Optimizing Transmission Lengths for Limited Feedback with Non-Binary LDPC Examples

Transcription

1 Optmzng Transmsson Lengths for Lmted Feedbac wth on-bnary LDPC Examples Kasra Valna, Sudarsan V. S. Ranganathan, Darush Dvsalar*, and Rchard D. Wesel Department of Electrcal Engneerng, Unversty of Calforna, Los Angeles, Los Angeles, Calforna 995 *Jet Propulson Laboratory, Calforna Insttute of Technology, Pasadena, Calforna 99 Abstract Ths paper presents a general approach for optmzng the number of symbols n ncrements pacets of ncremental redundancy n a feedbac communcaton system wth a lmted number of ncrements. Ths approach s based on a tght normal approxmaton on the rate for successful decodng. Applyng ths approach to a varety of feedbac systems usng non-bnary B low-densty party-chec LDPC codes shows that greater than 9% of capacty can be acheved wth average bloclengths fewer than 5 transmtted bts. One result s that the performance wth ten ncrements closely approaches the performance wth an nfnte number of ncrements. The paper focuses on bnarynput addtve-whte Gaussan nose BI-AWG channels but also demonstrates that the normal approxmaton wors well on examples of fadng channels as well as hgh-sr AWG channels that requre larger QAM constellatons. The paper explores both varable-length feedbac codes wth termnaton VLFT and the more practcal varable length feedbac VLF codes wthout termnaton that requre no assumpton of noseless transmtter confrmaton. For VLF we consder both a two-phase scheme and CRC-based scheme. I. ITRODUCTIO The classcal results from [] show that feedbac does not ncrease the capacty of dscrete memoryless channels. However, Polyansy et al. [] and Chen et al. [3] show that capacty can be approached n a smaller number of channel uses usng feedbac. Polyansy et al. [] ntroduce randomcodng lower bounds for varable-length feedbac codng wth termnaton VLFT and wthout termnaton VLF, whch approach capacty wth average bloclengths of hundreds of bts. A communcaton system wthout feedbac, on the other hand, requres thousands of bts to closely approach capacty [4]. Ths paper demonstrates practcal systems usng nonbnary low-densty party-chec B-LDPC codes that match or exceed the lower bounds of []. Most of the analyss n ths paper s not exclusve for B-LDPC codes, but B-LDPC codes are used for demonstraton because they perform well n the short-bloclength regme 5 to 6 bts that s of nterest. In VLFT analyss of [], the recever provdes full noseless feedbac to the transmtter. The transmtter sends addtonal ncremental bts untl t nows the recever has decoded the Ths materal s based upon wor supported by the atonal Scence Foundaton under Grant umbers 65 and 68. Any opnons, fndngs, and conclusons or recommendatons expressed n ths materal are those of the authors and do not necessarly reflect the vews of the atonal Scence Foundaton. Ths research was carred out n part at the Jet Propulson Laboratory, Calforna Insttute of Technology, under a contract wth ASA, and JPL Tas Plan message correctly, resultng n zero probablty of error. The T n VLFT stands for termnaton and corresponds to a noseless transmtter confrmaton TC bt that the transmtter uses to termnate the transmsson. The TC s transmtted through a channel dfferent from the man communcaton channel. In contrast, VLF wthout the T does not have the advantage of an TC. All VLF forward transmssons go over the same nosy channel. Thus, there s always a nonzero probablty of undetected error n VLF. VLF and VLFT are examples of hybrd automatc repeat request HARQ schemes. Pror to Polyansy et al. [] and Chen et al. [3], HARQ feedbac schemes had been studed n great detal n many papers ncludng for example [5] []. These papers provde an overvew of HARQ, dscuss how error correctng codes can be combned wth ARQ and demonstrate applcatons of HARQ. In partcular, [] shows that hybrd ARQ s especally useful n pont-to-pont scenaros. The codng schemes that are most commonly explored n HARQ systems [] [3] are based on convolutonal codes CCs or a concatenaton of turbo and bloc party-chec codes, where the Bahl-Coce-Jelne-Ravv BCJR algorthm s used to determne whch bt s unrelable and needs to be transmtted n the subsequent transmssons. These wors use a gene equvalent to TC n VLFT to termnate transmssons. In order to remove the gene and realze a more practcal system equvalent to VLF [7], [9], [4] [7] consder relablty-based HARQ usng convolutonal codes where the transmsson termnates when the probablty of havng a correctly decoded message s hgh enough. For example, n [9] the relablty metrc s based on the average magntude of the log-lelhood ratos of the source symbols. In [8], [9], Soljann et al. study VLFT HARQ usng ratecompatble bnary LDPC codes. They use maxmum lelhood ML decodng analyss to determne the sze of ncremental transmssons n case of decodng falure. In [], [] Soljann et al. extend ther analyss to tme-varyng bnary erasure channels. Some other hgh-throughput ARQ schemes use rateless spnal codes as n [], [3], where hash functons are used for the subsequent coded symbols. In [4], Romero uses cyclc redundancy chec CRC codes to study the performance of spnal codes n VLF settng. Use of polar codes wth HARQ s also studed n [5], [6]. These wors present polarcode-based HARQ schemes over bnary-nput addtve whte Gaussan nose BI-AWG and Raylegh fadng channels usng Chase combnng.

