ML Decoder for Decode-and-Forward Based Cooperative Communication System
Manav R.Bhatnagar,Member,IEEE and Are Hjørungnes,Senior Member,IEEE
Abstract—Decode-and-forward(DF)protocol based coopera-tive communication is vulnerable to the erroneous relaying by the relay.In this paper,we derive a maximum-likelihood(ML) decoder for the DF protocol utilizing arbitrary complex-valued constellations including M-PSK,M-PAM,and M-QAM.A set-up of a single pair of source and destination with one relay is studied.The source and the relay utilize orthogonal uncoded transmissions.The relay performs ML decoding and forwards the decoded symbol to the destination,and it might commit errors in decoding the data.The ML decoder at the destination is obtained by maximizing the probability density function(p.d.f.) of the data received during two orthogonal transmissions at the destination under the assumption that the average probability of error of the source-relay link is known at the destination. The proposed ML decoder is a generalized decoder which is applicable to arbitrary constellations,whereas,one existing DF cooperative decoder is applicable to the real valued constellations like BPSK and M-PAM.One existing decoder is also applicable to M2-QAM constellations.We also derive a low-complexity piecewise linear(PL)decoder for arbitrary complex-valued M-point constellations which performs similar to the ML decoder for all signal-to-noise ratio values.An approximate expression of the symbo
l error rate(SER)of the PL decoder for M-PSK constellation is derived.By using the approximate SER expressions,it is proved that the proposed ML and PL decoders achieve full diversity of two in the cooperative system studied. Index Terms—Decode-and-forward protocol,cooperative com-munication,low complexity decoding,ML decoding.
I.I NTRODUCTION
I N recent years,cooperative communication has evolved
to become a candidate for next generation technology for wireless communication systems.By utilizing the cooperation among the distributed nodes,many advantages of multiple-input multiple-output(MIMO)system can be exploited.A cooperative user/node assists the other user/node by allowing the usage of its own antenna and circuitry for improving the quality of the reception at the destination node/user[1],[2]. In the decode-and-forward(DF)protocol[3,Subsection III-A.2],a source sends information to a relaying node and the destination node.The relay decodes the data sent by the source and retransmits the decoded data to the destination.Hence, the destination has two received replicas of the sent data and the quality of reception is expected to improve.However,if Manuscript received July25,2010;revised November14,2010and May, 7,2011;accepted September5,
2011.The associate editor coordinating the review of this paper and approving it for publication was D.Tuninetti.
M.R.Bhatnagar(corresponding author)is with the Department of Electri-cal Engineering,Indian Institute of Technology Delhi,Hauz Khas,IN-110016 New Delhi,India(e-mail:manav@ee.iitd.ac.in).
A.Hjørungnes was with the UNIK–University Graduate Center,University of Oslo,Gunnar Randers vei19,P.O.Box70,NO-2027Kjeller,Norway. This work was supported by the Research Council of Norway projects 176773/S10called OptiMO and183311/S10called M2M,which belongs to the VERDIKT program.
Digital Object Identifier10.1109/TWC.2011.100611.101341the channel between the source and relay is corrupted with a lot of noise and the channel transfer function is small in absolute value,the relay cannot decode the data perfectly and relays erroneous data to the destination.This causes loss in the diversity performance of the destination receiver[3, Subsection IV-B.2].
In order to avoid the performance degradation in a DF cooperative system,a maximum-likelihood(ML)decoder at the destination receiver is proposed for BPSK constellation in[4].This decoder maximizes the conditional p.d.f.of the received data at the destination terminal for given ave
rage probability of error in the source-relay link and results into very low decoding complexity.In[5],a cooperative combining for decode-and-forward based cooperative communication is proposed,where one of the receiving weights is chosen to maximize the instantaneous signal-to-noise ratio(SNR)of the cooperative link between the source and the destination.A decoder of the M-PAM constellation is derived by extending the decoder obtained in[5,Eq.(10)]for BPSK constellation in[6].The decoder in[6,Eq.(5)]considers all instantaneous pairwise decoding possibilities in the relay and the destination for decoding the transmitted data.Since a rectangular M2-QAM constellation can be obtained by using two M-PAM constellations,the decoders of[6]are also applicable to rectangular M2-QAM constellations.In the decoding schemes of[5],[6],the destination requires to possess exact knowledge of the channel coefficient of the source-relay link along with the channel coefficients of the source-destination and relay-destination links.Therefore,the existing decoders[5],[6] result into very high decoding complexity.
