40120140501016

International Journal of ElectronicsJOURNAL OF ELECTRONICS AND
INTERNATIONAL and Communication Engineering & Technology (IJECET),
ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 1, January (2014), © IAEME

COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET)

ISSN 0976 – 6464(Print)
ISSN 0976 – 6472(Online)
Volume 5, Issue 1, January (2014), pp. 138-147
© IAEME: www.iaeme.com/ijecet.asp
Journal Impact Factor (2013): 5.8896 (Calculated by GISI)
www.jifactor.com

IJECET
©IAEME

OPTIMAL DECODING FOR WIRELESS RELAY NETWORKS WITH
DECODE-AND-FORWARD COOPERATION PROTOCOL
Taha Abdelshafy Abdelhakim Khalaf
Department of Electrical Engineering, Assiut University, Assiut, Egypt, 71516

ABSTRACT
In this paper, we derive an exact upper bound (UB) on bit error rate (BER) of the
decode-and-forward (DF) cooperation protocol. The destination uses the maximum a posterior (MAP)
decoder to estimate the data sent from the source. The MAP decoder is optimal in the sense that it
minimizes the the end-to-end error probability. The complexity of derivation comes from the fact that
however the source generates data with equal probability, the data received at the destination does not
have the same a priori probability. That is because of the error that occurs in the source-to-relay link.
Hence, the MAP decoding rule can not be simplified to the maximum likelihood (ML) decoding rule.
The results show that the upper bound is very tight. Therefore, the closed form expression of the upper
bound can be used to fully study and understand the diversity performance of the DF cooperation
protocol.
Keywords: Relay Network, Diversity, Decode-and-Forward, MAP Decoder.
I. INTRODUCTION
Cooperative relaying is gaining a significant attention in that multiple intermediate relay nodes
can collaborate with each other to enhance the overall network efficiency. It exploits the physical-layer
broadcast property offered by the wireless medium that transmitted signals can be received and
processed by any node in the neighborhood of a transmitter. The cooperative relaying approach has a
great potential to provide substantial benefits in terms of reliability (diversity gain) [1]-[4] and rate
(bandwidth or spectral efficiency)[5]-[7]. These benefits can extend the coverage, reduce network
energy consumption, and promote uniform energy drainage by exploiting neighbors’ resources. They
can be of great value in many applications, including ad-hoc networks, mesh networks, and next
generation wireless local area networks and cellular networks. Through cooperative diversity, relay
nodes forward the signal received from the source to propagate redundant signals over multiple paths
in the network. This redundancy allows the ultimate receiver to essentially average channel variations
resulting from fading, shadowing, and other forms of interference [2]. Several cooperation protocols
138

International Journal of Electronics and Communication Engineering & Technology (IJECET),

have been proposed in the literature such as amplify-and-forward (AF), decode-and-forward (DF), and
amplify
forward
compress-and-forward (CF) [10, 11].
CF)
The error performance of the cooperative communications with decode-anddecode
-forward protocol
with different combining techniques was investigated in the literature. The end-to-end performance of
end end
wireless communication systems with relays over Rayleigh-fading channels was studied in [12] when
fading
the direct link between the source and the destination does not exist. A general framework for ML
detection of both coherent and noncoherent uncoded cooperative diversity was presented in [13] where
the authors dervived a high SNR approximations based on the closed-form BER expressions. The
hors
closed form
performance of the maximum likelihood decoding for network coded cooperative communication is
studiied in [14].
In this paper, we drive a closed from expression for the upper bound on the bit error probability
of the decode-and-forward cooperation protocol with MAP decoder. Unlike all previous work, the
forward
receiver does not use any combining techniques. However, we consider the data received from the
consider
source together with the data received from the relay node as a codeword. The decoding rule is to find
the codeword that maximizes the a posterior probability. The MAP decoding rule is optimal in the
sense that is minimizes the end- end error probability. Since the error probability in the
-to-end
source-to-relay link is not necessarily 0.5 , the codewords received at the destination are not
relay
equaprobable. Therefore, the MAP decoding rule can not be simplified to the maximum likelihood
(ML) decoding rule. We used the closed form expression we derived to find an approximate upper
bound at high SNR regime. The approximate upper bound can be used to clearly understand the effect
of the SNR of the source-to-relay link on the system diversity.
relay
The remainder part of this paper is organized as follows. The system model is described in
Section II. The MAP decoding scheme is described in Section III. The derivation of the exact upper
.
.
bound on the bit error probability is provided in Section IV. Section V presents numerical results and
discussions. Finally, the conclusions are drawn in Section VI.
II. SYSTEM MODEL