2 The closest wor to the analyss presented here s by Pfletschnger et al. n [7] whch uses rate-adaptve, nonbnary LDPC codes n a HARQ scheme over Raylegh fadng channel n the VLFT settng. They present two algorthms that use channel statstcs and mutual nformaton to optmze the bloclengths for each transmsson to maxmze the throughput. Based on channel state nformaton at transmtter, the code rates, modulatons, and maxmum number of retransmssons are all optmzed pror to ntal transmsson. Chen et al. [3], [8] and Wllamson et al. [9] analyzed a VLFT scheme based on rate-compatble sphere-pacng wth an ML decoder RCSP-ML and smulated a VLFT scheme usng convolutonal codes. The approxmaton based on RCSP- ML extends sphere-pacng analyss from a sngle fxed-length code to a famly of rate-compatble codes, where each code n the famly acheves perfect pacng and s decoded by an ML decoder. For the -db BI-AWG channel wth feedbac, the convolutonal codes acheve about 95% of the dealzed RCSP- ML throughput R RCSP for average bloclengths up to 5 bts. In [3], Wllamson et al. also analyzed VLF systems for smlar bloclengths of up to bts. However, for average bloclengths of bts and larger, the throughput of the convolutonal code decreases because the frame-error rate performance of the convolutonal code degrades as the length of code ncreases. As Chen et al. menton n [8], codng schemes wth throughput performance close to RCSP-ML n VLFT stll reman to be dentfed for expected latences average bloclengths of to 6 bts. Ths bloclength regme s mportant because t s stll short enough that feedbac provdes a real advantage but also long enough that the system can be practcal. The prmary purposes of ths paper are to show how to optmze the lengths of ncremental transmssons and to demonstrate that B-LDPC codes wth optmzed ncremental transmssons can acheve throughputs close to theoretcal lmts for expected latences of 5 to 5 bts n the VLFT and VLF settngs. Most of the followng analyss s applcable to any codng scheme, but we use B-LDPC codes to demonstrate the possble performance motvated by [3], whch shows that B-LDPC codes wthout feedbac, perform well n ths short-bloclength regme. In our precursor conference papers [3], [33] we prelmnarly analyzed the performance of B-LDPC codes n VLFT for a BI-AWG channel wth an SR of db wth an unlmted number of transmssons and wth the number of transmssons m fxed to be fve. We also consdered two-phase VLF system wth m = 5. In VLFT, the non-bnary LDPC codes of [3] attan 9% to 93% of the predcted RCSP-ML throughput for average bloclengths of 5 to 45 bts. In a VLF scheme of [33] ncorporatng a confrmaton phase after each communcaton phase hence called two-phase, 9% of capacty s acheved n less than 5 bts wth a maxmum of fve transmssons. In ths paper, we extend the results of the prevous papers to consder a broader range of m, the number of possble transmssons. We also ntroduce a new VLF system that uses a stoppng crteron that ncorporates a cyclc redundancy chec CRC. Ths new system acheves better throughput performance than the schemes of [3], [33] for the example BI-AWG channel wth an SR of db n the bloclength regme of 5 to 6 bts. For ths channel, the CRC-based VLF scheme acheves about 94% of the capacty wth an unlmted number m of transmssons and about 9% of the capacty wth m =. We also extend these results to a hgher-sr 8 db channel and use a larger 6 quadrature ampltude modulaton QAM constellaton. The capacty of the 8 db 6-QAM AWG channel s.68 bts per symbol. The VLF-wth-CRC system wth an unlmted number of transmssons acheves a throughput of.37 bts per symbol wth a frame error probablty of less than 3. Ths throughput corresponds to 88% of the capacty n the bloclength regme of about 4 6-QAM symbols. Furthermore, we extend the results to a SR-5dB BI-AWG fadng channel wth the channel state nformaton CSI avalable at the recever. The capacty of ths channel s 7 bts. The VLF-wth-CRC system wth an unlmted number of transmssons acheves a throughput of wth a frame error probablty of less than 3. Ths throughput corresponds to 9% of the capacty n the bloclength regme of about 4 bts. The rest of the paper proceeds as follows: Secton II provdes an overvew of the VLFT system wth B-LDPC codes and the recprocal-gaussan approxmaton for the probablty mass functon of the cumulatve bloclengths. Secton III presents the sequental dfferental optmzaton algorthm SDO for optmzng the sze of each ncremental transmsson n VLFT. Secton IV presents a VLF system wth CRC and analyzes ths system wth an unlmted number of transmssons. Secton V extends the results of Secton IV to the system wth a lmted number of transmssons. Secton VI gves an overvew of the two-phase VLF scheme and uses SDO to optmze the cumulatve bloclength at each decodng attempt. Secton VII compares the throughput and the expected latency of B-LDPC and convolutonal codes n VLFT and VLF settngs. Secton VIII concludes the paper. II. VLFT WITH O-BIARY LDPC CODES Feedbac can facltate capacty-approachng performance at sgnfcantly shorter average bloclengths than systems wthout feedbac. Ths mprovement s made possble by captalzng on favorable nose realzatons to decode early. In case of a bad channel realzaton, the communcaton rate s lowered by transmttng addtonal nformaton untl the attempted rate matches the nstantaneous rate the channel supports. In ths paper, buldng on our precursor conference papers [3], [33], we use hgh-rate protograph-based B-LDPC codes for the ntal transmsson. See [3] for a dscusson of protograph-based LDPC desgn. These short-bloclength codes are rregular, havng mostly degree- and a few degree- varable nodes. Refer to [3] for more dscusson on the specfcaton of the codes. For most of the analyss, the operatng SR n ths paper s db, smlar to the wor of [8], [3], [33]. However, to emphasze the generalty of the approach n ths paper,

3 3 Secton II-C shows results for hgher-sr AWG and fadng channels. It s necessary that the ntal transmsson has a rate hgher than the capacty to tae advantage of good channel realzatons. The codng rate s lowered untl decodng s successful. For example for SR-dB BI-AWG channel, the ntal code can have a rate of.75 to.8 whle the capacty of the channel s 85. We wll consder feedbac systems that transmt ncremental redundancy one bt at a tme and also systems that transmt ncremental redundancy n multple-bt ncrements. For systems that use multple-bt ncrements, a practcal system may lmt the maxmum number m of ncrements. In the context of a specfed m, ths paper optmzes the lengths of the m possble ncrements to maxmze throughput. Secton II-A provdes a detaled descrpton of how we generate each bt of ncremental redundancy for the B-LDPC codes that we use. Then, Secton II-B shows that n the context of ths ncremental redundancy, the codng rate that frst produces successful decodng s closely approxmated by a normal dstrbuton. Knowng a dstrbuton that descrbes the codng rate of the frst successful decodng facltates optmzaton of the lengths of multple-bt ncrements, as descrbed n Secton III. A. Creatng a bt for ncremental transmsson In [3], Valna et al. use B-LDPC codes n a VLFT system wth -bt ncrements. After the ntal transmsson, the transmtter sends one bt at a tme untl the decoder decodes correctly. Tradtonally, rate-compatble codes are desgned by startng wth a low-rate mother code and ncreasng the rate by puncturng the code. The proposed B-LDPC codng scheme n [3] does not explctly nvolve puncturng. Rather, the desgn starts wth a short, hgh-rate B-LDPC code for whch all symbols are transmtted n the ntal transmsson. Each subsequent transmsson s a sngle bt carefully selected to help the decoder as much as possble gven ts current decodng state. The rate s gradually lowered by sendng these addtonal bts, each of whch s a functon of selected bts n the bnary representaton of the non-bnary symbols. A rate- K B-LDPC code over GF m used n a bnary communcaton ln encodes an nformaton sequence of sze Km bts nto a sequence of sze m bts. In order to use an B-LDPC code wth the prmtve element α over bnary-nput channels, each GF m = {, α, α,..., α m } symbol s converted to m bts. For example, consder GF 3 wth the prmtve element of α. Table I shows how each element of GF 3 can be unquely represented n 3 bts g 3, g, g. TABLE I: Bnary representaton of GF 8 elements α α α α 3 α 4 α 5 α 6 Poly. α α α+ α +α α +α+ α + g 3 g g The rate- K non-bnary LDPC codes proposed here ntally encode a sequence of Km bts K GF m symbols nto a codeword of length m bts. Through ncremental redundancy, the rate s lowered from Km m to Km m+b where b s number of addtonal ncremental bts. Each addtonal bt s created by an XOR combnaton summaton n GF of bts n the bnary representaton of one GF m symbol. For each varable node, the recever computes the relablty of each of the m possble combnatons of the bts n the bnary representaton s computed. For example, n GF 3 the relabltes of the seven possble combnatons g, g, g 3, g g, g g 3, g g 3, and g g g 3 are computed for each varable node. Fnally, the sngle combnaton bt that has the least relablty e.g. consderng all seven combnatons for all varable nodes and choosng the leastrelable combnaton for a sngle varable node s requested from the transmtter. Ths s a form of actve feedbac n whch relatvely extensve feedbac tells the transmtter what to transmt n contrast to non-actve feedbac n whch a sngle bt of feedbac ndcates whether to transmt. Ths s a generalzaton of the deas of actve hypothess testng [34]. In [3] Valna et al. compared the performance of a non-actve feedbac system and the actve feedbac system dscussed earler for B-LDPC codes and showed sgnfcantly better performance wth the actve feedbac system. The actve feedbac used n [3] tells the transmtter whch bt combnaton to be transmtted next. Ths actve feedbac scheme does not requre the recever to transmt bac the entre message, contrary to the analyss of []. In the non-actve feedbac scheme of [3] the addtonal bts are selected at random. Ths paper consders both actve and non-actve feedbac. The non-actve feedbac n ths paper corresponds to sendng the XOR of all bts representng one of the varable nodes of the orgnal rate-/ B-LDPC code. Ths predetermned non-actve feedbac system performs close to the system wth actve feedbac snce the actve feedbac of [3] usually ass for the XOR of all bts for the subsequent transmssons. The fgures and results n ths paper ndcate whether actve or non-actve feedbac scheme was used to generate them. The nput frame consstng of K GF m nformaton symbols s ntally encoded by the rate- K B-LDPC encoder nto a sequence of length GF m symbols. These GF m symbols are converted usng ther bnary representatons to bts. The m bts are modulated usng bnary phase shft eyng BPSK and transmtted over an AWG channel. The addtve nose s modeled as an ndependent, zero-mean Gaussan random sequence wth varance. As n [8], SR s calculated as, the rato of the transmsson power to the nose varance. B. Gaussan and recprocal-gaussan Approxmatons Consder a stream of ncremental redundancy as descrbed n Secton II-A arrvng one bt at a tme at the recever after an ntal transmsson of a hgh-rate B-LDPC code. We are nterested n the statstcal behavor of the random varable descrbng the bloclength of the frst successful decodng and the correspondng random varable descrbng the codng rate of that frst successful decodng.

4 4 Probablty Densty Functon p.d.f of bloclength untl the decoder converges to the correct codeword.4 VLFT Smulaton Actve.3 Inv Gaussan Approx Bloclength S Fg. : Emprcal probablty mass functon p.m.f. correspondng to the bloclength requred for successful decodng for the frst tme n VLFT usng GF 56 B-LDPC code over SR-dB AWG channel. Also shown s the recprocal- Gaussan approxmaton of 3 wth µ S = 374 and =.579. Smallest bloclength s = bts wth = 96 nformaton bts so that the ntal rate s R = =.8. Probablty Densty Functon p.d.f. of rate untl the decoder converges to the correct codeword VLFT Smulaton Actve Gaussan Approx Rate R S Fg. : Emprcal p.m.f. correspondng to R S = S computed from Fg. and Gaussan approxmaton of wth µ S = 374 and =.579. Complementary Cumulatve Dstrbuton Functon.8.4. VLFT Smulaton Actve Gaussan Approxmaton Rate R S Fg. 3: Emprcal c.c.d.f. and the approxmaton on the tal of a normal dstrbuton Q-functon correspondng to the shaded area of Fg.. For the system of [3], the VLFT smulaton actve plot n Fg. shows the emprcal p.m.f. of the bloclength of frst successful decodng. The total bloclength S ncludes the ntal bloc and all ncremental transmssons, wth actve feedbac requred for recever to decode the B-LDPC codeword correctly for the frst tme. The VLFT smulaton actve plot n Fg. shows the emprcal p.m.f. of the nstantaneous rate R S = at whch decodng s successful for the frst S tme. Fg. shows that R S s well-approxmated by a normal dstrbuton f RS r = π S e rs S wth mean µ S = ER S and varance S = VarR S. The ntuton behnd these approxmatons s consstent wth the normal approxmaton of the accumulated nformaton densty due to the law of large numbers LL n [4]. To maxmze throughput, the ntal code-rate of the B- LDPC code s chosen so that almost no codeword s successfully decoded n the ntal transmsson. Thus, the emprcal probablty mass functon p.m.f. of the number of addtonal ncrements requred to decode correctly does not have a spe at zero. Fg. 3 shows the complementary cumulatve dstrbuton functon c.c.d.f. for the dstrbuton of R S and the Gaussan approxmaton of Fg.. Fg. 3 confrms that the dstrbuton of R S s well approxmated by a Gaussan dstrbuton. As dscussed later, the emprcal c.c.d.f s used to show that the Gaussan approxmaton s vald for a varety of AWG channels ncludng the hgh SR ones usng larger constellatons and also for fadng channels. The VLFT smulaton actve plot n Fg. 3 shows the emprcal c.c.d.f. of the nstantaneous rate R S = S at whch decodng s successful for SR-dB BI-AWG of [33]. Ths c.c.d.f. plot shows the cumulatve probablty that the channel supports a rate hgher than the rate on the x axs. Ths hgher rate means that the decodng has been successful wth a lower number of transmtted bts. The c.c.d.f. plot corresponds to the shaded area of Fg.. The Gaussan Approxmaton plot of Fg. 3 corresponds to the tal probablty of the standard normal dstrbuton of Fg.. The parameters µ S and S n for a partcular code need to be determned through smulaton and curve fttng. Havng the p.m.f. of the S, the curve fttng process nvolves calculatng the p.m.f. and c.c.d.f. of R S and solvng a lnear regresson problem to obtan µ S and. ote that µ S s not the expected throughput but rather the average of the nstantaneous rates supported by the channel. The cumulatve dstrbuton functon c.d.f. of S s F S n = P S n, and we have F S n = P R S n = P R S = F RS. n n Tang the dervatve of F S usng the Gaussan approxmaton of F RS produces the followng recprocal-gaussan approxmaton for p.d.f. of S : n S f S n = n e πs S. 3 Ths p.d.f as shown n Fg. closely approxmates the emprcal dstrbuton of S. For <, the probablty of the decodng attempt beng successful at bloclength