In this paper,our main contributions are:1)We derive an ML decoder of the DF based cooperative system with arbitrary constellation by using uncoded transmission and knowledge of the average probability of error of the source-relay link in the destination.2)The proposed ML decoder is obtained by maximizing the conditional joint p.d.f.of the data received during two orthogonal transmissions giv
en that the exact channel coefficients of the source-destination and relay-destination links,and average probability of error of the source-relay link are known in the destination.Since the average probability of error is a function of the average SNR[7],the proposed ML decoder1avoids the need of 1The proposed ML decoder maximizes the conditional joint p.d.f.of the received data for given value of the variance of the source-relay channel in the destination,therefore,the derived decoder is optimal under the assumption that the destination has perfect knowledge of the exact relay-source and
destination-source coefficients,and the variance of the source-relay channel. Hence,the proposed ML decoder performs poorer than the ML decoder having the exact channel coefficients for the source-relay,source-destination,and relay-destination.The assumption made in this article is more practical than the assumption of exact source-relay channel coefficient in the destination.
1536-1276/11$25.00c⃝2011IEEE
< relay.
exact knowledge of the channel coefficient of the source-relay link in the destination node contrary to the existing decoders of[5],[6].Therefore,the proposed ML decoder is simpler for practical implementation as compared to the existing decoders.3)We derive a low complexity piecewise linear(PL)decoder of arbitrary complex-valued M-point constellation based DF cooperative system which performs similar to the proposed ML decoder with significant reduction in the decoding complexity as compared to the proposed ML decoder.4)For the proposed PL decoder,we derive an approximate symbol error rate(SER)expression for the M-PSK constellation.5)Based on this SER analysis,we prove that the diversity order of the proposed PL decoder is two, which also equals the diversity of the ML decoder.
II.S YSTEM M ODEL
We consider a cooperative communication system,which consists of one source,one relay,and one destination terminal as shown in Fig.1.Each of them can either transmit or receive at a time.The transmission of the data from the source to the destination terminal is furnished in two orthogonal phases.In thefirst phase,the source broadcasts data to the destination and the relay.The relay decod
es the received signal by utilizing a coherent receiver and transmits the decoded symbol to the destination,in the second phase.To avoid the interference,source and relay use orthogonal channels for transmission[3].For ease of presentation,we assume that in both phases,the source and relay transmit stream of data through time-division multiplexing(TDM).Hence,the source has to remain silent in the second phase in order to maintain the orthogonality between the transmissions.However,in the frequency-division multiplexing(FDM)or the code-division multiplexing(CDM)schemes,the source and the relay can transmit at the same time.We have assumed a coherent cooperative system,where the relay has perfect knowledge of the channel gain of the source-relay link.Moreover,the destination has perfect knowledge of the channel gains of the source-destination and the relay-destination links,average SNR of the source-relay link,and the noise variance for the ML decoding of the data transmitted by the source.Let the source transmits the symbol x∈A,where A is an arbitrary M-point constellation including M-QAM,M-PAM,and M-PSK containing the following points:{x1,x2,x3,...,x M}. The data received at the relay in thefirst phase can be written as
y s,r=ℎs,r x+e s,r,(1)whereℎs,r is the channel gain between the source and the
relay and e s,r is the zero mean complex additive white Gaussian noise(AWGN)with variance N s,r.W
e can write the data received at the destination in thefirst phase as follows:
y s,d=ℎs,d x+e s,d,(2) whereℎs,d is the channel gain between the source and the destination and e s,d is the zero mean complex AWGN noise with variance N s,d.
III.ML D ECODER IN THE R ELAY AND THE D ESTINATION In this section,we will derive the ML decoders of the transmitted data in the relaying node and the destination.The ML decoder of arbitrary constellations over non-cooperative links is a well established concept[8].However,we will review the derivation of an ML decoder in the relay to facilitate the derivation of the ML decoder in the destination in a cooperative communication system.
A.ML Decoder in the Relay
Since x∈{x1,x2,x3,...,x M},the decoding of x can be seen as M hypothesis testing problem[9].By going through the analysis given in[9,Section2.3],it can be shown that the relay needs tofind out the following log-likelihood-ratio (LLR):
Λr p,q=ln
(p
y s,r∣ℎs,r,x=x p
p y
s,r∣ℎs,r,x=x q
)
,p,q=1,2,...,M,(3) where p,q denote the pair of any two different symbols belonging to the M-PSK,M-QAM,or M-PAM constellation,
and p y
s,r∣ℎs,r,x=x k
,k=p,q is the conditional p.d.f.of y s,r given that the channelℎs,r and the transmitted symbol x are known.Since p y
s,r∣ℎs,r,x=x p
∼CN(ℎs,r x p,N s,r)and p y
s,r∣ℎs,r,x=x q
∼CN(ℎs,r x q,N s,r),where CN(⋅)represents complex normal distribution,it follows from(3)that
Λr p,q=
∣ℎs,r∣2
N s,r
(
∣x q∣2−∣x p∣2
)
+2Re
{
y s,r
ℎ∗s,r
N s,r
(x p−x q)∗
}
.
