A cooperative transmission scheme based on symbol constellation expansion over multipath fading channels

A cooperative transmission scheme based on symbol constellation expansion over multipath fading channels

Jong In Park^a, Youngpo Lee^a, Sun Yong Kim^b, Gyu-In Jee^b & Seokho Yoon^a*

aCollege of Information and Communication Engineering, Sungkyunkwan University, Suwon 440-746, Korea

bDepartment of Electronics Engineering, Konkuk University, Seoul 143-701, Korea Received 29 July 2012; Accepted 21 February 2013

In this paper, we propose a novel cooperative transmission scheme based on a symbol constellation expansion over Rayleigh and Rician multipath fading channels. The proposed scheme expands the symbol constellation through a unique combination of the original symbols, thus increasing the minimum distance among the single points. We also analyze the bit error rate (BER) performance showing that the proposed scheme achieves the fourth order diversity. Numerical results demonstrate that the proposed scheme not only achieves the fourth order cooperative diversity, but also has a better BER performance than those of the conventional schemes.

Keywords: Asynchronous cooperative communication systems, Constellation, Cooperative diversity order, Alamouti coding, Multipath fading channels

In practical communication systems, there are several impairments including noise, interference, and multipath fading^1,2. Although the multiple-input multiple-output (MIMO) system based on multiple antennas is an efficient tool to provide a spatial diversity for combating the effects of fading³, it may be impractical to accommodate multiple antennas on mobile devices due to the limitations of size, cost, and power⁴. The cooperative systems can overcome the limitations of MIMO systems, since they provide the spatial diversity referred to as the cooperative diversity by forming the virtual MIMO systems via distributed nodes with only a single antenna^5,6.

One of the main drawbacks of the cooperative systems is that the symbols from different relay nodes may arrive at the destination node at different time instants, causing a timing error. Although several cooperative transmission schemes have been proposed to provide the cooperative diversity for such asynchronous environments^7-9, they are difficult to implement since a symbol decoding step is required at the relay nodes. Performing time-reversal and conjugation operations at relay nodes and constructing the well-known Alamouti coded form¹⁰ at the destination node, Li and Xia¹¹ proposed a transmission scheme providing the second-order

cooperative diversity without the symbol decoding process. Unfortunately, the scheme of Li and Xia¹¹ performs poorly over multipath fading channels.

Li et al.¹² introduced a modified version of the scheme of Li and Xia¹¹, which overcomes the intersymbol interference in multipath fading channels by performing the time-reversal operation at only one of the relay nodes. Moreover, the scheme of Li et al.¹² achieves the fourth order cooperative diversity with two relay nodes by repeating the data symbols across the orthogonal frequency division multiplexing (OFDM) subcarriers. On the other hand, a cooperative transmission scheme providing the same cooperative diversity order and bit error rate (BER) as those of the scheme of Li et al.¹² was proposed by Lee et al.¹³, where an operation similar to the time-reversal at the relay node used by Li et al.¹² is performed at the source node, thus reducing the relay complexity.

In this paper, exploiting symbol constellation expansion through a unique combination of data symbols, a novel cooperative relay scheme is proposed and it is verified that the proposed scheme has the fourth order cooperative diversity with two relay nodes. Numerical results show that the proposed scheme achieves the fourth order cooperative diversity and at the same time has a better BER performance than those of the conventional schemes.

___________

*Corresponding author (E-mail:syoon@skku.edu)

System Model

We consider a cooperative system model with a source node S, two relay nodes R₁ and R₂, and a destination node D as shown in Fig. 1. As the impairments of the wireless communication systems, we consider the fading and white noise^14,15. The n^th impulse response coefficient is given by

1

, ,

=0

( ) = ( )

L

SRm SRml SRml

l

h n α δ n τ

−

∑

− ^{… (1)}

for the channel between the source and the m^th relay nodes, and

1

, ,

=0

( ) = ( )

Q

R Dm R D qm R D qm

q

h n α δ n τ

−

∑

− ^{… (2)}

Proposed Scheme

In this section, the operation of the proposed scheme is described in detail for source, relay, and destination nodes.