5 5 Complementary Cumulatve Dstrbuton Functon.8.4. VLFT Smulaton Actve Gaussan Approxmaton Rate R S Fg. 4: Emprcal c.c.d.f. and the approxmaton on the tal of a normal dstrbuton wth µ S =.63 and =.9 of the nstantaneous rate R S = S at whch decodng s successful for SR-8dB 6-QAM AWG channel. but not at usng ths approxmaton s n S f S ndn = n e πs S dn 4 = Q µ S Q. 5 The ncrease n bloclength from to reduces the rate from to. ote that 5 gves the probablty that the channel supports rate whle not supportng the hgher rate. The Q functons n 5 are due to the normally-dstrbuted hghest-rate-of-successful-decodng R S at and. C. General Applcablty of the ormal Approxmaton A smlar Gaussan analyss s obtanable for other channels and dfferent SR values. Fg. 4 shows a smlar complementary cumulatve Gaussan approxmaton for the same GF 56 B-LDPC code of Fg. wth an ntal bnary rate of.8 on SR-8dB 6-QAM AWG channel. Each nonbnary element of the B-LDPC code s mapped onto two 6-QAM symbols. Once agan, the dstrbuton of the nstantaneous rate that the channel supports s well approxmated by a normal dstrbuton. Furthermore, Fg. 5 shows the complementary cumulatve Gaussan approxmaton for the same GF 56 B-LDPC code of Fg. wth an ntal bnary rate of.8 on SR- 5dB BI-AWG fadng channel wth CSI nowledge at the recever. The output of the channel, Y = βx + where the nput X s a bnary phase-shft eyng BPSK modulated sgnal and s the Gaussan nose wth Var =. The average SR of ths channel s. The coeffcent β s a Raylegh dstrbuted random varable satsfyng E[β ] =. The value of β s nown at the recever. The dstrbuton of the nstantaneous rate that the channel supports s agan well approxmated by a normal dstrbuton. Snce the normal dstrbuton approxmaton s vald for varous channels, most of the analyses n the subsequent sectons of ths paper are also vald for varous channels wth dfferent SR values. To further dscuss the generalty of the Gaussan approxmaton on the rate that the channel supports n our feedbac Complementary Cumulatve Dstrbuton Functon.8.4. VLFT Smulaton Actve Gaussan Approxmaton Rate R S Fg. 5: Emprcal c.c.d.f. and the approxmaton on the tal of a normal dstrbuton wth µ S = 6 and =.5 of the nstantaneous rate R S = S at whch decodng s successful for SR-5dB AWG fadng channel. system, consder the accumulated nformaton densty X, Y at the recever at the tme of successful decodng. The expected value of X, Y s the capacty of the channel. For BI-AWG channel, the X, Y s derved as follows: X, Y =log f Y X y x f Y y e y x / =log e y / + e y+ / =log e z / e z / + e z+ / = log + e z+/. 9 For BI-AWG channel, X, Y s a functon only of the nose realzaton z = y x for x = ±, and hence X, Y = z. For each transmtted bt from the B-LDPC code over the channel, there s some amount of nformaton densty accumulated. The total amount of nformaton densty accumulaton I at the recever untl the recever decodes the message correctly s I = z. = The correspondng rate assocated wth the accumulated nformaton densty s R I = I s. As ponted out by [4], s a sum of ndependent random varables for whch the central lmt theorem wll converge qucly to a normal dstrbuton. An mportant consderaton for our approach s whether the rate at whch a practcal decoder succeeds also follows a normal dstrbuton. Ths hnges on the ablty of a rate-compatble code famly as n [35] to operate wth a small gap from capacty over the rate range of nterest. For the prevously dscussed SR-dB BI-AWG channel, Fg. 6 shows the c.c.d.f. of R I and the correspondng Gaussan approxmaton. The rate correspondng to the accumulated nformaton densty at the recever untl the decodng s successful also follows the Gaussan approxmaton.

6 6 Complementary Cumulatve Dstrbuton Functon.8.4. VLFT Smulaton Gaussan Approxmaton Rate R I Fg. 6: Emprcal c.c.d.f. and the approxmaton on the tal of a normal dstrbuton wth µ S = 4 and =.6 of the average accumulated nformaton densty R I = I S at whch decodng s successful for SR-dB AWG channel. average nformaton densty at decodng VLFT Smulaton Ideal Decoder Rate at whch decodng s successful Fg. 7: Average amount of the accumulated nformaton densty for decodng correctly at a partcular code rate for the GF56 B-LDPC of Fg. code over SR-dB AWG channel. Fg. 7 shows the average accumulated nformaton densty for decodng correctly at a partcular code rate for the B- LDPC code. Ths fgure shows on average, how much more nformaton n number of bts the B-LDPC code requres to decode the message correctly compared to the operatng rate. The deal decoder plot n Fg. 7 corresponds to the average accumulated nformaton densty beng equal to the rate the lne of equalty. III. OPTIMIZIG TRASMISSIO LEGTHS Consder the scenaro n whch the number of ncrements pacets of ncremental redundancy assocated wth a codeword that can be accumulated at the recever s lmted to m. Usng the p.d.f. of S from 3 we fnd the optmal bloclengths {,,..., m } to maxmze the throughput. The ntal bloclength satsfes where s the smallest possble bloclength of the orgnal B-LDPC code. Each of the addtonal bts beyond transmtted n the frst transmsson s the exclusve-or of all eght bts representng one of the varable nodes of the orgnal rate-/ GF56 B-LDPC code. The other transmssons use the scheme n Secton II-A to generate the subsequent bts. A. Throughput optmzaton through exhaustve search An accumulaton cycle AC s a set of m or fewer transmssons and decodng attempts endng when decodng s successful or when the m th decodng attempt fals. If decodng s not successful after the m th decodng attempt, the accumulated transmssons are forgotten and the process starts over wth a new transmsson of the frst bloc of symbols. From a strct optmalty perspectve, neglectng the symbols from the prevous faled AC s sub-optmal. However, the probablty of an AC falure s suffcently small that the performance degradaton s neglgble. eglectng these symbols greatly smplfes analyss. Defne the throughput as R T = E[K] E[], where E[] represents the expected number of channel uses n one AC and E[K] s the effectve number of nformaton bts transferred correctly over the channel n one AC. The expresson for E[] s E[] = Q + m [Q µs Q = + m [ Q ] ] m. 3 The rght hand sde of shows the contrbuton to expected bloclength from successful decodng on the frst attempt n the AC. Q S s the probablty of decodng successfully wth the ntal bloc of. Smlarly, the terms n are the contrbutons to expected bloclength from decodng that s frst successful at total bloclength at the th decodng attempt. Fnally, the contrbuton to expected bloclength from not beng able to decode even at m s Q m S whch s shown n 3. Even when the decodng has not been successful at m, the channel has been used for m channel symbols. The expected number of successfully transferred nformaton bts E[K] s E[K] = Q m, 4 m where Q S s the probablty of successful decodng at some pont n the AC. ote that E[K] depends only upon m. In fact, for large values of m, E[K] and thus not senstve to the choce of m Exhaustve search ES can be used to optmze {,,..., m } to maxmze R T = E[K]. The order of E[], where max s the complexty for ES s O max + m maxmum allowable overall bloclength for an AC. Snce E[K], maxmzaton of R T s equvalent to mnmzaton of E[].

