Bit-interleaved Rectangular Parity-Check Coded Modulation with Iterative Demodulation In a Two-Node Distributed Array

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1 Bt-nterleaved Rectangular Party-Check Coded Modulaton wth Iteratve Demodulaton In a Two-Node Dstrbuted Array Xn L, Tan F. Wong, and John M. Shea Wreless Informaton Networkng Group Department of Electrcal & Computer Engneerng Unversty of Florda Ganesvlle, Florda , USA Emal: lx@ecel.ufl.edu, twong, jshea}@ece.ufl.edu Abstract In ths paper, we propose a network-based dstrbuted antenna array approach, n whch a bt-nterleaved rectangular party-check coded modulaton wth an teratve demodulaton scheme s used. Dfferent from conventonal multplearray systems, ths dstrbuted array employs a par of physcally separated dentcal recevng nodes. These nodes receve the transmtted sgnal through ndependent channels. Then each node demodulates and decodes the receved sgnal n an teratve manner. At each teraton, the par of nodes exchange the relablty measure of a small porton of the symbols to obtan spatal dversty. Smulaton results show that sgnfcant dversty gan can be acheved at much lower traffc cost than maxmal rato combnng. I. INTRODUCTION The growng demand of hgh bt-rate data transmsson n wreless systems contnues to propel the research of usng antenna arrays to ncrease the capacty of wreless communcaton systems. A well-known array technque s to combne the receved sgnals from the array elements optmally, e.g., usng maxmal rato combnng (MRC) [], to gan spatal dversty at the recever. Ths technque offers capacty gans over sngle-antenna systems under the assumpton of ndependent fadng at dfferent antenna elements. However, fadng correlaton does exst when the elements are not spaced suffcently far apart n practce, whch can sgnfcantly reduce the capacty of a multple-antenna system []. However, too large a sze of the array may lmt ts applcablty n many practce scenaros. In a prevous work [3] [4], we proposed a network-based dstrbuted antenna array approach to obtan spatal dversty wthout the need of physcal connectons between the antenna elements. In ths approach the physcally separated antennas form a dstrbuted array va relable network connectons. The array nodes are far apart enough that the assumpton of ndependent and dentcally dstrbuted (..d.) fadng at dfferent nodes holds. Each antenna node may perform the recevng and decodng process ndependently whle t can communcate wth the other node. For smplcty, connectons between the nodes are assumed to be error free. Wth such a dstrbuted antenna array, spatal dversty can be obtaned through collaboraton and communcaton between the par of nodes n the cluster. The conventonal MRC technque s not desrable for the proposed dstrbuted antenna array system because of the requred excessve amount of traffc between the nodes. Our approach s to employ teratve soft-n/soft-out (SISO) decoder at each node to generate soft outputs for data bts, then exchange a small porton of these soft outputs between the nodes. Each node uses the addtonal nformaton from the other node as aprornformaton and restart the decodng process. Wth ths teratve decodng procedure, we can obtan a dversty gan close to that provded by MRC for BPSK [3] [4]. In ths paper, we consder employng coded modulaton (CM) n the dstrbuted array system to explore the possblty of obtanng spatal dversty wth hgher spectral effcency for bandwdth-constraned wreless channels. Bt-nterleaved coded modulaton (BICM) s a CM technque able to ncrease the dversty order of a code to ts mnmum Hammng dstance, thus leadng to a performance mprovement over fadng