COOPERATIVE relaying has been proposed as a promising

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1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL 31, NO 8, AUGUST Coding and Sytem Deign for Quantize-Map-and-Forward Relaying Vinayak Nagpal, Member, IEEE, I-Hiang Wang, Member, IEEE, Milo Jorgovanovic, Student Member, IEEE, David Te, Fellow, IEEE, and Borivoje Nikolić Senior Member, IEEE Abtract In thi paper we develop a low-complexity coding cheme and ytem deign framework for the half duplex relay channel baed on the Quantize-Map-and-Forward QMF) relaying cheme The propoed framework allow linear complexity operation at all network terminal We propoe the ue of binary LDPC code for encoding at the ource and LDGM code for mapping at the relay We expre joint decoding at the detination a a belief propagation algorithm over a factor graph Thi graph ha the LDPC and LDGM code a ubgraph connected via probabilitic contraint that model the QMF relay operation We how that thi coding framework extend naturally to the high SNR regime uing bit interleaved coded modulation BICM) We develop denity evolution analyi tool for thi factor graph and demontrate the deign of practical code for the half-duplex relay channel that perform within 1dB of information theoretic QMF threhold Index Term Relay channel, low denity parity check LDPC) code, low denity generator matrix LDGM) code, iterative decoding, modulation, interleaving, MIMO I INTRODUCTION COOPERATIVE relaying ha been propoed a a promiing technique to reolve the increaing demand for data throughput in wirele network Recently a lot of progre ha been made in etablihing the theoretical foundation of cooperative communication To apply thee principle toward the deign of practical wirele ytem, variou ytem deign tradeoff mut be taken into conideration Thi paper preent progre toward thi goal We propoe a ytem deign and coding framework for quantize-map-and-forward QMF) [1] relaying that ha low complexity and perform cloe to information theoretic bound S R Fig 1 Example relay network With multiple antenna at detination, ourcerelay cooperation provide additional degree of freedom for communication from the ource for ome fraction of total time, then forward a decription of their obervation in the remaining fraction There are everal apect involved in the deign of a cooperative relaying ytem Litening fraction and forwarding cheme mut be determined for each relay Suitable modulation and channel coding cheme mut be deigned for variou terminal Rate adaptation mechanim mut be conidered to account for change in availability of relay and channel trength Practical contraint mut be conidered eg minimizing the overall ytem complexity, reue of building block from traditional non-cooperative) ytem a much a poible, compatibility with protocol at higher layer and handling of ytem imperfection like ynchronization, channel etimation error etc In thi paper, we focu on the coding and ignaling apect of cooperative relaying Other component are dicued in brief toward the end of the paper D A Cooperative Sytem A cooperative wirele link typically conit of an information ource, a detination and one or more cooperating half duplex relay The relay are uually aumed to operate inband ie no additional channel reource are allocated for cooperation Without lo of generality, it i aumed that relay ue time-diviion-duplexing ie they liten to tranmiion Manucript received 1 Augut 2011; revied 1 May 2012 The material in thi paper wa preented in part at the Annual Allerton Conference on Communication, Control, and Computing, Monticello, Illinoi, USA, September 2010 The author are with the Department of EECS, Univerity of California at Berkeley, Berkeley, California, 94720, USA vinayaknagpal@nokiacom; i-hiangwang@epflch; {milo,dte,bora}@eecberkeleyedu) Digital Object Identifier /JSAC /13/$3100 c 2013 IEEE B Relaying Scheme Mot wirele ytem operate at moderate to high SNR ie in a regime where tranmit power i not the major limiting factor for link capacity At high SNR, the link capacity i limited by patial degree of freedom Relay cooperation i of pecial interet for practical ytem becaue it ha the potential to provide additional patial degree of freedom For illutration, conider a relay channel with ingle-antenna ource, a half-duplex ingle antenna relay and a detination with two antenna hown in Fig 1 Since the detination ha multiple antenna, the ource can patially multiplex traffic to the detination uing relay cooperation If the ource-to-relay channel i trong, thi network can approach the high-snr performance of the 2 2 MIMO channel [2] Variou trategie for relay cooperation are propoed in literature Among thee amplify-and-forward AF), decode-

2 2 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL 31, NO 8, AUGUST 2013 and-forward DF) and compre-and-forward CF) [3], [4] have received the mot attention Under DF, the relay decode the ource meage and forward a hard etimate of it, wherea under AF and CF it forward a oft etimate without explicitly decoding it In DF and CF, the relay map it etimate to a random codeword before forwarding, wherea in AF it forward an uncoded ignal The QMF cheme [1] alo ue oft etimate forwarding with random coding imilar to CF For the example network in Fig 1, the CF and QMF cheme are cloe to optimal at high SNR In fact they achieve within one bit/ec/hz to the information-theoretic capacity [1], [5] An intuitive explanation for why QMF/CF perform better than both AF and DF i given in the context of the example in Fig 1 below In the example network of Fig 1, the detination receive continuouly from the ource The relay receive a tronger verion, when it i litening Since the detination ha two antenna, it can reolve imultaneou tranmiion from the ource and relay In order to achieve patial multiplexing, the relay hould extract the le ignificant bit from it obervation), which the detination cannot reolve, and forward them Under AF, the relay only forward the more ignificant bit, which the detination can already reolve Therefore AF cooperation provide limited benefit Under DF, the relay decode the entire meage before it forward anything Since the litening time i limited, thi approach i inefficient The QMF/CF cheme implicitly extract the le ignificant bit from the relay obervation by uing quantization/compreion and random mapping Therefore thee cheme provide the mot cooperation gain Depite having imilar performance for the ingle relay network, the CF and QMF cheme have ignificant difference In the conventional CF cheme, the relay compree it oberved ignal and perform a random code mapping before forwarding The compreion rate i choen in order for the detination to perform two-tep decoding ie firt decode compreed ignal from relay and then ue it a ide information to decode the meage from ource For