(4) For M-PSK constellation∣x p∣2=∣x q∣2=C,where C is a constant,hence,
Λr p,q=2Re
{
y s,r
ℎ∗s,r
N s,r
(x p−x q)∗
}
.(5) The LLR of(4)is used as follows to decide about x p or x q:
Λr p,q
x p
≷
x q
0,(6) i.e.,ifΛr p,q is greater than zero,the decoder will decide that x p was transmitted by the source and ifΛr p,q is less than zero, x q will be decided as transmitted symbol by the decoder.The ML decoding based on(4)and(6)can be applied over the pairs of the symbols in the M-PSK constellation until thefinal estimate of the transmitted symbol is obtained.For example for QPSK constellation A={1,−1,j,−j},the decisions will be as follows:
ˆx=
⎧
⎨
⎩
1,ifΛr1,−1>0,Λr1,j>0,Λr1,−j>0,
−1,ifΛr−1,1>0,Λr−1,j>0,Λr−1,−j>0,
j,ifΛr j,1>0,Λr j,−1>0,Λr j,−j>0,
−j,ifΛr−j,1>0,Λr−j,−1>0,Λr−j,j>0.
(7)
The relay transmits the estimated symbolˆx in the second phase.
B.ML Decoder in the Destination
In the second phase,the data received in the destination can be written as
y r,d =ℎr,d ˆx +e r,d ,
(8)
where ˆx is the decoded symbol transmitted by the relay,ℎr,d is the channel gain between the relay and the destination,and e r,d is a complex zero-mean AWGN noise with variance N r,d .For deriving the ML decoder,we need to maximize the conditional p.d.f.of the data received in the destination in two orthogonal phases which is equivalent to maximize a likelihood ratio for decoding the data [9].In [10,Eq.(4)],an approximate ML decoder of multiple antennas based DF cooperative system is obtained by utilizing the exact channel state information (CSI)of the source-relay,relay-destination,and source-destination links for a complex-valued M -point constellation.However,in this paper,we aim at utilizing the average SNR of the source-relay link in the destination for finding the exact ML decoder of the data transmitted by the source and reducing the computational complexity of the ML decoding by using the PL approximation in a single antenna based DF cooperative system.
Let y =[y s,d ,y r,d ]be the vector containing the data received in both orthogonal phases by the destination.By using the analysis given in [9,Section 2.3],it can be shown that the destination needs to find out the following LLR for ML decoding of the data:
Λd
p,q =ln
(p y ∣ℎs,d ,ℎr,d ,x =x p ,ˆx ∈A p y ∣ℎs,d ,ℎr,d ,x =x q ,ˆx ∈A
),p,q =1,2,...,M,(9)where p y ∣ℎs,d ,ℎr,d ,x =x p ,ˆx ∈A is the conditional joint p.d.f.of the received vector y given that the channels of the source-destination and the relay-destination links,the symbol trans-mitted by the source x ,and the symbol transmitted by the relay ˆx are perfectly known in the destination.Since e s,d ,and e r,d are independent of each other,y s,d and y r,d are independent
of each other given ℎs,d ,ℎr,d ,x ,and ˆx
are perfectly known.Therefore,
p y ∣ℎs,d ,ℎr,d ,x =x p ,ˆx ∈A =p y s,d ∣ℎs,d ,x =x p p y r,d ∣ℎr,d ,x =x p ,ˆx ∈A .
(10)
If ϵis the average probability of error of decoding an M -point constellation in the relay,then we can write
p y r,d ∣ℎr,d ,x =x p ,ˆx ∈A =ϵp y r,d ∣ℎr,d ,ˆx =x p +(1−ϵ)p y r,d ∣ℎr,d ,ˆx =x p .
