Punctured Binary Turbo-Codes with Optimized Performance

Transcription

1 Punctured Binary Turbo-odes wit Optimized Performance I. atzigeorgiou, M. R. D. Rodrigues, I. J. Wassell Laboratory for ommunication Engineering omputer Laboratory, University of ambridge {ic1, mrdr, bstract ltoug rate-1/ turbo codes acieve an outstanding performance in additive wite Gaussian noise cannels, teir performance degrades for iger code rates. In tis paper it is illustrated tat puncturing bot te systematic as well as te parity bits can result in a better performance tan puncturing te parity bits only. In addition we suggest an interleaver design rule for punctured partially systematic turbo codes, wic can furter improve performance, witout increasing te encoding and decoding complexity. Keywords- turbo codes; punctured codes; interleaving. I. INTRODUTION Wen Berrou et al. [1, ] introduced binary turbo codes, tey considered puncturing of te parity bits in order to increase te code rate. agenauer et al. [] suggested a number of puncturing patterns wereas çikel et al. [] as well as Kousa and Mugaibel [5] presented a set of guidelines for constructing good puncturing vectors. owever tese autors focused on puncturing parity bits only. Babic et al. [] claimed tat scemes were parity bits are punctured only perform better tan scemes were parity and systematic bits are punctured but Land and oeer [] suggested tat performance can be improved by puncturing bot parity and systematic bits, in te case of rate-1/ turbo codes. In tis paper, we first describe te turbo encoder and decoder wen puncturing is applied. binary input aditive wite Gaussian noise (WGN cannel is assumed and we focus on punctured turbo codes aving rates of 1/ and /. We provide expressions for te calculation of teir input-redundancy enumerating function and we illustrate wit a specific example wy performance is improved by puncturing bot te systematic and parity bits as opposed to puncturing only te parity bits. Furtermore, we propose a design rule for te interleaver of partially systematic turbo codes, wic furter boosts te performance of te punctured turbo code. II. PUNTURED BINRY TURBO ODES Te binary turbo encoder is a parallel concatenation of two recursive systematic convolutional (RS encoders of rate 1/, as described in [1, ]. Te source bits are input to te first constituent encoder, wile te second encoder is fed by an interleaved version of te original data. Te interleaver is assumed to be pseudorandom and as a size of N bits. Te output of te turbo encoder consists of te systematic bits of te first encoder, te parity bits of te first encoder and te parity bits of te second encoder. In order to increase te code rate of 1/, puncturing is applied. value of 0 in a puncturing vector implies tat te corresponding bit is punctured. Eac puncturing vector of lengt R. arrasco ommunications and Signal Processing Group Dept. of EE& Engineering, University of Newcastle r.carrasco@ncl.ac.uk N is based on a pattern of lengt l, wic is repeated N/l times. Te systematic stream, te parity stream of te first encoder and te parity stream of te second encoder are punctured using puncturing vectors, based on patterns P u, P p and P p' respectively, and te resultant codewords are transmitted over te cannel. Te turbo decoder consists of two soft-input soft-output decoders. Bot decoders use a-priori information to produce soft estimates of te source bits by processing te log-likeliood ratios (LLRs of te receive systematic and parity bits. Wen te code rate is iger tan 1/, zero values are inserted at te positions were puncturing took place. Te first decoder uses a-priori information to produce soft estimates of te transmitted bits by processing te LLRs of te receive systematic bits and te LLRs of te receive parity bits of te first RS encoder. Extrinsic information is extracted from te soft estimates of te source bits and acts as a-priori information for te second decoder. Similarly, te second decoder processes te LLRs of te received interleaved systematic bits as well as te LLRs of te received parity bits of te second RS encoder, to produce better estimates of te source bits as well as extrinsic information, wic will be used as a-priori information by te first decoder at te next iteration. Decoding algoritms are usually based eiter on te Maximum -Posteriori algoritm, also known as te BJR algoritm [], or te soft output Viterbi algoritm (SOV [9]. III. PRTILLY SYSTEMTI TURBO ODES In tis work we focus on turbo codes in te form of parallel concatenated convolutional codes (Ps. Bot constituent codes are terminated and te termination bits are also punctured. s described in [], turbo codes can be systematic, partially systematic (some of te source bits are punctured or nonsystematic. agenauer et al. [], considered systematic turbo codes and suggested a number of puncturing patterns in order to acieve various code rates. We consider partially systematic turbo codes wose puncturing pattern is omogeneous, i.e., te puncturing bits are uniformly distributed among te systematic and te parity streams of te turbo encoder.. Weigt Enumerating Functions Before proceeding to te puncturing of rate-1/ turbo codes, we first present expressions describing teir performance [, 11]. Tese are applicable to terminated turbo codes, were te memory size is muc smaller tan te interleaver size. Te constituent convolutional codes are assumed to be identical. In order to analyze te performance of turbo codes, te expressions tat describe te constituent convolutional codes Tis work is sponsored by EPSR under Grant GR/S/01.

