LOW-DENSITY Parity-Check (LDPC) codes have received

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1 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 7, JULY Successve Maxmzaton for Systematc Desgn of Unversay Capacty Approachng Rate-Compatbe Sequences of LDPC Code Ensembes over Bnary-Input Output-Symmetrc Memoryess Channes Hamd Saeed, Member, IEEE, Hossen Pshro-Nk, Member, IEEE, and Amr H. Banhashem, Senor Member, IEEE Abstract A systematc constructon of capacty achevng owdensty party-check(ldpc) code ensembe sequences over the Bnary Erasure Channe (BEC) has been proposed by Saeed et a. based on a method, here referred to as Successve Maxmzaton (SM). In SM, the fracton of degree- nodes are successvey maxmzed startng from =2wth the constrant that the ensembe remans convergent over the channe. In ths paper, we propose SM to desgn unversay capacty approachng rate-compatbe LDPC code ensembe sequences over the genera cass of Bnary-Input Output-Symmetrc Memoryess (BIOSM) channes. Ths s acheved by frst generazng the SM method to other BIOSM channes to desgn a sequence of capacty approachng ensembes caed the parent sequence. The SM prncpe s then apped to each ensembe wthn the parent sequence, ths tme to desgn rate-compatbe puncturng schemes. As part of our resuts, we extend the stabty condton whch was prevousy derved for degree-2 varabe nodes to other varabe node degrees as we as to the case of ratecompatbe codes. Consequenty, we prove that usng the SM prncpe, one s abe to desgn unversay capacty achevng rate-compatbe LDPC code ensembe sequences over the BEC. Unke the prevous resuts n the terature, the proposed SM approach s naturay extendabe to other BIOSM channes. The performance of the rate-compatbe schemes desgned based on our method s comparabe to those desgned by optmzaton. Index Terms Low-densty party-check (LDPC) codes, ratecompatbe LDPC codes, capacty-achevng codes, capactyapproachng codes, systematc desgn of LDPC codes, stabty Paper approved by T.-K. Truong, the Edtor for Codng Theory and Technques of the IEEE Communcatons Socety. Manuscrpt receved Juy 14, 2010; revsed December 2, A premnary verson of ths work has been party presented at the Informaton Theory Workshop, June 2009, Voos, Greece, and n part at the Aerton Conference on Communcaton, Contro, and Computng, Sep Ths work was supported by the Natona Scence Foundaton under grants ECS and CCF H. Saeed was wth the Department of Eectrca and Computer Engneerng, Unversty of Massachusetts, Amherst, MA, USA. He s now wth the Wreess Innovatons Lab, Department of Eectrca and Computer Engneerng, Tarbat Modares Unversty, Tehran, Iran (e-ma: hsaeed@eee.org). H. Pshro-Nk s wth the Department of Eectrca and Computer Engneerng, Unversty of Massachusetts, Amherst, MA, USA (e-ma: pshro@ecs.umass.edu). A. H. Banhashem s wth the Department of Systems and Computer Engneerng, Careton Unversty, Ottawa, ON, Canada (e-ma: ahashem@sce.careton.ca). Dgta Object Identfer /TCOMM /11$25.00 c 2011 IEEE condton, bnary erasure channe (BEC), bnary symmetrc channe (BSC), addtve whte Gaussan nose (AWGN) channe. I. INTRODUCTION LOW-DENSITY Party-Check (LDPC) codes have receved much attenton n the past decade. Durng ths perod there have been great achevements n the area of desgnng LDPC code ensembes wth Beef Propagaton (BP) decodng whch exhbt an asymptotc performance practcay cose to the capacty over dfferent types of channes, ncudng the genera cass of Bnary-Input Output-Symmetrc Memoryess (BIOSM) channes [1]-[11]. In partcuar, for the Bnary Erasure Channe (BEC), the performance anayss and code desgn have been addressed n both the asymptotc regme [3]- [9] and for fnte bock engths [1], [2]. In [3] [5], Shokroah et a. proposed a scheme to desgn sequences of LDPC code ensembes over the BEC, whose performance s proved to acheve the capacty for suffcenty arge average check and varabe node degrees. A more genera category of capacty achevng sequences over the BEC were proposed n [12] [14]. Constructon and anayss of capacty achevng ensembe sequences of codes defned on graphs have aso been studed n [6] [9] for the BEC. A sequence of degree dstrbutons wth rate R s sad to be capacty achevng over the BEC f the threshods of the ensembes can be made arbtrary cose to 1 R, the capacty upper bound over the BEC, as the average check and varabe node degrees tend to nfnty. For BIOSM channes, t s easer to consder ensembes for a gven channe parameter nstead of a gven rate. The resuts however are easy extendabe to the case of fxed rate ensembes. We ca a sequence of degree dstrbutons capacty achevng over a BIOSM channe, f the rate of the ensembes wthn the sequence can be made arbtrary cose to the channe capacty whe mantanng the reabe communcaton. The desgn of provaby capacty achevng sequences over genera BIOSM channes s st an open probem. Another mportant probem of nterest n LDPC codes s to desgn rate-compatbe LDPC code schemes. In such a

2 1808 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 7, JULY 2011 scheme, startng from a gven prmary ensembe caed the parent code, we are nterested n obtanng a set of codes wth hgher transmsson rates, whch can provde reabe transmsson when the channe condton mproves, by puncturng the parent code. For rate-compatbty, the desgn must be such that for two consecutve rates, the code wth the hgher rate can be constructed by puncturng the code wth the ower rate. Startng from a parent code wth performance cose to capacty, the mportant chaenge n a rate-compatbe desgn s to aso keep the performance of the punctured codes cose to the capacty. More specfcay, f the parent code s chosen from a capacty achevng sequence, a punctured codes shoud be capacty achevng as average check node degree ncreases. To formuate the probem mathematcay, magne a parent code wth rate R n from a capacty achevng sequence whch can provde reabe transmsson over a channe wth parameter θ 0. Our am s to provde reabe transmsson over a set of channes wth parameters θ j, j = 1,..., J, whe ncreasng the rate by puncturng the parent code n a ratecompatbe fashon. For each θ j, j = 1,..., J, we need to choose a puncturng pattern that maxmzes the correspondng reabe transmsson rate R n,j.letc(θ j ) denote the capacty of the channe wth parameter θ j, and assume that θ <θ j and c(θ ) >c(θ j ) for >j. We ca a rate-compatbe scheme unversay capacty achevng, fm n R n,j = c(θ j ) for j =1,.., J. Anayss and desgn of rate-compatbe LDPC codes have been addressed asymptotcay n [15]-[19] and for fnte bock engths n [20]-[23]. It s worth mentonng here that Raptor codes [24] can aso acheve the capacty of the BEC at severa rates but n a dfferent framework than puncturng. Unke the BEC for whch amost a aspects of conventona (unpunctured) and rate-compatbe codes have been anaytcay nvestgated, for the genera famy of BIOSM channes, the contrbutons are mosty based on numerca methods and optmzaton. Ths usuay provdes tte nsght nto the desgn method. In ths respect, a fundamenta open probem s to prove the exstence of capacty achevng sequences of LDPC codes wth BP decodng over BIOSM channes as we as to systematcay construct such sequences. Ths can be seen as a sub-probem as we as a budng bock for the more genera probem of desgnng unversay capacty achevng rate-compatbe LDPC codng schemes. In [10], t has been shown that capacty approachng LDPC codes over BIOSM channes can be desgned usng optmzaton 1.Aess compex optmzaton-based desgn method over the Bnary- Input Addtve Whte Gaussan Nose (BIAWGN) channe has been proposed n [25]. Severa mportant anaytca propertes ncudng the so-caed stabty condton have been proven for BIOSM channes n [10], [11]. For rate-compatbe codes, t has been shown n [15], [17] that there s an upper bound on the puncturng rato of LDPC codes over BIOSM channes, above whch the code can not provde reabe transmsson for any channe parameter. Moreover, t has been shown that 1 We dstngush between capacty approachng and capacty achevng sequences. The former term s used when the performance of the ensembe sequence can be shown (probaby numercay) to approach capacty wthout any guarantee to acheve t. The atter term s used f the performance provaby tends to capacty as the average node degrees tend to nfnty. over the BEC, the random puncturng mantans the rato of rate to capacty at the same vaue as that of the parent code. Severa bounds on the performance of punctured LDPC codes have been derved n [19]. For the case of maxmum-kehood decodng, capacty achevng codes have been desgned based on puncturng n [18]. Among