Comparison of Convolutional and Turbo Coding For. Broadband FWA Systems

Transcription

1 Compariso of Covolutioal ad Turbo Codig For Broadbad FWA Systems Ioais A. Chatzigeorgiou, Miguel. D. odrigues, Ia J. Wassell ad olado A. Carrasco Digital Techology Group Computer aboratory, Uiversity of Cambridge Commuicatios ad Sigal Processig Group School of EE&C Eg., Uiversity of ewcastle Abstract It has bee demostrated that turbo codes substatially outperform other codes, e.g., covolutioal codes, both i the o-fadig additive white Gaussia oise (AWG) chael as well as multiple-trasmit ad multiple-receive atea fadig chaels. Moreover, it has also bee reported that turbo codes perform very well i fast fadig chaels, but perform somewhat poorly o slow ad block fadig chaels of which the broadbad fied wireless access (FWA) chael is a eample. I this paper, we thoroughly compare the performace of turbo-coded ad covolutioal-coded broadbad FWA systems both with ad without atea diversity uder the coditio of idetical compleity for a variety of decodig algorithms. I particular, we derive mathematical epressios to characterise the compleity of turbo decodig based o state-of-theart og-map ad Ma-og-MAP algorithms as well as covolutioal decodig based o the Viterbi algorithm i terms of the umber of equivalet additio operatios. Simulatio results show that turbo codes do ot offer ay performace advatage over covolutioal codes i FWA systems without atea diversity or FWA systems with limited atea diversity. Ideed, turbo codes oly outperform covolutioal codes i FWA systems havig sigificat atea diversity. Keywords Algorithms, Compleity theory, Commuicatio system performace, Cocateated Codig, Covolutioal Codes, Decodig, Fadig Chaels, Iterative Methods, Trellis Codes. This work was supported by EPSC uder Grat G/S67/0.

2 Itroductio Broadbad fied wireless access (FWA) systems eable high data rate commuicatios where traditioal ladlies are either uavailable or too costly to be istalled. These systems also eable operators i a competitive eviromet to roll-out broadbad services i a rapid ad cost effective maer []. I this cotet, broadbad FWA stadardisatio activities have bee performed uder the auspices of the IEEE 80.6 [] ad the ETSI HIPEMA [] workig groups. I particular, the IEEE 80.6a stadard proposes a umber of trasmissio techiques to combat multipath fadig i broadbad FWA systems, for eample orthogoal frequecy-divisio multipleig (OFDM). This stadard also proposes the use of turbo ad covolutioal chael codig techiques to further improve performace i broadbad FWA systems. Turbo codes have bee show to be very powerful i both the additive white Gaussia oise (AWG) chael [,5] as well as i multiple-trasmit ad multiple-receive atea ayleigh fadig chaels [6-8]. Turbo codes have also bee show to perform very well i rapidly fadig chaels [9], but to perform less well i slow ad block fadig chaels [0,], of which the broadbad FWA chael is a eample. I rapidly fadig chaels, codig together with iterleavig techiques are used to spread cosecutive code bits over multiple idepedetly fadig blocks to improve performace. However, i slow ad block fadig chaels codig together with iterleavig techiques caot i geeral be used i a effective maer because delay ad latecy cosideratios limit the depth of iterleavig. This situatio compromises i particular the performace of turbo codes because occasioal deep fades cause severe error propagatio i the iterative decodig process []. Accordigly, comparisos of the performace of turbo ad covolutioal codes i slow ad block fadig chaels costitutes a topic of practical research iterest. I particular, Hoshyar et al. have show that turbo ad covolutioal codes perform idetically i block fadig chaels with o atea diversity [0]. I additio, i et al. have show that turbo outperform covolutioal codes i ayleigh slow fadig chaels with atea diversity oly at a high sigal-to-oise ratio (S) [].

