Ilesanmi Banjo Oluwafemi

Transcription

1 Iteratioal Joural of Scietific & Egieerig Research, Volume 6, Issue, October-5 33 ISSN Improved Space-Time Codig Scheme over Nakagami Fadig Chaels Ilesami Bao Oluwafemi Abstract I this paper, the performace of space-time codig scheme is ivestigated over Nakagami fadig chaels. I the first part, the performace of super-orthogoal space-time trellis code is preseted while the performace of its cocateated versio is preseted i the secod part. The cocateatio codig scheme with iterative decodig ivolves covolutioal code as the outer code ad super-orthogoal space-time trellis code () as the ier code. The pairwise error probability (PEP) for the codig schemes were derived ad their performaces evaluated by computer simulatio. Simulatio results shows that the diversity order of the codig schemes guarateed by quasi static fadig chael icreases by m times i the presece of Nakagami fadig with iverse fadig parameter m ad the codig gai varies with the geeral Nakagami fadig chael. Idex Terms : Chael, codig gai, diversity, Nakagami, pairwise error probability, space-time, super-orthogoal space-time trellis code which are selected by iput bits ca be represeted by e, where l=,,,m- ad θ which is the rotatio agle ca take o the values θ =l /M, where l =,,,M-. The first row correspods to the symbols trasmitted i time slot ad the secod row correspods to the symbol i time slot. The first colum correspods to the symbols trasmitted by atea, while the secod colum to the symbol by atea. INTRODUCTION s are desiged based o the rak ad determiat HE iformatio capacity gai of a wireless system ca criteria ad its trellis structure has a large umber of parallel Tbe icreased by employig multiple trasmit atea trasitios. I [], a ew was desiged for fast ad/or receive atea i a commuicatio system [- fadig Rayleigh chael while i [] the pair-wise error 5]. Space-time codig (STC), which combie i its desig, probability (PEP) was obtaied for. The geerator chael codig, modulatio, trasmit diversity ad / or matrix otatio for was recetly itroduced i [3] to receive diversity, has bee itroduced as a power ad allow for systematic ad exhaustive computer search for badwidth efficiet method of commuicatio over fadig optimal codes with higher umber of states. Rules that gover chaels [6]. the compoets of the geerator matrix were give i [4]. A class of ew STCs kow as super-orthogoal space-time trellis codes (), was itroduced i [7-8].These codes The ivetio of turbo code with its astoishig are cocateatios of a super set of orthogoal space-time performace has attracted the iterest of researchers ito the block codes with space-time trellis codes. combies subect of cocateated codig scheme i recet times. Turbo the set partitioig priciple i [9] ad a super set of codes which are built from parallel cocateatio of orthogoal space-time block code (STBC) i a systematic covolutioal codes with iterative decodig perform close to way, to provide full diversity ad improved codig gai the Shao limit i AWGN chaels [5]. Serially over the earlier space time trellis code (STTC) schemes []. cocateated covolutioal codes were ivestigated i [6] are full rates ad full diversity STCs that provides with turbo priciples while i [7] hybrid cocateated improved codig gai. The trasmissio matrices of covolutioal codes were proposed with a Soft-Iput Soft- for two trasmit atea is give as [4] Output () maximum a posterior decodig module. θ x e x To improve the codig gai of STC, various cocateated C( x =, x, θ ) θ, () topologies have bee proposed i literature with reported xe x improve performace over covetioal ST codes [8-5]. I where for M-PSK sigal costellatios, the sigals x ad x [3], serial cocateated space-time trellis code (STTC) was l proposed while i [7], double cocateated scheme was M proposed which cosist of a serial cocateatio of a parallel cocateated covolutioal code with STTC. Ilesami Bao Oluwafemi is a lecturer i the Departmet of Electrical ad Electroic Egieerig, Ekiti State Uiversity, Nigeria. ibto75@gmail.com 5 The Nakagami fadig model is a more versatile fadig model that is based o Nakagami distributio also called the m distributio [7, 8]. The Nakagami distributio iclude the Rayleigh distributio ad oe sided Gaussia distributio as two special cases, ad ca model fadig chael that are more or less severe tha that of the Rayleigh distributio. Sice Rayleigh fadig caot accout for large - scale effect of shadowig, the Nakagami-distributed fadig may be