channels when hgh-order sgnal constellatons are used [5]. Although developed prmarly for fadng channel, by usng an teratve demodulaton (ID) algorthm, the mnmum free Eucldean dstance degradaton of BICM over addtve whte Gaussan nose (AWGN) channel can be overcome [6]. Ths teratve decodng approach of BICM s referred to as BICM- ID. Besdes, an advantage of BICM-ID s ts flexblty n desgn. Any code wth soft-output decoder can be used. In ths paper, we employ t wth rectangular party-check code (RPCC) [9] to construct our transmsson system. We also propose a dstrbuted decodng strategy sutable for BICM-ID. The remander of ths paper s organzed as follows. In Secton II, we present the two-node dstrbuted array system and channel model. In Secton III, the BICM teratve demodulaton for RPCCs and the desgn of dstrbuted decodng are descrbed n detal. Followng that, Monte Carlo smulaton results for dfferent sgnal constellatons are shown n Secton IV. Fnally, conclusons are gven n Secton V /03/$ IEEE 8

2 u ( n) u c v RPCC π Encoder L ( u^ ) RPCC Decoder Fg.. c L ( u) π π v Modulator µ / χ SISO Demod L ( v) x y Addtonal nfo from other node System model of BICM-ID wth RPCC II. SYSTEM MODEL Channel g We consder a smple dstrbuted array system wth two dentcal recevng nodes. A dstant transmtter sends a block of modulated sgnal to the two recever nodes. The two recever nodes are physcally separated far apart enough that fadng at each node s..d.. Each ndvdual node receves and decodes ts receved sgnal ndependently. For smplcty, we assume that the two nodes can communcate wth each other relably. We are only nterested n the communcaton lnk from the dstant transmtter to the two nodes. The transmtter adopts a typcal BICM approach [5], as shown n Fg.. A block of data bts u to be transmtted are encoded wth an RPCC encoder wth code rate R c. Then the coded bt stream c are fed nto a bt-wse random nterleaver π, generatng bt stream v = π(c). After that, the bt stream v s modulated onto a sgnal sequence x over a -dmenson sgnal set χ of sze χ = M = m by a M-ary modulator wth a one-to-one bnary map µ : 0, } m χ. Ths sgnal sequence s then sent through the channel. The overall spectral effcency of ths system s mr c bts/symbol. Here we use a memoryless fadng channel model that ncludes AWGN channel as a specal case. In ths model, the receved sgnal y at the two antenna nodes correspondng to the transmtted sgnal x χ can be expressed as y () = g () x + n (), y () = g () x + n (), where: ) g () and g () are channel fadng gans. For AWGN channels, g () = g () =. For Raylegh fadng channels, g () and g () are ndependent crcular-symmetrc complex Gaussan random varables wth E[g () ]=0and E[ g () ]=for =, ; ) n () and n () are ndependent zero-mean, crcularsymmetrc complex addtve Gaussan nose wth covarance E[ n () ]=σ for =,. We normalze the sgnal energy E[ x ]=. Thus, the average sgnal-to-nose rato (SNR) s /σ. In ths channel model, we assume that perfect channel state nformaton (CSI) (g (), g () ) s avalable at the recever nodes and hence coherent demodulaton s performed at each node. Wth ths model the pdf p(y () x), for =,, wth perfect CSI s gven by p(y () x) = πσ exp ( y() g () x /σ ). () n At each recever node, we treat the modulaton and code as two components of a concatenated codng system. By employng a maxmum a posteror (MAP) demodulator, we feed the extrnsc nformaton from the RPCC decoder back to the demodulator as the aprornformaton to carry out the demodulaton and decodng n an teratve manner. After some teratons, we exchange nformaton for a porton of symbols between the two nodes and restart the demodulaton and decodng