(11)
Since p y s,d ∣ℎs,d ,x =x p ∼CN (ℎs,d x p ,N s,d ),it follows that
p y ∣ℎs,d ,ℎr,d ,x =x p ,ˆx ∈A
=1πN s,d
e −1N s,d ∣y s,d −ℎs,d x p ∣2
×p y r,d ∣ℎr,d ,x =x p ,ˆx ∈A .(12)From (9)and (12),we have
Λd p,q =∣ℎs,d ∣2N s,d (∣x q ∣2−∣x p ∣2)+2Re {y s,d ℎ∗s,d N s,d
(x p −x q )∗
}+ln (p y r,d ∣ℎr,d ,x =x p ,ˆx ∈A
p y r,d ∣ℎr,d ,x =x q ,ˆx ∈A ).(13)
From [11,Section III],it follows that p y r,d ∣ℎr,d ,ˆx =x p denotes
the p.d.f.of a Gaussian mixture random variable.With this observation,we can express p y r,d ∣ℎr,d ,ˆx =x p as [11,Eq.(7)]
TABLE I
N UMBER OF COMPUTATIONS REQUIRED FOR THE PROPOSED ML
DECODER (17),PL DECODER (26),AND AN EXISTING DECODER [6,
E Q .(5)]FOR 16-QAM CONSTELLATION .Numerical Real summ Real multi Expo Loga computations -ation -plication -nential -rithm Existing decoder 536061445120[6,Eq.(5)]Proposed ML 2610393024015decoder Proposed PL 525
690
decoder
p y r,d ∣ℎr,d ,ˆx =x p
=1πN r,d (M −1)M ∑i =1,
i =p
e −1N r,d ∣y r,d −ℎr,d x i ∣2
.(14)From (11),(13),and (14),we have the following LLR:
Λd
p,q =∣ℎs,d ∣2N s,d (∣x q ∣2−∣x p ∣2)+2Re {y s,d ℎ∗s,d N s,d
(x p −x q )
∗}+ln ⎛⎜⎜⎜⎜⎜⎜⎜⎝ϵ
(M −1)∑M i =1i =p e −1N r,d (∣ℎr,d ∣2
∣x i ∣2−2Re {y r,d ℎ∗r,d x ∗i })+(1−ϵ)e −1N r,d (∣ℎr,d ∣2∣x p ∣2−2Re {y r,d ℎ∗r,d x ∗p })ϵ(M −1)∑M j =1j =q e −1N r,d (∣ℎr,d ∣2∣x j ∣2−2Re {y r,d ℎ∗r,d x ∗j })+(1−ϵ)e −1N r,d
(
∣ℎr,d ∣2∣x q ∣2−2Re {y r,d ℎ∗r,d x ∗q })⎞⎟⎟⎟⎟⎟⎟⎟
⎠.(15)
The ML decoding rule based on the LLR of (15)for QPSK constellation will be:ˆx =⎧ ⎨ ⎩1,if Λd 1,−1>0,Λd 1,j >0,Λd
1,−j >0,−1,if Λd −1,1>0,Λd −1,j >0,Λd −1,−j >0,j,if Λd j,1>0,Λd j,−1>0,Λd j,−j >0,−j,if Λd −j,1>0,Λd −j,−1>0,Λd −j,j >0.
(16)We should notice the following points from (15):
Observation 1:The ML decoder based on (15)works well with arbitrary constellations in contrast to the decoder in [6,Eq.(5)]which works for real valued constellations and regular M 2-QAM constellation only.
Observation 2:We can rewrite (15)as
Λd p,q =∣ℎs,d ∣2N s,d (∣x q ∣2−∣x p ∣2)+2Re {y s,d ℎ∗s,d N s,d
(x p −x q )∗
}+ln ⎛⎜⎜⎜⎜⎜⎝K +ϵ(M −1)e
−1N r,d (∣ℎr,d ∣2∣x q ∣2−2Re {y r,d ℎ∗r,d x ∗
q })+(1−ϵ)e −1N r,d
(∣ℎr,d ∣2
∣x p ∣2−2Re {y r,d ℎ∗r,d
x ∗p })K +ϵ
(M −1)e −1
N r,d (∣ℎr,d ∣2∣x p ∣2−2Re {y r,d ℎ∗r,d x ∗p })+(1−ϵ)e −1N
r,d (
∣ℎr,d ∣2∣x q ∣2−2Re {y r,d ℎ∗r,d x ∗q })
⎞⎟⎟⎟⎟⎟⎠,(17)where K =ϵ
(M −1)
M ∑i =1i =p,q
e −1N r,d (∣ℎr,d ∣2∣x i ∣2−2Re {y r,d ℎ∗r,d x ∗i })
.It can be seen from (17)that the ML decoding of the arbitrary
constellation is not very complicated contrary to what was predicted in [5],[12],[6].In Table I,we have shown the num-ber of computations (total number of real summations,real multiplications,and exponential and logarithmic calculations)required by the proposed ML decoder (17)and the decoder of [6,Eq.(5)]with the perfect knowledge of the average SNR of the source-relay link available at the destination.It can be seen from Table I that the proposed ML decoder (17)requires approximately half of the numerical computations compared to the decoder of [6,Eq.(5)]for decoding a symbol belonging to 16-QAM constellation,in addition to fifteen logarithmic
TABLE II
V ALUES OF T 1FOR DIFFERENT VALUES OF ϵUSED IN F IG .2.ϵ10−210−310−410−510−6T 1
±5.6937
±8.0054
±10.3089
±12.6115
±14.9141
calculations.
