Spatially Coupled Repeat-Accumulate Coded Cooperation

Transcription

1 Spatally Coupled Repeat-Accumulate Coded Cooperaton Naok Takesh and Ko Ishbash Advanced Wreless Communcaton Research Center (AWCC) The Unversty of Electro-Communcatons, Chofugaoka, Chofu-sh, Tokyo , Japan Emal: Abstract In ths paper, a new coded cooperaton scheme called spatally coupled repeat-accumulate coded cooperaton (SC-RA- CC) s proposed for multple access channels (MAC) wth a large number of users and one common destnaton. The proposed approach combnes two advantages of repeat-accumulate (RA) codes and spatal couplng. RA codes show the superor decodng performance wth the smple encodng and spatal couplng exhbts the so-called threshold saturaton behavor; the threshold of coupled codes wth sub-optmal belef propagaton (BP) decodng asymptotcally acheves the optmal maxmum a posteror (MAP) decodng threshold. Our SC-RA-CC can be consdered as the dstrbuted verson of spatally coupled repeat-accumulate codes whch provde the hgher spatal dversty and codng gan when both the number of users and the sze of packets are lmted. The performance of the proposed coded cooperaton s analyzed va densty evoluton (DE). Numercal results confrm that SC-RA- CC can effcently obtan even hgher dversty and codng gan than both drect transmsson and conventonal adaptve network coded cooperaton (ANCC). Index Terms Wreless sensor networks (WSNs), repeataccumulate codes, spatally coupled codes, coded cooperaton, densty evoluton. I. INTRODUCTION Multple access channels (MAC) wth a large number of users and one common destnaton have been actvely nvestgated to desgn many-to-one wreless communcatons systems such as wreless sensor networks (WSNs) for envronmental montorng, dsaster preventon, and so on [1]. The nformaton should be relably and effcently transmtted over the wreless channels whle the channels are typcally unrelable due to the multpath fadng. When the channel coherence tme s longer than a transmsson frame,.e., flat fadng, and the sgnal power at the destnaton s deeply faded, the channel codng does not help anymore. The use of multple antennas at the transmtter and/or recever sde can effectvely mtgate the effect of fadng even n flat fadng channels [2, 3]. However, for many applcatons such as WSNs, the deployment of multple antennas s too costly n terms of sze and power consumpton. To ths end, cooperatve dversty protocols such as amplfyand-forward (AF) and decode-and-forward (DF) protocols have been proposed, whch demonstrate a potental to acheve a dversty gan or enhance the capacty of wreless systems wthout deployng multple antennas at every nodes [4]. Furthermore, coded cooperaton has been proposed to obtan not only the spatal dversty gan but also the codng gan through the cooperaton [5]. Due to the half duplex constrant, the protocol of coded cooperaton s composed of two steps; at the frst step, all transmt nodes sequentally broadcast ther nformaton packets. At the second step, every node ndependently encodes a party packet based on the others packets correctly decoded at the frst step and transmts t to the destnaton. At the destnaton, every nformaton packet and correspondng party packet composes a codeword where party packets are transmtted over mutually ndependent wreless channels. Coded cooperaton, hence, can enoy both the spatal dversty gan and codng gan. Adaptve network coded cooperaton (ANCC) has been subsequently proposed for a large-scale network wth a large number of transmt nodes [6]. The dea of ANCC s to dynamcally map the subgraph of the network topology to the graph of codes by randomly XORng several packets out of the successfully receved ones. The resultng codes form the low-densty generator-matrx (LDGM) codes whch s a specal class of low-densty party-check (LDPC) codes [7]. Although the suffcently long codeword length s necessary to enhance the decodng performance of LDGM codes [8], the codeword length of ANCC s gven by the double of the number of transmt