configuration that involve multiple relay, two-tep decoding i ub-optimal and conventional CF i not within bounded gap from information-theoretic capacity [1] Even for the inglerelay configuration, conventional CF require that the relay have full knowledge of the quality of it forward channel Thi introduce a large etimation and feedback overhead for fading channel and increae the complexity of rate adaptation cheme Under QMF, the relay quantize it received ignal at noie level, randomly map it to a codeword and forward it Unlike CF, the quantization and mapping i performed without regard to the quality of forward channel at the relay Thi reduce the channel etimation and feedback overhead for the link It alo implifie rate adaptation protocol Additionally, QMF ue joint decoding a oppoed to ucceive decoding) and perform within bounded gap from capacity for network having an arbitrary number of relay [1] QMF ha played a key role in everal recent information theoretic reult on cooperative network [2], [5] [7] Due to thee favorable propertie, the QMF cheme i uperior to CF from the perpective of practical cooperative ytem Since mapping at a QMF relay i performed without any knowledge of forward channel trength, ide information from relay cannot be decoded at the detination independently QMF require joint decoding of the meage from ource) and ide information from relay) [1] Thi preent a unique challenge becaue joint decoding typically require higher complexity and make it harder to deign a practical cooperative coding cheme The key contribution of thi paper i to develop a low-complexity cooperative coding framework for QMF that ignificantly reduce the complexity of joint decoding and yet perform cloe to information theoretic bound C Related Work Majority of previou work on code deign for cooperative relaying i focued on the DF cheme DF relay fully decode the ource meage Therefore, DF coding cheme involve partitioning a large codebook into two part The ource tranmit one part of the codeword and the relay tranmit the remaining part [8] [10] Turbo code deign which perform 1dB away from the DF information theoretic threhold are demontrated in [11], [12] LDPC profile are developed for DF in [13] A bilayer LDPC tructure [14] and the protograph method [15] ha been ued to get LDPC deign 05dB from the DF threhold The bilayer tructure i extended for ue at high SNR uing bit-interleaved coded modulation BICM) [16] A for CF relaying, a coding cheme uing a combination of LDPC and irregular repeat accumulate IRA) code i preented in [17] Ratele coding cheme are developed in [18] A for QMF relaying, a coding cheme i propoed in independent work [19] baed on lattice trategie The cheme in [19] reduce the complexity of mapping at the relay to polynomial-time while the joint decoding complexity remain exponential-time D Summary of Reult In thi paper, a coding cheme for QMF relaying with linear complexity encoding at the ource, mapping at the relay and joint decoding at the detination i developed For a network with one relay, the propoed cheme perform within 05 1)dB gap from the information-theoretic QMF threhold For the code deign example conidered in Section V, the QMF threhold i 15dB better than DF The key technique ued in thi paper are ummarized a follow: 1) BICM: Deign of binary channel code with tandard higher order ignal contellation i conidered baed on the widely ued BICM technique [20] 2) LDPC-LDGM: The cheme ue low denity parity check LDPC) code at the ource for channel coding and low denity generator matrix LDGM) code at the relay for mapping 3) Joint Factor Graph: The joint decoding procedure at the detination i formulated a a belief propagation algorithm over a factor graph Thi graph contain the original channel code LDPC) and relay mapping function LDGM) a ubgraph connected via probabilitic contraint that model the QMF relay operation

3 NAGPAL et al: CODING AND SYSTEM DESIGN FOR QUANTIZE-MAP-AND-FORWARD RELAYING 3 Fig 2 S x S z R y R R x R Z Half-duplex binary input Gauian relay channel 4) Practical Decoding Algorithm: Uing a DBLAST pacetime architecture, calar quantization procedure at relay and pecific choice of component code, the reulting factor graph i greatly implified, making it uitable for practical decoder implementation 5) Code Deign: Denity evolution analyi tool [21], [22] are developed for the ytematic deign of joint LDPC- LDGM factor graph E Organization In Section II, the coding framework for QMF and correponding joint decoding algorithm i developed The treatment focue on a canonical ytem model with one relay and binary input In Section III, denity evolution and code deign tool are developed In Section IV, the framework i extended to the high SNR regime ie for high order modulation input uing BICM In Section V, the deign of code for an example cooperative link i demontrated Finally in Section VI a ketch i provided for extending the propoed framework to cenario with multiple relay II CODING FRAMEWORK A Sytem Model Initially, thi paper focue on the deign of code for a binary memoryle ymmetric BMS) relay channel a decribed below In SectionIV, thi model i extended to high order modulation input for ue at high SNR The BMS Gauian relay channel ha three half-duplex terminal: ource S), relay R) and detination D) with binary input additive white Gauian BIAWGN) channel between them, a hown in Fig 2 R liten for a fraction f [0, 1] of the total communication time and tranmit for the fraction 1 f) The block length for the tranmitted codeword at S and R are N S and N R repectively They atify the half-duplex contraint N R = 1 f)n S The codeword meage ent from S and R are b S {0, 1} NS and b R {0, 1} NR repectively The correponding tranmitted ignal are x S {± P S } NS and x R {± P R } NR where P S and P R are per-node ymbol contraint on average power ie E x 2 S,i P S and E x 2 R,i P R Bold-face lower cae letter are ued to denote a equence of ymbol Multiple M) receive antenna are aumed at the detination Thi permit conideration of network cenario where cooperative patial multiplexing i poible [23][2] An example cenario with M = 2 i dicued in the Appendix The received ignal at D and R are denoted a y i C M and y R,j for each ymbol time i {1, 2,,N S } Y D and j {1, 2,,fN S } repectively They are modeled a follow: y i = h 1 x S,i + h 2 x R,i + z i, y R,j = h R x S,j + z R,j Here h 1, h 2,h R denote the correponding channel gain x R,i = 0 and h 2 = 0 for i {1, 2,,fN S } when R i litening For the remaining time x R,i = x R,i fns, i {fn S +1,,N S } z i and z R,j are iid zero-mean Gauian noie vector with identity covariance matrice All the channel obervation at D are denoted by Y C M NS ie Y =[y 1 y 2 y NS ] Obervation at R are denoted a y R C 1 fns ie y R =[y R,1 y R,fNS ] The channel i characterized by the following parameter: SNR SR = P S h R 