
u
RKN to
of length N,
(1)()(
,, ,,
nN
bb bb

 
)
()n{0,1}
b


,,
. These coded bits are modulated
as 1SmM

,,
x
ss s, where M is the number of
constellation size. For example, when the source uses
QPSK modulation with Gray mapping then 4
M

.
Since every symbol m
s
corresponds to two bits,
we assign 1 2 3
accordingly. The received symbol at a relay
12
(, )bb
, s(1,1),s (0,0), s(1,0)
4
s (0,1)
,
p
s
R can be shown as
,,,1,
p
spsps p
yhxn


s
(1)
where ,
p
s
h is the channel gain between the relay ,
p
s
R
S D
R1, 1 R1, Ls
RLp, 1 RLp, Ls
Figure 1. Cooperative relay scheme with multiple relays.
and ,1
p
s
R

with the signal ,1
p
s originating from the
latter relay node in the p-th branch, and ,
x
p
s is
additive
Gaussian noise with unit variance. For simplicity, we
assume that all channels are Quasi-static Rayleigh fading
channels with zero mean and variance
n
2,
p
s

. Upon re-
ceiving ,
p
s, this relay estimates the symbol ,1
y
p
s
x

according to its relay function , ()
p
s
f
y which is de-
scribed in the following section, and forward the signal
,
p
s
x
to successive relays in one path i.e., ,
p
s to
,1
R
p
s
R

. We assume that the source and all relays operate
under the same average power constraints which means


2
S
Ex P
S

and


2
,
p
s
Ex PR , for some and
SR
PP
where


E

denotes the expectation operator. The av-
erage transmit SNR 0 for all links are assumed the
same 0SR
P
PPP

 such that 000 PN

, while the
instantaneous receive SNR at relay ,
p
s is represented
as R
22
hhPN  
,, 0, 0
0
ps psps. Finally, at the desti-
nation, all signals from the relays in the final hop are
combined by a maximum ratio combining (MRC) strat-
egy.
2.2. Baseline System
First, we summarize some of the relaying protocols de-
scribed in Section 1.
Amplify-and-Forward (AF): the relay amplifies the
received signal by a scaling factor. The received signal is
normalized so that the power constraint is satisfied. That
is,


,,,
p
spsps
x
fy y   (2)
where


2
,Rps
PEy   is a constant chosen so that


2
,
p
sR
Ex P

.
Decode-and-Forward (DF): the relay decides on the
hypothesis that minimizes the probability of error at the
relay node and forwards this decision with constant
power


,,
argmax

,
p
sps Sps
x
fy pxy   (3)

570 A. ABD AZIZ ET AL.
Soft Forwarding (SF): the relay forwards the MSE es-
timate subject to the power constraint amounts to linearly
scaled version of the conditional expectation
 
,,

,
p
sps Sp
xfy Exy   s (4)
With SF, the relay forwar ds reliability information of the
detected signal to the destination. The behavior of SF is
that when receive SNR at the relay is high, it behaves
like DF and when the SNR is low it behaves like AF.
The block diagram of BSF scheme is shown in Figure 2.
Next, to better motivate the proposed scheme, we
elaborate the baseline SF protocol which we rename here
as BSF to avoid confusion. From (1) after equalizing the
channel between relays ,
p
s and R,1
p
s
R (assuming
perfect CSI), ,
p
s
R receives the symbol
,
,1
,,
,
p
s
ps ps
p
sp
yn
sx
hh

 

s
(5)
where ,,
p
sps
nh is the equivalent zero mean complex
Gaussian white noise with the equivalent variance
2
22
,,,psps ps
h  
. Then, the relay will approximate bit
LLR values as


, () 0
, ()1
,
log ,
sMbs
b
sMbs
f
ss
f
ss











 (6)
where we define
 


22,
,exp 2
ps
fsss s  



and
from (6), the bit LLR values are passed through the
channel decoder (i.e., LDPC decoder in this paper) to
generate the a-posteriori probabilities (APP) which is
denoted as
D
ec

.
Then, if BPSK is considered, these LLR values are
forwarded to the destination based on the normalized
expectation value [2,5]



