Joint Source-Channel Coding for Asymmetric Slepian-Wolf Multiple ...

642

IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 6, NO. 5, OCTOBER 2017

Joint Source-Channel Coding for Asymmetric Slepian-Wolf Multiple Access Relay Channel Lei Liu, Ying Li, Chau Yuen, Yong Liang Guan, and Yue Sun

Abstract—This letter considers a code design for an asymmetric Slepian-Wolf multiple-access relay channel, in which the users and relay have different channel conditions. In this case, the conventional parity approach distributed source coding is no longer applicable. Hence, a new distributed joint source-channel superposition coding (DJSC-SC) is proposed. The proposed DJSC-SC matches the asymmetric channels, and makes full use of the good channel to obtain a better performance. Simulation results show that, compared with the conventional PA, the performance of the DJSC-SC scheme has a significant improvement and is within 1.4 dB from the capacity limit. Index Terms—Distributed source coding, superposition coding, joint source-channel coding, asymmetric MARC.

I. I NTRODUCTION N THIS letter, we discuss the design of a relaying scheme for a multiple-access relay channel (MARC) with two users, one relay and one destination. In such network, the users communicate with the destination with the help of the relay. When the relay cannot correctly recover the user messages, quantize-and-forward or compute-and-forward strategy are usually used [1]–[4]. Otherwise, decode-and-forward (DF) relaying [5]–[8] is considered for such MARC (DF-MARC).

I

A. Motivation and Related Work In the DF-MARC, joint source-channel-network coding (JSCNC) is usually used for compressing the sources and adding robustness against channel noise via a single network code, which improves the DF efficiency of the MARC [7], [8]. Distributed source coding (DSC) [9] is another key technique for the relay to improve the system rate, by compressing the messages from the users [10]–[12]. In the DF-MARC with DSC, the relay is treated as a new source, which can recover all the messages from the sources. The received messages from the user-destination (UD) links can be seen as a side information for the relay. This results in a Slepian-Wolf (SW) DSC problem for the MARC, and we call it SW-MARC. The syndrome approach (SA) [13] (that Manuscript received April 18, 2017; revised June 12, 2017; accepted July 5, 2017. Date of publication July 13, 2017; date of current version October 11, 2017. This work was supported in part by the National Natural Science Foundation of China under Grant 61671345, and in part by A*Star SERC under Project 142 02 00043. The associate editor coordinating the review of this paper and approving it for publication was J. Mietzner. (Corresponding author: Ying Li.) L. Liu, Y. Li, and Y. Sun are with the State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an 710071, China (e-mail: [email protected]; [email protected]; [email protected]). C. Yuen is with the Singapore University of Technology and Design, Singapore 487372 (e-mail: [email protected]). Y. L. Guan is with the School of EEE, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]). Digital Object Identifier 10.1109/LWC.2017.2726530

Fig. 1.

Asymmetric two-user half-duplex SW-MARC.

transmits syndrome bits as compressed signals) and the parity approach (PA) [14] (that transmits parity bits as compressed signals) are two main DSC schemes, which have been proved to be asymptotically optimal [15], [16]. In [17] and [18], Turbo code SAs are used for DSC. Liveris et al. [19] employed a low density parity check code (LDPC) PA for DSC. Uppal et al. [20] proposed an irregular repeataccumulate (IRA) code for the DSC at the relay, and achieved a capacity-approaching performance. However, these DSC approaches were designed for the symmetric SW-MARC, and few works focus on the practical design of DSC for the asymmetric SW-MARC. B. Contributions In this letter, as shown in Fig. 1, we consider a scenario that one user is close to the destination while the other user is far from the destination. Specifically, the distance between one of the users and the destination is far enough to make them out of range of each other. We assume the relay-destination (RD) link is better than the UD links. It results in an asymmetric SW-MARC, for which all relaying schemes designed for symmetric cases are no longer optimal, because the performance of the whole system is limited by the poor channel, and the good channel is exploited effectively. To improve the bit error rate (BER) performance of the asymmetric SW-MARC system, we propose a distributed joint source-channel superposition coding (DJSC-SC) method. The code design and power allocation are designed to match the two asymmetric channels. In addition, a soft iterative decoder is designed to obtain a capacity-approaching BER performance. Finally, numerical results are provided to verify the advantage of the proposed DJSC-SC algorithm. II. S YSTEM M ODEL Fig. 1 shows the MARC, which consists of two users, one relay and one destination. We assume that the two users have different channel conditions. There is no direct link between user 2 and the destination (UD2) as it is far from the destination. The user 1 has a direct link with the destination (called