2 need to be modified to describe te equivalent block codes. Te conditional weigt enumerating function (WEF of te equivalent block code can be expressed as: w, = were w, denotes te number of codewords generated by an input information word of amming weigt w, wose parity bits ave amming weigt. Te WEF of te P encoder using a uniform interleaver of lengt N is given by: P = ( N w From ( we obtain te input-redundancy weigt enumerating function (IRWEF of te P code: (1 ( P w P ( W, = W ( w from wic te bit error rate (BER can be derived. By setting W= and =, te weigt enumerating function (WEF assumes te form: P P B ( = ( W =, = = B ( were B is te number of codewords wit amming weigt (=w. Te WEF is related to te word error probability. In te case of a terminated turbo code of rate 1/, an input stream of N bits generates tree outputs of lengt Nν bits eac, were ν is te memory size of te constituent encoder. Expressions (1 and ( do not provide accurate results, since tey assume tat te lengt of eac output is equal to te interleaver size N. Terefore we need to extend tese expressions to take into account te terminating bits. Te conditional weigt enumerating function (WEF of te equivalent block code for te case of a terminated convolutional code can be rewritten as: = d w,d, were w,d, denotes te number of codewords generated by an input information word of amming weigt w, wose systematic bits (including te systematic terminating bits ave amming weigt d wereas te parity bits (including te parity terminating bits ave amming weigt. Te reason for te introduction of variable D is due to te fact tat te weigt of te systematic output of te code is not necessarily equal to te weigt of te input information word. Te additional ν terminating bits could furter increase te weigt of te systematic output (w d. Te P encoder transmits te Nν systematic bits of te first constituent code only (see fig.1. For rate-1/ turbo codes, te WEF of te first constituent code 1 (w, is given by (5. If we set D=1 in 1 (w, we obtain te WEF of te second constituent code, i.e., (w, = 1 (w,d=1,. owever, in te case of punctured turbo codes, te WEF of te second constituent code also needs to be calculated. Te WEF, IRWEF and WEF of te P encoder, using a uniform interleaver of lengt N, assume te form: = D d N w 1 P (5 ( P w P ( W, = W ( w P P B ( = ( W, D =, = = B ( were te codeword amming weigt is equal to d. Figure 1. Scematic of a turbo code in te form of P. B. Example: Puncturing a parallel concatenation of RSs(1, 5/ We illustrate te effect of puncturing by an example. Te P of our example uses two four-state, rate-1/, RS codes wit recursive polynomial and forward polynomial 5, i.e. RS(1, 5/. For tis example we assume tat te interleaver is only bits long for simplicity. Similar results are observed for larger interleaver sizes. Initially, we consider a rate-1/ systematic P. Te WEF of te equivalent rate-1/ block code for te first (0, (1, = D (, = D (, = D (, = D D ( D ( D ( D It is observed tat te free distance of te convolutional code is 5. By setting (w, (w,d=1, and substituting into equations (-( we obtain te WEF of te P code: B P N bits ( = (9 ( wose free distance is equal to. agenauer et al [], used te puncturing patterns P u =[11], P p =[] and P p' =[01], to obtain a systematic turbo code of rate 1/. Patterns P u and P p define te code rate of te first constituent code, wic is equal to /. Taking into account puncturing, te WEF of te rate-/ equivalent block code for te first Interleaver (0, (1, = D (, = D (, = D (, = D (1 onvolutional ode (1 onvolutional ode ( D (1 D 1st onstituent ode D D Nν bits Nν bits nd onstituent ode Nν bits (11

