ITERATIVE DECODING OF TURBO CODES

Transcription

1 Journal of Advanced College of Engineering and Manageent, Vol.3, 07 ITERATIVE DECODIG OF TURBO CODES Dhanehwar Sah Advanced College of Engineering and Manageent, T.U. Eail Addre: Abtract Thi paper preent a Thei which conit of a tudy of turbo code a an error-control Code and the oftware ipleentation of two different decoder, naely the Maxiu a Poteriori (MAP), and oft- Output Viterbi Algorith (SOVA) decoder. Turbo code were introduced in 993 by berrouet at [] and are perhap the ot exciting and potentially iportant developent in coding theory in recent year. They achieve near- Shannon-Liit error correction perforance with relatively iple coponent code and large interleaver. They can be contructed by concatenating at leat two coponent code in a parallel fahion, eparated by an interleaver. The convolutional code can achieve very good reult. In order of a concatenated chee uch a a turbo code to wor properly, the decoding algorith ut affect an exchange of oft inforation between coponent decoder. The concept behind turbo decoding i to pa oft inforation fro the output of one decoder to the input of the ucceeding one, and to iterate thi proce everal tie to produce better deciion. Turbo code are till in the proce of tandardization but future application will include obile counication yte, deep pace counication, teleetry and ultiedia. Finally, we will copare thee two algorith which have le coplexity and which can produce better perforance. Keyword: SOVA, MAP, SISO, Turbo Code, RSC, Channel Model, SR, BER, LLR, VA. Introduction One of the ai of thi paper will be to how that copriing and analyi for different decoding algorith of turbo code. There are variou iterative decoding technique. SISO: Soft inforation, or reliability, i crucial inforation type when turbo-lie (iterative) proceing of data i conidered. With the advent of turbo code in the area of inforation theory, a lot of attention i given to algorith that can provide uch oft reliability value while decoding the original inforation. There are two nown oft-input oft-output. The thei i propoed to wor on thee two SISO decoding Method: Maxiu a Poteriori (MAP) decoding algorith and SOVA (Soft Output Viterbi Algorith). Thi algorith i ued to iniize the probability of word or equence error.it wor by rejecting the leat liely path through the trelli at each node, and eeping the ot liely one. The reoval of unliely path leave u, uually, with a ingle ource path further bac in the trelli. Thi path election repreent a hard deciion on the tranitted equence. The Viterbi decoder etiate a axiu lielihood equence.. Turbo Code Turbo code were dicovered in 993 [] before that, Shannon liit on code perforance could only be approached with very long code word length. There wa the proble of decoder coplexity a well [9]. But we hall analyze in thi chapter how decoder coplexity can be reduced while ipleenting turbo code.. Encoder A parallel concatenated convolutional code i ued for encoding turbo code. In the Fig [] d i = (d, d, d 3..d ) repreent the binary input data equence which i paed into the input of 5 jace, Vol.3, 07 Iterative Decoding of Turbo Code

2 convolutional encoder [4][3] EC, a denoted in the original paper. A a reult, a coded bit trea P xk i generated which i then interleaved, often in a peudo-rando pattern. The interleaved data equence i paed to a econd convolutional encoder EC and another coded bit trea x i produced. Both of the code bit trea a yteatic code bit 5 x and parity bit x and x are ultiplexed (and poibly punctured) to for P K p x P K P K d K p x x Peudo- Rando Interleaver EC p x Channel EC p x Fig Turbo Encoder. Recurive Syteatic Convolutional (RSC) code The convolution (RSC) coder EC and EC ued turbo encoder are recurive yteatic convolutional (RSC) code. RSC code are the convolutional code that ue feedbac and the uncoded data bit are alo preent in the tranitted code bit equence. Fig how a RSC encoder. The hown RSC encoder i of rate /, with contraint length = 3, and a generator polynoial G = {g, g} = {7, 5}, where g i the feedbac connectivity and g i the output connectivity, in octal notation. An x = x x,..., x and parity RSC coponent encoder ha two output equence: data: equence (, ) p p p p equence x = ( x x,..., x ). d K, x + a a - a - + p x Fig Recurive Syteatic Convolution Code The linear nature of turbo code (at leat, thoe uing BPSK/QPSK odulation) ean that the iniu Haing ditance of the code can be deterined by coparing each Poible code wordwith the all zeroe codeword.thi proce iplifie analyi of the code oewhat and the iniu haing ditance i then equal to the iniu codeweight (nuber of ) which occur 6 jace, Vol.3, 07 Iterative Decoding of Turbo Code

