Can Punctured Rate-1/2 Turbo Codes Achieve a Lower Error Floor than their Rate-1/3 Parent Codes?

Transcription

1 Can Puncture Rate-1/2 Turbo Coes Achieve a Loer Error Floor than their Rate-1/3 Parent Coes? Ioannis Chatzigeorgiou, Miguel R. D. Rorigues, Ian J. Wassell Digital Technology Group, Computer Laboratory University of Cambrige, Unite Kingom {ic231, mrr3, ij24}@cam.ac.uk Rolano Carrasco Department of EE&C Engineering University of ecastle, Unite Kingom r.carrasco@ncl.ac.uk arxiv:cs/7165v1 [cs.it] 9 Jan 27 Abstract In this paper e concentrate on rate-1/3 systematic parallel concatenate convolutional coes an their rate-1/2 puncture chil coes. Assuming maximum-likelihoo ecoing over an aitive hite Gaussian channel, e emonstrate that a rate-1/2 non-systematic chil coe can exhibit a loer error floor than that of its rate-1/3 parent coe, if a particular conition is met. Hoever, assuming iterative ecoing, convergence of the non-systematic coe toars lo bit-error rates is problematic. To alleviate this problem, e propose rate-1/2 partially-systematic coes that can still achieve a loer error floor than that of their rate-1/3 parent coes. Results obtaine from extrinsic information transfer charts an simulations support our conclusion. I. ITRODUCTIO A puncture convolutional coe is obtaine by the perioic elimination of symbols from the output of a lo-rate parent convolutional coe. Extensive analyses on the structure an performance of puncture convolutional coes has shon that their performance is alays inferior than the performance of their lo-rate parent coes (e.g. see [1], [2]). The performance of puncture parallel concatenate convolutional coes (PCCCs), also knon as puncture turbo coes, has also been investigate. Design consierations have been erive by analytical [3] [5] as ell as simulation-base approaches [6] [8], hile upper bouns to the bit error probability (BEP) ere evaluate in [5], [9]. The most recent papers [7] [9] emonstrate that puncturing both systematic an parity outputs of a rate-1/3 turbo coe results in better high-rate turbo coes, in terms of BEP performance, than puncturing only the parity outputs of the original turbo coe. The aim of this paper is to explore hether rate-1/2 puncture turbo coes can eventually achieve better performance than their parent rate-1/3 systematic turbo coes on aitive hite Gaussian (AWG) channels. Assuming maximum-likelihoo (ML) ecoing, e emonstrate that, contrary to puncture convolutional coes, puncture turbo coes yieling loer error floors than that of their parent coe can be constructe. evertheless, e cannot be conclusive hen suboptimal iterative ecoing is use. For this reason, e also stuy the convergence behavior of iterative ecoing an e investigate hether the performance of the propose rate-1/2 puncture turbo coes converges toars the theoretical error floor, at lo bit error probabilities. This ork as supporte by EPSRC uner Grant GR/S46437/1. II. PERFORMACE EVALUATIO OF RATE-1/3 TURBO CODES Turbo coes, in the form of symmetric rate-1/3 PCCCs, consist of to ientical rate-1/2 recursive systematic convolutional encoers separate by an interleaver of size [1]. The information bits are input to the first constituent convolutional encoer, hile an interleave version of the information bits are input to the secon convolutional encoer. The output of the turbo encoer consists of the systematic bits of the first encoer, hich are ientical to the information bits, the parity check bits of the first encoer an the parity check bits of the secon encoer. It as shon in [11] an [12] that the performance of a PCCC can be obtaine from the input-reunancy eight enumerating functions (IRWEFs) of the terminate constituent recursive convolutional coes. The IRWEF for the case of a convolutional coe C assumes the form A C (W,Z) = A C,jW Z j, (1) herea C,j enotes the number of coeor sequences having parity check eight j, hich ere generate by an input sequence of eight. The overall output eight of the coeor sequence, for the case of a systematic coe, is +j. The conitional eight enumerating function (CWEF), A C (,Z), provies all coeor sequences generate by an input sequence of eight. Consequently, the relationship beteen the CWEF an the IRWEF is A C (W,Z) = j A C (,Z)W. (2) A relationship beteen the CWEF of a PCCC, P, an the CWEF of C, hich is one of the to ientical constituent coes, can be easily erive only if e assume the use of a uniform interleaver, an abstract probabilistic concept introuce in [12]. In particular, if is the size of the uniform interleaver ana C (,Z) is the CWEF of the constituent coe, the CWEF of the PCCC, A P (,Z), is equal to [ A A P C (,Z) ] 2 (,Z) = ( ). (3)

