Information-Coupled Turbo Codes for LTE Systems

Transcription

1 Informtion-Coupled Turbo Codes for LTE Systems Lei Yng, Yixun Xie, Xiowei Wu, Jinhong Yun, Xingqing Cheng nd Lei Wn rxiv: v [cs.it] 20 Sep 207 Abstrct We propose new clss of informtion-coupled (IC Turbo codes to improve the trnsport block (TB error rte performnce for long-term evolution (LTE systems, while keeping the hybrid utomtic repet request protocol nd the Turbo decoder for ech code block (CB unchnged. In the proposed codes, every two consecutive CBs in TB re coupled together by shring few common informtion bits. We propose feed-forwrd nd feed-bck decoding scheme nd windowed (WD decoding scheme for decoding the whole TB by exploiting the coupled informtion between CBs. Both decoding schemes chieve considerble signl-to-noise-rtio (SNR gin compred to the LTE Turbo codes. We construct the extrinsic informtion trnsfer (EXIT functions for the LTE Turbo codes nd our proposed IC Turbo codes from the EXIT functions of underlying convolutionl codes. An SNR gin upper bound of our proposed codes over the LTE Turbo codes is derived nd clculted by the constructed EXIT chrts. Numericl results show tht the proposed codes chieve n SNR gin of 0.25 db to 0.72 db for vrious code prmeters t TB error rte level of 0 2, which complies with the derived SNR gin upper bound. Index Terms Turbo codes, informtion coupling, HARQ, LTE I. INTRODUCTION In the long-term evolution (LTE stndrds, trnsport block (TB bsed hybrid utomtic repet request (HARQ is key fctor to provide low ltency nd high speed dt trnsmission []. In the TB bsed HARQ protocol, receiver uses only one bit cknowledgement (ACK or negtive cknowledgement (NACK to report the receiving sttus of TB to the trnsmitter. This mechnism minimizes the HARQ feedbck overhed. However, it results in wste of trnsmission power nd spectrum efficiency since ny code block (CB errors in TB will led to the retrnsmission of the entire TB [2]. In the current LTE stndrd, TB cn consist of tens of CBs [3]. This number will further increse in the coming5 th genertion (5G cellulr networks s the user pek throughput is expected to be incresed by times [4], but the mximum CB length cnnot be incresed proportionlly due to the high complexity for decoding long codes. Therefore, the problem of wsting trnsmission power nd spectrum efficiency ssocited with the TB bsed HARQ protocol will become worse in the future. To mitigte this problem, CB bsed HARQ scheme ws proposed nd investigted in [2] nd [5] [6] for the 802.6e stndrd nd the 5G new rdio ccess technologies, respectively. In the scheme, only erroneous CB (CBs is (re retrnsmitted. This sves trnsmission power nd improves spectrum efficiency, but leds to n excessive mount of overhed in downlink/uplink control chnnels in order to mnge lot of HARQ interlces even within one TB [6]. Another wy to solve the forementioned problem is to improve the TB error rte (TBER performnce while keeping the HARQ protocol unchnged. This cn be strightforwrdly chieved by using much longer CB. However, much longer CB mens much longer decoding ltency nd much higher decoder complexity. To exploit the benefits of long codes in terms of decoding threshold while keeping the decoding ltency nd decoder complexity low, sptilly-coupled (SC codes [7] - [4] with windowed decoders [5] - [7] re potentil cndidtes. Theoreticlly, codes with infinite length cn be constructed by coupling short codes in sptil domin. In prctice, low ltency nd low complexity windowed decoder of finite length is employed to decode terminted SC code such tht the decoding performnce of the code pproches the theoreticl limit of the SC codes of infinite length [5]. Inspired by the promising performnce of SC LDPC codes, uthors in [8] extended the sptil coupling technology to Turbo codes. Belief-propgtion (BP decoding threshold nlysis by density evolution under binry ersure chnnel (BEC hs shown tht BP thresholds of the SC Turbo codes sturte to the mximum posteriori (MAP thresholds. Finite length simultion results in [9] lso showed tht the SC Turbo codes with windowed decoder hve better decoding threshold thn their non-sc counterprts in BEC. Though the SC codes nd windowed decoders re potentil solutions to improve the TBER performnce for TB bsed HARQ schemes, the window size of the windowed decoder hs to be long enough to hve good decoding threshold performnce. Generlly speking, if the coupling memory is m, the window size should be t lest m +. Therefore, the windowed decoder hs much higher implementtion complexity thn the decoder for the underlying non-sc counterprt. It is understood tht SC codes show substntil coding gin compred to their non-sc counterprts. Such gin, nmely convolutionl gin, comes from the fct tht relible messges t two ends of terminted SC code re continuously decoded nd spred out, nd grdully improve the qulity of other messges s itertive decoding progresses [20]. Inspired by this relible messge spreding phenomenon, we propose to couple the CBs in TB into chin nd delibertely introduce better decoding threshold for the CBs t two ends of the chin. In [2], the uthors proposed clss of rtecomptible convolutionl codes (CCs by inserting dummy bits into informtion sequence, which is termed dummy bits inserting (DBI CCs. It hs shown tht the proposed codes hve comprble performnce to the optiml repetition CCs with the sme code rte. In [22], the uthors nlyzed the DBI Turbo codes by using extrinsic informtion trnsfer (EXIT chrt [23]. They showed tht the proposed Turbo codes outperform LTE Turbo codes in terms of frme error rte (FER nd convergence speed. Inspired by the relible messge spreding phenomenon of the SC codes [20] nd the improved FER performnce of

2 2 the DBI Turbo codes over the LTE Turbo codes in [22], we propose new clss of informtion-coupled (IC Turbo codes for LTE to improve the TBER performnce. Menwhile, we keep the TB bsed HARQ protocol nd the Turbo decoder for ech CB in the LTE unchnged. The min contributions re summrized below: We propose new clss of IC Turbo codes, which couple ll CBs in TB into chin by shring few common informtion bits between every two consecutive CBs. Dummy bits re inserted in the first nd the lst CBs of the coupled chin in order to chieve better decoding threshold nd initil decoding performnce of these two CBs thn tht of other CBs. The dvntges in these two CBs cn then be spred out to other CBs in itertive decoding through the coupled informtion. The proposed IC Turbo codes re different from the SC Turbo codes proposed in [8] nd [9], where ll CBs re coupled together to form much longer trellis. In our proposed IC Turbo codes, ech CB is terminted nd the trellis is exctly the sme s tht of the originl LTE Turbo codes. Therefore, we cn keep the Turbo decoder for ech CB unchnged. It is worth