3 in any codeword.. Thi i the relationhip between the codeweight and the nuber of codeword with that codeweight. An SC code, however, will return to the all zeroe tate after - input zeroe, where K i the contraint length of the encoder. The infinite ipule repone property of RSC code i copleented in turbo code by the interleaver between coponent encoder. The reult i a copoite codeword which will often have a high codeweight.. The peudo-rando nature of ot turbo code interleaver tend to reult in a apping uch that a few cobination of input bit poition which caue low codeweight equence in one RSC coponent code are peruted into cobination of poition which generate low codeweight equence in the econd RSC code. The reult in uch a cae are a low copoite codeweight. uch peudo-rando apping often lead to turbo code having a low iniu codeweight copared to ay, SC-baed convolutional code, reulting in a ared error floor at high SR.. The ditance pectru of the code a a whole becoe ignificant in deterining BER perforance, and that the cobination of RSC code and peudorando interleaving produce codeword with higher code weight ot of the tie. The low ultiplicity of low codeweiught equence aociate with turbo code oetie referred to a pectral thinning, lead to their good BER perforance at low SR..3 Interleaver An interleaver doe the wor of re-arranging a equence of ybol. One ue of interleaver in counication i that of the ybol interleaver which i ued after error control coding and ignal apping to enure that fading burt affecting bloc of ybol tranitted over the channel are broen up at the receiver by a de-interleaver, prior to decoding. Mot error control code wor uch better when error in the received equence are pread far apart. Another ue i to place an interleaver between coponent code in a erially concatenated code chee for exaple, between a Reed Soloon outer code and a convolutional inner code. In both cae, the interleaver i typically ipleented a a bloc interleaver. The original data equence i repreented by the equence of white quare, and the interleaved data equence i repreented by the grey quare. Berrou and Glavieux original paper [] featured reult uing a 56*56 interleaver. Turbo code BER perforance iprove with interleaver length-the o called interleaver gain- but the loading and unloading of the interleaver add a coniderable delay to the decoding proce. Thi would ae a 56*56 interleaver unuitable for ay real tie peech application which are delay enitive..6 Terination Convolutional coding i a continuou proce and code word do not have a fixed bloc length. The proce can pan the whole eage rather than a all group of bit. But the turbo code do have the fixed bloc length which i a deterined by the length of the interleaver. Uual procedure i to append tail bit to each bloc of data bit entering one or other of the coponent encoder, to return it to the all zero tate at the end of the trelli. Thi proce i called terination 3. Turbo Decoding The turbo decoder conit of two coponent decoder: DEC to decode the equence fro EC and DEC decode the equence fro EC. Both DEC are Maxiu a poteriori (MAP) decoder. DEC tae the received equence of yteatic value y and the received equence of parity value belonging to the firt encoder EC. The output of the decoder DEC i equence of oft etiate EXT of the tranitted data bit d. The EXT i called the extrinic data, in that it doe not contain any inforation which wa given to DEC to DEC. Thi inforation i interleaved, and then paed to the econd decoder DEC. The encoder i identical to the ued in the encoder. DEC tae a it y 7 jace, Vol.3, 07 Iterative Decoding of Turbo Code