,2tanh 2
tanh 2
BSF r
ps Dec
Dec
P
xE 



(7)
As for QPSK, odd and even LLR values are scaled and
forwarded over the in-phase and quadrature axes. This
result is shown to be optimal in terms of MSE at the re-
ceiver of the relay node. The normalization in (7) is done
to meet the power constraint at the relay. Nonetheless,
the problem in this scheme is that it does not consider the
probability of the entire bit sequence to approximate the
LLR values since it employs the expression (6) above.
Eventually, the likelihood function in (6) may contain
approximation errors and then, these errors would be
propagated to the destination along with the errors re-
sulted from the message passing algorithm for LDPC
decoder at the relay. The impact can be higher if more
relays or higher modulation schemes are employed. Fur-
thermore, the APPs obtained from the decoder are sub-
optimal because bits corresponding to one symbol are
not independent.
3. Symbol-Based Soft Forwarding (SSF)
In SSF as depicted in Figure 3, the relay utilizes a sim-
ple maximum likelihood detection (MLD) implementing
soft symbol estimation and expectation values
s
are
forwarded without decoding to the destination. At the
destination, all received signals from the relays and the
direct link are combined in order to recover the original
source data.
In this paper, we introduce a new method of signal
detection and forwarding at the relays which leads to a
more efficient use of the relay resources. Details of the
derivations are presented in the Appendix. From (24) in
the Appendix, the forwarded signal from a relay which is
in the form of the expectation values for one QPSK
symbol can be computed as

3 42
3 24
432
1
1
s
eseses
sEs eee
 
 



 (8)
where m

,


1, 2,3, 4m is the m-th symbol LLR in
QPSK modulation. Here, we propose a simple linear
combination scheme to combine all the possible constel-
lation points together with the corresponding posterior
probability as the weight. With this strategy, we can de-
crease the computational complexity at the relays and
avoid approximation errors in (6) of the baseline relay
detection method. Furthermore, unlike in the proposed
scheme, (6) has to do many search operations for optimal
detection and this requirement is proportional to the con-
stellation size of the modulation. From here onwards, we
drop the subscripts of node relations in (1) for simplicity.
In Table 1, we summarize the comparison between the
proposed scheme and the baseline.
Relay
Compute
bit LLR
b

Compute
s
LDPC
decoder
,ps
y,
B
SF
p
s
x
Figure 2. Block diagram of BSF scheme.
Relay
Compute
symb o l LLR
m

Compute
s
,
SSF
p
s
x
,
p
s
y
Figure 3. Block diagram of the proposed scheme.

A. ABD AZIZ ET AL.
571
Table 1. Comparison of the proposed protocols against BSF.
SSF BSF
Decoding No Yes (LDPC)
Transmission Equation (12) Equation (7)
Detection Symbol-wise Bit-wise
Another benefit of our proposed method in (8) is that
it provides an alternative method to the well known
soft-bit computations in [13].
Lemma: for special case of BPSK, (8) converges to
the well known

tanh 2
m
s  .
Proof: the proof of this convergence is easily shown as
follows. From (8), the expected value for BPSK signal is
 
2
2
2
112 211
s
es
ssPsysPsy e



  (9)
where we ignore the 3rd and the 4th term in the equation.
Then, inserting Equation (8) simply
becomes 12
1; 1;ss 


22 2
11tanhse e

  

2
)
(10)
which proves the convergence of the proposed scheme to
the formulation in [13]. Therefore, for QPSK (4M

,
we can re-write (8) as
11
m
MM
m
mm
m
s
se e



 (11)
As opposed to (6) in the baseline scheme, (11) is just a
simple linear combination of the probability of each
symbol in the QPSK modulation. As described above for
the baseline protocol, (11) also needs the power restric-
tion as in (7) before re-transmission. And we denote the
forwarded symbol from the relay as ,
SSF
p
s
x
.