7 7 B. Sequental dfferental optmzaton Sequental dfferental optmzaton SDO s an extremely effectve alternatve to ES. Over a range of possble values, SDO optmzes {,..., m } to mnmze E[] for each fxed value of by settng dervatves to zero as follows: {,..., m : E[] =, =,..., m }. 5 For each {,..., m}, the optmal value of s found by settng E[] =, yeldng a sequence of relatvely smple computatons. In other words, we select the that maes our prevous choce of optmal n retrospect. For example to fnd we compute the dervatve E[] = Q and solve for as where For >, follows: E[] =Q = Q S Q E[] + Q + Q S = 6 Q S, 7 = π e S S. 8 = depends only on {,, } as + Q Thus we can solve for as Q + Q = Q Q Q. 9 Actually, for each possble value of, SDO can be used to produce an nfnte sequence of values that solve 5. Each such sequence s an optmal sequence of ncrements for a gven densty of retransmsson ponts on the transmsson axs. As ncreases, the densty decreases. Usng SDO to compute the optmal m ponts s equvalent to selectng the most dense SDO-optmal sequence that when truncated to m ponts results n the hghest throughput. C. Applcaton to VLFT wth m transmssons Table II shows the optmzed {,,..., m }, resultng throughput R T, and expected bloclength λ = /R T for varous m. The values obtaned by SDO are very close to the values obtaned by ES. For m =, 5, 6, and 7, the optmzed bloclengths for both approaches are the same. For m = 3 and 4 the bloclengths dffer only n the value of m shown n bold and only by one bt. Ths small dfference n m causes a neglgble. TABLE II: Optmzed {,,..., m }, R T, and λ from ES and SDO for = 96 bts for VLFT on a db SR bnary-nput AWG channel usng µ S = 374 and =.579. Alg. m {,,..., m} R T λ ES, SDO 58, ES 3 5, 67, SDO 3 5, 67, ES 4 46, 58, 7, SDO 4 46, 58, 7, ES, SDO 5 43, 53, 63, 76, ES, SDO 6 4, 49, 57, 66, 79, ES, SDO 7 39, 47, 54, 6, 7, 8, R T λ VLFT R T for varous values of m VLFT R T m= VLFT R T for m=,3,..., m VLFT λ for varous values of m VLFT λ m= VLFT λ for m=,3,..., m Fg. 8: Throughput R T and the expected bloclength λ as a functon of the number of transmssons m acheved by non-bnary LDPC codes n the VLFT settng for = 96. dfference n the maxmum throughput R T and mnmum expected bloclength λ = R T. Snce the complexty of ES s exponental n m, t s nfeasble to obtan a globally optmal soluton for m > 7; whereas SDO, wth complexty O max, can fnd a soluton wthn seconds even for large m. Fg. 8 shows the optmum R T and λ for varous m usng SDO. The dashed lnes show the maxmum achevable R T and the correspondng mnmum achevable λ wth an unlmted m as n [3]. As a functon of m, R T qucly converges to the m = asymptote and even for m the throughput s close to the value achevable wth an unlmted number of ncrements. Correspondngly, the expected latency also converges qucly and for m the expected bloclength s close to the mnmum λ achevable by unlmted transmssons of one bt at a tme.

8 8 IV. VLF WITH CRC In ths secton, nstead of usng TC as a gene, cyclc redundancy chec CRC codes are used as error-detectng codes to detect whether there s an error n the decoded message. In systems ncorporatng CRCs, a certan number of chec bts, L crc, are computed and added to the nformaton message of length nf. At the recever, the B-LDPC decoder ntally attempts to decode the receved bloc. If decodng results n a codeword, the CRC chec determnes whether the chec bts agree wth the data by computng the checsum from the frst nf bts of the receved sequence and comparng ths checsum wth the last L crc receved bts. In order to acheve an undetected error probablty of ɛ, the CRC code length L crc s chosen so that the overall probablty of error resultng from the B-LDPC and CRC codes combned s smaller than ɛ. The transmtter termnates transmsson when the recever sends feedbac ndcatng that the decoded message passes the CRC chec. If the message s correctly decoded, t passes the CRC and the transmtter moves on to the next message. If the message s decoded ncorrectly and the decoded message fals to pass CRC, the transmtter sends more bts to ncrease relablty of the bts already transmtted. If the recever decodes the message ncorrectly and the erroneously decoded message passes the CRC chec, the transmtter moves on to the next message and the pacet s decoded n error. Ths error s undetected by the recever. In the case of unlmted transmssons m =, the transmtter transmts one bt at a tme untl the decoder ether decodes the message correctly or untl t decodes to a message that passes the CRC chec. Wth a lmted number of transmssons, the bloclength correspondng to each transmsson and the length of CRC are chosen to guarantee a probablty of undetected error of at most ɛ. If the message s not decoded correctly even after m transmssons and the ACKs are correctly receved, the recever deletes all receved symbols and a new transmsson cycle begns wth the transmtter sendng the orgnal bloc of symbols. Snce the CRC as an error detecton tool s used only when the decoder converges to a codeword, t s crucal to dfferentate between erroneous decodng and falure to converge to a codeword. Fg. 9 shows the emprcal p.m.f. of the requred cumulatve number of symbols E untl the recever wll never agan converge to an ncorrect codeword. ote that Fg. 9 s condtoned on the decoder ntally decodng to a wrong codeword at =. The probablty that the decoder decodes ncorrectly at s γ. For the experment that produced the p.m.f. n Fg. 9 γ =.65. For bloclengths larger than E, the decoder ether decodes correctly or fals to converge to any codeword. Ths s a dfferent condton than correct decodng, whch was modeled n Fgs. and. Fg. shows the emprcal p.m.f. of R E = E, the nstantaneous rate at whch the decoder stops decodng to the wrong codeword, and the correspondng Gaussan approxmaton. Fg. shows the state dagram representng all the scenaros that can happen based on our smulatons. Accordng to our Probablty Densty Functon p.d.f. of bloclength untl the decoder does not converge to a wrong codeword.4 VLFT Smulaton.3 Inv Gaussan Approx Bloclength E Fg. 9: Emprcal p.m.f. and