the resuts on the optmzatonbased desgn of rate-compatbe codes over BIOSM channes, we can menton [16] for the asymptotc regme and [20] [22] for fnte bock engths. In ths paper, we systematcay desgn sequences of unversay capacty approachng rate-compatbe LDPC code ensembes over BIOSM channes. We then provde some evdence suggestng that the desgned sequences coud n fact be unversay capacty achevng. Startng from the unpunctured case, we extend some of the propertes of capacty achevng sequences over the BEC [12], [13], to BIOSM channes. Among such propertes, ony the stabty condton [10] has been shown to be extendabe to BIOSM channes other than the BEC. We w anayze the case where the stabty condton s satsfed wth equaty,.e, the fracton of degree 2 edges (λ 2 ) s set equa to ts upper bound, and show that ths mposes an upper bound on the fracton of degree 3 edges (λ 3 ). Usng a smar approach for the other degrees, we propose Successve Maxmzaton (SM) of λ vaues as a systematc approach to desgn a sequence of LDPC code ensembes wth performance approachng the capacty as the average check node degree ncreases. For the rate-compatbe LDPC codes over BIOSM channes, we frst prove a property smar to the stabty condton. We show that for a gven parent code, there s an upper bound on the fracton of punctured degree-2 varabe nodes (Π 2 ) above whch the probabty of error of the punctured code s bounded away from zero and beow whch the probabty of error tends to zero f t s made suffcenty sma. We then consder the speca case of the BEC and show that smar upper bounds can be obtaned for varabe nodes of a degrees n addton to degree-2 nodes. Usng such upper bounds, we prove that appyng the SM prncpe resuts n a unversay capacty achevng rate-compatbe scheme over the BEC. Moreover, for such a scheme, f puncturng fractons are used to puncture the parent sequence (λ n,ρ n ) over the channe wth parameter θ j,wheres the varabe node degree, the vaues of Π n,j are ndependent of n. Ths resut s consstent wth the one obtaned n [15], [17] based on a competey dfferent approach. We then extend the resuts for the BEC to genera BIOSM channes, and show that the SM prncpe can be apped to puncturng fractons of varabe nodes to systematcay desgn a codng scheme whose performance unversay approaches the capacty n a rate-compatbe fashon. Ths proposes a sgnfcanty dfferent approach than the exstng optmzaton-based methods n the terature. Our numerca resuts ndcate that f the parent ensembe s chosen from the capacty approachng sequences desgned based on the SM prncpe, the performance of the resutng rate-compatbe schemes s smar to that of the exstng optmzaton-based resuts n the terature. Moreover, we show that for a sequence of parent code ensembes (λ n,ρ n ) desgned based on the SM prncpe, the vaues of puncturng Π n,j fractons Π n,j for degree 2 varabe nodes ( = 2) are ndependent from the parent ensembe (n) and ony depend

3 SAEEDI et a.: SUCCESSIVE MAXIMIZATION FOR THE SYSTEMATIC DESIGN OF UNIVERSALLY CAPACITY APPROACHING RATE-COMPATIBLE on the orgna channe parameter (θ 0 ) and the one for whch the puncturng pattern s desgned (θ j ). Our numerca resuts suggest that ths property may n fact hod for other vaues of. The mportance of ths property s that for a gven channe parameter θ j, the computed vaues of Π can unversay be apped to any ensembe desgned based on the SM method for a gven orgna channe parameter θ 0 wth an arbtrary check node dstrbuton. The paper organzaton s as foows. The next secton s devoted to notatons and some defntons. In Secton III, after a short revew on the constructon of capacty achevng sequences over the BEC, we expan the successve maxmzaton approach to devse capacty approachng sequences for other channes. Secton IV provdes some anaytca resuts reated to the proposed approach. In Secton V, we focus on the puncturng of a gven ensembe wthn a sequence that s desgned based on the SM methodoogy. We aso provde some propertes of rate-compatbe codes for the BEC and BIOSM channes. Moreover, we show that a smar SM prncpe to that of Secton III can be used to devse a unversay capacty approachng rate-compatbe scheme. In Secton VI, we show exampes of our desgns and Secton VII concudes the paper. The detas of the desgn agorthm and some numerca ssues are dscussed n the appendx. II. DEFINITIONS AND NOTATIONS In ths secton we present some defntons and propertes whch w be frequenty used throughout the paper. We many foow the notatons and defntons of [11], [17]. As our focus s on symmetrc channes and a BP decoder, throughout the paper, wthout oss of generaty, we assume that the aone code word s transmtted. Moreover, we assume that the messages n the BP agorthm are n the og-kehood rato doman. We represent a (λ, ρ) LDPC code ensembe wth ts edge-based check and varabe node degree dstrbutons as ρ(x) = D c =2 ρ x 1 and λ(x) = D v =2 λ x 1, wth constrants D c =2 ρ =1and D v =2 λ =1, where the coeffcent of x represents the fracton of edges connected to the nodes of degree +1, and D v and D c represent the maxmum varabe node degree and the maxmum check node degree, respectvey. Average check node and varabe node degrees are gven by: d c =1/( D c =2 ρ /) and d v =1/( D v =2 λ /), respectvey. The code rate R satsfes R =1 d v /d c. (1) We aso defne node-based degree dstrbutons as ρ(x) = Dc =2 ρ x 1 and λ(x) = D v =2 λ x 1, wth constrants Dc =2 ρ =1and D v =2 λ =1, where the coeffcent of x represents the fracton of nodes havng degree +1.Werepresent a BIOSM channe wth parameter θ by C(θ) and defne c(θ) as the Shannon capacty of that channe. We aso assume that the channe s physcay degraded when θ ncreases. For a sequence of degree dstrbutons (λ n (x),ρ n (x)), λ n and ρ n ndcate the th coeffcent of the nth member of the sequence for varabe node and check node degree dstrbutons, respectvey. Notaton T s used for the coeffcent of x 1 n the Tayor seres expanson of 1 ρ 1 (1 x) around x =0,.e., we have 1 ρ 1 (1 x) = =2 T x 1 (note that T 1 =0). Smar to [5], we mt our dscussons to check node degree dstrbutons for whch T s are postve. For exampe, check reguar ensembes exhbt such a property. Consder now the densty evouton n the beef propagaton agorthm for the channe C(θ), where we track the evouton of the nta channe densty P 0 throughout teratons n the asymptotc regme, where the bock ength tends to nfnty. 2 Based on [10], [11], Q, the probabty densty functon (densty) of the outgong message from check nodes at teraton can be wrtten as Q = Γ 1 ρ(γ(p 1 )), where P 1 s the densty of the message from teraton 1 enterng the check nodes and Γ s the check node operator defned n [10], [11]. Aso, P, the outgong densty from varabe nodes at teraton, can be wrtten as P = P 0 λ(q ),where s the convouton operaton, and the power of a densty n varabe node and check node degree dstrbutons has been defned n [11]. Note that there s a one-to-one correspondence between the densty P 0 and the channe parameter θ. Let P be a symmetrc densty (as defned n [10]). For such a densty, parameters P(P ) and P(P ) are defned by: and P(P )=0.5 P(P )= P (x)e ( x/2 +x/2) dx, P (x)e (x/2) dx. Parameter P(P ) s the probabty that the random varabe wth densty P s negatve. In the settng of ths paper, P(P ) s the probabty of error for the message wth densty P. Parameter P(P ) s caed the Bhattacharyya constant. For any densty P, P(P ) tends to zero f and ony f (ff) P(P ) tends to zero. Let p = P(P ) and q = P(Q ). Correspondng to the densty evouton equatons, we then have the foowng reatonshp [11]: p P(P 0 )λ(1 ρ(1 p 1 )). (2) It s mportant to note that for the BEC, (2) s satsfed wth equaty. Moreover, P(P 0 ) s equa to the channe erasure probabty. The foowng mportant nequates aso hod [11]: 2P(P ) P(P ) 2 P(P )(1 P(P )). (3) A gven degree dstrbuton (λ, ρ) s caed stabe ff there exsts ξ>0 such that f P(P ) <ξthen m P(P )=0. In that respect, t s proven n [10], [11] that f λ (0)ρ (1) > 1/P(P 0 ),thenp(p ) s bounded away form zero for every and f λ (0)ρ (1) < 1/P(P 0 ), then the ensembe s stabe. We ca an ensembe (λ, ρ) convergent over C(θ), f startng from the nta densty P 0, m P(P )=0.Thethreshod of an ensembe over C(θ) s the supremum vaue of θ for whch the ensembe s convergent. A sequence of degree dstrbutons (λ n,ρ n ) s caed capacty achevng over a BIOSM channe C(θ), f the correspondng ensembes are convergent over C(θ) and f ther rates R n tend to c(θ) for suffcenty arge average check node degrees as n tends to nfnty. 2 A the resuts presented n ths paper are for the asymptotc regme where the bock ength tends to nfnty.

