Joint Design of LDPC and Physical-Layer Network Coding for Bi-directional Relay System in the Presence of Insufficient Timing Synchronization Wenwen LIANG, Kui XU, Hua TIAN, Youyun XU Institute of Communications Engineering, PLA University of Science and Technology Nanjing 210007, China Email:
[email protected],
[email protected],
[email protected],
[email protected] Abstract—Physical-layer Network Coding (PNC) has shown its power for improving the throughput effectively of the bi-directional relay system. Most of the existing PNC schemes are carried out under the assumption of perfect synchronization. In this paper, we provide some new insights into the joint design of Low-Density Parity-Check (LDPC)-coded and PNC for asynchronous bi-directional relay system with BPSK signaling. In particular, a novel LDPC and PNC joint design method is proposed for mitigating the performance degradation caused by the different timing offsets between relay and two source nodes. Simulation results show that the proposed joint design of LDPC and PNC scheme outperforms traditional method for bi-directional relay system in the presence of insufficient timing synchronization.
Keywords-Physical-layer Network Coding; LDPC; Bi-directional Relaying; Timing Offset
I.
INTRODUCTION
Network coding (NC) has been shown to improve the network throughput significantly, which was first proposed in [1]. The network coding scheme was originally considered as a network-layer technique for wired networks. In bidirectional wireless relay system, the broadcast nature of the wireless physical medium is usually considered to cause enormous interference if two nodes transmit simultaneously. On the contrary, Physical-layer Network Coding (PNC) can employ this broadcast nature as a capacity-boosting approach for bi-directional relay system [2, 3]. To be more practical, channel coding should be employed to further improve the reliability of the system, a joint design scheme of Turbo codes and PNC is proposed in [4], which used the superimposition characteristic of electromagnetic wave and the linear characteristic of Turbo codes and PNC. In [5], a scheme of joint LDPC and PNC was presented. Joint channel decoding and physical layer network coding (JCNC) have been introduced in [6], it was recognized that the XOR of both source codeword is still a valid codeword with the same linear channel code at both source nodes. In [7], a practical scheme of joint PNC and channel coding based on QAM modulation is proposed. In [8], the authors derived the closed-
form expression for computing the log-likelihood ratios (LLR) of the network coded codeword for a complex multiple-access channel, and it was revealed that the equivalent channel observed at the relay is an asymmetrical channel. All the above PNC schemes are carried out under the assumption of perfect synchronization. For a bi-directional relay system in the presence of insufficient timing synchronization, there are different timing offsets between the two source nodes and the relay node. In [9], the BCJR formulation of the asynchronous multiple-access channel (MAC) is proposed, which can shed lights for various practical algorithms suitable for implementation. In [10], the design of channel coding to support the two-phase PNC is discussed. In [11], an optimal maximum likelihood (ML) decoding algorithm based on belief propagation (BP-PNC) for asynchronous PNC is proposed and investigated. In this paper, we provide some new insights into the joint design of LDPC and PNC for asynchronous bi-directional relay system with BPSK signaling. In particular, a novel LDPC and PNC joint design method is proposed for mitigating the performance degradation caused by the different timing offsets between relay and two source nodes. Simulation results show that the proposed joint design of PNC and LDPC scheme outperforms traditional method for bidirectional relay system in the presence of insufficient timing synchronization. The remainder of the paper is organized as follows. In SectionⅡ, a simple system model for asynchronous PNC is introduced. We then depict the joint LDPC and PNC over asynchronous MAC in Section Ⅲ. Simulation results are given in Section Ⅳ, and the conclusion is made in SectionⅤ. II.
SYSTEM MODEL FOR ASYNCHRONOUS BIDIRECTIONAL RELAY
The system model is shown in Fig.1. S1 and S 2 denote the source nodes, R is the relay node. It is assumed that the communication takes place in two phases-a multiple access phase and a broadcast phase. In the multiple access phase, the two source nodes send signals simultaneously to the relay. In the broadcast phase, the relay processes the superimposed
978-1-4577-1008-7/11/$26.00 ©2011 IEEE
signal of the simultaneous packets and maps them to a network-coded (XOR) packet for broadcast back to the source nodes. R
t
R
t + Δt
C=C1 ⊕ C 2
C2
C1 S1
S2
S1
S2
Fig.1. System model for asynchronous bi-directional relay depicting the MAC phase and the broadcast phase.