,2
SSF R
ps P
x
s
Es s

 (12)
where we define

2
R
P
Es
  as the amplification
factor such that ,
SSF
p
s
x
obeys the power constraint
R
P:

2
,
SSF
p
sR
Ex P.
3.1. Mean Square Error (MSE) at Relay
In fact, using (8), we minimize the MSE of the relayed
signals and preserve the soft information. It provides
sufficient reliability information which amounts to maxi-
mizing SNR at the relay output. MSE is described as the
variance of the equivalent noise term. Nonetheless, MSE
only provides a clue on the error rate performance since
BER does not rely on the variance of the error but on the
whole error distribution [5]. To prove this claim, we con-
sider the MSE of the input signal from the source and the
relay output for the first relay branch of the first relay
hop can be shown as



2
1, 1ps
MSEEssy   (13)
where
s
is the modulated symbols from the relay out-
put. Below we plot the MSE versus average SNR of the
relay node for SSF and the baseline BSF.
From Figure 4, it is obvious that the proposed scheme
achieves better performance in MSE especially at higher
SNR region. Although the difference is significant be-
tween these schemes, the performance of these schemes
can be further improved if accurate approximation on the
equivalent noise is found. In fact, Gaussian distribution
is assumed to model the equivalent noise in both sch-
emes. Such an accurate distribution is important for fur-
ther improvement for these schemes and is beyond the
aim of this paper. This means the proposed scheme re-
tains the reliability in order to reduce propagation errors
to the destination. In fact, if decoding is used at the relay,
wrong decisions are likely to happen and this may cause
further error propagation to the destination due to the
approximation in the message passing algorithm of the
LDPC decoder at the relay.
3.2. Relay Mutual Information
In this section, we analyze the channel capacity for the
relayed link against that of the baseline. For convenience,
we restrict this analysis to one relay node sch eme (Lp = 1;
Ls = 1). First, we have to evaluate the equivalent SNR of
Figure 4. Mean squared error at the output of the relay
node over source input signal.

572 A. ABD AZIZ ET AL.
the relayed link. The received signal at the destination
for one relay no de which is the second hop in series ( s =
2) can be shown in the following relation
1,21,2 1,11,2
1,2 1,2
11
,1
1,2 1,2
11
ee
e
e
ee
mm
j
c
mm
D
MM
m
mm
M
ps j
j
jc
c
MM
mm
yhxn
hs n
hs
hs n
























(14)
where c
s
in the first term is the symbol estimate of the
original symbol from the source while the second term is
the noise. After some algebraic manipulations, for one
relay node scheme 1,1 , the instantaneous SNR at the
destination can be expressed as
R



2
2
1,2
2
1,2 2
22
1,2 21,2
1
2
1,2
2
2
1,2
20
1
e
e
e
e
c
j
c
j
c
D
M
j
j
jc
S
M
S
j
jc
hEs
hEs En
hP
hPN























(15)
where we let 1 em
M
m



. Therefore, the average chan-
nel capacity for the proposed scheme considering the
relayed link is found by averaging over the channel gain
distribution as follows
 
1,221,2 1,2
0
log 1d
D
D
SSF
Cp D




 (16)
To illustrate the performance of the proposed scheme, we
simulate it through Monte-carlo simulations for one relay
node case which is shown in Figure 5.
4. Simulation Results
This section presents some results of simulations under-
taken to illustrate the performance of the proposed
schemes in BER against the average SNR per bit in
decibel (dB), 000 PN


1)
for the direct link in various
simulation setups. Receiving nodes are assumed to have
the same average receive SNR and perfect CSI of the
immediate links. All simulation works use the following
parameters in Table 2 unless otherwise stated. In this
simulation, for simplicity, we only consider blind coop-
erative relaying schemes where relay nodes always
re-transmit to the destination whether the signal is cor-
rectly detected or contains errors. No automatic repeat
request (ARQ) protocol is used to avoid the error propa-
gation from the relay nodes to the destination.
4.1. Multihop Setup (Lp = 1; Ls = 1, 2 and 3)
In this simulation, firstly we consider one relay branch
(
p
L

with multiple hops. Figure 6 shows the BER
versus average SNR in dB for the proposed schemes
against the baseline BSF. The simulation results validate
the derivation of the proposed symbol LLR using MLD
criterion and show the performance improvement by SSF
against the baseline BSF. In Figure 6, we observe that
the combination of the soft symbol estimation technique
and the proposed forwarding strategy outperforms the
rest and provides large performance improvement at no
decoding cost. When in ter-node channel is noisy, a relay
node without a decoder is sufficient to provide BER per-
formance improvement. SSF improves the BER curve
around 2 dB margin against the baseline BSF for case
s
(1
p
LL ;1)