Figure 1: System Model
We consider a relay network composed of one source (S), one relay (R), and one destination
(D) as shown in Fig. 1. The data transmission occurs over two phases. In the first phase, the source
sends its data to the destination. Due to the broadcast nature of the wireless medium, the relay may also
of
receive the source’s data (possibly with some errors). In the second phase, the relay node decodes the
data received from the source and then forwards the decoded data to the destination. We assume that
all data are sent using binary phase shift keying (BPSK) modulation scheme and the source generates
e
its bits with equal probability, i.e., p(bs = 0) = p (bs = 1) = 0.5 where bs is the source’s bit. We
assume that all channel are Rayleigh fading channels with additive white Gaussian noise (AWGN).
We also assume that each node is equipped with single antenna.
139


The signals received at the relay node and the destination in both phases are given by
−
yr = hsr d srm xs + nr ,

(1)

−
y1 = hsd d sdm xs + n1 ,

(2)

−
y2 = hrd d rdm xr + n2 ,

(3)

where
• yr is the signal received at the relay node in the first phase from the source.
• y1 is the signal received at the destination in the first phase from the source.
• y2 is the signal received at the destination in the second phase from the relay.
• xs ∈{+ Es ,− Es } is the BPSK modulated signal of the source’s bit, bs ∈ {0,1} , where Es is the
transmit energy per the source’s bit.
• xr ∈ {+ Er ,− Er } is the BPSK modulated signal of the relay’s bit, br ∈ {0,1} , where Er is the
transmit energy per the relay’s bit.
• hsr , hsd , and hrd are the channel fading gains of the S-R, S-D, and R-D links, respectively.
• d sr , d sd , and d rd are the lengths of the S-R, S-D, and R-D links, respectively.
• m is the path loss exponent.
• nr is the AWGN noise at the relay node which has zero mean and variance N 0 /2 .
r

• n1 and n2 are independent AWGN noise components at the destination which have zero mean and
variance N 0 /2 .
The bit br represents an estimate of the source’s bit bs at the relay node and it is given by

0
br = 
1

yr ≥ 0
yr < 0

(4)

Let the random variable e ∈{0,1} captures the error events on the source-to-relay channel, i.e,
br = bs ⊕ e , where “ ⊕ ” denotes the binary “xor” operation. Hence, e = 1 means br ≠ bs , i.e. the
source’s bit is received in error at the relay node. In the case of BPSK modulation over flat fading
channel with AWGN, the probability of error in the source-relay link, i.e. P(e = 1) , is given by [9]

γ sr 
1
1 −

2
1 + γ sr 
1
= [1 − Γsr ]
2

Pe =
sr

(5)

−
where γ sr = d srm Es /N 0 is the average received SNR of the link between the source and the relay and
r

Γsr = γ sr /(1 + γ sr ) . The received signals at the destination can be written in the matrix form as follows
140


y = Hx + n

(6)

where the received vector y = [ y1 y2 ]T , the transmitted vector x = [ xs xr ]T , the noise vector

n = [n1 n2 ]T , and the channel matrix H is given by
h d − m
H =  sd sd
0





−
d rdm 


0
hrd

(7)