Observation 3:The proposed ML decoder requires knowl-edge of the exact channel gains of the source-destination and the relay-destination links,and the variance of the source-relay channel.Since the channel statistics vary far more slower than the exact channel coef ficients,it is always practical that the relay can estimate the channel variance of source-relay channel and forward it to the destination.The decoders of [6,Eq.(5)]and [10,Eq.(4)]need perfect channel knowledge of the source-relay,source-destination,and relay-destination links at the destination for decoding the data.If the channel is varying fast,it is very dif ficult for the destination to track the source-relay channel exactly.Therefore,the proposed scheme has more realistic assumptions.
Observation 4:The ML decoding rule (15)can be simpli fied for M -PSK constellation as
Λd
p,q =2Re {y s,d
ℎ∗s,d N s,d
(x p −x q )∗}
+ln ⎛⎜⎜⎜⎜⎜⎜
⎜⎜⎜⎝
ϵ
(M −1)
∑M
i =1i =p
e
2Re {
y r,d ℎ∗
r,d
N r,d (x i
−x q )∗
}+(1−ϵ)e 2Re {
y r,d ℎ∗r,d N r,d (x p
−x q )∗}
ϵ(M −1)∑M j =1j =q
e 2Re {
y r,d ℎ∗r,d N r,d (x j −x q )∗}+(1−ϵ)
⎞⎟⎟
⎟⎟⎟
⎟⎟⎟⎟⎠.(18)
For M =2,the log-likelihood ratio decoder of (18)is same as the ML decoder found in [4,Eq.(11)].
C.Piecewise Linear Approximation Based Decoder For M -PSK constellation,we can rewrite (18)as
Λd p,q =2Re {y s,d
ℎ∗
s,d
N s,d
(x p −x q )∗}+ln ⎛⎜⎜⎜⎜⎜⎜
⎜⎜⎜⎝
ϵ
(M −1)e
2Re {
y r,d ℎ∗r,d N r,d x
∗
q
}
+(1−ϵ)e
2Re {
y r,d ℎ∗r,d N r,d x
∗
p
}
+ϵ
(M −1)∑M
i =1i =p,q
e
2Re {
y r,d
ℎ∗
r,d N r,d x ∗
i
}ϵ
(M −1)e 2Re {y r,d ℎ∗r,d N r,d x ∗p }+(1−ϵ)e 2Re {y r,d ℎ∗
r,d N r,d x ∗q }+ϵ
(M −1)
∑M
j =1
j =p,q
e
2Re {
y
r,d ℎ∗r,d N
r,d
x ∗
j
}⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(19)It is dif ficult to simplify (19)further.However,let us neglect the terms ϵ(M −1)∑M
i =1,i =p,q e 2Re {y r,d ℎ∗
r,d N r,d x ∗i
}
and
ϵ(M −1)
∑M
j =1,j =p,q
e 2Re {y r,d ℎ∗r,d N r,d
x ∗
j
}
in (19)to obtain a sub-optimal decoder.It will be shown by using QPSK,8-PSK,
and 16-PSK signaling schemes in Fig.5that neglecting these terms does not degrade the performance of the decoder signi ficantly at all SNRs.After some algebra,we get the following approximate LLR from (19):Λd p,q ≈t 0+ln (ϵ+(M −1)(1−ϵ)e
t 1
ϵe t 1+(M −1)(1−ϵ)
),(20)where t 0=2Re {y s,d
ℎ∗s,d N s,d
(x p −x q )∗
}
,t 1=2Re {y r,d
ℎ∗r,d
N r,d
(x p −x q )∗}
.(21)Let f (t 1)≜ln (ϵ+(M −1)(1−ϵ)e t 1
ϵe t 1+(M −1)(1−ϵ)
)
.(22)
It can be seen from (22)that when ϵ=0,f (t 1)=t 1and for very large and very small values of t 1,f (t 1)is clipped to
T 1=±ln [(M −1)(1−ϵ)
ϵ
].(23)
We have plotted f (t 1)for different values of ϵand t 1for QPSK constellation (M =4)in Fig.2.We have also listed the values of T 1for different values of ϵin Table II.It can be seen from Fig.2and Table II that when t 1>T 1,f (t 1)≈T 1and when t 1<−T 1,f (t 1)≈−T 1.Moreover,for −T 1≤t 1≤T 1,f (t 1)≈t 1.Therefore,we can approximate f (t 1)by a PL function as follows:f (t 1)≈f PL (t 1)≜⎧
⎨⎩
−T 1,if
t 1<−T 1,t 1,if −T 1≤t 1≤T 1,T 1,if t 1>T 1.