4 1810 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 7, JULY 2011 Consder the (k +2)-tupe (λ 2,λ 3,..., λ k,d v,ρ(x); θ) whch corresponds to a degree dstrbuton (λ(x),ρ(x)) = ( k =2 λ x 1 +(1 k =2 λ )x Dv 1,ρ(x)) over C(θ) where D v > k, and 0 λ < 1, {2,..., k, D v }. We ca ths settng a code-channe par. Wth sght neggence, we ca a code-channe par convergent f the ensembe s convergent over the channe. We defne Λ 2 (ρ(x),θ) = {λ 2 : D v ;(λ 2,D v,ρ(x); θ) s convergent}. Smary, Λ 3 (ρ(x),θ) = {λ 3 : λ 2 Λ 2 (ρ, θ), D v ;(λ 2,λ 3,D v,ρ(x); θ) s convergent} and so on. We show the correspondng compement sets wth respect to [0,1], wth Λ C (ρ(x),θ). It can be verfed that these sets are contnuous for BIOSM channes. III. CAPACITY ACHIEVING SEQUENCES AND THE PRINCIPLE OF SUCCESSIVE MAXIMIZATION Capacty achevng sequences of LDPC code ensembes over the BEC were frst desgned by Shokroah et a. n [3] [5]. In [12] [14], the authors proposed new sequences of capacty achevng LDPC code ensembes over the BEC whch contaned Shokroah s sequences as a speca case. Unke Shokroah s sequences n whch a varabe node degrees from 2 to D v have to be present, the sequences of [12] [14] ony contan varabe node degrees from 2 to k<d v and degree D v,wherek s a strcty ncreasng functon of D v (and utmatey a functon of ρ(x) and θ). 3 For a gven ensembe (λ n,ρ n ) wthn the sequences proposed n [12] [14], we have: λ n = T /θ = λ n, 2 k, and λ n D v =1 k =2 λn, where D v s determned by θ and ρ(x) such that D v 1 =2 D v T θ< T. (4) =2 In the foowng, we generaze the defnton of λ n type of BIOSM channe as to any λ n T /P(P 0 ). (5) In the method proposed n [12] [14], the vaues of λ n are computed based on the foowng prncpe: Startng from =2, and by sequentay gong through the vaues of 2, set the vaue of λ n to a maxmum vaue λ n such that the ensembe remans convergent for suffcenty arge D v.forthe BEC, ths process resuts n λ n = λ n [13], [14]. Intutvey, the method of [12] [14] works snce based on (1), maxmzng 1 the rate s equvaent to maxmzng d v, whch n turn mpes that one shoud assgn hgher percentages to the ower degree coeffcents as far as the constructed ensembe remans convergent. In ths paper, we generaze the method of [12] [14] to BIOSM channes and refer to t as successve maxmzaton. At the core of ths method are the successve upper bounds λ n on λ n vaues. These upper bounds shoud fuf a threshod property smar to that of λ n over the BEC as foows: Suppose that λ n = λ n for every <k, then for any ensembe wth λ n k > λ n k, the probabty of error s bounded away from zero. Moreover, for any ensembe wth λ n k < λ n k,fthe 3 Shokroah s sequences correspond to those of [12] wth k(d v)=d v 1. probabty of error s made suffcenty sma, then t w tend to zero as the number of teratons tends to nfnty, regardess of the vaues of λ n,>k. Unke the BEC case, there s no proof that such upper bounds exst for other BIOSM channes (wth the excepton of the bound on λ 2 ),andevenfthey do exst, ther vaues may not be easy obtaned anaytcay (except for =2, where we have λ n 2 = λn 2 ). It shoud be noted that for the BIAWGN channe, the exstence of such an upper bound on λ 3 was demonstrated emprcay n [25]. In the next secton, we prove the exstence of a postve upper bound on λ 3 that fufs the aforementoned propertes over any BIOSM channe, and conjecture that smar upper bounds exst for other λ vaues. Ths makes t possbe to appy the SM prncpe as a desgn too for ensembe sequences. Numerca evdence presented n Secton VI confrms that the resutng sequences are at east capacty approachng and may n fact be capacty achevng. IV. SOME ANALYTICAL RESULTS ON UNPUNCTURED LDPC CODES We frst derve an nequaty whch heps n provng the statements of ths secton. For the code-channe par E (λ 2,λ 3,..., λ k,d v,ρ(x); θ), defne f(y) =P(P 0 )λ(1 ρ(1 y)). Usng (2), t s easy to see that the nequaty f(y) <y,for 0 <y P(P 0 ),sasuffcent condton for the convergence of E. Defnng x = 1 ρ(1 y), ths condton can be reformuated as P(P 0)λ(x) 1+ρ 1 (1 x) < 0, 0 <x 1 ρ(1 P(P 0)) < 1. Usng the Tayor seres expanson of 1 ρ 1 (1 x) and rearrangng the terms, we thus have the foowng suffcent condton for convergence: k (P(P 0)λ T )x 1 +(P(P 0)λ Dv T Dv )x Dv 1 =2 =k+1, =D v T x 1 < 0, 0 <x 1 ρ(1 P(P 0)) < 1. (6) Reca that the stabty condton n [10], [11] remans sent about the case where λ (0)ρ (1) s exacty equa to 1/P(P 0 ). Here, we prove that for ths case, a smar upper bound exsts for λ 3. Before provng the man resut, we prove some auxary emmas and propostons. Proposton 1: Consder a convergent code-channe par E 1 = (λ 2,λ 3,.., λ a,..., λ b,..., λ k,d v,ρ(x); θ 0 ). Defne E 2 =(λ 2,λ 3,.., λ a ε,..., λ b + ε,..., λ k,d v,ρ(x); θ 0 ) such that 0 <λ a ε,λ b + ε<1. Then the code-channe par E 2 s aso convergent. Proof : The proof s smar to the proof of Lemma 1 of [30] and s thus omtted. Coroary 1: Consder a code-channe par (λ 2,.., λ k,λ Dv,ρ(x); θ) and the set Λ 3 (ρ, θ) as defned n Secton II. Seect a λ 3 Λ 3.Thenany0 λ 3 <λ 3 aso beongs to Λ 3. In other words, the set Λ 3 s contnuous. Lemma 1 (Suffcent condton for stabty): Let (λ, ρ)

5 SAEEDI et a.: SUCCESSIVE MAXIMIZATION FOR THE SYSTEMATIC DESIGN OF UNIVERSALLY CAPACITY APPROACHING RATE-COMPATIBLE be a degree dstrbuton over C(θ) and P 0 be the nta channe densty. Let λ 2 = λ 2 = 1/(P(P 0 )ρ (1)), and λ 3 <λ 3 = ρ (1)/(2P(P 0 )ρ (1) 3 ) (note that the rght hand sde s strcty postve), then the ensembe s stabe,.e., there exsts ξ>0 such that f P(P ) <ξ,thenm P(P )=0. Proof : Startng from (2) and wrtng the Tayor expanson of the rght hand sde of the nequaty at zero, we have: p P(P 0)[λ (0)ρ (1)p 1 +( λ (0)ρ (1) + λ (0)ρ 2 (1))p 2 1] +O(p 3 1 ). Let g(t) = ρ (1)/ρ (1) + P(P 0 )2tρ 2 (1). Usng λ (0)ρ (1) = 1/P(P 0 ), we thus have: p p 1 +P(P 0 )[ λ (0)ρ (1)+λ (0)ρ 2 (1)]p 2 1+O(p 3 1) = p 1 +[ ρ (1)/ρ (1) + P(P 0 )2λ 3 ρ 2 (1)]p O(p 3 1) = p 1 +g(λ 3)p 2 1+O(p 3 1) p 1 +(g(λ 3)+δ)p 2 1 Δ = h(p 1 ), where the ast nequaty s vad for sma enough p 1, and δ s a sma postve number. Snce λ 3 < ρ (1)/(2P(P 0 )ρ (1) 3 ),theng(λ 3 ) < 0. We choose a postve constant κ, such that f p 1 <κ,thenδ can be chosen sma enough such that g(λ 3 )+δ, the coeffcent of p 2 1, s negatve. Ths means that p h(p 1 ) <p 1 for 0 <p 1 <κ. Note that for t [0,κ], the ony fxed pont of functon h(t) s at t =0.Snceh(t) <tfor t (0,κ), we can see that m (p )=0. Now f we et ξ = κ 2 /4, usng (3), f P(P ) <ξ,thenm (p )=0. Ths means that m P(P )=0. Lemma 2: Consder the code-channe par (λ 2,λ 3,D v,ρ(x); θ) such that λ 3 <λ 3. Then for arge enough D v, the par s convergent. Proof : The suffcent convergence condton of (6) for the gven ensembe reduces to (P(P 0)λ 3 T 3)x 2 +(P(P 0)λ Dv T Dv )x Dv 1 =4, =D v T x 1 = x 2 [(P(P 0 )λ 3 T 3 )+(P(P 0 )λ Dv T Dv )x Dv 3 ] T x 1 < 0; 0 <x 1 ρ(1 P(P 0 )) < 1. Π. The overa puncturng fracton Π can then be expressed =4, =D v as Π= D v Note that the frst term n the coeffcent of x 2 =2 Π λ p p where {λ } s the node-based degree s negatve dstrbuton of varabe nodes for the parent ensembe. based on the emma assumpton and snce 0 <x<1, ths term can be made domnant for suffcenty arge D v,makng In many stuatons, t s necessary to obtan more than one rate by puncturng. In ths case, for a smpe mpementaton, the term wth x 2 negatve. A the other terms ncudng the puncturng pattern shoud be n such a way that for 2 x, 3 are aso negatve as T > 0,. Therefore, the consecutve rates, the punctured code wth a hgher rate can convergence condton hods for the gven ensembe. be constructed by puncturng the code wth the ower rate. A puncturng pattern wth ths property s caed rate-compatbe. Lemma 3: Let (λ, ρ) be a convergent degree dstrbuton Let the set of channe parameters θ j be ordered reversey over C(θ). Then we necessary have λ 3 λ U 3 where by channe degradaton (.e., θ 0 s for the worst channe λ U 3 =3/(d c(1 c(θ))) (3/2)λ Note that other (possby tghter) upper bounds have recenty been proposed n [27]. Proof : For any convergent ensembe, we must have 1 R c(θ). Usng (1), we thus have d v 1/(dc (1 c(θ))). Aso, λ 2 /2+λ 3 /3 D v 2 λ / = 1/d v. Therefore, λ 2 /2+λ 3 /3 1/(d c (1 c(θ))), whch reduces to λ 3 λ U 3. Theorem 1: Consder the code-channe par (λ 2,λ 3,D v,ρ(x); θ) where D v can be arbtrary arge. There exsts a threshod vaue λ 3 n the nterva [λ 3,λ U 3 ] such that f λ 3 < λ 3, the ensembe s convergent for a suffcenty arge vaue of D v and f λ 3 > λ 3, the probabty of error s bounded away from zero regardess of the vaue of D v. Proof : Defne λ 3 = nf(λ C 3 (ρ(x),θ)). Based on ths defnton, λ 3 > λ 3, the probabty of error of the resutng ensembe (λ 2,λ 3,D v,ρ(x)) over C(θ) s bounded away from zero no matter how arge D v s. Aso, there exsts an arbtrary sma ε 0, such that (λ 2, λ 3 ε,d v,ρ(x); θ) s convergent for a suffcenty arge vaue of D v.nowbasedon Proposton 1, f such a code-channe par s convergent, any other par (λ 2,λ 3,D v,ρ(x); θ) for whch λ 3 < λ 3 ε s aso convergent. Based on Lemma 2, we know that f λ 3 <λ 3, (λ 2,λ 3,D v,ρ(x),θ) s convergent for suffcenty arge D v. Therefore λ 3 λ 3. From Lemma 3, we know that λ 3 λ U 3. Ths proves the theorem. We expect the resut of Theorem 1 to be generazabe to λ k,k > 3, fλ = λ =nf(λ C (ρ(x),θ)), 2 k 1. Ths, however, remans to be proved. V. UNIVERSALLY CAPACITY APPROACHING RATE-COMPATIBLE LDPC CODES In Secton III, we dscussed the desgn of sequences of degree dstrbutons (λ n,ρ n ) whose rates approach the capacty as n tends to nfnty. In ths secton, we consder the probem of puncturng a degree dstrbuton for a gven n. For smpcty, we sometme drop the ndex n and refer to the ensembe as the parent ensembe. We use the notatons (λ p,ρ) and R p for the parent ensembe and ts rate, respectvey. We show the fracton of the punctured bts (varabe nodes) by Π. The resutng code rate n ths case s equa to R p /(1 Π). If the puncturng s performed randomy, we refer to t as random puncturng. Otherwse, the puncturng s caed ntentona [16]. In ntentona puncturng, varabe nodes of degree can potentay have dfferent puncturng fractons condton whch corresponds to the parent code). For any C(θ j ), consder the set Φ j = {Π j, 2 D v}. 5 For a 5 For λ =0, we assume Π =0.