recprocal-gaussan ft for the shortest cumulatve bloclength E after whch decodng never agan converges to an ncorrect codeword. The smallest bloclength for the GF56 B LDPC code s = bts wth = 96 nformaton bts. Thus, the ntal rate s R = =.8. Probablty Densty Functon p.d.f of rate untl the decoder does not converge to a wrong codeword VLFT Smulaton Gaussan Approx Rate R E Fg. : Emprcal p.m.f. and Gaussan approxmaton wth µ E = 6 and E =.56 of R E n VLFT settng. State Decoder Converges to a wrong codeword State Decoder does not converge to any codeword Decodng Attempt State 3 Decoder converges to the correct codeword Fg. : The state dagram correspondng to LDPC codng wth ncremental transmssons. smulatons, f the decoder converges to a wrong codeword, t contnues to decode to the same wrong codeword even wth addtonal ncremental transmssons. The ncreased relablty from ncremental transmssons never moves the decoder from one wrong codeword to another wrong codeword. It only helps the decoder ether to converge to the correct codeword or not to converge to any codeword at all. Fgs., correspond to the bloclength and rate of entry to state 3. Fgs. 9, correspond to the bloclength and rate of leavng state. In ths secton, smlar to the case of m = VLFT, the transmtter sends one bt of ncremental redundancy at a tme untl the decoder converges to the correct codeword

9 9 or converges to an ncorrect codeword that passes the CRC chec. We requre an undetected error probablty of smaller than ɛ. If the transmsson starts wth a bloclength of length, the total probablty of error s γ Lcrc, where Lcrc s approxmately the probablty of error that the CRC checs for a wrong codeword. Ths paper uses standard CRC codes. However, for the best error detecton, the CRC codes can be desgned specfcally for a partcular code as shown n [36]. For the error probablty constrant of ɛ, we choose the length of the CRC code so that γ Lcrc < ɛ. For example, f ɛ s set to be 3 and γ =.65, the length of the CRC code L crc = 8 s requred to guarantee the overall probablty of error, γ Lcrc = < ɛ = 3. As wll be llustrated n the results secton Secton VII, the throughput of ths scheme can be well predcted by the results obtaned from VLFT wth unlmted transmssons Secton III modfed by a factor of Lcrc that captures the bac-off n rate due to the CRC overhead. For example, n our prevous analyss from Table II for m =, the rate s 3 whle wth a CRC of length 8, for nf = 96 8 = 88 the rate s predcted to be =.579. As the smulaton results of Secton VII show, the actual acheved rate s.575 wth an undetected error probablty of We wll dscuss these results n more detal n Secton VII. V. VLF WITH CRC AD LIMITED TRASMISSIOS In VLF wth a lmted number of transmssons, the length of each ncremental transmsson should be selected to maxmze R T = E[K Lcrc] E[], where E[] s gven by and E[K L crc ] s the effectve number of transmtted nformaton bts, computed as [ K ] E[K L crc] = K L crc Q m P Lcrc, under the constrant that the probablty of undetected error P Lcrc < ɛ. P s the probablty of convergng to an ncorrect codeword at bloclength. An approxmaton technque smlar to the one used n optmzng the length of each ncremental redundancy bloc n VLFT s used here: [ Q ] m S P Lcrc. The optmzaton problem of maxmzng R T = E[K Lcrc] E[] reduces to mnmzng E[] for each L crc. The SDO technque used n Secton III can be used here under the addtonal constrant that P Lcrc < ɛ. For each L crc, the optmzed {,..., m } values for ths case are dentcal for SDO and ES and the values are gven n Table III. For small values of L crc we need to use a large value of to mae sure P Lcrc < ɛ. As a larger value of L crc s selected, and consequently {,..., 5 } decrease whle the error probablty constrant s stll satsfed. For L crc = 7 the set of {,..., 5 } = {43, 53, 63, 76, } mnmzes the expected latency λ and maxmzes R T. For larger values of L crc > 7, the set of optmum bloclengths does not change and only the overall probablty of error decreases as the CRC length s ncreased. The optmal set of bloclengths for L crc 7 and m = 5 s the same as the set for VLFT and m = 5 from Table TABLE III: Optmzed {,..., m } for m=5 n VLF-wth- CRC usng SDO for dfferent values of L crc. The exact same values were obtaned by ES. L crc {,,..., 5 } λ R T ɛ 93, 98, 5, 6, , 9, 99,, , 85, 9, 3, , 8, 87, 98, , 7, 8, 9, , 64, 7, 84, , 53, 63, 76, , 53, 63, 76, , 53, 63, 76, , 53, 63, 76, II. The ntuton for ths s that once L crc s large enough that decodng decsons are extremely relable, the optmal bloclengths for VLF-wth-CRC should match those of VLFT. Because the bloclengths are dentcal, the throughput R T for m = 5 wth L crc = 7 can be computed by reducng the R T n Table II to account for the overhead of the CRC. The reducton from the m = 5 VLFT rate R T = 3 s where =.559 whch corresponds to the R T from Table III for L crc = 7. Whle both SDO and ES gve the same values for dfferent L crc values, the order of complexty for SDO s OL crc max whle wth ES algorthm the complexty has the much larger order of O L max crc m. As the smulaton results of Secton VII show, the actual acheved rate s.54 wth an undetected error probablty of For the smulatons n Secton VII the CRC code used for L crc = 7 has a polynomal representaton of x9 x 7 +x 3 +. Ths CRC code has been used by Telecommuncaton Standardzaton Sector of the Internatonal Telecommuncatons CCITT whch sets nternatonal communcatons standards. The CRC code used for L crc = 8 has a polynomal representaton of x7 x 8 + x + x + and s used n MultMeda Cards MMC and Secure Dgtal SD cards. VI. TWO-PHASE VLF ow we consder the two-phase VLF model n whch the transmtter source uses the prmary communcaton channel to confrm whether the recever destnaton has decoded to the correct codeword. As n [37], the two-phase ncremental redundancy scheme has a communcaton phase followed by a confrmaton phase. Fg. shows a bloc dagram for the two-phase communcaton scheme. Startng at the left, a message bloc of sze s transmtted communcaton phase. If the destnaton decodes correctly, the source sends a coded forward ACK on the same forward nosy channel to confrm the successful decodng confrmaton phase. If the destnaton decodes ncorrectly, the source sends a coded forward ACK. The ACKs and ACKs are repetton codes of length A symbols and are transmtted over the same forward nosy channel from the transmtter source to the recever destnaton. If the decoder does not converge to any codeword wth symbols,