nodes, regardless of the sze of nformaton packets. Therefore, f the number of transmt nodes s lmted, the decodng performance of ANCC would have a large gap from the theoretcal lmt of the underlyng codes. Moreover, n order to decode these coded packets, the destnaton should have the knowledge of entre underlyng graph. In practce, the overhead nformng such nformaton becomes consderably large whch leads to the loss of bandwdth effcency of the system. The queston that may arse here s how to acheve the decodng performance approachng to the channel capacty wth practcal encodng and decodng complexty when both the number of users and the length of nformaton packets are lmted. In ths paper, we propose a new coded cooperaton scheme named spatally coupled repeat-accumulate coded cooperaton (SC-RA-CC). Repeat-accumulate (RA) codes have been proposed n [9] as turbo-lke codes, whch show the decodng performance comparable to turbo codes [10] whle ts encodng complexty s lower. More recently, spatally coupled LDPC code has been proposed n [11]. The remarkable nventon of ths code s spatal couplng that exhbts the so-called threshold saturaton behavor; the threshold of coupled codes wth sub-optmal BP decodng asymptotcally acheves the

2 3,1 1,2 1! 2! 3! N! N,3 2,N+1 N,N+1 N+1! Fg. 1. System model of cooperatve network consstng of N transmt nodes and a common destnaton. Fg. 2. Trangle dstrbuton for evaluate spatal dversty. optmal maxmum a posteror (MAP) decodng threshold. Based on these semnal works, spatally coupled RA codes have been proposed to combne two advantages of RA codes and spatal couplng, namely low complexty encodng and sgnfcantly superor decodng performance [12]. SC-RA-CC can be consdered as a dstrbuted verson of spatally coupled repeat-accumulate codes whle the proposed approach stll shows the superor decodng performance close to the channel capacty even when both the number of users and the length of nformaton packets are lmted. Ths paper s organzed as follows. In Secton II, we brefly descrbe the system model assumed n ths paper. The encodng protocol of proposed SC-RA-CC s descrbed n Secton III. In Secton IV, the threshold of SC-RA-CC wth BP decodng s theoretcally analyzed by usng the densty evoluton. In Secton V, computer smulatons are performed to confrm the analytcal results and compare the performance of SC-RA-CC wth the drect transmsson and conventonal ANCC. Fnally, Secton VI concludes ths work. II. SYSTEM MODEL In ths paper, we consder a network wth N transmt nodes and a common destnaton as llustrated n Fg. 1. Each node s equpped wth a sngle antenna and subect to the halfduplex constrant. Thus, we assume that whle lstenng to the channel, the node may not transmt. Each node sequentally transmts ther own packets to the common destnaton based on the tme dvson multple access (TDMA). We here defne a transmsson block as a tme perod that N nodes transmt ther own packets and assume that every node transmts T dfferent packets through T transmsson blocks. In ths paper, wreless channels are modeled as bnary erasure channel (BEC). Ths model smplfes the mathematcal analyss of error-correctng codes. Moreover, n order to model the effect of slow fadng, we assume that the bt erasure probablty does not change durng T consecutve transmsson blocks. In typcal cooperatve communcatons, the transmt nodes are closely-placed and thus the average receved sgnal power at the transmt nodes s even hgher than that at the destnaton. Therefore, we assume that the channels between the transmt nodes are deally error-free for smplcty. Then, bt erasure probablty from node 1, 2,, N} to 1, 2,, N + 1} at transmsson block t = 1, 2,, T can be wrtten as ϵ (t), = 0 (1 N) ϵ ( = N + 1). (1) Here (N + 1) s a ndex of the destnaton node and ϵ denotes the bt erasure probablty of the channel between node to the destnaton node. Note that, n coded cooperaton, (T + 1) transmsson blocks are needed to transmt T dfferent nformaton packets of every user, whch wll be explaned n detal n the next secton. To evaluate the codng gan and dversty gan obtaned by our proposed approach, we consder the followng two dfferent channel models. 