III. MAP DECODING
In this section, we present the MAP decoding scheme for cooperative communication networks
with decode-and-forward cooperation protocol. The MAP decoding scheme is optimal in the sense that
it minimizes the error probability. Let b = [bs br ]T denote the transmitted vector where br = bs ⊕ e .
This vector takes one of four possible values b1 = [0 0]T , b2 = [11]T , b3 = [0 1]T , and b4 = [1 0]T based
on the values bs and e . Since P (e = 1) is not necessarily equal to P(e = 0) , These four vectors are
not equaprobable. The probability of transmitting a specific vector bi is given by

1 − Pesr

 2
P(bi ) = 
 Pe
 sr
 2

i = 1,2
(8)
i = 3,4

The MAP decoder finds the transmitted vector bi that maximizes the a posterior probability P(bi | y )
. An estimate of the transmitted vector is calculated as follows
b = arg maxb P (b | y )
i

i

(9)

and the maximization process is over the 4 possible vectors. Applying Bayes theorem to (9) yields

b = arg maxb

i

P( y | bi ) P(bi )
P( y )

= arg maxb P( y | bi ) P (bi )
i

(10)

where
P ( y | bi ) =

1
−|| y − Hxi || 2 /N 0
e
(πN 0 )

(11)

and xi is the modulated vector that corresponds to the vector bi , e.g., x3 = [+ Es − Er ]T .
Substituting from (11) into (10) yields
141


b = arg maxb

i

1
−|| y − Hxi || 2 /N 0
e
P (bi )
(πN 0 )

= arg minb (|| y − Hxi || 2 − N 0 log ( P (bi )) )
i

(12)

IV. PROBABILITY OF BIT ERROR
In this section, we derive the upper bound on the bit error probability for the MAP decoding
scheme. For mathematical traceability, we assume that the source and the relay node adjust their
transmit power such that all signals are received at the destination with the same SNR γ = E/N 0 . This
m
m
can be achieved by choosing E s = d sr E and Er = d rd E .
The bit error probability is given by

PE ≤ (P(b1 → b2 ) + P(b1 → b4 ) )P(b1 ) + (P (b2 → b1 ) + P (b2 → b3 ) )P(b2 ) (13)

+ (P(b3 → b2 ) + P(b3 → b4 ) )P(b3 ) + (P(b4 → b1 ) + P(b4 → b3 ) )P(b4 )
where P (bi → b j ) is the pairwise error probability of confusing bi with b j when bi is transmitted
and when these are the only two hypothesis.
When the vector bi is transmitted, the received vector would be

yi = Hxi + n

(14)

and the probability P (bi → b j ) is given by

P(bi → b j ) = P || yi − Hx j || 2 − N 0 log( P(b j )) < || yi − Hxi || 2 − N 0 log( P (bi )) 



|| H ( xi − x j ) || 2 N 0
P(bi ) 


= P (< n, H ( xi − x j ) >) >
+
log
(15)
2
2
P(b j ) 




(

When h = [ hsd hrd ]T is given, < n, H ( xi − x j ) > would be a Gaussian random variable with zero
mean and variance

N0
|| H ( xi − x j ) ||2 . Accordingly,
2

 || H ( xi − x j ) ||
N 0 /2 log(α ij ) 

P (bi → b j | h) = Q
+

|| H ( xi − x j ) || 
2N0



(16)

where α ij = P (bi )/P (b j ) . From (16), we notice that the probability P (bi → b j ) depends on: 1- The
Hamming distance between bi and b j , wij , (i.e., the number of positions at which bi and b j are
different); 2- Whether bi and b j have the same probability.
Let P (α ij ) = P (bi → b j ) when wij = 1 and P2 = P (bi → b j ) when wij = 2 . From (16), we notice
1
that P2 doesn’t depend on α ij . Hence, substitute from (8) into (13), yield
142


(

PE ≤ 2 P2 + [P (α 41 ) + P (α 32 )]Pe + [P (α14 ) + P (α 23 )] 1 − Pe
1
1
1
1
sr

sr

)

(17)