6 1812 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 7, JULY 2011 rate-compatbe puncturng scheme, we must have Π m Π n for any m<nand any. To anayze the asymptotc behavor of a punctured ensembe, we mode the puncturng of LDPC codes over a channe C(θ) as the transmsson of the unpunctured bts over C(θ) whe sendng the punctured bts on an erasure channe wth erasure probabty of 1. Let E be the set of a edges n the graph. Aso et E punc be the set of edges n the graph whch are connected to the varabe nodes of degree whch are punctured. Aso, et E punc be the unon of sets E punc. Smary, defne E un and E un for unpunctured edges. We defne λ punc (x) = λ punc x 1, where λ punc = Epunc E punc. Notaton. denotes the cardnaty of the set. We aso defne φ punc as the fracton of punctured edges: φ punc = Epunc E. The poynoma λ un (x) and the fracton φ un can be defned smary for unpunctured edges. Based on the above defntons, we have: Π = + E un = φ punc λ punc E punc E punc E punc E λ p = λpunc φ punc λ p, (7) + φ un λ un = λ p. (8) We can now derve the densty evouton equatons for our settng. Smar to the prevous secton, et Q be the probabty densty functon of the outgong message of the check nodes at teraton. We defne P punc and P un as the densty at the output of the punctured and unpunctured varabe nodes at teraton, respectvey. We then have P punc P un = P punc o λ punc (Q ), = P un o λ un (Q ), (9) P = φpunc P punc + φ un P un, Q =Γ 1 (ρ(γ(p 1))), n whch P punc 0 = Δ 0 where Δ x s the Drac deta functon at x [11]. Consder a sequence of degree dstrbutons (λ n (x),ρ n (x)). Consder aso a set of channes wth parameters θ j,j = 0, 1,..., J, ordered ncreasngy by ther quaty. Now assume that the parent ensembe sequence (λ n (x),ρ n (x)) s punctured by the set φ n,j = {Π n,j, 2 D v } to create hgher rate ensembe sequences that are convergent over the correspondng channes. Ths scheme s unversay capacty achevng f m n R n,j = c(θ j ) for a vaues of j. A unversay capacty achevng scheme s caed rate-compatbe f the puncturng patterns φ n,j are ratecompatbe for every vaue of n. In the foowng, we prove a theorem for puncturng a gven degree dstrbuton. Let (λ p,ρ) be a convergent parent ensembe over a channe wth parameter θ 0. The code-channe par (λ p,ρ; θ) s convergent for any θ θ 0.LetP 0 be the channe densty assocated wth θ. Wedefne parameter Π 2, correspondng to the parent code-channe par, as: Π 2 = [1 P(P 0)ρ (1)λ p 2 ] [1 P(P 0 )]ρ (1)λ p. (10) 2 Note that f the par s stabe,.e., f λ 2 <λ 2,then Π 2 > 0. The foowng emma can be easy proved based on (7) and (8). Lemma 4: Let θ = ρ (1)(φ punc p punc 0 λ punc 2 + φ un p un 0 λun 2 ), (11) where p un 0 = P(P0 un ) and ppunc 0 = P(P punc 0 )=1.Wehave Π 2 Π 2 ff θ 1. Theorem 2: Let (λ p,ρ) be a parent ensembe convergent over C(θ) wth λ p 2 =0. Suppose that ths code s punctured basedonthesetφ={π ; =2,...,D v } (note that C(θ) has a one-to-one correspondence wth the channe densty P 0 ). There exsts a threshod vaue Π 2, gven by (10), such that f Π 2 > Π 2, then for any, P(P punc ) and P(P un ) are bounded away from zero and f Π 2 < Π 2, there exsts a strcty postve constant ξ such that f P(P punc ) <ξ,and P(P un ) < ξ for some, thenm P(P punc ) = 0 and m P(P un )=0. Proof :[Suffcency] (Π 2 < Π 2 ): Let p punc,p un,p punc 0,p un 0,p and q denote P(P punc ), P(P un ), P(P punc 0 ), P(P0 un ), P(P ) and P(Q ), receptvey. Now for the densty evouton equatons, we have: = p punc 0 λ punc (q ), p punc p un p = φ punc p punc = p un 0 λun (q ), (12) + φ un p un, q 1 ρ(1 p 1 ). Combnng the equatons, we obtan: q +1 1 ρ(1 φ punc p punc 0 λ punc (q ) φ un p un 0 λun (q )). (13) By expandng the above formua nto Tayor seres at zero we have: q +1 ρ (1)(φ punc p punc 0 λ punc 2 q + φ un p un 0 λun 2 q )+O(q 2 ), or q +1 θq + O(q 2 ), where θ s defned n (11). Based on Lemma 4, θ<1 and thus we can fnd η>0 such that θ + η<1. Note that snce P(P punc ) <ξ,andp(p un ) <ξforanarbtrary sma ξ, basedon(3),p un and p punc are aso arbtrary sma. Ths makes p arbtrary sma based on (12). Snce 1 ρ(1 x) s a strcty ncreasng functon and m x 0 1 ρ(1 x) =0, based on (12), we can make q arbtrary sma f we choose sma enough ξ. Forsuffcenty sma q, we can see that q +1 < (θ + η)q <q. Wth an argument smar to the one used n the proof of Lemma 1, we thus have m q =0. Based on (12), m p punc = m p un =0, whch mpes m P(P punc )=0,andm P(P un )=0. [Necessty](Π 2 > Π 2 ): To prove the necessty, t s enough to show that the probabty of error of Q n,(p(q n )) s bounded away from zero as n tends to nfnty. Suppose that P(Q n ) tends to zero. Then there exsts an teraton for whch P(Q )=ε/2, where we can choose ε arbtrary sma. For the next teraton, smar to the proof of Theorem 5 of [10], assume that we repace ths densty wth the BEC densty of the same probabty of error,.e.,

7 SAEEDI et a.: SUCCESSIVE MAXIMIZATION FOR THE SYSTEMATIC DESIGN OF UNIVERSALLY CAPACITY APPROACHING RATE-COMPATIBLE Υ 0 = εδ 0 +(1 ε)δ. Now, f we ndcate the resutng densty, at the output of varabe nodes, for the next teraton by Υ 1,wehave: Υ 1 = ε(λ punc 2 φ punc ρ (1)P punc 0 + λ un 2 φun ρ (1)P0 un )+ Δ (1 ε(λ punc 2 φ punc ρ (1) + λ un 2 φun ρ (1))) + O(ε 2 ). It can be verfed that after teratons, we obtan: Υ = ε[λ punc 2 φ punc ρ (1)P punc 0 + λ un 2 φ un ρ (1)P0 un ] + Δ (1 ε[λ punc 2 φ punc ρ (1) + λ un 2 φun ρ (1)] )+O(ε 2 ), where denotes the -fod convouton. Defne the sequence of denstes D =[λ punc 2 φ punc ρ (1)P punc 0 + λ un 2 φun ρ (1)P0 un ] /D, where D = [λ punc 2 φ punc ρ (1)P punc 0 + λ un 2 φ un ρ (1)P0 un ] dx. Usng the propertes of P( ) n Secton II, t s easy to verfy that the vaue of P(D ) s equa to θ /D. We cam that there exsts an n for whch, f n, wehave1 < θ2 4D. To see ths, note that f 4D 1, snce based on Lemma 4 θ>1, the nequaty s trva. Otherwse, to satsfy the nequaty, we n 4D set n =. Based on (3), 2nθ ( θ D )2 4P(D )(1 P(D )) < 4P(D ). Therefore, for n, we obtan 1 D < P(D ). For = n, we can wrte P(Υ n ) = εp([λ punc 2 φ punc ρ (1)P punc 0 + λ un 2 φ un ρ (1)P0 un ] n )+O(ε 2 ). For suffcenty sma ε, wethenhavep(υ n )=εdp(d n )+ O(ε 2 ) > ε > ε/2 = P(Υ 0 ). Now snce Q s physcay degraded wth respect to Υ 0 (see [11], Lemma 4.78), Q +n s physcay degraded wth respect to Υ n (see [11], Lemma 4.80). Therefore, P(Q +n ) P(Υ n ) > P(Υ 0 )=P(Q ).Ths contradcts the fact that the probabty of error s a decreasng functon of the number of teratons. In other words, ε can not become arbtrary sma. Ths competes the proof. Ths property s smar to the stabty condton [10] for parent LDPC codes whch provdes an upper bound on λ 2. Coroary 2 (Independency of Π 2 from n for puncturng schemes wth Π 2 = Π 2 ): Consder a sequence of ensembes (λ n (x),ρ n (x)) whch are convergent over C(θ 0 ) and et P0 0 be the assocated channe densty. Now consder an mproved channe C(θ j ),j>0 and et P j 0 be the assocated channe densty. If for any ensembe wthn the sequence, the vaue of λ 2 satsfes the stabty condton correspondng to θ 0 wth equaty,.e., f λ n 2 = λn 2, then the vaue of the upper bound Π 2 correspondng to θ j obtaned n Theorem 2, s ndependent of n (n fact, t s ndependent of the parent ensembe sequence (λ n,ρ n )). Proof : We have λ n 2 vaue n (10), we obtan Π 2 = [1 P(P j 0 )/P(P 0 0 )] ndependent of n. = 1/(P(P p 0 )ρ n (1)). Repacng ths [1 P(P j 0 )]/P(P 0 0 ), whch s Coroary 3 (Rate-compatbty of Π 2 for puncturng schemes wth Π 2 = Π 2 ): Consder a sequence of ensembes (λ n (x),ρ n (x)) whch are convergent over C(θ 0 ) and et P0 0 be the assocated channe densty. Now consder an mproved channe C(θ j ),j 0 and et P j 0 be the assocated channe densty. If for any ensembe wthn the sequence, the vaue of λ 2 satsfes the stabty condton correspondng to θ 0 wth equaty,.e., f λ n 2 = λn 2, then the vaue of the upper bound Π 2 s a decreasng functon of θ j. Proof : From Coroary 2, we have Π 2 = [1 P(P j 0 )/P(P 0 0 )] [1 P(P j 0 )]/P(P 0 0). It s easy to check that functon f(x) = [1 x/p(p 0 0 )] [1 x]/p(p0 0) s a decreasng functon of x for x (0, 1). Ths together wth the fact that P(P j 0 ) s an ncreasng functon of θj proves that Π 2 s a decreasng functon of θ j. In [12], [13], smar upper bounds to that of stabty condton were obtaned for other varabe node degrees over the BEC. In the foowng, we prove a smar resut for the case of rate-compatbe codes over the BEC. Equaton (13) can be rewrtten as foows for the case of the BEC: q +1 =1 ρ(1 φ punc p punc 0 λ punc (q ) φ un p un 0 λ un (q )) = g(q ). (14) Smar to the unpunctured case, the convergence condton can be rewrtten as g(y) <y,0 <y 1 ρ(φ punc p punc 0 + φ un p un 0 ). Note that the above equaton s a necessary and suffcent condton for convergence. Modfyng the nequaty we obtan: φ punc p punc 0 λ punc (y)+φ un p un 0 λ un (y) 1+ρ 1 (1 y) < 0. By expandng 1 ρ 1 (1 y) nto Tayor seres and rearrangng the terms we have D v =2 (φ punc p punc 0 λ punc =D v+1 + φ un p un 0 λun T )y 1 T y 1 < 0. (15) Now based on Theorem 2, there s an upper bound Π 2 on the vaue of Π 2. Settng Π 2 = Π 2 s equvaent to havng θ =1. Puttng ths together wth T 2 =1/ρ (1), makes the coeffcent of y n (15) equa to zero. Now to have convergence, the nequaty must hod for a vaues of y ncudng those cose to zero. For those vaues, the domnant term s the term wth degree 2. Ths mpes that for convergence, we must have (φ punc p punc 0 λ punc 3 + φ un p un 0 λ un 3 ) T 3. Ths mposes an upper bound on the vaue of Π 3 above whch the probabty of erasure s bounded away from zero and beow whch the ensembe s convergent f the probabty of erasure s made suffcenty sma. The same method can be apped for other vaues of. Ths s expaned n the foowng proposton. Proposton 2: Let(λ p,ρ) be a convergent parent ensembe over the BEC wth channe parameter ε 0. Suppose that the parent ensembe s punctured to be used over a channe wth parameter ε j < ε 0. Let p un 0 be the Bhattacharyya

8 1814 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 7, JULY 2011 constant for ths channe,.e., p un 0 = ε j. Aso assume that λ p =0, 2 n D v.defne Π = 1 pun 0 λp /T (1 p un 0 )λp /T. (16) Then f Π = Π for 2 < n, there exsts an upper bound Π n = 1 pun 0 λp n /Tn (1 p un 0 )λp n/t n on Π n above whch the resutng punctured ensembe s not convergent over C(ε j ) and beow whch the ensembe s convergent over C(ε j ) f the probabty of erasure can be made suffcenty sma. 6 Proof : As dscussed before, to have a threshod upper bound on Π n, terms wth degree <nn (15) have to be removed. In order to do so, one has to set (φ punc p punc 0 λ punc + φ un p un 0 λ un )=T, 2 <n. Usng (7) and (8) we can see that ths s equvaent to havng Π = 1 pun 0 λp /T (1 p un =Π 0 )λp /T for 2 <n. Ths competes the proof. We now woud ke to prove that the constructon of unversay capacty achevng rate-compatbe LDPC codes over the BEC can be acheved by appyng the SM prncpe to the vaues of Π,.e., startng from a parent sequence and for each ensembe member of the sequence, we maxmze Π 2 as far as the ensembe remans convergent and contnue ths procedure successvey for other Π vaues. Ths w be performed for each of the J target channe parameters and we demonstrate that the resutng puncturng patterns are n fact rate-compatbe. We aso show that f the orgna parent sequence s capacty achevng, so w be a the J sequences of punctured ensembes. Theorem 3: Consder a capacty achevng parent ensembe sequence (λ n,ρ n ) over the BEC wth parameter ε 0, constructed based on the method of [12]. For the set of channe erasure vaues ε j (ε 1 >ε 2 >... > ε J ), we puncture each ensembe wthn the parent sequence based on the SM prncpe. The resutng scheme s then unversay capacty achevng rate-compatbe. Proof : For a gven n, assume that the ensembe ncudes consttuent varabe node degrees 2 to k and the maxmum varabe node degree D v.for2 k, the vaues of Π n,j resutng from the SM prncpe can be obtaned based on Proposton 2. Aso note that snce the ensembe s desgned based on [12], we have λ n n = T /ε0 for 2 k. Repacng the vaues of λ n n (16), we have: Π n,j = 1 εj /ε 0 (1 ε j )/ε = ε0 ε j 0 (1 ε j ) P (j) ;2 k, 1 j J, n, (17) whch s ony a functon of j.weasosetπ n,j D v = P (j), n, j. In ths case, we have (for smpfcaton, the ndces n and j are dropped for the puncturng-reated parameters): φ punc p punc 0 λ punc D v +φ un p un 0 λ un D v = λ n D v P (j) +ε j λ n D v (1 P (j) ) = ε 0 λ n D v = ε 0 (1 k T /ε 0 )=ε 0 =2 6 Note that for =2,wehaveΠ 2 = Π 2. k T, (18) =2 where we use (7), (8) and (17) for the frst equaty and (17) for the second equaty. To demonstrate that the resutng scheme s convergent over C(ε j ), t s enough to show that (15) hods. In (15), after appyng the SM prncpe, the terms wth =2to k w a be equa to zero, and the terms wth >D v are equa to T y 1 and w be negatve. We show the sum of the remanng terms by S and w have: for the gven puncturng fractons Π n,j D v 1 S =( D v 1 ( =k+1 =k+1 T +ε 0 T )y 1 +((ε 0 k T ) T Dv )y Dv 1 < =2 k D v T T Dv )y Dv 1 =(ε 0 T )y Dv 1 < 0, =2 where the frst equaty s obtaned based on (18), the frst nequaty s a resut of y 1 >y Dv 1 for <D v and 0 <y< 1, and the ast nequaty hods based on (4). Ths competes the convergence proof. Now we prove that Note that Π n,j = D v J. Wethenhave =2 R n,j /c(ε j )=R n,0 /c(ε 0 ). (19) =2 λn Πn,j =Π n,j ;2 D v, 1 j R n,j c(ε j ) = Rn,0 /(1 Π n,j ) 1 ε j = Rn,0 1 ε 0 = Rn,0 c(ε 0 ), where the 2nd equaty s obtaned based on (17). Ths proves (19). Now snce the parent ensembe s capacty achevng, m n R n,0 = c(ε 0 ). Based on (19), ths mpes that m n R n,j = c(ε j ) for any j. Ths proves the unversay capacty achevng property. To see the rate-compatbty, smar to the argument n Coroary 3, one can see that Π n,j n (17) s a decreasng functon of ε j (ncreasng functon of channe quaty) for any n. Therefore, for ε m <ε k,wehave Π n,k > Π n,m ; {2,..., k, D v }. Ths competes the proof. Ths resut s consstent wth the one obtaned n [17] statng that random puncturng of a parent ensembe over the BEC preserves the dstance to capacty. The approach taken n [17] s, however, dfferent and s based on the fact that one can mode the puncturng of an ensembe over the BEC as the concatenaton of the orgna BEC channe wth another BEC channe wth erasure rate equa to puncturng. Smar to the fatness condton, the approach of [17] s not extendabe to other BIOSM channes. The mportance of our approach s that n prncpe, t may be extendabe to other BIOSM channes where we can expect that appyng the SM prncpe to compute Π vaues, mght aso resut n (a scheme performng cose to) a unversay capacty achevng rate-compatbe scheme. Unke the BEC case, however, the upper bounds on Π have to be estmated numercay (smar to the procedure we use to compute the upper bounds of λ, > 2, for the unpunctured case) except for Π 2 whose upper bound s gven by Theorem 2. Appyng ths procedure to the capacty approachng ensembes desgned based on the method of Secton III as parent ensembes, we have

