Incremental Redundancy: A Comparison of a Sphere-Packing Analysis and Convolutional Codes

Transcription

1 Icremetal Redudacy: A Compariso of a Sphere-Packig Aalysis ad Covolutioal Codes Tsug-Yi Che Departmet of Electrical Egieerig Uiversity of Califoria, Los Ageles Los Ageles, Califoria tyche@eeuclaedu Nambi Seshadri Broadcom Irvie, Califoria ambi@broadcomcom Richard D Wesel Departmet of Electrical Egieerig Uiversity of Califoria, Los Ageles Los Ageles, Califoria wesel@eeuclaedu Abstract Theoretical aalysis has log idicated that feedback improves the error expoet but ot the capacity of memoryless Gaussia chaels Che et al [1] demostrated that a modified icremetal redudacy scheme ca use oiseless feedback to help short covolutioal codes deliver the bit-error-rate performace of a log blocklegth turbo code, but with much lower latecy This paper presets a code-idepedet aalysis based o sphere-packig that approximates the throughput-vs- latecy achievable regio possible with feedback ad icremetal redudacy for a specified AWGN SNR Simulatio results idicate that tail-bitig covolutioal codes employig feedback ad icremetal redudacy perform close to the sphere-packig approximatio util the throughput reaches the limit of the system s ability to approach the chael capacity I INTRODUCTION While feedback caot icrease the capacity of a memoryless chael [2], it ca sigificatly reduce the complexity of ecodig ad decodig at rates below capacity, as show i the works by Elias [3] ad Chag [4] i 1956 The error-expoet results of [5] [8] suggest that feedback ca be used to reduce latecy As a practical demostratio, [1] showed that usig modified icremetal redudacy with feedback (MIRF) allowed short covolutioal codes to deliver bit-error-rate performace comparable to a log blocklegth turbo code, but with lower latecy The demostratio of [1] qualitatively agrees with the error expoet aalysis i [5] [8] However, the error-expoet theory does ot provide a crisp predictio of the quatitative latecy beefit possible with MIRF at a specific throughput This paper provides the eeded quatitative, codeidepedet aalysis of latecy vs throughput that describes the beefit of MIRF over a baselie system The baselie system has a feedback lik but uses it oly for ACK/NACK; i other words, it s the simple ARQ scheme The MIRF system is similar to the oe studied i [1] The aalysis uses spherepackig ad bouded-distace decodig to model the behavior of a good code for the AWGN chael While the sphere-packig aalysis is code-idepedet, it turs out to match well with simulatios usig good short-blocklegth codes Specifically, this paper compares the This research was supported by a gift from the Broadcom Foudatio Dr Wesel has also cosulted for the Broadcom Corporatio o matters urelated to this research aalysis with simulatios of tail-bitig covolutioal codes The excellet agreemet betwee the aalysis ad simulatio results idicates both that the aalysis provides a accurate characterizatio ad that the short-blocklegth codes curretly available perform similarly to sphere-packig with boudeddistace decodig The rest of the paper is orgaized as follows: Sectio II reviews the sphere-packig approximatio of decodig error for a good code Sectios III ad IV use the sphere-packig approximatio of Sectio II to aalyze simple ARQ ad MIRF, respectively, ad compare with simulatios Sectio V cocludes the paper II SPHERE-PACKING This sectio reviews the sphere-packig aalysis i [2] for a memoryless AWGN chael ad shows that the probability of codeword error of a sphere-packig code with boudeddistace decodig is the complemet of the cdf of a chi-square distributio Cosider a (2, ) chael code that ecodes bits of iformatio ito a legth- codeword with rate The iput ad output of the chael ca be writte as Y X(i) + Z, i 1, 2,, 2 where Y is the received word, X(i) is the codeword for the i-th message, ad Z is a -dimesioal iid Gaussia vector Let the SNR be η ad assume without loss of geerality that the oise has uit variace The average power of a received word Y is P (1 + η) Sphere-packig seeks a codebook that has 2 equally separated codewords withi the -dimesioal sphere with radius r outer (1 + η) Oe ca visualize a large outer sphere that cotais 2 decodig spheres, each with the same radius r ier A upper boud o the ier sphere radius perfectly packs 2 code spheres ito the outer sphere With this ideal sphere-packig i mid, a coservatio of volume argumet yields the followig

