A Comparison between Weight Spectrum of Different Convolutional Code Types

Transcription

1 A Comparson between Weght Spectrum of fferent Convolutonal Code Types Baltă Hora, Kovac Mara Abstract: In ths paper we present the non-recursve systematc, recursve systematc and non-recursve non-systematc / rate memory convolutonal code spectrums. On the bases of ths spectrum t s made a qualty comparson on the performance of ths codes.. Introducton The hstory of channel codng or Forward Error Correcton (FEC) codng dates back to Shannon s poneerng work n whch he predcted that arbtrarly relable communcatons are achevable by redundant FEC codng. Convolutonal FEC codes were dscovered by Elas [EI], n 955. ue to the smplcty of the codes and the possblty to use the decodng algorthm such as SISO (Soft Input Soft Output), the convolutonal codes are the most used component codes of the turbo codes. A rate Rk/n convolutonal code s an applcaton from the sem-nfnte set of bnary matrx wth a k lne numbers towards sem-nfnte set of bnary matrx wth a number of n lnes, where n > k : Thus, usng C transformaton, each matrx I s assocated a matrx 0 0 0k k k C : Mk M n (.) M k j j jk, of the form: I { 0} j 0,, s, k V M n a0 a0 a0k a0n a a ak an, of the form: a a ak an a j a j a jk a jn V a { 0} j 0,, s, n, (.), (.) The matrx I contans the nformaton bts, n order 0 0 K 0k K, and the matrx V the coded sequences: a0 a0 K a0n ak. The code s systematc f a, j, and s k. Usng a polynomal representaton, the matrx I and V can be wrtten as: s as s s 0 s s s 0 as s s 0 I s, V M s 0 s M ask s s 0 sk M s 0 s asn s 0 Wth these notatons, the codng relaton, C, can be wrtten as: where G() s the code generator matrx: G I (.4) V (.5). Unv. "Poltehnca" Tmşoara, Facultatea de ETC. emal: balta@etc.utt.ro; kmara@etc.utt.ro.

2 G g gn g g g g n k nk (.6) The convolutonal codes (CCs) can be classfed as: a) Systematc or non-systematc codes. If the frst k lnes from G() s the unt matrx of order k, I k, then the codes are systematc, [HY]. In ths case the bts from I can be fnd agan n V. Wth another words, n systematc codes the orgnal nformaton bts or symbols consttute part of the encoded codeword and hence they can be recognsed explctly at the output of the encoder. For non-systematc codes, the bts from V are lnear combnatons of bts from I, consequently, there aren t any nformaton and control bts, as n prevous case. b) Recursve or non-recursve codes. If all generator polynomals that compose G() are fnte, then the result g can be wrtten as the form: code s non-recursve. If not, the generator polynomals a g b (.7) where the polynomals a and b are fnte. So, f there s a polynomal b ( d), at least, then the code s recursve. The constrant length, K s a very mportant parameter of CCs. The defnton of constrant length s: K + maxgrad j, s { a, b } The problem of recursve and non-systematc CCs s that they can be catastrophc. The generator matrx for ths type of code s: a ' G G ; a b (.9) (.8) Suppose a convoluton codes, rate G R and K wth the generator matrx: a0 + a + a + b + b a0 + a + a b b + + ; a { 0, } (.) The encoder of ths code has the general scheme represented n Fg.. If the coeffcent of ths branch s, then there s physcal connecton, otherwse there s not. a a 0 a a b b b a 0 a b a a Fg.. Convolutonal encoder wth R/, K. We present one (representatve) encoder for non-recursve systematc code (NRSC), recursve systematc code (RSC), non-recursve non-systematc code (NRNSC), R and K, n Fg..

