Turbo Packet Combining for Broadband Space-Time BICM Hybrid-ARQ Systems with Co-Channel Interference

PUBLISHED IN IEEE TRANSACTIONS ON WIRELESS COMMUNICA TIONS, MA Y 2010 (PRINT A V AILABLE @ HTTP://DX.DOI.ORG/10.1109/TWC .2010.05.090441) 1 T urbo P acket Combining for Broadb and Space–T ime BICM Hybrid–ARQ Systems with Co–Channel Interfere nce T ari k Ait-Idir , Memb er , IEEE, Houda Chafnaji, a nd Samir Saou di, Senior Member , IEEE Abstract —In this paper , efficient turbo packet combinin g for single c arrier (SC) broadband multiple-input–multip le-output (MIMO) hybrid–automatic r epeat request (ARQ) transmission with unknown co-channel interfere nce (CCI) is studied. W e pro- pose a new frequency domain soft mini mum mean square err or (MMSE)-b ased signal lev el combining t echnique where receiv ed signals and channel frequency responses (CFR)s corresponding to all retra nsmissions are used to decode the data packet. W e prov ide a recursiv e implementation algorithm f or the in troduced scheme, and show that both its computational complexity and memory requirements are q uite insensitive to t he ARQ delay , i.e., maximum number of ARQ rounds. Fu rthermore, we analyze the asymptotic performa nce, and show that un der a sum-rank condition on the CCI MIMO ARQ chann el, the proposed packet combining scheme is not i nterference-limited. Simulation results are provided to demonstrate the gains offered by the proposed technique. Index T erms —A utomatic repeat request (ARQ) mechanisms, multiple-in put–multiple-ou tput (MIMO), single carrier (SC), un- known co-channel interference (CCI), intersymbol interference (ISI), fr equency doma in methods. I . I N T R O D U C T I O N S P AC E–TIME– BIT -I NTERLEA VED coded modulatio n (ST – BICM) with iterative decod ing is an a ttractiv e sign al- ing schem e that o ffers high spectral efficiencies over m ultiple- input–m ultiple-ou tput (MIMO)-intersymb ol interference (ISI) channels [1], [2], [3], [ 4], [5]. T o combat ISI in single carrier (SC) broadb and ST –BICM transmission, frequen cy domain equalization , initially introduced for single a ntenna systems [6], [7], [8], [9], has b een propo sed using iterati ve (turb o) processing [10]. It is a rece i ver sch eme tha t allows high ISI can cellation cap ability at an a ffordable complexity cost. In practical systems, unknown co-chann el interferen ce (CCI) caused by other tran smitters (d istant user s and/o r neigh boring cells) who simultan eously use the same r adio resource can dramatically degrade the link per forman ce. This limitation can be ov ercome by using the so-called hyb rid–auto matic repeat The Associate Editor coordina ting the revi e w of this paper and approving it fo r pu blicat ion is Dr . M. C. V alenti. Manusc ript rec ei ved March 26, 2009; revi sed December 24, 2009; accept ed February 21, 2010. T his paper was prese nted in part at the 19th Annual IEEE Symposium on Personal Indoor and Mobile Radio Communication s (PIMRC 2008), Cannes, France, September 2008, and in part at the IE EE Global Communication s Conference (Globeco m’09), Honolulu, Hawaii , Nov-Dec 2009. T . Ait-Idir and H. Chafnaji are with the Communicati on Systems Depart- ment, INPT , Madinat Al-Irfane, Rabat, Morocco. T hey are also with Institut T elecom / T eleco m Breteg ne/LabSti cc, Brest, France (email : aitidir@i eee.or g). S. Saou di is with Instit ut T elecom / T elecom Bret egne /LabSticc , Brest, France. He is also with Univ ersité Européenne de Bretagne. request (ARQ) pro tocols, where ch annel coding is combined with ARQ [1 1], [12]. In hy brid–ARQ, erro neous data packets are kept in the rec eiv er and used to detect/deco de the re- transmitted frame [13], [14], [15], [16], [17], [1 8], [1 9]. This technique is of ten refer red to as “pa ck et comb ining” . Practical packet combin ing schemes h av e been addressed in [ 20]. In [21], an elegant info rmation- theoretic framework has been introdu ced to analy ze the through put and delay of hybr id– ARQ und er random user be havior . In terestingly , the author s have sho wn that h ybrid– ARQ systems are not in terference limited, i.e., arbitra rily h igh throug hput can b e achieved by simply increasing the transmit power of all users even wh en multi-user detection (MUD) tech niques are not used at the receiver . Moti vated by the above conside rations, we i n vestigate efficient low-complexity turbo frequ ency domain reception technique s for SC broadban d ST –BICM signaling with hybrid– ARQ o perating over CCI-limited MIMO chann els. The powerful d iv ersity–multiplexing tradeoff too l, initially introdu ced by Zheng an d Tse f or co herent delay-limited , i.e., quasi-static, MIM O chan nels [22], has been elegantly extended by El Gamal et al. to MIMO ARQ channe ls with flat fading, a nd referred to as div ersity–multip lexing–delay tradeoff [ 23]. The autho rs have pr oved that the ARQ d elay , i.e., maximum nu mber o f ARQ proto col ro unds, im proves th e outage prob ability 1 perfor mance for la rge classes o f MIMO ARQ chan nels [23]. In particular, they have demo nstrated tha t the diversity or der can be increased due to ARQ even wh en the MIMO ARQ channel is long -term static, i.e. , the MIMO channel is rand om but fixed for all ARQ roun ds. The di versity– multiplexing–d elay tradeoff has then been cha racterized in the case of block-fadin g MI MO ARQ channels, i.e., multiple fading b locks are allowed within the same A RQ round [25]. In [26], the outage probab ility of MIMO- ISI ARQ chan nels has been ev aluated und er th e assumption s of short-te rm static channel dynamic 2 , an d Chase-type ARQ [27], i.e., th e data packet is entirely retransm itted. It has been shown that, as in the flat fading case, ARQ presents an important sou rce o f div ersity , but its influence become s only minimal wh en the 1 In non-ergodic , i.e., block fading quasi-static channels, the outage prob- abili ty is a meaningfu l measure that provides a lower bound on the block error probabili ty . It is defined as the probabili ty that the mutual information, as a function of the channel realizati on and the av erage signal-to-noi se ratio (SNR), is below the transmission rate [24]. 2 In the case of short-term static dynamic, the ARQ channel realizat ions are independe nt from round to round. This dynamic applies to slow ARQ protocol s where the del ay between two rounds is la rger than the channel coheren ce time. 2 PUBLISHED IN IEE E T RANSACTIONS ON WIRELESS COMMUNICA T IONS, MA Y 2010 (PRINT A V AILABL E @ HT TP://DX.DOI.ORG/10.1109/TWC. 2010.05.090441) ARQ d elay is increased. Th is observation suggests that th e design of practical packet co mbining sch emes sho uld target a high d iv ersity or der for early ARQ ro unds. Supplemen tary retransmissions ar e then u sed to correct r are erro neous data packets, when they occur . More recen tly , packet com bining for MIMO ARQ sys- tems has b een inves tigated (e.g. [28], [29], [30], [31], [32], [33], [34], [35], [36]). T urbo comb ining techniques, wher e decodin g is iter ativ ely performe d throu gh the exchange o f soft information b etween th e soft-input–sof t-outpu t (SISO) packet comb iner and the SISO decoder, have been p roposed for the MIMO-ISI ARQ channel using unconditional minimum mean squ are erro r (MMSE)-aided co mbining [37], [26]. T hese approa ches hav e then been extended to broad band MIMO code division multiple access (CDMA) systems with ARQ [38]. T ime doma in turbo p acket com bining for CCI-limited MIMO- ISI ARQ channels h as b een in troduce d in [ 39]. In this pap er , we inv estigate efficient turbo r eceiv er tech- niques for SC ST –BICM transmission with Chase-type ARQ over broadb and MIMO channel with u nknown CCI. W e in- troduce a frequen cy dom ain MMSE-ba sed turbo packet com- bining sch eme, where all ARQ rou nds are used to deco de the data packet. By using an iden tical c yclic prefix ( CP) w o rd for multiple retran smissions of a symbol block , we perform transmission com bining at the signal level. The frequ ency domain soft MMSE packet co mbiner performs so ft ISI can- cellation an d re transmission comb ining in th e p resence of unknown CCI jointly over all re ceiv ed signal blo cks. W e