A Sudy on Non-Binay Tubo Codes Hoia BALTA, Maia KOVACI Univesiy Polyechnic of Timişoaa, Faculy of Eleconics and Telecommunicaions, Posal Addess, 3223 Timişoaa, ROMANIA, E-Mail: hoia.bala@ec.u.o, maia.kovaci@ec.u.o Absac: - In his pape a sudy of unpuncued non-binay ubo codes, NBTCs fom poin of view of ellis closin is made. Non-binay ubo codes, aes R=n/(n+2), n> (whee n is he inpus numbe) ae buil fom wo Recusive Sysemaic Convoluional (RSC) componen codes. The ellis closin mehods of he convoluional codes, wih moe inpus, wee sudied and new closin mehods wee poposed. The simulaions wee made fo consain lenhs K=3, K=4 and fo n=2 (duo-binay) and n=3 inpus numbe, and fo block lenhs aound 8 bis. The AWGN and he BPSK modulaion wee employed. The ubo decode was used 5 ieaions fo each block. A Lo Likelihood Raio (LLR) sop cieion was seleced. The BER pefomances of he non-binay ubo codes cases wih: boh unclosed ellises, he ellis of he fis componen code is closed and he ellis of he second componen code is unclosed, boh ellises ae closed and he cicula ellises cases ae compaed. Keywods: - non-binay ubo codes, ellis I. Inoducion The non-binay o muli-binay ubo codes have moe han one inpu, []. Thus, he (convoluinal) codes pesen he same inpus numbe. The eneal scheme of he muli-inpus convoluional code (muli-binay named in he followin) is shown in he Fi.. The code has binay inpus, noed u, u -, u and + oupus, consuced by he same bis plus one conol bi, noed u. The code elaions ae: m s m + 2 s + =,m u + -,m u +... +,m u +,m c =,m- u + -,m- u +... +,m- u +,m- c +....... s + =, u + -, u +... +, u +, c + c =, u + -, u +... +, +. u s sm s () u u u - u u - u,m -,m,m,m- -,m-,m-, -,,, -,, c = u,m s m-,m-, s, = Fi. Muli-inpus convoluional code
The cicui (code) funcionin analyze of he Fi. can be pefomed usin ansfom. Thus, ansfein in he fequency domain he equaions of he elaion () and muliplyin by uns, he fis equaion wih m, he second wih m-, and so on, he las bu one equaion wih and summin (all of hem) we obain he equaion: wih: () c() = U () ()+ U - () - ()+ + U () (), (2) c() = U () + U - () + + U () The k () polynomials, wih k=, have idenical coefficiens wih he eneain maix coefficiens, G: m k () = k, j, k. (4) j = j The k () polynomials, wih k=, coespond o he inpus, and () is he eacion polynomial. The polynomial coefficiens U j (), cu j=, consiue he daa sequences povided o he muli-binay code houh he inpus of hem: N U j () = u, j. (5) = j. (3) II. The sae diaam. The ellis diaam. Sain fom he equaion sysem () we can consuc he sae diaam of he mulibinay code. The diaam will conain 2 m nodes, suiable o he 2 m possible saes. Fom each node will leave 2 banches, suiable o he 2 inpus vecos possible. Thus, he diaam will conain 2 m+ banches o ansiions. Each ansiion is associaed of he one symbol duaion. On he symbol duaion inpu bis ae ake ove and + bis ae eneaed a he oupu. The duobinay code suiable o he eneain maix (made by ij coefficiens, fom he sysem equaions ()), is pesened on Fi. 2: G = [5 3 7] 8 = [ ; ; ], (6) and he coespondin sae diaam and ellis, [2] ae pesened on Fi. 3 a) and b). u 2 u c=u Fi.2 uo-binay code associaed o he maix G = [5 3 7] 8 ;
Fi.3 An example of he duo-binay code: a) sae diaam, b) ellis diaam. III. The ellis closin of he muli-binay convoluional codes As he classic ubo codes, [3], wih one inpu, he muli-binay ubo code semens he dae sequences on blocks, which ae coded and decoded, he ubo code iself becomin a code block. Thus, i appeas he necessiy o finalize he codin by he ellis closin. The ellis closin suppose he inseion, on daa sequences, of he some edundan bis, wih he oal o foce on decode a ceain sae, a he end of he codin. Excep he cicula codes, his sae will be he null sae. In his paaaph he ellis closin mehods of he muli-binay convoluional code ae descibed, independenly by his posue as componen code of he ubo code. Also, he ellis closin poblems of he ubo code componen codes is descibed. A. The cicula ellis Simila o he cicula binay codes, he cicula muli-binay codes pefom he codin of he each daes block sain and endin wih he same code sae (diffeen, fom a block o he ohe). To ealize his hin, he muli-binay code pefoms peviously a pe-codin, fac ha allow i o find he sae fom whee i mus be sain he codin of he especive block. B. The closin ellis a he null sae The ellis of he muli-binay code can be closed a he null sae, m, wih he inseion pice of he k edundan bis in daes block. To find hese bis, we suppose ha beween m he code saes numbe (he code memoy) and he inpus numbe, hee is he elaion: (q ) < m q. (7) Obviously, no mae in wha sae will be he code wih q pahs befoe he codin finalizin, hee ae 2 q m pahs houh ellis, which leads, a he final, o he null sae (noed wih on he Fi.3 diaams). If, q=m, hen hee is only one pah which leads a he null sae. Thus, he ellis closin supposes o add he affeens m bis, of his pah, on daa sequence. If, q>m, hen fom he q bis which mus o be pefomed fo he especive block, q m bis ae allocae o he daes and m bis will specify he pah wha leads o he null sae.