pointing out tht we do not try to construct much longer code s tht in the SC-LDPC codes. We propose feed-forwrd nd feed-bck (FF-FB decoding scheme nd windowed (WD decoding scheme for our proposed IC Turbo codes to exploit the coupled informtion between every two consecutive CBs. In the FF-FB decoding scheme, the feed-forwrd (FF decoding process trverses from the first CB in the coupling chin to the lst CB, nd the feed-bck (FB decoding process conducts in the opposite direction. These two decoding processes spred the relible messges from two ends of the coupled chin to other CBs nd improve the TBER performnce. The WD decoding scheme goes through the coupled chin only once from the first CB to the lst CB, where ech decoding window consists of only two consecutive CBs. The proposed IC Turbo codes with both decoding schemes chieve considerble TBER performnce gin over the LTE Turbo codes. The WD decoding scheme hs lower decoding complexity thn tht of the FF-FB decoding scheme. Moreover, it keeps the on-the-fly decoding feture, i.e., CBs cn be decoded in first-rrivl-first-decode mnner. We propose method to construct EXIT functions for the CC decoders of the LTE Turbo codes with repetition from the EXIT functions of the underlying CC. In the proposed method, the communiction chnnel is modeled by two independent prllel chnnels with different effective signl-to-noise-rtios (SNRs. This is different from the method proposed in [24], where ech sub chnnel is modelled by n AWGN chnnel cscded with n ersure chnnel nd the AWGN chnnel for ll sub chnnels hs sme effective SNR. We derive n upper bound on the SNR gin of our IC Turbo codes over the LTE Turbo codes nd show tht the proposed EXIT functions re well fitted with simultion results. We lso show tht the derived upper bound cn effectively estimte the SNR gin of our proposed codes over the LTE Turbo codes. We evlute the TBER performnce of our proposed IC Turbo codes through intensive Monte Crlo simultions. Simultion results demonstrte tht our proposed codes hve SNR gin rnges from 0.26 db to 0.72 db compred to the LTE Turbo codes for vrious code prmeters t TBER level of 0 2. II. TB BASED HARQ IN LTE AND PROBLEM STATEMENT The TB bsed HARQ process in LTE is illustrted in Fig.. A TB u of length L is fed to the physicl lyer for trnsmission. The physicl lyer of trnsmitter firstly ttches 24-bits TB cyclic redundncy check (TB CRC t the end of u. If L + 24 is lrger thn the pre-defined mximum CB length, which is 644 in LTE [26], the TB is segmented into N CBs nd ech CB is ttched with CB CRC of 24 bits. Otherwise, segmenttion nd CB CRC ttchment re omitted nd the TB consists of only one CB, i.e.,n =. The resultnt CBs{u n },n =,,N, re fed to systemtic Turbo encoder of rte = 3 sequentilly. The Turbo encoder consists of two 8-stte prllel conctented CCs (PCCCs with octl genertor polynomils (, 3 5 nd one internl interlever. Denote the output codewords of the Turbo encoder by {v n },n =,,N. The Turbo codewords v n re sent to rte-mtching device to obtin the required code rte. The resultnt codewords re denoted by v n. After dptive modultion, signls x re trnsmitted. The length of x is determined by the TB length L, the segmenttion rules in LTE nd the modultion nd coding scheme used for the TB. We omit these detils here becuse they re not essentil to our proposed IC Turbo codes. We refer the interested reders to [26]. At the receiver side, noisy signls y = x+n ( re received, where n is n dditive white Gussin noise (AWGN vector with i.i.d components. Ech component hs zero men nd vrince σ 2 ch = N0 2, where N 0 is the doublesided noise power spectrum density. Here, y nd n hve the sme length s x. SNR is defined s ρ = Es N 0 nd E s is the verge symbol energy. Upon receiving the noisy signls y, soft demodultor clcultes the log-likelihood rtio (LLR for ech coded bit by ( L v n,m ( log 2 Pr y m v n,m = 0 Pr ( y m v n,m =. (2 Here, v n,m represents the m-th coded bit in the n-th codeword v n nd y m denotes ( the chnnel observtion ( which contins v n,m. Then, L v n collects ll L v n,m nd it is sent to de-rte-mtching device to clculte the LLRs for coded bits in v n. In LTE, the rte-mtching mechnisms include puncturing nd repetition. For the punctured bits in v n, we set the ssocited LLRs to zeros. For repeted coded bit v n,m in v n, if it is repeted by Q times nd the ssocited LLRs for

3 3 u û TBCRC ttchement TBCRC check CB segmenttion& CB CRC ttchement CBCRC&TB conctention u n ACK/NACK uˆ n Turbo Encoding Turbo Decoding v n ( n Rtemtching De-rtemtching ' v n ' L v L( v n Adptive Modultion x Å n y Demodultion Fig.. Block digrm of the HARQ process in LTE. { ( ( } (Q the Q+ observtions re L v n,m (0,,L v n,m, the LLR for this coded bits is clculted by L(v n,m = Q (q. (v L n,m (3 q=0 Here, we consider tht the observtions for the coded bit re independent. After de-rte-mtching, L(v n collects ll L(v n,m for ech CB, nd then it is fed to Turbo decoder. The Turbo decoder consists of two constituent BCJR decoders [27] nd n interlever/deinterlever. By using n itertive decoding process, estimted CBs {û n },n =,,N, re given. For ech CB, if CB CRC detects n error, subsequent CBs in this TB will not be processed nd n NACK bit is sent by the receiver to its peer trnsmitter. Otherwise, the estimted CBs {û n },n =,,N, re conctented nd used to clculte the TB CRC. If TB CRC detects n error, n NACK bit is sent by the receiver to its peer trnsmitter, which triggers retrnsmission process. Otherwise, n ACK bit is sent nd the TB is received successfully. Note tht, s only one bit feedbck per TB is used in the LTE HARQ protocol, the whole TB hs to be retrnsmitted if ny CB in the TB is in error. Tht is why we cll it TB bsed HARQ. Obviously, when TB consists of severl or tens of CBs, TB bsed feedbck my result in wste of trnsmission power nd reduced trnsmission efficiency [2] [5] [6]. We lso note tht ech CB hs 24-bits CRC if TB consists of multiple CBs. The CB CRC cn be used s CB-level erly stopping criterion to reduce decoding itertions of ech CB. It cn lso be used s TB-level erly stopping criterion to reduce the number of CBs to be decoded in n erroneous TB. Both of them cn sve the receiver s computtionl resources [28]. However, to enjoy the benefits of TB-level erly stopping, CBs must be decoded on-the-fly. In this pper, we propose new clss of IC Turbo codes for LTE to improve the TBER performnce, i.e., decrese Pr(û u. At the sme time, we keep the TB bsed HARQ protocol nd the Turbo decoder for CB in the current LTE stndrds unchnged. We minly consider the IC Turbo codes tht hve code rtes R IC lower thn the mother code rte = 3. In this cse, repetition is used s the rte-mtching mechnism by the trnsmitter nd chse-combining is used by the receiver in LTE. Higher code rtes cn be relized by using the sme puncture