2 iequality: Vol(Ier sphere) K rier Vol(Outer sphere) 2 ( (1 ) K + η) 2 where K is the spherical volume costat that depeds oly o Solvig for the radius of the code spheres yields (1 + η) r ier (1) 2 Now cosider the bouded-distace decodig rule: if the received word is withi r ier of codeword X(i), the declare the output of the decoder to be message i If the received word is ot withi r ier of ay codeword or is withi that distace of multiple codewords, the a error is declared Nearesteighbor decodig outperforms bouded-distace decodig, but is more difficult to aalyze The total oise power is a chi-square with degrees of freedom Assumig the largest theoretically possible code sphere radius of (1), the probability of decodig error P e is ( ) 2 (1 + η) P e Pr zi 2 > 2 i1 ( ) (1 + η) 1 F χ2 () 2 2Rc where F χ2 ()(t) is the CDF of the chi-square distributio with degrees of freedom For the rest of our aalysis, we always assume the radius is equal to the upper boud of (1) III ANALYSIS OF SIMPLE ARQ Cosider the simple ARQ protocol o a AWGN chael with oiseless feedback The trasmitter seds the codeword over the oisy chael ad waits for the feedback from the receiver The receiver will sed a ACK/NACK over the oiseless feedback chael to the trasmitter if the codeword is decoded successfully/usuccessfully If NACK is received at the trasmitter, the trasmitter will resed the same codeword util a ACK is received Oce a ACK is received, the trasmitter ecode ad trasmit a ew codeword I the previous sectio we computed the probability of decodig error P e based o the sphere-packig aalysis With P e i had, the expected umber of trasmissios τ required to commuicate a sigle message usig the simple ARQ scheme is as follows: τ 1 1 P e Defie the throughput as the umber of bits trasmitted correctly per chael use The expected throughput R t of the simple ACK/NACK scheme is give by: R t τ (1 P e ) F χ2 ()(r 2 ier) (2) R t Sphere packig aalysis SNR 2dB, 6851 N 6 N 12 N 74 N 16 N Fig 1 Sphere-packig aalysis of throughput vs iitial code rate for simple ARQ o a AWGN chael with SNR2 db ad blocklegths ragig from 6 to 1, Defie the latecy λ as the umber of forward chael uses required to commuicate a message The expected latecy λ is give by the product of expected umber of trasmissios τ ad the codeword block legth : λ τ 1 P e F χ2 () ( (1+η) ) (3) 2 2Rc Figure 1 shows throughput vs iitial code rate for a spherepackig aalysis of the simple ARQ with SNR2 db ad blocklegths ragig from 6 to 1, Figure 2 compares the sphere-packig aalysis of simple ARQ to 64-state tailbitig covolutioal code simulatios of the simple ARQ with block legth 64 This short block legth is where the 64-state covolutioal code is most effective relative to the spherepackig limit of performace Pseudo-radom pucturig (circular buffer rate matchig [9]) provides the high rate codes i these simulatios As show i Figure 3, the ecoder geerates a rate-1/3 codeword The the output of each costituet ecoder passes through a sub-block iterleaver The iterleaved bits of each ecoder are collected i a buffer ad a proper umber of coded bits is set to the trasmitter Our aalysis ad simulatios fix the iitial coded blocklegth ad vary the code rate Hece the iitial blocklegth remais costat ad the umber of iformatio bits per block icreases as the iitial rate grows Therefore, the blocklegth of the rate-1/3 mother code icreases as the iitial rate grows The power of the covolutioal code, however, remais the same as the blocklegth of the mother code icreases (We ll discuss this further i sectio IV-C) This practical restrictio differs from our assumptio of sphere-packig i which the code becomes more powerful as the blocklegth (umber of symbols received) icreases This accouts for the disagreemet i the high-rate regime of Figure 2 Through a computatioal parametric aalysis, latecy ca be examied as a fuctio of throughput i the cotext of sphere-packig ad bouded-distace decodig Equatios (2) ad (3) respectively express expected throughput ad latecy as fuctios of the iitial code rate ad block legth