4 input the interleaved yteatic received value y and the equence of received parity value fro the econd encoder EC along with the interleaved fro of the extrinic inforation EXT, provided by the firt decoder. The econd decoder DEC produce a it output a et of value which when de-interleaved uing the invere for of interleaver, contitute oft etiate EXT of the tranitted data equence d. Thi extrinic data, fored without the aid of parity bit fro the firt code, i feedbac to DEC. Thi procedure i repeated in an iterative anner. The iterative decoding proce add greatly to the BER perforance of turbo code for exaple, Berrou and Glavieux Eb achieved BER = 0 5 at within 0.7 db of the Shannon liit, uing a rate / turbo code and 8 o decoding iteration. However, after everal iteration, the two decoder etiate of d will tend to coverage. At thi point, DEC output a value Λ( d ) a log lielihood repreentation of the etiate of d.thi log- lielihood value tae into account the probability of a tranitted 0 or baed on the yteatic inforation and parity inforation fro both coponent code. More negative value of Λ( d ) repreent a trong lielihood that the tranitted bit wa a 0 and ore poitive value repreent a trong lielihood that it wa tranitted bit. Λ( d ) i de- interleaved o that it equence coincide with that of the yteatic and firt parity trea. Then a iple threhold operation i perfored on the reult, to produce hard deciion etiate, d for the tranitted bit. The decoding etiate EXT and EXT, do not necearily converge to a correct bit deciion. If a et of corrupted code bit for a pair of error equence that neither of the decoder i able to correct, then EXT and EXT ay either diverge, or converge to incorrect oft value. Deinterleaver y EXT EXT Interleaver DEC DEC EXT y p y interleaver Λ( d ) Deux y p y Deinterleaver =, Fig 3 Turbo Decoder Structure 8 jace, Vol.3, 07 Iterative Decoding of Turbo Code

5 3. The MAP Algorith The decoding algorith ipleented in DEC and DEC for iterative decoding need to be analyzed. The firt one under dicuion i the Maxiu A poteriori (MAP) algorith preented in Berrouet. al original paper []. We decribe here a derivation of the MAP decoding algorith for yteatic convolutional auing an AWG channel odel, a preented by pietrobo [6]. 4. Channel odel In additive white Gauian noie channel, the received ignal i the u of the tranitted (attenuated in oe way) and noie with a Gauian probability denity function (pdf) given by: p (n) = exp n (3.6) The effect of AWG i to hinder a detector in the etiate of the tranitted ignal baed on a poibly very wea received ignal. Becaue AWG affect all electronic circuitry, it alot alway added to a iulation channel odel. 4. Perforance of Turbo Code for Log-MAP A iulation progra ha been written and it give the perforance for and length data. However, it tae uch tie depending on data length, punctured or unpunctured pattern, the Eb/0 ratio provided and the channel odel ued.an analyi wa done on an AWG channel for rate /, length 04-bit and log MAP Turbo decoder. We oberve the gain achieved by the turbo code relative to convolutional code of relative coplexity. We can clearly ee the iterative power of turbo code. Fig 4 Perforance of 400 bit, rate /, log-map Turbo Code veru ratefour State Convolutional Code over AWG 9 jace, Vol.3, 07 Iterative Decoding of Turbo Code

6 Fig 5 Perforance of 04 bit, rate / log-map Turbo Code veru rate / fourtate convolutional Code over AWG. Log-Map over AWG Fig 6 Perforance of 04 bit, rate /, log-map Turbo Code veru rate/fourtate convolutional Code over AWG. 4. Soft Output Virtebri Algorith (SOVA) Thi algorith i ued to iniize the probability or word or equence error. It will wor by rejecting the leat liely path through the trelli at each node, and eeping the ot liely one. The reoval of unliely path leave u, uually, with a ingle ource path further bac in the trelli. Thi path election repreent a hard deciion on the tranitted equence. The Viterbi decoder etiate a axiu lielihood equence. 0 jace, Vol.3, 07 Iterative Decoding of Turbo Code