processes. III. DISTRIBUTED DECODING FOR BICM-ID WITH RECTANGULAR PARITY-CHECK CODE One mportant component n our bt-nterleaved coded modulaton system s the rectangular party-check code [8]. It conssts of sngle party-check codes that operate on rows and columns of a square matrx that contans the nformaton bts. RPCCs wth large block szes are very hgh-rate systematc codes that can be decoded by a low-complexty teratve SISO algorthm. More detals of RPCC can be found n [9] [0] [3] and [4]. Another mportant component n the BICM-ID system s the teratve demodulaton module. Based on the dea that performng demodulaton and decodng n an teratve manner s a key to mprove the performance of BICM [6] [7], we employ the recever model as llustrated n Fg.. To smplfy the teratve decodng process, we frst modfy the demodulator to work n the log-lkelhood rato (LLR) doman. Suppose that each m-bt vector v =(v,v,,v m ) from the nterleaver are mapped nto one sgnal x out of the m sgnals n the set χ by mappng rule µ,.e., x = µ(v) χ, at the modulator, and that the receved sgnal correspondng to x s y. Letl (x) denote the th ( =,,,m) bt of the label of x. For convenence, we assume that l (x) =b s n the GF() wth the elements +, }. In our soft demodulator, we wll consder the MAP rather than maxmum-lkelhood (ML) bt metrc. It s easy to see that the MAP bt metrc of v = b +, } s gven by λ(v = b, y) = log P (v = b, y) = log p(y z)p (z v = b)p (v = b), () z χ where p(y z) s gven n () accordng to our channel model. We assume a perfect bt-nterleaver π such that v,v,,v m } are ndependent to each other. Wth ths assumpton, we have m P (z)=p(z =µ(v,v,,v m ))= P (v j = l j (z)). (3) Hence, the MAP bt metrc can be smplfed to λ(v = b, y) max log p(y z)+ log P (v j = l j (z)) z χ b j + log P (v = b)+c }, (4) where χ b denotes the subset of all sgnal z χ wth l (z) =b, and C s a constant. Above, the approxmaton 83

3 log( a ) max (log a ) s used. For convenence we choose the constant as C = m ( log P (vj = +) + log P (v j = ) ). (5) Then the metrc becomes λ(v = b, y) = max log p(y z)+ z χ b m l j (z)l(v j ) }, (6) where L(v j ) = log ( P (v j =+)/P (v j = ) ) s the apror LLR of bt v j. Thus the soft value of bt v n LLR form s computed by L(v y)=l(v, y) =λ(v =+, y) λ(v =, y) = L(v ) + max log p(y z)+ l j (z)l(v j ) } z χ + max z χ log p(y z)+ j l j (z)l(v j ) }. (7) Subtractng the aprorllr of v, L(v ), from (7) we can obtan the extrnsc nformaton of v L e (v ) = max log p(y z)+ l j (z)l(v j ) } z χ + max z χ j log p(y z)+ j l j (z)l(v j ) }. (8) We treat ths extrnsc nformaton as the output of the soft demodulator. From (8), we can see that n order to obtan the extrnsc LLR of a bt of a sgnal, we need to use the apror LLRs of the other m bts and the channel observaton of the sgnal as nput. Wth the modfcaton above, the demodulaton and decodng procedure can perform n an teratve way convenently. In Fg. we use L (n) ( ) to denote the LLR at the nth teraton. Frst, we ntalze all the aprorllrs L (n) (v) and L (n) (u) to zeros for n =0.Atthenth teraton, when the channel observaton y of the transmtted sgnal sequence s receved, we demodulate t usng (8) to produce L (n) e (v). After denterleaver π, L (n) e (c) =π (L (n) e (v)) s fed nto the RPCC decoder for decodng. Snce the RPCC decodng s an SISO teratve algorthm, we shall use the extrnsc nformaton L e (n ) (u), produced n the (n )th teraton, as the a pror nformaton L (n) (u) of decoder n the nth teraton. The extrnsc nformaton L (n) e (u) generated by the RPCC decoder s then passed thought the nterleaver π and fed back as the aprornformaton L (n+) (v) for the soft demodulator agan. After a