Optmzng HARQ Feedback and Incremental Redundancy n Wreless Communcatons Ma Zhang, Andres Castllo, Borja Peleato Electrcal and Computer Engneerng Purdue Unversty West Lafayette IN 797 Emal: {maz,castl,bpeleato}@purdue.edu Abstract Wreless networks have been adjustng ther transmt power, modulaton order, and codng rate based on the channel condtons for a long tme. Such adaptve protocols am to maxmze the overall throughput by strkng a trade-off between transmttng as much nformaton as possble and mnmzng the probablty of loosng such nformaton. However, there s lttle lterature on adaptve retransmssons when falures occur. Ths paper studes the trade-offs that hybrd automatc repeat request (HARQ) protocols face when choosng the type and amount of ncremental redundancy (IR) that should be sent when the decodng of a data block fals at the recever. It proposes a method to optmze such choce for a system wth non-deal error correctng codes (ECC) and lmted feedback capabltes. Addtonally, t s shown through smulatons that the overall data rate can be sgnfcantly ncreased by bundlng the acknowledgements. I. INTRODUCTION The densty of wreless devces and ther traffc requrements have experenced an exponental growth durng the past two decades, and t does not seem lke ths trend wll slow down anytme soon. In order to accommodate such demand, researchers are pushng the lmts of the avalable systems to squeeze as much throughput as possble. Instead of usng conservatve confguratons capable of supportng the requred data streams n a wde range of condtons, most systems are now usng adaptve schemes to maxmze the bandwdth effcency n every scenaro. A sgnfcant amount of work has been devoted to desgnng algorthms for adaptng physcal layer parameters such as the transmt power, modulaton, codng rate, etc. [], [], [3] However, there s lttle lterature on adaptve retransmssons when falures occur. Tradtonal automatc repeat request (ARQ) schemes operate as follows: when packets are correctly decoded the recever acknowledges them wth a short ACK packet, and when the decodng fals (e.g., the LDPC decoder does not converge or the CRC for the decoded packet s ncorrect), the recever can ether send a NACK packet or wat for the transmtter s tmeout to expre as f the packet had been lost. If the transmtter receves a NACK or fals to receve an ACK before the tmeout, the whole packet s retransmtted, assumng that ts data s stll relevant. Retransmttng the whole packet s justfed when the prevous one has been completely lost, but n many cases the In some tme senstve applcatons, such as VOIP, the data needs to be receved wthn a certan tme nterval, otherwse t s useless. recever s able to recover sectons of the packet or obtan nformaton that can be leveraged n subsequent transmssons. In those cases, t s often enough for the transmtter to send a few addtonal bts, referred to as ncremental redundancy (IR), whch the recever can use to recover the whole packet. Ths s commonly known as Type-II hybrd automatc repeat request (Type-II HARQ) []. Hybrd ARQ s not a new concept. Its throughput has been upper-bounded under the assumpton of unlmted sngle bt IR and perfect feedback [5] and several methods have been proposed to optmze the IR block lengths under a fnte number of retransmssons [6], [7], [8], [9]. However, most of the work has focused on dealzed error correctng codes (ECC), known channels, and ether nfnte or sngle bt feedback. Ths paper proposes a method to optmze the length and type of the IR bts n a more realstc scenaro wth mperfect ECC codes, lmted feedback, acknowledgement bundlng, and an overhead cost for each round of ncremental redundancy. Sendng an ndvdual ACK/NACK for each transmtted data packet can be rather neffcent, so many systems have started bundlng the packets nto groups that can be acknowledged wth a sngle bt []. Ths paper uses bundlng, but for a dfferent purpose. A sngle bt of feedback can only convey ACK/NACK nformaton. If the packet (or bundle) has been successfully receved (ACK), the transmtter can delete t from ts memory, otherwse t wll send a fxed block of IR bts or retransmt the whole packet. The recever does not have enough capacty to specfy whether t needs the whole packet to be retransmtted or just a few addtonal party bts to correct the remanng errors. We wll use bundlng to ncrease the effectve feedback capabltes. Instead of sendng a small number of feedback bts for each packet, we wll send a larger number of bts to convey nformaton about the bundle as a whole. It s obvous that, for a fxed number of feedback bts per bundle, smaller bundles wll provde hgher throughput than large ones. Large bundles have a hgher probablty of sufferng a decodng falure and, even f only one codeword fals, the transmtter wll need to send ncremental redundancy for all of the others. However, n practce the number of feedback bts s lmted, e.g. to one per codeword. Ths paper wll seek a tradeoff between havng small bundles wth lmted feedback capabltes or large bundles wth the ablty of makng multple dfferent requests.