7 It i decribed here a derivation of MAP decoding algorith for yteatic convolutional code auing an AWG channel odel, a preented by pietrobon [6]. We tart with the ratio of the APP, nown a the lielihood Ratio Λ( ), or it logarith, called the LLR, a hown below. Where Λ ( d ) = λ λ Ld ( ) = log λ i, binary equence i, 0, λ λ i, 0, (4.) (4.) the joint probability that data = and tate = conditioned on the received R oberved fro tie = through oe tie, i decribed a λ = pd ( = is, = R ) i, (4.3) R Repreent a corrupted code bit equence after it ha been tranitted through the channel, deodulated, and preented to the decode in oft deciion for. In effect, the MAP algorith require that the output equence fro the deodulator be preented to the decoder a a bloc of bit at a tie. Let R be written a follow, {,, + } R = R R R To facilitate the ue of Baye theore, Equation (4.) i partitioned uing the letter A, B, C, D and Equation (4.3). Equation (4.) can be written in thi for: (4.4) λ = pd ( = is, = R, R, R ) i, (4.5) A = p( d = i, S = ) B = R C = R and D = R + Fro Baye theore p ( A, B, C, D ) p ( B A, C, D ) P ( A, C, D ) p ( A B, C, D ) = = p ( B, C, D ) p ( B, C, D ) P ( B A, C, D ) P ( D A, C ) P ( A, C ) = P ( B, C, D ) Hence, application of thi rule to Equation (4.5) yield (4.6) y = p( R d = i, S =, R ) p( R d = i, S =, R ) i, + x pd ( = is, = R, ) pr ( ) (4.7) jace, Vol.3, 07 Iterative Decoding of Turbo Code

8 Where { R R } RK =, K + Equation (4.7) can be expreed in a way that give greater eaning to the i, probability ter contributing to λ. The three nuerator factor on the right ide of Equation (4.7) will be defined and developed a the forward tate etric, the revere tate etric, and the branch etric. 4.3 State etric and the Branch Metric We define the firt nuerator factor on the right ide of Equation (4.7) a the forward tate etric at tie K and tate, and denote it a a Thu for i =,0. =, =, = = (4.8) R otice that d = i and are deignated a irrelevant, ince the auption the S = iplie that event before tie are not influenced by obervation after tie K. In other word, the pat i not P R i independent of the fact that d = i and equence R. affected by the future, hence ( ) However, ince the encoder ha eory, the encoder tate S = i baed on the pair, o thi ter i relevant and ut be left in the expreion. The for of Equation (4.8) i intuitively atifying, ince it preent the forward tate etric a at tie a being a probability of the pat equence; that i, dependent only on the current tate induced by thi equence, and nothing ore. Thi hould be failiar fro the econd nuerator factor on the right ide of Equation (4.7) repreent a revere tate etric, β at tie and tate, decribed below =, =, =PR (,) S = (i, ) β (4.9) Where ( i, ) i the next tate, given an input I and tate, and β ( i, ) + i the invere tate etric at tie + and tate ( i, ). The for of Equation (4.9) i intuitively atifying ince it preent the revere tate etric, ( i, ) β + at future tie +, a being a probability of the future equence, which depend on the tate (at future tie ). Thi hould be failiar becaue it create the baic definition of a finite-tate achine [7]. We define the third nuerator factor on the right ide of Equation (4.7) i, a the branch etric at tie and tate. denotedδ Thu we write ( =, =, ), (4.0) Subtituting Equation (4.8) through (4.0) into Equation (4.7) yield the following ore copact expreion for the joint probability: i, (, i) i, aδ βl+ λ = pr ( ) Equation (4.) can be ued to expre Equation (4.) and (4.) a follow: Λ d = a a, (, ) δ β+ 0, (0, ) δ β+ (4.) (4.) jace, Vol.3, 07 Iterative Decoding of Turbo Code

