Variable Rate Superorthogonal Turbo Code with the OVSF Code Tree Insah Bhurtah, P. Clarel Catherine, K. M. Sunjiv Soyjaudah

Transcription

1 Variable Rae Superorhogonal Turbo Code wih he OVSF Code Tree Insah Bhurah, P. Clarel Caherine, K. M. Sunjiv Soyjaudah Absrac When using modern Code Division Muliple Access (CDMA) in mobile communicaions, he user mus be able o vary he ransmission rae of users o allocae bandwidh efficienly. In his work, Orhogonal Variable Spreading Facor (OVSF) codes are used wih he same principles applied in a low-rae superorhogonal urbo code due o heir variable-lengh properies. The inroduced sysem is he Variable Rae Superorhogonal Turbo Code (VRSTC) where puncuring is no performed on he encoder s final oupu bu raher before selecing he oupu o achieve higher raes. Due o bandwidh expansion, he codes ouperform an ordinary urbo code in he AWGN channel. Simulaions resuls show decreased performance compared o hose obained wih he employmen of Walsh- Hadamard codes. However, wih OVSF codes, he VRSTC sysem keeps he orhogonaliy of codewords whils producing variable rae codes conrary o Walsh-Hadamard codes where puncuring is usually performed on he final oupu. Keywords CDMA, MAP Decoding, OVSF, Superorhogonal Turbo Code. I. INTRODUCTION HE discovery of urbo codes in 1993 by Berrou, Glavieux Tand Thiimajshima provided a breakhrough in channel coding such ha error-correcing codes wih relaively simple srucures could permi consisen ransmission raes close o he channel capaciy wih achievable compuaional complexiy. Turbo codes approached he channel capaciy by 0.7 db [1]. In a urbo code sysem, wo or more consiuen encoders are involved o differen inerleaved versions of he same informaion sequence [2]. Cellular mobile phones are being used every day by millions of people across he world. The number of cusomers needing services such as shor messaging, voice, daa, and video is growing. To accommodae he high demand, providers mus increase sysem capaciy wihou degrading he qualiy of service o an unaccepable level hrough CDMA. I employs signaure codes (raher han ime slos or frequency bands) o arrange simulaneous and coninuous access o a radio nework by muliple users. In oher words, i allows many uncoordinaed users o ransmi simulaneously on he same frequency channel wih differen spreading codes [3]. A Insah Bhurah is an MPhil/PhD suden a Elecrical and Elecronic Engineering Deparmen, Universiy of Mauriius, Mauriius ( P. Clarel Caherine is a lecurer a Indusrial Sysems Engineering, School of Innovaive Technologies and Engineering (SITE), Universiy of Technology, Mauriius ( c.caherine@ieee.org). K.M. Sunjiv Soyjaudah is a professor of Communicaions Engineering a Elecrical and Elecronic Engineering Deparmen, Universiy of Mauriius, Mauriius ( ssoyjaudah@uom.ac.mu). user is given a unique spreading code sequence o spread his daa sream before ransmission. As all spreading codes are orhogonal o each oher, all users can ransmi a he same ime and channel wihou any inerference. The daa of a paricular user is recovered by he correlaion of received signal wih he spreading sequence, and all oher inerfering signals are rejeced as hey are orhogonal [3]. In 2009, Ramsawock invesigaed in a Variable Rae Orhogonal Convoluional (VROC) code which is a variable low rae convoluional code wih orhogonal oupu codewords [3]. This was achieved by he inroducion of a puncuring block beween he shif regiser oupu and he OVSF ree block. The shif regiser oupus ener a puncuring block which selecs he bis o be puncured. Following puncuring, he remaining bis ener he OVSF ree where a codeword is