Source node

The source node first generates two complex- valued OFDM data symbol blocks

= [ (0), (1), , ( 1)]

for = 1 and 2,

T

d Xd Xd Xd N

d

−

X … (3)

where X_d( )k and [ ]⋅^T denote the phase shift keying (PSK) data symbol on the k^th subcarrier of X_d and the transpose operation, respectively, and N is the number of the subcarriers. Then, the symbol constellation is expanded as follows:

= [ (0), (1), , ( 1)]

for = 1, 2,3, and 4,

T

d Cd Cd C Nd

d

−

C … (4)

where

1 2

* *

2 1

1 2

* *

2 1

( ) =

1/ 2{ ( ) ( )}, when = 1 1/ 2{ ( ) ( )}, when = 2 1/ 2{ ( ) ( )}, when = 3 1/ 2{ ( ) ( )}, when = 4 C kd

X k jX k d

X k jX k d

X k jX k d

X k jX k d

 +



− +



 −

 −



… (5)

denotes the complex-valued data symbol on the k^th subcarrier of C_d satisfying

* 1

* 1

, when = 1 and 3

= , when = 2 and 4

d d

d

j d

d

+

−





C C

C … (6)

with ( )⋅^*denoting the conjugation operation. After inserting the cyclic prefix (CP), finally, we transmit the d^th OFDM symbol

1 1

1

= [ (0), (1), , ( 1)]

= [ ( ), ( 1),

, ( 1), (0), , ( 1)] ,

T

d d d d s

d G d G

T

d d d

P P u u u L

P c N N c N N

c N c c N

−

− − +

− −

u

… (7)

to the relay nodes, where P₁ is the transmit power per an OFDM data symbol and L_sN+N_G with the length N_Gof CP and

1

2 /

=0

( ) = IDFT { }

= 1 ( ) ,

for = 0, , 1,

d N d

N

j kn N d

k

c n

C k e N

n N

− π

−

∑

C

… (8)

where IDFT ( )_N ⋅ denotes the N-point inverse discrete Fourier transform (IDFT). It is assumed that N_G is longer than the sum of the total propagation path delay from the source node to the destination node and the maximum relative timing difference between symbols arriving at the destination node from the relay nodes.

Relay nodes

Four consecutive OFDM symbols received at the m^th relay node can be expressed as

,

1 ,

1 ,

=

/ 2 , when = 1 and 2

/ 2 , when = 3 and 4,

d m

d SRm d m

d SRm d m

P d

P d

 ⊕ +



⊕ +



v

u h n

u g n

… (9)

Fig. 1  A cooperative system model

where _SR [ _{SR ,0}, _{SR ,1},

m = α m α m

h , _{SR ,L 1}] ,^T

α m ₋

nd m, is the additive white Gaussian noise (AWGN) vector for the d^th received symbol of the m^th relay node with zero mean and unit variance, ⊕ denotes the linear convolution, and _SR = [ _{SR ,0}, _{SR ,1},

m β m β m

g , _{SR ,L 1}] .^T β m ₋

Here,

{

β^SRm^,l

}

l^L−₀¹

= are assumed to follow the same distribution as that of

{

^SRm^,l

}

l^L₀^1:

α ⁻

= A different notation is used for the channel coefficient in the last two transmit symbol intervals between the source and the m^th relay nodes since the channels are assumed to be constant during the two transmit symbol intervals.

Then, each relay node helps destination node to construct the Alamouti coded form by transmitting symbol u_{d m,} as shown in Table 1 to the destination node during four consecutive OFDM symbol durations, where u_{d m,} and P₂ denote the d^th transmit symbol of the m^th relay node and the transmission power at the relay nodes, respectively.