9 , (, ) aδ β + = log 0, (, ) aδ β + L d (4.3) Where Λ i the lielihood ratio of the th data bit and L, the logarith of Λ,i the LLR of the th data bit, where the logarith i generally taen to the bae e. 4.4 Calculating the Forward State Metric Starting fro Equation (4.8), a can be expreed a the uation of all poible tranition probabilitie fro tie -, a follow = ( =, = ', = ) j= 0 a p d j S R S R a{ R R } We can rewrite,, and fro Baye theore, (4.4) = = = = j= 0 a p( R S, d j, S ', R ) pd ( = js, = ', R S = ) x j= 0 ( ) = p ( R S = b( j, ) p( d = j, S = b( j, ), R ) (4.5a) (4.5b) Where b(j,) i the tate going bacward in tie fro tate, via the previou branch correponding to input j. Equation (4.5b) can replace Equation (4.5a) ince nowledge about the tate and the input j, at tie -, copletely, define the path reulting in 4. State S =. Uing Equation (4.8) and Equation (4.0) to iplify the notation of Equation (4.5) yield the following: (,) =,(,) (4.6) Equation (4.6) indicate that a new forward tate etric at tie and tate i obtained by uing two weighted tate etric fro tie -. The weighting conit of the branch etric aociated with the tranition correponding to data bit 0 and. Figure 3. illutrate the ue of two b( j, ) different type of notation for the paraeter alpha. We ue a for the forward tate etric at tie -, when there two poible underlying tate (depending upon whether j=0 or ). And we ue a for the forward tate etric at tie, when the two poible tranition fro the previou tie terinate on the ae tate tie. 3 jace, Vol.3, 07 Iterative Decoding of Turbo Code

10 b (0, a ) J=0 0, b(0, ) δ K a β J=0 0, δ (0, ) a + J= b(, ) a J=, b(, ) δ, δ (, ) β + K- K K K+ (a) Forward tate etric (b) Revere tate etric a = a δ + a b(0, ) 0, b(0, ) b(, ), b(, ) δ β β δ β δ (0, ) 0, (, ), = Where b (j,) i the tate going bacward in where ( i, ) i the next tate going an tie correponding to an input j. input j and tate. 4.5 Branch Metric: δ ( xu yv ) = π exp + i, i i i, Graphical repreentation for calculating a and β []. 4.6 Calculating the Revere State Metric Starting fro Equation (4.9) where = R S = (, i ), (, i) β β + = pr ( S = ) = pr (, R S = ) we how β a follow: (4.7) We can expre β a the uation of all poible tranition probabilitie to tie +, a follow: β + + ' j= 0 Uing Baye theore, = pd ( = js, = R ',, R S = ) (4.8) β ' j= 0 = p( R S =, d = j, S = ', R ) x p( d = j, S = ', R S = ) (4.9) S = and d =j in the firt ter on the right ide of Equation (4.9) copletely define the path reulting in = ( j, ) S K + ) the next tate given an input j and tate. thu, thee condition allow replacing S + = ' with S = in the econd ter of Equation (4.9), yield the following: j, ( j, ) = pr ( S ) ( jpd, ) ( js, R, ) + + = = = = + j= 0 j= 0 β δ β (4.0) 4 jace, Vol.3, 07 Iterative Decoding of Turbo Code