3 a a) Fg.. Encoder for: a) Non-recursve systematc code, G [ + + ] T G C T C b) c), ; b) Recursve systematc code,, ; c) Non-recursve non-systematc code, [ ] T G +, + + a. State dagram The convolutonal encoders presented n Fg. can be assmlated wth some fnte-state machnes, [OU], characterzed by the state transton dagrams. Gven that there are two bts n the shft regster (blocks ) at any moment, there are four possble states (,. 0and ) n the state machne and the state transtons are governed by the ncomng bt. So, the nodes of each dagram represent the possble states, and the labels of each branch gve the correspondng output bts sequences. The state dagrams for the encoders from Fg. are shown n Fg.. A state transton due to a logcal zero s ndcated by contnuous lne n the fgure, whle a transton actvated by a logcal one s represented by a dotted lne a) b) c) Fg.. The state dagrams for CCs: a) NRSC; b) RSC; c) NRNSC from Fg... Transfer functon The practcal nformaton sequences, I, are fnte length N, so, a convolutonal code s codng a sequence of nformaton that starts from state and returns to that state, after decodng of N+K- bts. To return to zero state the K- bts are used. So, any emssble sequence (obtaned from a possble codng) corresponds to a path, made up by a successon of branches through state dagram, the path begns and ends n the zero state, [VUY]. An error of the decoder supposes the selecton of another path, other than the emtted one, too. ue to lnearty of the CCs (relaton.5) the dfference between the two paths (receved, w, and emtted, v) obtaned by the relaton: ε w v (.) s, also, a possble emssble sequence, whch corresponds to a path through state dagram, from to states. In others words, the paths that correspond to a mnmum number of branches ndcate the possblty of the error. More exactly, the weght sequences (the number of nformaton s of the branches) represent the number of errors resulted. So, t s useful to fnd the weght spectrum of dfferent paths, [WA]. Ths spectrum wll be a measure of decoder error probablty. The node s splt n two parts, n the startng state and fnshed state respectvely, to fnd all paths that leave and return back to zero state, as well as ther weghts. The state dagram can be understood lke a graph of nodes and transmttances. The state dagram labelled wth varous monomals n the letters δ, β and λ s: 0

4 δ βλ w λ x δ βλ u v y Fg. 4. State dagram labelled wth dstance, length, and number of nput ones, of Fg. a) (NRSC) The branches transmttances are products of three arguments: δ,β and λ. The exponents of these arguments ndcate, [VIT], for: δ - the ndetermnate assocated wth the Hammng weght of the encoded output sequence, β - the ndetermnate assocated wth the Hammng weght of the nformaton sequence (0 or ), λ - the ndetermnate assocated wth each branch (for a branch s ). In ths case we can wrte the equatons: u δ βλ x + v w u + δ βλ w v u + λ w y v (.) whch descrbe the state transtons n Fg.4. Solvng the set of equatons (.) we fnd the transfer functon: δ β λ T a 4 (.) δ βλ δ βλ δ β λ + δ β λ 4 ( δ βλ δ βλ) α, β, λ Expandng T ( δ, β, λ, after the powers of λ (powers that ndcate the branches of a certan path), we can fnd: a ) a α, β, λ δ βλ + δ β λ + δ β λ + δ β λ + K T (.4) The obtaned result (the equaton.4) can be understood as follows. The term δ 4 βλ ndcates a path that leaves from and arrves to ; t has branches (the exponent of λ) and ts weght s 4 (the exponent of δ). Ths path s: 0 (Fg.. a)). The correspondng encoded output sequence s v 0 wth the weght 4. Ths path has also branches, one wth broken lne and two wth contnuous lnes, correspondng to an nput sequence. 4 4 The term δ β λ ndcates a path (the path 0 ) wth 4 branches and the weght 4, whch correspond to the nput sequence wth the weght (the path has two branches wth broken lnes). We can nterpret all others terms of expandng (.4) n the same way. In Fg. 5, we present the state dagrams labelled wth varous monomals n the letters δ, β and λ of the Fg. b, c. w w x δ βλ u λ v δ βλ y a) b) Fg. 5. State dagram labelled wth dstance, length, and number of nput ones, of Fg..b) (RSC) and.c) (NRNSC). x δ βλ u βλ v δ λ y Ther transfer functons are: δ β λ δ β λ + δ β λ T b( α, β, λ) (.5) δ λ + δ β λ