9 SAEEDI et a.: SUCCESSIVE MAXIMIZATION FOR THE SYSTEMATIC DESIGN OF UNIVERSALLY CAPACITY APPROACHING RATE-COMPATIBLE λ 3 U λ ~ 3 * λ 3 TABLE I PERFORMANCE OF A CHECK-REGULAR SEQUENCE DESIGN BASED ON THE SM METHOD OVER A BIAWGN CHANNEL WITH σ =.9557 D c R AW GN /c(.9557) k Fg. 1. The vaues of λ U 3,λ 3 and λ 3 vs. D c for a check-reguar ensembe over the BIAWGN channe wth σ = n-fact been abe to desgn unversay capacty approachng rate-compatbe ensembes over other BIOSM channes. It s mportant to note that the vaues of Π j,n n Theorem 3 do not depend on and n. Whe ndependency of s a speca property for the BEC, based on Coroary 2 these vaues are ndependent from n for =2over any BIOSM channe. Our numerca resuts show that for a gven >2, = D v and j, the vaues of Π j,n are very cose for dfferent vaues of n, suggestng a genera ndependency from n. VI. DESIGN EXAMPLES We frst present a pot of the upper and ower bounds on λ 3 reated to Theorem 1, namey λ U 3 and λ 3 as we as the numercay cacuated vaues for λ 3 for dfferent vaues of D c. The resuts are for check-reguar ensembes over a BIAWGN channe wth the nose standard devaton of σ =.9574 (capacty=1/2). As can be seen n Fg. 1, λ 3 s a strcty decreasng functon of D c smar to λ 2. Exampe 1: We appy our method to desgn ensembes for a BIAWGN channe wth capacty 1/2 (σ =.9574). As a pont of reference, we consder the 4th ensembe of Tabe II n [10] wth maxmum varabe node degree of 50, refereed to as C AW GN. Ths ensembe has the best performance among the rate one-haf ensembes desgned n [10] for the BIAWGN channe. The check node degree dstrbuton of ths ensembe s ρ AW GN (x) =.33620x x x 10. The threshod of ths ensembe s σ = The capacty of a channe wth σ =.9718 s equa to.5045, mpyng that the rate of ths ensembe s 99.1% of the capacty. Keepng the same check node dstrbuton, and settng k, the number of consttuent varabe node degrees to 24, we desgn the foowng varabe node degree dstrbuton based on the SM method: 7 λ(x) =.1826x x x x x x x x x x x x x x x x x x x x The detas of the desgn agorthm and some numerca ssues are dscussed n the appendx. D c.0027x x x x x 59. The rate of ths ensembe s whch s 99.0% of the capacty, showng amost the same dstance to capacty as C AW GN. The dsadvantage of ths ensembe compared to C AW GN, s havng a arger maxmum varabe node degree and arger number of consttuent varabe node degrees. The average check node degree for ths ensembe s about We apped the SM method to desgn a check-reguar ensembe wth D c = 10 (wth the same vaues for σ and k), and we were abe to desgn an ensembe whose rate was aso 99.0% of the capacty. Ths suggests that at east for the desgns based on SM, the mportant factor that determnes the ensembe performance, s the average check node degree rather than the actua check node degree dstrbuton. In other words, no optmzaton on the check node sde woud be necessary. Exampe 2: Consder the foowng sequence desgn of check-reguar ensembes for channe parameter σ = We start wth D c =5and k =3,andforD c > 5, weset k =2 Dc Ths means that the number of consttuent varabe node degrees for an ensembe wth check node degree D c s roughy twce that of an ensembe wth check node degree D c 1. As can be seen n Tabe I, the performance of the ensembes consstenty mproves as the average check node degree ncreases. The performance of the ensembe wth D c =10n Tabe I s sghty ess than 99% of the capacty. Exampe 3: For the Bnary Symmetrc Channe (BSC), we consder the ensembe C BSC desgned based on optmzaton n Exampe 2 of [10]. Ths rate one-haf ensembe has check node degree dstrbuton ρ BSC (x) =.25x x 10 and threshod θ =.106. Ths mpes that for check node degree dstrbuton ρ BSC (x) and channe parameter θ =.106, the best achevabe rate based on optmzaton s 0.5. We now appy the SM method to desgn an ensembe wth the same check node degree dstrbuton and channe parameter. The desgned ensembe has the foowng varabe node degree dstrbuton where we have mted the number of consttuent varabe node degrees k to 22: λ(x) =.1666x x x x x x x x x x x x x x x x x x x x x x 65. The rate of ths ensembe s.4988 whch s very cose

10 1816 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 7, JULY 2011 R Number of consttuent varabe nodes (k) Fg. 2. The rate of the ensembe desgned based on the SM method over the BSC wth channe cross over probabty.106 and check node dstrbuton ρ BSC (x) versus the number of consttuent varabe node degrees. R/c σ Fg. 3. The rato of rate to capacty (R/C) for the rate-compatbe ensembe sequence of Tabe II. to.5, the best achevabe rate based on optmzaton. The vaues of k and D v for the desgned ensembe are aso cose to (and smaer than) that of ensembe C BSC. Ths code s abe to acheve 97.4% of the capacty. Ths exampe suggests that the speed of convergence to capacty wth respect to the average check node degree s faster for the BIAWGN channe compared to the BSC. The speed of convergence to capacty s consderaby hgher for the BEC. For the capacty vaue of 1/2 as an exampe, wth D c =10, one s abe to acheve 99.8% of the capacty of the BEC [12]. One advantage of our method s ts smpe mpementaton. Takng advantage of ths property, we nvestgate the effect of the number of consttuent varabe node degrees k on the achevabe code rate. We agan consder the check node degree dstrbuton ρ BSC (x) and for dfferent vaues of k from 3 to 24, desgn ensembes usng the SM method. In Fg. 2, we have potted the rate of the desgned ensembes versus k. As can be seen, for k > 22, the curve starts to saturate, mpyng that there s not any advantage of choosng k greater than about 22. The vaues of maxmum varabe node degrees for the desgned ensembes range from 23 to 81. For the rate-compatbe codes, we consder the sequence of Tabe I and puncture the frst three ensembes for a set D c = 7 D c = 6 D c = of four channes wth nose powers smaer than that of the parent ensembe. The detas are provded n Tabe II, where we have the puncturng poynoma Π(x) = D v =2 Π x 1 to represent the puncturng fractons n the ast four coumns. In Fg. 3, we have potted the rato R j /c(θ j ) of the ensembes of Tabe II versus the channe nose standard devaton σ. As can be seen, the performance of the punctured codes for a gven parent ensembe s smar to or better than the parent ensembe. In fact, we expect the punctured ensembe to perform amost the same as the parent ensembe, smar to the case of the BEC. To justfy the mprovement resutng from puncturng, we note that athough the parent ensembes have been constructed based on the SM method, for fnte vaues of D c, they are not necessary optma n that they may not provde the best possbe rate for the gven channe parameter. Ths eaves the door open for further mprovement wth puncturng. From Fg. 3, t s aso observed that for any gven channe parameter, the performance of punctured ensembes approaches the capacty as the average check node degree ncreases. Based on Tabe II, the desgned sequence aso fufs the rate-compatbty property. Note however that unke, for exampe the approach of [16], we dd not mpose any constrant to guarantee rate-compatbty and our emprca resuts suggest that ths property s nherent n the proposed method. For the case of Π 2, we anaytcay proved ths fact n Coroary 3. To compare the performance of the schemes desgned based on the SM prncpe and those obtaned by optmzaton, we consder the ensembe used n [16] as a reference. Ths ensembe (C) has been optmzed for the rate one haf and has a threshod of σ = We can assume that ensembe C has been optmzed for the hghest rate when the channe parameter σ s set to The degree dstrbuton of C s: λ C (x) =.25105x x x x 9, ρ C (x) =.63676x x 7. Keepng the check node degree dstrbuton of ensembe C ntact, we desgn an ensembe C SM wth the same number of consttuent varabe nodes usng the SM method: λ SM (x) =.2717x x x x 9. We then appy the SM method agan, ths tme to puncture C SM. The puncturng poynomas for the same four channes consdered n Tabe II are gven n Tabe III. The dstance to capacty (n db) for the parent ensembe and ts punctured versons s reported n Fg. 4. As can be seen n Fg. 4, the scheme performs very cosey to the scheme obtaned by optmzaton-based puncturng of the ensembe C. In fact, the proposed scheme even sghty outperforms the scheme of [16] on channes wth σ =.6300 and σ = The proposed scheme performs nferor ony on the best channe parameter (σ =.4675) and even for ths channe parameter, the performance gap s ess than.08 db. 8 We have aso demonstrated the performance of random puncturng of the ensembe C for comparson. Aso note agan that unke [16], we dd 8 Note that our parent code tsef performs cose to.1db worse than C and the gap n performance s aways ess than ths gap for dfferent puncturng rates.