11 Equation (4.0) indicate that a new revere tate etric at tie and tate i obtained by uing two weighted tate etric fro tie +The weighting conit of the branch etric aociated with the tranition correponding to data bit 0 and Figure 3..b illutrate the ue of two different type of notation for the paraeter beta. We ue ( i, β ) + for the revere tate etric at tie + when there are two poible underlying tate (depending on whether j=0 or ). And we ue β for the revere tate etric at tie, where the two poible tranition arriving at tie + te fro the ae tate at tie. Fig 3 repreent a graphical illutration for calculating the forward and revere tate etric. Ipleenting the MAP decoding algorith ha oe iilaritie to ipleenting the Viterbi decoding algorith [3]. In the Viterbi algorith, we add branch etric to tate etric. Then we copare and elect the iniu ditance (axiu lielihood) in order to for the next tate etric. The proce i called add-copare-elect (ACS). In the MAP algorith, we ultiply (add, in the logarithic doain) tate etric by branch etric. Then, intead of coparing the, we u the to for the next forward (or revere) tate etric, a een in figure. The difference hould ae intuitive ene. With the Viterbi algorith, the ot liely equence i being the bet path. With the MAP algorith, oft nuber (lielihood or Log-lielihood) i being ought; hence the proce ue all the etric fro all the poible tranition within a tie interval, in order to coe up with the bet overall tatitic regarding the data bit aociated with that tie interval. 4.7 Calculating the Branch Metric We tart with Equation (4.0), which i rewritten below: δ = pd ( = is, = R, ) = pr ( d = is, = ps ) ( = d = i) pd ( = i) i, (4.) Where R = x, y, x i the received data bit, and y i the correponding noiy received parity bit. Since the noie affecting the data and the parity are independent, the current tate i independent of the current input, and can therefore be any one of the 0 tate, where V i the nuber of eory eleent in the convolution code yte. That i, the contraint length,, of the code i equal to V+. Hence, ps ( = d = i) = v and i δ = p( x d = i, S = ) p( y d = i, S = ) π v (4.) i Where π i defined a p(d =i), the a priori probability of d. The probability p(x =x ) of a rando variable. X taing on the value x i related to the probability denity function (pdf) p x,(x ) a follow [7]. P(X =x )=px,(x )d (4.3) For notational convenience, the rando variablex, which tae on value x, i often tered the rando variable x,which repreent the eaning of X and Y in Equation (4.). Thu, for an AWG channel where the noie ha zero ean and variance, we ue Equation (4.3) in order to replace the probability ter in Equation(4.) with their pdf equivalent, and we write σ 5 jace, Vol.3, 07 Iterative Decoding of Turbo Code

12 δ i i i, i, π x u y v = exp dx exp 0 dy πσ σ πσ (4.4) Where u and v repreent the tranitted data bit and parity bit, repectively (in bipolar for), dx and dy are the differential of x, y and get aborbed into the contant A below. ote that the i i paraeter u repreent data that ha no dependence on the tate. However, the paraeter v, repreent data, which doe depend on the tate, ince the code ha eory. i, i i i, δ = Aπexp ( xu + yv ) σ (4.5) If we ubtitute Equation (4.4) into Equation (), we obtain, yv a exp β x σ Λ ( d ) = π exp 0, σ yv a exp β σ x πexp σ and π e (, ) + (0, ) + (4.6a) (4.6b) (4.6c) L d = L( d) + Lc( x) + Le d 0 Where, π = π / π i the input priori probability ratio (prior lielihood) and π e i the output e extrinic lielihood each at tie. In equation (4.6b), one can thin of π a a correction ter (due to the coding) that change the input prior nowledge about a data bit. In a turbo code, uch correction ter are paed fro one decoder to the next, in order to iprove the lielihood ratio for each data bit, and thu iniize the probability of decoding error. Thu the decoding proce entail the ue of e Equation (4.6b) to Copute Λ for everal iteration. The extrinic lielihood π, reulting fro a particular iteration replace the a priori lielihood ratio π + for the next iteration. Taing the logarith of ( ) in Equation (4.6b) yield Equation (4.6c) which how that the final oft nuber ( )i ade up of three LLR ter: the priori LLR, the channel eaureent LLR, and the extrinic LLR [7]. The MAP algorith can be ipleented in ter of lielihood ratio ( ) a hown in Equation (4.6a) or (4.6b). However, ipleentation uing lielihood ratio i very coplex becaue of the ultiplication operation that are required. By operating the MAP algorith in the logarithic doain [6, 8] a decribed by the LLR in Equation (4.6b) or (4.6c), the coplexity can b greatly reduced by eliinating the ultiplication operation. 4.8 Perforance of Turbo Code for SOVA A iulation progra ha been written and it give the perforance for any length data. However, it tae uch tie depending on data length, punctured or unpunctured pattern, The E b / 0 ratio provided and the channel odel ued. At firt, analyi wa done on an AWG channel for rate / length 400-bit and SOVA turbo decoder. We oberve the gain achieved by the turbo code relative to 6 jace, Vol.3, 07 Iterative Decoding of Turbo Code