2 The IRWEF of P, A P (W,Z), can be then compute from the CWEF, A P (,Z), in a manner ientical to (2). The input-output eight enumerating function (IOWEF) provies the number of coeor sequences generate by an input sequence of eight, hose overall output eight is, in contrast ith the IRWEF, hich only consiers the output parity check eight j. If P is a systematic PCCC, the corresponing IOWEF assumes the form B P (W,D) = B,W P D, (4) here the coefficients B P, can be erive from the coefficients A P,z of the IRWEF, base on the expressions B P, = A P,j, an = +j. (5) The IOWEF coefficients B, P can be use to etermine a tight upper boun on the BEP for ML soft ecoing for the case of an AWG channel, as follos ( ) P B 1 B P 2RP,Q, (6) here R P is the rate of the turbo coe, hich in our case is equal to 1/3. The upper boun can be reritten as P B P(), (7) here P() is the contribution to the overall BEP of all error events having information eight, an is efine as P() = ( ) BP, Q 2RP. (8) Beneetto et al. shoe in [12] that the upper boun on the BEP of a PCCC using a uniform interleaver of size coincies ith the average of the upper bouns obtainable from the hole class of eterministic interleavers of size. For small values of, the upper boun can be very loose compare ith the actual performance of turbo coes using specific eterministic interleavers. Hoever, for 1, it has been observe that ranomly generate interleavers generally perform better than eterministic interleaver esigns [13]. Consequently, the upper boun provies a goo inication of the actual error rate performance of a PCCC, hen long interleavers are consiere. III. PERFORMACE EVALUATIO OF RATE-1/2 PUCTURED O-SYSTEMATIC PCCCS Rates higher than 1/3 can be achieve by puncturing the output of a rate-1/3 turbo encoer. Puncture coes are classifie as systematic (S), partially systematic (PS) or non-systematic (S) epening on hether all, some or none of their systematic bits are transmitte [7]. In this section e concentrate specifically on rate-1/2 S-PCCCs, because their eight enumerating functions can be easily relate to the eight enumerating functions of their parent coes, as it ill no be emonstrate. A symmetric rate-1/2 S-PCCC, P, can be obtaine by puncturing the systematic output of a rate-1/3 PCCC, P, hich consists of to ientical rate-1/2 recursive systematic convolutional coes. Hoever, P can also be seen as a PCCC constructe using to ientical rate-1 non-systematic convolutional coes, each one of hich has been obtaine by puncturing the systematic bits of a rate-1/2 systematic convolutional coe, ientical to the one use in P. If C is the puncture rate-1 non-systematic convolutional coe an C is the parent rate-1/2 systematic convolutional coe, their IRWEFs, A C (W,Z) an A C (W,Z) respectively, are ientical, i.e., A C (W,Z) = A C (W,Z), (9) since, by efinition, the IRWEF oes not provie information about the eight of the systematic bits of a coeor. Thus, puncturing of the systematic bits of C ill not cause any change in its IRWEF. Either by applying the same reasoning or by consiering (2) an (3), e fin that the IRWEF of the rate-1/2 S-PCCC, A P (W,Z), is ientical to the IRWEF of its parent coe, A P (W,Z), i.e., A P (W,Z) = A P (W,Z). (1) Puncturing has an effect only hen calculating the IOWEF ofp,b P (W,D). We use the notation to enote the overall eight of a coeor sequence after puncturing as oppose to, hich refers to the overall eight of the same coeor sequence before puncturing. Therefore the IOWEF of P can be expresse as B P (W,D) =, W D. (11) B P Since all systematic bits are puncture, the eight of the information bits oes not contribute to the overall eight of the puncture coeor sequences, an hence it follos that B P, = AP,j, an = j. (12) From (5) an (12) e fin that the relationship beteen the IOWEF coefficients, B, P an BP,, is as follos since B P, = BP,( +) or B P,( ) = BP, (13) = +. (14) This is equivalent to saying that if all information sequences of eight are input to both P an P, the overall eight of the generate coeor sequences follos the same istribution in both cases, but is shifte by in the case of P. The upper boun on the BEP for ML soft ecoing for the case of an AWG channel takes the form P B P (), (15) here P () is given by P () = ( ) BP, Q 2RP (16)