Destination node

Now, we describe the demodulation and decoding steps at the destination node. Without loss of generality, we consider the two received symbols during the first two OFDM symbol durations. The received symbols during the last two OFDM symbol durations can be demodulated and decoded in the same manner. After removing the CP, the last τ₁^'=

1,

(max( ) 1)

G SR l

N − τ − samples of the received symbols are shifted to the front of the received symbols. Then, the received symbols can be expressed as

1

'

1 1 1 1,1

' '

1 2

1 2

'

2,2 2 1

[{ /2(IDFT ( )) }

{ /2(IDFT ( ))

} ]

N SR

R D N SR

R D

P P

τ

γ

=

⊗ +

⊗ + ⊗

+ ⊗ ⊗ +

y

C h n

h C h

n h G w

… (10)

for the first received symbol and

2

'

1 1 1 1,1

' '

1 2

1 2

'

2,2 2 2

=

[{ /2( IDFT ( )) }

{ /2( IDFT ( ))

} ] ,

N SR

R D N SR

R D

P j

P j

τ

γ − ⊗ +

⊗ + ⊗

+ ⊗ ⊗ +

y

C h n

h C h

n h G w

… (11)

for the second received symbol, where

( )

2 1

= P P 2 ,

γ + ⊗ denotes the circular

convolution, ^'_SR = [ _{SR ,0}, _{SR ,1}, ,

m α m α m

h

, 1,0, , 0] ,^T

SRmL

α ₋ ^'_{R D} = [ _{R D,0}, _{R D,1},,

m α m α m

h

, 1

, _{R D Q} ,0, ,0] ,^T α m ₋

G_τ = [0_τ,1,0, , 0] ^T with 0_τ and τ denoting all-zero vectors with dimension 1×τ and a relative timing difference between the symbols arriving at the destination node from the relay nodes R₁ and R₂, respectively, and w_d denotes an AWGN vector with zero mean and unit variance added to the d^th received symbol at the destination node. Thus, the DFT outputs are obtained as

1 1

1

2 /

1

=0

1 1 1, 1 ,

1 2 2, 2 ,

2 /

1,1 1 ,

2 /

2,2 , 1

2

( ) = DFT { }

1 ( )

[ /2 ( ) ( ) ( )

/2 ( ) ( ) ( )

( ) ( )

( ) ( ) ] ( )

N N

j kn N n

SR p R Dp

SR p R Dp

j k N

R Dp j k N R Dp

Y k

y n e N

P C k H k H k P C k H k H k

e N k H k

N k H k e W k

π

π τ

π τ

γ

−

−

−

−

=

= +

× +

+ +

∑

y

… (12)

and

Table 1  Processing at the relay nodes of the proposed scheme

Relay node 1 Relay node 2

OFDM

Symbol 1 _1,1 ²

1,1 1

= 2

P P+

u v _1,2 ²

2,2 1

= 2

P P+

u v

OFDM

Symbol 2 _2,1 ²

1,1 1

= 2

j P

− P

u + v _2,2 ²

2,2 1

= 2

j P P+

u v

OFDM

Symbol 3 _3,1 ²

3,1 1

= 2

P P+

u v _3,2 ²

4,2 1

= 2

P P+

u v

OFDM

Symbol 4 _4,1 ²

3,1 1

= 2

j P

− P

u + v _4,2 ²

4,2 1

= 2

j P P+

u v

2 2

1

1 1, 1 ,

( ) = DFT { }

[ { ( )} ( ) ( )

2

N

SR p R Dp

Y k

P jC k H k H k γ

= −

y

1

2 , ,

2 2

2 /

1,1 1 ,

2 /

2,2 2 , 2

{ ( )} ( ) ( )

2

( ) ( )

( ) ( ) ] ( )

SR R D

p p

j k N

R Dp j k N R Dp

P jC k H k H k

e N k H k

N k H k e W k

π τ

π τ

−

−

+

× + +

+

… (13)

for n= 0, ,N−1, where DFT ( )_N ⋅ denotes the -point

N DFT and N_{d m,} ( ),k _{SR ,} ( ),

H m p k _{R D,} ( ),

m p

H k

and W k_d( ) are the DFT outputs of n_{d m,} , ^'_SR , h m ^'_{R D},

h m

and w_d, respectively. Using the property in Eq. (6), we can rewrite Eq. (12) and Eq. (13) in the following matrix form:

1

* 2

1,

1 1

1 2 2,

( ) =

( )

( ) /2 ( )