(24)The PL based approximate decoder is given as
Λd p,q =t 0+f PL (t 1).
(25)A PL decoder for the DF based cooperative communication system utilizing uncoded BPSK data
is proposed in [4,Sub-section III-B.2].It can be seen from (23),(24),and (25),that the proposed PL decoder reduces to the existing PL decoder of the BPSK constellation [4,Subsection III-B.2]for M =2.Therefore,the proposed PL decoder is a generalization of the PL decoder for the BPSK constellation in [4,Subsection III-B.2]to M -PSK constellation.
By following a similar procedure as given above,we can derive a PL decoder for an arbitrary M -point constellation from (17)as follows:
Λd p,q =v 0+f PL (v 1),(26)where
v 0=
ℎ∗s,d
2N s,d (∣x q ∣2−∣x p ∣2
)
+2Re {
y s,d
ℎ∗s,d
N s,d
(x p −x q )∗
}
,v 1=
ℎ∗r,d 2N r,d (∣x q ∣2−∣x p ∣2)+2Re {y r,d ℎ∗r,d N r,d
(x p −x q )∗
},(27)
and f PL (v 1)is de fined in (24).It can be seen from Table I that the proposed PL decoder (26)requires approximately five times less numerical computations as compared to the proposed ML decoder in (17)for 16-QAM constellation.Moreover,no exponential or logarithmic calculations are needed for the proposed PL decoder.
An intuitive reason why the proposed PL decoder is working is as follows:Let x p and x q ,where p,q =1,...,M,p =q ,be the symbols belonging to an M -point constellation.The average probability of error at the relay terminal in decoding x p as x i ,where i =1,...,M,i =p ,is ϵ/(M −1).For finding the approximate LLR corresponding to x p and x q in the destination,the proposed PL decoder only considers the error in decoding of x p as x q in the relay.Hence,the PL decoder in the destination discards the remaining M −2possibilities (other than x q )of the error in the relay.It further simpli fies the decision making procedure by utilizing a clipping operation as can be seen from (24).
It can be seen from (15),(18),(25),and (26),that the proposed ML and PL decoders use the average probability of error ϵwhich can be calculated analytically based on the average SNR of the source-relay link.Therefore,the transmitted signal can be decoded in the destination without instantaneous CSI of the source-relay link.
IV.P ERFORMANCE A NALYSIS OF THE PL D ECODER FOR
M -PSK C ONSTELLATION
SER analysis of the ML decoders in (15)and (18)is prohibitively complex due to their non-linear nature.Instead,we derive the average SER of the PL detector in (25)which is not only simpler to impl
ement but also closely matches the performance of the ML detector.We obtain the SER of the PL decoder utilizing M -PSK constellation.
A.SER of M -PSK Constellation
Let x q ∈A ,where A is an M -PSK constellation,be the symbol transmitted by the source.There will be an error in the decision if the destination wrongly decides that x p ∈A is more probable to be transmitted than x q ,on the basis of the LLR function calculation for the pair {x p ,x q }.From (2),(8),and (21),we can write t 0and t 1as t 0=2Re {∣ℎs,d ∣2N s,d x q (x p −x q )∗+ℎ∗
s,d
N s,d (x p −x q )∗e s,d }
,
t 1=2Re {∣ℎr,d ∣2N r,d ˆx (x p −x q )∗+ℎ∗
r,d
N r,d
(x p −x q )∗e r,d }
.