11 SAEEDI et a.: SUCCESSIVE MAXIMIZATION FOR THE SYSTEMATIC DESIGN OF UNIVERSALLY CAPACITY APPROACHING RATE-COMPATIBLE TABLE II THE VALUES OF Π USED TO PUNCTURE THE FIRST 3ENSEMBLES OF THE SEQUENCE OF TABLE I D c σ = σ = σ = σ = σ = λ(x) =.4322x Π(x) =.2947x+ Π(x) =.4115x+ Π(x) =.4703x+ Π(x) =.5308x x x x x x x x x x x 5 6 λ(x) =0.3457x Π(x) =.2947x+ Π(x) =.4115x+ Π(x) =.4703x+ Π(x) =.5308x x x x x x x x x x x 5 7 λ(x) =0.2881x Π(x) =.2947x+ Π(x) =.4115x+ Π(x) =.4703x+ Π(x) =.5308x x x x x x x x x x x x x x x x 9 TABLE III THE VALUES OF Π S USED TO PUNCTURE THE ENSEMBLE C SM σ = Π(x) =0.2947x x x x 9 σ = Π(x) =0.4115x x x x 9 σ = Π(x) =0.4703x x x x 9 σ = Π(x) =0.5308x x x x 9 not mpose any constrant to guarantee rate-compatbty. Ths reduces the desgn compexty sgnfcanty. It s nterestng to see that based on Tabes II and III, except for = D v, the vaues of Π are amost ndependent (for Π 2 provaby ndependent based on Coroary 2) of the parent ensembes and ony depend on the channe parameter for whch the puncturng s apped. In other words, for a gven channe parameter θ j, the computed vaues of Π can unversay be apped to any ensembe desgned based on the SM method for a gven orgna channe parameter θ 0 and any arbtrary check node dstrbuton. Dstance to capacty (db) Optmzed puncturng on optmzed ensembe C Random puncturng on optmzed enesembe C SM puncturng on C SM σ Fg. 4. Performance comparson of schemes constructed based on the proposed SM method and those constructed based on optmzaton method of [16] VII. CONCLUSION In ths paper, we proposed the method of successve maxmzaton (SM) for the systematc desgn of unversay capacty approachng rate-compatbe LDPC code ensembe sequences over BIOSM channes. The SM method was frst apped to desgn a sequence of capacty approachng parent ensembes. It was then apped to each parent ensembe, ths tme to desgn rate-compatbe puncturng schemes. As part of our resuts, we were abe to extend the stabty condton whch was prevousy derved for degree-2 varabe nodes to other varabe node degrees as we as to the case of ratecompatbe codes. Consequenty, we rgorousy proved that usng the SM prncpe, one s abe to desgn unversay capacty achevng rate-compatbe LDPC code ensembe sequences over the BEC. Unke the prevous resuts on such schemes over the BEC n the terature, the proposed SM approach can be naturay extended to other BIOSM channes. Usng such an extenson, we desgned rate-compatbe codes over the BIAWGN channe and the BSC whose performance unversay approaches the capacty as the average check node degree ncreases. We demonstrated that for fnte vaues of D c, the performance of the ensembes desgned by our method s comparabe to those desgned based on optmzaton. One mportant drecton n the contnuaton of ths work s to anaytcay compute the vaues of λ or to devse desgn agorthms whch are more robust aganst numerca errors. Ths can pave the road for demonstratng that the proposed sequences can n fact acheve the capacty of BIOSM channes. APPENDIX DESIGN ALGORITHM AND NUMERICAL STABILITY ISSUES The desgn method of Secton III can be formuated nto the foowng agorthm whch w be referred to as Agorthm 1. Let k be the number of consttuent varabe node degrees. Startng from =2, we need to fnd λ =sup{λ (ρ(x),θ 0 )} for dfferent vaues of successvey. We know that λ 2 = λ 2, and thus start from =3. The cacuaton of λ can be performed by contnuousy ncreasng the vaue of λ from zero and n each step, checkng f the ensembe wth suffcenty arge D v s convergent usng densty evouton. In each step, we aways set λ Dv =1 j=2 λ j. We repeat ths process for successve vaues of unt ether λ Dv < 0 for a gven vaue of, or>k+1. Then, we decrease the vaue of D v as far as the ensembe remans convergent. We, however, remnd the reader that snce the computatons are performed n crtca vaues of λ (.e., at the border of stabty/nstabty), ths agorthm s very senstve to numerca errors. To mtgate the effect of such errors, one has to use densty evouton wth very hgh precson as we as very sma ncrements n the vaues of λ n the vcnty of the threshod λ.thsn turn, ncreases the computatona compexty. Consequenty, reducng the precson may resut n numerca errors whch usuay propagate to other steps. Ths ssue w be dscussed n ths appendx. Concentratng on the computaton of λ 3, we note that ths computaton drecty depends on the vaue of λ 2 and s performed under the assumpton that λ 2 = λ 2. The mportant

12 1818 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 7, JULY 2011 queston, however, s that what happens f the vaue of λ 2 s sghty dfferent from λ 2 due to numerca errors. Is the upper bound on λ 3 a contnuous functon of λ 2 such that the computed upper bound for λ 3 tends to λ 3 f λ 2 tends to λ 2? Based on our numerca resuts, the answer to ths queston s postve. In fact, we have been abe to prove ths for the case of the BEC n the next proposton. We conjecture that such contnuty aso exsts for other channes as we as other varabe node degrees. Proposton 3: Over the BEC, consder a code-channe par E = (λ 2,λ 3,D v,ρ(x); ε) where D v can be made arbtrary arge. For any gven vaue of λ 2 n Λ 2 (ρ(x),ε), defne the set A = {λ 3 : D v,e s convergent} and I(λ 2 ) = sup(a). Then, I s a contnuous functon of λ 2.In partcuar, I s contnuous at λ 2 = λ 2. Proof : Usng the convergence condton (6) for k =3and rearrangng the terms, we have T λ 3 < ε x 3 + λ 2 λ 2 1 λ 2 λ 3 x Dv 3. (20) x ε =3 Now note that f λ 3 A for a certan D v, for any greater vaue of D v, we st have λ 3 A. Therefore the vaue of I(λ 2 ) does not change f D v tends to nfnty. Therefore we have: I(λ 2 )= mn 0<x<x 0 { =3 T ε x 3 + λ 2 λ 2 x }, where x 0 =1 ρ(1 P(P 0 )), and we have negected the ast term of (20) assumng that D v tends to nfnty. Frst, we prove that m λ2 λ 2 I(λ 2 )=I( λ 2 )= λ 3.Wedefne f 1 (x) = T =3 ε x 3 + b x,whereb = λ 2 λ 2, 0 b λ 2.For b =0(λ 2 = λ 2 ), the mnmum of ths functon n the nterva [0, ) s at x = 0. Now assume that b > 0. The second dervatve of ths functon s strcty postve n the nterva (0, ). In other words, ths functon has a oca mnmum n ths nterva whch s the root of equaton f 1(x) =0where f 1 (x) = T =4 ε ( 3)x 4 b x. Ths equaton has ony 2 one root n the nterva of (0, ). It can easy be seen that f b tends to zero, the root of the equaton aso tends to zero. In other words, argmn(f 1 (x)) tends to zero as b tends to zero. Thus we can concude that I(λ 2 ) s contnuous at λ 2 = λ 2. The proof of contnuty for other ponts s straght forward. As prevousy mentoned, Agorthm 1 requres very hgh precson to mtgate the effect of numerca errors. The reason s that the vaue of λ s very senstve to the vaue of λ 1. For exampe, athough we proved n Proposton 3 that for the BEC, functon I s contnuous at λ 2 = λ 2, one can verfy that d dλ 2 I(λ 2 ) λ2= λ 2 can be very arge, mpyng that any sma devaton from λ 2 w cause a sgnfcant devaton from the vaue of λ 3. Moreover, the computaton of the exact vaue of λ 1 by densty evouton requres very hgh precson. For exampe, usng a smar method to that of Agorthm 1 to compute λ 2 wth a reasonabe compexty (n our case, dynamc range of [-50,50] and 13-bt quantzaton) woud resut n a vaue for λ 2 whch s non-neggby dfferent from (and usuay greater than) λ 2 (the dfference can some tmes be as hgh as haf a percent of λ 2 ). Now f we set λ 2 = λ 2 = λ 2 and compute the vaue of λ 3 and subsequenty the vaue of λ 4, the computed vaue for λ 4 w appear to be cose to zero. Ths, however, s not the correct vaue, at east for the case of the BEC where we aready know that λ 4 = λ 4. The cose to zero vaue of λ 4 s caused by the fact that the computed vaue of λ 3 s sghty arger than ts true vaue (ths s confrmed for the case of the BEC where we aready know the true vaue of λ 3 ). To prevent ths, we need to sghty reduce the vaue of λ 3 from ts computed upper bound. At the same tme, the amount of reducton n the vaue of λ 3 s very crtca and may make the computed vaue for λ 4 too arge. Ths n turn w cause the computed vaue for λ 5 n the next step to be cose to zero. In genera, a numerca error at one step propagates to the foowng steps. One way to prevent λ +1 to tend to zero at step +1 due to an over-estmated vaue of λ n the prevous step (step ), s to et λ +1 to aso ncrease to a fracton of λ whe ncreasng λ. Ths jont ncrement can aso be apped to more than 2 consecutve degree coeffcents. Thsmeansthatatstep, whe ncreasng λ, we aso ncrease λ +1,.., λ +K, to a fracton of λ. We then ncrease by 1, and set a coeffcents wth ndex greater than to zero, and repeat the process. At each step, after the maxmzaton s performed, we st mutpy the resutng vaue by a constant α ess than or equa to 1. 9 Usng ths jont ncrement technque, the vaues of λ to λ +K at step become ess dependent on the vaue of α used at step 1. REFERENCES [1] C. D, D. Proett, I. E. Teatar, T. J. Rchardson, and R. L. Urbanke, Fnte-ength anayss of ow-densty party-check codes on the bnary erasure channe, IEEE Trans. Inf. Theory, vo. 48, no. 6, pp , June [2] M. Luby, M. Mtzenmacher, M. A. Shokroah, D. Speman, and V. Stemann, Practca oss-resent codes, n Proc. 29th ACM Symposum Theory Computng, 1997, pp [3] A. Shokroah, Capacty-achevng sequences, n Codes, Systems, and Graphca Modes (IMA Voumes n Mathematcs and Its Appcatons), B. Marcus and J. Rosentha, edtors. Sprnger-Verag, 2000, vo. 123, pp [4] A. Shokroah, New sequences of near tme erasure codes approachng the channe capacty, n Proc. 13th Internatona Symposum Apped Agebra, Agebrac Agorthms, Error Correctng Codes (M. Fossorer, H. Ima, S. Ln, and A. Po), no. 1719, Lecture Notes n Computer Scence, pp , [5] P. Oswad and A. Shokroah, Capacty-achevng sequences for the erasure channe, IEEE Trans. Inf. Theory, vo. 48, no. 12, pp , Dec [6] I. Sason and R. Urbanke, Party-check densty versus performance of bnary near bock codes over memoryess symmetrc channes, IEEE Trans. Inf. Theory, vo. 49, no. 7, pp , Juy [7] I. Sason and R. Urbanke, Compexty versus performance of capactyachevng rreguar repeat-accumuate codes on the bnary erasure channe, IEEE Trans. Inf. Theory, vo. 50, no. 6, pp , June [8] H. D. Pfster, I. Sason, and R. Urbanke, Capacty-achevng ensembes for the bnary erasure channe wth bounded compexty, IEEE Trans. Inf. Theory, vo. 51, no. 7, pp , Juy [9] H. D. Pfster and I. Sason, Accumuate-repeat-accumuate codes: capacty-achevng ensembes of systematc codes for the erasure channe wth bounded compexty, IEEE Trans. Inf. Theory, vo. 53, no. 6, pp , June [10] T. J. Rchardson, A. Shokroah, and R. L. Urbanke, Desgn of capacty approachng rreguar ow densty party check codes, IEEE Trans. Inf. Theory, vo. 47, no. 2, pp , Feb In our smuatons, we have used α =0.95 and K=2 whch s apped for >3.