mechnism s tht used in the LTE Turbo codes. In this pper, we minly focus on the cse N 2. For N =, our codes degrde to DBI Turbo codes s proposed in [22]. III. ENCODING OF OUR PROPOSED IC TURBO CODES In this section, we first present the encoding scheme of our proposed IC Turbo codes. Then, the determintion of code prmeters for given TB length nd required code rte is given. At lst, the effective code rtes of the proposed codes re expressed w.r.t the mother code rte. A. Encoding Scheme of Our Proposed IC Turbo Codes The block digrm of the encoding scheme is shown in Fig. 2. In big picture, the encoding process tkes in TB u of length L nd two dummy bit sequences d H nd d T of length D H nd D T respectively, nd genertes N systemtic Turbo codewords {v n }, n =,,N. Ech codeword v n consists of two prity check sequences vn nd vn, 2 nd systemtic bit sequenceu n, wherev n ndv2 n re generted by two constituent CC encoders in the Turbo code, respectively. In the proposed encoder, severl common informtion bits re shred between two consecutive Turbo CBs, which cn be exploited in the itertive decoding to spred extrinsic informtion between CBs. We cll this type of codes informtion coupled Turbo codes. This coupling method results in coupling memory m =, which enbles low complexity windowed decoding scheme in Section IV-B2. We cn lso increse the coupling memory, i.e, m >, however, it will result in higher implementtion complexity becuse decoding window of size m+ is required to chieve good decoding performnce. In ddition, dummy bits re inserted in the first nd the lst Turbo CBs, which provide more relible decoding outputs for these two CBs thn tht of other CBs. These relible messges grdully spred out in the itertive decoding process nd eventully improve the overll TBER performnce. The effect of these inserted dummy bits in the FF-FB decoding scheme will be discussed in Section IV-B. The detiled encoding process consists of three steps: CB segmenttion; 2 Informtion coupling nd DBI; nd 3 Turbo encoding, which re described s below: Step CB segmenttion: Segment TB u into N informtion blocks, i.e., u = [u,,u N ]. Here, [u,,u N ] represents the conctention opertion of N vectors u, u 2 to u N. For n =,,N, let D n represent the number of shred informtion bits between CBsu n ndu n+. Denote the length of the n-th CB u n by K n, n {,,N}. Then, TB is segmented in such wy: let the first informtion block u hve length of K D H ; let the n-th informtion block u n, n {2,,N }, hve length of K n D n ; nd let the lst informtion blocku N hve length of K N D N D T.

4 4 u v v u u d H u d T u u 2 st CB Encoding 2 nd CB Encoding v 2 v u u C u u 2 u N u C2 u C N - u N N-th CB Encoding v 2 2 v 2 u 2 v N 2 v N u N Φ. B. Code Prmeters Determintion nd Effective Code Rte The encoding scheme described bove introduced set of code prmeters N, K n, D n, D H nd D T. Now we first present how to determine these prmeters for given TB length L nd given trget effective code rte R IC. Then, we will discuss the reltionship between the effective code rte of n IC Turbo code to the mother code rte. Firstly, the segmenttion process in Step of the encoding scheme gurntees L = (K D H + N (K n D n n=2 +(K N D N D T. (4 Fig. 2. codes. Block digrm of the encoding scheme for the proposed IC Turbo Step 2 Informtion coupling nd DBI: Construct N CBs {u n },n =,,N, through informtion coupling nd DBI. Let u Cn be the D n coupled informtion bits between CBs u n nd u n+, n =,,N, i.e., the informtion bits in informtion block u n shred by CBs u n nd u n+. We cll D n s coupling length between CBs u n nd u n+. For the first CB u, let u = {d H,u }, where d H is the dummy bits inserted into the first CB. For n =,,N, let the n-th CB u n = { { u Cn,u n}. For the lst CB un, let u N = ucn,u N,d } T, where dt is the dummy bits inserted into the lst CB. Step 3 Turbo Encoding: A Turbo encoder, which consists of two CC encoders nd n interlever, encodes the N CBs. For CB u n,n {,,N}, the encoder genertes the prity check sequences vn nd v2 n by two CC encoders seprtely, nd outputs u n, vn nd vn 2 s codeword. Note tht, s the dummy bits d H nd d T re known by the receiver in dvnce, they do not need to be trnsmitted through communiction chnnels. As well s this, the coupled informtion sequence u Cn,n {,,N }, is only trnsmitted in the n-th codeword v n nd re not trnsmitted in the (n+-th codeword v n+. Remrk : As shown in [2] tht the distnce spectrum of DBI CCs depends on the number nd the positions of these dummy bits in CB. It is not ffected by the vlue of the dummy bits. Therefore, ll-zero dummy bits re considered in this pper, i.e., d H = d T = 0. In ddition, we consider tht the dummy bits re eqully spced or nerly eqully spced in CB. The optimiztion of distributing the dummy bits in CB is not considered in this pper. Remrk 2: The positions of coupled informtion bits in CB will ffect the decoding performnce of the proposed scheme. In this pper, we consider tht the coupled informtion bits re eqully spced or nerly eqully spced in the ssocited CBs. The optimiztion of distributing the coupled informtion bits in the ssocited CBs is not considered in this pper. We lso consider tht the coupled informtion bits of CB with its previous nd next CBs re independent, i.e., u Cn ucn = Then, recll tht the dummy bits d H nd d T re not trnsmitted in the output codewords nd the coupled informtion sequences {u Cn },n =,,N, re only trnsmitted once in Step 3 of the encoding scheme, the totl output length of the proposed IC Turbo codes is written s N IC = + ( K + N D H R ( 0 KN D N D T n=2 ( Kn D n. (5 To stisfy the trget effective code rte R IC for given TB length L, ( L K = N IC = D H + ( N Kn D n R IC n=2 R ( 0 KN + D N D T (6 should be stisfied. Let Z + be the set of positive integers nd Ω be the set of vlid CB lengths in the LTE stndrd. Theoreticlly, ny vlues of N, D n, D H, D T Z + nd K n Ω cn be selected. In prctice, we prefer to hvek n = K nd mximize K to exploit the best vilble coding gin. In ddition, we consider D n = D H = D T = D < K 2 in this pper. Setting D n = D H = D T = D leds to simple encoding structure nd similr CB error rte (CBER for ll CBs in TB, which will be shown in Fig. 4 in Section IV-B. Setting D < K 2 gurntees the coupled informtion of CB with its previous nd next CBs cn be independent. By considering these simplifictions, (4 nd (6 become L = (N (K D+(K 2D (7 nd ( ( L K K = N IC = (N D + 2D. (8 R IC In prctice, one my not be ble to select pproprite vlues for N, D Z + nd K Ω to stisfy (7 nd (8 for given L nd R IC. In tht cse, zero pdding to u is conducted to obtin new TB of length L to gurntee tht n pproprite set of K, N nd D cn be selected to stisfy (7 nd (8. In ddition, L L should be minimized to sve rdio resources. Here, we ssume (7 nd (8 re stisfied for simplicity.