3 an R P = 1/2. Taking into account (13) an (14), e can rerite (16) as a function of an B, P, as follos P () = ( ) BP, Q 2RP ( ) (17) IV. PERFORMACE AALYSIS Before continuing to the erivation of a conition hich nees to be met so that a rate-1/2 S-PCCC can achieve a better ML boun than its parent rate-1/3 PCCC, e first enumerate a number of results, erive an justifie in [14]: 1) The minimum information eight min for recursive convolutional encoers is min = 2. 2) For recursive constituent coes an long interleavers, the contribution to the overall BEP of all error events ith o information eight is negligible. Furthermore, as the interleaver size increases, the contribution to the overall BEP of all error events ith information eight min is ominant. 3) The upper boun of a PCCC hich uses recursive constituent coes, epens on its free effective istance free.eff, hich correspons to the minimum overall output eight hen the information eight is min. It is also straightforar to verify that, although puncturing of the systematic bits of a rate-1/3 PCCC affects the upper boun on the BEP, the previously highlighte trens still apply. Base on (7) an (15), the rate-1/2 S-PCCC, P, ill achieve a better boun than its parent, if 2P () < P(). (18) 2 For large interleaver sizes, the ominant terms ill be P (2) an P(2), thus (18) reuces to P (2) < P(2), or equivalently Q(f ()) < Q(f ()), (19) free.eff free.eff here f () an f() are efine as f 2RP () = ( 2), 2RP f() =, (2) accoring to (8) an (17). Function Q(ξ) is a monotonically ecreasing function of ξ, here ξ is a real number. Consequently, if ξ 1 an ξ 2 are real numbers ith ξ 1 > ξ 2, it follos that Q(ξ 1 ) < Q(ξ 2 ), an vice versa, i.e, Q(ξ 1 ) < Q(ξ 2 ) ξ 1 > ξ 2. (21) Therefore, inequality (19) is satisfie if f () > f(), for every free.eff. (22) Both f () an f() are monotonically increasing functions, therefore (22) hols true if only f ( free.eff ) > f( free.eff ), or free.eff > 2R P R P R P. (23) Upper boun, Rate 1/3 PCCC(1,5/7,5/7) P(2), Rate 1/3 PCCC(1,5/7,5/7) P(3), Rate 1/3 PCCC(1,5/7,5/7) Upper boun, Rate 1/2 S PCCC(1,5/7,5/7) P (2), Rate 1/2 S PCCC(1,5/7,5/7) P (3), Rate 1/2 S PCCC(1,5/7,5/7) b Fig. 1. Upper bouns an contributions to the BEP of all error events ith information eight of 2 an 3, for the rate-1/2 S-PCCC(1,5/7,5/7) an its parent rate-1/3 PCCC. The interleaver size is 1, Upper boun for parent rate 1/3 PCCC Upper boun for rate 1/2 S PCCC PCCC(1,7/5,7/5) PCCC(1,5/7,5/7) PCCC(1,2/3,2/3) PCCC(1,17/15,17/15) b Fig. 2. Comparisson of upper bouns for various rate-1/2 S-PCCCs an their parent rate-1/3 PCCCs. The interleaver size is 1,. After substituting R P an R P ith 1/2 an 1/3 respectively, e fin that a rate-1/2 S-PCCC can achieve a loer upper boun on the BEP, over an AWG channel, than its rate-1/3 parent PCCC only if the effective free istance of the parent PCCC, free.eff, meets the conition free.eff > 6. (24) The contributions of the error events ith eight 2 an 3, for the previous case, are examine in Fig.1. We observe that up to a certain value of, the rate-1/2 PCCC(1,5/7,5/7) exhibits a loer upper boun than its parent coe. ote that for this range of values, the inequality P (2) < P(2) is satisfie. Although P(3) an P (3) o not significantly affect the bouns at lo values, they play an important role at higher values. Hoever, for a larger interleaver size (e.g., = 1,), P (2) is ominant, as explaine previously. More specifically, P (2) etermines the upper boun of the rate-1/2 S-PCCC(1,5/7,5/7), hich is alays loer than the upper boun of its parent rate-1/3 PCCC for the range of values investigate, as e observe in Fig.2.