3 We observe tat te free distance of te code is reduced from 5 to. Te WEF of te equivalent block code for te second (0, (1, = (, 5 (, = (, = (1 onsequently te WEF of te rate 1/ systematic P code is: B P ( = (1 so puncturing as reduced te free distance of te systematic turbo code from to. We observe tat te free distance of te turbo code is mainly determined by te free distance of te first constituent code, i.e. d free =, since puncturing of te second constituent code leads to parity words wit zero amming weigt. We also observe tat te puncturing rate of te first constituent code is 1/, i.e. 1 in every bits is punctured (P u =[11], P p =[], wile te puncturing rate of te second constituent code is 1/ (P p' =[01]. We now consider te case of partially systematic turbo codes. If te puncturing patterns P u =[1], P p =[1] and P p' =[011] are used to obtain a turbo code of rate 1/, te first constituent code as a code rate of / and its WEF as te form: (0, (1, = (, = D (, = D (, = D D ( ( ( D (1 ltoug te puncturing rate of te first constituent code was increased from 1/ (systematic punctured code to 1/ (partially systematic code, te free distance of te code did not cange. Furtermore, as seen in table I, te puncturing rate of te first constituent code as been increased by 1 bit every 1 codeword bits but te puncturing rate of te second code as been reduced by bits every 1 parity bits. Terefore, it is more likely tat te number of parity words wit zero weigt, generated by te second encoder, is reduced or even eliminated. Indeed, te WEF of te equivalent block code for te second (0, (1, = (, = 5 (, = (, = (15 wic contains no parity words of zero weigt. Te WEF of te rate 1/ partially systematic P code is equal to: B P ( = (1 Te free distance of te partially systematic turbo code is, wic is equivalent to a better word error performance, compared to te rate 1/ systematic turbo code. Te puncturing rate of eac constituent code for te case of a rate k/n systematic or partially systematic P obtained from a rate 1/ P code, are illustrated in table II. TBLE I. PUNTURING RTES OF TE ONSTITUENT ODES OF RTE-1/ P onstituent ode Rate 1/ Systematic P First 1 = 1 Second = 1 TBLE II. onstituent ode First Second. Simulation Results Puncturing Rate Rate 1/ Part. Sys. P = 1 1 = 1 1 Variation PUNTURING RTES OF TE ONSTITUENT ODES OF RTE-k/n P Rate k/n Systematic P k k k k Puncturing Rate Rate k/n Part. Sys. P k k k k Variation k 1 1k k 1k In fig., te performances of rate 1/ systematic and partially systematic Ps are compared. Bot use a random interleaver of size 00 bits. Te systematic P uses te puncturing patterns P u =[11], P p =[] and P p' =[01], wereas te partially systematic P applies te puncturing patterns P u =[1], P p =[1] and P p' =[011]. Te increase of te puncturing rate of te first encoder causes degradation to te performance of te first decoder, wic affects te overall P performance. owever, as te number of iterations increases, te turbo decoder approaces te performance of a maximum likeliood (ML decoder and te benefit of reducing te puncturing rate of te second encoder is noticeable. Figure. Bit error rate of rate 1/ systematic and partially systematic Ps after 1,, and iterations. Te interleaver size is N=1,000. In fig., te performance of rate / systematic and partially systematic P after iterations as been included. Te systematic code uses te puncturing patterns P u =[1111], P p =[00] and P p' =[00], wereas te partially systematic code applies te puncturing patterns P u =[01], P p =[] and P p' =[01]. Te rate / systematic P performs better at low E b /N 0 values, owever te partially systematic P finally outperforms te systematic P wen te E b /N 0 ratio is greater tan.5 db.