seleced. Wih his mehod, codewords of variable lenghs are obained [3]. One imporan aspec of his scheme is ha he same rellis is used for decoding. In [4], a Telescopic Proecion Code makes use of he variaion of he rellis o suppor he user s inpu rae variable, insead of puncuring. However, he decoding process is simpler when a fixed rellis is used [3]. The echnique in [3] has been aemped in superorhogonal urbo codes. In 1996, Pehkonen and Komulainen designed a superorhogonal urbo code combining principles of low-rae convoluional coding and parallel concaenaion using codewords from he Walsh- Hadamard marix [5]. The same design srucure has been used bu wih OVSF codewords. Ieraive decoding is performed wih he MAP, log-map and max-log-map algorihms. The simulaion resuls ouperform an ordinary urbo code in [5] bu show decreased performance compared o codes employing Walsh-Hadamard codes in [5]. However, he advanage of OVSF codes is ha variable-lengh codes can be obained in he OVSF ree compared o Walsh-Hadamard codes where a single Walsh-Hadamard marix of fixed lengh is used for codeword assignmen. The selecion of OVSF variable codewords is performed over a single OVSF ree. To achieve he same wih Walsh-Hadamard codes, marices having variable lenghs should be sored in memory compared o he sorage of a fixed OVSF ree requiring less size. Also, he orhogonaliy of codewords is kep during puncuring. Wih Walsh-Hadamard codes, puncuring is ofen performed on he encoder s final oupu resuling in desrucion of orhogonaliy of he codewords. The paper is organized such ha in Secion II, superorhogonal urbo coding is described and he new code employing OVSF codes is inroduced wih appropriae descripions of he encoder and decoder. Secion III presens 258

2 he variable rae superorhogonal urbo code. In Secion IV, simulaion resuls in he AWGN channel are given followed by he conclusion in Secion V. II. SUPERORTHOGONAL TURBO CODING Superorhogonal convoluional codes (SCC) are low-rae codes having good disance properies due o bandwidh expansion. Hence, hey are used for spread-specrum applicaions [6]. The codes are described in [7] and [8]. An SCC is made up of a K c -2 sage shif regiser where K c is he consrain lengh and K c -2 sages operae on an orhogonal block encoder consising of a Walsh-Hadamard orhogonal sequence generaor. The K c -2 sages mean ha he firs and las sages are excluded and hey do no conribue in driving he orhogonal sequence generaor. These sages are hen modulo-2 added o every bi of he Walsh-Hadamard sequence [3]. The SCC forms he core of he Variable Rae Superorhogonal Convoluional (VRSC) code. Ramsawock showed he VRSC code by inroducing a puncuring block [3] as seen in Fig. 1. The block orhogonal encoder was replaced by he OVSF ree so ha he code becomes boh variable rae and orhogonal [3]. Fig. 1 VRSC Code The VRSC code has been exended and inegraed ino a superorhogonal urbo code engaging OVSF codes o provide variable rae and a he same ime using he same rellis for decoding all derived raes while mainaining orhogonaliy beween codewords. A. Moivaion for OVSF Code Assignmen Fig. 2 depics an OVSF code ree where OVSF codes of variable spreading facors (lenghs) are obained. OVSF codes have advanages of accommodaing variable-rae services which are imporan o emerging mulimedia applicaions [9]. This is achieved by he use of variable spreading facors generaed from an OVSF code ree. Orhogonal Walsh funcions from Hadamard marices canno suppor rae variaion due o he fac ha he marices have fixed lengh. Wih he OVSF code ree, however, differen sequence lenghs can be obained in he same ree. In his