The paper wll be organzed as follows. Secton II wll ntroduce the system model and communcaton lnk parameters that wll be used throughout the rest of the paper. Secton III wll dscuss the dfferent types of IR bts that wll be consdered and how each of them wll be used to mprove the avalable nformaton on a gven bundle. Secton IV wll propose a method for adaptvely choosng the number and type of IR bts requested by the recever based on the current state (number of decodng falures, codng rate, and SNR of the prevously receved transmsson). Fnally, Secton V wll provde smulaton results llustratng the performance of the proposed scheme and Secton VI concludes the paper. II. SYSTEM MODEL In order to optmze the aforementoned parameters for a gven communcatons system, we wll need to characterze some of the components of such system. A. Channel Wreless systems probe the channel perodcally, adaptng the code rate and modulaton to reman wthn a certan frame error rate (FER) range. However, t s mpossble to obtan a perfect characterzaton of the channel for every codeword. Sudden obstructons or beam msalgnment n mmwave lnks can temporarly degrade the qualty of the channel, untl the next channel soundng cycle. There exsts a certan amount of correlaton n the channel experenced by adjacent codewords (adjacent n tme or frequency). For smplcty, the channel wll be modeled as an nterference-free AWGN wth varable SNR. It wll be assumed that all the codewords n a gven bundle experence the same SNR, but t s possble that subsequent transmssons of ncremental redundancy are receved wth dfferent qualty. B. Modulaton When the SNR s hgh, the transmtter can group multple bts n a sngle modulaton symbol so as to ncrease throughput. Most communcaton systems are lmted by the average transmt power, so ampltude modulatons are typcally chosen snce they offer lower probablty of error than phase modulatons. There are multple ways n whch bts can be mapped to the symbol values. Fg. shows a 6-QAM constellaton wth two mappngs: Gray and superposton. Gray mappng offers lower bt error rate, but t s slghtly harder to mplement. Gray Superposton FIGURE : Mappng of bts to 6-QAM constellaton. Unfortunately, ampltude modulatons wth more than two bts per symbol yeld dfferent probabltes of error for dfferent bts, regardless of the mappng. Ideally, these symbols could be encoded wth non-bnary error correcton codes, whch are not affected by the dfferences between ndvdual bts. The performance wth jont demodulaton and decodng could then approach capacty [], but the computatonal requrements of non-bnary decoders render them mpractcal for many applcatons. Bnary error correctng codes cannot capture the correlaton between bts n the same symbol, havng to deal wth ther margnal dstrbutons nstead. The decoder therefore operates as f the bts came from ndependent BPSK modulatons wth constant SNR throughout each codeword. Hgh-order modulaton symbols wll be constructed usng bts from dfferent codewords, wth dfferent codng rates accordng to the probablty of error that they wll suffer. That s, n a stream of 6-QAM symbols, good bts (MSB n Fg. ) wll be grouped n codewords wth hgh rate and bad bts (second bt n Fg. ) n codewords wth lower rate. By splttng the bts n ths way nstead of mxng good and bad ones n the same codeword, we obtan a slght performance gan []. C. Error Correcton Most standards and prototype systems have chosen to use bnary QC-LDPC codes due to ther outstandng performance and parallel archtecture, whch allows very effcent encodng and decodng usng parallel shft regsters [3], []. Furthermore, LDPC codes can be easly punctured or extended to ncrease or decrease ther nformaton rate, respectvely. LDPC codes are characterzed by a sparse party check matrxh {,} (n k) n, such thathx = for all codewords x. Receved channel values (.e. matched flter output) are processed to obtan a log-lkelhood rato (LLR) for each ndvdual bt b as l = LLR(b r ) = log p ( r ) p ( r ), () where p ( r ) and p ( r ) respectvely represent the probablty of b beng or, gven the correspondng receved value r. These LLRs are then progressvely refned by teratve message passng over the Tanner graph of the H matrx, untl they converge to a feasble codeword or the algorthm reaches a maxmum number of teratons. LDPC decoders very rarely converge to a wrong codeword; t s much more lkely that they smply fal to converge by the maxmum number of teratons. Our smulatons wll focus on the QC-LDPC code of length n = 68 and k = 3 (rate /3) proposed n the 3GPP standard for 8.n [5], but the technques proposed here could be appled to any other code by adjustng the FER characterstcs. As shown n [], the FER for ths code (and extensons) can be well approxmated by ( ) µ R P e (R,SNR) = Q, () σ where µ =. SNR.7 +.86,σ =. SNR..8, SNR s n lnear scale, and R represents the code rate.