2, SNR SD = P S h 1 2 and SNR RD = P R h 2 2 B Quantize-Map-Forward Scheme The quantize-map-and-forward cheme [1] i ummarized a follow S ha a equence of meage m k {1,,2 NSR }, k =1, 2,to be tranmitted At both S and R, codebook C S and C R are created repectively S map each meage to one of it codeword and tranmit it uing N S ymbol reulting in an overall tranmiion rate of R Relay liten to the firt fn S time ymbol of each block It quantize it obervation at noie level ie the quantization ditortion i equal to the noie power at the relay Relay map the quantized bit to a codeword in C R It tranmit thi codeword uing 1 f)n S ymbol The detination D attempt to decode the meage ent by S from received ignal Y) In order to decode, D mut know all channel parameter SNR SD, SNR RD and SNR SR, the relay litening fraction f and both codebook C S and C R It i aumed that SNR SD,SNR RD are meaured at D and SNR SR i meaured at R uing pilot ymbol It i further aumed that SNR SR i forwarded to D by R The etimation and forwarding overhead of thee tep i ignored for the analyi preented in thi paper C Factor Graph for Joint Decoding In the context of the ytem model and cooperation cheme outlined above, let u focu on binary linear codebook CS b and CR b Thee can be repreented a bipartite Tanner graph uing repective parity check matrice In uch a repreentation bit variable) node repreent the codeword and check function) node repreent parity contraint that mut be atified in order for the codeword to be valid Let u conider the maximum a poteriori MAP) rule for joint decoding at D In thi ubection, joint decoding i expreed a a umproduct algorithm over a factor graph that contain the Tanner graph of component code CS b, Cb R ) a ub-graph connected via probabilitic contraint that repreent the QMF relaying operation [24][25][26] Joint decoding involve earching for the codeword b S CS b that maximize the a poteriori probability p b S Y) An efficient way to do thi earch i to conider the bitwie maximum a poteriori MAP) decoder, where the aim i to compute

4 4 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL 31, NO 8, AUGUST 2013 p b S,i Y) = b S,i p b S Y) for all i =1, 2,N S p b S Y) = b R f Y b S, b R ) p b S, b R ) f Y) b R f Y b S, b R ) p b S, b R ) For the firt fn S bit, R i litening and D oberve an interference-free ignal from S During the remaining tranmiion, D oberve a uperpoition of ignal from S and R Therefore, the firt term f Y b S, b R ) factorize a follow: fy b S, b R ) = fn S N R fy i b S,i ) fy fns+j) b S,fNS+j),b R,j ) j=1 The code C b S and Cb R have characteritic function 1b S C b S ) and 1b R C b R ) repectively p b S, b R )=p b S ) p b R b S ) 1 b S C b S) p br b S ) a) = 1 b S C b S) 1 br C b R) p br b S ) a) i due to the fact that b R mut be a codeword in CR b The reulting factor graph in Fig 3 how that in addition to node repreenting channel obervation ie fy i b S,i,b R,j ) the ubgraph 1 b S CS) b and 1 br CR) b are connected by p b R b S ) that repreent the quantization operation at R If the component code CS b and Cb R are pare, the overall factor graph i alo pare A um-product algorithm for decoding over uch a factor graph ha complexity that grow linearly with the length of component code However, the um-product update rule at the function node p b R b S ) i very complex due to it high degree N S + N R ) Moreover, it introduce very hort cycle in the graph, which deteriorate the performance of um-product decoding In order to get reaonably cloe to MAP performance and low decoding complexity, the p b R b S ) node mut be factorized further In the following ubection, choice for component code CS b and CR b and pecific technique for factorization are dicued D Choice of Component Code In the dicuion above, general binary linear code CS b and CR b are conidered A natural choice i to ue pare graph code like LDPC) that are known to have good performance and linear complexity decoding/encoding operation In a previou communication [24], preliminary reult for uch factor graph were preented uing off-the-helf LDPC code at both S and R A oberved in [24], off-the-helf point-to-point) LDPC code do not allow cloe-to-optimal performance over cooperative channel An informationtheoretic undertanding of thi obervation i preented in [27] The author point out that capacity-achieving code for the point-to-point channel exhibit higher etimation error whenever the SNR i below the Shannon limit Therefore, in cooperative network where the operating SNR i below the point-to-point Shannon limit, uch off-the-helf code are no longer uitable to utilize ide information from the relay at the detination A a conequence, pecialized code are required for cooperative channel For pare graph code, pecialized code profile that are optimized for relaying can be deigned uing tandard tool uch a denity evolution analyi [21], [22] However, for the LDPC-LDPC combination [24] denity evolution doe not extend readily to QMF joint factor graph In thi paper, the ue of LDPC code at S and LDGM code at R i propoed LDPC code are known to perform very cloe to information theoretic limit when ued for channel coding Similarly LDGM code are commonly ued for loy data compreion [28] and the LDPC-LDGM combination i a good fit for the QMF relay channel Moreover, denity evolution analyi tool can be extended to LDPC-LDGM joint factor graph Such an extenion i developed in Section III Thi permit explicit contruction of code profile optimized for relaying Baed on the LDPC-LDGM choice, let u introduce auxiliary variable node b Q = {b Q,i } KR in the factor graph b Q repreent the K R bit after quantization at R Theeare mapped to the codeword b R of length N R obtained after paing through a low denity generator matrix having K R row, N R column and characteritic function 1b R CR b ) Since b R i a determinitic function of b Q, p b R b S ) can be factorized a follow Fig 4): pb R b S )=pb R, b Q b S ) = pb R b Q, b S )pb Q b S ) = 1b R CR)pb b Q b S ) The LDGM mapping can either compre or expand the K R quantized bit ie the LDGM coding rate can be greater than 1 The b R node alway have degree 2 andtheyimplyperform forwarding of meage under the um-product algorithm E Scalar Quantizer In general, a vector quantizer can be ued at R However,it i hown [1] that QMF perform within bounded gap of capacity even with a calar quantizer Under calar quantization, the obervation for every bit from S i quantized independently If each y R,i i quantized into b Q [A i ] for i =1, 2,fN S, then the p b Q b S ) function node factorize into fn S eparate node each repreenting a calar quantization operation p b Q b S )= fn S fn S p b Q [A i ] b S,i ), where A i = {1, 2,,K R }, A i A j = i j where A i denote the ubet of indice in b Q that obervation y R,i i quantized into A a reult, the variable node of the two Tanner graph are connected by function node repreenting the tochatic relation p b Q [A i ] b S,i ) among them Henceforth, thee are called quantize Q) node, a they are induced by the quantization procedure at the relay An example factor graph howing the LDPC-LDGM contruction i illutrated in Fig 4 where each ymbol obervation y R,i i quantized into one bit ie K R = fn S and A[i] ={i}) A hown, there are four kind of node in the reulting factor graph: obervation