3 I this paper, we thoroughly compare the performace of turbo ad covolutioal codes i broadbad FWA systems both with ad without atea diversity. However, this work differs from that i [0] ad [] i that the comparisos are carried out uder the coditio of idetical compleity for a variety of decodig algorithms, icludig the widely used log-domai maimum a posteriori algorithm (og-map) [] as well as the simplified Ma-og-MAP algorithm [] for turbo decodig ad the covetioal Viterbi algorithm [5] for covolutioal decodig. This paper is orgaised as follows: Sectio itroduces the system model ad gives a brief descriptio of the decodig algorithms used for turbo decodig ad covolutioal decodig, whist sectio characterises their compleity. Sectio compares the performace of turbo ad covolutioal codig uder the coditio of idetical compleity for a variety of decodig algorithms i broadbad FWA systems both with ad without atea diversity. Fially, sectio 5 summarises the mai cotributios of this paper. System Model. Geeral Overview I this work, we cosider systems based o OFDM trasmissio, which lies at the heart of curret broadbad FWA stadards. We also cosider sigle atea FWA systems, which do ot eploit space diversity, as well as a multiple atea FWA systems, which do eploit space diversity. Figure depicts the system block diagram, where trasmit ad receive ateas, respectively. T ad represet the umber of At the trasmitter, the iformatio bits are ecoded ad block iterleaved. We cosider both turbo ad covolutioal ecoders. For turbo codig, the ecoder cosists of the parallel cocateatio of two recursive systematic covolutioal (SC) ecoders with rate /, as described i [,5]. Alterate pucturig of the parity bits trasforms the covetioal rate turbo code to a rate turbo code. For covolutioal codig, the ecoder cosists of a SC ecoder with rate /. The mapper maps groups of log M s bits ito oe of s M comple symbols from a uit power M s -QAM costellatio.

4 I sigle atea systems ( = T ), the space-time processig block does ot further process the modulatio symbols; istead, the modulatio symbols are passed directly to the OFDM block. However, i multiple trasmit atea systems ( > T ), the space-time processig block will further process the modulatio symbols before passig them to the OFDM block. I particular, the space-time processor geerates for each particular OFDM sub-carrier a space-time block code (STBC) accordig to the geerator matrices G, G or G give by [6-8] = G, () = G, () = G, () Here, we cosider space-time coded OFDM systems where redudacy spas space ad time domais [9], rather tha space-frequecy coded OFDM systems where redudacy spas space ad frequecy domais [0,].

5 where,, ad deote modulatio symbols. The rows of the matrices represet symbols trasmitted i differet time slots by a particular OFDM sub-carrier. The colums of the matrices represet symbols trasmitted by differet ateas agai by the particular OFDM subcarrier. Essetially, a total of K T symbols obtaied from the origial K modulatio symbols are trasmitted durig K separate time slots by T trasmit ateas by each particular OFDM sub-carrier. ote that G, G ad G are appropriate for two, three ad four trasmit ateas, respectively, ad for a arbitrary umber of receive ateas. ote also that G is rate K K =, whereas G ad G are rate K K =. Sigle atea systems (where T = ad K = K = ) are a special case of multiple trasmit atea systems (where > ad K, K > ). Thus, i the sequel both sigle as well as multiple atea systems are treated uder the same framework. Fially, at each trasmit atea chai, comple symbols correspodig to the elemets for a particular time slot for the differet STBC are imposed oto orthogoal sub-carriers by meas of a IFFT, a cyclic prefi is iserted with duratio loger tha the impulse respose of the chael to combat itersymbol iterferece (ISI) ad itercarrier iterferece (ICI), ad fially the sigal is digital-to-aalogue coverted. The OFDM sigal is distorted by a broadbad FWA chael as well as AWG. The broadbad FWA chael is time-dispersive but ot sigificatly time-varyig. Hece, we assume that the chael is essetially costat durig the trasmissio of a frame of data. At the receiver, at each receive atea chai the sigal is aalogue-to-digital coverted, the cyclic prefi is removed, ad the comple symbols correspodig to the elemets for a particular time slot for the differet STBC are removed from the orthogoal sub-carriers by meas of a FFT. T 5

6 6 The relatio betwee the comple receive symbols ad the comple trasmit symbols associated with the STBC coveyed by the th OFDM sub-carriers ca be writte as follows H S + =, () where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = K K K M O M M, (5) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = K S S S K S S S K S S S T T T M O M M S, (6) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = K K K M O M M, (7) ad = t r r r t t H H H H H H H H H,,,,,,,,, M O M M H. (8) Here, we focus without loss of geerality o the first space-time block code frame.

7 ow, j ( k) deotes the comple receive symbol associated with the th OFDM sub-carrier at time slot k ad receive atea j, S i ( k) deotes the comple trasmit symbol associated with the th OFDM sub-carrier at time slot k ad trasmit atea i, i j H, is the uit power radom chael frequecy respose at the th OFDM sub-chael from trasmit atea i to receive atea j (ote that i j H, is idepedet of time slot k ), ad j ( k) deotes the oise radom variable at the th OFDM sub-chael at time slot k ad receive atea j. The oise radom variables are ucorrelated circularly symmetric comple Gaussia with mea zero ad variace /S orm, where S orm =S/ T ad S deotes the average sigal-to-oise ratio per receive atea. et, the comple symbols are demapped ito soft bits. I particular, the soft demapper computes the log-likelihood ratio () give by D m ( b ( k) ) m { b ( k) = } m b ( k) = 0 Prob = l, (9) Prob { } where b m ( k) is the mth bit coveyed by the kth modulatio symbol associated with the STBC coveyed by the th OFDM sub-carrier. The i (9) is also give by D m ( b ( k) ) = l + S S S S p p ( S ) Prob{ S } ( S ) Prob{ S } m { b ( k) = } m b ( k ) = 0 Prob = l + l Prob{ } a priori iformatio, A m ( b ( k )) = l 7 + S S S S p log M K m ( S ) Prob{ b ( k )} + S S m = k = p log M K m ( S ) Prob{ b ( k )} S S m = k = p p log M K m ( S ) Prob{ b ( k )} m = k = m, k m, k log M K m ( S ) Prob{ b ( k )} m = m, k m, k etrisic iformatio, E k = m ( b ( k ) ), (0)