IV. The ubo code closin saeies The muli-binay ubo code, MBTC, simila wih TC classic, conains a leas wo componen codes (muli-binay). Because of he ineleavin of he infomaion sequence, povided o he second code, he ellis closin of his decode canno be made by he inseion of he infomaion bis beween of he N bis fom he ineleaved block. The only possibiliy is ha hese bis canno be included beween he bis povided o he fis code. Fou possibiliies of he MBTC ellis closin and hei codin aes ae pesened on Table.. The vaian of he unclosed of he boh ellises is he mos easily o be implemened, and alonside cicula vaian, i povides he hihe codin ae.. The cicula vaian is aacive fom poin of view of he codin ae, bu i supposes a double volume of compue a codin. The decode does no know exacly he sae fom i can sa, in he codin pocess, bu i knows ha i mus sa and end wih he same sae. This fac is used in iniializaion of he fowad and backwad ecuences fom he decodin aloihm (Maxim A Poseioi).. The closin of he fis ellis is made wih he edundancy-inceasin pice, which implies he codin ae deceasin. The advanae is he fim knowin of he sa coefficiens in ecuence by he fis decode, bu no by he second decode.. Boh decodes know he beinnin saes and how o iniialize hei fowad and backwad ecuences coefficiens. Howeve, he complicaions appea in he muliplexin of he added bis by he second code fo he closin of his ellis. Moeove, hese bis ae no ubo coded and hey canno seve o he ubo decodin pocess. V. Expeimenal esuls The MBTC simulaion esuls, fo he fou ellis closin saeies descibed in he peceden paaaph, ae pesened in Fi. 4 a) and b). The implemenaion is done fo wo pais of he RSC componens codes. In he fis case he memoy 2 duo-binay code is used, havin he eneain maix G=[,, 5/7, 3/7], and in he second, he 3 memoy duo-binay code is used, havin he eneain maix G=,, 3/5, /5]. In all he cases, a decodin, he ubo decode was used 5 ieaions fo each block, a LLR sop cieion was seleced, [4], and a MAP aloihm was used. The AWGN and he BPSK modulaion wee employed. The obained BER and FER cuves wee compaed wih he mos pefomin classic codes, wih he same codin ae (appoximaed a /2) and wih he same memoy (memoy 2 fo Table Vaian Closin fom Codin ae Rc Code Code 2 unclosed unclosed /(+2) cicula cicula /(+2) closed unclosed (N-m)/((+2) N) closed closed (N-m)/((+2) (N+m))
he code wih G=[, 5/7] and memoy 3 fo he code G=[, 5/3], boh puncued wih he puncued maix P=[; ]). Thee is an essenial diffeence beween MBTC pefomances and he puncued one, fo he memoy 2. Pacically, hee is no his diffeence beween 5/3 classic-puncued code and 3//5 MBTC, fo memoy 3. Fom all he ellis closin vaians poposed, as i was expeced, he vaian pesens he wos pefomances. Also, a hihe discepancy beween BER and FER pefomances can be emaked. If a 3//5 code he BER pefomances pacically ae he same, he FER cuves diffe essenially. This siuaion appeas because of he speadin of he eos houh moe blocks. The adequae unclosin of he ellis eneaes hese eos. uncoded BER FER 5/3 5/3 a) uo-binay TC 3//5 vesus classic puncued TC 5/3; uncoded BER FER 5/7 5/7 b) uo-binay TC 5/3/7 vesus classic puncued TC 5/7. Fi. 4 The ubo code pefomances fo he closin saeies pesened in Table.
VI. Conclusions In his pape a sudy on he ellis closin poblems of he unpuncued MBTC is appoached. The ellis closin possible saeies and hei pefomances (obained by simulaions), fo wo cases of he duo-binay codes, ae pesened. Takin in accoun he implemenaion complexiy, he codin ae and he pefomances obained, he mos indicaed closin vaian is (he fis ellis closed, he second ellis unclosed). The sudy mus be coninued, o veify also he conclusions fo he ohes muli-binay componens codes, pobably wih bee pefomances hen he classic TC a he same ae. Acknowledemen The auhos wan o hank hei Ph.. dieco, Pofesso Mianda Nafoniţă, fo he suesion o wie his pape and fo he coninuous help. Refeences [] C. Beou, M. Jézéquel, C. ouillad, S. Keouédan, The Advanaes of Non-Binay Tubo Codes, Infomaion Theoy Wokshop ITW2 Cains, Ausalia, Sep 2-7, 2, pp. 6 63; [2] M. Jézéquel, C. Beou, C. ouillad, Tubo codes (convoluifs), semina, Timisoaa, Romania, 5-8 Mach, 24, hp://hemes.ec.u.o/ceceae/cai.hml; [3] C. Beou, A. Glavieux, P. Thiimajshima - Nea Shannon limi eo-coecin codin and decodin: Tubo-codes, Poc.I 93, Geneva, Swizeland, May 993, pp. 64 7; [4] L. Tifina, H.G.Bala, A. Ruşinau, eceasin of he Tubo MAP ecodin Time by Usin an Ieaions Soppin Cieion, pape acceped onn he IEEE confeence, Iasi, 4-5 July.