5 NAGPAL et al: CODING AND SYSTEM DESIGN FOR QUANTIZE-MAP-AND-FORWARD RELAYING 5 CHK pb R b S ) QM CHK b S,1 b S,2 b S,NS b R,1 b R,2 b R,NR VAR VAR OBS fy 1 b S,1 ) fy 2 b S,2 ) SOURCE RELAY OBS fy fns +1) b S,fNS +1),b R,1 ) fy NS b S,NS,b S,NR ) Fig 3 Factor graph for joint decoding CHK CHK Q Q Q b S,1 b b R,1 S,2 bq,2 bq,fns b S,NS b Q,1 b R,NR VAR OBS fy 1 b S,1 ) fy 2 b S,2 ) SOURCE VAR RELAY LDGM OBS fy fns+1) b S,fNS+1),b R,1 ) fy NS b S,NS,b S,NR ) Fig 4 Factor graph: LDPC code at S, LDGM code at R with 1 bit calar quantizer Fig 5 S x S z SR y SR R Channel model uing DBLAST x R Z RD Y RD Z SD Y SD OBS) node, variable VAR) node, check CHK) node, and quantize Q) node Some VAR node in the CS b ubgraph hare OBS node with VAR node in the CR b ubgraph Thi i becaue of multiple acce at D F DBLAST Scheme The factor graph hown in Fig 4 can be implified further uing the Diagonal Bell Lab Space-Time architecture DBLAST) [29] A dicued in the Appendix, the degree 2 OBS node repreenting multiple acce) are factorized uing DBLAST Under DBLAST, the detination oberve two D orthogonal et of obervation ee Fig 5) The factorization i hown in equation 1) where Y := [Y SD Y RD ] An example of the implified factor graph i depicted in Fig 6 In thi graph, VAR node in the two Tanner graph are connected only through Q node Since y RD,i = 0 for i =1,,fN S, we rename y RD,fNS+j) y RD,j,forj = 1,,N R The reulting graph ha a tructure imilar to an irregular LDPC code but with pecial Q contraint In Section II-G, um-product update for thi graph are derived following the general principle outlined in [25] It i hown that for a imple one-bit quantizer each Q node further factorize into a CHK contraint and a dummy VAR node Thi reduce the factor graph to a Tanner graph that doe not have any pecial node Such a property i ueful to leverage exiting technique ued for the deign of low-power, high-throughput LDPC decoder G Decoding Algorithm For the point-to-point ytem, belief-propagation i an iterative algorithm that compute the a poteriori probability to decode meage bit The algorithm compute thi exactly if the factor graph ha no cycle Otherwie, it compute the approximate a poteriori probability for each bit [25] For the

6 6 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL 31, NO 8, AUGUST 2013 fy b S, b Q )= = fn S fn S N R fy SD,i b S,i ) fy SD,fNS+j), y RD,fNS+j) b Q,j,b S,fNS+j)) j=1 N R fy SD,i b S,i ) fy SD,fNS+j) b S,fNS+j))fy RD,fNS+j) b Q,j ) j=1 N S N R = fy SD,i b S,i ) fy RD,fNS+j) b Q,j ) 1) j=1 LDPC Check Contraint LDGM Check Contraint Q Q Q b S,1 b S,2 b Q,2 b R,NR b S,NS b Q,1 b Q,fNS b R,1 b R,2 Dummy Variable Node OBS fy 1 b S,1 ) fy 2 b S,2 ) fy S,NS b S,NS ) fy R,1 b R,1 ) fy R,NR b R,NR ) Fig 6 Simplified factor graph with one-bit calar quantizer and DBLAST factor graph in Fig 6, meage being paed on the edge of the factor graph and the update rule at the variable/check node tay unchanged The only new ingredient in the mix i the Q node introduced by our framework Let the ubcript V, C, andq denote VAR node, CHK node, and Q node repectively For F {C, Q}, letω l) VF denote the meage ent from variable node V to function node F in the l th iteration Every edge in the graph i connected to exactly one variable node and a meage on the edge repreent the a poteriori probability for the repective variable The meage can be repreented a LLR, but for the ake of implicity we repreent the meage a a twodimenional vector in thi ubection 1 ω VF := [p 0 p 1 ],where ω VF 1) = p 0 [0, 1] repreent the probability that the bit i 0 and ω VF 2) = p 1 [0, 1] repreent the probability that the bit i 1p 0 + p 1 =1) The meage ent from V to F {C, Q} i the normalized product of all incoming meage into V except for the meage from F The normalization enure that p 0 + p 1 =1 for the outgoing meage The meage ent from C to V i the indicator function that the check i atified, marginalized on the bit repreented by V The meage ent from Q to V i the marginalization of the function p b Q [A i ] b S,i ) on the ymbol repreented by V b Q i computed from a noiy obervation of b S,i, the node Q impoe a probabilitic contraint on the variable Since the quantization i calar: u {0, 1} Ai and v {0, 1}, gu,v):=p b Q [A i ]=u b S,i = v) Thi function i fully repreented by a lookup table with 1 Later we will replace ω by w, the commonly ued meage log p 0 LLR) p 1 in belief propagation 2 Ai +1 value, which i ued to derive the update rule for Q A an example, let u conider a one-bit calar quantizer at the relay and derive the update rule For thi cae, the Q node can be further factorized into a CHK node and a dummy VAR node that end a contant meage Note that A i = {i}, i= 1, 2,,fN S and K R = fn S The factor graph i depicted in Fig 6 Conider u {0, 1} and v {0, 1}, gu, v) :=p b Q,i = u b S,i = v) =1 p f )1{u = v} + p f 1{u v} here p f := 1 2 erfc SNR SR 2 denote the probability of bit error for calar one-bit quantization over a BIAWGN channel Since the function g i ymmetric in u and v, it can be aumed that the VAR node i of the ource, and the marginalization i on v Let the other VAR node be V Thi lead to the following update rule: ω QV 1) = 1 p f )ω V Q1) + p f ω V Q2) ω QV 2) = 1 p f )ω V Q2) + p f ω V Q1), Thi take the ame form of a CHK node update with incoming meage ω V Q and [1 p f p f ] Therefore, the Q node in thi et-up pecialize to a CHK node with additional dummy VAR node ending contant meage [1 p f p f ] that depend on SNR SR The reulting factor graph i depicted in Fig 7 III CODE DESIGN In thi ection, deign of pecific code for QMF relaying i dicued Typically pare graph code like LDPC and LDGM are drawn randomly from enemble, which are decribed uing degree profile In the point-to-point cae, if the