1) Constant Channel Model: In ths model, the erasure probablty of every channel lnk between node to the destnaton s gven by a constant erasure probablty ϵ, namely ϵ = ϵ. Ths channel model s sutable to evaluate the pure codng gan obtaned by the proposed approach. 2) Trangle Dstrbuton Channel Model: As mentoned above, coded cooperaton ams to explot not only the codng gan but also the spatal dversty gan. To evaluate the potental of spatal dversty enhancement of proposed cooperaton, the constant channel model s nadequate snce the model does not have any dfference between dfferent channel lnks. Thus, we further ntroduce a trangle dstrbuton channel model whch can be consdered as a smplfed fadng model. In ths model, erasure probablty s characterzed by the trangle dstrbuton of whch probablty densty functon (pdf) f(x) s gven by 2x f(x) = 3ϵ 1 0 x 3ϵ 1 2(x 1) 3ϵ 3ϵ 1 < x 1. (2) Note that the average erasure probablty ϵ n ths model should be 1 3 ϵ 2 3. Fgure 2 llustrates the pdf wth ϵ = 1.3/3 as an example. III. SPATIALLY COUPLED REPEAT-ACCUMULATE CODED COOPERATION In ths secton, the encodng of SC-RA-CC s explaned. Moreover, two dfferent graph expressons of SC-RA-CC are

3 Demodulator u (t) 1 u (t) u (t) N π (t),1 π (t), π (t),n 1 N Combner v (t) Source D s (t) p (t) t = 1 t = 2 = 1 = 2 = 3 = 4 = 1 = 2 π (1) 4,1 π(1) 4,2 π(1) 4,3 π (2) 1,2 π(2) 1,3 π(2) 1,4 Fg. 3. Encodng block dagram of SC-RA-CC for the transmt node at the tth transmsson block. gven so as to elaborate the structure of SC-RA-CC achevng the superor performance. A. Encodng Encodng block dagram of transmt nodes at the tth transmsson block s llustrated n Fg. 3. Let s (t) denote K- bt nformaton packet generated by transmt node at the tth transmsson block. A K-bt bnary vector u (t) ndcates the latest nformaton packet transmtted by node at the tth transmsson block and thus t can be rewrtten by u (t) s (t) ( < ) = s (t 1) ( > ). (3) The vector u (t) s nterleaved by correspondng nterleaver π (t),. Note that and the resultng bnary vector s denoted by the nterleaver π (t), s mutually and stochastcally ndependent of ones wth dfferent t,, and. As observed n the fgure, every transmsson block, the transmt node has (N 1) nterleaved vectors. These (N 1) vectors are combned nto a K-bt vector v (t) by the followng equaton: v (t) = N =1, (4) where expresses bt-by-bt XOR operaton. Fnally, a K- bt party packet p (t) s obtaned by dfferentally encodng the vector v (t) as llustrated n Fg. 3. The node transmts s (t) and p (t) as ts own message to the destnaton at the tth transmsson block. Note that, at the frst transmsson block.e., t = 1, frst (N 1) transmt nodes do not have (N 1) nformaton packets from the others to calculate (4). In ths case, node ust encodes ( 1) packets. Smlarly, at the last transmsson block.e., t = T + 1, transmt nodes do not have new nformaton packets anymore but the transmt node has (N ) packets to be encoded. Also note that the last user = N at the last transmsson block t = T + 1 does not have any packets to transmt. Therefore, the resultng codng rate of SC-RA-CC R s calculated as R = T N 2T N+N 2. Fg. 4. Tanner-graph of SC-RA-CC wth K = 3 and N = 4 where party bts of user 4 at t = 1 and user 1 at t = 2 are only llustrated. t = 1 t = 2 t = 3 = 1 = 2 = 3 = 4 = 1 = 2 = 3 = 4 = 1 = 2 = Fg. 5. Protograph of SC-RA-CC wth N = 4 and T = 2. B. Graph Expressons of SC-RA-CC To clearly show the structure of SC-RA-CC, we here ntroduce a representaton of bpartte graph called Tannergraph. Fgure 4 s a Tanner-graph of SC-RA-CC wth N = 4 and K = 3 at the arbtrary transmsson block t where crcles ndcate varable nodes correspondng to coded bts and squares ndcate check nodes correspondng to a sngle party check constrant. Also, whte and black crcles respectvely ndcate nformaton and party bts. Note that, n Fg. 4, party bts