c 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2162-2345  See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

LIU et al.: JOINT SOURCE-CHANNEL CODING FOR ASYMMETRIC SLEPIAN-WOLF MARC

Fig. 2.

k

643

k

DJSC-SC for asymmetric SW-MARC, where the relay always decodes [X11 , X22 ] correctly.

UD1). As the relay is closer to the destination than the user 1, the RD link is better than the UD1 link. The relay works in half-duplex mode. Firstly, the two users transmit their respective independent source signals Xk11 and Xk22 to the relay and destination (solid lines). Secondly, the relay recovers Xk11 and Xk22 , and then compresses (by a joint source-channel code) them into Zn−k1 ,1 n > k1 , which is forwarded to the destination (dashed line). The superscript denotes the length of vectors, and we assume that k2 < n − k1 as user 2 has a worse channel condition. The destination receives Yk1 and Un−k1 in two phases respectively. As in SW-MARC the relay always decodes correctly, we treat the relay as a new source with message X = [Xk11 , Xk22 ], and treat Yk1 as a side information of X. This results in a DSC problem given in Fig. 2. The decoder jointly decodes Yk1 and Un−k1 , and outputs ˆ k2 ]. The side information Yk1 here is ˆ = [X ˆ k1 , X the estimate X 1 2 k1 generated from X1 . Thus, 1 [Yk1 , Un−k1 ] = [Xk11 , Zn−k1 ] + [NkC1 , Nn−k ], A

(1)

1 where NkC1 and Nn−k are Gaussian noises. Let C1 and C2 be A the capacity of correlation channel and actual channel respectively, and let R be the sum rate. From the Shannon theory and the Slepian-Wolf theory [1], we have Proposition 1: When n and k1 are large enough, X can be recovered with arbitrary small error probability if and only if

R≤

n − k1 k1 C1 + C2 . n n

(2)

In this letter, we assume C1 < C2 . III. DJSC-SC D ESIGN For the asymmetric SW-MARC, the PA is no longer applicable since its performance is limited by the poor channel. Therefore, we propose a new DJSC-SC in this section.

DJSC-SC.2 The codewords C12 and C2n−k1 are interleaved and n−k1 1 modulated into Xn−k respectively. The powers of c12 and Xc2 k1 n−k1 n−k1 k1 1 X1 , Xc12 ,Xc2 , NC , and Nn−k are Px1 , Pc12 , Pc2 , σC2 and σA2 A respectively. The relay transmits the superposition code (SC) n−k1 1 through the actual channel. In the DJSC-SC Xn−k c12 + Xc2 1 + scheme, by power allocation, we let the interference Xn−k c2 n−k1 k1 NA be equivalent to the interference NC , i.e., the whole C1n passes into an equivalent symmetrical channel. The joint decoder first decodes the codeword C1n by treating the signal n 1 1 Xn−k as noise. Then, we remove Xn−k c2 c12 with the recovered C1 n−k k k from Un−k1 , and then decode C2 1 . Finally, X11 and X22 are ˆ Thus, recovered from C1n and C2n−k1 , and get the estimate X.  Yk1 = Xk11 + NkC1 , (3) n−k1 1 1 + Nn−k . Un−k1 = Xn−k c12 + Xc2 A Let R1 and R2 be the code rates of C1n and C2n−k1 respectively. According to capacity theory of point-to-point channels and multi-access channels, Proposition 2 is proposed. Proposition 2: When n and k1 are large enough, X can be recovered in the DJSC-SC scheme with arbitrary small error probability if and only if R1 ≤ C1 ,

and R2 ≤ C2 − C1 .