(28)
Let us de fine the following intermediate variables:
K ≜
∣ℎs,d ∣2N s,d x q (x p −x q )∗
,K 1≜
ℎ∗s,d N s,d
(x p −x q )∗
.(29)From (28)and (29),the conditional p.d.f.of t 0can be written
as
p t 0∣ℎs,d ,N s,d ,x p ,x q =1√4π∣K 1∣2
N s,d
e −12(t 0−2Re {K })22∣K 1∣2N
s,d .(30)
The probability of error of the PL decoder can be expressed using three mutually exclusive events.The conditional SER,given that the channel gains of all involved links are known,is
P e (ℎs,d ,ℎr,d )=Pr {t 0−T 1>0∣t 1<−T 1,x =x q }×Pr {t 1<−T 1∣x =x q }+Pr {t 0+T 1>0∣t 1>T 1,x =x q }×Pr {t 1>T 1∣x =x q }+Pr {t 0+t 1>0,−T 1≤t 1≤T 1∣x =x q },
(31)
where Pr {⋅}represents probability of an event.By observing the Gaussian nature of t 0from (30)we can deduce that Pr {t 0−T 1>0∣t 1<−T 1,x =x q }=Q
(T 1−2Re (K )
√
2N s,d ∣K 1∣
),Pr {t 0+T 1>0∣t 1>T 1,x =x q }=Q
(
−T 1−2Re (K )√2N s,d ∣K 1∣
)
.(32)
Next,by using the fact that the relay can decode the data erroneously,we have:
Pr {t 1<−T 1∣x =x q }=(1−ϵ)Pr (t 1<−T 1,x =x q ,ˆx =x q )
+ϵPr (t 1<−T 1,x =x q ,ˆx =x q ),Pr {t 1>T 1∣x =x q }=(1−ϵ)Pr (t 1>T 1,x =x q ,ˆx =x q )+ϵPr (t 1>T 1,x =x q ,ˆx =x q ).(33)Let us now de fine the following intermediate variables:decoder
J q ≜∣ℎr,d ∣2
N r,d x q (x p −x q )∗
,
J i ≜∣ℎr,d ∣2
N r,d x i (x p −x q )∗
,i =1,2,...,M,
J 0≜ℎ∗r,d N r,d
(x p −x q )∗
.(34)
The conditional p.d.f.of t 1given that the relay makes correct decision can be expressed as
p t 1∣ℎr,d ,N r,d ,ˆx =x q ,x p ,x q =1
√
4π∣J 0∣2
N r,d
e −12
(t 1−2Re {J q })2
2∣J 0∣2N r,d
.
(35)
When the relay makes a wrong decision,then ˆx =x q and the conditional p.d.f.of t 1is given as [11,Eq.(7)]
p t 1∣ℎr,d ,N r,d ,ˆx =x q ,x p ,x q =
1(M −1)√4π∣J 0∣2
N r,d
×
M ∑
i =1,i =q
e
−12
(t 1−2Re {J i })2
2∣J 0∣2N r,d
.(36)
From (35),it follows that
Pr {t 1<−T 1∣x =x q ,ˆx
=x q }=1−Pr {t 1>−T 1∣x =x q ,ˆx =x q }
=1−Q
(
−T 1−2Re {J q }√2N r,d ∣J 0∣
)(37)and,from (36),we have
Pr {t 1<−T 1∣x =x q ,ˆx =x q }=
1
M −1×
M ∑i =1
i =q
[1−Q (−T 1−2Re {J i }
√2N r,d ∣J 0∣)].(38)
Hence,from (33),(37),and (38),we get Pr {t 1<−T 1∣x =x q }=1−(1−ϵ)Q
(
−T 1−2Re {J q }
√2N r,d ∣J 0∣)
−ϵM −1M ∑i =1i =q
Q
(−T 1−2Re {J i }
√2N r,d ∣J 0∣
).(39)On the similar lines,we find that
Pr {t 1>T 1∣x =x q }=(1−ϵ)Q
(T 1−2Re {J q }
√
2N r,d ∣J 0∣
)
+ϵM −1M
∑i =1i =q
Q (T 1−2Re {J i }√2N r,d ∣J 0∣
).(40)Let us now evaluate the third term of (31).By the fact that the relay can decode the data wrongly,we get Pr {t 0+t 1>0,−T 1≤t 1≤T 1∣x =x q }
=(1−ϵ)Pr {t 0+t 1>0,−T 1≤t 1≤T 1∣ˆx =x q ,x =x q }+ϵPr {t 0+t 1>0,−T 1≤t 1≤T 1∣ˆx =x q ,x =x q }.(41)When the relay makes correct decision,the probability of joint event can be expresses as [13]
Pr {t 0+t 1>0,−T 1≤t 1≤T 1∣ˆx =x q ,x =x q }
=∫T 1−T 1∫∞−w
p t 1(w )p t 0(z )d w d z
=∫T 1
−T 1
p t 1(w )(∫∞−w
p t 0(z )dz )dw.(42)From (30),(35),and (42),
Pr {t 0+t 1>0,−T 1≤t 1≤T 1∣ˆx =x q ,x =x q }=
1
2∣J 0∣√πN r,d ×∫T 1−T 1
e −12(w −2Re {J q })
22∣J 0∣2N r,d
Q
(−w −2Re {K }√2N s,d ∣K 1∣)dw.(43)On the similar lines it follows that Pr {t 0+t 1>0,−T 1≤t 1≤T 1∣ˆx =x q ,x =x q }
=12(M −1)∣J 0∣√πN r,d ∫T 1−T 1Q
(
−w −2Re {K }
√2N s,d ∣K 1∣
)×
M ∑
i =1
i =q
e
−12
(w −2Re {J i })2
2∣J 0∣2N r,d
dw.(44)
It is dif ficult to solve the integrals of (43)and (44)analytically.