7 NAGPAL et al: CODING AND SYSTEM DESIGN FOR QUANTIZE-MAP-AND-FORWARD RELAYING 7 LDPC Check Contraint [1 p f p f ] LDGM Check Contraint b S,1 b b R,1 S,2 b b Q,1 S,NS b Q,2 bq,fns b R,2 b R,NR OBS fy 1 b S,1 ) fy 2 b S,2 ) fy S,NS b S,NS ) Dummy Variable Node fy R,1 b R,1 ) fy R,NR b R,NR ) Fig 7 Equivalent factor graph of that in Fig 6 The Q node are factorized into a CHK node and a dummy variable block-length i ufficiently large, the decoding performance of uch code converge to the enemble average [22] Let u conider degree profile λ S,ρ S ) and λ R,ρ R ) for the LDPC and LDGM code at ource and relay repectively λ S and ρ S are polynomial repreenting variable and check degree ditribution for the LDPC code: λ S x) = λ S,i x i 1, ρ S x) = ρ S,i x i 1 i=2 Here λ S,i and ρ S,i denote the fraction of edge with degree i at a variable node and at a check node repectively For the LDGM code, we have the imilar definition for λ R and ρ R except that thee are regarding the edge connecting check node and variable node for b Q not b R ) Thee profile mut atify the following contraint: 1 0 R =1 ρ Sx)dx 1 0 λ Sx)dx, K R = N 1 R λ S 1) = λ R 1) = ρ S 1) = ρ R 1) = 1 i=2 1 0 ρ Rx)dx 0 λ Rx)dx = f 1 f, In addition to the ub-graph repreenting the two component code, the joint factor graph hown in Fig 7 alo include edge connecting them via Q node) A dicued previouly in Section II-C, let u conider a fixed one bit calar quantizer The edge connecting with Q node are conidered fixed in the ret of thi ection In contrat, the edge in LDPC and LDGM ubgraph are drawn randomly from the enemble uing contruction procedure decribed in [22] In point-to-point channel, the typical method to analyze and deign pare graph code i to compute the enemble average performance for given degree profile auming infinite block length convergence to computation tree) The enemble average performance decoding error probability for given SNR) i calculated uing denity evolution developed in [21], [22] The two key element of claical denity evolution, namely, concentration around enemble average and convergence to computation tree channel for ufficiently large block length hold for the propoed QMF relaying ytem a well The proof can be readily extended from thoe of point-to-point channel [22] [21] Without lo of generality it i aumed that the all-zero codeword i tranmitted from S Thi i a reult of the ymmetry of the relay channel However, R doe not tranmit the all-zero codeword becaue the ource-to-relay channel i noiy For ufficiently large block length and a given value of SNR SR there i a typical equence b Q that i mapped to a typical b R baed on the LDGM code The probability of occurrence for atypical codeword vanihe a the block length become large Therefore, it i ignored for computing the enemble average performance A typical b Q comprie of K R 1 p f )0 and K R p f 1 p f i defined in Section II-G For a given degree profile, λ R,ρ R ), each bit of the typical b R i iid Bernoulliq), whereq i the probability of having an odd number of 1 in a column of the generator matrix drawn randomly from the LDGM enemble) q = ) ρr j)/j 1 1 2pf ) j j i ρ Ri)/i 2 To develop denity evolution rule for QMF relaying, we conider the belief-propagation algorithm with log-likelihood ratio a the meage paed among variable node and variou function node Let w l) FV and wl) VF denote the meage ent from the function node F to the variable node V and vicevera, at the l-th iteration F {C S, C R, Q, O S, O R } repreent the LDPC CHK node, LDGM CHK node, Q node, OBS node at S and R repectively V {V S, V Q, V R } repreent the VAR node correponding to b S, b Q and b R repectively The um-product update rule in term of the commonly ued LLR are written a follow: w l) VF = w l) w l+1) FV w l+1) FV F N V)\{F} w l) F V 2) OV = w V, O, V) ={O S, V S ), O R, V R )} =2tanh 1 tanh V N F) if F = C S, C R =2tanh 1 1 2p f ) if F = Q 1 2 wl) V F) 3) V N F) ) 1 tanh 2 wl) V F Here N ) here denote the et of neighboring node and w V repreent the LLR from channel obervation Compared to the point-to-point cae where there i only one kind of variable node V) and one kind of CHK node C) the update rule can be expreed imply by 2) and 3)

8 8 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL 31, NO 8, AUGUST 2013 where F = C Denity evolution analyi tracking the denity of thee meage in each iteration For the point-to-point cae with degree ditribution λ, ρ), there i only one type of edge and the evolution i expreed uing a pair of coupled recurive equation a follow: )) j 1) P l+1) CV =Γ 1 j ρ j Γ VC VC = P V ) i 1) λ i CV i Here Γ ) denote a tranformation on the denity a defined in [21], denote the convolution operator and { } denote the denity of meage w l) { } P V repreent the conditional denity of the LLR of the point-to-point channel For the QMF relaying cae, there are 4 type of edge and denitie for meage along all of them mut be tracked The recurive denity update are derived imilarly: Function Node to Variable Node: P l+1) C S V S =Γ 1 j j ) P l+1) QV S =Γ Γ 1 V QQ ) P l+1) C RV Q =Γ 1 P l+1) QV Q =Γ 1 Γ ρ S,j Γ V SC S )) j 1) )) j 1) ρ R,j Γ V QC R Γ V S Q Γ Γ δ log 1 p f p f δ log 1 p f p f )) )) Variable Node to Function Node: V SC S =P VS ) i 1) {fλ S,i l) C S V S P QV S + i ) } i 1) 1 f)λ S,i C SV S V QC R = i V RC R =P VR V =P SQ V S i V QQ = i λ R,i C RV Q ) i 1) P l) QV Q λ S,i C SV S ) i) λ R,i C RV Q ) i) ) V RC R Here δ r ) denote the Dirac delta function at point r R how up in the expreion becaue the Q node δ log 1 p f p f i equivalent to a CHK node connected to a contant The difference in evolution rule between the QMF relaying and point-to-point channel arie due to the probabilitic Q contraint in the joint factor graph A in the point-to-point cae, P VS i the conditional denity of the LLR of the ource to detination channel, given that an all-zero codeword i ent from S P VR i the marginal denity of the LLR of the relay to detination channel under the marginal law that b R i iid Bernoulliq) The denity evolution rule derived above are ued to compute the probability of error in decoding of b S For ucceive interference cancellation uing DBLAST, b R mut alo be reliably decoded Denity evolution rule to compute probability of decoding error for b R can be imilarly derived IV BIT INTERLEAVED CODED MODULATION So far, the dicuion ha focued on the BMS Gauian relay defined in Section II-A For the high SNR regime, input alphabet x S A NS and x R A NR where A repreent contellation point in a high-order modulation cheme mut be conidered In practice many ytem ue BICM [20] to combine channel code deigned for binary alphabet with high-order ignal contellation BICM ha alo been propoed for variou cooperative channel cenario [30][31][32][16] In thi ubection, a procedure i dicued for extending the coding framework from the BMS relay channel to a relay channel with input from high-order alphabet Under claical BICM [20], a point to point Gauian channel i decompoed into parallel independent memoryle ub-channel Every ub-channel p Y B,S y b, ) ha binary input b {0, 1} and depend on tate {1, 2,,L} which i choen uniformly and known to both the terminal 2 L i the cardinality of the choen ignal contellation) At the receiver, LLR for a bit that wa mapped to tate i calculated from ymbol obervation y C in cae of MIMO receiver y C M ) LLRy, ) = log P B Y,Sb =0 y, ) P B Y,S b =1 y, ) However, thi binary channel i not guaranteed to be outputymmetric ie the croover probability