D. System FIGURE : Types of ncremental redundancy. Ths paper consders a pont to pont lnk wth a noseless low-rate feedback channel used for acknowledgng packets or requestng ncremental redundancy. It wll be assumed that n the deal case there s one bt of feedback avalable for each codeword, gvng us 6 possble feedback messages for a bundle of. However, t may turn out that our polcy does not requre that many feedback messages per bundle, n whch case the requred number of feedback bts can be even lower. Furthermore, our smulatons wll perform an unlmted number of ncremental redundancy rounds, wth a constant overhead penalty for each of them. That means, f any codeword n a bundle fals, the recever wll keep requestng addtonal bts untl the whole bundle can be successfully decoded, but each addtonal request wll have an addtonal cost on top of that ncurred by the requested bts. III. INCREMENTAL REDUNDANCY The term Incremental Redundancy (IR) ncludes any addtonal bts requested by a recever so as to attempt the decodng of a codeword (or bundle of codewords) that had prevously faled. As shown n Fg., these bts can be of dfferent types, dependng on how they are constructed: ) Chase combnng [6]: retransmt some of the bts prevously sent. The recever can use maxmal rato combnng to mprove ther effectve SNR. It has the advantage of smplcty, snce there s no need to change the decoder, but t offers suboptmal performance. ) Codeword party (or extenson) bts: generate new party for a specfc codeword by extendng the matrx H wth new rows and columns representng combnatons of bts not prevously used. There are ways to avod complcatng the decoder, but t cannot take advantage of correlaton between codewords. 3) Bundle party bts: construct a btwse erasure code across multple bundled codewords [7]. The decodng s slghtly more complcated, but t couples multple codewords together, makng t possble to explot ther correlaton and leverage the receved values. Practcal LDPC decoders have lmted memory, often nsuffcent to handle a jont decodng of all the codewords and IR n a bundle. Therefore, we wll assume that each codeword s decoded ndependently, after refnng the nput LLR values wth Chase combnng IR and bundle party bts. Codeword party bts, however, are part of the codeword and can be drectly fed to the LDPC decoder. If the same bt b s receved twce (Chase combnng) wth ndependent nose components, r () = b+n () and r () = b+ n (), thenp ( r (),r () ) s proportonal top ( r () )p ( r () ). The same apples for b =. Hence, the jont LLR can be found by smply addng the ndvdual LLR values for each transmsson. In the AWGN paradgm, LLR(b r ) = SNR r, so Chase combnng effectvely ncreases the SNR of the transmtted bts to SNR CC = SNR +, (3) SNR IR where SNR and SNR IR represent the SNR of the orgnal and ncremental redundancy transmsson. Smlarly, we can model the effect of the bundle party bts as an ncrease n SNR for the faled codewords n the bundle. Suppose that a vector of n bts from dfferent codewords, b = [b,b,,b n ], and ther XOR, x = b b n, are transmtted through AWGN channels wth dfferent SNR, resultng n receved valuesrandr x, respectvely. The probablty of a specfc bt b beng condtoned on these receved values can be found as ( ) n p ( r,r x ) = p j (d j r j ) p x ( d r x ) j= ( ), d {,} n d = v {,} n n p j (v j r j ) j= p x ( v r x ) () wherep x ( v r x ) represents the probablty thatx = v v n gven the receved value r x. Unfortunately, the number of terms n Eq. () ncreases exponentally wth the bundle sze, whch makes t mpractcal. In our smulatons, we use a mnsum approxmaton smlar to that used n a mn-sum LDPC decoder [8]. Specfcally, the updated LLR s calculated as n+ lk new = l k + sgnl mn l, (5) = k =...n+ k where l n+ represents the LLR value for x = b. Fg. 3 shows the SNR resultng from such an update, as a functon of the number of faled codewords and assumng that both the orgnal transmsson and the IR experenced dentcal SNR. LDPC decoders can fal to converge, but t s extremely rare for them to converge to the wrong codeword. Successfully decoded codewords can therefore be substtuted n the bundle party equatons, reducng them to combnatons of the faled codewords. The LLR values for the faled codewords wll then be updated usng Eq. (5) before attemptng to decode them. If the decodng of any prevously faled codeword succeeds, the LLRs can be updated agan. The decoder does multple rounds of updatng LLRs and decodng, untl ether all codewords n the bundle are successfully decoded, or some of the codewords repeatedly fal regardless of the effort of updatng LLRs. We