In practice, the asynchronism between source nodes S1 and
S 2 can not be controlled elegantly. The bit asynchronism should also be seriously considered. Let Ci = (ci,1 ,...ci, j ,...ci,N ) be the codeword vector after LDPC encoding, where i = 1, 2 , and i is the sequence of source nodes, j = 1,2...N , N denotes the length of codeword, T is the period of one symbol， +t (+t < T ) is the timing offset between the two signals sent by the source nodes S1 and S 2 . Hence, we can consider the practical situation which is shown in Fig.2.
T
C1
c1,1 c1,2 c1,3 c2,1 c2,2 c2,3
C2
c1,N −1 c1,N
y j = x 1, j + x 2, j cos θ + n (2) From Eq. (2), the signal sent by source node S 2 is impacted by cos θ . Such as, when θ is ±0.5π or closes to ±0.5π , apparently we can see that y j = x 1, j + n (3) and it is easily validated that there is large performance degradation when we used the LLR designed by [8] to decode. Asynchronous situation is a very familiar practical scene, how to mitigate the performance degradation caused by the different timing offsets is the intention of this paper, and the scheme proposed in this paper will be described in the next section. III.
PROPOSED JOINT DESIGN SCHEME OF LDPC AND PNC
We propose a novel design scheme of LDPC and PNC in order to mitigate the performance degradation caused by the different timing offsets between relay and two source nodes. The proposed scheme can be divided to two steps A and B as follows.
A.
Design of demodulator for the received signals During the MAC phase, the two source node sends signals to relay node simultaneously. Let x 1 and x 2 be the signals sent by S1 and S 2 , the practical received superimposed signal one symbol period of relay node can be expressed as
c2,N −1 c2,N
Δt
y(t ) = x 1 cos ωct + x 2 cos ωc (t ++t ) + n(t )
= x 1 cos wct + x 2 cos(wct + θ) + n(t )
Fig.2. Bit asynchronism between sources S1 and S 2 .
(4)
Assumed that BPSK modulation is considered in the system, let X i = (x i,1,...x i, j ,...x i,N ) be the signal vector sent by
S i after BPSK modulation, Y = (y1,...y j ,...yN ) denotes the signal vector received by the relay node R , where i = 1, 2 , j = 1, 2...N . Also we assume that the channel between S i and R is AWGN channel, so the received superposition of bandpass signal in one symbol period can be described as
y j (t ) = x 1, j cos ωct + x 2, j cos ωc (t ++t ) + n(t ) = x 1, j cos wct + x 2, j cos(wct + θ) + n(t )
As shown in Eq. (2), in order to mitigate the performance degradation caused by θ , the demodulation process we design is shown in Fig.3. We use cos ωct and cos(ωct + θ) to demodulate the two received signals respectively, and then get the sum of the demodulated signals. It can be detailedly described as follows. T
∫ (⋅)dt
(1)
0
y(t )
cosωct
where cos ωct denotes carrier wave, wc is carrier frequency,
T
n(t ) is the expression of AWGN, θ (θ = ωc Δt ) denotes the phase difference, we assume that θ is known, and we have θ ∈ [−0.5π, 0.5π ] .