4 1 1 =4.5 B 1.8 =1.5 B.8.8 =1.5 B.6.6 =1. B.6 =.2 B =.8 B =.2 B, =1 6 =1.5 B, = =1. B, =1 6 =4.5 B, = =.8 B, =1 6 =1.5 B, = (a) Rate-1/3 PCCC(1,5/7,5/7) (b) Rate-1/2 S-PCCC(1,5/7,5/7) (c) Rate-1/2 PS-PCCC(1,5/7,5/7) Fig. 3. Extrinsic information transfer characteristics of iterative ecoing for various turbo coes using an interleaver size of 1 6 bits. To ecoing average trajectories for each case are also plotte; one for an interleaver size of 1 3 bits an the other for an interleaver size of 1 6 bits. In Fig.2, e also observe that in all cases except for PCCC(1,2/3,2/3), the rate-1/2 S-PCCCs achieve better bouns than their parent PCCCs. The reason is that the free effective istance of the parent rate-1/3 PCCC(1,2/3,2/3) is free.eff = 4, thus conition (24) is not met. For all other turbo coes investigate in Fig.2, free.eff > 6 hols true. V. COVERGECE COSIDERATIOS The upper boun on the BEP for ML soft ecoing provies an accurate estimate of the suboptimal iterative ecoer performance at high values, for an increasing number of iterations [12]. Since the performance of rate-1/3 PCCCs graually converges to the ML boun, this boun can be use to preict the BEP error floor region of the corresponing coe. Hoever, hen puncturing occurs, e nee to explore hether the performance of the iterative ecoer eventually converges to the ML boun. For this reason, e use extrinsic information transfer (EXIT) chart analysis [15], hich can accurately preict the convergence behavior of the iterative ecoer for very large interleaver sizes (e.g., =1 6 bits). An iterative ecoer consists of to soft-input/soft-output ecoers. Each ecoer uses the receive systematic an parity bits as ell as a-priori knolege from the previous ecoer to prouce extrinsic information on the systematic bits. Ten Brink escribe the ecoing algorithm process using EXIT chart analysis [15]. To this en, the information content of the a-priori knolege is measure using the mutual information I A beteen the information bits at the transmitter an the a-priori input to the constituent ecoer. Mutual information I E is also use to quantify the extrinsic output. The extrinsic information transfer characteristics are then efine as a function of I A an, i.e., I E = T (I A, ). By plotting the mutual information transfer characteristics of both constituent ecoers in a single EXIT chart, evolution of the iterative ecoing process can be visualise. During the first iteration, the first ecoer oes not have any a-priori knolege, thus I A1,1=, hile the secon ecoer uses the extrinsic output,1 = T (, ) of the first ecoer as a-priori knolege, i.e.,1=,1. The extrinsic output of the secon ecoer, I E2,1 = T (I A2,1, ), is forare to the first ecoer to become a-priori knolege uring the next iteration, i.e., I A1,2=I E2,1, an so on. ote that convergence to the (I A,I E )=(1,1) point, i.e., toars lo BEPs, occurs if the transfer characteristics o not cross. As an example, e consier the rate-1/3 systematic PCCC(1,5/7,5/7) to be the parent coe. Fig.3(a) shos the transfer characteristics of the constituent ecoers for the parent PCCC using an interleaver of size = 1 6. We see that for =.2 B, the ecoer characteristics o not intersect an the average ecoing trajectory [15] manages to go through a narro tunnel. Fig.3(b) epicts the ecoer characteristics for the rate-1/2 S-PCCC employing an interleaver of the same size. For = 1. B, the average trajectory just manages to pass through a narro opening, hich appears close to the starting point (, ). Therefore, for long interleavers, the performance of the suboptimal iterative ecoer for the rate-1/2 S-PCCC converges toars the error floor region, efine by the upper boun, an eventually outperforms its rate-1/3 parent PCCC. Hoever, convergence begins at a higher value an a larger number of iterations is require. In Fig.3(a) an Fig.3(b) the average trajectories for the more practical interleaver size of 1, bits are also epicte. For =1.5 B the trajectory for rate-1/3 PCCC quickly converges toars lo BEPs. Hoever, the trajectory for rate-1/2 S-PCCC ies aay after 2 iterations, even for an value of 4.5 B, ue to the increasing correlations of extrinsic information. We attribute this problem to the absence of receive systematic bits, hich causes erroneous ecisions. As a result, error propagation prohibits the iterative ecoer from converging. Thus, for small an more practical interleaver sizes, the performance of rate-1/2 S-PCCC oes not graually approach the upper boun for ML ecoing.