SNR after XOR (db) AWGN channel, LDPC code wth n=68 and rate /3 5.9.8 Number of falures (f) 6 8 6 3 3 3 SNR (db) FIGURE 3: SNR after updatng the LLRs of f bts based on a transmsson of ther XOR. call the former case a successful bundle, and the latter a faled bundle or a bundle error. Fnally, codeword extenson bts reduce the rate of the code. The effect that ths has on the probablty of decodng success s ntrnsc to the code beng used. Typcally, ths performance s measured under the assumpton that both the transmtted codeword and ts assocated IR are receved wth the same SNR, but ths may not be the case n practce. Decodng falures often happen due to sudden falls n SNR, whereas IR s generally transmtted wth hgher SNR by adjustng the modulaton order, transmt power, re-algnng the beam, re-estmatng the channel, etc. Our dervatons can be greatly smplfed by defnng the effectve SNR of a codeword as SNR eff = ( E [ SNR 3 ]). (6) Assumng that the energy per bt E b s constant throughout the codeword, the effectve SNR s just E b dvded by the average nose power. Fg. shows the probablty of decodng falure for dfferent nose power mxtures wthn a codeword. It shows that the probablty of falure manly depends on the effectve SNR, not on the SNR varance. The sold curves, whch can barely be dstngushed, all have the same effectve SNR whle the dashed ones llustrate the effect of a 5% ncrease and decrease n SNR. Summarzng, Chase combnng and bundle party bts can be used to ncrease the SNR for some bts n the faled codewords, accordng to Eq. (3) and Fg. 3. Ths wll n turn ncrease the effectve SNR of those codewords, whle extenson IR bts reduce ther code rate. IV. OPTIMIZATION Ths secton descrbes a method to optmze the type and number of IR bts requested based on the nformaton avalable FER.7.6.5..3.. =.5σ = σ =.5σ for 5%, =.75σ other =.5σ for %, =.95σ other =.5σ for %, =.5σ other =.75σ..3..5.6.7.8.9 σ FIGURE : Probablty of decodng falure as a functon of nose power, for a sgnal wth E b =. Sold curves, correspondng to cases wth the same SNR eff, are very close. at the recever, so as to mnmze a gven cost functon. In order to make the problem manageable, the SNR and rate R are quantzed to take a fnte number of values. Furthermore, the number of IR bts requested s also restrcted to a small pre-defned dscrete set, so as to lmt the number of feedback bts requred to make such request. The HARQ protocol for a bundle of codewords can then be modeled as a Markov Decson Process (MDP) wth a fnte set of actons and states: State: s = (f,snr,r), where f represents the number of codewords n the bundle whose decodng faled, SNR ther average sgnal to nose rato, and R ther codng rate. If the faled codewords have dfferent SNR, we take the lowest one. Acton: A(s) = (α, β), where α represents the number of extenson bts requested (for each codeword n the bundle) and β represents the number of bundle party bts requested. No Chase combnng bts wll be requested snce extenson bts always offer better performance for our system model [6]. Cost: C = bα+β+i {α+β} K, where b denotes the number of codewords per bundle, and I x represents an ndcator functon takng value when x and otherwse. Ths cost counts the number of IR bts sent n a certan round, plus a penalty K to account for the overhead. The objectve s to mnmze the total cost untl success, whch n effect maxmzes the overall throughput,.e. we wsh to fnd A(s) = argmne{total cost s,α,β} (7) α,β for all s. Acton (α,β) wll reduce the rate and ncrease the SNR, takng the system from state s = (f,snr,r ) to s = (f,snr,r ), wth f f, SNR gven by Fg. 3,