0
We use coherent detection mode to demodulate the received signals and cos ωct is used. Then we can get
∫ (⋅)dt
y1
+ +
∑
y
y2
cos(ωct + θ ) Fig.3. Design of demodulator for the received signal from source node
S1 and S 2
⎧⎪0 x = 1, x = −1 or x = −1, x = 1 ⎪⎪ 1 2 1 2 ⎪⎪ ξ = ⎨2h x 1 = 1, x 2 = 1 ⎪⎪ y ' (t ) = y(t ) cos ωct = (x 1 cos ωct + x 2 cos(ωct + θ) + n(t )) cos ωct ⎪⎪−2h x 1 = −1, x 2 = −1 ⎪⎩ = x 1 cos2 ωct + x 2 cos(ωct + θ) cos ωct + n(t ) cos ωct (11) (c1 ⊕ c2 ) denotes the network-coded bit, 1 + cos 2ωct cos(2ωct + θ) + cos θ = x1 + x2 + n(t )cos ωct L denotes the LLR of (c1 ⊕ c2 ) about received signal y . 2 2 According to the definition of LLR, we have (5) ⎫ ⎪⎧ P (c1 ⊕ c2 = 1 | y ) ⎪ ⎪ Let y ' (t ) passes the low pass filter, we have (12) L = log ⎪⎨ ⎬ ⎪⎪P (c ⊕ c = 0 | y )⎪ ⎪ 1 2 ⎩ ⎭ y1 = (x 1 + x 2 cos θ + n1 ) / 2 (6) From the relation between x and c , (c ⊕ c ) can be
cos(ωct ) is used to demodulate the received signals, the process of demodulation is shown as follow
i
cos θ 1 T ] , Eq.(6) can be Let X = [x 1 x 2 ] , F1 = [ 2 2 expressed as
i
1
2
denoted as
⎧⎪1 x1 = 1, x2 = −1 or x1 = −1, x2 = 1 c1 ⊕ c2 = ⎪⎨ ⎪⎪0 x1 = −1, x 2 = −1 or x1 = 1, x2 = 1 ⎩ (13)
y1 = X ⋅ F1 + n1 / 2
Then Eq. (13) can be transformed as
(7) Similarly cos(ωct + θ) is used to demodulate the received signals, then we can get
y2 = (x 1 cos θ + x 2 + n2 ) / 2
⎧⎪1 ξ = 0, P = 0.5 c1 ⊕ c2 = ⎪⎨ ⎪⎪0 ξ = 2h ,P= 0.25 or ξ = −2h, P = 0.25 ⎩ (14) By substituting Eq. (13) in (12), we can get LLR as
(8) Let F1 = [
1 cos θ T ] , Eq.(8) can be expressed as 2 2
y2 = X ⋅ F2 + n2 / 2 (9) From Eq. (7) and (9), we can see that the signals vector X sent by S1 and S 2 are impaired by F1 and F2 respectively. The sum of y1 and y 2 can be expressed as y = y1 + y 2 = X ⋅ F1 + n1 / 2 + X 2 ⋅ F2 + n2 / 2 = X ⋅ (F1 + F2 ) + (n1 + n2 )/ 2 = X ⋅ E ⋅ h + n'
⎪⎧ ⎪⎫⎪ P(ξ = 0 | y ) L = log ⎪⎨ ⎬ ⎪⎩⎪P (ξ = 2h or ξ = −2h | y )⎪⎭⎪ ⎪⎧ ⎪⎫⎪ P (y | ξ = 0)P (ξ = 0) = log ⎪⎨ ⎬ ⎪⎩⎪ P (y | ξ = 2h )P (ξ = 2h ) + P (y | ξ = −2h )P (ξ = −2h )⎪⎭⎪ ⎪⎧ ⎪⎫⎪ 2P (y | ξ = 0) = log ⎪⎨ ⎬ ⎪⎩⎪ P (y | ξ = 2h ) + P (y | ξ = −2h )⎪⎭⎪ (15) When the input of the additive white Gaussian noise channel is X, and the noise variance is σ 2 / 2 , the output can be expressed as
= hX + n '
P (y | x ) =
(10)
1
πσ
e
−
(y −x )2 σ2
where n ' = (n1 + n2 )/ 2 , n ' ~ N (0, σ 2 / 2) , h = (1 + cos θ)/ 2 ,
(16)
E = [1 1]T now the received signals x 1 and x 2 have the same coefficient h , then we can calculate LLR according to [8].
Substituting Eq. (16) in (15), then we can get LLR about network-coded bit (c1 ⊕ c2 ) as
B.
LLR for joint decoding
Let ξ = h(x 1 + x 2 ) , x 1 and x 2 are ambipolar signals, and they are sent equiprobably, than ξ can be expressed by
L= (17)
4h 2 4yh − log(cosh 2 ) σ2 σ
So we can get (c1 ⊕ c2 ) from LDPC decoding [12] for using Eq. (17) to decode.