 The total rate is R = R1 + αR2 , where α = (n−k1 ) n.

B. DJSC-SC for Asymmetric Binary Input AWGN Channels For an AWGN channel with binary phase shift keying (BPSK) modulation, we have the following proposition. Proposition 3: When n and k1 are large enough, X can be recovered in the DJSC-SC scheme with arbitrary small error probability if and only if     R1 ≤ CBA Px1 σC2 ,   Pc Px (5) R2 ≤ CBM Pc12 , Pc2 , σA2 + CBA ( 22 ) + CBA ( 21 ), σA

A. DJSC-SC for Asymmetric SW-MARC To match the two channels, we consider superimposing a new codeword on the transmission of the good channel. Fig. 2 shows the process in detail. The relay encodes Xk11 and Xk22 into k1 n−k1 C1n = [C11 , C12 ] and C2n−k1 with two channel codes, where k1 C1n is a systematic codeword with C11 = Xk11 . The codes used at the relay not only compress the received Xk11 but also add redundancy for X to resist the channel noise. Thus, we call it 1 As this process is joint source-channel coding, it is possible to obtain a sequence of arbitrary length as it contains both source coding (by removing redundancy) and channel coding (by adding redundancy).

(4)

σC

where +∞ −τ 2 √ 2 e CBA (s) = 1 − log(1 + e−2τ s−2s )dτ √ 2π

(6)

−∞

2 Conventionally, Xk1 should be compressed first by a DSC to remove the 1 redundancy related to the side information Yk1 , and then the compressed

sequence is encoded by a channel code to add redundancy to combat the channel noise in the RD link. Correspondingly, the source coding and channel coding are performed separately. However, in this letter, the proposed scheme performs the source compression and channel coding using a single code C1 , i.e., the relay performs a joint source-channel coding.

644

IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 6, NO. 5, OCTOBER 2017

is the capacity of a binary-input AWGN (BIAWGN) channel [20], and

Algorithm 1 Numerical Algorithm for Power Allocation

C

+∞ p− (τ ) (P, Pi , σ )= 1− √ p+(τ )log(1+ + )dτ (7) p (τ ) 2 2π σ

BM

2

1

−∞

is the capacity of a binary input mixture Gaussian noise (BIMGN) channel [20]. In (7), p+ and p− are p± (τ ) = e

√ √ 2 −(τ ∓ P− Pi ) 2σ 2

+e

√ √ 2 −(τ ∓ P+ Pi ) 2σ 2

.

(8)

Proof: Let C1∗ and C2∗ be the achievable rates of the correlation channel and the actual channel respectively. From (3), the correlation channel meets the BIAWGN capacity (6)   C1∗ = CBA Px1 σC2 . (9) The calculation of C2∗ has two steps. Firstly, C1n is transmitted n−k1 into a BIMGN actual channel. Thus, from (7) of C12 , the   1 Xn−k is recovered with C21 = CBM Pc12 , Pc2 , σA2 . Besides, c12 C2n−k1 is transmitted    into a BIAWGN actual channel. We get C22 = CBA Pc2 σA2 of C2 with P = Pc2 and σ 2 = σA2 . Thus,    C2∗ = C21 + C22 = CBM Pc12 ,Pc2 ,σA2 + CBA Pc2 σA2 . With C1∗ and C2∗ , we get Proposition 3 according to (4). C. Code Design and Power Allocation 1 1 To ensure that the interference Xn−k + Nn−k be equivalent c2 A k1 ∗ to the interference NC , we have C1 = C21 , i.e.,    (10) CBA Px1 σC2 = CBM Pc12 , Pc2 , σA2 .