work, puncuring was done in he encoder iself o mainain he orhogonaliy of he OVSF codewords. In CDMA, he loss of orhogonaliy beween spreading codes resuls in muliple access inerference [10]. If Walsh-Hadamard sequences were employed, i would have been impossible o perform puncuring by he deleion of bis from he shif regiser oupus, as he remaining bis would no have been able o selec a differen Walsh sequence lengh in a fixed lengh marix. For hese codes, puncuring is usually performed by he deleion of bis in he codeword resuling in he desrucion of orhogonaliy. Fig. 2 OVSF Code Tree In superorhogonal urbo coding, he calculaion of he branch meric using MAP algorihm involves a correlaion of he received word wih he corresponding Walsh or OVSF sequence as will be seen in sub-secion II C. Wih puncuring of he codeword a he final oupu for Walsh-Hadamard codes, a dummy bi is usually insered during decoding. Hence, during correlaion, he value of he branch meric which is fundamenal for calculaion of alpha and bea and overall decoding, will be differen. Variable-rae services are also used in Unequal Error Proecion (UEP). The necessiy of UEP arises in applicaions involving speech, audio and video. In CDMA, users need o ransmi audio and video where some of he message posiions are more sensiive o channel errors while ohers show less sensiiviy [11]. To proec cerain pars of he message, he code rae mus be varied leading o an imporan applicaion of codes uilizing OVSF sequences wihou inerference. B. Encoder In he recursive superorhogonal urbo encoder, each componen consiss of a K c -sage shif regiser. Fig. 3 shows an encoder having a feedback polynomial 23(ocal) wih K c = 5. The original sequence of size N-k (where k is he number of ail bis) is fed o he upper encoder. For each inpu bi, he inner K-2 sages are excluded (firs and las sages) and he remaining bis in he shif regisers drive he OVSF code ree o obain a codeword wih lengh, n = 8. This sequence undergoes a series of module-2 addiion operaions o give he final codeword. Le he inpu daa sequence be u = (u 0,u 1,,u (N-k)-1 ). A ime = 0, u 0 produces he oupu P 0 1 = {p 0 1, p 1 1,,p n-1 1 } where p is a member of he OVSF sequence. 259

3 The way in which he rellis is erminaed affecs he performance of he code. The procedure in [12] was used, ha is by erminaing only he upper encoder and leaving he oher open. k = 4 ail bis were added o u o drive he encoder sae o zero. The oupu of he upper encoder becomes P 1 ={P 1 0,P 1 1,,P 1 (N-1) }. The original bis u wih k ail bis are passed hrough a pseudorandom inerleaver giving u' which eners he lower encoder producing P 2 = {P 2 0,P 2 1,,P 2 (N-1) }. Compared o an ordinary urbo code, since he sysemaic bis are no sen hrough he channel, here is no common channel oupu sream for consiuen decoders. For each consiuen encoder in Fig. 3, he code rae is 1 / n, ha is 1 / 8. The overall code rae hen becomes 1 / 16. Fig. 3 Superorhogonal Turbo Encoder wih he OVSF Code Tree C. Decoder Fig. 4 Superorhogonal Turbo Decoder The pariy bis P 1 and P 2 are muliplexed and modulaed using Binary Phase-Shif Keying (BPSK). They are corruped in an AWGN channel giving ˆP 1 and ˆP 2. Decoding of urbo codes is done in an ieraive fashion. The decoding algorihm used is he modified BCJR or MAP algorihm [13]. Suppose ha he received sequence from an AWGN channel is y, he decoder obains he sequence which is used by he MAP algorihm o esimae he original ransmied bi sequence u for which he algorihm calculaes he a-poseriori log-likelihood raio LLR, a real number defined by he raio [2]: P u LLR = log P u ( =+ 1 y) ( = 1 y) The