8 where S S + = = + S is the set of matrices of trasmit symbols m { : b ( k) = } S ), m { : b ( k ) = 0} S is the set of matrices of trasmit symbols S ), ad the probability desity fuctio p( S ) p ( S ) S such that ( k) = b m (i.e., S such that ( k) = 0 is give by H ( H S ) ( H S ) T S = e K ( π T S ) b m (i.e.,. () ote that the log-likelihood ratio is the sum of the a priori iformatio ad the etrisic iformatio, i.e., D m m ( b ( k) ) b ( k) A m ( ) + ( b ( k) ) =. () E The a priori iformatio is equal to zero, i.e., m ( b ( k) ) = 0 A. () The etrisic iformatio ca be further simplified for particular modulatio schemes as well as STBC by virtue of the orthogoal properties of G, G ad G. For eample, i the sigle atea case ( = ) with o STBC ( K = K = ) ad with Gray coded QPSK modulatio T = ( log M = ) it follows that E E,, j H ( b ( ) ( ) ) S H e e ( ) 8 { } =, () { },, j H ( b ( ) ( ) ) S H Im e ( ) =. (5) I the multiple atea case ( T =, > ) with the STBC specified by G ( K = K = ) ad with Gray coded QPSK modulatio ( log M = ) it follows that E E, r r, r r ( b ( ) ) S e ( H ) ( ) + H ( ) = ( ), (6) r = = ( ), (7) r =, r r, r r ( b ( ) ) S Im ( H ) ( ) + H ( )

9 E ( ) =, (8) r =, r r, r r ( b ( ) ) S e ( H ) ( ) H ( ) E = ( ). (9) r =, r r, r r ( b ( ) ) S Im ( H ) ( ) H ( ) ote that similar etrisic iformatio epressios ca also be determied for other particular modulatio schemes ad STBCs. Fially, the soft bits (the s) are block de-iterleaved ad decoded. For turbo codig, the costituet soft-iput soft-output decoders use either the optimal log-map algorithm [] or the ma-log-map algorithm []. For covolutioal codig, the decoder uses the covetioal Viterbi algorithm [5].. Decoders Overview We ow describe the basic ideas behid the various decodig algorithms that are ecessary for the compleity computatios. We iitially cosider the Viterbi algorithm used for systems based o covolutioal codes. Subsequetly, we cosider both the log-map ad the ma-log- MAP algorithms used for systems based o turbo codes... Viterbi Algorithm The Viterbi algorithm [5] estimates the most probable sequece of states for a received sequece of soft bits. A brach i the trellis diagram of the covolutioal code correspods to a trasitio from a memory state s at time t- to aother state s at time step t. The brach metric ( s, s) t BM correspods to the sum of the ier products betwee the codeword bits associated with the brach ad the received soft bits at time step t. Moreover, a path i the trellis diagram correspods to a series of itercoected braches. The path metric correspods to the sum of the brach metrics of the braches that compose the path. As the path progresses through the trellis, 9

10 subsequet braches joi the path so that the path metric chages accordigly. If two paths merge to a state s at a time step t, the Viterbi algorithm selects the path with the highest metric, the survivor path, ad disregards those with lower metrics. The path metric of the survivor path at a time step t for a state s, PM (s) PM t, is give by ( s) ( s ) ( s, s) ( s ) ( s, s) t t t t + t = ma( PM + BM, PM BM ), (0) where s ad s correspod to the states of the competig paths at time step t. This addcompare-ad-select process yields maimum likelihood (M) decisios... BCJ algorithm Although the Viterbi algorithm yields M decisios, it ca either produce reliability values (s) associated with the output decoded bits or it ca eploit a priori iformatio associated with the iput iformatio bits. However, these two processes are of utmost importace to eable the costructive iformatio echage betwee the two compoet decoders for successful iterative decodig of turbo codes. Berrou et al. [] have proposed the use of a maimum a posteriori (MAP) decodig algorithm based o the widely kow BCJ algorithm [] for each compoet decoder i a turbo decoder I particular, the BCJ algorithm yields the followig reliability values for a decoded bit at time step t D ( bt ) = l α t ( s ) γ t ( s, s) βt ( s) l αt ( s ) γ t ( s, s) βt ( ), () b (, : b (, : t =+ t = s where the terms α ( ) ad β (s) are derived by meas of a forward ad a backward recursio, t s respectively, based o t a ( s) = t β t s ( s ) = a s t ( s ) γ ( s, s), β ( s) γ ( s, s), t t t () 0