Therefore,for the conditional SER calculations we can use the results of (43)and (44)by calculating the integrals numerically.From (31),(39),(40),(42),(43),and (44),we can calculate the conditional (conditioned over channels)pairwise probability of error when the receiver in the destination chooses wrong symbol x p in place of the original symbol transmitted by the source x q .
The constellation diagram of an arbitrary M -PSK constella-tion is shown in Fig.3.The decision boundaries corresponding to the symbol x 1are also shown.From the equiprobability of the constellation points it can be deduced that
ݔݔଵݔଶFig.3.Constellation diagram of M -PSK signals.Z i ,i =1,2,...,M
represents the region of correct decision.
Pr [error ]=1M M
∑
i =1
Pr [error ∣x i ]=Pr [error ∣x 1].(45)
Let x q =x 1=1be the transmitted symbol.The SER of the M -PSK constellation can be approximated as [8,Eq.(5.2.25)]
P e (ℎs,d ,ℎr,d )≈P x 2,1e
(ℎs,d ,ℎr,d )+P x M ,1
e (ℎs,d ,ℎr,d ),(46)where P x 2,1e
(ℎs,d ,ℎr,d )and P x M ,1
e (ℎs,d ,ℎr,d )are the condi-tional probabilities o
f error of decodin
g x 2and x M ,respec-tively,in place of the originally transmitted symbol x 1=1.We have obtained (46)by using the union bound approac
h [8]and truncating to the two dominant terms.Therefore,the approximate analytical SER follows the exact SER tightly at high SNRs.
The average approximate SER of the proposed PL decoder utilizing M -PSK constellation can be obtained by averag-ing (46)over the channels ℎr,d and ℎs,d .Let us now de-fine the following instantaneous signal-to-noise ratios (SNRs)
γs,d ≜∣ℎs,d ∣2/N s,d and γr,d ≜∣ℎr,d ∣2
/N r,d .Since ℎr,d and ℎs,d are Gaussian distributed,γs,d and γr,d will be Rayleigh distributed as p γs,d (γs,d )=1¯γs,d
e −γs,d
¯γ
s,d ,
p γr,d (γr,d )=1¯γr,d
e −γr,d
¯γ
r,d ,(47)
where ¯γs,d and ¯γ
r,d are average SNRs of the source-destination and the relay-destination links,respectively.The approximate average SER for M -PSK can be obtained by solving the following integral [7,Eq.(5.1)]:E γs,d ,γr,d [P e (γs,d ,γr,d )]=
∫∞0∫
∞
P e (ℎs,d ,ℎr,d )p γs,d (γs,d )
×p γr,d (γr,d )dγs,d dγr,d .
(48)
It is dif ficult to arrive at a closed form solution of (48),however,one can numerically calculate the integrals of (48)to find the SER of the proposed PL decoder with M -PSK constellations.
Let us take an example of QPSK constellation to demon-strate the SER calculations based on the above analysis.For QPSK constellation,the source transmits a symbol in {1,−1,−j,j }.Let us assume that it has transmitted x q =1.From (48)the approximate average SER for QPSK constella-tion will be given as
版权声明：本站内容均来自互联网，仅供演示用，请勿用于商业和其他非法用途。如果侵犯了您的权益请与我们联系QQ:729038198，我们将在24小时内删除。
发表评论