2 Iteratioal Joural of Scietific & Egieerig Research, Volume 6, Issue, October ISSN ecoutered i practical situatios especially i mobile Ω wireless commuicatios [7]. m =, m /. (4) I [7], the performace of STTC i Nakagami fadig chael was ivestigated ad a upper boud was obtaied for the pairwise error probability. It was show that the diversity order of STTC guarateed by quasi static fadig chael icreases by m times i the presece of Nakagami fadig with iverse fadig parameter m ad that the codig gai varies with the geeral Nakagami fadig chael. I [9], the performace of STTC desiged based o the Euclidea distace criteria (EDC) was also ivestigated ad it was show that STTC desiged for Rayleigh fadig chael are suitable for Nakagami fadig chael. I [3], the performace of STBC over Nakagami-m fadig chael ad a closed form expressio for the exact symbol error rate for orthogoal space time block code over idepedet idetically distributed Nakagami-m fadig chael was obtaied. Also i [3], the PEP for space-time codes i Ricia- Nakagami chaels was obtai while i [3], the performace of MIMO systems through Nakagami Fadig chaels with arbitrary fadig parameter was coducted. I this paper, the performace of space-time codig scheme is preseted over Nakagimi fadig chael. I the first part, the performace of desiged 6, 3 ad 64-states is ivestigated over Nakagami fadig chael. I order to improve the performace of the codig scheme, two cocateated codig scheme with costituet code of ad covolutioal code is preseted i the secod part. The first cocateated scheme cosists of a serial cocateatio of a covolutioal code with a (CC-) while the secod ivolves parallel cocateatio of two serially cocateated covolutioal ad codes (HC- distributio with idetical m ad Ω ad Es is the eergy per symbol. The t symbol is the oise term modeled as idepedet samples of a zero mea complex Gaussia radom process with variace N/ per dimesio.. System Model : Fig. shows a codig scheme with T trasmit atea ad R receive atea. The sigal at each receive atea is a mixture of the faded sigal ad oise. The iformatio sigal is ecoded before trasmissio by ). The two schemes are from [33], but are ow the T trasmit atea. At the receiver, matched filterig is ivestigated over Nakagami-m fadig chael. Simulatios performed ad the received sigal is decoded usig Vertibi results are preseted for the case of two trasmit ad oe decodig algorithm. receive atea i quasi-static Nakagami-m fadig chaels. The rest of the paper is orgaized as follows. Sectio II describes the system model cosistig the chael model, the ecoder ad the decoder structure. I sectio III, the PEP of the codig schemes is preseted. The performace of the cocateated scheme is evaluated by computer simulatios i sectio IV, while coclusios are preseted i sectio V. SYSTEM MODEL. Chael Model We cosider a quasi-static Nakagami fadig chael with the fadig amplitude of α. The PDF of α is give by m m m α m / Ω [34] p( α ) = α e, () Γ( m) Ω where Г(x) deotes the Gamma fuctio of x, ad [ ] Ω = E α, (3) 5 E [( α Ω )] The otatio E[x] deotes the expected value of x ad m is the iverse fadig parameter. Whe m=, we have Rayleigh fadig ad whe m= we have o fadig chael. Cosider a codig scheme with T trasmit atea ad R receive atea. The sigal at each receive atea is a mixture of the faded sigal ad oise. I this paper, a quasistatic fadig chael is assumed i which the fadig is costat over a frame of legth L ad vary from frame to frame. Assumig that the trasmitted symbol from the i-th atea at time t is receive atea is r t, the r t T = h i= x x, ad the receive symbol at time t of the i t E + i i, t s t, (5) where h i, is the complex path gai from trasmit atea i to receive atea, whose evelopes obey the Nakagami m- R Iformatio source Matched Filter ecoder Fig..: system model decoder Pulse Shaper Iformatio sik T