of user 4 at t = 1 and user 1 at t = 2 are only llustrated for smplcty. As explaned above, every transmt node utlzes (N 1) nformaton packets to encode the party packet. The party of node 4 s obtaned by XORng the nterleaved nformaton bts of three users. Subsequently, node 1 would combne three nformaton packets wth correspondng nterleavers to encode the party packet at the next transmsson block. To further elaborate the structure of SC-RA-CC, we ntroduce a protograph [13] of SC-RA-CC. Fgure 5 shows the protograph wth N = 4 and T = 2. The graph has whte and black crcles respectvely ndcatng K-bt nformaton packets and party packets,.e., varable nodes. Also, square nodes ndcate check nodes. Note that the edges between

4 whte varable nodes and check nodes nclude the operaton of random permutaton (cf., Fg 4). As obvous from the fgure, check nodes are connected wth spatally-close varable nodes and check nodes on both ends of the graph have less edges than those n the center. The correspondng varable nodes of check nodes wth less edges would be successfully decoded wth hgh probablty and ths phenomenon would be propagated to nodes n the center wth the ad of BP decoder [14], whch results n the superor performance of SC-RA-CC. Also, transmt node 1 at the frst transmsson block t = 1 does not have the party packet and transmt node 4 at the last transmsson block t = 3 do not have any packets to transmt as descrbed n Secton III-A. Ths protograph representaton clearly explans how the cooperaton explots the network structure to construct the code even when the number of users and the length of nformaton packet are lmted. Dfferent from ANCC, spatally coupled party packets of SC-RA-CC contrbute the decodng of N T nformaton packets and thus SC-RA-CC can effcently construct the long codeword enough. It s also worth notng that the SC-RA-CC does not requre the overhead to nform the XOR operaton of every transmt node. Dfferent from ANCC, every nodes always combne all the receved packets wthout randomly choosng them. IV. DENSITY EVOLUTION ANALYSIS In ths secton, the decodng threshold of SC-RA-CC s analyzed by densty evoluton [15]. 1) Constant Channel: Densty evoluton tracks average bt erasure probablty of every packets upon BP decodng. As obvous from the protograph of SC-RA-CC descrbed n Secton III-B, varable nodes correspondng to nformaton packets and party packets have the dfferent number of edges. Thus, we separately derve the bt erasure probablty of these nodes. Upon l teratons of BP decodng, bt erasure probablty of nformaton packet x (l+1) t, respectvely calculated by x (l+1) and party packet y (l+1) t, are ( ( N 1 1 t, = ϵ 1 (1 µ(y, + )) 2 N 1 =1 )) N 2 (5) N 1 k=1, k (1 µ(x, + k)) and ) N 1 y (l+1) t, = ϵ (1 (1 µ(y, )) (1 µ(x, k)), (6) k=1 where the functon µ(α, β) calculates the ndex of α by α (l) t 1,β+N (β 0) µ(α, β) = α (l) t,β (0 < β N). (7) α (l) t+1,β N (N β) Fg. 6. BER versus bt erasure probablty ϵ graph on constant channel. The decodng threshold of BP decodng ϵ th s defned as } ϵ th sup ϵ > 0 lm t, = 0 (8) l x(l) For the erasure probablty below ths threshold, decodng erasure probablty goes to zero. Fnally, the bt erasure probablty ϵ s smply used nstead of ϵ for the constant channel. 2) Trangle Dstrbuton Channel: To evaluate the achevable dversty gan of cooperaton, we evaluate the outage probablty rather than ϵ th averaged over the channel realzaton where the outage event occurs when the decodng erasure probablty does not go to zero upon decodng. Thus outage probablty can be defned as P o E [ Pr ( )] lm l x(l) t, > 0. (9) The outage probablty of drect communcaton s smply calculated by [ ] ϵ th P o = E 0 f(x)dx, (10) where the functon f(x) s gven by (2) and ϵ th s calculated by (8). Also, the outage probablty of SC-RA-CC s numercally calculated usng (5) and (6). V. NUMERICAL RESULTS In ths secton we shows numercal results of computer smulaton and densty evoluton analyss. In the followng, we assume that T = 20 and the maxmum number of teratons n every BP decodng s 300. A. Constant Channel Fgure 6 shows the bt erasure rate (BER) of drect communcaton, SC-RA-CC, and conventonal ANCC [6] where the length of nformaton packets s assumed to be K = 100, 300, and 500 and the number of transmt nodes s assumed to be N = 4 and 8. For ANCC, the degree D s set to be 2 that s optmzed va computer smulatons. Theoretcal lmts