number of teratons the estmate of data bts û s obtaned from the hard decson on L (n) (û). The mappng µ has a sgnfcant effect on the performance of BICM-ID. For BICM, Gray code mappng outperforms set parttonng (SP) mappng [5]. However, when assocated wth the teratve demodulaton, SP mappng outperforms Gray mappng at hgh SNR [6], [7]. Ths can be seen from (8) that, due to the property of the Gray mappng that the label of a j symbol has only one bt dfferent form ts nearest neghbors, the effect of aprorllrs can be weakened sgnfcantly. However, ths s not the case for SP mappng. Thus the MAP demodulator can make a more effectve use of the apror nformaton for SP mappng than for Gray mappng. The presented BICM-ID scheme s readly applcable to a dstrbuted array. As we ponted out n [3], a decoded data bt wth a small soft output magntude from the RPCC decoder s more lkely to be n error. However, f the bt-based strategy n [3] s used here to gan dversty from other recevng node, we wll lose the advantage aganst MRC n term of savng nformaton exchange traffc when the modulaton order M ncreases. Hence, we develop a symbol-based strategy for BICM-ID to reduce the nformaton exchangng traffc. At frst, we defne the symbol relablty measure out of the decoder as P (ˆx) L(ˆx) = log P (ˆx) = log P (ˆx) (9) P (z), z ˆx where ˆx = µ(ˆv,, ˆv m ) χ s the estmate of transmtted sgnal x. For convenence, we defne symbol metrc for each constellaton pont z χ as λ(z) = m l j (z)l(ˆv j ), (0) where L(ˆv j ) s the soft output of the coded bt v j. Ths symbol metrc reflects the probablty P (x = z) gven the LLRs L(ˆv j ) for j =,,,m. In fact, ˆx should be the constellaton pont} that has the largest relablty,.e., ˆx = arg max z χ λ(z). Smlar to (3)-(5), (9) can be smplfed to } L(ˆx) λ(ˆx) max λ(z) = mn L(ˆvj ) }. (),,m z ˆx,z χ Snce the LLR magntude of a bt can be used as the measure of ts relablty, () ndcates that the relablty of a decoded symbol s determned by the soft value of ts least relable bt, whch s bascally n agreement wth the bt-based dea n [3]. Wth ths defnton, the dstrbuted decodng procedure works as follows. After every I (I ) teratons of demodulaton and decodng, each node computes the symbol relablty L(ˆx) and rank the symbols accordng to ther relablty. Then each node requests addtonal nformaton from the other node for symbol x that L(x) ranks n the lowest a%. We denote the addtonal nformaton for x as L a (x). Suppose that the estmate correspondng to symbol x at the other node s x = µ(ṽ,, ṽ m ), whch may be dfferent from ˆx snce the assumpton of ndependent channels. Upon recevng the request, a node sends: ) the relablty of the requested symbols generated n ts own decodng process as the addtonal nformaton,.e., L a (x) =L( x); ) the hard decson of x, l j ( x) for j =,,,m, whch s also the hard decson of (ṽ, ṽ,, ṽ m ). Heren, we adopt followng strategy, a node does not request addtonal nformaton for the symbol f a request has been made for t n all prevous exchanges. In ths case, the node wll request nformaton for the next symbol n the rankng 84

4 order to make sure that the request for a total of N a% symbols wll be made for the current exchange, where N s the symbol block sze. The advantage of ths strategy s that the addtonal nformaton can cover more symbols for a number of exchanges. After the exchange, as shown n Fg., each node wll use L a (x) and the hard decson l j ( x) (j =,,,m) obtaned from the other node to reconstruct an addtonal symbol metrc λ a (z) smlar to (0) for each possble constellaton pont z χ. Snce l j ( x) s the hard decson of bt ṽ j,wehave L(ṽ j )=l j ( x) L(ṽ j ). () From () we can