also provide an effi cient recursive implementatio n algorithm for the propo sed scheme, an d show that both the compu tational load and mem ory req uiremen ts are quite insen siti ve to the ARQ delay . The com plexity order is only cu bic in terms of the number of transmit anten nas. Received signals and channel frequen cy responses ( CFR)s co rrespon ding to all ARQ ro unds are used without being required to be stored in the recei ver . W e analyze the asy mptotic per forman ce of the p ropo sed co mbin- ing scheme . Interestingly , we show that under a rank -conditio n on th e MIMO ARQ chann el correspo nding to unkn own CCI, the proposed combining schem e is n ot interference-limited , i.e.,unk nown CCI can be comple tely rem oved. Finally , we provide numer ical simulation results fo r some scenarios to validate our finding s. The remain der of the paper is organized as follows. In Section I I we d escribe the ARQ system u nder con sideration, along with the communicatio n mo del in the presence of unknown C CI. In Section III, we present the freque ncy domain turbo packet c ombinin g scheme we propose in this p aper, and analyze both its com plexity an d m emory require ments. In Section IV, we carry o ut th e asympto tic per forman ce analysis, and provid e representa ti ve n umerical results that demo nstrate the gains achieved by the pro posed sche me. Fin ally , we p oint out c onclusions in Section V. Notation: • Superscr ipts ⋆ , ⊤ , an d H denote conju gate, transpose, and Hermitian transpose, respectively . E [ . ] is the math emati- cal expectation of the argument ( . ) . • Let X be a squar e matrix, diag { X } denotes the row vector correspondin g to the diago nal of X , an d tr { X } denotes the trace of X . When X 1 , · · · , X M ∈ C N × Q , diag { X 1 , · · · , X M } denotes the M N × M Q matrix whose diagona l blocks are X 1 , · · · , X M . diag { x } is the N × N diagonal m atrix whose d iagonal entries are the elements of the complex vector x ∈ C N . ( X ) m,m denotes the m th d iagonal e ntry of m atrix X . • I N is th e N × N identity matrix, and 0 N × Q denotes an all z ero N × Q matrix. For i = 0 , · · · , T − 1 , E i,N is a N × N T zero matrix wh ere the i th N × N block is equ al to I N . • Operato r ⊗ denotes the Krone cker produ ct, and δ m,n is the Kr onecker sym bol, i.e., δ m,n = 1 f or m = n and δ m,n = 0 fo r m 6 = n . • For each sequence of matrices X 0 , · · · , X T − 1 (respec- ti vely , scalars x 0 , · · · , x T − 1 ), ˜ X , 1 T P T − 1 i =0 X i denotes its time average (respec ti vely , ˜ x , 1 T P T − 1 i =0 x i ). • U T is a T × T unitary m atrix whose ( m, n ) th ele ment is ( U T ) m,n = 1 √ T exp  − j 2 π m n T  for m, n = 0 , · · · , T − 1 , wh ere j = √ − 1 . U T ,N is T N × T N defined as U T ,N , U T ⊗ I N . • For each vector x ∈ C Q , x f denotes the discrete Four ier T ransform (DFT) of x , i.e., x f = U Q x . • The acr onym i.i.d. m eans “in depend ent and identically distributed”. I I . A R Q S Y S T E M M O D E L A. SC–MIMO ARQ T ransmission Scheme W e consider an SC multi-antenna- aided t ransmission scheme wher e the tran smitter and th e receiver ar e eq uipped with N T transmit (index t = 1 , · · · , N T ) and N R receive (index r = 1 , · · · , N R ) antenn as, r espectively . The MIMO channel is fr equency selective and is comp osed of L sym bol- spaced taps (index l = 0 , · · · , L − 1 ). The en ergy of eac h tap l is denoted σ 2 l , an d the total energy is no rmalized to one, i.e. , P L − 1 l =0 σ 2 l = 1 . Each inf ormation block is initially encod ed then interleaved with the a id of a semi- random interleaver Π . Th e resulting frame is serial to parallel con verted an d mapp ed over the elements o f the con stellation set S to prod uce symb ol matrix S ∈ S N T × T , wher e T is the numbe r of chan nel use (c .u). A CP word, wh ose length is T C P ≥ L − 1 , is then append ed to S , thereb y yield ing m atrix S ′ ∈ S N T × ( T + T C P ) . T his allows th e prevention of inte r-block in terference (IBI) and the exploitation of the multipath diversity of the MIMO broadband channel. W e suppose tha t no ch annel state in formatio n (CSI) is av ailab le at the tra nsmitter an d assume infinitely deep interleaving. The refore, tra nsmitted symbols a re independent and have equ al transmit power , i.e., E  s t,i s ⋆ t ′ ,i ′  = δ t − t ′ ,i − i ′ . (1) At the upp er layer, an ARQ protoc ol is used to help cor rect erroneo us frames. An ack nowledgment message is g enerated after the de coding of eac h in formation bloc k. T herefor e, when the decod ing is successfu l, the r eceiv er send s b ack a positive acknowledgment (ACK) to the transmitter, wh ile the feedbac k of a negative acknowledgment (NA CK) ind icates that the AIT -IDIR et al. : T URBO P ACKET C OMBINING FOR BRO ADBAND SP ACE–TIME BICM HYBRID–ARQ SYSTEMS WITH CO–CHANNEL INTERFER ENCE 3 Figure 1. SC–MIMO ARQ communicatio n scheme at ARQ round k . decodin g o utcome is erroneo us. L et K d enote th e ARQ delay , and k = 1 , · · · , K denote the ARQ rou nd index. When the transmitter recei ves an A CK feedb ack, it stops the tr ansmission of th e cur rent block an d moves o n to the next info rmation block. Reception of a N AC K message incurs supplementar y ARQ round s u ntil th e p acket is cor rectly de coded or the ARQ delay K is reach ed. W e focus on Chase-ty pe ARQ, i.e., the symbol ma trix S ′ is co mpletely r etransmitted. In additio n, we suppose perfec t packet err or detection, and assume th at the one bit A CK/NA CK feedback is error-free. B. Communica tion Model in the Pr esence o f Un known CCI The broadban d MIMO ARQ channel is assumed to be short- term static fading , i.e., the channel indep endently changes from round to round . Note that this ch annel dynamic applies to slow ARQ pr otocols whe re the delay between two c onsecutive ARQ round s is larger th an the chann el coher ence time. It a lso applies to ortho gonal frequency division multiplexing ( OFDM) systems where freq uency hop ping is used to m itigate ISI. Let H ( k ) 0 , · · · , H ( k ) L − 1 ∈ C N R × N T denote channel matrices at the k th ARQ ro und, and whose entries a re i.i.d. zero- mean circu larly symm etric Gaussian, i.e., h ( k ) r,t,l ∼ C N  0 , σ 2 l  , where h ( k ) r,t,l denotes the fading chan nel corr espondin g to path l and co nnecting the t th tran smit and the r th r eceive antenn as at the k th ARQ rou nd. Theref ore, the ch annel energy at each receive antenn a r is L − 1 X l =0 N T X t =1 E     h ( k ) r,t,l    2  = N T . (2) The cha nnel pr ofile, i.e., power d istribution σ 2 0 , · · · , σ 2 L − 1 and number of taps L , is supp osed to b e identical for at least K consecutive rou nds. This is a reason able assum ption becau se the chan nel pro file dy namic main ly d epends o n th e shad owing effect. T ransmitted data blocks are cor rupted b y an unkno wn CCI signal cau sed by a co–chan nel transmission that u ses N ′ T transmit anten nas (index t ′ = 1 , · · · , N ′ T ) and T c. u. The link between the in terferer transmitter and the r eceiver is composed of L ′ taps, where the channel matrix of each tap l ′ = 0 , · · · , L ′ − 1 at r ound k is H CCI ( k ) l ′ ∈ C N R × N ′ T and its energy is σ 2 u l ′ 3 . W e sup pose that the r eceiver has no knowledge e ither about the interf erer CSI o r ab out its chan nel profile an d nu mber of transmit antenn as (i,. e., parameters N ′ T , L ′ , H CCI ( k ) l ′ , and σ 2 u l ′ ∀ l ′ are complete ly unknown at the receiver). As the desired user, the in terferer e mploys a CP- aided transmission strategy . Its transm itted symb ols s CCI ( k ) t ′ ,i at each rou nd k verify the indepe ndence/en ergy-normalization condition ( 1) as usefu l symbo ls. Ther efore, the signal-to- interferen ce ratio (SIR) at each rece i ve antenn a is given as SIR = N T N ′ T P L ′ − 1 l ′ =0 σ 2 u l ′ . (3) W e assume per fect frame synchr onization between the in- terferer and the d esired user . They can differ in terms of the CP word len gth, which depen ds on the delay o f the multipath channel, but are synchro nized in terms of the useful symbol frames. Und er this assumption , CP deletion yields th e following baseband received N R × 1 signal at roun d k an d channel u