10 Fg. : Two-phase VLF bloc dagram and the forward transmsson stages n two-phase VLF systems. the transmtter sps the unnecessary confrmaton phase and mmedately transmts the second ncrement of bts. In the two-phase VLF settng, we use the probablty dstrbutons of S, R S, E and R E from Fgs.,, 9, and. The optmzaton problem s to maxmze R T = E[K] E[] where E[K] = wth P EE m = P SS Q m S m = P EE representng the probablty the recever decodes both the message and the ACK erroneously and P SS s the probablty the recever decodes both message and ACK successfully. ote that assumes consstent wth our smulaton results that once the decoder s n state 3 of Fg., t does not return to state even f a forward ACK s ncorrectly receved as a forward ACK. In any case, as n Secton II we assume E[K]. The expected number of symbols transmtted n an AC s, [ ] [ P SS = Q S Q ] S A Q 4 c [ ] [ ] P EE = γ Q E A Q 5 E c [ ] [ P SE = Q S Q ] S A Q 6 c [ ] [ ] P ES = γ Q E A Q. 7 c E In 4 the probablty of decodng correctly at and not at bloclengths smaller than or equal to s Q S Q S and Q A c s the probablty that the ACK s decoded as a ACK, where [ c s the standard devaton of the ] channel nose. In 5, γ Q E E s the probablty of decodng erroneously at. We optmze the bloclengths for two-phase VLF to maxmze R T under the constrant that m = P EE < ɛ, usng both ES and SDO approaches from Secton III for fxed values of {A,..., A m }. For ES we consdered values of m and constraned m to be no larger than the bloclength correspondng to a rate-. code m. For SDO we consdered values rangng from the ntal codng length to 3, whch was the range that gave useful values of ɛ. Table IV shows two sets of {,..., m } wth m = 5 obtaned for dfferent n SDO wth ɛ 3. The optmzed {,..., m } wth ɛ 3 from ES s close to the SDO optmzed bloclengths. The optmzed bloclengths from SDO can also be used as optmzaton lmts for ES algorthm and sgnfcantly reduce the ES optmzaton space. VII. RESULTS E[] = m = +A [ P SS + m m = +P EE P SS + ] +A [ P SE m = P EE +P ES ], 3 where P SE s the probablty of decodng the message successfully but decodng the ACK as a ACK. Conversely, P ES s the probablty of decodng the message erroneously but decodng the ACK successfully. The term multplyng m n 3 s the probablty that an AC ends wthout satsfyng ether of the stoppng condtons. 3 s also approxmated to m m Q S + Pm SE. The probabltes P SS, P EE, P SE, and P ES are computed as follows: Fg. 3 shows R T versus λ for B-LDPC and convolutonal codes usng VLFT. In VLFT wth an unlmted number of transmssons -bt ncrements, convolutonal codes wth ML decoders perform very well at short average bloclengths of up to bts. VLFT schemes have throughputs greater than capacty at short bloclengths because of the TC. Convolutonal codes follow the margnal RCSP-ML wth unconstraned nput plot closely at short-bloclength wth a small gap that s due to the bnary nput for convolutonal codes. TABLE IV: Optmzed {,..., m } for m=5 two-phase VLF usng SDO and ES wth {A,..., A 5 } = {5, 4, 3, 3, 3}. Alg. {,,..., 5 } λ R T ɛ SDO 96 45, 56, 67, 8, E-3 SDO 96 46, 58, 7, 88, E-4 ES 96 46, 58, 7, 84, E-4

11 Expected Throughput R t =3 =6 VLFT AWG Channels, SR = db, Capacty=8 =96 =9 =88 =64 Margnal RCSP ML R RCSP AWG capacty BI AWG capacty VLFT m= B LDPC Actve VLFT m= B LDPC Actve VLFT m=5 B LDPC Actve VLFT m= 64 CC on-actve R t =/λ =6, 3, 64, 96, 9, Average Bloclength λ Fg. 3: R T vs. λ for B-LDPC and 4-state convolutonal codes for VLFT wth m =, m =, and m = 5. % of RCSP ML rate for VLFT Percentage of RCSP ML rate for VLFT VLFT m= B LDPC Actve VLFT m= B LDPC Actve VLFT m=5 B LDPC Actve VLFT m= 64 CC on-actve Expected Latency λ Fg. 4: Percentage of VLFT R T that B-LDPC acheves wth m =, m =, and m = 5. At longer bloclengths of about bts, margnal RCSP- ML rate approaches the capacty. B-LDPC codes outperform convolutonal codes at longer bloclengths because the codeword error rate of convolutonal codes ncreases once the bloclength exceeds twce the tracebac depth [38] whereas the B-LDPC code performance contnues to mprove wth bloclength. The gaps between the throughputs for m =, m = 5, and m = B-LDPC codes are smlar to the gap observed n Fg. 8. For m = the performance of B-LDPC codes n VLFT s much closer to the case of m =. The B- LDPC codes of Fg. 3 are over GF 56. The shortest code for = 96 bts has an ntal bloclength of 5 GF 56 symbols bts, correspondng to an ntal rate of.8. The B-LDPC codes for = 9 and = 88 have ntal bloclengths of 56 and 384 bts, respectvely. Some results for the fnte-m systems follow the non-actve feedbac scheme descrbed n Secton II-A. Fg. 4 shows the percentage of RCSP-ML rate for VLFT acheved by B-LDPC and convolutonal codes n VLFT. In the expected-bloclength range of 5-6 bts, B-LDPC TABLE V: Optmzed {,..., 5 } for two-phase VLF and VLF-wth-CRC wth m=5 at SR db, and correspondng R T and λ values acheved n smulatons. {A,..., A 5 } = {5, 4, 3, 3, 3} for two-phase VLF usng B-LDPC codes. For the convolutonal codes, A = 6, 8, and 9 for = 96, 9, and 88 bts, respectvely. Code {,,..., 5 } λ R T % CRC B 89 43, 53, 63, 76, Phase B 96 46, 58, 7, 84, Phase CC 96 38, 53, 66, 8, CRC B 85 93, 39, 35, 346, Phase B 9 3, 3, 344, 369, Phase CC 9 87, 39, 33, 35, CRC B 8 459, 487, 58, 55, Phase B , 487, 58, 55, Phase CC 88 46, 44, 463, 488, codes acheve a throughput of about 9% of RCSP-ML throughput and about 9% and 96% of unconstraned and bnary-nput capacty, respectvely wth an unlmted number of transmssons. When the number of the transmssons s lmted to and 5, the throughput percentage decreases to about 9% and 85%, respectvely. RCSP-ML analyss s appled to the unconstraned-nput AWG channel at SR - db, for whch the