. The relative margin increases as the
number of relay nodes slightly increases in comparison
Figure 5. Average capacity of the relayed link at the desti-
nation for the proposed scheme.
Table 2. Simulation parameters.
Information Bits 504 bits/packet
Modulations QPSK
Channel Model Quasi-static Rayleigh Fading Cha nnel
Error Correct ing Code Regula r (3, 6) LDPC (1008, 504)
Sum-Product Iteration 20 times

A. ABD AZIZ ET AL.
573
with the baseline scheme. For comparison, in Figure 7,
we also compare the proposed schemes against the per-
formance of DF protocol. As expected, the gap is larger
in DF protocol due to the absence of reliability informa-
tion and decoding errors at relays.
Another observation from these figures is that at low
SNR, the proposed scheme yields poorer performance
than that of the baseline schemes. This implies that the
use of LDPC decoder can be effective to combat error
propagation in the baseline scheme. In fact, the crossed
points are shifted further along the x-axis when more
hops are taken for the relayed link. This result is consis-
tent with our intuition since when there are more hops,
the detection errors at the relays are propagated along the
relay path resulting in the poor end-to-end performance
Figure 6. BER comparison of SSF (blue) and BSF (red) in a
multihop coded relay scheme.
Figure 7. BER comparison of SSF (blue) and DF (green) in
a multihop coded relay scheme.
of the scheme. Although the simulation results are lim-
ited to three relays only in series, we expect that the BER
performance will decrease as the number of relays in-
crease further.
4.2. Multibranch Only (Lp = 1, 2; and Ls = 1)
Next, Figure 8 illustrates the BER performance when 1,
2, or 3 relays in parallel are available to assist the source.
It is clear that the proposed SSF can also improve effec-
tively the BER performance for multiple relays. Interest-
ingly, the gain demonstrated in this result increases re-
markably as the number of relays increases. For instance,
performance gain of more than 1 dB each can be
achieved easily for all cases at BER of 10–4. Intuitively,
we expect that the effectiveness can be more evident if
this proposed strategy is applied to larger network con-
figurations. The degradation of the overall system for the
baseline BSF is due to two main factors as follows
1) Lack of diversity gain due to the use of LDPC
codes in block fading channel [10].
2) Due to the approximation error since in the baseline
scheme, relays need to approximate the bit LLR as fol-
lows
 
, () 0 , ()1
log
bsMbs sMbs
f
ssf ss 
 







 (17)
BSF utilizes the expression above to approximate the
bit LLR values. This expression does not consider the
probability of the entire bit sequence as opposed to our
proposed scheme in (11). Note that the performance
achievement of SSF is topped with better resource effi-
ciency which simplifies the symbol LLR calculations and
forwards reliability information of the detected symbols
to the destination.
4.3. Hybrid Multihop and Multibranch Setup
(Lp = 2; Ls = 2)
Here, we consider a more general setup with parallel and
serial relays in Figure 9. For simplicity, we only use
(2; 2
sp
LL )

 and we name this setup as hybrid mul-
tihop and multibranch scheme. Like in the previous re-
sults, SSF outperforms BSF with considerable margin
around 2 dB and 4 dB against DF respectively. The loss
in the baseline scheme is due to the constraints explained
in the previous simulation result.
From these simulation results, we observe that at low
SNR region, BSF and DF outperform the proposed
scheme significantly in the multihop setting. This can be
attributed to the channel coding which is used at the re-
lays. At high SNR region, SSF tends to be like AF pro-
tocol which is optimal when SNR goes to infinity. As a