se i , y ( k ) i = L − 1 X l =0 H ( k ) l s ( i − l ) mo d T + L ′ − 1 X l ′ =0 H CCI ( k ) l ′ s CCI ( k ) ( i − l ′ ) mo d T + n ( k ) i | {z } w ( k ) i = CCI+noise , (4) where n ( k ) i ∼ C N  0 N R × 1 , σ 2 I N R  denotes th e receiver thermal noise. Th e SC–MIMO ARQ communicatio n scheme at rou nd k is depicted in Fig. 1. In the following, we a ssume perfect channel estimation at each ARQ roun d k (i.e., H ( k ) l ∀ l are perfec tly known) wh ile CCI ch annel matr ices H CCI ( k ) l ′ ∀ l ′ , k are comp letely unk nown at the receiver side. 1) Single-Ro und Communica tion Model: T o derive th e block commu nication mod el c orrespon ding to ARQ ro und k , we consider th e fo llowing blo ck sign al vector, y ( k ) , h y ( k ) ⊤ 0 , · · · , y ( k ) ⊤ T − 1 i ⊤ ∈ C N R T , (5) that grou ps sign als co rrespon ding to the entire symbo l f rame. V ector y ( k ) can b e expressed as, y ( k ) = H ( k ) s + w ( k ) , (6) 3 The ARQ processes corresponding to the desired user and the interfere r are not necessarily synchronized. Therefore, the round index k appeari ng in the CCI channel matri ces only refer s to the inde x of a real izati on of the interfe rer channel at ARQ round k . The same remark holds for CCI symbols in (4). Also, note that P L ′ − 1 l ′ =0 σ 2 u l ′ 6 = 1 in order to account for the path-loss betwee n the interferer and the recei ver . 4 PUBLISHED IN IEE E T RANSACTIONS ON WIRELESS COMMUNICA T IONS, MA Y 2010 (PRINT A V AILABL E @ HT TP://DX.DOI.ORG/10.1109/TWC. 2010.05.090441) where s ,  s ⊤ 0 , · · · , s ⊤ T − 1  ⊤ ∈ S T N T , (7) w ( k ) , h w ( k ) ⊤ 0 , · · · , w ( k ) ⊤ T − 1 i ⊤ ∈ C N R T , (8) H ( k ) ,                H ( k ) 0 0 N R × N T · · · 0 N R × N T . . . H ( k ) 0 . . . H ( k ) L − 1 . . . . . . 0 N R × N T H ( k ) L − 1 0 N R × N T . . . 0 N R × N T H ( k ) L − 1 . . . . . . . . . 0 N R × N T 0 N R × N T · · · H ( k ) 0                N R T × N T T (9) is a bloc k circu lant matr ix that can be block diagonalized in a Fourier basis as H ( k ) = U H T ,N R Λ ( k ) U T ,N T , (10) where Λ ( k ) , diag n Λ ( k ) 0 , · · · , Λ ( k ) T − 1 o ∈ C N R T × N T T . (11) Exploiting (10) a nd th e block cir culant structure of H ( k ) , we get Λ ( k ) i = L − 1 X l =0 H ( k ) l exp  − j 2 π il T  . (12) Applying the DFT U T ,N R on signal vector y ( k ) yields th e single-rou nd freque ncy domain com munication model y ( k ) f = Λ ( k ) s f + w ( k ) f , (13) where y ( k ) f , s f , and w ( k ) f denote th e DFT o f y ( k ) , s , an d w ( k ) , respectively . 2) Multi-Ro und Communication Model: Let us suppose that received signals and channel matrices co rrespon ding to ARQ ro unds 1 , · · · , k ar e av ailable at the receiver . First, we introdu ce the sign al vector notatio n y ( k ) i , h y (1) ⊤ i , · · · , y ( k ) ⊤ i i ⊤ ∈ C kN R , (14) where received signals corresp onding to mu ltiple ARQ ro unds are grouped in such a way to construct k N R virtual receiv e antennas. Similar ly , we d efine, H ( k ) l , h H (1) ⊤ l , · · · , H ( k ) ⊤ l i ⊤ ∈ C kN R × N T , (15) w ( k ) i , h w (1) ⊤ i , · · · , w ( k ) ⊤ i i ⊤ ∈ C kN R . (16) The block sign al vector that serves for jointly perfo rming, at ARQ r ound k , packet combin ing and equalizatio n in the presence o f CCI is constructed similarly to (5), y ( k ) , h y ( k ) ⊤ 0 , · · · , y ( k ) ⊤ T − 1 i ⊤ ∈ C kN R T , ( 17) and can b e expressed as, y ( k ) = H ( k ) s + w ( k ) , (18) . . . . . . . . . . . . . . . . . . . . . ARQ round # k (round # k −1) DFT CP DFT CP ARQ round #1 interleaver SISO (To transmitter) ACK/NACK CRC check. soft with CCI packet combining LLRs deinterleaver deletion deletion decoder a priori Figure 2. Block diagram of the recei ver scheme at ARQ round k where w ( k ) , h w ( k ) ⊤ 0 , · · · , w ( k ) ⊤ T − 1 i ⊤ ∈ C kN R T . (19) Matrix H ( k ) has the same structure as (9), wh ere its first T k N R × N T block column is eq ual to h H ( k ) ⊤ 0 , · · · , H ( k ) ⊤ L − 1 , 0 N T × ( T − L ) kN R i ⊤ . (20) H ( k ) can be factorized similarly to (10) as, H ( k ) = U H T ,kN R Λ ( k ) U T ,N T , (21) where Λ ( k ) , diag n Λ ( k ) 0 , · · · , Λ ( k ) T − 1 o ∈ C kN R T × N T T , (22) Λ ( k ) i , h Λ (1) ⊤ i , · · · , Λ ( k ) ⊤ i i ⊤ ∈ C kN R × N T , (23) and matrice s Λ ( k ′ ) i , k ′ = 1 , · · · , k , are giv en b y (12). T he multi-rou nd fr equency dom ain commun ication m odel at ARQ round k is then expressed as, y ( k ) f = Λ ( k ) s f + w ( k ) f , (24 ) where y ( k ) f and w ( k ) f denote the DFT of y ( k ) and w ( k ) , respectively . I I I . F R E Q U E N C Y D O M A I N T U R B O P A C K E T C O M B I N I N G I N T H E P R E S E N C E O F U N K N O W N C C I A. General Descriptio n At ea ch ARQ rou nd, the decodin g of a data packet is perfor med by iterati vely exchangin g soft inf ormation in the form of log -likelihood r atio (LLR) values between th e soft packet co mbiner , i.e., the join t transm ission com bining and equalization unit, and the soft-input–so ft-outpu t (SISO) de- coder . Let us su ppose that, at ARQ ro und k , all recei ved signals and c hannel matrices co rrespon ding to previous rounds k − 1 , · · · , 1 are available at the receiver . Note that this assumption could n ot be f easible in practice since the receiver will requ ire a huge memo ry . In Su bsection II I-D, we show that the pr oposed turbo packet comb ining algo rithm requ ires little memo ry while it u ses sign als and CSIs co rrespon ding to all ARQ roun ds 1 , · · · , k . The b lock diagram o f the frequen cy domain tur bo p acket co mbining r eceiv er at ARQ r ound k is depicted in Fig . 2. AIT -IDIR et al. : T URBO P ACKET C OMBINING FOR BRO ADBAND SP ACE–TIME BICM HYBRID–ARQ SYSTEMS WITH CO–CHANNEL INTERFER ENCE 5 First, the m ultiple ARQ r ounds frequen cy d omain block signal vector y ( k ) f , U T ,kN R y ( k ) and CFR Λ ( k ) are con- structed. Second, the soft packet co mbiner estimates th e co - variance of unkn own CCI p lus noise, th en comp utes the mu lti- transmission MMSE filter that takes into a ccount b oth co- antenna interference (CAI) and ISI while suppressing unknown CCI. These two elemen ts are then used with a prio ri in forma- tion to comp ute extrinsic LL Rs correspon ding to co ded and interleaved bits. The generated soft in formatio n is transferred to th e SISO deco der to compu te a posteriori L LRs about b oth coded and useful bits. Only e xtrinsic information is fed back to the soft packet combiner to help perform transmission combin- ing an d equalization in the next tur bo iteration. The iterative soft p acket comb ining and deco ding p rocess is stop ped af ter a preset numb er of tur bo iterations and decision abo ut the d ata packet is per formed . The A CK/NA CK message is then sent back to the transmitter depending on the d ecoding ou tcome. Note that durin g the first iteratio n a priori LLR values are the output of the SISO d ecoder ob tained at the last iteration o f previous round k − 1 . B. Pr op erties o f CCI p lus Noise Covarian ce In this subsectio n, we focus on covariance properties of CCI plus noise present in both the sing le-roun d and multi-ro und commun ication mo dels gi ven by (6) and (18 ), respectively . These p roperties present an impor tant in gredient in th e turb o packet comb ining algor ithm we introd uce in Subsection III-C. Let Θ k denote the covariance of CCI plus noise w ( k ) i present in rec eiv ed signal (4) at rou nd k , Θ k , E h w ( k ) i w ( k ) H i i ∈ C N R × N R . (25) Let us group covariance matrices cor respond ing to roun ds 1 , · · · , k in th e block diag onal matr ix Ξ k , diag { Θ 1 , · · · , Θ k } ∈ C kN R × kN R . (26) Pr opo sition 1: Th e cov ariance Ξ k , E h w ( k ) w ( k ) H i of the CCI plus noise block vector w ( k ) present in the multi-rou nd commun ication model ( 18) after k roun ds is exp ressed as Ξ k = I T ⊗ Ξ k ∈ C T kN R × T kN R . (27) Pr oof: The expression in (27) is easily obtained b y calculating th e mathematical expec tation of w ( k ) w ( k ) H . In the deriv ation , we on ly exploit the independ ence between the entries of H CCI ( k ) l and H CCI ( k ′ ) l ′ ∀ l , l ′ , k , and k ′ (i.e., short-term static block fading dy namic of the CCI M IMO ARQ channel), and the fact that CCI symbo ls satisfy (1). No