and R = k k +αr R (8) Polcy Alpha, Rate =. 5 wherek s the number of nformaton bts n a codeword.snr and R are assumed to be determnstc, but the number of falures f follows a bnomal dstrbuton wth pmf ( ) f P(f s,α,β) = p f ( p) f f, (9) f where p represents the probablty of decodng falure wth the IR. It s mportant to realze that ths decodng wth IR cannot be treated as an ndependent attempt; the probablty of falure should be condtoned on the fact that the decodng faled wthout IR. Therefore, p = P e(snr,r ) P e (SNR,R ), () where P e (SNR,R) represents the probablty of decodng falure for a sngle codeword wth the gven effectve SNR and rate, approxmated n Eq. (). The optmal number of bts to be requested when the system s n state s can be found as A(s) = (α (s),β (s)) = argmne{total cost s,α,β} α,β = argmn α,β bα+β + s P(s s,α,β)v(s ), () Falures Falures 3 3 -.5 - -.5 - -.5.5.5.5 SNR (db) FIGURE 5: Polcy obtaned for α at rate 9 Polcy Beta, Rate =. -.5 - -.5 - -.5.5.5.5 SNR (db) FIGURE 6: Polcy obtaned for β at rate 9 5 6 5 3 where P(s s,α,β) denotes the probablty of transtonng from s to s by sendng (α,β) bts, and V(s) denotes the expected total cost to get from state s to success: V(s) = E{Total cost s,a(s)} = bα (s)+β (s)+i {A(s)} K + s P(s s,a(s))v(s ). () The transton probablty P(s s,a(s)) s gven by Eq. (9) f Fg. 3 and Eq. (8) hold, and s otherwse. Wth the states dscretzed to a fnte number of values, we can use the value teraton algorthm [9] to optmze V(s) and A(s) for alls. Essentally, t starts wth a random value functon and alternates between updatng the polcy A(s) accordng to Eq. () and the value V(s) accordng to Eq. (), untl convergence. At that ponta(s) stores the polcy to be followed whenever the system s n state s. V. NUMERICAL RESULTS Ths secton provdes smulatons llustratng the polces obtaned wth the proposed method and comparng ther data rate wth that obtaned by sendng fxed amounts of IR for each ndvdual codeword. For ths purpose, the codewords were bundled n groups of b = and the overhead penalty was set at K = 3. Ths overhead can nclude actual bts sent (e.g. packet headers) as well as vrtual penaltes for the addtonal latency or complexty. Fgs. 5 and 6 show the polces obtaned for α and β at rate R = 9,.e. the number of extenson bts (α) and bundle party bts (β) to be requested after havng already receved 3 extenson bts, as a functon of the number of faled codewords remanng n the bundle and the effectve SNR of those codewords. It can be observed that, as the SNR decreases, the number of bts to be requested ncreases. Ths makes sense, snce hghly corrupted bundles wll requre more IR for successful recovery. Also, when the number of falures s small our polcy suggests requestng bundle party bts nstead of extenson bts. Agan, ths makes sense snce the recever cannot convey to the transmtter whch codewords faled. If t requests extenson bts, the transmtter wll have to send them for every codeword n the bundle, even for those that have already been successfully decoded. Furthermore, the gan provded by bundle party bts s much hgher when the number of falures s low, as shown n Fg. 3. The polces we obtaned only have 6 possble combnatons of (α,β), so bts of feedback are suffcent to specfy the retransmsson strategy. Therefore our proposed polcy requres feedback bt per codeword, whch s comparable to tradtonal fxed IR schemes wth ndvdual acknowledgments. Fg. 7 compares the data rate of these schemes. The data rate s computed by dvdng the number of nformaton bts correctly delvered by the total number of bts transmtted, ncludng an overhead ofk = 3 bts for each IR round. It can be observed that our polcy provdes sgnfcantly hgher data