5 5 Now, we propose to select the code prmeters in such wy to mximize K (equivlent to minimize N nd stisfy (7 nd (8 N = min {N : N Z +} s.t. (7 nd (8 nd D < K 2,D Z+,K Ω, K = L( R IC, (9 N ( R IC NK L D = N +. For given set of code prmeters N, K nd D, the effective code rte of n IC Turbo code is clculted from (7 nd (8 s (N (K D+(K 2D R IC = ( ( K K (N D + 2D N (K D D = N (K D D. (0 When the number of CBs N, the effective code rte R IC K D K R D 0; when the length of the coupled informtion sequence D 0, R IC. IV. DECODING OF THE PROPOSED IC TURBO CODES In the encoding scheme presented in Section III-A, we introduced coupled informtion between every two consecutive CBs. In this section, we first present the decoding scheme for one CB, nmely intr-cb decoding. Bsed on tht, we propose two inter-cb decoding schemes to decode the whole TB by exploiting the coupled informtion between every two consecutive CBs. A. Intr-CB Decoding We cn see from Fig. 2 tht ech CB u n, n {,,N}, consists of three prts of informtion, which re illustrted in Fig. 3(. The first prt is the coupled informtion bits from the previous CB or the inserted dummy bits in the first CB, i.e., u Cn for CB u n, n {2,,N}, or d H for CB u. We cll this prt of informtion s pre-coupled informtion or hed dummy bits, respectively. The second prt is the informtion bits tht re not coupled with other CBs, i.e., u n \u C n for CBs {u n }, n =,,N, oru N for the lst CB. Here u n\u Cn represents the elements in u n exclude the elements in u Cn. This prt of informtion is termed un-coupled informtion. The third prt is the coupled informtion bits to the next CB or the inserted dummy bits in the lst CB, i.e., u Cn for CB u n, n {,,N }, or d T for CB u N. This prt of informtion is nmed post-coupled informtion or til dummy bits. As mentioned in Section III-B tht we consider D < K 2 in this pper, it gurntees ech CB is composite of ll three informtion prts. According to the composition of CB u n, its ssocited decoding block digrm is shown in Fig. 3(b. Let CH Cn nd L C n be the chnnel informtion nd the priori LLR informtion ssocited with the pre-coupled informtion u Cn, CH n be the chnnel informtion bout the un-coupled u C or n d H pre-coupled info. orheddbs CH C Cn L L / n e LC n d H n-thcbu n u n \ uc n or u N ( CH n n-th CB Decoding (b CH C n L / L C n u Cn or d T un-coupled Info. post-coupled info. ortildbs d T e L C n Fig. 3. ( Composition of CB nd (b its ssocited decoding block digrm. informtion, nd CH Cn nd L C n be the chnnel informtion nd the priori LLR informtion ssocited with the post-coupled informtion u Cn. Denote L e C n nd L e C n the extrinsic LLR informtion bout the pre-coupled informtion u Cn nd the post-coupled informtion u Cn, respectively. In ddition, let L d H nd L d T represent the priori informtion of the hed nd the til dummy bits. Recll tht we consider d H = d T = 0 in this pper, L d H = L d T = since the decoder hs perfect knowledge bout these dummy bits. To decode CB u n, the Turbo decoder tkes in chnnel informtion CH Cn, CH n nd CH Cn, nd tkes in the priori informtion L C n (L d H nd L C n (L d T to perform intr- CB itertive decoding. When predefined intr-cb decoding stopping criterion is stisfied, such s predetermined mximum number of Turbo itertions hs been reched or the CB CRC does not detect n error, the decoder outputs û n s n estimtion of CB u n. It lso outputs the extrinsic informtion L e C n (except the first CB nd L e C n (except the lst CB, which will be used in the inter-cb decoding process between CBs. Note tht this intr-cb decoder is lmost the sme s tht for the LTE Turbo codes. The only difference is tht the proposed decoder utilizes priori informtion from djcent CBs nd outputs extrinsic informtion to them. This only ffects the initiliztion nd the outputs of the decoder. Therefore, we cn lmost keep the LTE Turbo decoder for CB unchnged. B. Inter-CB Decoding Bsed on the intr-cb decoding scheme presented bove, we propose two inter-cb decoding schemes to exploit the extrinsic informtion between coupled CBs to improve the TBER performnce. The first decoding scheme, nmely FF- FB decoding scheme, decodes serilly from the first undecoded CB to the lst undecoded CB nd then, if it is necessry, decodes serilly from the lst undecoded CB to the first

6 6 undecoded CB. This FF-FB decoding process cn be repeted for few times to chieve n excellent TBER performnce. However, it suffers from high decoding ltency. As well s this, for long coupling length D, it hs high decoding complexity (more detils will be shown in Section V. To decrese the decoding ltency nd the decoding complexity, we propose WD decoding scheme for the proposed codes. It utilizes window which consists of only two consecutive CBs nd slides from the first CB to the lst CB. It retins good TBER performnce nd hs low decoding ltency nd decoding complexity. Next, we present the proposed inter-cb decoding schemes. FF-FB Decoding Scheme: Let ( L e C n L Cn, n {,, N } denote the extrinsic ( priori informtion ssocited with u Cn, which is forwrded from the n-th CB to the n+-th CB. Let ( L e C n L C n denote the extrinsic ( priori informtion sent from the n+-th CB to the n-th CB. The decoding scheme is described s below: Step Initilize: Let L d H = L d T =. Set the mximum intr-cb nd inter-cb decoding itertions to I CB nd I TB, respectively. Set the current inter-cb itertions to i TB = 0. Step 2 FF Decoding: The inter-cb decoder decodes CBs serilly from the first CB to the lst CB. The forwrd informtion ( L e C n L Cn, n {,,N }, is clculted nd pssed down from the n-th CB to the n+-th CB. The detils re s below: If i TB = 0, for the n-th CB u n, the decoder uses the ssocited chnnel informtion nd the priori informtion L C n bout the pre-coupled informtion bits to estimte the CB. It lso outputs the extrinsic informtion L e C n nd L e C n ssocited with the post- nd the pre-coupled informtion bits u Cn nd u Cn, respectively. Note tht, in this decoding itertion, the bckwrd priori informtion L C n ssocited with the post-coupled informtion bits u Cn is not vilble for the n-th CB, i.e., L C n = 0. Therefore, only one round FF decoding cnnot fully exploit the coupled informtion between CBs. If i TB 0, the inter-cb decoder decodes from the first undecoded