for a bit i not independent of it value Let f Λ λ) repreent the PDF of LLRy, ) The channel i output ymmetric if the following condition hold: f Λ B λ b =0)=f Λ B λ b =1) Conventional method for deigning linear coding cheme uch a denity evolution etc cannot be ued with aymmetric channel Thi iue i reolved by adding random dither at every bit to make the channel output-ymmetric a propoed in [20], [33], [34] Dither are iid Bernoulli ) 1 2 variable known to both the tranmitter and receiver For a dither d {0, 1} the channel p Y B,S,D y b,, d) i binary, memoryle and ymmetric BMS) LLRy,, d) = 1) d LLRy, ) Thi method i called parallel BICM PBICM) in [34] and Fig 8 how the architecture for a PBICM point to point link having L tate ie ignal contellation of ize 2 L {m i } L repreent meage and b i and b i the tranmit codeword before and after dithering The equivalent BMS channel can be characterized by L, the SNR of the underlying AWGN channel and the ymbol mapping in modulation In the ret of thi paper we conider that Gray mapping i ued In order to ue PBICM with the relay channel, a definition of quantize-and-map operation under PBICM i required With a PBICM modulator at ource S, the obervation at relay

9 NAGPAL et al: CODING AND SYSTEM DESIGN FOR QUANTIZE-MAP-AND-FORWARD RELAYING 9 d 1 D S, S ) D R, R ) m 1 m L ENC 1 ENC L b 1 b 1 b L d L b L Interleaver B Mapper x y SR PBICM Demod n SR,1 n SR,L Qu 1 Qu L m R,1 m R,L ENC 1 ENC L b R,1 br,l PBICM Mod x R PBICM Modulator a) PBICM Tranmitter Fig 9 QMF relaying with PBICM y Ñ LLR Calculation Deinterleaver PBICM Demodulator n 1 n L d 1 d L b) PBICM Receiver n 1 n L DEC 1 DEC L Fig 8 PBICM architecture {d i } L are dither, and d i =1 2d i m 1 R y R ) repreent L interleaved codeword If R perform quantization at the ymbol level, then the decompoition into independent binary ub-channel will be lot A an alternative, it i propoed that R perform quantization at the bit level S and R both ue PBICM modulator block with contellation ize 2 L having tate and dither vector given by S, R, D S and D R repectively The QMF operation at R i decribed below depicted in Fig 9): 1) For oberved ymbol equence y SR := {y SR,j } fns j=1 perform PBICM demodulation The output i repreented a {n SR,i } L where each n SR,i := {n SR,i,j } fns j=1 repreent LLR for the i th codeword 2) Quantize every LLR in {n SR,i } L A an example, for a one bit calar quantizer thi imply involve oberving the ign of LLR 3) Encode the quantizer output {m R,i } L uinganldgm code 4) Tranmit the reultant codeword {b R,1 } L uing a PBICM modulator Uing thi definition of QMF, the Gauian relay channel i decompoed into parallel BMS relay channel The BMS relay channel i hown in Fig 10 It i characterized by contellation ize at S and R and the SNR of the underlying AWGN link ie SNR SR, SNR SD, SNR RD V LINK DESIGN EXAMPLE In Section III an extenion of denity evolution tool wa developed [21][22] for joint LDPC-LDGM factor graph baed on QMF relaying In thi ection, a link deign example with contruction of explicit code i hown for a DBLASTequivalent channel hown in Fig 5 BMS relay channel The performance of deigned code i preented uing imulation with high order modulation baed on PBICM principle decribedinsectioniv m L A Sytem Parameter The capacity advantage of cooperative relaying i mot pronounced when the ource to relay link i ignificantly better than the direct link between ource and detination We therefore conider an example cenario where the S to R link i 10 db tronger than the other SNR SD = SNR RD, SNR SR =10 SNR SD 4) 1) Modulation Order: A a guideline for ytem deign ue the following information-theoretic bound on maximal achievable rate uing QMF relaying with continuou Gauian input x S and x R and a vector Gauian quantizer at the noie level R QMF,G = 5) { 1 f)cg SNR min SD )+fc SNRSR ) } G 2 + SNR SD, 1 f)c G SNR RD )+C G SNR SD ) f Here C G x) := log1+x) i the AWGN point-to-point capacity at ignal-to-noie ratio x If the input are contrained to tructured contellation uch a 16 QAM, 64 QAM, then the achievable rate with 2 2n -QAM modulation and BICM i computed a follow: R QMF,n = 6) { 1 f)cn SNR min SD )+fc SNRSR ) } n 2 + SNR SD, 1 f)c n SNR RD )+C n SNR SD ) f Here too we ue a vector Gauian quantizer at the noie level Note that n {2, 3, 4} and C n x) denote the 2 2n -QAM contellation-contrained point-to-point capacity at ignal-tonoie ratio x under BICM 2) Litening-Time Fraction: For QMF, the litening-time fraction f at R can be independently optimized to maximize ytem throughput [2], [5], [35] The optimal f i found by balancing the two term in the minimization of 5): ) 1 f )C G SNR SD )+f SNRSR C G + SNR SD 2 =1 f )C G SNR RD )+C G SNR SD ) f Alternatively a ub-optimal litening fraction f can be ued baed on reduced channel knowledge at relay It i hown in [2] that thi doe not have a ignificant impact on throughput For ytem parameter in Eq 4), R QMF,G and R QMF,n are plotted for n =2, 3, 4 in Fig 11 v SNR SD For each point, the optimized litening fraction f i ued To deign a link with throughput of 54 information bit per ymbol, both 64 QAM and 256 QAM are potentially good choice for

10 10 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL 31, NO 8, AUGUST 2013 S b S PBICM Broadcat c) y SR, S, d S ) b R R PBICM b) Y D b d BICM ub-channel L SNR d y b d L SNR 2 BICM Subchannel L SNR 1 BICM Subchannel d y 2 y 1 d a) Equivalent relay channel b) Equivalent point-to-point channel c) Equivalent broadcat channel Fig 10 Equivalent binary-input ytem Maximum Achievable Rate Bit/ec/Hz) Bit/ec/Hz 1347 db 256 QAM SNR db) 1418 db 64 QAM Gauian Input 256 QAM 64 QAM 16 QAM Fig 11 Maximum achievable rate for QMF relaying with modulation contraint on channel input plotted v SNR SD for SNR relationhip in Eq 4) modulation having QMF information theoretic threhold at 1418 db and 1347 db repectively Let u chooe 64 QAM 6 coded bit per ymbol) for the example deign, which mean that S hould ue an LDPC code of rate R = 54 6 =09 The optimal litening fraction correponding to SNR SD =1418 db i f 2 3 Thi determine the LDGM coding rate K R = f N R 1 f 2 B Code Deign Code CS b and Cb R optimized for the above ytem parameter can be deigned uing denity evolution tool [21] Thi involve finding good degree profile that have the lowet poible decoding SNR threhold and randomly generating finite block length code from them In order to reduce the computational complexity of denity evolution we ue the Gauian approximation to denity evolution developed in [36] Additionally, we ue the following heuritic to reduce the earch pace for profile 1) For CS b we conider check degree profile that are concentrated [36] ie all check degree from edge perpective) are either k or k+1 for ome integer k ) For CS b we conider variable degree profile with maximum degree of 8 3) For CR b we