see that L(ṽ j ) L a (x). Ths means each bt n x has as least a relablty of L a (x). Now we replace L(ṽ j ) wth L a (x) for j =,,,m n (), whch s equvalently to set the relablty of all ts bts the same as the relablty of a symbol. Thus, we can construct the addtonal symbol metrc as λ a (z) = m l j (z)l j ( x)l a (x). (3) Ths addtonal symbol metrc s then used as the apror nformaton for demodulaton, and (6) becomes λ(v =b,y)= max log p(y z)+ m } l j (z)l(v j )+δλ a (z), (4) z χ b where δ< s a scalng factor used to reduce the effect of error propagaton. Usually, δ can be set to In the followng I teratons, the whole process then repeats wth addtonal exchange of symbol relablty and ts hard decson between the two nodes. Note that n ths strategy we just need to exchange one real number L( x) and m bts l j ( x) (j =,,,m) for each symbol. However, for MRC, one needs to exchange a complex number y (channel observaton) and a real number g (magntude of fadng gan, for AWGN channel no need to exchange t snce g =) for each symbol. Ths means we just requre less than /3 (for Raylegh fadng channel) of or equal (for AWGN channel) to the exchangng traffc of MRC for each symbol, meanwhle we only need to exchange nformaton for a porton of symbols. Hence wth ths symbol-based strategy, we can reduce the requred nformaton exchange traffc sgnfcantly. IV. SIMULATION RESULTS In ths secton, we examne the performance of the proposed dstrbuted array scheme by Monte Carlo smulatons. In the smulatons, we set the packet sze to 04 data bts,.e., the data bts are arranged nto a 3 3 matrx for the RPCC encodng. Wth ths block sze, the RPCC gves a code rate of In the decodng procedure, a node requests addtonal nformaton for 5% of the symbols wth the smallest relablty at the begnnng of every 0 teratons after the frst 0. For nstance, 3 exchanges cause an overall traffc approxmately equal to 30% (for Raylegh fadng channel) or 45% (for AWGN channel) of what requred by MRC. BER Sngle recevng node nodes wth 45% traffc nodes wth MRC Gray mappng SP mappng E b /N 0 (db) Fg.. BER for BICM-ID wth 3 RPCC and 8PSK n the two-node dstrbuted array over AWGN channel. In the case of MRC, we assume that each node passes all ts channel observatons and fadng gans to the other node and maxmally combnes the channel observatons before demodulaton. Smulatons show the bt error rates (BER) at the two nodes are almost the same as each other. So we take the average of them as the performance of the dstrbuted array system. Fg. shows the BER performance of BICM-ID wth 3 RPCC n the dstrbuted array over AWGN channels when 8PSK wth Gray and SP mappng are used. In the fgure, E b s the receved energy per bt per antenna. Wth MRC, about 3dB spatal dversty gan can be acheved for both mappngs. Wth our dstrbuted array approach, we obtan a.4db and.4db gan for Gray and SP mappng at the traffc cost of 45% (.e., 3 exchanges n total) of MRC. Fg. 3 shows the BER curves for Raylegh fadng channels. The spatal dversty gan provded by MRC s about 8.5dB for both Gray and SP mappng. By exchangng a total of 0% (.e., exchanges n total) of the nformaton amount requred for MRC, our dstrbuted BICM-ID system obtan a 8.3dB and 7.3dB gan at the BER of 0 5 for Gray and SP mappng, respectvely. In Fgs. 4 and 5, we show the the average SNR (E s /N 0 ) at BER of 0 5 versus spectral effcency for the two-node dstrbuted array system for dfferent constellatons wth Gray mappng and SP mappng over AWGN channels and Raylegh fadng channels, respectvely. The average SNR can be computed approxmately by SNR = mr c E b /N 0, where m s the number of bts per symbol carryng, and R c s the code rate of RPCC. We can see that for both AWGN and Raylegh fadng channels the proposed dstrbuted BICM-ID approach can acheve almost the dversty gan provded by MRC, but wth only exchangng 0% ( exchanges n total) and 45% For 3QAM, quas-gray mappng s used because Gray mappng s mpossble n ths case. 85