CB to the lst undecoded CB by using both priori informtion L C n nd L C n ssocited with the pre- nd the post-coupled informtion bits to decode u n. Other detils re identicl to tht of i TB = 0. Increse the current inter-cb itertions by one, e.g., i TB = i TB +. Step 3 FF TB CRC: Clculte TB CRC bsed on the estimtions {û n },n =,,N. If TB CRC detects n error nd i TB < I TB, go to Step 4; Otherwise, go to Step 6. Step 4 FB Decoding: The inter-cb decoder decodes CBs serilly from the lst undecoded CB to the first undecoded CB. The decoding of ech CB is identicl to the FF decoding with i TB 0. Increse the current inter-cb itertions by one, e.g., i TB = i TB +. Step 5 FB TB CRC: Clculte TB CRC bsed on the estimtions {û n },n =,,N. If TB CRC detects n error nd i TB < I TB, go to Step 2; Otherwise, go to Step 6. Step 6 Output decoded CBs: Output {û n },n =,,N, s the finl estimtions of {u n }. Remrk 3: The proposed FF-FB decoding scheme spreds relible messges from the first nd the lst CBs to other CBs vi FF nd FB decoding processes, respectively. With sufficiently lrge number of inter-cb decoding itertions, ll CBs grdully pproch the decoding performnce of the first nd the lst CBs, which in turn improves the TBER performnce. To show this relible messges spreding phenomenon, we plot the CBER ginst the CBs indices nd the inter- CB itertions i TB in Fig. 4. The TB consists of 7 CBs, the coupling length D = 024, the mximum number of inter-cb itertions is set to I TB = 20 nd the chnnel SNR ρ = 5.5 db. Clerly, it cn be seen tht the first nd the lst CBs hve much better CBER performnce thn tht of other CBs in the first few inter-cb itertions. As the decoding process progresses, ll CBs grdully pproch similr CBER s tht of the first nd the lst CBs. CB error rtes for ll CBs Indices of CBs in TB i TB of FF FB Decoding Fig. 4. CBER performnce of the proposed FF-FB inter-cb decoding scheme ginst CBs indices nd the decoding itertions i TB. The prmeters for the IC Turbo code re K = 644, N = 7, D = 024, L = 4K nd R IC = The proposed FF-FB decoding scheme hs lrge TB decoding ltency s the decoding process hs to go through ll CBs for few itertions to chieve good TBER performnce. This lso prevents using CB CRC s TB-level erly stopping criterion to sve the receiver s computtionl resources. To decrese the decoding ltency nd the decoding complexity, we next propose WD decoding scheme. 2 WD Decoding Scheme: The proposed WD decoding scheme uses window which consists of two consecutive CBs to decode CB nd slides from the first CB to the lst CB. The n-th decoding window, which is used to decode the n-th CB, includes the n-th nd the (n+-th CBs. The decoding process inside the n-th window is described s below: Step Initilize: Obtin L C n, the priori informtion ssocited with the pre-coupled informtion of the n-th CB, from the (n -th decoding window. Let L e C n = 0, i.e., the (n + -th CB does not generte extrinsic informtion bout u Cn yet. Let L C n+ = 0, i.e., without priori informtion

7 7 from the (n+2-th CB. Set the mximum itertions in the decoding window to I WD. Set the current decoding itertions i WD = 0. Step 2 Decode the n-th CB: Let L C n = L e C n, i.e., obtin the priori informtion ssocited with the post-coupled informtion u Cn from the (n+-th CB. Decode the n- th CB using the priori informtion L C n nd L C n, nd the chnnel informtion CH Cn, CH n nd CH Cn. Output the estimted n-th CB û n nd the extrinsic informtion L e C n ssocited with the post-coupled informtion u Cn. Increse the current decoding itertions by one, i.e., i WD = i WD +. Step 3 CB CRC for the n-th CB: Clculte CB CRC bsed on û n. If CRC detects n error nd i WD < I WD, go to Step 4; Otherwise, go to Step 5. Step 4 Decode the (n+-th CB: Let L C n = L e C n, i.e., obtin the priori informtion bout u Cn from the n-th CB. Decode the (n+-th CB using the priori informtion L C n nd L C n+, nd the chnnel informtion CH Cn, CH n+ nd CH Cn+. Output extrinsic informtion L e C n nd L e C n+. Go to Step 2. Step 5 Output the estimtion of the n-th CB: If CB CRC for n-th CB does not detect n error, output û n s the finl estimtion of the n-th CB. Let L C n = L e C n. Move decoding window one CB to the right, i.e., move to the (n+-th CB. If CB CRC for the n-th CB detects n error, stop decoding. Other CBs will not be decoded. Remrk 4: In the decoding process for the n-th CB, the priori informtion L C n+ is lwys zero since the (n+-th CB does not hve the priori informtion bout u Cn+ from the (n+2-th CB. Therefore, the extrinsic informtion L e C n generted by the (n+-th CB is inferior to tht generted by the FB decoding process in the FF-FB decoding scheme. This in turn results in decoding performnce loss compred to the FF-FB decoding scheme. This performnce loss will be shown in Section VII-A. However, the WD decoding scheme lso chieves considerble SNR gin compred to the LTE Turbo codes. Moreover, it decodes CBs on-the-fly, which leds to low decoding ltency nd low decoding complexity. We will discuss the decoding complexity of the proposed decoding schemes in the next section. V. COMPUTATIONAL COMPLEXITY OF THE PROPOSED DECODING SCHEMES In this section, we discuss the decoding complexity of the proposed decoding schemes. As discussed in Section IV-A, the intr-cb decoder for our proposed scheme is lmost the sme s tht for the LTE Turbo codes. Therefore, we count the decoding complexity for length-k CB s n unit nd consider length L TB for the discussion of computtionl complexity herefter. In the LTE stndrd, the TB is segmented into L K length-k CBs 2, which results in mximum computtionl 2 In the LTE stndrd, the TB segmenttion rules result in L K CBs with similr lengths, but not necessrily exct the sme length of K. Here, we ssume ll CBs re of the sme length for simplicity. complexity of L K. Here, x rounds up rgument x to the smllest positive integer. We use this mximum computtionl complexity s reference for tht of our proposed decoding schemes. A. Computtionl Complexity of the FF-FB Decoding Scheme In our proposed encoding scheme, the TB is segmented L+D into K D CBs ccording to (7. Therefore, the mximum computtionl complexity for one FF-FB decoding itertion is L+D K D. Consider mximum number of inter-cb itertions of I TB, the overll computtionl