In order to derive P (α ij ) , we consider the case when i = 1 and j = 4 . Substitution in (16) yields
1

log(α14 ) 
2

P (b1 → b4 | hsr ) = Q 2hsr γ +
2

2 2hsr γ 



(18)

2
where γ = E/N 0 . In order to find P (α14 ) , we average (18) over the distribution of hsr . Since all
1
channels have the same distribution, {∀i, j : wij = 1} , we have

∞ 
log (α ij ) 
 f X ( x)dx
P (α ij ) = ∫ Q 2 x +
1

0
2 2x 



(19)

where

f X ( x) =

1

γ

e − x/γ

(20)

Substituting from (20) into (19) with considering Q (u ) = 0.5erfc(u/ 2 ) yields

1
2γ

P (α ij ) =
1

∞



∫ erfc

0

x+

b  − x/γ
e dx
x

(21)

where b = log (α ij )/4 . Integrating (21) using integration by parts where w = erfc( x + b/ x ) ,

dv = e − x/γ dx , dw =

− 1  b  − ( x +b ) 2 /x
dx , and v = −γe − x/γ , yields
1 −  e
πx  x 


b  − x/γ γ

e −
− γerfc x +
x
π


b≥0
 I
= 1
1 + I1 b < 0

1
P (α ij ) =
1
2γ

∞

∫

1  b  − x/γ −( x + b )2 /x 
dx 
1 − e
x  x
0

(22)

where
I1 =

−1
2 π

∫

∞

0

1  b  − x/γ − ( x + b ) 2 /x
dx
1 − e
x  x

(23)

After some mathematical manipulations, we can write (23) as follows
2

(u + d )
 γ
d −

− e u du
(24)
∫0  1 + γ u 


where r = 2b(1 − 1 + 1/γ ) , d = b 1 + 1/γ , and u = x(1 + 1/γ ) . We can write (24) as follows

− e −r
I1 =
2 π

∞

1
u

143


− e −r
I1 =
2 π

∫

∞

0

1
u

 I
=  −r 2
− e + I 2

2

(u + d )

γ
d −
 −1 +
+ 1 − e u du

1+ γ
u


b≥0

b<0

(25)

where

I 2 = 1 −



 e −r

1+ γ  2 π


γ

∫

1 − (u
e
u

∞

0

+ d )2
u

du

(26)

By changing variables, let y = u and dy = dy/(2 u ) . After substitution in (26) we have

I 2 = 1 −



γ  e −r

1+ γ  π



γ
= 1 −

1+ γ


∫

∞

0

e

 e − (2 d + r )


π


 y2 +d
−

y


∫

∞

0






2

− y2 −

e

dy
d2
y2

dy

(27)

Using integration tables to find the integral in (27) yields [8]

1 
γ  −4 d −r
e
 1 −

1+ γ 
2 

I2 = 

 1 1 − γ e − r

 2
1+ γ 

 

b≥0
(28)

b<0

After substituting for the d, r and b , we have
1
1 
γ  − 2 (1+
 1 −
α ij
1+ γ 
2 
I2 = 
1
γ  − 2 (1−
1 
1−
α ij

2 
1+ γ 
 

1+

1+

1

γ

1

γ

)

α ij ≥ 1

(29)
)

α ij < 1

Substituting from (29) into (25) then into (22) yields
 1 − Γ − (1+1/ Γ )/2
α ij
if α ij ≥ 1

2

P1 (α ij ) = 
 1 + Γ − (1−1/ Γ )/2
if α ij < 1
1 − 2 α ij


γ

(30)

In order to derive P2 , we consider the case when i = 1 and j = 2 . Substituting
1+ γ
α ij = 1 into (16) yields
where Γ =

144


(

2
2
P(b1 → b2 | h) = Q 2(hsr + hrd )γ

)

(31)

2
2
Averaging (31) over the joint distribution of hsr and hrd yields [9]

2

 1 − Γ  1  m + 1 1 + Γ 

P2 = 
 ∑



 2  m = 0 m  2 
1
1 
2
= (1 − Γ ) 1 + Γ 
2
 2 

m

(32)