5 0 0 0 Sngle recevng node nodes wth 0% traffc nodes wth MRC Gray mappng SP mappng QAM 5 BER Spectral Effcency (bts/symbol) PSK 6QAM 3QAM E b /N 0 (db) QPSK Sngle recever recevers wth 0% traffc.5 recevers wth MRC Unflled symbol: Gray mappng Sold symbol: SP mappng Average SNR (db) Fg. 3. BER for BICM-ID wth 3 RPCC and 8PSK n the two-node dstrbuted array over Raylegh fadng channel. Fg. 5. Average SNR at 0 5 BER versus spectral effcency for BICM-ID wth 3 RPCC n a two-node dstrbuted array over Raylegh fadng channels. Spectral Effcency (bts/symbol) QPSK 8PSK 6QAM 3QAM 64QAM Sngle recever recevers wth 45% traffc recevers wth MRC Unflled symbol: Gray mappng Sold symbol: SP mappng Average SNR (db) Fg. 4. Average SNR at 0 5 BER versus spectral effcency for BICM-ID wth 3 RPCC n a two-node dstrbuted array over AWGN channels. (3 exchanges n total) of the amount of nformaton requred by MRC between the two nodes for AWGN channel and Raylegh fadng channel, respectvely. Ths especally shows the advantage of our approach for fadng channels. V. CONCLUSION In ths paper, we have nvestgated a network-based dstrbuted array approach, n whch hgh-order constellatons wth teratve demodulaton and RPCCs are used. A sgnfcant dversty can be obtaned wth a relatvely small amount of nformaton exchange between the ndependent and physcally separated recevng nodes. Ths nvestgaton motvates us to consder employng other more powerful codes to replace the RPCC n the dstrbuted array. Another consderaton s to extend the two-node array to the case of more than two recevng nodes. Ths wll rse many jont network coordnaton and physcal layer decodng ssues. These ssues wll be our future research drectons. ACKNOWLEDGMENT Ths work was supported n part by the Natonal Scence Foundaton under Grant ANI-0087 and n part by the Offce of Naval Research under Grant N The authors would also lke to thank Yuguang Fang and Arun Avudanayagam for ther many helpful suggestons on ths paper. REFERENCES [] J. G. Proaks, Dgtal Communcatons, 4th ed., McGraw-Hll, 000. [] C. Chuah, D. N. C. Tse, J. M. Kahn and R. A. Valenzuela, Capacty scalng n MIMO wreless systems under correlated fadng, IEEE Trans. Inform. Theory, vol. 48, no. 3 pp , March 00. [3] T. F. Wong, X. L and J. M. Shea, Iteratve decodng n a two-node dstrbuted array, n Proc. IEEE Mlcom 0, Anahem, CA, Oct. 00. [4] T. F. Wong, X. L and J. M. Shea, Dstrbuted decodng of rectangular party-check code, Electroncs Letters, vol. 38, no., pp , Oct. 00. [5] G. Care, G. Tarcco, and E. Bgler, Bt-nterleaved coded mdulaton, IEEE Trans. Infom. Theory, vol. 44, pp , May 998. [6] X. L, and J. A. Rtcey, Bt-nterleaved coded modulaton wth teratve decodng, n Proc. IEEE ICC 99, Vancouver, BC, Cananda, pp , June 999. [7] X. L and J. A. Rtcey, Trells-coded modulaton wth bt nterleavng and teratve decodng, IEEE J. Select. Areas Commun., vol. 7, no. 4, pp , Apr [8] J. Hagenauer, E. Offer, and L. Papke, Iteratve decodng of bnary block and convolutonal codes, IEEE Trans. Inform. Theory, vol. 4, pp , Mar [9] T.F.Wong and J.M.Shea, Mult-dmensonal party check codes for bursty channels, n Proc. 00 IEEE Int. Symp. Informaton Theory, Washngton, D.C., pp. 3, June 00. [0] T. F. Wong, J. M. Shea, and X. L, Usng mult-dmensonal partycheck codes to obtan dversty n Raylegh fadng channels, n Proc. IEEE Globecom, San Antono, TX, pp. 0 4, Nov