3 R t SNR 2dB, 6851, N64 Sphere Packig 64 States Cov Code 124 States Cov Code Fig 2 Throughput R t vs iitial code rate for sphere-packig aalysis ad 64-state, 124-state covolutioal code simulatios at blocklegth 64, SNR 2 db for simple ARQ Ifo Bits Rate 1/3 Ecoder Bit Collectio ad Selectio To trasmitter Fig 3 Pseudo-radom pucturig (or circular buffer rate matchig) of covolutioal code At the bit selectio block, a proper amout of coded bit is selected to match the desired code rate IV INCREMENTAL REDUNDANCY WITH FEEDBACK This sectio exteds the sphere-packig aalysis to examie the latecy vs throughput curve possible with the MIRF scheme of [1] for usig icremetal redudacy ad feedback A A Icremetal Redudacy Scheme Sphere-packig aalysis of the MIRF scheme assumes a log ad low-rate mother code The rate of this code ca be arbitrary low ad its blocklegth L ca be arbitrarily large We the pick 2 codewords out of the L-dimesioal sphere described i Sectio II The MIRF scheme trasmits a iitial block legth < L with iitial code rate If the decodig is ot successful, the trasmitter will receive a NACK ad will sed s extra symbols The decoder the attempts to decode agai usig all received symbols for the curret codeword The process cotiues util the decodig is successful or the maximum codeword legth L is reached The MIRF scheme ca also be iterpreted as a rateless codig scheme; the trasmitter ca sed out additioal redudacy bits cotiuously util it receives a ACK message from the receiver Let B i be the vector of the symbols received at the i-th trasmissio, B 1 R ad B i R i where i +s() Let the power of the oise i the B i be N i Defie the evet ζ i {i-th block caot be decoded} {N i : N i > ri 2}, i (1+η) where r i is the correspodig ier sphere radius 2 Rc/ i at the i-th trasmissio The expected latecy is computed as follows: λ IR + spr [ζ 1 ] + spr [ζ 1 ζ 2 ] + m [ i ] + s Pr i1 ζ j where m is the maximum umber of trasmissios allowed, which is costraied by L The joit probability ca also be expressed as the product of coditioal probabilities Thus [ i ] i [ ] Pr P e,j, where P e,i Pr ζ i ζ j The sphere-packig aalysis gives i P e,i Pr zj 2 > ri 2 ζ j ζ j P e,i is challegig to evaluate sice the regio of i ζ j is difficult to characterize We will approximate P e,i as described below B Approximatio of Noise i Successive Decodigs Suppose that the decoder is at the i-th trasmissio ad tryig to decode B i There are two importat mechaisms at play The first mechaism is that sice B was decoded usuccessfully, we kow that B has a oise power larger tha r 2 This icreased oise power makes P e,i larger tha if these + s(i 1) symbols were decoded as a iitial trasmissio The secod mechaism is that B i has the advatage of the s extra symbols received at the i-th trasmissio, which will icrease the radius r i ad thus icrease the probability of successful decodig I short, the code becomes more powerful as the umber of symbols received icreases accordig to the secod mechaism but decodig becomes more challegig as the previously trasmitted symbols are discovered to be oisier tha origially hoped accordig to the first mechaism The mixture of these two mechaisms must be captured i our aalysis A optimistic approximatio igores the first mechaism ad assumes that every attempt of decodig sees a ew istace of oisy symbols with loger blocklegth but at the origial oise variace Figures 4 ad 5 show plots of this optimistic approximatio to compare with the coditioal aalysis preseted below The difficulty with properly accoutig for the first mechaism is that coditioed o previous decodig failures, the oise is o loger iid Gaussia However, we ca make a worst-case aalysis based o the followig two observatios As show i [2] ad the refereces therei, the Gaussia distributio is the worst memoryless oise possible give a specified oise power Lapidoth [1] further showed that irrespective of the oise distributio ad eve regardless of whether the oise is iid, the capacity assumig iid Gaussia oise is achievable with earest-eighbor decodig ad o rate