574 A. ABD AZIZ ET AL.
Figure 8. BER comparison of SSF (blue), BSF (red) and DF
(green) in a multibranch topology.
Figure 9. BER comparison of SSF (blue), BSF (red) and DF
(green) in a hybrid multihop multibranch relay scheme.
result, SSF performs the best in all simulations setups.
These results reflect the performance improvement in
the proposed scheme for all simulation setups due to
better reliability information and forwarding strategy we
have employed. Nonetheless, even without decoding at
the relays, the proposed schemes do not lose the coding
gain entirely. The proposed schemes reduce the impact
of error propagation from decoding errors since decoding
is only done at the destination. Such a simple strategy is
beneficial for low-complexity networks like sensor net-
work which allows the possibility to deploy a large
number of low-complexity relays.
5. Conclusions
This paper proposes a novel soft forwarding protocol in
LDPC coded scheme. SSF implements symbol-wise de-
tection (but no decoding) based on a ML criterion. This
strategy minimizes the impact of propagation error at
relays and thus, provides better reliability information.
We introduce a unified framework which features two
key strategies in these schemes: detection based on sim-
ple symbol LLR estimation at the relay and soft-for-
warding strategy based on transmission of the expected
values of signal point. This strategy sums up the prob-
abilities of each modulated symbol and hence, avoids
unnecessary approximation errors. A relay can be further
simplified if the signals are treated symbol-wise since the
signals are not originally intended for the relay use. Our
main motivation is that LDPC decoder in Quasi-static
Rayleigh fading environment gives little impact in SISO
scheme and bit-wise analysis requires high computation
and thus, consumes many resources at the relay node.
Through simulation results, we prove that our simple
strategy of SSF presents significant gains than the base-
line BSF scheme.
6. Acknowledgements
This study has been supported by the Scientific Research
Grant-in-aid of Japan No. 21560396 and the A-STEP by
JST (Japan Science and Technology agency). The au-
thors also acknowled ge the sup por ts b y Dr. Eiji Ok amoto
and Mrs. Kazumi Ueda.
7. References
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[3] J. Laneman, D. Tse and G. Wornell, “Cooperative Diver-
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[4] K. Gomadam and S. A. Jafar, “Optimal Relay Function-
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tions, Vol. 25, No. 2, 2007, pp. 390-401.
[5] P. Weitkemper, D. Wübben, V. Kühn and K. D. Kam-
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Channel Coding, Ulm, 14-16 January 2008.
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[8] T. Wang, A. Cano, G. Giannakis and N. Laneman, “High
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[10] E. Biglieri, J. Proakis and S. Shamai, “Fading Channels:
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[11] A. A. Aziz, Y. Iwanami and E. Okamoto, “Efficient
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[12] A. A. Aziz, Y. Iwanami and E. Okamoto, “On the Im-
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[13] J. Hagenauer, E. Offer and L. Papke, “Iterative Decoding
of Binary Block and Convolutional Codes,” IEEE Trans-
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[14] V. Aue and R. Nuessgen, “Method for Generating Soft-
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[15] K. S. Kim, K. Hyun, C. W. Yu, Y. O. Park, D. Yoon and
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and Its Implementation Algorithm for Gray-Coded QAM
Signals,” ETRI Journal, Vol. 28, No. 3, 2006, pp. 291-
300.

576 A. ABD AZIZ ET AL.
Appendix
Derivation of Symbol LLR based on MLD and Expected
value of Transmit Signal Point for the proposed relay
function
In this sub-section, we present the proposed detection
and forwarding strategies employed at relay nodes.
Without loss of generality, we consider a scheme with
one relay node only ; thus, we remove
the subscripts of the node relation for simplicity. Since
calculating the exact bit LLR by using the conventional
MLD is excessively exorbitant, there are a few ways
proposed to approximate bit LLR values like in [14,15]
whose aim is to avoid the high computation from the
exact bit LLR expression. Although by using the con-
ventional MLD the optimum performance can be
achieved, the approach requires computation which
grows exponentially with the constellation size of the
modulation schemes. In this paper, we propose another
simple approximation technique in order to avoid such
complexity. Our proposed scheme requires symbol-wise
computations and thus, avoids higher computations to
evaluate every bit of the received signals. We define that
the capital denotes the probability, and is
the Probability Density Function (PDF) denoted in the
following relation . For simplicity, QPSK
modulation is considered which utilizes the Gray map-
ping. We represent the complex symbols,
( 1;1)
sp
LL
() ()Psps s
()Ps ()ps
12
(, )
m
s
bb