limit ourelve to regular LDGM profile Uing thee heuritic we deign the following degree profile for the ytem parameter in thi example λ S x) =028x +032x x x x 7 ρ S x) =004x x 29 λ R x) =x 4,ρ R x) =x 9 Simulation reult for the bit error rate in decoding of b S uing code with block length 10 4 and 10 5 )drawn from above profile are hown in Fig 12a) uing PBICM with 64QAM modulation, one bit calar quantizer and an ideal interleaver A hown the BER performance i 1dB of the QMF threhold For the ingle relay cenario, the informationtheoretic threhold for QMF and CF are identical, therefore a a reference for comparion threhold for DF, AF, and the no-cooperation cae are alo hown The DF and the AF threhold are computed uing the following expreion Derivation follow tandard analyi of the cheme and are omitted here R DF,n = max f [0,1] min {fc n SNR SR ), 1 f)c n SNR RD )+C n SNR SD )} R AF,n = 1 2 C n SNR SD )+ 1 2 C n SNR eff ) SNR SR SNR RD SNR eff = SNR SD + 1+SNR SR + SNR RD The optimal litening time for DF i determined by the channel parameter, while that for AF i alway 1/2 For the DBLAST architecture, b R mut alo be reliably decoded at or below the target SNR for ucceive interference cancellation to work) Fig 12b) how the BER for b R which i alo within 1dB of the QMF threhold for both of the block-length VI CONCLUSIONS The QMF relaying cheme ha the following key advantage over other known relaying cheme uch a AF, DF, and CF 1) For the ingle relay network, it outperform AF and DF at high SNR

11 NAGPAL et al: CODING AND SYSTEM DESIGN FOR QUANTIZE-MAP-AND-FORWARD RELAYING 11 log BER) Block Length ~ 10^5 QMF Threhold 1418dB Block Length ~ 10^4 DF Threhold AF Threhold 157dB 161dB 177dB No Cooperation SNR_SDdB) a) BER for b S uing deign rate of 54bit/ec/Hz with 64QAM Block Length ~ 10^5 Block Length ~ 10^4 it repective chedule The joint factor graph would include multiple LDGM ub-graph The DBLAST architecture propoed in thi paper extend naturally to network with one level of multiple noninterfering relay eg the diamond network A dicued previouly, DBLAST ignificantly reduce the complexity of the factor graph DBLAST require that all codeword from relay are decoded correctly at detination in order to permit ucceive interference cancellation Thi additional contraint doe not lead to a reduction in the QMF information-theoretic achievable rate In fact, uch a requirement i explicitly conidered in the probability of error analyi for the QMF cheme in [6] However, ome challenge for multiple relay network remain to be addreed When the relay can hear one another or the ource can reach the detination via multiple hop, it i unclear how the DBLAST architecture can be applied In uch cenario, an alternate pace-time architecture mut be conidered Moreover a the number of relay increae, the channel knowledge overhead required to compute optimal litening chedule become large Practical technique at the phyical and MAC layer are required to addre thi complexity Thee are conidered a direction for future work logber) QMF Threhold 1418dB DF Threhold 157dB SNR_SDdB) AF Threhold 161dB 177dB No Cooperation b) BER imulation for b R uing deign rate of 54bit/ec/Hz with 64QAM Fig 12 Code deign imulation reult 2) For the ingle relay network, it achieve the ame performance a CF but reduce channel feedback overhead Unlike CF, QMF doe not require knowledge of forward channel trength at the relay 3) For arbitrary relay network with multiple relay, QMF achieve better high SNR performance than AF, DF and CF In thi paper, a low-complexity channel coding framework i developed for QMF relaying For the ingle relay network, the framework perform within 05 1)dB of fundamental limit The technique preented here can be extended to complex ytem cenario, which are dicued below A Multiple Relay When there i more than one relay in the ytem, the propoed factor graph extend in a traightforward manner Optimal litening chedule can be computed for each of the relay A propoed, the ource would ue an LDPC code and each relay would ue an LDGM code baed on 18 B Rate Adaptation and Hybrid ARQ In the link deign example, uitable coding rate, contellation and litening fraction are computed for a given et of operating channel condition However, optimizing code baed on intantaneou channel condition i not feaible in practice Under commonly ued rate adaptation mechanim, terminal witch between a few candidate code and a few candidate contellation baed on channel condition Cooperative link need to conider multiple channel parameter to determine tranmiion rate ie for a ingle relay three SNR parameter are required a oppoed to jut one for a pointto-point link Thi make rate adaptation chedule for relay network more complex An advantage of QMF relaying i that rate adaptation chedule depend only on the ability of the detination to decode a oppoed to DF, where adaptation mut conider decoding at relay a well Modern adaptation mechanim like hybrid automatic repeat requet HARQ) can be incorporated into the propoed framework Additional parity bit for refinement ent from the ource after receiving a repeat requet from the detination It can be cooperatively delivered to the detination uing QMF relaying The joint decoding factor graph i expanded to incorporate thee refinement parity bit and the decoding algorithm remain unchanged ACKNOWLEDGEMENTS The author acknowledge Prof Rüdiger Urbanke for fruitful dicuion leading to the choice of LDPC-LDGM tructure We alo acknowledge the tudent, faculty and ponor of the Berkeley Wirele Reearch Center and upport of the Center for Circuit & Sytem Solution C2S2) Focu Center, one of ix reearch center funded under the Focu Center Reearch Program, a Semiconductor Reearch Corporation program

12 12 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL 31, NO 8, AUGUST 2013 APPENDIX The QMF relaying cheme introduce correlation between x S and x R, which can be thought of a coding acro tranmit antenna in a MIMO channel A natural pace-time architecture for uch a channel i DBLAST Uing DBLAST for the relay channel ha alo been propoed in [12][11][29][30] It relie on introducing a delay of one block at the relay and uing ucceive interference cancellation SIC) at the detination At the k-th block the detination receive the uperpoition of the following: ignal from the ource containing the codeword ent at block k, namely, x S m k ) ignal from the relay containing the ide information about the ource codeword at block k 1, namely, x R q k 1 ) Meage ent from the ource are independent acro block At the k-th block, the detination jointly decode block k 1 meage m k 1 and ide information x R q k 1 ))by treating x S m k ) a Gauian noie The receiver ubtract relay codeword x R q k 1 ) from it received ignal Y[k] and keep the reidual Ỹ[k] for decoding the next block Thi architecture allow the ue of