13 convolutional code of relatively coplexity. We can clearly ee the iterative power of turbo code figure. Fig 7 Perforance by Siulation of Length 400 bit rate /, and GeneratorPolynoial G = {7,5},Turbo code over AWG Channel. Fig 8 Perforance by iulation oflength 04, Rate / and GeneratorPolynoial G = {7,5}, Turbo Code over AWG Channel. Fig 9 Perforance by Siulation of Length 04, Rate /3 and GeneratorPolynoial G = {7,5} Turbo Code over AWG Channel. 7 jace, Vol.3, 07 Iterative Decoding of Turbo Code

14 4.9 Coparion between the MAP and SOVA The MAP algorith i unlie the Viterbi algorith (VA), where the APP for each data bit i not available. Intead, the VA find the ot liely equence to have tranitted. However, there are iilaritie in the ipleentation of the two algorith. When the decoder bit error probability, PB, i all, there i very little perforance difference between the MAP and Viterbi algorith. However, at low value of bit-energy to noie power pectral denity, Eb/o and high value of PB, the MAP algorith can outperfor decoding with a oft-output Viterbi algorith called SOVA [5] by 0.5 db or ore [6]. For turbo code, thi can be very iportant, ince the firt decoding iteration can yield 49. Poor error perforance: The ipleentation of the MAP algorith proceed oewhat lie perforing a Viterbi algorith in two direction over a bloc of code bit. Once thi bidirectional coputation yield tate and branch etric for the bloc, the APP and the MAP can be obtained for each data bit repreented within the bloc. The oft-input/oft-output (SISO) decoder i the critical part of the decoder, uing the oft output Viterbi algorith (SOVA) [5], [7] or the log-axiu a poteriori algorith (log-map) [7]. Log-MAP give better perforance than SOVA, but SOVA ha leer coplexe. For real tie application, we want the lowet BER, while latency i not a priority. The MAP algorith i not conidered becaue it ha high coplexity and uffer fro uerical proble. For an encoder eory M=3 the nuber of operation [8] uing MAP, Log-MAP and SOVA. The Log-MAP i.8tie ore coplex than SOVA. Thu, fro a latency point of view SOVA i bet of MAP turbo decoding algorith, for a perforance of view Log-MAP i the bet.the perforance of both algorith for the tandard rate / four tate yteatic convolutional code (g = 7 and g = 5) i given in Table. The BER value for both cae are decreaed, when SR i increaed. Table BER for a four tate code uing MAP decoder & SOVA decoder Eb/0(dB) Log-MAP SOVA Fig 0 Perforance by Siulation of Length 04, Rate / and GeneratorPolynoial G = {7,5}, Turbo Code over AWG Channel. 8 jace, Vol.3, 07 Iterative Decoding of Turbo Code