as 1, , 3 4
where 1 and 2 correspond to the first and second bit
of a symbol in the constellation.
(0s
b,0) b2
s(1, 0)s
m
(1,1), (0,1)s
s
is the m-th transmit
signal, where is the constella-
tion size for QPSK. For this system, the optimal detector
will search the symbol such that it maximizes the
a-posterior probability
1, ,mM(4)M
()
m
Ps yy, the probability of
receiving the transmit signal m
s
given that is re-
ceived in the small region . We successively describe
below two main steps of the proposed relaying strategy:
y
y 
Step 1: Calculate Symbol LLR Values
The symbol LLRs can be shown as


 



 
111
221
331
441
log0,
log ,
log ,
log
e
e
e
e
Psyy Psyy
Ps yyPsyy
Ps yyPsyy
Ps yyPsyy














(18)
where m

, is the m-th symbol LLR in
QPSK modulation, the transmit symbol 1

1, 2,3, 4m

s
is taken as
the reference and its LLR value 1

is always equal to
zero. Thus, there are three LLR values of 23 ,

 and
4

. When is received, the symbol LLRs 23 y,

 and
4

are calculated to be plotted on the real straight line.
This method simplifies the MLD rule to one-dimensional
space only. If 4

is the largest on the real straight line,
then 4
s
is detected and if 23 ,

 and 4

all have
minus values on the real straight line, then 1
s
is de-
tected. For example, 4

is evaluated as follows

















44
41
1
4
1
4
4
11
4
4
1
,
glog ,
,
g log
,
g log
log
ee
ee
ee
e
Ps yyPs yyPyy
Ps yyPyy Psyy
Ps Pyys
Ps yy
Ps yyPs Pyys
pys
Ps
Ps pys
pys
pys
























 
 





 



44
11
lo
lo
lo


(19)
where we let


44
loge1
P
sPs




. If the transition
probability density is defined as

2
2
2
1exp 2
2
m
yhs
py







m
s




, then we can re-write
(19) as the following


44
4
 4 1
22
41
22
22
414
2
log
log expexp
22
1
2
e
e
pys pys
yhs yhs
yhs yhs











  








 


(20)
When the probability of the two symbols are the same
41
() ()14,Ps Ps

 then 441 log [()()]0.
ePs Ps
Thus, 2

and 3

are also evaluated in the same way
as 4

.
Step 2: Compute Expected Values from Symbol LLR
Next we elaborate the transmission method from the
relay to the destination. Since the computation of expec-
tation values involves the soft symbols, we term this
technique as soft modulation. In this paper we introduce
another method of computing the expectation values
which is another extension of our work in [12]. From
Step 1, after the symbol LLR 123 ,,

 and 4

are
calculated, the expected valu e of transmit sig nal poin t
s
can be evaluated in what follows. From (18)-(20), after
some simplifications, we can obtain that





  
3 42
3 42
11
, ,
Ps y Ps yPs y
eee
Psy Psy Psy
 

1
(21)
Since

A. ABD AZIZ ET AL.
577
   
3 24
1111
1Psy ePsy ePsy ePsy
 










3 42
3 24
1122 33 44
4321
1
11 11
1
1
mmmm
M
mm
m
MM MM
mm
mm mm
s
sPsy sPsy sPsysPsy
sesese ssPs y
eee
seese e
 
 


 
 





 
(24)
 (22)
Then,








3 24
3 22
33 24
3 42
1
2
3
4
11
1
1
1
Psyee e
Ps yeeee
Ps yeeee
Ps yeeee
 
 
 
 




4
4



(23) As opposed to the earlier methods, we propose a sim-
ple strategy to utilize soft forwarding at the relay node.
After computing the symbol LLR values by using
(18)-(20), th e expected value of signal point from (2 4) is
sent by relays to the destination according to the power
limitations at the relays.
Therefore, the expectation values for one QPSK sym-
bol can simply be computed as