3 Iteratioal Joural of Scietific & Egieerig Research, Volume 6, Issue, October ISSN CC- Ecodig: The ecoder block diagram of the CC- is show i Fig... I the system, a block of N idepedet data bits is ecoded by the covolutioal outer ecoder ad the output block of the coded bits is iterleaved by usig a radom bit iterleaver ( ). The iterleaved sequeces are the passed to the ecoder to geerate a stream of QPSK symbols which is trasmitted from the ateas. Source bit CC Ecoder Ecoder TX TX Fig..: Ecoder block diagram of the CC- system [34] CC- Decodig: The simplified block diagram of the CC- decoder is show i Fig. 3.. The subscript of the c or u specifies the decoder where st is used for the ecoder ad cc is used for the covolutioal ecoder. The coded itrisic LLR for the module is computed as λ( c st = σ, R T i i r ρ s + = i= (6) R T i, i r ρ s, σ = i= σ is the variace of i where s is the referece symbol ad the AWGN. The takes λ ( c st ad the a priori iformatio from the CC- which is iitially set to zero to compute the extrisic iformatio ˆ λ ( c st. This extrisic ( ) iformatio is de-iterleaved ad fed to the CC- to become its a priori iformatio λ (. The a priori iformatio is the used to compute the extrisic LLR ˆ λ ( for the covolutioal code (CC-). The c cc extrisic LLR is the iterleaved to become the a priori iformatio λ ( u st for the for the ext iteratio. Durig the first iteratio, λ ( u st is set to zero as o a priori iformatio is available at the -. The trasmitted source symbols are assumed to be equally likely ad therefore the iput LLR λ ( to the CC- is permaetly set to zero. The process is iterated several times ad o the fial iteratio, a decisio is take o the extrisic iformatio λ ( c cc to obtai the estimate of the origial trasmitted bit stream. c cc c cc Rx λ ( u st λ ( ccc, O) + λ( c, O ) cc + λ( ust, O) λ( c st λ ( u st λ C, I cc λ( u cc CC λ( u Fig. 3.: The block diagram of the CC- decoder [34] HC- Ecodig: I Fig. 4., the trasmittig block diagram of the HC- system is show. The HC- topology cosists of a parallel cocateatio of two serially cocateated schemes. Each of the serial cocateated schemes cosists of a outer covolutioal code cocateated via a iterleaver with a ier ecoder. I the system, a block of N idepedet bits is ecoded by the covolutioal outer ecoder (CC) of the upper serial part of the scheme. The output of the upper covolutioal ecoder is the passed through a radom bit iterleaver ( ). The permuted bits from the iterleaver are the fed to the upper ecoder to geerate a stream of complex data that are trasmitted from each of the trasmit ateas usig the trasmissio matrix. I the lower serial part of the ecodig, the lower covolutioal ecoder (CC) receives the permuted versio of the block of N idepedet bits ad geerates blocks of coded bits which are passed through aother iterleaver ( ) to the lower ecoder. The complex data from the output of the lower ecoder are trasmitted from the trasmit ateas. It should be oted that the same covolutioal ad codes are used i the upper ad lower ecodig of the systems. Each of the ecoders is termiated usig appropriate tail bits. All the four trasmit ateas are well separated by at least half of the wavelegth of the sigal. bits CC CC Ecoder Ecoder TX TX TX3 TX4 Fig. 4.: Ecoder block diagram of the HC- system [34] Decodig of HC-: The HC- decoder cosists of two serial arms ad oe parallel sector as show i Fig. 5.. The decoder is specified by the subscript of the c or u, where for the upper ecoder st is used, ad st is used for the lower ecoder, is used for the upper covolutioal ecoder CC while is used for the lower covolutioal ecoder CC. The coded itrisic LLR for the module is computed as i (5). cc u 5