( ) ,

/2 ( ) ( )

p p

p

Y k Y k

G k P C k

H k

G k P C k

γ

 

 

 

   

  + 

   

 

… (14)

where G_1,p and G_1,p denote the noise term of Y k₁( ) and Y k₂( ), respectively. H_p( )k is the channel matrix defined as

1, 2,

* *

2, 1,

( ) ( )

( ) = ,

( ) ( )

p p

p

p p

H k H k

H k

H k H k

 

 

 −  … (15)

where _{1, , ,}

1 1

( ) = ( ) ( )

p SR p R D p

H k H k H k and

2 /

2,_p( ) = _SR2,_p( ) _{R D p}2 , ( ) ^{j k N}.

H k H k H k e^{− π τ} Clearly, the channel matrix H_p( )k is the Alamouti coded form, and thus, we can obtain the estimates c k₁( ) and c k₂( ) as

1 1

*

2 2

( ) ( )

= ( )

( ) ( )

H p

c k Y k

H k

c k Y k

   

   

    … (16)

for C k₁( ) and C k₂( ), respectively. It should be noted that the estimates c k₃( ) and c k₄( ) of C k₃( ) and

4( )

C k can be obtained in the same manner. Lastly, we can obtain the estimates x k₁( ) and x k₂( ) as

1

1 3

2 4

1 3

2 4

( ) =

1[Re{ ( ) ( )}

2

Im{ ( ) ( )}]

[Im{ ( ) ( )}

2

Re{ ( ) ( )}]

x k

c k c k c k c k j c k c k

c k c k +

− +

+ +

− +

… (17)

and

2

1 3

2 4

1 3

2 4

( ) =

1[Im{ ( ) ( )}

2

Re{ ( ) ( )}]

[Re{ ( ) ( )}

2

Im{ ( ) ( )}]

x k

c k c k c k c k j c k c k

c k c k

−

− −

− −

− −

… (18)

for the data symbols X k₁( ) and X k₂( ), respectively, where Re{}⋅ and Im{}⋅ denote the real and imaginary parts, respectively.

Analysis on the Diversity Order

With the assumption of Rayleigh distributed channels, the expected value of BER P_b is known to be inversely proportional to the transmit signal-to- noise ratio (SNR) to the power of diversity order .ν

[ ] (

b

)

^,

E P ∝ S N ^−ν … (19)

where ν and ^E

[ ]

i denote the diversity order and the expectation operation, respectively¹⁶. Based on the relationship in Eq. (19), we can evaluate the cooperative diversity order of the proposed scheme.

From Eq. (14)-(18), the estimate x k₁( ) of the proposed scheme can be expressed as

( ) ( )

{ (

( ) ( ) ) } ^{( )}

( ) ( ) ( ) ( )

(

( ) ( ) ( ) ( ) )

1

2 2

1 1, 2,

2 2

3, 4, 1

* *

1, 1, 2, 2,

* *

3, 3, 3, 3,

( ) =

2 1

2

p p

p p

p p p p

p p p p

x k

P H k H k

H k H k X k

H k G k H k G k H k G k H k G k

γ +

+ +

+ +

+ +

… (20)

where H3,p

( )

k and H4,p

( )

k denote DFT outputs of the channel coefficients for the third and the fourth received symbols at the destination node, respectively. Also, G3,p

( )

k and G4,p

( )

k are DFT outputs of the noise terms for the third and the fourth received symbols at the destination node, respectively. For simplicity, we denote the variances of the DFT outputs of the channel coefficient and noise by ρ and σ_N², respectively.