5 54 δ βλ T c ( α, β, λ) (.6) 4. Weght spectrum Obvously the relaton (.4) wll nclude a number of paths that ncrease exponentally wth the weght path. In order to make the relaton useful t should be reformulated n a compact manner [OU]. Ths can be done by constructng a functon, named weght spectrum, that gves for each value of exponental δ (weght path, Pc) the correspondng number of paths (the numbers of terms from relaton (.4)). We present the possble weghts spectrum (the column N) for CCs wth rate R/ and K, n table, except the RNSC ones. The column Sp, gves for all the N paths, wth the same Pc, the weghts of the nput sequences. If, there are errors after decodng, the erroneous decode sequence s a wordcode, conformable to the relaton (.). In another words, the spectrum dstances gve a measure of the errors probablty. When the Sgnal/Nose Rato (SNR) s hgh enough, the most probable errors words are to be looked for n the paths wth small weghts. If, nstead, for small SNR, the errors can arse n bgger groups, then, we should take n consderaton the paths wth bgger weght (respectvely, the bottom secton of Table, that corresponds to the hgh value of Pc). The comparson n the frst bass of Table can be made from the pont of vew of the mnmum dstance code. So, there are two codes wth mnmum dstance d mn, the NRSC [,5] and RSC[,/5], fve codes wth d mn 4 (NRSC [,7], RSC[,7/], RSC[,/7], RSC[,/7] and NRNSC [,7]), three codes wth d mn 5: RSC[,7/5], RSC[,5/7] and NRNSC[5,7]. We noted n parenthess the generator matrx n octal. For examples, to G()[,/+ ] correspond G[,/5], n octal. The superor mnmum dstance of the last three code, specfed above ndcates a superorty from the pont of vew of the correcton capacty, fact that t wll be demonstrated, especally, for the great SNR, where the words (the paths) wth small weghts are mportant. Table Cod Pc Cod Pc NRSC () [,5] RSC (6) [,/7] NRSC () [,7] RSC (7) [,/7] RSC () [,7/] RSC (8) [,5/7] RSC (4) [,/5] NRNSC (9) [,7] RSC (5) [,7/5] NRNSC () [5,7]

6 where: Pc - the path weght; N the numbers of paths wth Pc weght, Sp the weght sum of the nformaton sequences correspondng to N paths. We present n Fg. 6 the dstance spectrums of RSC[,5/7]: a) Fg.6 stance spectrums for RSC[,5/7]: a) log(n+) / Pc; b) log(sp+) / Pc. Another remark s the resemblance of spectrum codes RSC[,7/5] and RSC[,5/7]. Ths means that these codes wll gve the same performances. On the bass of the prevous remarks and because the functon Sp(5) takes the value for NRNSC[5,7] and for RSC[,7/5] we can conclude that the most performant code of the bg sgnal/nose ratos wll be NRNSC[5,7], [FOR]. For the small SNR the paths wth bg weghts are valuable. In another words, here we recommend the codes NRSC[,7] and RSC[,/7] or even NRSC[,5] and RSC[,/7]. 5. Conclusons and perspectves The paper presents an analogy between CCs wth rate R/ and constrant length K, on the bass of the weght spectrum. All encoders have memory (the same numbers of cellules, two) and there are four possble states for each of them. Because the convolutonal encoders have the same numbers of states (four) they have the same complexty. The comparson shows that, for bg sgnal/nose ratos, the non-recursve codes, are better then the recursve ones. The best weght spectrum was found for NRNSC[5,7]. The best recursve codes are RSC[,7/5] and RSC[,5/7] that have the same performances. Ths analyss can be extended at CCs wth another codng rate and another constrant length. The analyss can be contnued n the study of the performances of turbo-codes. Acknowledgement The authors want to thanks ther Ph.. drector, Professor Mranda Nafornţă, for the suggeston to wrte ths paper and for her contnuous help. b) References [EI] P. Elas, Codng for nosy channels, IRE Conventon Record, pt.4, pp.7-47, 955 [FOR] G.. Forney Jr, Convolutonal Codes I: Algebrac Structure, IEEE Transacton on Informaton Theory, nov.970, pp [HY].Hanzo, T.H.ew, B..Yeap, Turbo Codng, Turbo Equalsaton and Space-Tme Codng for Transmsson over Fadng Channels, John Wley & Sons td, England, [OU] C.oullard, Turbo codes (convolutfs), semnar Tmşoara, 5-8 marte 4 [VUY] B.Vucetc, J.Yuan, Turbo Codes Prncples and Applcatons, Kluwer Academc Publshers, USA, 0 [VIT] A.J. Vterb, CMA. Prncples of Spread Spectrum Communcaton, Addson-Wesley Publshng Company [WA] G.Wade, Codng Technques, An Introducton to Compresson and Error Control, Creatve Prnt and esgn, Ebbw Vale, Great Brtan, 0