a implified equivalent channel model Note that the one-block delay introduced at R ha the added benefit of allowing time for QMF proceing at R 1) Simplified Channel Model: The equivalent channel model i hown in Fig 5 For decoding the block k 1 meage m k 1, the decoder take two input Y[k] and Ỹ[k 1] We can think of Y[k] and Ỹ[k 1] a two orthogonal link with independent Gauian noie Therefore, for the purpoe of code deign we can alternatively invetigate a impler model depicted in Fig 5 In thi model, Y ij = h ij x i + Z ij, i, j) =R, D), S, D), y SR = h SR x S + z SR A an example, let u conider a cenario where D ha two receive antenna M =2) In that cae, the DBLAST equivalent channel become [37]: where, y ij = h ij x i + z ij, i, j) =R, D), S, D), h SD = h 1, h RD = h h P S h 1 2 h 2 1 and h 2 1 denote the perpendicular and parallel component of h 2 with repect to h 1, repectively The ignalto-noie ratio of the three link are SNR SR = h SR 2 P S, SNR SD = h SD 2 P S,andSNR RD = h RD 2 P R repectively Remark 1: Conider the original channel and the DBLASTequivalent channel Note that the capacitie of thee two channel are within two bit of each other Thi i baed on the following obervation: 1) The min-cut upper bound for both channel are within one bit of each other for any litening fraction f [0, 1]) The mutual information acro cut {S}, {R, D} remain unchanged between the two channel Conider the mutual information acro the cut {S, R}, {D} Iti known that SIC achieve the um capacity of multipleacce channel In the original channel Fig 2) S and R have unlimited cooperation A a reult, the min-cut bound for DBLAST incur a power-gain lo of at mot 1 f) bit 2) QMF relaying cheme achieve the min-cut upper bound to within one bit for the two channel [1] REFERENCES [1] A Avetimehr, S Diggavi, and D Te, Wirele network information flow: A determinitic approach, IEEE Tran Inf Theory, 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Inf Theory, vol 47, no 2, pp , Feb 2001 [37] L Zheng and D Te, Diverity and multiplexing: A fundamental tradeoff in multiple-antenna channel, IEEE Tran Inf Theory, vol 49, no 5, pp , May 2003 Vinayak Nagpal received the B Eng degree from the Univerity of Pune, India, in 2003, and the MS degree from Chalmer Univerity of Technology, Sweden, in 2006 He received the PhD degree from the Univerity of California at Berkeley, USA, in 2012 under the guidance of Prof Borivoje Nikolić Since then, he i affiliated with the Nokia Reearch Center, Berkeley, USA Previouly, he held poition at Conexant Sytem, Pune, India 2003), the National Radio Atronomy Obervatory, Charlotteville, VA 2005), and the Harvard Smithonian Center for Atrophyic, Cambridge, MA 2006) Hi reearch interet include wirele network and real time ignal proceing 2006 I-Hiang Wang received the BS degree in electrical engineering from National Taiwan Univerity, Taiwan, in 2006 He received a PhD degree in electrical engineering and computer cience from the Univerity of California at Berkeley, USA, in 2011 Since 2011, he ha been affiliated with the École Polytechnique Fédérale de Lauanne, Switzerland, a a potdoctoral reearcher Hi reearch interet include network information theory, wirele network, coding theory, and network coding Dr Wang received a 2-year Vodafone Graduate Fellowhip in Milo Jorgovanovic received hi Dipl Ing degree in electrical engineering from the Univerity of Belgrade, Serbia, in 2007, and the MSc degree from the Univerity of California at Berkeley in 2010 He i currently working toward hi PhD degree at the Univerity of California at Berkeley under the guidance of Prof Borivoje Nikolić He ha held internhip poition with the Kodak European Reearch Center in Cambridge, UK 2006), the Technical Univerity of Berlin, Germany 2009), and Samung Mobile in Richardon, TX 2010) Hi reearch interet include MIMO detection algorithm and architecture, wirele communication ytem deign, ignal proceing for digital communication, and digital integrated circuit deign David Te received the BASc degree in ytem deign engineering from the Univerity of Waterloo, Waterloo, ON, Canada, in 1989, and the MS and PhD degree in electrical engineering from the Maachuett Intitute of Technology, Cambridge, in 1991 and 1994, repectively From 1994 to 1995, he wa a Potdoctoral Member of Technical Staff at the Department of AT&T Bell Laboratorie Since 1995, he ha been with the Department of Electrical Engineering and Computer Science at the Univerity of California at Berkeley, where he i currently a Profeor Dr Te received a 1967 NSERC 4-year graduate fellowhip from the government of Canada in 1989, a NSF CAREER award in 1998, Bet Paper Award at the Infocom 1998 and Infocom 2001 conference, the Erlang Prize in 2000 from the INFORMS Applied Probability Society, the IEEE Communication and Information Theory Society Joint Paper Award in 2001, the Information Theory Society Paper Award in 2003, and the 2009 Frederick Emmon Terman Award from the American Society for Engineering Education He ha given plenary talk at international conference uch a ICASSP in 2006, MobiCom in 2007, CISS in 2008, and ISIT in 2009 He wa the Technical Program Co-chair of the International Sympoium on Information Theory in 2004 and wa an Aociate Editor of the IEEE TRANSACTIONS ON INFORMATION THEORY from 2001 to 2003 He i a co-author, with P Viwanath, of the text Fundamental of Wirele Communication, which ha been ued in over 60 intitution around the world Borivoje Nikolić received the DiplIng and MSc degree in electrical engineering from the Univerity of Belgrade, Serbia, in 1992 and 1994, repectively, and the PhD degree from the Univerity of California, Davi in 1999 He lectured electronic coure at the Univerity of Belgrade from 1992 to 1996 He pent 2 year with Silicon Sytem, Inc, Texa Intrument Storage Product Group, San Joe, CA, working on dik-drive ignal proceing electronic In 1999, he joined the Department of Electrical Engineering and Computer Science, Univerity of California at Berkeley, where he i now a Profeor Hi reearch activitie include digital and analog integrated circuit deign and VLSI implementation of communication and ignal proceing algorithm He i a co-author of the book Digital Integrated Circuit: A Deign Perpective 2nd ed, Prentice-Hall, 2003) Dr Nikolić received the NSF CAREER award in 2003, the College of Engineering Bet Doctoral Diertation Prize and Anil K Jain Prize for the Bet Doctoral Diertation in Electrical and Computer Engineering at Univerity of California, Davi in 1999, a well a the City of Belgrade Award for the Bet Diploma Thei in 1992 For work with hi tudent and colleague, he ha received bet paper award at the IEEE International Solid- State Circuit Conference, Sympoium on VLSI Circuit, IEEE International SOI Conference, and the ACM/IEEE International Sympoium of Low-Power Electronic