Average rate.7.6.5..3.. Comparson of ncremental schemes, K = 3 Orgnal codeword, wthout IR Fxed IR = Fxed IR = Fxed IR = Our polcy, bundle = -.5 - -.5 - -.5.5.5 SNR (db) FIGURE 7: Average data rate for dfferent retransmsson schemes. rate than fxed IR ones, regardless of what ths fxed number s and the SNR. The dfference s specally large n the low SNR regme, snce our polcy can request many IR bts n the frst round whereas fxed IR ones requre multple IR rounds ncurrng n large overhead penaltes. VI. CONCLUSION Ths paper proposes modelng the allocaton of IR bts n a HARQ protocol as a Markov Decson Process seekng to mnmze a pre-determned cost functon. It descrbed how the problem should be formulated and solved, resultng n a set of polces parameterzed by the number of falures per codeword bundle, effectve SNR of the receved codewords, and codng rate. Smulaton results show that the proposed method provdes a sgnfcant ncrease n throughput wth respect to tradtonal fxed-length HARQ schemes, where codewords are ndvdually acknowledged nstead of bundled. The ncreased flexblty n requestng dfferent numbers and types of IR bts more than makes up for the fact that the recever cannot specfy whch codewords have faled and whch ones have been successfully decoded. ACKNOWLEDGMENT Ths work was supported by AFRL and DARPA under grant 888 and by Noka Networks. REFERENCES [] F. Peng, J. Zhang, and W. E. Ryan, Adaptve modulaton and codng for IEEE 8. n, n Wreless Communcatons and Networkng Conference, 7. WCNC 7. IEEE. IEEE, 7, pp. 656 66. [] C. U. Castellanos, D. L. Vlla, C. Rosa, K. I. Pedersen, F. D. Calabrese, P.- H. Mchaelsen, and J. Mchel, Performance of uplnk fractonal power control n UTRAN LTE, n Vehcular Technology Conference, 8. VTC Sprng 8. IEEE. IEEE, 8, pp. 57 5. [3] B. Furht and S. A. Ahson, Long Term Evoluton: 3GPP LTE rado and cellular technology. Crc Press, 6. [] S. Ln, D. J. Costello, and M. J. Mller, Automatc-repeat-request errorcontrol schemes, IEEE Communcatons magazne, vol., no., pp. 5 7, 98. [5] Y. Polyansky, H. V. Poor, and S. Verdú, Feedback n the non-asymptotc regme, IEEE Transactons on Informaton Theory, vol. 57, no. 8, pp. 93 95,. [6] K. Vaklna, A. R. Wllamson, S. V. Ranganathan, D. Dvsalar, and R. D. Wesel, Feedback systems usng non-bnary LDPC codes wth a lmted number of transmssons, n Informaton Theory Workshop (ITW), IEEE. IEEE,, pp. 67 7. [7] A. R. Wllamson, T.-Y. Chen, and R. D. Wesel, A rate-compatble sphere-packng analyss of feedback codng wth lmted retransmssons, n Informaton Theory Proceedngs (ISIT), IEEE Internatonal Symposum on. IEEE,, pp. 9 98. [8] R. D. Wesel, K. Vaklna, S. V. Ranganathan, and D. Dvsalar, Resource-aware ncremental redundancy n feedback and broadcast, n Internatonal Zurch Semnar on Communcatons, 6, p. 63. [9] K. Vaklna, S. V. Ranganathan, D. Dvsalar, and R. D. Wesel, Optmzng transmsson lengths for lmted feedback wth nonbnary LDPC examples, IEEE Transactons on Communcatons, vol. 6, no. 6, pp. 5 57, 6. [] R. Sustaval and M. Meyer, LTE coverage mprovement by TTI bundlng, n Vehcular Technology Conference, 9. VTC Sprng 9. IEEE 69th. IEEE, 9, pp. 5. [] M. C. Davey and D. MacKay, Low-densty party check codes over GF (q), IEEE Communcatons Letters, vol., no. 6, pp. 65 67, 998. [] J. Song, B. Peleato, D. J. Love, T. Luo, D. Ogbe, and A. Ghosh, Optmzng ncremental redundancy for mllmeter wave wreless communcaton usng low densty party check codes, https://engneerng.purdue.edu/ bpeleato/mmwave LDPC.pdf, 7. [3] Z. L, L. Chen, L. Zeng, S. Ln, and W. H. Fong, Effcent encodng of quas-cyclc low-densty party-check codes, IEEE Transactons on Communcatons, vol. 5, no., pp. 7 8, 6. [] Z. Wang and Z. Cu, Low-complexty hgh-speed decoder desgn for quas-cyclc LDPC codes, IEEE Transactons on Very Large Scale Integraton (VLSI) Systems, vol. 5, no., pp., 7. [5] IEEE, IEEE 8.n Wreless LAN Medum Access Control MAC and Physcal Layer PHY specfcatons., IEEE 8.n-D. Std., 6. [6] P. Frenger, S. Parkvall, and E. Dahlman, Performance comparson of HARQ wth Chase combnng and ncremental redundancy for HSDPA, n Vehcular Technology Conference,. VTC Fall. IEEE VTS 5th, vol. 3. IEEE,, pp. 89 833. [7] T. A. Courtade and R. D. Wesel, Optmal allocaton of redundancy between packet-level erasure codng and physcal-layer channel codng n fadng channels, IEEE Transactons on Communcatons, vol. 59, no. 8, pp. 9,. [8] T. Rchardson and R. Urbanke, Modern codng theory. Cambrdge unversty press, 8. [9] D. P. Bertsekas, Dynamc programmng and optmal control. Athena scentfc Belmont, MA, 995, vol., no..