assumption on the structur e of the CCI b lock matrix is used. A d etailed pr oof of ( 27) in the case of sliding-w indow aided time-dom ain detection can b e found in [39, Subsectio n III.C]. Pr opo sition 2: Th e covariance Θ k , E h w ( k ) w ( k ) H i of the sing le-roun d CCI p lus no ise blo ck vector w ( k ) at ARQ round k is Θ k = I T ⊗ Θ k . (28) Pr oof: The pr oof f ollows b y sim ply in voking Prop osition 1 fo r o ne ro und. Pr opo sition 3: Covariance ma trices of f requen cy domain CCI plu s noise vectors w ( k ) f and w ( k ) f (correspo nding to the DFTs of w ( k ) and w ( k ) , respectively) are Ξ k and Θ k , respectively . Pr oof: Th e proof of Prop osition 3 follows from the fact that Ξ k and Θ k are block c irculant a nd block diag onal matrices. Proposition 1 in dicates that the cov a riance of the multi- round CCI p lus noise vector can be obtained b y separately computin g single- round covariances u sing Propo sition 2. This result g reatly impac ts the compu tational complexity of the propo sed alg orithm as it w ill b e shown in Su bsection III -D. C. Pr opo sed Scheme In this sub section, we derive the f requency d omain MMSE- based soft packet combiner that cancels both CAI and ISI jointly over multiple ARQ round s in the pre sence o f u nknown CCI. T o co mbine sig nals co rrespond ing to A RQ r ounds 1 , · · · , k , we use c onv entional soft parallel interf erence cancellation (PIC) (of both multi-rou nd CAI an d I SI) an d u ncond itional MMSE filtering tech niques [3]. Ther efore, at each turbo it- eration of ARQ round k , the MM SE-based soft packet com - biner prod uces a co mplex scalar d ecision z ( k ) t,i that serves f or computin g extrinsic LLR values co rrespond ing to cod ed and interleaved bits map ped over symb ol s t,i . Let ϕ t,i denote the vector of a p riori LLRs of bits correspo nding to sym bol s t,i , and av ailable at the inp ut of the soft comb iner at a par ticular turbo iteration . σ 2 t,i , E h | s t,i | 2 | ϕ t,i i −  E  s t,i | ϕ t,i  2 denotes the conditional variance of s t,i . By in voking either the ortho gonal projection theor em or Lagra ngian methods, and using (11) and (27), soft MMSE-based packet combining at ARQ rou nd k , ca n be perfor med in the freq uency do main a s, z ( k ) f = Γ ( k ) y ( k ) f − Ω ( k ) ¯ s f , (29) where z ( k ) f is the DFT of z ( k ) , h z ( k ) 1 , 0 , · · · , z ( k ) N T ,T − 1 i ⊤ ∈ C N T T , i.e., z ( k ) = U H T ,N T z ( k ) f , ¯ s f ∈ C N T T denotes the DFT of th e soft sym bol vecto r ¯ s , E  s | ϕ t,i : ∀ ( t, i )  , and    Γ ( k ) = diag n Λ ( k ) H 0 B ( k ) − 1 0 , · · · , Λ ( k ) H T − 1 B ( k ) − 1 T − 1 o , Ω ( k ) = C ( k ) − I T ⊗ diag n ( ˜ C ( k ) ) 1 , 1 , · · · , ( ˜ C ( k ) ) N T ,N T o , (30)            B ( k ) i = Λ ( k ) i ˜ Σ Λ ( k ) H i + Ξ k , C ( k ) i = Λ ( k ) H i B ( k ) − 1 i Λ ( k ) i , C ( k ) , diag n C ( k ) 0 , · · · , C ( k ) T − 1 o , Σ i , diag  σ 2 1 ,i , · · · , σ 2 N T ,i  ∈ R N T × N T . (31) Matrices ˜ C ( k ) and ˜ Σ deno te time averages of C ( k ) i and Σ i , respectively , as defined in Section I. The input f or the soft demapp er can b e extracted f rom z ( k ) f as z ( k ) t,i = e ⊤ t,i U H T ,N T z ( k ) f where e t,i is th e ( iN T + t ) th vector of th e canon ical basis. As it can be seen fro m the fo rward–back ward filtering stru cture in 6 PUBLISHED IN IEE E T RANSACTIONS ON WIRELESS COMMUNICA T IONS, MA Y 2010 (PRINT A V AILABL E @ HT TP://DX.DOI.ORG/10.1109/TWC. 2010.05.090441) T able I S U M M A RY O F F R E Q U E N C Y D O M A I N M M S E - B A S E D T U R B O P AC K E T C O M B I N I N G I N T H E P R E S E N C E O F C C I 0. Initialization Initialize { D (0) i } T − 1 i =0 and ˜ y (0) f with 0 N T × N T and 0 N T × 1 , respectiv ely . 1. Packet Combini ng at ARQ round k 1.1. Compute the DF T of CSI and receiv ed signals at round k , i.e., Λ ( k ) 0 , · · · , Λ ( k ) T − 1 and y ( k ) f , respectiv ely . 1.2. For each turbo iteration 1.2.1. Compute the soft symbol vector ¯ s and varian ces σ 2 t,i , then deduce the DFT ¯ s f and ˜ Σ . 1.2.2. Estimate the CCI plus noise cova riance Θ k at round k using (32), then compute Θ − 1 k . 1.2.3. Compute { Λ ( k ) H i Θ − 1 k Λ ( k ) i } T − 1 i =0 and Λ ( k ) H ( I T ⊗ Θ − 1 k ) y ( k ) f . 1.2.4. Deduce { D ( k ) i } T − 1 i =0 and ˜ y ( k ) f using recursions (34) and (38), respectiv ely . 1.2.5. Compute the i n verses of { ˜ Σ + D ( k ) i } T − 1 i =0 , then construct { C ( k ) i } T − 1 i =0 and F ( k ) using (35) and (47), respectiv ely . 1.2.6. Compute forward and backward parts Γ ( k ) y ( k ) f and Ω ( k ) ¯ s f using (36) and (30), respectiv ely . 1.2.7. Calculate extrinsic LLRs. 1.2.8. Perform SISO decoding. 1.3. 1.2.9. End 1.2. 1.4. If “frame err or” then store { D ( k ) i } T − 1 i =0 and ˜ y ( k ) f computed at the last turbo iteration, and send “N A CK”, Otherwise, send “ ACK”. (29), the freq uency domain MMSE filter explicitly cancels soft CAI and ISI while it only require s th e covariance of unk nown CCI p lus no ise. Note tha t b oth Prop ositions 1 and 2 are used to derive (29). T o obtain estimates of u nknown CCI p lus noise covariance matrices Θ 1 , · · · , Θ k , req uired by (3 1), let us co nsider the single-rou nd frequency domain communica tion model (13). Proposition 3 indicates that the covariance of w ( k ) f is Θ k = I T ⊗ Θ k . Therefor e, w ith r espect to the blo ck diag onal structure o f (1 3), un known CCI plus noise cov ar iance Θ k can directly be estimated in the freque ncy do main at each turbo iter ation, with the aid of a p riori LLRs, accord ing to the following average, Θ k = 1 T T − 1 X i =0 n y ( k ) f i − Λ ( k ) i ¯ s f i o n y ( k ) f i − Λ ( k ) i ¯ s f i o H . (32 ) y ( k ) f i and ¯ s f i denote th e DFTs o f y ( k ) and s at fr equency bin i , respectively , i.e., y ( k ) f i = E i,N R y ( k ) f and ¯ s f i = E i,N T ¯ s f . Cov ariance matrices Θ 1 , · · · , Θ k − 1 are similarly estimated at ARQ roun ds 1 , · · · , k − 1 , respectively , and co rrespon d to estimates obtaine d at the last tur bo iter ation. In other words, when th e decod ing ou tcome is er roneou s, a N ACK m essage is fed b ack to the tran smitter , a nd the unknown CCI plu s noise covariance estimate obtained at th e last iteration is saved in the receiver to help perform p acket co mbining at the next ARQ round . D. Implementa tion Aspects W e first provide an efficient implementation of th e prop osed scheme since turbo combin ing req uires at each turbo iteration the compu tation of matrix in verses B ( k ) − 1 0 , · · · , B ( k ) − 1 T − 1 ∈ C kN R × kN R giv en by (31). Second , we analyze the comp uta- tional complexity and mem ory requirem ents of th e pro posed implementatio n alg orithm. 1) An Efficient Implemen tation Algorithm: The special structure of the frequ ency do main ARQ c hannel matrix (23) together with the matrix in version lemma [4 0] allo w us to express the in verse of B ( k ) i as, B ( k ) − 1 i = Ξ − 1 k − Ξ − 1 k Λ ( k ) i  ˜ Σ + D ( k ) i  − 1 Λ ( k ) H i Ξ − 1 k , ( 33) where D ( k ) i is o btained acco rding to the f ollowing r ecursion, ( D ( k ) i = D ( k − 1) i + Λ ( k ) H i Θ − 1 k Λ ( k ) i , D (0) i = 0 N T × N T . (34) Therefo re, m atrices C ( k ) 0 , · · · , C ( k ) T − 1 are simply comp uted as, C ( k ) i = D ( k ) i − D ( k ) i  ˜ Σ + D ( k ) i  − 1 D ( k ) i , (35) while the f orward filterin g part of (29) is calculated at each ARQ r ound k as, Γ ( k ) y ( k ) f = F ( k ) ˜ y ( k ) f , (36) where F ( k ) = diag  I N T − D ( k ) 0  ˜ Σ + D ( k ) 0  − 1 , · · · , I N T − D ( k ) T − 1  ˜ Σ + D ( k ) T − 1  − 1  , (37) and ˜ y ( k ) f is given by the following recur sion, ( ˜ y ( k ) f = ˜ y ( k − 1) f + Λ ( k ) H ( I T ⊗ Θ − 1 k ) y ( k ) f , ˜ y (0) f = 0 N T × 1 . (38) The pro posed tur bo packet co mbining algorithm is su mma- rized in T able I. No te that, du ring the first iteration of ro und k , the anti- causal parts in recursion s (3 4) and ( 38), i.e., D ( k − 1) i and ˜ y ( k − 1) f , respectiv ely , co rrespon d to the output of these recursions at th e last iter ation o f p revious ro und k − 1 . AIT -IDIR et al. : T URBO P ACKET C OMBINING FOR BRO ADBAND SP ACE–TIME BICM HYBRID–ARQ SYSTEMS WITH CO–CHANNEL INTERFER ENCE 7 T able II S U M M A RY O F