4 above the iid Gaussia capacity is achievable with radom Gaussia codig ad earest-eighbor decodig Hece, give that our sphere-packig aalysis is similar i structure to a Gaussia codebook ad that our decodig is similar to earest eighbor decodig, modelig the oise as iid Gaussia (with a appropriately computed variace) is a reasoable approximatio Thus, to accout for the first mechaism, we calculate the coditioal expectatio of the oise power i B We the model the oise vector of B as iid Gaussia oise with this coditioal expected oise power To simplify the calculatio, we further approximate by coditioig o ζ istead of ζ j With that approximatio, we are able to calculate the coditioal decodig error ad aalyze the throughput ad latecy of the MIRF scheme Whe the step size is large eough (s N/1), the approximatio matches well with the simulatios o modified MIRF scheme usig tail-bitig covolutioal codes ad ML decodig Let I (r) be the itegral of the product of the oise power ad the probability desity over the complemet of the - dimesioal sphere with radius r The ew expected oise power i B, deoted as N, is E zj 2 ζ N i, i 2, 3 I (r ) 1 F χ2 ( )(r 2 ) The details of calculatig I (r) ca be foud i [11] The error probability of the code coditioed o i ζ j is approximated by the error probability of the same code but with a ew oise vector Z [z 1, z 2,, z i ], z i N(, σ2 c ), i(1+η) where σc 2 N +s i The radius of the code is r i 2 / i This error probability is the same if we ormalize the oise to uit variace ad cosider a code with radius r i i (1+η ) 2 / i, where η iη This ormalizatio allows us s+n to calculate the probability with chi-square CDF We summarize the procedure of computig the coditioal error probability for each trasmissio as follows 1) Calculate the expected oise power of the block B coditioed that B caot be decoded ad deote it as N 2) Update the ew total oise power by N + s 3) Normalize the oise variace to uit variace ad update ew SNR η iη s+n 4) Update the equivalet ier sphere r i accordig to the ew SNR η 5) The coditioal probability is approximated by P e,i 1 F χ2 ( i )(r i 2) where r i i(1+η ) 2 Rc / i C Aalytic Depth of Covolutioal Code This sectio explais the performace degradatio of the simulatio at the high rate regime i terms of the decisio depth (or traceback depth) of covolutioal code The aalytic decisio depth [12] [13] is the pathlegth at which the survivor path icidet o the zero state has a path metric that is the uique miimum distace over all survivor path metrics (excludig the all zero path) The optimal decisio depth of fiite traceback Viterbi algorithm is usually determied by simulatio The aalytic decisio depth, however, gives a good lower boud o the decisio depth For example, the aalytic decisio depth of the stadard rate 1/2, 64-state feedforward covolutioal ecoder is 28 Simulatio results show that a decisio depth of 35 gives a oticeable performace improvemet over 28, ad decisio depth larger tha 35 give oly egligible improvemet Table I shows the profiles of some covolutioal codes with differet umber of states ν is the umber of memory elemet, d free is the free distace ad D decisio is the aalytic decisio depth Suppose the system is operatig uder SNR of 2dB ad starts out with a 124-state covolutioal code, iitial blocklegth 128 ad iitial code rate 9 (iformatio bits) To get the overall code rate below the capacity (6851), say 6, the decodig blocklegth have to icrease up to 174 This blocklegth far exceeds the aalytic decisio depth of 31 for the 124-state covolutioal code Oce the blocklegth is far greater tha the aalytic decisio depth, performace does ot improve, ulike the steady improvemet provided by our sphere packig assumptio Table II is the RCPC code profile whe pseudo-radom pucturig is used to obtai high rate codes As the rate icreases, the d free decreases ad the D decisio icreases The performace of the covolutioal code will also degrade if the block legth caot support the miimum decisio depth required For example, a blocklegth 64 covolutioal code with iitial code rate 9 caot support the eeded decisio depth of 182 The above two restrictios o the practical codig system are two causes of the disagreemet at the high rate regime for both simple ARQ ad MIRF TABLE I PROFILE OF DIFFERENT RATE 1/3 CONV CODES ν Ecoder (Octal) d free D decisio 6 (133, 171, 165) (365, 353, 227) (561, 325, 747) (1735, 163, 1257) (3645, 2133, 3347) TABLE II PROFILE OF RATE COMPATIBLE CONV CODES Mother Code: ν 1 (3645, 2133, 3347) Rate d free D decisio