11 ad the term γ ( s, s) is calculated by cosiderig both the brach metric at time step t ad the a t priori iformatio for the decoded bit, as described i more detail i []. The BCJ algorithm is cosidered to be etremely comple owig to the various multiplicatio operatios as well as the logarithmic operatios required to compute the a-posteriori for each decoded bit. However, two simple modificatios were proposed to reduce its compleity without severely compromisig performace... og-map ad Ma-log-MAP algorithms The first modificatio to the BCJ algorithm yields the ma-log-map algorithm proposed by Koch ad Baier i 990 []. This modificatio is based o the calculatio of the a-posteriori by usig the approimatio λ λ ( + e ) ma( λ, λ ) l e. () Cosequetly, epressios () ad () are cosiderably simplified, sice the overall umber of operatios decreases ad moreover multiplicatios are trasformed ito additios i the logdomai. However, this modificatio results i cosideratio of oly the M path i the trellis through a particular state, rather tha every path i trellis through this state []. Therefore, the performace of the ma-log-map algorithm is iferior to that of the BCJ algorithm. Aother modificatio yields the log-map algorithm proposed by obertso et al. i 995 []. This modificatio is based o the correctio of the approimatio by usig the Jacobia logarithm, that is λ ( ) ( ) λ ma(, ) l λ λ + e = λ λ + + e l e. () ote that sice the correctio term takes oly a limited umber of values, look-up tables ca be used to reduce the compleity of the computatios Otherwise, if the correctio term is computed eactly, this (eact) log-map algorithm is etirely equivalet to the BCJ algorithm.

12 Compleity Cosideratios We ow cosider the characterizatio of the compleity of the various decodig algorithms. We will follow the covetioal approach i the field of codig theory, where the compleity of a decodig algorithm is measured i terms of the total umber of computatioal operatios [,], such as additios, subtractios, multiplicatios ad divisios. I particular, similarly to [], we epress the compleity of the various basic operatios i terms of that of a additio operatio. Hece, we ultimately epress the compleity of log-map, ma-log-map ad the Viterbi decodig algorithms i terms of the total umber of equivalet additios eecuted. This approach delivers results with wider applicability, sice the compleity measure is ot tied to specific hardware implemetatios. The basic operatios performed by the various decodig algorithms iclude additio (ADD), subtractio (SUB), multiplicatio by ± (MU), divisio by (DIV), compariso (CP), ma(,y) or mi(,y) (MAX) ad table look-up (KUP). The ADD, SUB, MU, DIV ad CP operatios correspod to oe equivalet additio, whilst the MAX operatio correspods to two equivalet additios, sice it first uses a CP operatio to compare the two iput values ad the stores the result i a register []. The KUP operatio correspods to three equivalet additios because o more tha three CP operatios are required to map a iput value to oe of the eight values stored i the look-up table [] for the close approimatio of the epoetial factor i ().The procedures performed by the log-map ad the ma-log-map algorithms ca be classified as follows [,]: Brach Metrics Calculatio (Proc. A) Forward Metrics Calculatio (Proc. B) Backward Metrics Calculatio (Proc. C) Soft Decisio of the decoded bit (Proc. D)

13 I the case of ma-log-map, procedures B, C ad D require implemetatio of the MAX fuctio. I the case of log-map, these procedures also require the implemetatio of the MAX fuctio plus oe ADD, oe SUB ad oe KUP operatios. The procedures performed by the Viterbi algorithm ca be classified as follows [5]: Brach Metrics Calculatio (Proc. A) Path Metrics Update (Proc. E) Hard Decisio Geeratio (Proc. G) Moreover, i this case procedure A does ot eploit ay a priori iformatio. Tables - summarize the computatioal requiremets of the various decodig algorithms as a fuctio of the ecoder memory order M. ote that here we assume that the costituet SC ecoders for turbo codig, as well as the SC ecoder for covolutioal codig are rate /. ote that we also take ito accout the additioal compleity associated with the brach metrics calculatios due to a priori iformatio eploited by the turbo decoder. Fially, Table summarizes the overall compleity (i terms of the umber of equivalet additio operatios) of the various decodig algorithms. As a eample, let us cosider i detail the computatioal requiremets of the Viterbi algorithm for a rate / covolutioal code (see Table ). Calculatio of a brach metric requires MU operatios for the computatio of the two ier products betwee the codeword bits associated with the brach ad the received soft bits, ad ADD operatio for the summatio of the two products. Hece, procedure A requires M MU ad M ADD operatios, give that two braches emerge from each of the M states per time step. Moreover, calculatio of a path metric requires ADD ad MAX operatios (see (0)). Cosequetly, procedure E requires M ADD ad M MAX operatios per time step. Fially, procedure G requires oly KUP operatio for the geeratio of a hard bit per time step, as eplaied i [].