M E M O RY A N D A R I T H M E T I C A D D I T I O N S R E Q U I R E D B Y T H E P RO P O S E D A N D L L R - L E V E L C O M B I N I N G S C H E M E S , A N D R E L AT I V E C O S T E V A L UAT I O N Combining scheme Memory Arithmeti c Additions Relati ve Costs QPSK 8-PSK 16-QAM LLR-Lev el Proposed T N T log 2 |S | 2 T N T ( N T + 1) T N T N it ( K − 1) log 2 |S | 2 T N T N it ( K − 1) ( N T + 1) N T 2 3 N T − 1 3 N T − 1 2 R N R =    1 δ Rx . . . δ Rx 1    N R × N R , R N ′ T =    1 δ Tx . . . δ Tx 1    N ′ T × N ′ T , (42) 2) Compu tational Complexity and Memo ry Requirements: The propo sed r ecursive implemen tation algorith m av oids stor- ing received signals and CFRs co rrespond ing to multiple ARQ round s. It also prevents the computation of k N R × k N R matrix in verses. Th is dram atically redu ces the imp lementation cost since the comp lexity order of directly comp uting B ( k ) − 1 i is cubic against k N R , and is greatly in creased from round to round . In the fo llowing, we analyze b oth the com plexity an d memory r equireme nts of the pr oposed schem e, and co mpare them with those o f the L LR-lev el combin ing techn ique 4 . First, n ote that in th e case of LLR-level packet comb ining, frequen cy domain MMSE equalization is separately performed for each ARQ roun d. Therefo re, T inversions o f N R × N R matrices are required to compute the forward and bac kward filters. Since in gen eral it is re quired to ha ve more receive than transm it antenn as, espec ially when CCI is present in the system, an imp lementation similar to that intro duced in the previous subsection is beneficial b ecause o nly T inversions of N T × N T matrices will be req uired. In th is case, the two variables in recu rsions (34) and (38) ar e co mputed at ARQ round k as, D ′ ( k ) i = Λ ( k ) H i Θ − 1 k Λ ( k ) i ∈ C N T × N T , an d ˜ y ′ ( k ) f = Λ ( k ) H ( I T ⊗ Θ − 1 k ) y ( k ) f ∈ C N T , while all the other step s in T a- ble I remain the same (includ ing the CCI plus noise cov ariance estimation proced ure in step 1.2 .2.). Th erefor e, b y letting N it denote the num ber of turbo iterations at eac h ARQ roun d, both combinin g algorithms ha ve similar computation al complexities since the pro posed scheme an d th e LLR-level sch eme r equire at most C + new scheme = 2 T N T N it ( K − 1) ( N T + 1) and C + LLR − lev el = T N T N it ( K − 1) lo g 2 |S | a rithmetic additions to perf orm (34) an d ( 38), an d to com bine LLRs co rrespon ding to multiple ro unds, re spectiv ely . LLR-level packet combining perfo rms the c ombinatio n of extrinsic L LR values generate d by freq uency dom ain soft equalizers at mu ltiple ARQ roun ds. Therefore, a storage capacity of T N T log 2 |S | real values is r equired to store accumulated LLR values corr espondin g to all ARQ round s. 4 In this paper , LLR-le vel combining refers to the iterat i ve (turbo) packet combining and S ISO decoding recei ver , where transmissions corresponding to k ARQ rounds are separatel y turbo equalized using k frequency domain MMSE soft equaliz ers. T o perform packet combining at each iteratio n of ARQ round k , extrinsic LLR value s generat ed by the s oft MMSE equalizer at round k and those obtained at the last iteration of previous rounds 1 , · · · , k − 1 are added together , then SISO decodi ng is performed. The p ropo sed schem e combin es m ultiple transmissions at the signal level u sing signa ls and CFRs co rrespon ding to all ARQ round s, withou t being require d to be exp licitly stor ed in the receiver . Th is is perf ormed with the aid of the two variables D ( k ) i and ˜ y ( k ) f in recu rsions (34) and (38), re spectiv ely . Th is translates into a me mory size o f 2 T N T ( N T + 1) real values. Therefo re, th e com putationa l com plexity an d storag e requir e- ments are less sensitive to the ARQ delay . The techniqu e requires only a few m ore addition s and a bit more memory compare d to LLR-level comb ining. T able I I summarizes im- plementation req uiremen ts an d repo rts the relative co sts 5 for some m odulation schem es. I V . P E R F O R M A N C E E V A L U A T I O N A. Asymptotic P erformance Ana lysis In the following, we provide a frame-basis a nalysis where we der iv e system cond itions u nder wh ich p erfect CCI cance l- lation holds. W e su ppose that the interferer CSI is perfectly known, and investigate the influ ence of its channel p roperties on the interference cancellation capability o f the propo sed packet com bining scheme in the high SNR regime. Theor em 1: W e consider a CCI-limited MI MO ARQ sy s- tem with N T transmit and N R receive antenn as, and ARQ delay K . Le t Θ CCI k denote the CCI covariance at ARQ r ound k = 1 , · · · , K , i.e., th e covariance of th e glob al noise at th e receiver is Θ k = Θ CCI k + σ 2 I N R , an d ρ k be th e rank of Θ CCI k . W e assume perfect LLR feedba ck from the SISO decod er . The freq uency d omain soft MMSE packet comb iner provides perfect CCI su ppression f or a symptotically h igh SNR if k X u =1 ρ u < k N R − N T . (39) Pr oof: See the Append ix. W e now proceed to derive an upper bo und on ρ k , where we incorpo rate the rank o f the CCI fading c hannel. Un der the assumption that CCI symbols satisfy (1), i.e., infinitely d eep 5 Relat i ve costs refer to the relati ve number of arithmetic additi ons △ C and memory △ M required by the proposed scheme compared to L LR-le vel combining . Wit h respect to storage require ments and number of arithmetic additi ons in T able II, we hav e △ C = △ M = 2 N T +1 log 2 |S | . 8 PUBLISHED IN IEE E T RANSACTIONS ON WIRELESS COMMUNICA T IONS, MA Y 2010 (PRINT A V AILABL E @ HT TP://DX.DOI.ORG/10.1109/TWC. 2010.05.090441) interleaving, we get Θ CCI k = L ′ − 1 X l ′ =0 H CCI ( k ) l ′ H CCI ( k ) H l ′ . (40) Let us write eac h CCI c hannel m atrix as H CCI ( k ) l ′ = R 1 / 2 N R A CCI ( k ) l ′ R 1 / 2 N ′ T ∀ l ′ , (41 ) where A CCI ( k ) l ′ ∈ C N R × N ′ T characterizes the scatterin g en vi- ronmen t between the CCI transmitter and receiver [41], and R N R and R N ′ T are the correlatio n matrices controlling the receive an d transmit a ntenna arrays, an d ar e in general given by (42), where 0 ≤ δ Rx , δ Tx < 1 [42]. Note that (41) correspo nds to a g eneral mo del of correlated fading MIMO channels, where the scattering ra dii at tr ansmitter an d r eceiver sides is taken into ac count, and A CCI ( k ) l ′ is not n ecessarily a fu ll rank matrix, i.e ., ra nk n A CCI ( k ) l ′ o ≤ min ( N ′ T , N R ) [41]. Noting tha t R N R and R N ′ T are fu ll ra nk matrices, and with respect to th e fact that CCI tap chan nel matrices are indepen dent, and using (40) a nd ( 41), we g et ρ k ≤ min    N R , L ′ − 1 X l ′ =0 rank  H CCI ( k ) l ′ H CCI ( k ) H l ′     = min    N R , L ′ − 1 X l ′ =0 rank  A CCI ( k ) l ′ R N ′ T A CCI ( k ) H l ′     ≤ min    N R , L ′ − 1 X l ′ =0 rank n A CCI ( k ) l ′ o    . (43 ) A closer look at Th eorem 1 and u pper bou nd (43) provid es interesting system in terpretation s. • Impact of CCI Fading Channel: First, note that the CCI cancellation capability of the frequency domain MMSE packet combine r is related to the CCI chan nel rank. When the interferer h as a rank-d eficient chann el m atrix at a certain ARQ ro und, in terferenc e can completely be removed (at subseq uent rounds) if the sum-ran k condition in Theorem 1 is satisfied. In practice, the channel rank can dr amatically drop in the case of the so- called pinho le channel, wher e the tra nsmitter and r eceiver are largely separated an d are surrou nded by multip le scatterers [ 41]. In th is scenar io, the channel can even pr ev ent multipath from building up since the thin air pipe co nnecting tran s- mitter and receiver scatter ers is very long. For instance, in a system with N R = 3 re ceiv e and N T = 2 tra nsmit antennas, and an unknown interfer er wh o is experiencin g one path ( L ′ = 1 ) chan nel realizations with ra nk eq ual to two, CCI can be removed at the second ARQ round because the sum -rank c ondition (39) h olds f or k ≥ 2 . • Impact of the Number of T ransmit Antennas and ARQ Dela y: Co ndition (39) sugge sts how , for a giv en CCI ch annel pro file, the nu mber o f tr ansmit an tennas N T and ARQ rounds K