numeraor and denominaor of (1) consis of he a- poseriori condiional probabiliies, ha is probabiliies obained when y is known. The posiive and negaive value of LLR shows which bi +1 or -1 was coded a ime. To calculae he branch meric for any possible sae ransiion, a correlaion of he received word and is corresponding OVSF sequence is achieved. The branch probabiliy becomes [6]: (1) ( apr) 2 γ = c exp U L.exp ( 2 p 1) ˆ ξ σ i p (2) i i where c' is a consan for all possible parallel ransiions, daa bis and code symbols, ξ = (S, S -1 ) is any possible sae ransiion of he encoder a ime, U ξ { 0,1} is he daa bi relaed o ξ, L (apr) is he a priori esimae given by he previous decoder, σ is he sandard deviaion of he noise, = {0,1,,N-1} and i = {0,1,,n-1}. p is he symbol of he codeword (OVSF sequence) and p ˆ i is is corruped version. The a-poseriori log-likelihood raio LLR can also be compued as LLR = log R1 R0 i ( S ) ( S, S ) ( S ) α γ β ( S ) ( S, S ) ( S ) α γ β R 1 and R 0 denoe ransiions associaed wih inpus +1 and - 1 respecively. The probabiliies α and β are calculaed recursively by ( S ) ( S ) ( S S ), S 1 (3) α = α γ (4) wih iniial condiions: α 0 (S ) = 0 for S 0 and α 0 (S ) = 1 for S = 0 ( S ) ( S ) ( S S ) β = β γ (5), S wih iniial condiions: β N (S ) = 0 for S 0 and β N (S ) = 1 for S = 0. Ieraive decoding is achieved as shown in Fig. 4. Two Sof-In Sof-Ou (SISO) decoders are used. The oupu of SISO decoder 1 is expressed as [6]: 260

4 LLR = L + L (6) ( apr ) ( ex) (apr) where L 1 is he a-priori informaion provided by he previous decoder. This informaion is removed from LLR 1 o (ex) obain he exrinsic informaion, L 1 used as a-priori informaion by he second decoder. D. Log-MAP and Max-Log-MAP Algorihms The MAP algorihm suffers from high complexiy, as i needs o perform many muliplicaions. Simpler versions have evolved such as he log-map and he max-log-map algorihm which reduce ha complexiy [14]. Muliplicaions are replaced by addiions. The applicaion of log in (2) yields Γ = ( apr) 2 = c U L ( 2p 1) ˆ ξ + σ i pi i ( ) ( apr ) 2 log c exp Uξ L.exp σ 2 pi 1 pi i α and β are replaced by A and B. Therefore, ( ) = logα ( ) = max ( ) +Γ (, ) A S S A S S S S 1 wih iniial condiions: A 0 (S ) = 0 for S = 0 and A 0 (S ) = - for S 0 ( ) = log β ( ) = max ( ) +Γ (, ) B S S B S S S S wih iniial condiions: B N (S ) = 0 for S = 0 and B N (S ) = - for S 0 For a non-erminaed rellis, B N (S ) = log( 1 / 2 m ) for S = 0 and B N (S ) = log( 1 / 2 m ) for S 0 wih m being he overall encoder memory. For he log-map algorihm, max (, ) max(, ) log 1 a b ( ) ˆ (7) (8) (9) ab = ab + + e (10) The omission of he correcion funcion a b log 1+ e resuls in he max-log-map algorihm where ( ) max ( ab, ) max( ab, ) =. The expression for LLR becomes ( 1) ( 1 ) ( ) ( ) ( ) ( ) LLR = max A S +Γ S, S + B S R1 max 1 R A S 0 S 1, S B S +Γ + (11) Superorhogonal Convoluional (VRSC) code insead of puncuring a bi ou of an OVSF codeword. In his work, his mechanism has been applied in a VRSTC sysem. Fig. 5 depics a VRSTC encoder. The only difference beween his encoder and he one in Fig. 3 is ha puncuring block is inroduced beween he shif regisers and he OVSF code ree block. The puncuring marix used in his work is P M 1 1 = The VRSTC encoder operaes almos in he same manner as he superorhogonal urbo encoder in Fig. 3. Wih he inroducion of he puncuring block, he code becomes boh variable and orhogonal. Regarding he VRSTC encoder in Fig. 5 wih K c = 5, excluding he firs and las sages, he remaining bis in he shif regisers ener he puncuring block a ime = 0. Considering he firs column of P M, no puncuring occurs and