13 SAEEDI et a.: SUCCESSIVE MAXIMIZATION FOR THE SYSTEMATIC DESIGN OF UNIVERSALLY CAPACITY APPROACHING RATE-COMPATIBLE [11] T. Rchardson and R. Urbanke, Modern Codng Theory. Cambrdge Unversty Press, [12] H. Saeed and A. H. Banhashem, New sequences of capacty achevng LDPC Code ensembes for the bnary erasure channe, n Proc. Internatona Symp. Inf. Theory, Juy [13] H. Saeed and A. H. Banhashem, New sequences of capacty achevng LDPC code ensembes over the bnary erasure channe, IEEE Trans. Inf. Theory, vo. 56, no. 12, pp , Dec [14] H. Saeed and A. H. Banhashem, Systematc desgn of LDPC codes for bnary erasure channes, IEEE Trans. Commun., vo. 58, no. 1, pp , Jan [15] H. Pshro-Nk and F. Fekr, Resuts on punctured ow-densty partycheck codes and mproved teratve decodng technques, IEEE Trans. Inf. Theory, vo. 53, no. 2, pp , Feb [16] J. Ha, J. Km, and S. W. McLaughn, Rate-compatbe puncturng of ow-densty party-check codes, IEEE Trans. Inf. Theory, vo. 50, no. 11, pp , Nov [17] H. Pshro-Nk, N. Rahnavard, and F. Fekr, Nonunform error correcton usng ow-densty party-check codes, IEEE Trans. Inf. Theory, vo. 51, no. 7, pp , Juy [18] H. Chun-Hao and A. Anastasopouos, Capacty achevng LDPC codes through puncturng, IEEE Trans. Inf. Theory, vo. 54, no. 10, Oct [19] I. Sason and G. Wechman, On achevabe rates and compexty of LDPC codes over parae channes: bounds and appcatons, IEEE Trans. Inf. Theory, vo. 53, no. 2, pp , Feb [20] J. Ha, J. Km, D. Knc, S. W. McLaughn, Rate-compatbe punctured ow-densty party-check codes wth short bock engths, IEEE Trans. Inf. Theory, vo. 52, no. 2, pp , Feb [21] B. N. Veamb and F. Fekr, Fnte-ength rate-compatbe LDPC codes: a nove puncturng scheme, IEEE Trans. Commun., vo. 57, no. 2, pp , Feb [22] M. R. Yazdan and A. H. Banhashem, On constructon of ratecompatbe ow-densty party-check codes, IEEE Commun. Lett., vo. 8, no. 3, pp , Mar [23] Y. Guosen Yue, W. Xaodong, and M. Madhan, Desgn of ratecompatbe rreguar repeat accumuate codes, IEEE Trans. Commun., vo. 55, no. 6, pp , June [24] A. Shokroah, Raptor codes, IEEE Trans. Inf. Theory, vo. 52, no. 6, pp , June [25] H. Saeed and A. H. Banhashem, On the desgn of LDPC codes ensembes for BIAWGN channes, IEEE Trans. Commun., vo. 58, no. 5, pp , May [26] C. Measson, A. Montanar, and R. Urbanke. Why we can not surpass capacty: the matchng condton, n Proc. Aerton Conf. Commun., Contro Computng, Oct [27] I. Sason, On unversa propertes of capacty-approachng LDPC code ensembes, IEEE Trans. Inf. Theory, vo. 55, no. 7, pp , Juy [28] E. Arkan, Channe poarzaton: a method for constructng capactyachevng codes, n Proc. Internatona Symp. Inf. Theory, Juy [29] A. Ashkhmn, G. Kramer, and S. ten Brnk, Extrnsc nformaton transfer functons: mode and erasure channe propertes, IEEE Trans. Inf. Theory, vo. 50, no. 11, pp , Nov [30] H. Pshro-Nk and F. Fekr, Performance of ow-densty party-check codes wth near mnmum dstance, IEEE Trans. Inf. Theory, vo. 52, no. 1, pp , Jan Hamd Saeed (S 01-M 08) receved the B.Sc. and M.Sc. degrees from Sharf Unversty of Technoogy, Tehran, Iran, n 1999 and 2001, respectvey, and the Ph.D. degree from Careton Unversty, Ottawa, ON, Canada, n 2007, a n eectrca engneerng. In , he was a postdoctora feow n the Department of Eectrca and Computer Engneerng, Unversty of Massachusetts, Amherst. In 2010, he joned the Department of Eectrca and Computer Engneerng, Tarbat Modares Unversty, Tehran, Iran, where he s now an Assstant Professor. He s aso affated wth the Advanced Communcatons Research Insttute (ACRI), Sharf Unversty of Technoogy, Tehran, Iran. Hs research nterests ncude codng and nformaton theory, wreess communcatons, and sgna processng for broadband communcatons. Dr. Saeed s the recpent of severa awards ncudng the Careton Unversty Senate Meda for Outstandng Academc Achevement, a Natura Scences and Engneerng Counc of Canada (NSERC) Industra Research and Deveopment Feowshp, and an Ontaro Graduate Schoarshp. Hossen Pshro-Nk (S 01-M 06) s an Assocate Professor of eectrca and computer engneerng at the Unversty of Massachusetts, Amherst. He receved a B.S. degree from Sharf Unversty of Technoogy, and M.Sc. and Ph.D. degrees from the Georga Insttute of Technoogy, a n eectrca and computer engneerng. Hs research nterests ncude the mathematca anayss of communcaton systems, n partcuar, error contro codng, wreess networks, and vehcuar ad hoc networks. Hs awards ncude an NSF Facuty Eary Career Deveopment (CAREER) Award, an Outstandng Junor Facuty Award from UMass, and an Outstandng Graduate Research Award from the Georga Insttute of Technoogy. Amr H. Banhashem (S 90-A 98-M 03-SM 04) receved the B.A.Sc. degree n eectrca engneerng from Isfahan Unversty of Technoogy, Isfahan, Iran n 1988, and the M.A.Sc. degree n communcatons engneerng from the the Unversty of Tehran, Tehran, Iran, n 1991, wth the hghest academc rank n both casses. From 1991 to 1994, he was wth the Eectrca Engneerng Research Center and the Department of Eectrca and Computer Engneerng, Isfahan Unversty of Technoogy. Durng , he was wth the Department of Eectrca and Computer Engneerng, Unversty of Wateroo, Wateroo, Ontaro, Canada, workng towards the Ph.D. degree. He hed two Ontaro Graduate Schoarshps for nternatona students durng ths perod. In 1997, he joned the Department of Eectrca and Computer Engneerng, Unversty of Toronto, Toronto, Ontaro, Canada, where he worked as a Natura Scences and Engneerng Research Counc of Canada (NSERC) Postdoctora Feow. He joned the Facuty of Engneerng at Careton Unversty n 1998, where at present he s a Professor n the Department of Systems and Computer Engneerng. Hs research nterests ncude codng and nformaton theory, dgta and wreess communcatons, theory and mpementaton of communcatons agorthms, and compressve sensng and sampng. He has pubshed more than 130 papers n refereed journas and conferences. Dr. Banhashem served as an Assocate Edtor for the IEEE TRANSAC- TIONS ON COMMUNICATIONS from 2003 to He s a member of the Board of Drectors for the Canadan Socety of Informaton Theory and the Drector of the Broadband Communcatons and Wreess Systems (BCWS) Centre at Careton Unversty. He has been nvoved n many nternatona conferences as char, member of technca program commttee, and member of organzng commttee. These ncude co-char for the Communcaton Theory Symposum of Gobecom 07, TPC co-char of the Informaton Theory Workshop (ITW) 2007, member of the organzng commttee for the Internatona Symposum on Informaton Theory (ISIT) 2008, and co-char of the Canadan Workshop on Informaton Theory (CWIT) Dr. Banhashem s a recpent of Careton s Research Achevement Award n 2006, and s awarded one of the NSERC s one hundred 2008 Dscovery Acceerator Suppements (DAS).