Assuming Px1 = Pc12 + Pc2 = Psum , Pc2 and Pc12 can be achieved by solving (10) with the power constraint. It is easy to see that the proposed DSJSC-SC method is equivalent to the parity approach if the MARC is symmetric. Differences compared to index coding and JSCNC [7], [8]: 1) In the DJSC-SC, the superposition is performed in the real field. However, the index coding and JSCNC superimpose in a finite field. 2) The index coding focuses on the network layer, i.e., some of the transmissions are noiseless. However, the proposed DJSC-SC deals with the physical layer, i.e., all the channels are noisy. 3) It is the different channel noises that result in the “asymmetry” in the MARC. However, “asymmetry” does not exist in the index coding as it contains a noiseless channel. 4) In the DJSC-SC, power allocation is used at the relay. However, there is no power allocation in the network coding. Remark: As C1∗ and C2∗ can only be numerically calculated by (10). Algorithm 1 presents the detailed design of the DJSCSC for the asymmetric MARC given Psum , σC2 and σA2 . D. Achievable Rate Analysis In the BPSK PA, the rate is calculated by (6). In the BPSK DJSC-SC, the rate is derived by Proposition 3. Proposition 4: When n and k1 are large enough, the achievable rates of PA RPA and the achievable rate RDJSC−SC of the DJSC-SC scheme for the actual channel are given by

   RPA = CBA Psum σC2 ,     RDJSC−SC = CBM Pc12 , Pc2 , σA2 + CBA Pc2 σA2 .

1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15:

Input: Psum ,  > 0, σC2 , σA2 and calculate C1∗ by (9). Initialization: P(0) = 0, P(1) = Psum , C(0) = 0. Do  P∗ = (P(0) + P(1) )/2 and C∗ = CBM P∗ , Pc2 , σA2 ; if C∗ ≤ C1∗ P(0) = P∗ ; else P(1) = P∗ ; end  = C(0) − C∗ and C(0) = C∗ ; While |  | >  or |P(1) − P(0) | >  − P∗ , Pc12 = P∗ and Pc2 = P sum  C21 = C∗ , C22 = CBA Pc2 σA2 and C2 = C21 + C22 , R1 = C1 and R2 = C2 − C1 . Output: Pc12 , Pc2 , R1 and R2 .

The Pc2 and Pc12 can be obtained by numerically solving Eq. (10) via Algorithm 1. We can see that (based on the numerical results) the DJSC-SC has a higher rate than the PA in the high signal-to-noise ratio (SNR) region. However, the gain will disappear with the decrease of SNR. E. Soft Iterative Decoding Design of the Decoder In this section, we consider the soft iterative decoder for the proposed DJSC-SC. Fig. 3 shows the block diagram of the soft iterative decoder. Firstly, the received Yk1 and Un−k1 are demodulated to lxc12 +xc2 and lx1 respectively. Decoder 1 (DEC 1) then decodes with the input lxc12 +xc2 , lxc2 (initialized with zero), and lx1 , and outputs ec12 . Interleaver 1 interleaves ec12 into lxc12 , which is fed into decoder 2 (DEC 2). Decoder 2 decodes with lxc12 and lxc12 +xc2 , and gets ec2 . Interleaver 2 interleaves ec2 into lxc2 , which is fed into decoder 1. Repeat this process above until the number of iterations is completed. In the last iteration, we output Lc11 , Lc12 and Lc2 . Hard decisions are made for Lc11 , Lc12 and Lc2 to get the ˆ k1 and X ˆ k2 . Let yi , i ∈ {1, 2, . . . , k1 } and decoded sequences X 1 2 ui , i ∈ {1, 2, . . . , n − k1 } be the received symbols in (3). The n−k1 1 lx1 , lc12 and lc2 of information sequences Xk11 , Xn−k c12 and Xc2 are calculated by  2yi Px1 lx1 ,i = , i ∈ {1, 2, · · · , k}, (11) σC2 lc12 ,i = ln

e

2ui (Ac12 +Ac2 ) σA2

e

+ lxc2 ,i

2Ac2 (Ac12 +ui ) σA2

+e

2Ac12 (ui +Ac2 ) σA2

+ lxc2 ,i

,

(12)

,

(13)

+1

for i ∈ {1, 2, . . . , n − k1 }, and similarly lc2 ,i = ln

e

2ui (Ac12 +Ac2 ) σA2

e

+ lxc12 ,i

2Ac12 (ui +Ac2 ) σA2

+e

2Ac2 (Ac12 +ui ) σA2

+ lxc12 ,i

for i ∈ {1, . . . , n − k1 }, where Ac2 =



+1

Pc2 and Ac12 =



Pc12 .