4 TBLE III. WEF OF TE ND E NODER OF RTE-1/ PRTILLY SYSTEMTI P USING N INTERLEVER OF SIE Figure. Bit error rate of rate 1/ and / systematic and partially systematic Ps after iterations. Te interleaver size is N=1,000. IV. INTERLEVER DESIGN Te turbo decoder consists of two soft-input soft-output decoders tat excange information in order to produce better estimates of te transmitted bits. Te performance of a constituent decoder depends on te IRWEF of te corresponding constituent code, wic is a function of bot te systematic and te parity weigt (W,, and consequently te WEF B (= (W=1,D=,=.. Puncturing Pattern of te onstituent odes ssume tat (P u, P p and (P u, P p' are two pairs of puncturing patterns tat produce good punctured convolutional codes. In te case of punctured turbo codes, pair (P u, P p is applied to te first constituent code and P p' to te second constituent code. t te receiver, te receive systematic bits and te receive parity bits of te first constituent decoder conform to te puncturing pair (P u, P p, wereas te interleaved receive systematic bits and te receive parity bits of te second constituent decoder conform to a puncturing pair (P u', P p'. If te turbo code is systematic, P u = [11 1]. Since no systematic bits are punctured, te second constituent decoder conforms to puncturing pair (P u, P p' since P u' = P u, wic guarantees tat performance of te second constituent code will also be good. In te case of a partially systematic turbo code, te second decoder uses te parity bits selected by te puncturing pattern of te second encoder P p' as well as te interleaved systematic bits. ltoug te receive systematic bits conform to pattern P u, te interleaved systematic bits conform to P u', wic is not necessarily equal to P u. Terefore, it is not guaranteed tat te second decoder will converge as quickly as it would, if te puncturing pair was (P u, P p'. Returning to our previous example, te WEF of te second encoder of a rate-1/ systematic P, using an interleaver of size bits, is: 5 B ( 5 (1 More specifically, te WEF of te second encoder of a rate- 1/ partially systematic P, using an interleaver of size bits is illustrated in table III. Te puncturing patterns used are P u =[1] and P p' =[011]. Te random interleaver could take one of! = possible configurations. configuration of te form 1 maps te first bit to position 1, te second bit to position, etc. ccording to P u, te rd systematic bit sould be punctured, owever te bit at position is eventually punctured. In our example, te six last configurations would lead to poor performance. Interleaver onfigurations 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1, 1, 1, 1, 1, 1 B. Interleaver Design Rule Weigt Enumerating Function In order to tackle tis problem, te interleaver needs to be redesigned in suc a way tat pattern P u eventually punctures te same positions of te systematic and te interleaved systematic streams. Tis can be acieved if te interleaver of size N restricts bit u k to lie in positions tat differ by multiples of l from position k, i.e., k±l, k±l and so on, were l is te lengt of te puncturing pattern. We call tis interleaver an l-bit periodic random interleaver. Figure. Example of a -bit periodic random interleaver In our example, te lengt of eac puncturing pattern is, terefore a -bit periodic random interleaver is used. In fig., a scematic representation of suc an interleaver is given. Te bits tat occupy positions 1, and in te systematic stream are mapped to one of positions 1, and in te interleaved systematic stream wit equal probability. Te same logic applies to all positions. n interleaver following tis design rule could only use configurations 1 and 1 (see table III. complete example of two rate-1/ turbo codes is presented in fig.5. In fig.5(a te effect of a random interleaver is sown wile in fig.5(b te effect of a -bit periodic random interleaver is illustrated. t te receiver, zeroes are inserted at te punctured positions of te receive