4 Iteratioal Joural of Scietific & Egieerig Research, Volume 6, Issue, October ISSN The takes the itrisic LLR λ ( c, I st ) ad Serial part oe the a priori iformatio from the CC which is iitially set to zero ad computes the extrisic LLR ˆ( λ c st, O ). This extrisic LLR from the is passed through the iterleaver ( ) to obtai λ ( c. The LLR s output of the CC module which are λ ( c ad λ ( u are calculated. The LLR λ ( c is subtracted from λ ( c to obtai the LLR ˆ( λ c which is the set via iterleaver to obtai the itrisic λ( c, I + st ) λ( u, O ) iformatio λ ( c, I st st ) λc,i + λ( for the - for the ext u, O st ) CC From chael - iteratio. λu,o For the lower parallel arm, the takes the λu,i itrisic LLR λ ( c, I λ( u st - st ) ad the a priori iformatio from λ c, O the CC, which is also iitially set to zero, ad computes Serial part two + λ c,o the extrisic LLR ˆ( λ u st, O ). This extrisic LLR from the is passed through the iterleaver ( ) to Fig. 5.: Decodig block diagram of the HC- system obtai λ ( c. The LLRs λ ( c ad λ ( u from [34]. the output of the CC module are the calculated. The LLR λ ( c is subtracted from λ ( c to obtai the LLR ˆ( λ c 3 PAIRWISE ERROR PROBABILITY which is the passed through the iterleaver Let the trasmitted codeword ad the erroeously to obtai the itrisic iformatio λ ( c, I st ) for the decoded codeword be deoted by Ĉ ad C respectively. -. If we deote the symbol-wise Hammig distace betwee C For the parallel itercoectio compoet of the iterative ad Ĉ by d ( C, C ˆ ) ad assume maximum likelihood decodig process, the LLR ˆ( λ u, O (ML) decodig, the coditioal PEP that the receiver will ) obtaied by subtractig the LLR λ ( u from the LLR λ ( u is set via the de-iterleaver to obtai the LLR λ ( u which is the ucoded a priori iformatio from the CC ito the CC. Also the LLR ˆ( λ u obtaied by subtractig LLR λ ( u from the LLR λ ( u is set via the iterleaver to obtai the LLR λ ( u which is the ucoded a priori iformatio from the CC ito the CC. The process is iterated several times ad the bit with the maximum APP is chose by the decisio device i the last iteratio usig the summed values of the output ucoded LLRs of both the CCad CC decoders. λ(, I) c st From chael λ( u st + λ( u st - λ(, I) u st λ c, I λ c O, - + CC λ u,i Parallel part λc,o -,O λ u λ u, O - + λu,o Decisio λ u, O select Ĉ over C coditioed o the chael gais assumig perfect chael state iformatio (CSI) at the receiver is give by [6] T E R sd P( C Cˆ H ) = Q hi,, (7) i= = N where d = d ( c, cˆ) l= distace of the outer code. C( l) Cˆ( l) is the squared Euclidea x. By usig Q( x) = exp( ), we have ˆ E ( ) = exp sd C C H 4N T R P (8) The amplitude of i distributed ad the pdf of i= = h i, h, are idetically idepedet m- i, h is give by 5

5 Iteratioal Joural of Scietific & Egieerig Research, Volume 6, Issue, October ISSN m m From the performace curve, it is observed that the m m h ( ) i, Ω P( hi, ) = hi, e, (9) diversity order of the system i a Rayleigh fadig chael is a Γm Ω multiple of the fadig parameter m i a Nakagami fadig chael. Also, the codig gai achieved by the system is also where Ω = E[ h i, ] ad see to icrease with a multiple of m. The 6-state is observed to maitai its superior codig gai advatage over Ω the STTC couterpart over Nakagami-m fadig with o m =, m /. E[ ( hi, Ω )] diversity order advatage. If (8) is average with respect to the distributio of PEP ca be approximated at high SNR as P( C Cˆ) = fm d where f ( m) Esd fm 4N Es ( C, Cˆ) 4N ( m / Ω) m m = R T R T, R h i,, the () I Rayleigh fadig chael, m= ad Ω= ad () is simplified as P C Cˆ) = ( () Equatio () shows that the diversity order of codig scheme guarateed by slow fadig chael icrease by m times i the presece of geeral Nakagami fadig with iverse fadig parameter, ad the codig gai is multiplied / mr by a factor of f ( m). - - m =.5 m mrt T m =. Γ( m). 