14 Figure compares the compleity of turbo decodig ad covolutioal decodig for particular cofiguratios. As a eample, we ote that the compleity of a turbo decoder with memory order M = applyig the log-map algorithm with 7 iteratios, is comparable to that of a similar turbo decoder applyig the ma-log-map algorithm with iteratios, or to that of a covolutioal decoder with memory order M = 8 applyig the covetioal Viterbi algorithm. Fially, we ote that Wu [] has also previously aalysed the compleity of various decodig algorithms i terms of the umber of equivalet additio operatios. However, our aalysis differs from that preseted i [] i oe fudametal aspect. We take the compleity of a look-up operatio to be equivalet to equivalet additio operatios, rather tha the 6 equivalet additio operatios cosidered i []. Hece, our results are less pessimistic i terms of the umber of equivalet additio operatios tha those i []. We also ote that obertso et al. [] have also previously aalysed the compleity of a variety of decodig algorithms, but for simplicity mathematical ad logical operatios were assumed to ehibit idetical compleity. Simulatio esults I our simulatios, the covolutioal ecoder uses a SC ecoder with rate, geerator polyomial (,75 56) ad memory order M = 8. The umber of iformatio bits fed to the covolutioal ecoder is 06, so that the umber of ecoded bits is 08. The turbo ecoder uses two idetical termiated SC ecoders with rate, octal geerator polyomial (,5 7) ad memory order M =, ad a radom iterleaver with size either = 0 or = 09. Alterate pucturig of the parity bits trasforms the covetioal rate turbo code to a rate turbo code. I this case, the umber of iformatio bits fed to the turbo ecoder is either 0 (for = 0 ) or 09 (for = 09), so that the umber of ecoded bits is 08 or 89, respectively. The covolutioal decoder uses the Viterbi algorithm. The turbo decoder uses either the log-map algorithm with 7 iteratios or the ma-log-map algorithm with iteratios. ote that these cofiguratios have idetical decodig compleity. The depth of the

15 block iterleaver ad de-iterleaver is set to be equal to 6. I our simulatios, we also use OFDM/QPSK sigals with OFDM symbol duratio T =.8µ s, cyclic prefi duratio T CP =.µs, ad = 56 sub-carriers. Furthermore, i the simulatios we focus o sigle atea as well as multiple atea systems based o STBCs specified by G, G ad G. Si iterim broadbad FWA chael models have bee adopted by the IEEE 80.6a stadard [5]. We cosider the SUI model, which correspods to average suburba coditios. This model icludes three fadig taps with delays 0 µs, 0.5 µs ad.0 µs, with relative powers 0 db, 5 db ad 0 db, ad with K-factors, 0 ad 0, respectively. The delay spread is 0.6 µs ad the Doppler spread per tap is 0. Hz. The SUI chael model specifies a atea correlatio coefficiet value equal to 0.. However, i the simulatios we will assess systems both with ad without atea correlatio. Figure compares the performace of various turbo-coded ad covolutioal-coded systems for both sigle ad multiple atea cofiguratios for the case of frames havig 08 ecoded bits. Here, we set the atea evelope correlatio coefficiet to be equal to the omial value of 0.. We ote that turbo codes substatially outperform covolutioal codes i the AWG chael. However, the performace of turbo codes is similar to that of covolutioal codes i sigle atea broadbad FWA systems. Moreover, the performace of turbo codes is also similar to that of covolutioal codes i multiple atea broadbad FWA systems. I particular, we ote that this is essetially the case for turbo codig based o both the log-map as well as the ma-log-map algorithms. These results are due to the limited diversity offered both by sigle atea as well as multiple atea FWA chaels. I sigle atea FWA chaels there is o time diversity due to the very slow time variatio ature of the chael, ad there is oly mild frequecy diversity due to the mild time-dispersive ature of the chael. I multiple atea We assume that the chael is essetially costat durig the trasmissio of a frame of data by virtue of the low Doppler spread value. The error rate results are averaged over 0000 chael realisatios. 5