he bis move o he OVSF ree block where a codeword of lengh n = 8 bis is seleced. A ime = 1, he second column of P M indicaes ha he middle bi mus be deleed (ou of he nex hree remaining bis in he shif regiser), and he remaining wo unpuncured bis selec a codeword of lengh n = 4 in he OVSF ree block. A ime = 2, he firs column of P M is again seleced resuling in no puncuring. In his way, codewords of variable lenghs are obained, and he orhogonaliy of he codewords is kep. The code rae of he upper encoder is 1 / 8 and ha of he lower encoder is ¼. Hence, he overall code rae of he VRSTC sysem is 1 / 12. R 1 and R 0 denoe ransiions associaed wih inpus +1 and - 1 respecively. III. VARIABLE RATE SUPERORTHOGONAL TURBO CODE (VRSTC) In [3], a echnique was designed o mainain he orhogonaliy of he codewords in a Variable Rae Fig. 5 VRSTC Encoder Fig. 6 illusraes he VRSTC rellis. As he overall memory is m = 4, he rellis consiss of 16 saes. Inpu bis 0 and 1 are 261

5 associaed wih doed and solid lines respecively. Branches corresponding o he no puncuring case are labeled wih eigh-bi codes (lef hand side) whereas branches corresponding o he puncured case are marked wih four-bi codes. The srucure of he rellis is he same for unpuncured and puncured codewords, bu only he size of he codeword changes. This scheme no only varies he rae bu i also uses he same rellis for decoding while mainaining orhogonaliy beween he codewords. The rellis srucure is kep fixed o simplify he decoding process. The superorhogonal urbo encoder and he VRSTC encoder in Figs. 3 and 5 respecively are recursive encoders. Alhough i was menioned in sub- Secion II B ha he sysemaic bis are no sen, he recursiveness of he encoders leads o a sysemaic code. In Fig. 6, i can be observed ha he sysemaic bi is presen in each codeword a he fifh underlined posiion for he unpuncured codewords. As for he puncured ones, he sysemaic bis are found in he hird underlined posiion of each codeword. IV. PERFORMANCE EVALUATION Compuer simulaions were carried ou o evaluae he performance of superorhogonal urbo codes employing OVSF codes wih differen numbers of blocklengh in he AWGN channel. Resuls as bi error rae and block error rae curves are illusraed in Figs. 7 o 14 for N = 200, 1000, 4000 and For all graphs, bi error rae curves gave beer performance han he block error rae curves due o he fixed number of block errors used o sop he decoding. Consrain lengh K c = 5 was used in he simulaions and he feedback polynomial in he superorhogonal urbo code was 23 (ocal). The overall code rae of he unpuncured and puncured codes were 1 / 16 and 1 / 12 respecively. Only he upper encoder was erminaed while he oher one was lef open. Decoding was performed wih he MAP, log-map and max-log-map algorihms wih 16 ieraions. The resuls obained for unpuncured codes were compared o [5] where Walsh-Hadamard codes were employed. In small blocklenghs, such as N = 200, he bi error rae is slighly worse a E b N o = 0.5 and E b N o = 1.0, and slighly beer a E b N o = 1.5 and E b N o = 2.0 in [5]. When compared wih N =1000 bis in [5], here is a loss of 0.3 db a a bi error rae of Despie he loss of performance, he superorhogonal urbo codes wih he OVSF code have advanages of accommodaing variable rae ransmission while keeping he orhogonaliy of he codewords wih he VRSTC sysem. Wih OVSF codes, differen sequence lenghs in he same ree can be obained as opposed o a Walsh-Hadamard marix described in Subsecion II A. Fig. 6 VRSTC Trellis Figs. 7 and 8 demonsrae bi error rae and block error rae resuls respecively, obained wih N = 200 bis wih he MAP, log-map and max-log-map