complexity is written s L+D ITB K D p i TB. ( i TB=0 Here, p itb is the frction of undecoded CBs in TB in itertion i TB. Obviously, p 0 = becuse ll CBs need to be decoded in the first itertion. We cn see from ( tht the overll computtionl complexity of the proposed FF-FB decoding scheme is determined by p itb,i TB {0,,I TB }, for given I TB, which is in turn determined by the chnnel qulity, the FER performnce of the underlying Turbo code nd the number of CBs in TB. Unfortuntely, closed-form expression for the FER performnce of the LTE Turbo code is not vilble in generl. As result, we investigte the verge number of decoding for one CB nd the normlized computtionl complexity by simultions for vrious TB lengths L nd vrious coupling lengths D. Some results re shown in Fig. 5. Averge number of decoding for one CB WD FF FB D = 384, N = 5, L = 4K D = 768, N = 7, L = 6K D = 024, N = 7, L = 4K TBER Aprox Es/N0 (db Normlized computtionl complexity WD FF FB D = 384, N = 5, L = 4K D = 768, N = 7, L = 6K D = 024, N = 7, L = 4K TBER Aprox Es/N0 (db Fig. 5. Averge number of decoding for one CB (left hnd side nd normlized overll computtionl complexity (right hnd side for coupling length D = 384,768,024 nd TB length L = 4K,6K,4K. K = 644. It cn be seen from the left hnd side of Fig. 5 tht for coupling length D = 384 to 024, the verge number of decoding for one CB of the FF-FB decoding scheme grows from.05 to 3.3 t TBER level of 0 2. This is becuse when the coupling length D increses, we need more inter-cb decoding itertions to fully exploit the benefits provided by the coupled informtion. However, we will see in Section VII-B tht the incresed verge number of decoding itertions lso increses the SNR gin of our proposed IC Turbo codes over the LTE Turbo codes. The normlized overll computtionl complexity, which is defined s the verge number of decoding for one CB multiplied by L+D K D / L K,

8 U ch V L(U 8 is shown in the right hnd side of Fig. 5. The normlized overll computtionl complexity rnges from.3 to 4 times of the mximum computtionl complexity of the LTE Turbo codes when D increses from 384 to 024. B. Computtionl Complexity of the WD Decoding Scheme As in the FF-FB decoding scheme, the lck of closed-form FER expression of the LTE Turbo code prevents us obtining closed-form expression of computtionl complexity for the WD decoding scheme. Therefore, We lso investigte its computtionl complexity by simultions nd show some results in Fig. 5. We cn see from the left hnd side of Fig. 5 tht for coupling length D = 384 to 024, the verge number of decoding for one CB of the WD decoding scheme grows from.08 to 2 t TB error rte level of 0 2. When D = 384, the WD nd FF-FB decoding schemes hve similr verge number of decoding per CB. However, when D = 768 nd D = 024, the WD decoding scheme hs much lower verge number of decoding itertions. The normlized overll computtionl complexity for the WD decoding scheme is lso shown in the right hnd side of Fig. 5. Generlly speking, for the coupling length D = 384 to 024, the overll computtionl complexity of the WD decoding scheme is bout.35 to 2.43 times of tht of the LTE Turbo codes. Moreover, it cn be seen tht for vrious TB lengths nd reltively lrge coupling length, e.g., D > 384, the WD decoding scheme hs much lower computtionl complexity thn tht of the FF-FB decoding scheme. In prticulr, the WD decoding scheme sves 40% computtionl resources compred to the FF-FB decoding scheme when D = 024. VI. EXIT CHART ANALYSIS In this section, we develop EXIT functions for the LTE Turbo codes nd our proposed IC Turbo codes. We lso propose n upper bound for the SNR gin of our proposed codes over the LTE Turbo codes, which cn be clculted by the developed EXIT functions. A. Decoder Y- Model nd EXIT Chrt Y- SNR Preliminries Y- EXIT chrt is widely used to nlyze the convergence behvior of itertive decoders SNR [23]. It evlutes the reltionship between the verge extrinsic informtion I E generted by soft-input-soft-output (SISO decoder nd its input verge priori informtioni A. Plotting the EXIT functions of two constituent SISO decoders together, one cn clerly nd intuitively predict the itertive decoding behvior of n itertive decoder nd observe the decoding trjectory. U V L( V Convolutionl Encoder Comm. Chnnel Extrinsic Chnnel SISO (BCJR decoder L( U L ( U L e ( U Fig. 6. Decoding model for component CC in Turbo codes. To construct the EXIT function of SISO decoder, generl informtion theoretic decoder model ws introduced in [29]. We dopt this decoder model for component CC in Turbo codes s shown in Fig. 6. Let U nd V represent the rndom vribles with reliztions of u n nd v n. The SISO decoder, which is BCJR decoder, uses chnnel observtion LLRs L(V from communiction chnnel nd priori LLRsL (U delivered by extrinsic chnnel to compute the extrinsic LLRs L e (U s well s posteriori LLRs L(U. The LLRs re defined s in (2. The communiction chnnel models the rel physicl chnnel. It lso tkes into ccount the rtemtching device nd dptive modultor t the trnsmitter side, nd the soft demodultor nd de-rte-mtching device t the receiver side. The extrinsic chnnel is n rtificil chnnel which conveys the priori LLRs L (U from other sources, such s nother constituent decoder nd the decoders for the coupled CBs in our proposed IC Turbo codes. B. EXIT Functions for CCs in LTE Turbo Codes In LTE, code rtes tht lower thn the mother code rte = 3 re chieved by repetition. Denote R REP < the effective code rte fter repetition. According to the LTE rtemtching mechnism [26], the mximum number of repetitions for coded bit in V is written s R0 Ψ =. (2 R REP Let P = [p 0,,p ψ,,p Ψ ] nd p ψ, ψ {0,,Ψ}, be the frction of coded bits in V tht re repeted by ψ times. Then 0, ψ < Ψ p ψ = +Ψ R0 R REP, ψ = Ψ. (3 R REP Ψ, ψ = Ψ Eq. (3 mens tht there re t most two kinds of bits in V which re repeted by Ψ nd Ψ times. Denote them by V Ψ ndv Ψ respectively. Consider physicl AWGN chnnel with SNR ρ, it is known tht the effective SNR for coded bit is proportionl to the number of repetitions for this bit, i.e., the effective SNR is (ψ + ρ for coded bit repeted by ψ times. Assume lrge enough chnnel interlever, V Ψ nd V Ψ cn be considered to be trnsmitted through two independent communiction chnnels with different effective SNRs [24] of ρ Ψ = Ψ ρ nd ρ Ψ = (Ψ+ ρ, respectively. Now, let us consider the verge informtion delivered by the communiction chnnel. Let σ 2 ch = Es 2ρ, σ2 Ψ = Es 2ρ = Ψ E s 2Ψρ nd σ2 Ψ = Es E 2ρ = s Ψ 2(Ψ+ρ be the noise vrinces of the physicl AWGN chnnel, the equivlent communiction chnnel for V Ψ nd the equivlent communiction chnnel for V Ψ, respectively. We convert these vrinces to the vrinces of their ssocited output LLRs [23], i.e., σ 2 ch = 4, σ 2 ch σ 2 Ψ = 4 nd σ 2 σ 2 Ψ = 4. Then we hve Ψ σ 2 Ψ σ Ψ = Ψ σ ch nd σ Ψ = Ψ+ σ ch. (4 Let I ch,ψ be the mutul informtion between V Ψ nd L(V Ψ, nd I ch,ψ be the mutul informtion between V Ψ