are chosen to achiev e p erfect CCI cancellation. For instance, if transmission is corrupted by CCI with q uasi-static channe l rank 6 , and if the ARQ delay allo wed by the upper layer is K , then only N T < K ( N R − ρ 0 ) transmit antenn as can be allocated to the user o f interest to achieve interf erence su ppression at th e latest at ARQ round K , where ρ 0 is the rank of Θ CCI k , i.e., ρ k = ρ 0 ∀ k . In creasing the ARQ delay will relax the condition o n the number o f transmit anten nas and therefore allow f or an increase in the div ersity and/or multiplexing gains depen ding on the diversity- multiplexing-d elay trade-off opera ting p oint [23]. Note that wh en N ′ T ≪ N T , the CCI channel rank d ramatically drops, and therefore CCI s uppre ssion is achiev ed even when a shor t ARQ d elay K is r equired . • Interactio n with the Scheduling Mechanism: In the case of op portun istic commu nications, inte rference with co-chan nel u sers who have h igh c hannel ranks can be prevented. For instance, when a retransmission is required on the rev erse link, the base station (BS) can c hoose the timin g o f the next ARQ round in such a way that transmission simultan eously occurs with that o f a user with lo w channel r ank. This is feasible since the BS has comp lete knowledge ab out user CSIs in the rev erse link. The same schedu ling mechan ism can be used in th e forward link if all users provide the BS with feedb ack informa tion ab out their chan nel ran ks. Whe n the system suffers from CCI cau sed by neighbo ring cells, the sum- rank con dition (3 9) can b e ach iev ed by simply inc reasing the number of ARQ ro unds because the CCI ch annel rank tends to be co nstant over time. B. Numerical Results In this sub section, we p rovide block error r ate (BLER) perfor mance results for the prop osed combinin g technique. Our fo cus is to demonstrate the superior perfo rmance of th e introdu ced sch eme compa red to LLR-le vel combining . W e also evaluate BLER p erforma nce fo r scenar ios where the interferer has r ank de ficient chann el matrices to corrob orate the the oretical a nalysis in Sub section IV -A. In all simulations, we con sider a BICM scheme where the en coder is a 1 2 -rate conv olutional code with polyn omial generato rs (35 , 23) 8 , and the modulatio n scheme is qu adratur e phase shift keying (QPSK). The leng th o f the code b it f rame is 1032 bits in cluding tails. The ARQ de lay is K = 3 , and th e E b / N 0 ratio ap pearing in all figures is th e SNR per useful bit per receive antenn a. W e consider a L = 2 path MIMO-ISI ch annel profile whe re σ 2 0 = σ 2 1 = 1 2 . In practical wir eless systems, the wir eless ch annel may hav e more than two path s due to severe freq uency selective fading. In this p aper, we restrict our selves to L = 2 for the sake of simu lation si mplicity . Performance in the case of severe frequen cy selectiv e fadin g ch annels can be fo und in [38]. W e use bo th the matched filter bo und (MFB) per ARQ r ound and the outag e pr obability [ 26] of the CCI-free MIMO-ISI channel as absolute perf ormanc e bo unds to evaluate the CCI cancellation capability and diversity ord er ach iev ed by the 6 In this case, CCI with quasi-static channel rank refers to an interferer whose channel rank is constant over m ultipl e ARQ rounds. AIT -IDIR et al. : T URBO P ACKET C OMBINING FOR BRO ADBAND SP ACE–TIME BICM HYBRID–ARQ SYSTEMS WITH CO–CHANNEL INTERFER ENCE 9 −8 −6 −4 −2 0 2 4 6 8 10 10 −4 10 −3 10 −2 10 −1 10 0 E b /N 0 per Rx ant. (dB) BLER CC(35,23) 8 , QPSK, N T =N R =2, L=2, SIR=3dB k=1 LLR−level, k=2 LLR−level, k=3 Proposed, k=2 Proposed, k=3 MFB, k=1 MFB, k=2 MFB, k=3 Outage, R=2, K=3 Figure 3. BLER performance for CC (35 , 23) 8 , QPSK, N T = N R = 2 , L = L ′ = 2 equal energy paths, and SIR = 3 dB. −8 −6 −4 −2 0 2 4 6 8 10 −4 10 −3 10 −2 10 −1 10 0 E b /N 0 per Rx ant. (dB) BLER CC(35,23) 8 , QPSK, N T =N R =2, L=2, SIR=5dB k=1 LLR−level, k=2 LLR−level, k=3 Proposed, k=2 Proposed, k=3 MFB, k=1 MFB, k=2 MFB, k=3 Outage, R=2, K=3 Figure 4. BLER performance for CC (35 , 23) 8 , QPSK, N T = N R = 2 , L = L ′ = 2 equal energy paths, and SIR = 5 dB. propo sed com bining sch eme. The number o f turbo itera tions is set to five a nd the Max-L og-MAP version of the m aximum a posteriori ( MAP) alg orithm is used f or SISO d ecoding . W e first in vestigate perfor mance fo r scenarios where the user of interest an d the interfere r hav e the same numb er of transmit an tennas ( N T = N ′ T ) and id entical chann el profiles, i.e., L = L ′ , eq ual power taps, and CCI fading cha nnel co effi- cients are i.i.d. In Fig. 3, we co mpare the BLER per forman ce of the p roposed scheme with that of LLR-level co mbining for a ST –BICM code with rate R = 2 , i.e. , N T = 2 . Th e numb er of receive antennas is N R = 2 , and SIR = 3 dB . W e ob serve that the pr oposed scheme significan tly o utperfo rms LLR-level combinin g. The performan ce gap at ARQ roun d k = 3 is ab out 1dB for BLER ≤ 10 − 2 . No te that both co mbining schemes fail to perfe ctly cancel CCI sin ce per forman ce cu rves ten d to −4 −2 0 2 4 6 8 10 12 10 −3 10 −2 10 −1 10 0 E b /N 0 per Rx ant. (dB) BLER CC(35,23) 8 , QPSK, N T =4, N R =2, L=2, SIR=5dB k=1 LLR−level, k=2 LLR−level, k=3 Proposed, k=2 Proposed, k=3 MFB, k=1 MFB, k=2 MFB, k=3 Outage, R=4, K=3 Figure 5. BLE R performance for CC (35 , 23) 8 , QPSK, N T = 4 , N R = 2 , L = L ′ = 2 equal energy paths, and SIR = 5 dB. −8 −6 −4 −2 0 2 4 6 8 10 10 −4 10 −3 10 −2 10 −1 10 0 E b /N 0 per Rx ant. (dB) BLER CC(35,23) 8 , QPSK, N T =N R =2, L=2, N’ T =1, L’=1, SIR=3dB k=1 Proposed, k=2 Proposed, k=3 MFB, k=1 MFB, k=2 MFB, k=3 Outage, R=2, K=3 Figure 6. BLER performance for CC (35 , 23) 8 , QPSK, N T = N R = 2 , L = 2 equal energ y paths, N ′ T = 1 , L ′ = 1 , and ρ k = 1 , k = 1 , · · · , K , SIR = 3 dB. saturate f or high E b / N 0 values. Fig. 4 rep orts p erform ance of b oth techniques when SIR is incre ased to SIR = 5dB . In this case, the perf ormance gap between the two schemes is reduced . The CCI cancellation capab ility is also impr oved as can be seen from the steeper slopes of BLER cu rves. In Fig. 5, we ev aluate the per forman ce fo r a high ra te ST –BICM code where R = 4 , i.e. , N T = 4 . Only N R = 2 receive antennas are considered, and SIR = 5dB . The propo sed scheme dr amatically outper forms LLR-level com bining, i.e., the p erforma nce gap at ARQ r ound k = 3 is ab out 4dB at 7 ∗ 10 − 3 BLER. The prop osed schem e also offers high er cancellation cap ability an d diversity order th an LL R-lev el combinin g. W e now turn to scenarios where the interfe rer h as a rank- 10 PUBLISHED IN IEE E T RANSACTIONS ON WIRELESS COMMUNICA T IONS, MA Y 2010 (PRINT A V AILABL E @ HT TP://DX.DOI.ORG/10.1109/TWC. 2010.05.090441) −8 −6 −4 −2 0 2 4 6 8 10 10 −3 10 −2 10 −1 10 0 E b /N 0 per Rx ant. (dB) BLER CC(35,23) 8 , QPSK, N T =4, N R =4, SIR=1dB Scenario 1, k=1 Scenario 1, k=2 Scenario 1, k=3 Scenario 2, k=1 Scenario 2, k=2 Scenario 2, k=3 MFB, k=1 MFB, k=2 MFB, k=3 Outage, R=4, K=3 Figure 7. BLER performance for CC (35 , 23) 8 , QPSK, N T = N R = 4 , L = 2 equal energ y paths, and SIR = 1 dB. Scenario 1: N ′ T = 4 , L ′ = 4 , Scenari o 2: N ′ T = 2 , L ′ = 1 , and ρ k = 2 , k = 1 , · · · , K . deficient uncorrelated MIMO channel, i.e., rank n A CCI ( k ) l ′ o < min ( N ′ T , N R ) ∀ l ′ , δ Tx = δ Rx = 0 , and assume tha t the ra nk is constant o ver all ARQ rounds. In Fig. 6, we repor t the BLER perfor mance of the prop osed scheme fo r a CCI-limited MIMO system with settings similar to Fig . 3, i.e., N T = N R = 2 , and SIR = 3dB . The interfer er exp eriences flat fading, i.e., L ′ = 1 , and only has N ′ T = 1 transmit antenn a. T herefo re, with respect to ( 43), ρ k = 1 ∀ k . Note th at in th is in terferenc e scenario, the perf ect CCI cancellation co ndition (3 9) ho lds for k ≥ 2 . W e observe that bo th the CCI can cellation c apability and the diversity order of the p roposed scheme are impr oved. The perfor mance g ain with resp ect to the case of N ′ T = 2 and L ′ = 2 is ab out 1 . 