algorihms. The unpuncured codes for all algorihms give beer performance han he puncured ones. The bes performance is obained wih he log-map algorihm alhough evenually i gives he same performance as he MAP one. The max-log-map algorihm exhibis he wors performance (puncured max-log-map code) wih he replacemen of he erm max wih max. Hence, he omission of he correcion erm resuls in performance degradaion. The benefi of log-map decoding over max-log- MAP decoding is ha performance gains of 0.25dB and 0.2dB are obained a bi error rae and block error rae of 10-3 respecively for unpuncured codes. For N = 1000 bis, performance gains of 0.27dB and 0.25dB (unpuncured codes) have been achieved in Figs. 9 and 10 respecively wih he 262

6 log-map algorihm over he max-log-map one for error raes of The advanage of using random inerleaving is ha he performance increases when N becomes larger. The reason behind is ha a large inerleaver has he endency of bringing furher apar he successive bis of he original daa. When comparing bi error rae resuls of Figs. 11 and 12 for N = 4000 and N = 5400 respecively wih previous graphs of smaller blocklenghs, i can clearly be observed ha beer performance is obained wih all hree decoding algorihms as N is increased. For N = 4000 bis, performance gains of 0.32dB and 0.31dB have been achieved in Figs. 11 and 12 respecively wih he log-map algorihm over he max-log- MAP one for error raes of For N = 5400 bis, performance gains of 0.325dB and 0.30dB were repored in Figs. 13 and 14 respecively for he same error raes. In [5], he resuls were compared o an ordinary rae 1 / 3 urbo code wih generaor polynomials of 23 and 35 (ocal) where a performance advanage was observed. Unpuncured superorhogonal urbo codes wih he OVSF code also demonsrae he same behavior when compared in [5], and his is achieved a he expense of bandwidh expansion resuling in appropriae variable rae applicaions in CDMA. The ime aken by he simulaions for N = 4000 and N = 5400 were recorded and ploed in Fig. 15. They were achieved wih Quad-Core processors on Ubunu Operaing Sysem. Obviously, he puncured codes ake less ime han he unpuncured ones as bis have been deleed in he shif regisers during he encoding process according o P M. I is known ha Log-MAP decoding minimizes compuaional complexiy compared o MAP. I also exhibis almos he same performance han he MAP algorihm. As N is increased, beer performance is observed. As simulaions were sopped when a fixed number of block errors were obained, he Log- MAP algorihm ook he longes ime o complee he simulaions as here were more error-free blocks, and more ime was required o achieve he required number of block errors, hence explaining he increase simulaion ime. The max-log-map algorihm akes he leas decoding ime, as i is simpler and faser o implemen bu a he expense of performance degradaion. All puncured codes are also suiable for implemenaion as decoding akes less ime despie he loss in performance. Moreover, variable rae is obained wih he orhogonaliy of he codewords being kep. The puncured version could find is use in real-ime applicaions such as in Hybrid Auomaic Repea reques (HARQ) sysems where forward error correcion is uilized. Fig. 7 Bi Error Rae Agains E b N o N = 200 bis Fig. 8 Block Error Rae Agains E b N o N = 200 bis 263

7 Fig. 9 Bi Error Rae Agains E b N o N = 1000 bis Fig. 10 Block Error Rae Agains E b N o N = 1000 bis Fig. 12 Block Error Rae Agains E b N o N = 4000 bis Fig. 13 Bi Error Rae Agains E b N o N = 5400 bis Fig. 11 Bi Error Rae Agains E b N o N = 4000 bis Fig. 14 Block Error Rae Agains E b N o N = 5400 bis 264