6 states STTC m=. -3 m= T R Fig. 6: performace for a 6-state ad a 6-state Esd N STTC [6] i a Nakagami fadig chael with oe receive 4 atea - 4 RESULTS AND DISCUSSIONS I this sectio, we evaluate the performace of the spacetime codig scheme ad its cocateated versio by computer simulatio uder a arrow bad frequecy flat Nakagami fadig chaels. Narrow bad trasmissio is assumed. Therefore, the results illustrate the performace i time divisio multiple access (TDMA) type systems, like the global system for mobile commuicatio (GSM), IS 36, or ehace data rates o GSM Evolutio (EDGE). For all the simulatios, 3 symbols per frame are trasmitted from each trasmit atea ad CSI is assumed at the receiver. It is assumed that the fadig chael is quasi-static i.e. the fadig chael coefficiet is costat over oe frame but varies from oe frame to aother. Figs. 6, 7 ad 8 preset performace curves for 6-, 3- ad 64-state QPSK for oe receive atea over Nakagami fadig chaels with fadig parameters m =.5,, ad, respectively. I Fig. 6, the simulatio result for 6-state STTC from [6] for m = usig receive atea is also show m=.5 m=. -4 m= Fig. 7 performace for a 3-state i a Nakagami fadig chael with oe receive atea 5

6 Iteratioal Joural of Scietific & Egieerig Research, Volume 6, Issue, October ISSN m=.5 m=. -4 m= m =.5 m =. m = Fig. 8: performace for a 64-state i a Nakagami fadig chael with oe receive atea For the cocateated part, two trasmit ateas ad a sigle receive atea are used for each part of the coectio. For the outer code, the RSC rate-/, 4-state covolutioal codes are employed for the CC- ad the HC-. The 6 states preseted i [3] are used as the ier code. Fig. 9 shows the performace of the CC- over Nakagami fadig chaels with m =.5,, ad, respectively, where the costat m deote the fadig parameter with m = ad m = correspodig to the Rayleigh ad o fadig chael, respectively. As ca be observed from the plot, the diversity order of the cocateated grows liearly with the fadig parameter m, which agrees with the observatio from the pairwise error probability aalysis. The codig gai is also observed to icrease with a icrease i m value. The CC- topology achieves a diversity order of for m =.5, diversity order of for m = ad diversity order of 4 for m =. Fig. shows the performace of the HC- over Nakagami fadig chaels with m =.5,, ad, respectively. As ca also be observed from the figure, the diversity order of the cocateated icreases liearly with the fadig parameter m, which is cosistece with the pairwise error probability aalysis. The topology achieves a diversity order of for m =.5, diversity order of 4 for m = ad diversity order of 8 for m =. The codig gai is also observed to icrease with a icrease i m value. 5 Fig. 9: performace of CC- over Nakagami fadig chaels m =.5 m =. m = Fig. : performace of the HC- over Nakagami fadig chaels 5 CONCLUSION I this paper, the performace of space-time time codig scheme is evaluated over Nakagami-m fadig chael. The first cosisted the performace of 6, 3, ad 64-state while the secod part cosisted its cocateated versio. The CC- system cosisted of a serial cocateated covolutioal code ad a ier, while the HC- system cosisted of parallel cocateatio of two serially cocateated covolutioal ad codes. The ecodig ad the iterative decodig of the two topologies were discussed. Simulatio results are preseted for the case of quasi-static Nakagami fadig chael ivolvig 3

7 Iteratioal Joural of Scietific & Egieerig Research, Volume 6, Issue, October ISSN symbols per frame legth from each trasmit atea. The error correctig codig: Turbo-codes, i proc. IEEE Iter. Cof. PEP for the codig schemes was preseted ad it was show Commu. (ICC). pp. 64-7, 993. that the diversity order of the codig scheme is a multiple of [6] S. Beedetto ad G. Motorsi. Serial cocateatio of iterleaved the fadig parameter m. Results shows that the diversity codes: performace aalysis, desig ad iterative decodig,. IEEE order of the schemes guarateed by quasi static fadig Tras. If. Theory. pp , 998. chael icreases by m times i the presece of Nakagami [7] S. Beedetto, D. Divsalar G.Motorsi ad F. Pollara, A Soft-Iput fadig with iverse fadig parameter m ad the codig gai Soft-Output APP Module for iterative decodig of cocateated varies with the geeral Nakagami fadig chael. codes,. IEEE Commu. Letter. vol., o., pp. -4. Ja 997. [8] X. 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