5dB at 3 ∗ 10 − 3 BLER and rou nd k = 3 , and the slop e of th e BLER curve at ro und k = 3 is similar to that o f the MFB cu rve. Fig. 7 compares th e perform ance of the prop osed scheme for two scenario s with he avy CCI, i.e., SIR = 1 dB . The ST –BICM code h as rate R = 4 , i.e., N T = 4 , a nd the nu mber of receive antennas is set to N R = 4 . In the first scenario (Scen ario 1), the interf erer has N ′ T = 4 transmit an tennas, L ′ = 2 eq ual p ower taps, and i.i.d. fading coefficients, while in the secon d scen ario ( Scenario 2 ), N ′ T = 2 , L ′ = 1 , and the C CI channel rank is equal to two. Therefo re, ρ k = 2 ∀ k , and co ndition (39) hold s for k ≥ 2 . It is clear that in the second scenario , better CCI can cellation capability is achieved for k ≥ 2 . For instance , the p erform ance gap fo r k = 3 is more than 2dB at 2 ∗ 10 − 2 BLER. Also, the diversity order o f the CCI-f ree MI MO-ISI cha nnel is alm ost a chieved. V . C O N C L U S I O N In this pa per, we inves tigated efficient iter ativ e tur bo packet combinin g for broadb and ST – BICM transmission with hyb rid ARQ over CCI-limited MIMO-ISI chann els. W e have intro- duced a frequency domain turbo c ombinin g scheme where signals and CFRs correspo nding to all ARQ round s ar e com - bined in a M MSE fashion to decode th e data packet at each round . The covariance of the overall (over all ARQ rounds) CCI plus noise requir ed b y the frequen cy domain MMSE soft packet com biner is constructed by separately co mputing the covariance related to ea ch ro und. The pro posed tech nique has a complexity o rder cub ic ag ainst the product of the number of receive anten nas an d ARQ delay . This limitation is overcome by an optimized recu rsiv e implementation algorithm where com plexity is on ly cubic in term of the nu mber of transmit antennas. W e evaluated the compu tational load an d memory requ irements, an d fou nd that the intro duced recursive technique only re quires few arithm etic add itions and memo ry compare d to c onv entional LLR-level combinin g schemes. W e analyzed the effect of CCI chan nel rank o n perfo rmance. I n- terestingly , und er a sum-ran k conditio n, th e f requen cy domain MMSE soft pa cket combin er can completely r emove CCI for asym ptotically high SNR. Fina lly , we p rovided simu lation results where we showed that the pro posed tech nique a chieves BLER perfo rmance sup erior to LLR-le vel combining , and offers high CCI cance llation capability and d iv ersity o rder for many interfer ence scenario s. A P P E N D I X P R O O F O F T H E O R E M 1 Under the assumptio n of p erfect LLR feed back fr om the SISO deco der, the frequen cy d omain soft packet combiner output ( 29), at ARQ rou nd k , can b e expressed as, z ( k ) f perfect LLR = As f + x ( k ) f , (44) where A is the diago nal matrix of fr equency domain symbol gains, A = diag n ( G ( k ) 0 ) 1 , 1 , · · · , ( G ( k ) 0 ) N T ,N T , · · · , ( G ( k ) T − 1 ) 1 , 1 , · · · , ( G ( k ) T − 1 ) N T ,N T o , (45) with G ( k ) i = Λ ( k ) H i Ξ − 1 k Λ ( k ) i , and x ( k ) f is the filtered CCI plus thermal noise a t th e outpu t o f the packet com bining filter . Its covariance matrix is G ( k ) = diag n G ( k ) 0 , · · · , G ( k ) T − 1 o . (46) Now , let u s examin e the stru cture of m atrix G ( k ) i for asymp - totically h igh SNR, i.e ., σ 2 → 0 . Let Π 1 Π H 1 , · · · , Π k Π H k be the low-rank decomp ositions of matrices Θ CCI 1 , · · · , Θ CCI k , where Π 1 ∈ C N R × ρ 1 , · · · , Π k ∈ C N R × ρ k . F or the s ake of notation simplicity , we write P k u =1 ρ u = ρ . It follows that th e r ank of Π = diag { Π 1 , · · · , Π k } is ρ , and Ξ k = ΠΠ H + σ 2 I kN R is a square in vertible m atrix. Therefore, it h as an eigen value decomp osition ( E.V .D) that can b e expressed as, Ξ k =  P ρ P kN R − ρ  | {z } P  Υ + σ 2 I ρ σ 2 I kN R − ρ  ×  P H ρ P H kN R − ρ  , (47) AIT -IDIR et al. : T URBO P ACKET C OMBINING FOR BRO ADBAND SP ACE–TIME BICM HYBRID–ARQ SYSTEMS WITH CO–CHANNEL INTERFER ENCE 11 where PP H = P H P = I kN R since Ξ k is symmetric. This condition yields th e following set of eq ualities, P H ρ P ρ = I ρ , (48a) P H kN R − ρ P kN R − ρ = I kN R − ρ , (48 b) P H ρ P kN R − ρ = 0 , (48c) P ρ P H ρ + P kN R − ρ P H kN R − ρ = I kN R . (4 8d) Therefo re, a T aylo r expan sion of Ξ − 1 k when σ 2 → 0 , is g iv en as, Ξ − 1 k = P ρ Υ − 1 P H ρ + σ − 2 I kN R + O  σ 2  . (49) Note that Υ doe s not ha ve any null diagon al element, i.e., Υ is in vertible. In deed, m ultiplying th e left an d rig ht sides of (4 7) by P H and P , respe ctiv ely , an d with respe ct to ( 48a), we ge t, P H ρ ΠΠ H P ρ = Υ . By no ting that P H ρ Π is ρ × ρ a nd has ran k equal to ρ , it follows that Υ − 1 =  Π H P ρ  − 1  P H ρ Π  − 1 . Therefo re, wh en σ 2 → 0 , we have, G ( k ) i = Λ ( k ) H i P ρ Υ − 1 P H ρ Λ ( k ) i + σ − 2 Λ ( k ) H i Λ ( k ) i + O  σ 2  . (50) Since th e time domain chan nel coefficients are i.i.d ., it follows that the k N R × N T matrix Λ ( k ) i has full- column rank unless all fading coe fficients are eq ual to zero . If ρ + N T < k N R , i.e., ρ < k N R − N T , then all the first ρ co lumns o f Ξ k (column vectors o f P ρ ) a re in th e kernel of Λ ( k ) H i , i.e ., Λ ( k ) H i P ρ = 0 N T × ρ . It fo llows that, wh en σ 2 → 0 , G ( k ) i = σ − 2 Λ ( k ) H i Λ ( k ) i + O  σ 2  . (51) Therefo re, wh en SNR → ∞ , we g et SINR = 1 σ 2 T T − 1 X i =0 tr n Λ ( k ) H i Λ ( k ) i o + O  σ 2  = 1 σ 2 L − 1 X l =0 k X u =1 tr n H ( u ) H l H ( u ) l o | {z } SNR MF + O  σ 2  , (52) where SNR MF correspo nds to the instantaneo us matched filter (MF) SNR in the case of k rou nds CCI-free MIMO-ISI ARQ channel.  A C K N O W L E D G M E N T The auth ors would like to tha nk Dr . Matthew V alenti f or coordin ating the review pro cess, a nd the three ano nymous revie wer s for their very help ful comm ents an d sug gestions. R E F E R E N C E S [1] S. L. Ariyavisita kul, “T urbo space-time processing to improv e w irele ss channe l capaci ty , ” IEEE Tr ans. Commun. , vol . 48, no. 8, pp. 1347-1359, Aug. 2000. [2] A. M. T onello, “MIMO MAP equali zation and turbo dec oding in interl eav ed space time coded systems”, IEEE T rans. Commun. , vol. 51, no. 2, pp. 155-160, Feb . 2003. [3] X. W autel et, A. Dejonghe, and L. V andendorpe, “MMSE-based frac- tional turbo recei ver for s pace-t ime BICM ov er frequency selecti ve MIMO fading channels, ” IEEE Tr ans. Sig. P r oc., vol. SIG-52, pp. 1804- 1809, Jun. 2004. [4] R. V isoz, A. O. Bert het, and S.Chtou rou, "A ne w class of ite rati ve equali zers for space-time BICM ov er MIMO block fading multipath A WGN channel," IEE E T rans. Commun. , vol. 53, no. 12, pp. 2076 - 2091, Dec. 2005. [5] T . Ait-Idir , S. Saoudi, and N. Naja, “Space-t ime turbo equalizat ion with successi ve interferenc e cance llati on for frequenc y selecti ve MIMO channe ls, ” IEEE T rans. V eh. T echn ol. vol. 57, no. 5, pp. 2766-2778, Sep. 2008. [6] H. Sari, G. Karam, and I. J eancl aude, “Frequenc y domain equali zatio n of mobile radio and terrestrial broadc ast channe ls, ” in Pro c. IEEE GLOBECOM, San Francisco, CA, Nov . –Dec. 1994. [7] M. Tüchler and J. Hagenaue r , “Tur bo equaliza tion using frequenc y domain equaliz ers, ” in P r oc. Allerton Conf., Monticello, IL, Oct. 2000. CD-R OM. [8] D. Falconer , S. L. Ariyav isitakul , A. Benyamin-Se eya r , and B. Eidson, “Frequen cy domain equalizati on for single-ca rrier broadband wireless access systems, ” IEE E Commun. Mag ., vol. 40, no. 4, pp. 58–66, Apr . 2002. [9] F . Pancald i and G. M. Vi tetta , “Block channel equa liza tion in th e frequenc y domain, ” IE EE T rans. Commun., vol. 53, no. 3, pp. 463–471, Mar . 2005. [10] R. V isoz, A. O. Berthet, S. Chtourou, ”Frequenc y-domain block turbo- equali zatio n for single-carrie r transmission over MIMO broadband wire- less channel, ” vol. 54, no. 12, pp. 2144–2149, Dec. 2006. [11] S. Lin and D. Costello, E rror Control Coding: Fundamenta ls and Applica tions. Engle wood Cliffs, NJ: Prentice-Hall , 1983. [12] D. Costello, J. Hagenauer , H. Imai, and S. W icker , “ Applicati ons of error-contro l coding, ” IEEE T rans. Inf orm. Theory , vol. 44, pp. 2531–2560, Oct. 1998. [13] P . S. Sindhu, “Retra nsmission error control with memory , ” IEEE T rans. Commun., vol. 25, pp. 473–479, May 1977. [14] G. Benelli, “An ARQ scheme with memory and soft error detecti on, ” IEEE T rans. Commun., vol. 33, pp. 285–288, Mar . 