capacty s 84. The capacty of BI-AWG channel at -db SR s 4 whch s about 6% lower than the unconstraned-nput AWG capacty. Table V summarzes the bloclengths that maxmze the throughput n the two-phase VLF and VLF-wth-CRC settngs wth ɛ= 3, for both B-LDPC codes and for comparson tal-btng convolutonal codes. Bloclengths for the B- LDPC codes are obtaned from -3 usng ES on an optmzaton space lmted by ntal SDO results. Bloclengths for the convolutonal codes are based on the coordnate-descent algorthm n [39] usng the assumpton of rate-compatble sphere-pacng. Table V also shows the percentage of BI- AWG capacty obtaned n the two-phase VLF settng wth m = 5 transmssons. For = 9 and 88, the B-LDPC code obtans throughputs greater than 9% of BI-AWG capacty wth an average bloclengths λ of less than 5 bts n the -phase settng. B- LDPC codes n the VLF-wth-CRC settng wth m = 5 acheve throughputs slghtly lower than the ones n the -phase settng wth m = 5. However, smlar to Fg. 8 f m s ncreased to, VLF-wth-CRC results n hgher throughputs. Large values of m lead to a degradaton n throughput performance for two-phase VLF due to the overhead assocated wth the more frequent forward ACK and ACK messages n the confrmaton phase. The rate-/3 convolutonal codes n Table V have octal generator polynomals 7, 7, 55 for the 64-state code and 35, 73, 3747 for the 4-state code [3]. The B- LDPC codes are descrbed completely onlne. Fg. 5 shows the throughput obtaned n the VLF settng for B-LDPC codes, 64-state and 4-state tal-btng UCLA Communcaton Systems Laboratory CSL webste at

12 Expected Throughput R t =3 =6 BI AWG Channel, SR = db, Capacty=4, ε= 3 =88 =84 =8 =96 =64 BI AWG capacty VLF random codng lower bound VLF CRC m= B LDPC Actve VLF CRC m= B LDPC on-actve VLF CRC m=5 B LDPC on-actve VLF Two phase m=5 4 CC VLF on-actve Two phase m=5 64 CC on-actve VLF Two phase m=5 B LDPC Actve Max. rate for fxed length code, no feedbac R t =/λ =6, 3, 64, 88, 84, Average Bloclength λ Fg. 5: R T vs. λ for B-LDPC wth m = n VLFwth-CRC and 64 and 4-state convolutonal codes and B- LDPC codes wth m = 5 n VLF. % Throughput of BI AWG Capacty VLF BI AWG Channel, SR = db, Capacty=4 VLF Lower bound VLF CRC m= B LDPC Actve VLF CRC m= B LDPC on-actve VLF CRC m=5 B LDPC on-actve VLF Two phase m=5 B LDPC Actve VLF Two phase m=5 64 CC on-actve Average Bloclength λ Fg. 6: Percentage of BI-AWG capacty that B-LDPC and convolutonal codes acheve n VLF. convolutonal codes wth m = 5, m =, m = for ɛ = 3. As the bloclength ncreases, as mentoned n [8], the performance of the codes n VLF gets closer to the performance n VLFT. The plots for m = 5 are from Table V. Wth m =, the = 89 the B-LDPC code acheves a throughput greater than the random codng lower bound obtaned from the analyss n []. Fg. 6 shows the percentage of the capacty of the BI- AWG channel at -db SR acheved by B-LDPC and convolutonal codes usng VLF. In the expected bloclength range of 3-5 bts, B-LDPC codes wth CRC acheve a throughput of about 94% of capacty wth an unlmted number of transmssons. When the number of the transmssons s lmted to, the throughput percentage decreases to about 93%. For m = 5, B-LDPC codes perform slghtly better n two-phase VLF settng than n VLF-wth-CRC. ote that for m = or even m = two-phase VLF wll not perform well because of the overhead assocated wth the confrmaton messages. As dscussed n Secton II-B smlar Gaussan approxmaton analyss can be done for hgher-sr AWG channels. for nstance, for SR-8dB AWG channel whch uses a larger 6-QAM constellaton, the VLF-wth-CRC system wth an unlmted number of transmssons acheves a throughput of.37 bts per symbol wth a frame error probablty of less than 3. Ths throughput corresponds to 88% of capacty n the bloclength regme of 4 6-QAM quadrature ampltude modulaton symbols. Furthermore, the VLF-wth-CRC system on 5-dB BI-AWG fadng channel wth an unlmted number of transmssons acheves a throughput correspondng to 9% of capacty n the bloclength regme of about 4 bts. VIII. COCLUSIO Ths paper uses the recprocal-gaussan approxmaton for the bloclength of frst successful decodng to optmze the sze of each ncremental transmsson to maxmze throughput n VLFT and VLF settngs. For feedbac wth a lmtaton on the number of transmssons, the sequental dfferental optmzaton SDO algorthm can be used qucly and accurately to fnd the optmal transmsson lengths for a wde range of channels and codes. In ths paper we appled SDO to nonbnary LDPC codes for a varety of feedbac systems. We focused on the bnary-nput AWG channel but verfed the effectveness of the Gaussan approxmaton and SDO on the standard AWG channel wth a 6-QAM nput and on a fadng channel. In the 3-5 bt average bloclength regme, ths paper reports the best VLFT and VLF throughputs yet. VLFT throughputs are hgher than VLF, but VLF s more practcal because t does not assume a noseless transmtter confrmaton symbol. For VLF-wth-CRC wth m =, B-LDPC codes wth optmzed bloclengths acheve about 94% of the capacty of -db BI-AWG channel for an average bloclength of 3-5 bts. In the same bloclength regme, for VLF-wth-CRC wth m =, B-LDPC codes wth optmzed bloclengths acheve about 93% of the capacty. The performance results can also be consdered n terms of SR gap. In Fg. 6, the random-codng lower bound for a system wth feedbac s.7 db from the Shannon lmt for = 8 wth a bloclength of less than 5 bts. Loong at the VLF-CRC B-LDPC codes for = 8 n Fg. 6, the m = B-LDPC code s.53 db from Shannon lmt. The B-LDPC non-actve feedbac system n Fg. 6 uses ten rounds of sngle-bt feedbac to operate wthn 5 db of the Shannon lmt wth an average bloclength of less than 5 bts. Smlar analyss can also be done for hgher-sr AWG and fadng channels. REFERECES [] C. E. Shannon, The zero error capacty of a nosy channel, IRE Trans. Inf. Theory, vol., no. 3, pp. 8 9, Sep [] Y. Polyansy, H. V. Poor, and S. Verdú, Feedbac n the nonasymptotc regme, IEEE Trans. Inf. Theory, vol. 57, no. 8, pp , Aug.. [3] T.-Y. Chen,. Seshadr, and R. Wesel, A sphere-pacng analyss of ncremental redundancy wth feedbac, n IEEE Int. Conf. Commun. ICC, June, pp. 5.