IV.
SIMULATION RESULTS
In this section, some simulations are given to illustrate the performance of the proposed scheme. In simulations, the AWGN channel and BPSK modulation are used. LDPC encoder uses (115, 230) and (2505, 5010), the code rate is 1 / 2 , and it uses 100 iterations. θ varies from −0.5π to 0.5π ,and we get 0.1π , 0.3π and 0.5π in simulations. Synchronous PNC is considered in [8], at the asynchronous PNC (APNC), the traditional LLR is not suitable in asynchronous situation, and we compare the traditional LLR and the LLR proposed in this paper. The performance comparison is shown in Fig.4. , Fig.5. and Fig.6.
Fig.5. shows the relation between the two schemes when SNR changes and the (2505, 5010) LDPC code is used. From Fig.4.we can get the same conclusion as Fig.3. Fig.6. shows the relation between θ and BER when SNR is 1, 2 and 3dB. The (115, 230) LDPC code is used. From Fig.5. we can see that bit error rate becomes worse with θ increasing. This conclusion is the same as Fig.3. and Fig.4. Besides, the proposed scheme in this paper has obvious gain compared to asynchronous scheme. Bit Error Rate
0
10
-1
10
Bit Error Rate
0
10
-2
10 -1
BER
10
-3
10
-2
10
-4
BER
10
-3
-5
10
10 APNC• =0.1 PNC• =0.1 APNC• =0.3 PNC• =0.3 APNC• =0.5 PNC• =0.5
-4
10
-5
10
0
1
2
3
4 Eb/No (dB)
5
-6
10
6
7
8
Fig.4. The bit error rate performance of the (115, 230) LDPC code.
Fig.4. shows the relation between the two schemes when SNR changes and the (115, 230) LDPC code is used. When θ is 0.5π , we can see that traditional scheme can not decode correctly, but the proposed scheme can encode correctly when SNR is over 4dB. With θ being 0.3π , the proposed scheme obtains 1dB gain compared to synchronous scheme when BER is 10−3 . As θ is 0.1π , we can continue to observe the outperformance of APNC over PNC by 0.3dB when the BER is 10−3 . Also we can see from Fig.3. that bit error rate becomes worse with the increasing of θ . Bit Error Rate
0
10
-1
10
-1.5
-1
-0.5
0 /rad
0.5
1
1.5
2
Fig.6. The bit error rate performance of the (115, 230) LDPC code when is changed
V.
θ
CONCLUSION
This paper we have proposed a new joint design scheme of and LDPC while considering the asynchronous situation. In the MAC phase, the relay node decodes the codeword (c1 ⊕ c2 ) from the received signals for using BP algorithm which was depicted in [12]. Then during the broadcast phase, the relay node broadcasts (c1 ⊕ c2 ) to source nodes S1 and S 2 , and the proposed scheme supports asynchronous PNC compared to the traditional scheme. Channel coding is widely employed to improve the system performance. There are a lot of productions for the investigations of joint LDPC and NC over asynchronous bidirectional relaying. In the next work we will transfer our interests into asynchronous PNC which is suitable for practical implementation. ACKNOWLEDGE
-2
10 BER
This paper is supported in part by the National Science Foundation of China under Grants 60972050. The work was also supported by the National Key S&T Project under Grant 2010ZX03003-003-01and supported by the National Hi-Tech Research and Development Program (863) of China (No. 2009AA01Z249), and also supported by the Project supported by the Jiangsu Province National Science Foundation under Grant BK2011002.
-3
10
NPC• =0.5 APNC• =0.5 PNC• =0.3 APNC• =0.3 PNC• =0.1 APNC• =0.1
-4
10
-5
10
-2
PNC,SNR=1dB APNC,SNR=1dB PNC,SNR=2dB APNC,SNR=2dB PNC,SNR=3dB APNC,SNR=3dB
0
1
2
3 4 Eb/No (dB)
5
6
7
Fig.5. The bit error rate performance of the (2505, 5010) LDPC code.
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