15 5. Reult A four tate, rate/, 400- bit perforance over the AWG channel after 5 Iteration, we get BER zero when E b / 0 i ore than db in SOVA decoder and.75db in Log-MAP decoder.at the ae bit rate /, 04 perforance over AWG channel after five iteration the Log-MAP decoder how the better perforance than SOVA decoder.at the ae bit rate /3, 04 perforance over AWG channel after five iteration the Log-MAP decoder how earlier BER zero than that of SOVA decoder. 6. Concluion Fro latency point of view, SOVA decoder i better, a it i le coplex, than Log-MAP decoder. On the other hand, the perforance of Log-MAP decoder i ore ound than SOVA decoder. But it i ore coplex than SOVA. 7. Recoendation Reearch on the optiu decoding trategie of the MAP decoder and SOVA decoder over the Raleigh Channel i recoended. Reference. Berrou G., Glavieuc A., and Thitajhia P., ear Shannon liit error-coding: Turbo code, in proc.993, int. conf. co., Geneva, Seitzerland, May 993,pp Bahl L R., Coce J., Jeline F., and Racic J., Optial decoding of linear code for iniizing ybol error rate, IEEE Tran, Infor. Theory, Vol, IT-0, pp, 84-87, Benedetto S., Divalar D., Montori G., and Pollara F., A oft-input oft-output Maxiu A poteriori (MAP) odule to decode parallel and erial concatenated code,tda progre report 4-7, oveber 5, Eroz Mutafa, Roger Haon A. Jr., On the deign of prunableinterleaver for Turbo code, in proc. IEEE VTC 99, Houton, TX, May 5-9, Ungerboec G., Channel Coding with Multilevel/phae Syte, IEEE Tran, on inforation Theory, vol, 8no,, pp, 55-67, January Berrou C. and Glavieux A., Reflection on the prize paper: ear Optiu error correctin coding and decoding turbo code, IEEE Inforation Theory Society ewletter, vol. 48 no., june Perez L., Sgher J., and Cotello D., A Ditance Spectru Interpretation of turbo Code, IEEE Tranaction on Inforation Theory, vol, 4, 6, pp, , ov, Barbulecu A. S. and Pietrobon S.S., On terinating the trelli of Turbo Code in the Sae State, IEE Electronic Letter, vol.3, no. I, Jan Blacert W. J., Hall E. K., and Wilon S. G., Turbo Code Terination and InterleaverConditon, IEE Electronic Letter, vol, 3, no. 4, pp, , ov Slar B., Digital Counication: Fundaental and Application, Chapter 6, Prentice- Hall International, Inc Viterbi A.J, Convolutional Code and their perforance in Counication Syte. IEEE Tran Counication Technology, vol.co-9, no.5, pp 75-77, October 97.. Hagenauer J., Hoeher P., A Viterbi Algorith with Soft-Deciion Output and it Application, proc. GLOBECOM 89, Dalla, Texa, oveber 989, pp Pietrobon S.S., Ipleentation and perforance of a Turbo/MAP decoder, Int t, J Satellite Counication vol. 5, Jan-Feb 998, pp jace, Vol.3, 07 Iterative Decoding of Turbo Code

16 4. Slar B., Digital Counication Fundaental and Application, Second Edition (Upper Saddle River, J: Prentice-Hall, 00) 5. Roberton P., Villebru E., and Hoeher P., A Coparion of Optial and Sub-Optial MAP Decoding Algorith Operating in the Log Doain, proc, of ICC 98, Seattle, Wahington, June 995, pp, Berrou C., Glavieux A., ear Optiu Error Correcting Coding and Decoding Turbo Code, IEEE Tran. On Counication, vol. 44, no. 0, October 99, pp Roberton P., Hoeher P., and Villbrun E., Optial and Sub-Optial MAP Algorith Suitable for Turbo Decoding,Europeon Tranaction on Telecounication. Vol.8, no., pp, 9-5, March-April 997, paper ued in writing Yor turbo coder. 8. Jung P., Coparion of Turbo Code Decoder Applied to Short Frae Traniion, IEEE Journal on elected Area in Counication, vol. 4, no.3, pp, , April Roberton P., Iproving Decoder and Code Structure of Parallel Concatenated Recurive Syteatic (Turbo) Code, in IEE Tran of International Conference on Univeral Peronal Counication, San Diego, Sept. 994, pp, Hagenauer J., Roberton P., and Pape L., Iterative (Turbo) Decoding of Syteatic Convolutional Code with the MAP and SOVA Algorith, in ITG-Fachbericht 30, Oct 994, pp, -9.. Erfanin J., Paupathy S., and Gula G., Reduced Coplexity ybol Detector with Parallel Structure for ISI Channel, IEEE Tran Counication, vol.4, pp , Feb, Mar. Apr Koch W. and Baier A., Optiu and Sub=Optiu Detection of Coded Data Diturbed by Tie Varying ISI, in proceeding GLOBECOM 90, San Diego, Dec.990, pp Roberton P., Illuinating the Structure of Code and Decoder for parallel Concatenated Recurive Syteatic (Turbo) Code, in proceeding of GLOBECOM 94, San Francico, Deceber 994, pp Woodard J.P. and Hanzo L., Coparative Study of Turbo Decoding Technique: An overview, IEEE Tran, on vehicular Technology, ov. 000, vol. 49, o. 6, pp jace, Vol.3, 07 Iterative Decoding of Turbo Code