IV. S IMULATION R ESULTS In this section, we give some simulation results of the DJSC-SC scheme and the conventional PA for SW-MARC. Fig. 4 shows the performances of LDPC PA in AWGN and

LIU et al.: JOINT SOURCE-CHANNEL CODING FOR ASYMMETRIC SLEPIAN-WOLF MARC

Fig. 3. Block diagram of the soft iterative decoder for DJSC-SC. DEC 1 and DEC 2 denote the two decoders, and π1 , π2 , π1−1 and π2−1 are the interleavers and deinterleavers.

645

V. C ONCLUSION We proposed a DJSC-SC scheme for the asymmetric MARC. The code design and power allocation that match the two asymmetric channels were discussed. A soft iterative decoder was also designed to get a capacity-approaching BER performance. Simulation results showed that the proposed DJSC-SC has a significant performance improvement over the conventional PA scheme. Theoretical evaluation like extrinsic information transfer chart can be used for the code design and performance analysis of the proposed scheme, which is a promising direction for future work. R EFERENCES

Fig. 4. Performance comparison between the conventional LDPC parity approach and the turbo superposition coding for the AWGN and fading channel. (i) The two blue lines are for the symmetric SW-MARC; (ii) the three red lines are for the asymmetric SW-MARC.

fading channel for the symmetric and asymmetric MARC. The relay adopts a 1/2-rate LDPC code with code length 12192 for the joint source-channel coding. The respective degree distribution of the variable node and check node are v(x) = 0.455x + 0.3433x2 + 0.1603x7 + 0.0407x29 and c(x) = 0.1003x7 + 0.8997x8 . The Eb /N0 is calculated by Eb /N0 = Psum /(2Rsum σA2 ), where Rsum is the sum rate and Psum = Px1 = PC2 + PC12 = 1. The maximum number of iterations is set to 30. It illustrates that the performance of LDPC PA is very close (1.2dB or 3.9dB) to the SW limit of the symmetric MARC. However, for the asymmetric MARC, the performance of PA is far away (3.3dB or 6dB) from the SW limit. Therefore in this case, the PA is not a good choice. Fig. 4 also presents the simulations of DSC for the asymmetric MARC. A 1/2-rate LDPC PA and DJSC-SC with two 1/3-rate turbo codes (whose generation polynomial is 1+D2 +D3 ) are considered respectively at the relay. The 1+D+D2 +D3 source length of user 1 is k1 = 4064 and that of user 2 is k2 = 2032, and the total code length is n = 12192. The right two curves in Fig. 4 show the performance comparison of the 1/2 code rate LDPC PA and two 1/3 code rate turbo DJSC-SC scheme. The two schemes have the same total code rate 1/2. We get PC12 = 0.6135 and PC2 = 0.3865. To keep asymmetry the variances of the two  of the system,  noises satisfy 1 σC2 = PC12 σA2 + PC2 with σC2 > σA2 . It shows that compared to the PA, the DJSC-SC has a huge performance gain and is very close (1.4dB or 3.6dB) to the SW limit.