5 To alleviate this problem, the turbo encoer coul sen some systematic bits, hile keeping the rate equal to 1/2 by puncturing some parity bits. In [9] e have presente a technique for eriving goo puncture coes an e have ientifie a rate-1/2 PS-PCCC that achieves the secon best performance boun for ML ecoing, after the rate-1/2 S-PCCC. Fig.3(c) shos the transfer characteristics of the constituent ecoers as ell as the average trajectory for =1 6. A comparison ith Fig.3(b) for the S-PCCC case, reveals that a loer is require an less iterations are neee, in orer for the rate-1/2 PS-PCCC to converge. Furthermore, the trajectory for an interleaver size of 1, bits reaches the top corner of the EXIT chart for the same as the parent rate-1/3 PCCC, guaranteeing that the iterative ecoer ill converge toars lo BEP values. VI. SIMULATIO RESULTS The process for selecting rate-1/2 PS-PCCCs that lea to superior performance than their parent rate-1/3 PCCCs can be summarize in to steps. We first implement the technique e have propose in [9] to erive goo puncture PCCCs that exhibit lo error floors. We then use EXIT charts an average trajectories to ientify the puncture PCCC hose performance converges toars the error floor region, hen iterative ecoing is use. We consier PCCC(1,5/7,5/7) an PCCC(1,7/5,7/5) to emonstrate the effectiveness of this process. The iterative ecoer applies the BCJR algorithm [16] an performance is plotte after 1 iterations. A ranom interleaver of size 1, bits is use. In Fig.4 e see that the performance of both parent rate-1/3 PCCCs coincies ith the corresponing upper bouns for ML ecoing, at high values. As expecte, the performance of iterative ecoing for rate-1/2 S-PCCCs oes not converge toars lo BEPs, thus they o not outperform their parent coes, although they exhibit a loer upper boun. evertheless, rate-1/2 PS-PCCCs that achieve a loer error floor than their parent rate-1/3 PCCCs, can be foun base on [9]. ote that, in both cases, the encoer of the selecte rate-1/2 PS-PCCC transmits 7 parity bits an only 1 systematic bit for every 4 input information bits. VII. COCLUSIO In this paper e have emonstrate that, if a certain conition is met over the AWG channel, puncturing of the systematic output of a rate-1/3 turbo coe using a ranom interleaver leas to a rate-1/2 non-systematic turbo coe that achieves a better performance than its parent coe, hen ML ecoing is use. In the case of iterative ecoing, the absence of systematic bits makes convergence toars lo bit-error rates ifficult for the rate-1/2 non-systematic turbo ecoer. evertheless, e can assist convergence by transmitting some systematic bits, hile keeping the rate 1/2 by puncturing more parity bits. Thus, e can combine the techniques escribe in [9] an [15] to ientify goo puncturing patterns, improve banith efficiency by reucing the rate of a PCCC from 1/3 to 1/2 an, at the same time, achieve a loer error floor PCCC(1,7/5,7/5) b PCCC(1,5/7,5/7) b Upper boun for rate 1/3 PCCC Upper boun for rate 1/2 S PCCC Simulation for rate 1/3 PCCC (1 iterations) Simulation for rate 1/2 S PCCC (1 iterations) Simulation for rate 1/2 PS PCCC (1 iterations) Fig. 4. Comparison beteen the upper bouns an simulation results for PCCCs using an interleaver size of 1, bits. REFERECES [1] J. Hagenauer, Rate compatible puncture convolutional coes an their applications, IEEE Trans. Commun., vol. 36, pp , Apr [2] D. 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