Denoting the transmit SNR per a transmitted OFDM symbol by SNR_T =P₁/ 2σ_N², we can express the received SNR (SNR )_R of the estimate

₁( ) X k as

( ) ( )

( ) ( )

2 2

1, 2,

2 1

2 2

3, 4, 2

SNR

2 2

2 2

p p

R

p p

N

H k H k

P

H k H k

γ

σ



=  +

  + +  

… (21)

The BER p_{b p,} of the proposed scheme can be expressed as

( )

( ) ( )

,

2 2

1, 2,

2 1

SNR

2 2

b p R

p p

P Q

H k H k

Q γ P

 

 

=   +

( ) ( )

( ) ( )

(

( ) ( ) ) )

2 2

3, 4, 2

2 2 2

1

1, 2,

2

2 2

3, 4,

2 2

exp 4

p p

N

p p

N

p p

H k H k

P H k H k

H k H k

σ

γ σ



 

+ +  

≤ − +



+ +

… (22)

By using Eq. (22), we can obtain expectation

,

E P b p of the BER p_{b p,} as

( ( )

( ) ( )

( ) )

( ( ) ^{( )}

( ) ( ) ) )

( ( ) ) ( ^{( )} )

( ( ) ) ( ^{( )} )

( ( ) ) ( ^{( )} )

( ( ) ) ( ^{( )} )

,

2 1 2

2 1,

2 2

2, 3,

2 4,

2 1

0 0 0 0 2

2 2

1, 2,

2 2

3, 4,

2 2

1, 2,

2 2

3, 4,

2 2

1, 2,

2 2

3, 4,

exp 4

exp 4

b p

p N

p p

p

N

p p

p p

p p

p p

p p

p p

E P

E P H k

H k H k

H k

P

H k H k

H k H k

p H k p H k p H k p H k d H k d H k d H k d H k

γ σ

γ σ

∞ ∞ ∞ ∞

 

 

 

≤  −

 



+ +

  +  

= −



× +

+ +

×

×

×

×

∫ ∫ ∫ ∫

( )

( ( ) ) ( ^{( )} )

( )

( ( ) ) ( ^{( )} )

( )

( ( ) ) ( ^{( )} )

( )

( ( ) ) ( ^{( )} )

( )

2 2

1 2 1, 0

2 2

1, 1,

2 2

1 2 2, 0

2 2

2, 2,

2 2

1 2 3, 0

2 2

3, 3,

2 2

1 2 4, 0

2 2

4, 4,

2 1 2

0 2

exp 4

exp 4

exp 4

exp 4

exp 4

p N

p p

p N

p p

p N

p p

p N

p p

N

P H k p H k d H k

P H k

p H k d H k

P H k

p H k d H k

P H k

p H k d H k

P H k γ

σ

γ σ

γ σ

γ σ

γ σ

∞

∞

∞

∞

∞

 

= − 

 

×

 

× − 

 

×

 

× − 

 

×

 

× − 

 

×

  

= − 

 

∫

∫

∫

∫

∫

( ( )

²

) ( ^{( )}

²

) }

⁴

p H k d H k



×

… (23)

where ( )p z is a probability density function (PDF) of z and ^{H k}

( )

is a complex Gaussian random variables with mean zero and variance .ρ As ^{H k}

( )

^{2 follows}

the Chi-square distribution, the PDF ^{p H k}

( ( )

²

)

^can

be expressed as

( ( )

²

)

^12exp

( ^{( )}

^{2 2}

)

p H k H k ρ

= ρ − … (24)

Then, by using Eq. (24), we can rewrite Eq. (23) as

( )

( ( ) ) ( ^{( )} ) }

( )

,

2 1 2

0 2

2 2 4

2 1 2

0 2

exp 4

exp 4

b p

N

N

E P

P H k

p H k d H k P H k γ

σ

γ σ

∞

∞

 

 

  

≤ − 

  



×

  

= − 

  



∫

∫

( ( ) )

( ( ) ) }

2 2

2 2 4

1 exp H k

d H k ρ ρ

× −

×

( ) ) ( ^{( )} ) }

( )

2 2 2

1

2 0 2 2

2 2 4

2 2 2

2 1

2 2

4

2 2 2

1 2

2 2

0 4 2 4

2 2

2 2 2

1 1

2

4

2 2

1 2

4 1 exp

4

1 1

4 4 exp 4

4

4 1

4 1

4

1 4

N N

N N

N N

N N

N

N

P

H k d H k

P

P H k

P P

P

γ ρ σ

ρ σ ρ

γ ρ σ

ρ

σ ρ

γ ρ σ

σ ρ

σ

γ ρ

γ ρ σ

σ

γ ρ σ

∞

∞

  +

= −

 