5 Fg. 7. DER versus average bt erasure probablty ϵ over trangle dstrbuted channels. Fg. 8. BER versus average bt erasure probablty ϵ over trangle dstrbuted channels. of SC-RA-CC wth N = 4 and 8 are also presented, whch are calculated by (8). Note that the codng rate of SC-RA-CC wth N = 4 and 8 and conventonal ANCC s 0.494, 0.491, and 0.5, respectvely. Comparng wth the drect transmsson and conventonal ANCC, SC-RA-CC sgnfcantly mproves the decodng performance because of hgh codng gan. Also, SC-RA-CC provdes the lower BER as K ncreases snce the codeword length of SC-RA-CC s proportonal to both the length of nformaton packets and the number of users whle that of ANCC only depends on the number of users. However, gaps between smulaton results and decodng thresholds obtaned by densty evoluton are stll observed, whch are about Ths s because, n the calculaton of densty evoluton, the nfnte codeword length s assumed whle the codeword length n smulatons s lmted. The performance of N = 4 exhbts the error floor even when K = 500. SC-RA-CC couples (N 1) nformaton packets as explaned n Secton III-B. Ths structure effcently retreves the erased bts from the end ponts to the center of the protograph. However, f the couplng s too weak (namely N s small), the performance mprovement va BP decodng would be lmted and the SC-RA-CC would exhbt the error floor. B. Trangle Dstrbuton Channel Fgure 7 shows the decodng error rate (DER) of drect communcaton, conventonal ANCC, and SC-RA-CC wth N = 4 and 8 versus average bt erasure probablty ϵ of the trangle dstrbuton channel where the decodng error s defned that the decoded codeword ncludes at least one bt erasure. In the followng, the length of nformaton packet K s 100. The outage probabltes of drect communcaton and SC-RA-CC are also drawn n the fgure. As obvous from the fgure, owng to the dversty gan, the slope of SC-RA-CC s steeper than that of drect communcaton and conventonal ANCC meanwhle the drect communcaton and ANCC acheve the better performance at the hgh erasure probablty regon. In SC-RA-CC, the BP decoder retreves the nformaton from the end ponts to the center. The BP decodng, however, would stop when facng wth burst erasures [16]. Ths phenomenon results n worse DER performance at the hgh erasure probablty regon. Also, as smlar to the case wth constant channel, the hgher codng gan can be obtaned by ncreasng N. Fnally, Fg. 8 shows the BER performance of drect communcaton, conventonal ANCC, and SC-RA-CC wth N = 4 and 8 over trangle dstrbuton channels. Dfferent from all the results shown above, the SC-RA-CC wth small N provdes the lower BER performance. When facng wth the burst erasures, the BP decodng of SC-RA-CC causes more bt erasures n the codeword. When large N s utlzed, the resultng protograph becomes senstve to burst erasures. To overcome ths ssue, an attractve opton s the use of mult-dmensonal spatal couplng [16] or rregularly spatal couplng lke Luby transform (LT) codes [17], whch stll remans as the future work. VI. CONCLUSION In ths paper, we proposed SC-RA-CC and numercal results confrmed that SC-RA-CC can sgnfcantly mprove the decodng performance even when the number of nodes and the length of nformaton packets are lmted. ACKNOWLEDGMENT Ths work was supported n part by JSPS KAKENHI Grant Number REFERENCES [1] I.F. Akyldz, W. Su, Y. Sankarasubramanam, and E. Cayrc, A survey on sensor networks, IEEE Commun. Mag., vol. 40, no. 8, pp , Aug [2] L. Zheng and D.N.C. Tse, Dversty and multplexng: a fundamental tradeoff n multple-antenna channels, IEEE Trans. Inf. Theory., vol. 49, no. 5, pp , May 2003.