It is clear from (5) that if Es ≥ 0 or equivalently 0 ≤ Γsr ≤ 1 then the error probability in the S-R link
would be 0 ≤ Pe ≤ 0.5 . Therefore, substitution from (30) and (32) into the right hand side of (17)
sr

results in the following upper bound



 1 − Γsr
1
1  1
2
PUB = (1 − Γ ) 1 + Γ  + (1 − Γsr )1 − Γ
1+ Γ
2
 2  2
sr




1− Γ

 2Γ 
 

 


(33)

At higher receive SNR’s γ sr and γ , the upper bound of (33) can be approximated as follows

PUB ≈

γ
1 
1 +
2 
4γ  γ sr






(34)

V. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we present numerical results for both analysis and simulations.
0

10

-1

10

-2

Bit Error Rate BER

10

-3

10

-4

Upper Bound γSR = 0 dB

10

Simulations γSR = 0 dB
-5


10

-6

10


-7

10

0

5

10
15
20
Receive SNR at the destination, γ

25

30

Figure 2: Bit error probability versus γ = E/N 0 based on simulations and upper bound
145


Fig. 2 compares the bit error probability obtained from simulations with the exact upper bound
given by (33). The BER is plotted against the receive SNR γ = E/N 0 of S-D and R-D links at different
values of the the receive SNR of the S-R link γ sr or equivalently different values of the distance d sr
. We found that the derived upper bound is very tight. Therefore, the closed form expression of the
upper bound can be used to fully study and understand the diversity performance of the DF
cooperation protocol. We also found that the SNR of the S-R link highly affects the end-to-end error
rate of the system. At low values of γ sr the error rate of the S-R link Pe would be high and
sr

accordingly the system does not provide the expected diversity gain. We notice that the diversity gain
increases as the value of γ sr increases.
2
1.9
1.8

Diversity Order (L)

1.7
1.6
1.5
1.4
1.3

γ1 = 20 dB

1.2

γ1 = 30 dB
γ1 = 40 dB

1.1

γ1 = 50 dB
1

0

10

20

30
40
50
60
70
Receive SNR at the relay, γsr

80

90

100

Figure 3: Diversity gain versus γ sr at different values of γ 1
Since the diversity order is related to the slope of the BER curve with respect to γ and in order to
the study the effect of the γ sr on the diversity, let’s define the diversity order L as follows

L=

log10 PUB (γ 2 ) − log10 PUB (γ 1 )
(γ 2 (dB) − γ 1 (dB))/10

(35)

where PUB (γ i ) is the upper bound at γ i and i = 1,2 . The definition of (35) measures the change of
exponent of the BER with respect to one decade of the SNR in dB. Fig. 3 shows the diversity order
defined in (35) against the receive SNR of the S-R link γ sr at different values of γ 1 where

γ 2 = γ 1 + 10 dB. We found that full diversity of 2 is achieved at higher values of γ sr . We also found
that at fixed value of γ sr , higher diversity order is achieved at lower values γ , e.g., at γ sr = 40 dB,
the diversity order is ≈ 1.7 when γ 1 = 20 dB, and the diversity order is ≈ 1 when γ 1 = 50 dB. This
result is consistent with the approximation of (34) where the diversity order is close to 1 when
γ > γ sr and close to 2 when γ < γ sr and both of them are high. The intuitive explanation for this
phenomenon is that the enhancement of BER performance is dominated by the link with the lower
SNR. In other words, the link with the lower SNR is considered as the bottleneck of the BER
improvement. Therefore, when γ is small it will be the bottleneck and the BER is proportional to

1/γ 2 .
146


VI. CONCLUSIONS
In this paper, we presented the MAP decoding scheme for the wireless relay network that
adopts the decode-and-forward as a cooperation protocol. We derived a closed form expression for the
upper bound on the bit error probability. The numerical results show that the upper bound is very tight.

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