16 systems, atea correlatio will also substatially limit the advatage owig to space diversity. Hece, the presece of frequet deep fades sigificatly impairs the performace of turbo codes owig to severe error propagatio i the iterative decodig process []. Figure also compares the performace of various turbo-coded ad covolutioal-coded systems for both sigle ad multiple atea cofiguratios agai for the case of frames havig 08 ecoded bits. However, here we set the atea evelope correlatio coefficiet to be equal to zero, i.e., the ideal situatio. I this case, as the umber of ateas is icreased (i.e. as atea diversity is icreased), turbo codes evetually substatially outperform covolutioal codes. I fact, as the umber of ateas is icreased the uderlyig fadig chael will approach a o-fadig AWG chael, where turbo codes are kow to substatially outperform covolutioal codes. Figure 5 ad Figure 6 compare the performace of turbo-coded systems for various sigle atea ad multiple atea system cofiguratios for frame legths of 08 ad 89 ecoded bits. Figure 5 applies to systems with a atea evelope correlatio coefficiet of 0., whereas Figure 6 applies to systems with zero atea evelope correlatio coefficiet. I AWG chaels a icrease i the legth of the turbo code frame, i.e., a icrease i the legth of the radom iterleaver employed by the turbo ecoder, gives rise to substatial performace improvemets. I cotrast, a icrease i the legth of the covolutioal code frame does ot geerally result i performace improvemets [5]. Yet, we ote that i broadbad FWA chaels the legth of the frame does ot chage the ature of the previous treds. I particular, i low diversity FWA systems (i.e., systems with a low umber of ateas) turbo codes with differet frame legths perform idetically. I high diversity FWA systems (i.e., systems with a high umber of ateas) turbo codes with a loger frame outperform turbo codes with a shorter frame, ad cosequetly also outperform covolutioal codes. To coclude, we observe that very high order diversity systems are required for turbo-coded systems to outperform covolutioal-coded systems. However, this may be difficult to achieve i 6

17 FWA systems for various practical ad ecoomic reasos. Specifically, the FWA chael is ot sigificatly time-dispersive or time-varyig ad cosequetly caot offer much frequecy or time diversity. Moreover, atea correlatio severely limits spatial diversity. Additioal results (ot preseted here) also suggest that the treds observed for the specific turbo ad covolutioal codes cosidered i this work also apply to other turbo ad covolutioal codes with idetical compleity. 5 Coclusios I this paper, we have compared the performace of turbo-coded ad covolutioal-coded broadbad FWA systems both with ad without atea diversity uder the coditio of idetical compleity for a variety of decodig algorithms. We have show that turbo codig does ot offer ay performace advatage over covolutioal codig for FWA systems without atea diversity or for FWA systems with limited atea diversity. We have also show that turbo codig oly outperforms covolutioal codig i FWA systems havig sigificat atea diversity. These results are of practical iterest for the deploymet ad desig of high performace broadbad FWA systems. 7

18 efereces [] H. Bölcskei, A. J. Paulraj, K. V. S. Hari,. U. abar ad W. W. u, Fied broadbad wireless access: State of the art, chaleges ad future directios, IEEE Commuicatios Magazie, vol. 9, pp.00-08, Jauary 00. [] IEEE Stadard, Part 6: Air iterface for fied broadbad wireless access systems Amedmet : Media access cotrol modificatios ad additioal physical layer specificatios for - GHz, IEEE 80.6a, Jauary 00. [] ETSI Stadard, Broadbad radio access etworks (BA); HiperMA; Physical (PHY) layer, ETSI TS 0 77 V.., Jauary 005. [] C. Berrou, A. Glavieu ad P. Thitimajshima, ear Shao limit error-correctig codig ad decodig: Turbo-codes, Proceedigs of the IEEE Iteratioal Coferece o Commuicatios, vol., pp , May 99. [5] C. Berrou ad A. Glavieu, ear optimum error correctig codig ad decodig: Turbo codes, IEEE Trasactios o Commuicatios, vol., pp. 6-7, October 996. [6] A. Stefaov ad T. M. Duma, Turbo coded modulatio for wireless commuicatios with atea diversity, Proceedigs of the IEEE Vehicular Techology Coferece-Fall, vol., pp , September 999. [7] A. Stefaov ad T. M. Duma, Turbo coded modulatio for systems with trasmit ad receive atea diversity, Proceedigs of the IEEE Global Telecommuicatios Coferece, vol. 5, pp. 6-0, ovember 999. [8] A. Stefaov ad T. M. Duma, Turbo-coded modulatio for systems with trasmit ad receive atea diversity over block fadig chaels: system model, decodig approaches, ad practical cosideratios, IEEE Joural o Selected Areas i Commuicatios, vol. 9, pp , May 00. [9] J. P. Woodard ad. Hazo, Comparative study of turbo decodig techiques: A overview, IEEE Trasactios o Vehicular Techology, vol. 9, pp. 08-, ovember