1985. [15] S. B. W icke r , “ Adapti ve err or control through t he use of di versity combining majority logic decoding in hybrid ARQ protocol, ” IEEE T rans. Commun., vol. 39, pp. 380–385, Mar . 1991. [16] E. Yli-Juuti, S. S. Chakraborty , and M. Liinaharja , “An adapti ve ARQ scheme with packet combi ning, ” IE EE Commun. Lett. , v ol. 2, pp. 200–202, July 1998. [17] M. Liinaharj a, S. S. Chakraborty , and E. Yli-Juuti, “An adapti ve ARQ scheme with packe t combining for time varying channel s, ” IEEE Com- mun. Lett., vol. 3, pp. 52–54, Feb . 1999. [18] B. A. Harve y and S. B. Wic ker , “Packet combining system based on the Vi terbi decode r , ” IE EE Tr ans. Commun., vol. 42, pp. 1544–1557, Feb ./Mar ./Apr . 1994. [19] S. Kallel , “ Analysis of a type-II hybrid ARQ scheme with code com- bining, ” IEEE T rans. Commun., vol. 38, pp. 1133–1137, Aug. 1990. [20] M. J. McTif fin, A. P . Hulbert, T . J. Ketseogl ou, W . Heimsch, and G. Crisp, “Mobile access to an A T M network using a CDMA air interface , ” IEEE Jou rnl. Select. A re as Commun., vol . 12, no. 5, pp. 900–908, Jun. 1994. [21] G. Caire, and D. Tu ninett i, “ARQ protocols for the Gaussian collision channe l, ” IEE E Tr ans. Inf. Theory , vol. 47, no. 4, pp. 1971–1988, Jul. 2001. [22] L. Zheng, and D. N. C. T se, “Div ersity and multiple xing:A fundamental tradeof f in multiple antenna channe ls, ” IEEE T rans. Inf. Theory , vol. 49, no. 5, pp. 1073–1096, May 2003. [23] H. El Gamal, G. Caire, and M. O. Damen, “The MIMO ARQ channel: di ver sity–multi plex ing–delay tradeof f, ” IEE E T rans. Inf., Theory , vol. 52, no. 8, Aug. 2006, pp. 3601–3621. [24] D. Ts e, and P . V iswanath, “Fundamental s of Wi reless Communicati on, ” Cambridge Uni versit y Press, May 2005. [25] A. Chuang, A. Gui llen i Fabre gas, L .K. Rasmussen, I.B. Collings, “Optimal throughput-di versit y-delay tradeoff in MIMO ARQ block- fadi ng channels, ” IE EE T rans. Inf., Theory , vol. 54, no. 9, Sep. 2008, pp. 3968–3986. [26] T . Ait-Idir , and S. Saoudi, “Tu rbo packet combining strate gies for the MIMO-ISI A RQ channel, ” IEE E T rans. Commun. , vol. 57, no. 12, pp. 3782–3793, Dec 2009. [27] D. Chase, “Code combining–a maximum-likeli hood decoding approach for combining an arbitrary number of noisy packet s, ” IEEE T rans. Commun., vol. COM-33, no. 5, pp. 385-393, May 1985. [28] H. Z heng, A. L ozano, and M. Haleem, “Multiple ARQ processes for MIMO systems, ” in Pr oc. 13th IEE E Intern. Symp. P ersonal Indoor and Mobile Radio Commun. (PIMRC), Lisbon, Portugal, Sep. 2002. 12 PUBLISHED IN IEE E T RANSACTIONS ON WIRELESS COMMUNICA T IONS, MA Y 2010 (PRINT A V AILABL E @ HT TP://DX.DOI.ORG/10.1109/TWC. 2010.05.090441) [29] E. N. Onggosanusi, A. G. Dabak, Y . Hui, and G. Jeong, “Hybrid ARQ transmission and combining for MIMO systems, ” in Pr oc. IEE E Int. Conf . Commun., (ICC), Anchorage, AK, May 2003. [30] Zhihong Ding, and M. Rice, “T ype-i hybrid-ARQ using MTCM spatio- temporal vector coding for MIMO s ystems, ” in Proc. IEEE Int. Conf. on Commun. (ICC), Anchorage, AK, May 2003. [31] A. Hotti nen, and O. T irkko nen, “Matrix modulat ion and adapti ve retrans- mission, ” in Proc . 7th IE EE Intern. Symp. Sig. Proc. and Applicatio ns (ISSP A), Paris, France, Jul. 2003. [32] H. S amra, and Z . Ding, “Sphere decoding for retransmission di versit y in MIMO flat-fadi ng channels, ” in P r oc. IEEE Int. Conf. on Acoustics, Speec h, and Signal Proce ssing (ICASSP) , Montreal , Canada, May 2004. [33] T . Koik e, H. Murata, and S. Y oshida, “Hybrid ARQ scheme suitable for coded MIMO transmission, ” in Proc. IEEE Int. Conf. on Commun. (ICC), Paris, France, Jun. 2004. [34] H. Samra, and Z. Ding, “New MIMO ARQ protocols and joint detection via sphere decoding, ” IEEE T rans. Sig. Proc . vol. 54, no. 2, pp. 473-482, Feb . 2006. [35] D. Krishnaswamy , and S. Kalluri , “Multi-le vel weighted combining of retransmit ted vectors in wireless communications, ” in Proc. IEEE V eh. T echnol . Conf., (VTC) , Montreal, Canada , Sep. 2006. [36] E. W . Jang, J. Lee, H.-L. Lou, and J. M. Cioffi, ”On the combining schemes for MIMO systems with hybrid ARQ, ” IEEE T rans. W ireless Commun., vol. 8, no. 2, pp. 836-842, F eb . 2009. [37] T . Ait-Idir , H. Chafnaji, and S. Saoudi, "Joint hybrid ARQ and Iterati ve Space-T ime Equalizat ion for Coded Transmission over the MIMO-ISI Channel ," in Proc. IEEE W irel ess Commun. Net. Conf. (WCNC), Las V ega s, NV , Mar-Apr . 2008. [38] H. Chafnaji, T . Ait-Idir , and S. Saoudi, “Pack et combining and chip lev el frequenc y domain turbo equalizat ion for multi-code transmission over multi-an tenna broadban d channel, ” in Proc , 19th Annual IEEE Symp. P ersonal Indoor Mobile Radio Commun. (PIMRC’08) , Cannes, France, Sep. 2008. [39] T . Ait-Idi r , and S. Saoudi, “T urbo packe t combining for MIMO-ISI channe ls with co-channel interfere nce, ” in Pr oc, 19th Annual IEEE Symp. P ersonal Indoor Mobile Radio Commun. (PIMRC’08) , Cannes, France, Sep. 2008. [40] S. Haykin, Adaptive Fi lter Theory , 3rd Ed. Upper Saddle Riv er, NJ: Prentic e-Hall, 1996. [41] D. Gesbert, H. Bölcskei, D. A. Gore, and A. J. Paulraj, “Outdoor MIMO wireless channels: m odels and per formance pre dictio n, ” IEEE T rans. Commun., vol. 50, no. 12, pp. 1926–1934, Dec. 2002. [42] M. Sellathurai , and S. Haykin, “T urbo-BLAST : Performance ev aluat ion in correla ted Rayleigh-f ading en vironment , ” IEEE J. Sel. Areas Com- mun., vol. 21, no. 3, pp. 340–349, Apr . 2003. T arik Ait-Idir (S’06-M’07) was born in Rabat, Mo- rocco, in 1978. He recei ved the "diplôme d’ingéni eur d’état " in telecommunica tions from INPT , Rabat, and the Ph. D. degree in electric al engineering from T elecom Bretagne, Brest, France, in 2001, and 2006, respect i vely . He is currently an A ssistant Professor of wireless communicat ions at the Communicat ion Systems department, INPT , Rabat. He is also an adjunct researcher with Institut T elecom / T elecom Brete gne/LabSticc. From July 2001 to F ebruary 2003 he was with Ericsson. His research interests include PHY and cross-layer aspects of MIMO systems, rela y communi- catio ns, and dynamic spectrum m anagemen t. Dr . Ait-Idir has been on the techni cal progra m committee of sev eral IEEE conference s, includ ing ICC, WCNC, PIMRC, and VTC, and chaired some of their sessions. He has been a technical co-chair of the MIMO Systems Symposium at IWCMC 2009, and IWCMC 2010. Houda Chafnaji recei ved the "diplôme d’ingénie ur d’état " in telecommunic ations in 2004 from INPT , Rabat, Morocco, and the M.Sc. in telecommunic a- tions in 2006 from INRS, Montreal, Canada. She is current ly a PhD. candidate at T elecom-Bre tagne, Brest, France. Since 2006, she has been a lecturer at the department of Communicati ons Systems, INPT . Her research interests include w irele ss communi- catio ns with a focus on physical and MAC layers design, Hybrid ARQ, cooperati ve communicat ion, pack et combining, and performance ev aluation of wireless communicatio ns systems. Samir Saoudi (M’01-SM’09) was born in Rabat, Morocco, on November 28, 1963. He recei ved the "diplôme d’ingénie ur d’état" from ENST Bretagne, Brest, France, in 1987, the Ph.D. degree in telecom- municati ons from the ’Univ ersité de Ren nes-I’ in 1990, and the "Habilitati on à Diriger des Recherches en Sciences" in 1997. Since 1991, he has been with the Signal and Communications department, Institut T elecom / T elecom Brete gne/LabSti cc, where he is current ly a Professor . He is also with Univ ersité Européenne de Bretagne. In summer 2009, he has visited Orange Labs-T okyo. His research intere sts include speech and audio coding, non parametri c probabili ty density function estimati on, CDMA tech- niques, multiuser detectio n and MIMO techniques for UMTS and HSP A ap- plica tions. His teachi ng interests are signal processing, probabili ty , s tochast ic processes and speech processing. Dr . Saoudi supervised more than 20 Ph.D. Students. He is the author and/or coauthor of around eighty public ations. He has been the general chairman of the second International Symposium on Image/V ideo Comm unicat ions ov er fixed and mobile networks (ISIVC’04), and technic al co-chair of the MIMO systems symposium at IWCMC 2010.