9 9 nd L(V Ψ. The verge informtion delivered by L(V bout V is written s 0.9 I ch,rep = I ch,ψ p Ψ +I ch,ψ p Ψ = J ( σ Ψ p Ψ +J ( σ Ψ p Ψ, ( where [23] 0.6 ( + e J (σ = 2/2σ ε σ πσ log 2 ( +e ǫ dǫ. (6 I E Code Rte=/3,/4,/5,/6 Inserting (3 nd (4 into (5 results in ( ( I ch,rep = J Ψ σch +Ψ R 0 R REP ( ( +J Ψ+ σch Ψ. (7 R REP With the verge chnnel informtion I ch,rep, we now construct EXIT functions for the LTE Turbo codes with repetition from tht of the LTE mother Turbo code. As the BCJR decoder is identicl for both codes for the sme CB length, the decoder genertes identicl extrinsic informtion if the chnnel informtion nd the priori informtion received by the BCJR decoder re of the sme vlue nd hve the sme distribution. Therefore, the EXIT functions for the LTE Turbo codes with repetition cn be derived by REP (I (U, σ ch = (I (U, σ ch. (8 Here, REP (I (U, σ ch is the EXIT function for the CC in n LTE Turbo code with repetition, (I (U, σ ch is the EXIT function for the CC in the LTE Turbo mother code, 0 I (U is the priori informtion conveyed by the extrinsic chnnel, nd σ ch is clculted by σ ch = J (I ch,rep = J ( J ( Ψ σch p Ψ +J ( Ψ+ σch p Ψ. (9 In (8 nd (9, we only consider tht the vlue of chnnel informtion is identicl for the LTE Turbo codes with repetition nd the LTE mother Turbo code. Strictly speking, the distribution of the chnnel informtion should lso be identicl for the BCJR decoder to generte the sme output extrinsic informtion. We generte the EXIT functions for the CCs in the LTE Turbo codes with repetition by using (8 nd (9. We lso obtin the EXIT functions through Monte Crlo simultions bsed on rndom repetition 3, i.e., V Ψ nd V Ψ re selected rndomly from V. The results re shown in Fig. 7. It cn be seen tht the EXIT functions constructed by our proposed method re well fitted with the simulted results for vrious code rtes. 3 It hs been shown in [24] tht the EXIT chrts obtined from rndom repetition cn be used s close pproximtion for the deterministic repetition scheme used in LTE Exit Functions from (8 nd (9 Simulted Exit Functions I A Fig. 7. EXIT functions for CCs in LTE Turbo codes with repetition. The code rtes of CCs re 3, 4, 5 nd 6. C. EXIT Functions for CCs in our Proposed IC Turbo Codes Recll tht the intr-cb decoding scheme for the proposed IC Turbo decoder tkes in three prts of informtion to estimte CB. Therefore, the extrinsic chnnel for the CC decoder of the proposed IC Turbo codes consists of three informtion chnnels: the extrinsic chnnel tht conveys the priori informtion I, (U from the other constituent BCJR decoder; the pre-coupled informtion chnnel tht delivers the priori informtion I,2 (U from the previous CB (or dummy bits d H ; nd the post-coupled informtion chnnel tht delivers the priori informtion I,3 (U from the postcoupled CB (or dummy bits d T. Note tht, I,2 (U nd I,3 (U re fixed in the intr-cb decoding process nd evolve in the inter-cb decoding process since we only exchnge informtion between CBs in the inter-cb decoding process. As I,2 (U nd I,3 (U evolve with the inter-cb decoding progress, the EXIT function for the CC decoder in our proposed IC Turbo code becomes series of EXIT functions. This results in big chllenge to clculte n exct EXIT function for the CC decoder in our proposed scheme. Insted, we propose to construct the EXIT function for the cse where the pre-coupled informtion bits nd the post-coupled informtion bits re ssumed to be perfectly known by the decoder. Though this my over estimte the priori informtion I,2 (U nd I,3 (U, it simplifies the construction of EXIT functions for our proposed codes. Moreover, it provides decoding threshold lower bound for our proposed IC Turbo codes. We propose to use this decoding threshold lower bound to estimte the mximum SNR gin of our proposed IC Turbo codes over the LTE Turbo codes. We cll this mximum SNR gin s SNR gin upper bound. Simultion results show tht the proposed SNR gin upper bound hs n ccurcy within 0. db for set of code prmeters, which will be shown in Tble I nd Fig. 0. Now, we construct the EXIT function for the CC decoder in our proposed IC Turbo codes by ssuming perfect priori

10 0 informtion I,2 (U nd I,3 (U. In [22], the uthors proposed n effective wy to construct the EXIT chrt for the CC decoder in DBI Turbo codes by shifting the EXIT chrt for the underlying CC decoder to n opertion point with higher priori informtion. It hs been confirmed by simultion results tht in the AWGN chnnels, the EXIT functions constructed by the proposed method re well fitted with the simultion results when the frction of dummy bits is not higher thn 40%. By ssuming perfect priori informtion I,2 (U nd I,3 (U, our proposed IC Turbo codes cn be viewed s kind of DBI Turbo codes with frction of dummy bits of 2D K. Thus, we dopt this method to construct the EXIT function for the CC in our proposed IC Turbo codes. As the pre-coupled nd post-coupled informtion bits re ssumed to be perfectly known by the decoder, the priori informtion I,2 (U nd I,3 (U re written s I,2 (U = I,3 (U = D K. (20 Since both I, (U nd I,2 (U convey the informtion bout the pre-coupled informtion bits, the redundnt prior informtion D K I, (U should be removed from the totl priori informtion I (U. By the sme token, the redundnt prior informtion D K I, (U in both I, (U nd I,3 (U bout the post-coupled informtion bits should be removed. Recll tht, we consider in this pper tht the pre-coupled informtion bits nd the post-coupled informtion bits of CB re independent. Thus, the priori informtion for the CC decoder is written s I (U = I, (U+I,2 (U+I,3 (U D K I, (U D K I, (U = I, (U+ 2D ( I, (U (2 K Now, the EXIT function for the CC in our proposed IC Turbo codes cn be constructed by IC ( I, (U, σ ch = (I (U, σ ch, (22 ( where IC I, (U, σ ch is the EXIT function for the CC in n IC Turbo code nd (I (U, σ ch is the EXIT function for the CC in the Turbo mother code. 0 I, (U,I (U re relted by (2 nd σ 2 ch is the vrince of L(V. We generte the EXIT functions for the CCs in our proposed IC Turbo codes from tht of the underlying CC by using (2 nd (22. We lso obtin the EXIT functions through Monte Crlo simultions bsed on rndom DBI. The EXIT functions for vrious coupling percentges re shown in Fig. 8. It cn be seen tht the EXIT functions constructed by our proposed method gree with the simultion results well when the percentges of coupled informtion bits re 2.5%, 25%, 33.3% nd 50%. VII. NUMERICAL RESULTS In this section, we present the numericl results of our proposed IC Turbo codes nd the LTE Turbo codes under AWGN chnnels. We first compre the TBER performnce of the FF-FB nd the WD decoding schemes for vrious IC Turbo I E %, 25%, 33.3% 50% 0. Simulted EXIT Functions EXIT Functions From (2 nd ( I A Fig. 8. EXIT functions for CCs in our proposed IC Turbo codes with 2.5%, 25%, 33.3% nd 50% coupled informtion bits. codes. Then the SNR gins of our proposed codes under the FF-FB decoding scheme over the LTE Turbo codes for vrious TB lengths L nd coupling lengths D re investigted. We compre the simulted SNR gins with the proposed SNR gin upper bounds to vlidte our proposed EXIT chrt functions. At lst, we compre the TBER performnce of our proposed codes with the WD decoding scheme to tht of the LTE Turbo codes for vrious TB lengths L nd coupling lengths D. In ll simultions, the CB length K = 644 nd the mximum number of intr-cb decoding itertions I CB = 8 re considered. The mximum number of inter-cb itertions is set to I TB = 20 for the FF-FB scheme nd I WD = 6 for the WD decoding scheme, respectively. A. Comprison of the FF-FB nd WD Inter-CB Decoding Schemes Fig. 9 depicts the TBER performnce of our proposed FF- FB nd WD inter-cb decoding schemes for three IC Turbo codes with D = 384,768,024 nd L = 4K,6K,4K, respectively. It cn be seen from Fig. 9 tht for ll codes, the FF-FB decoding scheme hs better TBER performnce thn tht of the WD decoding scheme t the sme SNR level. As discussed in Section IV-B Remrk 4, this is becuse the extrinsic informtion from the post-coupled CB for the WD decoding scheme is inferior to tht for the FF-FB decoding scheme. Moreover, it cn be seen tht the SNR gin of the FF- FB decoding scheme over the WD decoding scheme increses with the coupling length D. This mens the effect of lcking priori informtion from the (n + 2-th CB for the n-th decoding window becomes significnt s the coupling length D increses. As the FF-FB decoding scheme hs better TBER t the sme SNR level thn tht of the WD decoding scheme, we will use the FF-FB decoding scheme to evlute the performnce limit of our proposed codes next. On the other hnd, the WD decoding scheme hs lower decoding ltency nd lower decoding complexity thn tht of the FF-FB decoding scheme,

11 D= TBER D=024 #$! " D=768 D = 384, N = 5, L = 4K D = 768, N = 7, L = 6K D = 024, N = 7, L = 4K Es/N0 (db Fig. 9. TBER performnce of the proposed FF-FB nd WD decoding schemes with coupling length D = 384, 768, 024 nd TB length L = 4K, 6K, 4K. TBER db 0.52 db 0.26 db Proposed Code: N = 5, D = 384, R IC = 0.38 LTE Turbo code: N = 4, R REP = 0.38 Proposed Code: N = 7, D = 768, R IC = 0.3 LTE Turbo code: N = 6, R REP = 0.3 Proposed Code: N = 5, D = 024, R IC = LTE Turbo code: N = 4, R REP = Es/N0 (db Fig. 0. SNR gins of our proposed IC Turbo codes with coupling length D = 384,768,024 over LTE Turbo codes for the sme code rtes. we will use the WD decoding scheme in Section VII-C to investigte the SNR gins of our proposed codes over the LTE Turbo codes for prcticl purposes. B. Simulted SNR Gins vs the Proposed SNR Gin Upper Bound We use the EXIT chrts developed in Section VI to clculte the decoding thresholds of the LTE Turbo codes nd our proposed codes for vrious code rtes. The proposed SNR gin upper bound in Section VI-C is clculted s the gp between the decoding thresholds of proposed code nd LTE Turbo code with the sme code rte. The clculted results re shown in Tble I. It cn be seen tht the SNR gin upper bound increses with the coupling length D. The simulted TBERs of the LTE Turbo codes nd the proposed codes with the FF-FB decoding scheme re demonstrted in Fig. 0. We cn lern from the simultion results tht for D = 384,768 nd 024, the simulted SNR gins for vrious code rtes re within 0. db from the proposed SNR gin upper bound t TBER level of 0 2. This confirms tht the developed EXIT chrts in Section VI re vlid nd the proposed SNR gin upper bound is tight for vrious coupling lengths. In ddition, our simultion results show tht the proposed FF-FB decoding scheme for our proposed codes cn effectively exploit the benefits introduced by the coupled informtion becuse it chieves the SNR gin upper bound within 0. db. C. TBER Performnce of the Proposed IC Turbo Codes under WD Decoding In this section, we evlute the TBER performnce of our proposed codes with the WD decoding scheme nd compre it to tht of the LTE Turbo codes with the sme code rtes. In prticulr, we investigte the effect of the TB length L nd the coupling length D on the TBER performnce of the proposed codes. Fig. shows the TBER performnce of the proposed codes with D = 024 nd L = 4K, 9K, 4K, nd tht of the corresponding LTE turbo codes. We cn see tht our proposed codes hve considerble SNR gins over the LTE Turbo codes for vrious TB lengths. When L increses from 4K to 4K, the SNR gin increses from 0.44 db to 0.53 db t TBER of 0 2 nd increses from 0.43 db to 0.5 db t TBER of 0. TBER L = 4K Proposed Codes D=024 L = 4K L = 4K L = 4K, R REP =R IC = L = 9K, R REP =R IC = L = 4K, R REP =R IC = 0.29 LTE Turbo Codes L = 4K Eb/N0 (db Fig.. TBER of the proposed codes nd the corresponding LTE Turbo codes for TB length L = 4K,9K,K nd coupling length D = 024. Fig. 2 shows the TBER performnce of the proposed codes with L = 4K nd D = 384,024, nd tht of the corresponding LTE turbo codes. It cn be seen from Fig. 2 tht our proposed codes hve significnt SNR gins for