19 [0]. Hoshyar, S. H. Jamali ad A.. S. Bahai, Turbo codig performace i OFDM packet trasmissio, Proceedigs of the IEEE Vehicular Techology Coferece-Sprig, Tokyo, Japa, vol., pp , May 000. []. i,. J. Cimii ad C. I. Chuag, Compariso of covolutioal ad turbo codes for OFDM with atea diversity i high-bit-rate wireless applicatios, IEEE Commuicatios etters, vol., pp , September 000. [] H. El Gamal ad A.. Hammos, Jr., Aalyzig the turbo decoder usig the Gaussia approimatio, IEEE Trasactios o Iformatio Theory, vol. 7, pp , February 00. [] P. obertso, E. Villebru ad P. Höeher, A compariso of optimal ad sub-optimal MAP decodig algorithms operatig i the log domai, Proceedigs of the IEEE Iteratioal Coferece o Commuicatios, vol., pp , Jue 995. [] W. Koch ad A. Baier, Optimum ad sub-optimum detectio of coded data distributed by time-varyig itersymbol iterferece, i Proc. IEEE Coferece o Global Commuicatios, Sa Diego, CA, pp , December 990. [5] J. G. Proakis. Digital Commuicatios, th ed. ew York: McGraw-Hill, 00. [6] S. M. Alamouti, A simple trasmitter diversity scheme for wireless commuicatios, IEEE Joural o Selected Areas i Commuicatios, vol. 6, pp. 5-58, October 998. [7] V. Tarokh, H. Jafarkhai ad A.. Calderbak, Space-time block codes from orthogoal desigs, IEEE Trasactios o Iformatio Theory, vol. 5, pp , July 999. [8] V. Tarokh, H. Jafarkhai ad A.. Calderbak, Space-time block codig for wireless commuicatios: Performace results, IEEE Joural o Selected Areas i Commuicatios, vol. 7, pp. 5-60, March 999. [9] D. Agrawal, V. Tarokh, A. aguib ad. Seshadri, Space-time coded OFDM for high data-rate wireless commuicatio over widebad chaels, Proceedigs of the IEEE Vehicular Techology Coferece, vol., pp. -6, May 998. [0] H. Bölcskei amd A. J. Paulraj, Space-frequecy coded broadbad OFDM systems, Proceedig of the IEEE Wireless Commuicatios ad etworkig Coferece, vol., pp. -6, September

20 [] B. u ad X. Wag, Space-time code desig i OFDM systems, Proceedigs of the Global Telecommuicatios Coferece, vol., pp , ovember-december 000. [].. Bahl, J. Cocke, F. Jeliek ad J. aviv, Optimal decodig of liear codes for miimizig symbol error rate, IEEE Trasactios o Iformatio Theory, vol. 0, pp. 8-87, March 97. [] M. P. C. Fossorier, Iterative eliability-based Decodig for ow Desity Parity Check Codes, IEEE Joural o Selected Areas i Commuicatios, vol. 9, pp , May 00. [] P. H.-Y. Wu, O the compleity of turbo decodig algorithms, Proceedigs of the IEEE Vehicular Techology Coferece-Sprig, vol., pp. 9-, May 00. [5] V. Erceg et al., Chael Models for Fied Wireless Applicatios, IEEE 80.6a cot. IEEE 80.6.c- 0/9r, Jue 00. 0

21 Biographies IOAIS A. CHATZIGEOGIOU is curretly a Ph.D. cadidate i commuicatio egieerig at the Uiversity of Cambridge. From 000 to 00, he held positios i Marcoi Commuicatios ad Imarsat td. He received his Dipl.-Ig i electrical egieerig from Democritus Uiversity of Thrace, Greece, i 997 ad his M.Sc. i satellite commuicatio egieerig from the Uiversity of Surrey, U.K., i 000. His research iterests iclude chael codig, space-time codig ad equalizatio techiques for fied wireless access systems. He is a Member of the IEEE ad the Techical Chamber of Greece. MIGUE. D. ODIGUES (mrdr@cam.ac.uk) was bor i Porto, Portugal o May 0, 975. He received the iceciatura degree i electrical egieerig from the Faculty of Egieerig of the Uiversity of Porto, Portugal i 998 ad the Ph.D. degree i electroic ad electrical egieerig from Uiversity College odo, U.K. i 00. He has held postdoctoral research appoitmets at Cambridge Uiversity, U.K., ad at Priceto Uiversity, U.S.A., i the period from 00 to 006. He joied the faculty of the Departmet of Computer Sciece, Faculty of Scieces of the Uiversity of Porto i 007. His research iterests iclude iformatio theory, commuicatios theory ad sigal processig ad their applicatios to wireless systems. He has over 50 publicatios i iteratioal jourals ad coferece proceedigs i these areas. He is also a Visitig esearcher at Uiversity College odo, U.K. Dr. odrigues has bee the recipiet of doctoral ad postdoctoral fellowships from the Portuguese Foudatio for Sciece ad Techology, a postdoctoral fellowship from Fudação