×



=

 +

 −



 

 +  

× −  

   

 

 

 

 

=  =

+  

   + 

 

 

 

 

≤ =

 

 

 

∫

( )

⁴

2 1 2

1 2

4 1

N P . PP σ

ρ

 + 

 

  … (25)

Finally, assuming that P₂=P₁/ 2, E P _{b p,}  can be expressed as

2 4

, 2

2

4 4

2 2

2 2

1 1

4

8 8

,

N b p

N N

E P P

P P

σ ρ

σ σ

ρ ρ

 

 ∝ 

 

 

   

=  = 

   

… (26)

when P₁ is a large value. Then, Eq. (26) can be rewritten as

( )

2 4 ,

4

4 SNR SNR

b p T

T

E P ρ ⁻

−

 

 ∝ 

 

 

… (27)

in terms of the transmit SNR. Thus, the cooperative diversity order of the proposed scheme is 4.

Results and Discussion

In this section, the performance of the proposed scheme is compared with those of the conventional schemes^12,13 in terms of BER. In evaluating the performance, we assume the following parameters: A DFT size of N= 64, a CP length of N_G= 16 samples, binary PSK data modulation, and P₁= 2 .P₂ It is also assumed that the multipath fading channels have a two path equal-power delay profile with a relative delay of 3 samples between the two paths, and the relative timing difference τ between the symbols arriving at the destination node from the relay nodes is distributed uniformly over [0,6]

samples. Figures 2 and 3 show the BER performances of the proposed and conventional schemes over the Rayleigh and Rician multipath fading channels with K-factors of 2, 4, 6, and 8, respectively, where the BER performance of the 2 2× Alamouti system is also plotted for reference. From Figs 2 and 3, it is clearly observed that the proposed and conventional schemes exhibit the same BER curve slope as that of the Alamouti scheme for 2 2× MIMO systems.

However, the BER performance of the proposed scheme is better than those of the conventional schemes for all K-factors (Rayleigh fading channel is a special case of Rician fading when K=0). This can be explained as follows: The proposed scheme

expands the original symbol constellation as shown in Eq. (5) and performs the associated operations at relay and destination nodes increasing the minimum distance among the single constellation points.

Therefore, the proposed scheme can reduce the error probability more effectively, eventually improving the overall system performance.

Figure 4 demonstrates the BER performances of the conventional and proposed schemes over Rayleigh flat fading channels in the presence of residual frequency offset (RFO) normalized to the subcarrier spacing of 0.01, 0.05, and 0.1 at the destination node. From Figs 3 and 4, we can see that the RFO degrades the BER performance of proposed and conventional schemes.

However, the proposed scheme still has a better BER performance than those of the conventional schemes. It is also shown that the proposed and conventional schemes have the same slope of the BER curves in the presence and absence of the RFO, which means that RFO does not degrade the diversity order.

Fig. 3 – BER performances of the proposed and conventional schemes as a function of P₁ over Rician fading channels with K-factors of 2, 4, 6 and 8

Fig. 2  BER performances of the proposed and conventional schemes as a function of P₁ in the Rayleigh fading channels.

Conclusions

In this paper, we have proposed a transmission scheme based on the constellation expansion for asynchronous cooperative communication systems over multipath fading channels. It has been confirmed from numerical results that the proposed scheme achieves a better BER performance than those of the conventional schemes while maintaining the same diversity order.

Acknowledgement

This research was supported by the National Research Foundation (NRF) of Korea under Grants 2012R1A2A2A01045887, and 2012R1A1A2004944 with funding from the Ministry of Science, ICT & Future Planning (MSIP), Korea, by the Information Technology Research Center (ITRC) program of the National IT Industry Promotion Agency under Grants NIPA-2013- H0301-13-1005 and NIPA-2013-H0301-13-2005 with funding from the MSIP, Korea, and by National GNSS Research Center program of Defense Acquisition Program Administration and Agency for Defense Development.

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Fig. 4  BER performances of the proposed and conventional schemes as a function of P₁ without RFO and with RFO of 0.01, 0.05 and 0.1