6 [3] V. Tarokh, H. Jafarkhan, and A.R. Calderbank, Space-tme block codes from orthogonal desgns, IEEE Trans. Inf. Theory., vol. 45, no. 5, pp , Jul [4] C. Deqang and J. Laneman, Modulaton and demodulaton for cooperatve dversty n wreless systems, IEEE Trans. Wreless Commun., vol. 5, no. 7, pp , July [5] T.E. Hunter and A. Nosratna, Dversty through coded cooperaton, IEEE Trans. Wreless Commun., vol. 5, no. 2, pp , Feb [6] X. Bao and J. L, Adaptve network coded cooperaton (ANCC) for wreless relay networks: matchng code-on-graph wth network-ongraph, IEEE Trans. Wreless Commun., vol. 7, no. 2, pp , February [7] R.G. Gallager, Low-densty party-check codes, IRE Trans. Inf. Theory., vol. 8, no. 1, pp , January [8] J. Hou, P.H. Segel, and L.B. Mlsten, Performance analyss and code optmzaton of low densty party-check codes on raylegh fadng channels, IEEE J. Sel. Areas Commun., vol. 19, no. 5, pp , May [9] D. Dvsalar, H. Jn, and R. J. McElece, Codng theorems for turbolke codes, Proc.36th Allerton Conf. on Commun. Control and Comput., pp , September [10] C. Berrou, A Glaveux, and P. Thtmashma, Near Shannon lmt error-correctng codng and decodng: Turbo-codes. 1, n Proc. IEEE Int. Conf. Commun., Geneva, Swtzerland, May 1993, vol. 2, pp vol.2. [11] A. Jmenez Felstrom and K.S. Zgangrov, Tme-varyng perodc convolutonal codes wth low-densty party-check matrx, IEEE Trans. Inf. Theory., vol. 45, no. 6, pp , Sep [12] S. Johnson and G. Lechner, Spatally coupled repeat-accumulate codes, IEEE Commun. Lett., vol. 17, no. 2, pp , February [13] J. Thorpe, Low-densty party-check (LDPC) codes constructed from protographs, JPL INP, Tech. Rep , Aug [14] F.R. Kschschang, B.J. Frey, and H.-A Loelger, Factor graphs and the sum-product algorthm, IEEE Trans. Inf. Theory., vol. 47, no. 2, pp , Feb [15] T.J. Rchardson and R.L. Urbanke, The capacty of low-densty party-check codes under message-passng decodng, IEEE Trans. Inf. Theory., vol. 47, no. 2, pp , Feb [16] R. Ohash, K. Kasa, and K. Takeuch, Mult-dmensonal spatallycoupled codes, n Inf. Theory Proc. (ISIT), 2013 IEEE Int. Symposum on, July 2013, pp [17] M. Luby, LT codes, n Proc. 43rd Annual Symp. on Fundatons of Computer Scence (FOCS), 2002, pp