5 R t SNR 2dB, 6851, Step 1 bits Legth 64 Simple ARQ Legth 64 MIRF Aalytic Legth 64 MIRF Optimistic 64 state Legth 64 MIRF simulatio 124 state Legth 64 MIRF simulatio Fig 4 Throughput versus code rate for a MIRF scheme for step size 1 bits Forward Chael Use SNR 2dB, 6851, Step 1 bits Legth 64 Simple ARQ Legth 64 MIRF Aalytic Legth 64 MIRF Optimistic 64 state Legth 64 MIRF simulatio 124 state Legth 64 MIRF simulatio R t Fig 5 Latecy versus throughput for step size 1 bits D Compariso with Simulatios Figure 4 shows throughput vs iitial code rate, ad Figure 5 shows latecy vs throughput Both figures show results for sphere-packig aalysis ad simulatios of a tail-bitig covolutioal code from the [9] ad [14] The lowest code rate for each simulatio is 1/3 The agreemet betwee aalysis ad simulatio i the lowrate regime is strikig I the high-rate regime, the covolutioal codes fall short of the aalysis because the throughput has reached the limit of the system s ability to approach the chael capacity Figure 4 shows that maximum throughput icreases from below 5 with simple ARQ to above 5 with icremetal redudacy Figure 4 also shows that the iitial code rate should be higher whe usig icremetal redudacy tha whe usig simple ARQ Qualitatively, this is obvious However, Figure 4 idicates how much higher the iitial code rate should be Figure 5 shows that icremetal redudacy allows latecy to remai low eve i the throughput rage betwee 4 ad 5, where simple ARQ does ot V CONCLUSION It is ot surprisig that icremetal redudacy with feedback ca reduce latecy The key result of this paper is a code-idepedet aalysis that is able to accurately determie how much latecy reductio ad throughput improvemet is possible with icremetal redudacy of various step sizes This is a useful tool i system desig that was ot previously available This paper presets a sphere-packig aalysis of latecy vs throughput for a baselie simple ARQ scheme ad a modified icremetal redudacy scheme This powerful aalysis quatifies the latecy beefit possible with icremetal redudacy ad closely predicts the performace of Che s modified icremetal redudacy scheme, validatig it as a extremely efficiet use of icremetal redudacy For the simple ARQ scheme, the throughput curves based o the sphere-packig aalysis match up well with the simulatio results of the covolutioal codes whe a ML decoder is used The simulated throughput curves for covolutioal codes are match well with the aalysis The suboptimal decoder i our aalysis accouts for the disagreemet at the maximum throughput regio, ad iadequate stregth of covolutioal code whe the blocklegth icreases explais the disagreemet at high rate regime For the MIRF scheme, further approximatios o the coditioal probability were made to simplify the aalyses Simulatios of the MIRF scheme usig covolutioal codes show that the approximatios are ot too far away from practice REFERENCES [1] T-Y Che, B-Z She ad N Seshadri, Is Feedback a Performace Equalizer of Classic ad Moder Codes? i ITA Workshop, Sa Diego, CA, USA, Feb 21 [2] T M Cover ad J A Thomas, Elemets of Iformatio Theory Wiley- Itersciece, 1991 [3] P Elias, Chael Without Codig, i MIT Res Lab of Electroics, Cambridge, MA, USA, Sep 1956 [4] S Chag, Theory of iformatio feedback systems, IEEE Tras If Theory, vol PGIT-2, pp 29 4, Sep 1956 [5] J Schalkwijk ad T Kailath, A codig scheme for additive oise chael with feedback I: No badwidth costrait, IEEE Tras If Theory, vol IT-12, o2, pp , Apr 1966 [6] J Schalkwijk, A codig scheme for additive oise chael with feedback II: Bad-limited sigals, IEEE Tras If Theory, vol IT- 12, o2, pp , Apr 1966 [7] A Kramer, Improvig commuicatio reliability by use of a itermittet feedback chael, IEEE Tras If Theory, vol IT-15, o1, pp 52 6, Ja 1969 [8] K S Zigagirov, Upper bouds for the error probability for chaels with feedback, Probl Pered Iform, vol 6, o1, pp [9] 3rd Geeratio Partership Project ( 3GPP TS Multiplexig ad chael codig [1] A Lapidoth, Nearest Neighbor Decodig for Additive No-Gaussia Noise Chaels, IEEE Tras If Theory, vol 42, No 5, pp , Sep 1996 [11] T-Y Che, N Seshadri ad R D Wesel, Sphere-Packig Aalysis of Icremetal Redudacy with Feedback, i Proc It Cof Commu (to appear), Kyoto, Japa, Ju 211 [12] R D Wesel, Ecyclopedia of Telecommuicatios - Covolutioal Codes Joh Wiley & Sos Ic, 23 [13] J B Aderso ad K Balachadra, Decisio Depth of Covolutioal Codes, IEEE Tras If Theory, vol 35(2), pp , Mar 1989 [14] S Li ad D J Costello, Error Cotrol Codig Pearso Pretice Hall, 24