22 Calouste Gulbekia, the Prize Egeheiro Atóio de Almeida, the Prize Egeheiro Cristiao Spratley, ad the Merit Scholarship from the Uiversity of Porto. IA J. WASSE was bor i Wolverhampto, Eglad o October, 960. He received the B.Sc., B.Eg. (Hoours) degree (First Class) i electrical ad electroic egieerig from the Uiversity of oughborough, UK i 98 ad the Ph.D. degree i electroic ad electrical egieerig from the Uiversity of Southampto, U.K. i 990. He is a Seior Uiversity ecturer at the Computer aboratory, Uiversity of Cambridge. Prior to this he has held positios at the Uiversity of Huddersfield, Hutchiso Persoal Commuicatios td., Multiple Access Commuicatios td. ad Marcoi td. His research iterests iclude fied wireless access systems, radio propagatio ad sigal processig. Dr. Wassell is a Member of the Istitutio of Egieerig ad Techology. OADO A. CAASCO (r.carrasco@ewcastle.ac.uk) received the B.Sc. (Hoours) degree from the Uiversity of Satiago, Chile, ad the Ph.D. degree for his work o implemetig digital filters usig several processors, from the Uiversity of ewcastle-upo-tye, U.K. He was awarded the IEE Heaviside Premium i 98 for his work i multiprocessor systems. Betwee 98 ad 98 he was employed by Alfred Peters imited, Sheffield (ow Meditech) ad carried out research ad developmet i sigal processig associated with cochlear stimulatio ad respose. He has bee with Staffordshire Uiversity sice 98 ad is ow Professor of Mobile Commuicatios at the Uiversity of ewcastle-upo-tye. His priciple research iterests are digital sigal processig algorithms for mobile ad etwork commuicatio systems ad speech processig/recogitio. Professor Carrasco has over a hudred scietific publicatios, five chapters i telecommuicatio

23 referece tets ad a patet to his ame. He is a member of several orgaizig committees, a member of the EPSC College ad a member of the EPSC assessmet pael. He is a Fellow of the Istitutio of Egieerig ad Techology.

24 Figures Figure : Commuicatios system model.

25 Figure : Compleity compariso betwee turbo decodig ad covolutioal decodig. 5

26 Figure : Error rates for various turbo-coded ad covolutioal-coded OFDM systems for both sigle ad multiple atea FWA cofiguratios for frames havig 08 code bits. Atea evelope correlatio coefficiet is set to 0.. 6

27 Figure : Error rates for various turbo-coded ad covolutioal-coded OFDM systems for both sigle ad multiple atea FWA cofiguratios for frames havig 08 code bits. Atea evelope correlatio coefficiet set to zero. 7

28 Figure 5: Error rates for turbo-coded OFDM systems for both sigle ad multiple atea FWA cofiguratios for frames havig 08 or 89 code bits. Atea evelope correlatio coefficiet is set to 0.. 8

29 Figure 6: Error rates for turbo-coded OFDM systems for both sigle ad multiple atea FWA cofiguratios for frames havig 08 or 89 code bits. Atea evelope correlatio coefficiet is set to zero. 9

30 Tables Table : Computatioal requiremets of the log-map algorithm. ADD SUB MU DIV MAX KUP Procedure A M 6 M M Procedure B M M M M Procedure C M M M M Procedure D 6 M M ( M ) ( M ) Table : Computatioal requiremets of the ma-log-map algorithm. ADD SUB MU DIV MAX KUP Procedure A M 6 M M Procedure B M M Procedure C M M Procedure D M ( M ) Table : Computatioal requiremets of the Viterbi algorithm. ADD SUB MU DIV MAX KUP Procedure A M M Procedure E M M Procedure G Table : Compleity of the decodig algorithms. umber of Equivalet Additios og-map algorithm Ma-log-MAP algorithm 8 M 8 M Viterbi algorithm 0 M + 0