8 Fig. 15 Time (log scale) agains E b N o for Unpuncured and Puncured Codes N = 4000 and N = 5400 V. CONCLUSION I was seen how variable-rae in he VRSTC could be achieved in superorhogonal urbo coding wih OVSF codes. Simulaion resuls demonsrae beer performance han higher rae (rae 1 / 3 ) ordinary urbo codes a he expense of bandwidh expansion. Alhough, he MAP simulaions resuls show decreased performance compared o hose using Walsh- Hadamard codes, he uilizaion of OVSF codes have many advanages including he sorage of a single OVSF ree o produce variable-lengh codewords compared o he sorage o many marices of various lenghs o achieve he same for Walsh-Hadamard codes. Moreover, he orhogonaliy of codewords is kep wih he VRSTC sysem. The decoding algorihms used show very good performance a lower E b N o wih increasing blocklenghs. The VRSTC scheme is hus suiable for use in CDMA and in fuure mobile communicaions sysems o suppor many ypes of applicaions where variable-rae is a prerequisie. ACKNOWLEDGMENT The assisance and guidance of Dr. G. Ramsawock in his research work, and he financial suppor of he Teriary Educaion Commission of Mauriius are graefully acknowledged. REFERENCES [1] C. Berrou, A. Glavieux, and P. Thiimajshima, Near shannon limi error-correcing coding and decoding: Turbo-codes, IEEE Inernaional Conference on Communicaions ICC 93, Geneva, Swizerland May 1993, vol. 2, pp [2] B. Sklar, Turbo code conceps made easy, or how I learned o concaenae and reierae, MILCOM 97, vol. 1, pp [3] G. Ramsawock, Combined channel coding and modulaion using variable rae, Ph.D. disseraion, Universiy of Mauriius, Mauriius, [4] A. Lienz and J. Villasenor, Very low variable rae convoluional codes for unequal error proecion in DS-CDMA sysems, IEEE Transacions on Communicaions, July 1997, vol. 45, no. 7, pp [5] K. Pehkonen and P. Komulainen, "A superorhogonal urbo-code for CDMA applicaions," IEEE 4h Inernaional Symposium on Spread Specrum Techniques and Applicaions Proceedings, Sep 1996, vol. 2, pp [6] P. Komulainen and K. Pehkonen, Performance evaluaion of superorhogonal urbo codes in AWGN and fla rayleigh fading channels, IEEE Journal on Seleced Areas in Communicaions, vol. 16, no. 2, 1998, pp [7] A.J. Vierbi, Mehod and apparaus for generaing superorhogonal convoluional codes and he decoding hereof, US A, March 9, [8] A.J. Vierbi, CDMA: Principles of Spread Specrum Communicaion. Addison-Wesley Publishing Company, ISBN , [9] L. Yen and M. Tsou, An OVSF code assignmen scheme uilizing muliple RAKE combiners for W-CDMA, Compuer Communicaions, vol. 27, no.16, 2004, pp [10] G. Suchira and M.L. Valarmahi, BER performance of modified walsh hadamard codes in a DS-CDMA and cogniive underlay sysem, European Journal of Scienific Research, vol. 64, no. 4, 2011, pp [11] M.H. Le and R. Liyana-Pahirana, Unequal error proecion codes for wavele image ransmission over W-CDMA, AWGN and rayleigh fading channels, 10h Inernaional Conference on Telecommunicaions, ICT 2003, 23 Feb - 1 March 2003, vol. 2, pp [12] P. Roberson, Illuminaing he srucure of code and decoder of parallel concaenaed recursive sysemaic (urbo) codes, IEEE Global Telecommunicaions Conference, GLOBECOM 94, San Francisco, CA, 28 Nov- 2 Dec 1994, vol. 3, pp [13] R. L. Bahl, J. Cocke, F. Jelinek and J. Raviv, Opimal decoding of linear codes for minimizing symbol error rae, IEEE Transacions on Informaion Theory, vol. 20, no.2, 1974, pp [14] P. Roberson, E. Villebrun and P. Hoeher, A comparison of opimal and sub-opimal MAP decoding algorihms operaing in he log domain, Proceedings of he Inernaional Conference on Communicaions, vol. 2, 1997, pp