systematic stream. In te case of random interleaving, zeroes in te interleaved receive systematic stream appear at random positions. In te case of periodic random interleaving, zeroes in te interleaved receive systematic stream appear in te same positions as in te receive systematic stream. Terefore it is guaranteed tat pattern P u specifies te puncturing period and te puncturing positions in bot te systematic and te interleaved systematic streams.. Simulation Results Te performance of a rate-1/ partially systematic turbo code using a -bit periodic random interleaver is sown in fig.. It can be observed tat it outperforms te partially systematic turbo code using a random bit interleaver rigt from te first iteration. In fig. te performance of rate 1/, 1/ and / turbo codes is

5 presented. t a rate 1/, te proposed sceme performs only 0. db worse tan te rate 1/ turbo code at a BER= -. V. ONLUSIONS ND FUTURE WORK We illustrated tat terminated partially systematic Ps perform better tan terminated systematic Ps for rates 1/ and /. Furter investigation needs to be made in order to explore weter tere is a rate after wic te performance of partially systematic Ps degrades, compared to systematic Ps. It is suspected tat te increased puncturing rate of te first encoder considerably reduces te free distance of te code. Since it is te caracteristics of te first constituent code tat mainly affect te overall performance of te turbo code, it is expected tat, for ig code rates, te additional parity bits provided by te second encoder will not suffice to keep te value of te free distance of te turbo code as ig as possible. n optimized metod was also presented according to wic, partially systematic turbo codes of rate 1/ and / wit improved performance are generated. Te proposed metod outperforms te systematic turbo codes as well as te partially systematic turbo codes, wic use a random bit interleaver, after a number of iterations. Future work will include investigating optimal puncturing patterns so as to acieve iger rate codes wit good performance. REFERENES [1]. Berrou,. Glavieux, Near Optimum Error orrecting oding and Decoding: Turbo-odes, IEEE Trans.ommun., vol.. no., pp , Oct []. Berrou,. Glavieux, P. Titimasima, Near Sannon limit errorcorrecting coding and decoding: Turbo-odes, in Proc. I 9, Geneva, Switzerland, pp. -0, May 199. [] J. agenauer, E. Offer, L. Papke, Iterative Decoding of Binary Block and onvolutional odes, IEEE Trans. Inform. Teory, vol., no., pp. 9-5, Marc 199. [] Ö. çikel, W. E. Ryan, Punctured Turbo-odes for BPSK/QPSK annels, IEEE Trans. ommun., vol., no. 9, pp , September [5] M.. Kousa,.. Mugaibel, Puncturing Effects on Turbo odes, IEE Proc. ommun. vol. 19, no., pp. 1-1, June 00. [] F. Babic, G. Montorsi, F.Vatta, "Design of rate-compatible punctured turbo (RPT codes " I 00, New York, vol., pp. 1 5, 00. [] I. Land, P. oeer, Partially Systematic Rate 1/ Turbo odes, Proc. Int. Symp. Turbo odes, Brest, France, pp. -90, September 000. [] L. R. Bal, J. ocke, F. Jelinek and J. Raviv, Optimal Decoding of Linear odes for Minimising Symbol Error Rate, IEEE Trans. Information Teory, pp. 9-, Mar. 19. [9] J. agenauer, P. oeer, Viterbi lgoritm wit Soft-Decision Outputs and its pplications, Proc. Globecom 9, vol., pp. -1, November 199. [] S. Benedetto, G. Montorsi, Unveiling Turbo odes: Some Results on Parallel oncatenated oding Scemes, IEEE Trans. Inform. Teory, vol., no., pp. 09-, Marc 199. [11] S. Benedetto, G. Montorsi, Design of Parallel oncatenated onvolutional odes, IEEE Trans. ommun.., vol., no. 5, pp , May 199. Figure 5. Effect of interleaving to te receive systematic bits of a rate 1/ partially systematic turbo code. In (a a random bit interleaver is used wile in (b a random -bit periodic interleaver is used. Figure. BER comparisson of rate 1/ Ps after 1,, and iterations. Te interleaver size is N=1,000. Figure. Performance comparison of rate 1/, 1/ and / turbo codes after iterations. Te interleaver size is N=1,000.