[1] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York, NY, USA: Wiley, 1991. [2] B. Nazer and M. Gastpar, “Compute-and-forward: Harnessing interference through structured codes,” IEEE Trans. Inf. Theory, vol. 57, no. 10, pp. 6463–6486, Oct. 2011. [3] L. Liu, Y. Li, Y. Su, and Y. Sun, “Quantize-and-forward strategy for interleave-division multiple-access relay channel,” IEEE Trans. Veh. Technol., vol. 65, no. 3, pp. 1808–1814, Mar. 2016. [4] L. Liu, Y. Liang, and Y. Li, “A new upper bound on the achievable rate of relay channel with MIP-QF strategy,” IEEE Trans. Veh. Technol., to be published, doi: 10.1109/TVT.2017.2651106. [5] M. Hernaez, P. M. Crespo, and J. Del Ser, “On the design of a novel joint network-channel coding scheme for the multiple access relay channel,” IEEE J. Sel. Areas Commun., vol. 31, no. 8, pp. 1368–1378, Aug. 2013. [6] Y. Zhang and Z. Zhang, “Joint network-channel coding with rateless code over multiple access relay system,” IEEE Trans. Wireless Commun., vol. 12, no. 1, pp. 320–332, Jan. 2013. [7] F. P. S. Luus and B. T. Maharaj, “Joint source-channel-network coding for bidirectional wireless relays,” in Proc. IEEE ICASSP, Prague, Czechia, 2011, pp. 3156–3159. [8] S. El Rouayheb, A. Sprintson, and C. Georghiades, “On the index coding problem and its relation to network coding and Matroid theory,” IEEE Trans. Inf. Theory, vol. 56, no. 7, pp. 3187–3195, Jul. 2010. [9] Z. Xiong, A. D. Liveris, and S. Cheng, “Distributed source coding for sensor networks,” IEEE Signal Process. Mag., vol. 21, no. 5, pp. 80–94, Sep. 2004. [10] X. Zhou, M. Cheng, K. Anwar, and T. Matsumoto, “Distributed joint source-channel coding for relay systems exploiting source-relay correlation and source memory,” EURASIP J. Wireless Commun. Netw., vol. 2012, pp. 1–13, Dec. 2012. [11] Y. Murin, R. Dabora, and D. Gündüz, “Joint source-channel coding for the multiple-access relay channel,” in Proc. IEEE Int. Symp. Inf. Theory, Cambridge, MA, USA, Jul. 2012, pp. 1937–1941. [12] Y. Murin, R. Dabora, and D. Gündüz, “Source-channel coding theorems for the multiple-access relay channel,” IEEE Trans. Inf. Theory, vol. 59, no. 9, pp. 5446–5465, Sep. 2013. [13] S. S. Pradhan and K. Ramchandran, “Distributed source coding using syndromes (DISCUS): Design and construction,” IEEE Trans. Inf. Theory, vol. 49, no. 3, pp. 626–643, Mar. 2003. [14] J. Garcia-Frias and Y. Zhao, “Compression of correlated binary sources using turbo codes,” IEEE Commun. Lett., vol. 5, no. 10, pp. 417–419, Oct. 2001. [15] A. Wyner, “Recent results in the Shannon theory,” IEEE Trans. Inf. Theory, vol. IT-20, no. 1, pp. 2–10, Jan. 1974. [16] P. L. Dragotti and M. Gastpar, Distributed Source Coding: Theory, Algorithms and Applications. Amsterdam, The Netherlands: Academic Press, 2009. [17] Z. Tu, J. Li, and R. S. Blum, “An efficient SF-ISF approach for the Slepian–Wolf source coding problem,” EURASIP J. Appl. Signal Process., vol. 6, pp. 961–971, May 2005. [18] A. Aaron and B. Girod, “Compression with side information using turbo codes,” in Proc. IEEE Data Compression Conf. (DCC), Snowbird, UT, USA, Apr. 2002, pp. 252–261. [19] A. D. Liveris, Z. Xiong, and C. N. Georghiades, “Compression of binary sources with side information at the decoder using LDPC codes,” IEEE Commun. Lett., vol. 6, no. 10, pp. 440–442, Oct. 2002. [20] M. Uppal, Z. Liu, V. Stankovic, and Z. Xiong, “Compress-forward coding with BPSK modulation for the half-duplex Gaussian relay channel,” IEEE Trans. Signal Process., vol. 57, no. 11, pp. 4467–4481, Nov. 2009.