Performance Comparison of LDPC Codes using SRK Equalisation and. OFDM Techniques for Broadband Fixed Wireless Access Systems

Transcription

1 erformace Compariso of LDC Codes usig SRK Equalisatio ad OFDM Techiques for Broadbad Fied Wireless ccess Systems M K Kha, R.. Carrasco,.J.Wassell, J..easham. Commuicatios ad Sigal rocessig Group, School of EE&C Egieerig, Uiversity of ecastle-upo-tye {m.k.kha, r.carrasco, j.a.easham}@ ecastle.ac.uk. Digital Techology Group, Computer Laboratory, Uiversity of Cambridge [email protected] bstract - Broadbad Fied Wireless ccess (BFW) systems eable services such as high-speed data commuicatio, high quality voice/video coferecig ad high-speed teret access i areas here a ired lik is ot possible. Hoever, the BFW chael is a slo fadig chael havig deep frequecy selective fadig caused by clusters of scatterers i the eviromet that itroduce itersymbol iterferece (S) at the receiver. this paper, Lo Desity arity Check (LDC) codes, optimised for the SSO BFW chael, are desiged usig the structured Balace complete Block Desig (BBD) method. The use of both SK ad 6-M modulatio are ivestigated theoretically. To help overcome the S effects of the chael, equalisatio techiques are employed separately ith LDC decodig for a system employig SK ad 6-M modulatio schemes. The equaliser sigle carrier approach is the replaced ith orthogoal frequecy divisio multipleig (OFDM) ad the performace of these to approaches is evaluated i terms of bit-error rate. The simulatio results sho that equalisatio ith LDC codig has a measurable performace gai over LDC codig ith OFDM. de Terms - LDC codes, BFW chael, Equalisatio, BBD Desig, OFDM, WiM or EEE 8.6 This ork has bee fuded by the ESRC, Grat o. GS/ S6 /.

2 . troductio Broadbad fied ireless access (BFW) [] systems offer a solutio to the problem of providig iepesive broadbad services from the local echage to the customer. t ca be deployed quickly, coverig a large area ith lo cost compared to cable istallatios. BFW system ca provide traditioal voice, data, voice over iteret protocol VO, remote educatio, video coferece, highspeed iteret access etc. chievig high data rates i ireless commuicatio eviromets is limited by multipath fadig betee trasmitter ad receiver degradig the system performace. The trasmitted data through the broadbad fied ireless chael is subject to atteuatio ad distortio by various factors such as foliage, buildigs, precipitatio, ad vehicles etc. Therefore, there is a eed to use a model hich is parameterised accordig to the various terrai ad eviromets eperieced by BFW systems. The Staford Uiversity terim (SU) chael model [] has bee developed specifically for fied ireless systems ad takes ito accout a umber of parameters for eample, Doppler effect, path loss, multipath delay spread ad fadig characteristics. Cosequetly it has bee chose to model the BFW chael i this paper. The EEE orkig group o Broadbad Wireless ccess Stadards are developig EEE 8.6 [3], hich provides the stadards for broadbad ireless system implemetatio. s part of the specificatio the use of OFDM has bee proposed i order to combat frequecy selective fadig i the BFW chael. lso, the use of space diversity for capacity improvemet usig OFDM has bee aalysed i []. This paper also discusses the physical layer research challeges i MMO-OFDM BFW systems, icludig physical chael measuremets ad modellig, aalogue beam formig techiques usig adaptive atea arrays, space-time techiques for MMO-OFDM, error cotrol codig techiques etc. [5], a broadbad multiple-iput multiple-output (MMO) OFDM system has bee desiged ad evaluated for a fied ireless lik betee tall buildigs i a urba area.

3 Moreover, there has bee a lot of emphasis o joit equalisatio ad decodig of coded systems. The applicatio of the turbo priciple [6] to iterative decodig ad demodulatio has provided sigificat gais [7]. Such systems are essetially serial cocateatio schemes [8] ith the S chael as the ier code. The aim of the research described i this paper is to desig ad aalyse the performace of to sigle iput sigle output (SSO) BFW systems. The first system uses a sigle carrier approach ith a Square Root Kalma (SRK) equaliser i cojuctio ith some ely geerated LDC codes usig Balaced complete Block Desig (BBD) [9]. For a BFW system, it is importat to desig LDC codes ith very high code rates to miimise the redudat iformatio that is itroduced ad hece maitai a high data rate. t the same time, the code must also have a very good error-correctig capability hich implies that it must have a large code legth. The BBD costructio method produces LDC codes ith large legths ad high code rates greater tha.9 that outperform radomly costructed LDC codes ith similar parameters. secod advatage of usig this scheme is its reduced ecodig compleity, hich is importat i reducig the overall delay of the BFW system. The secod scheme replaces the sigle carrier approach ith oe employig OFDM i.e., a SSO OFDM BFW system. The scheme ca easily be eteded to MMO BFW systems. The mai aim of this paper to aalyse the performace of high rate, lo compleity LDC codes ith varyig legths ith iterferece reductio techiques, such as equalizatio ad OFDM, to improve the performace of the system. The high rate LDC codes have bee costructed by the authors usig the BBD costructio method: the (75, 535) LDC code of rate.93 ad the (35, 3) LDC code of rate.98. The theoretical aalysis of the SSO BFW system usig o-coheret [6] detectio for SK/ 6-M is also preseted ith simulatio results. For coheret detectio readers ca refer to [] hich preset the theoretical aalysis of SSO/MMO BFW systems. This paper is orgaised as follos: Sectio presets the overall descriptio of the fied WiM (EEE 8.6d) or BFW system. The theoretical aalysis of the BFW chael by employig o- 3

4 coheret detectio usig SK ad 6-M is preseted i sectio 3. Sectio, e briefly discuss the costructio method, ko as BBD, of LDC codes used for simulatio i this paper. The, a overvie of the SRK equalisatio techique is described i Sectio 5 alog ith the descriptio of the sum-product decodig algorithm of LDC codes. Sectio 6 compares the simulatio results for the equalised system ad u-equalised system. t also discusses the performace compariso of various LDC codes usig SK ad M modulatio schemes. Fially, sectio 7 summarises the mai cotributios of this paper.. System Model The overall geeral system diagrams are sho i Figure. Figure (a), sigle carrier trasmissio ith SRK equalisatio has bee used ad i Figure (b), OFDM has bee used ith the aim of overcomig the S itroduced by the BFW chael. the trasmitter the data bits b k, geerated by the source, are ecoded by the LDC ecoder ito ecoded bits c. The block iterleaver re-orders the ecoded bits ad the modulator maps them ito SK or M symbols s. Figure (a) this sigal (represeted by ) is modulated oto a sigle carrier for use i the equalised system, hereas i Figure (b) the sigal is fed ito a OFDM modulator. the iitial stage of OFDM modulatio, the symbol stream from the mapper is coverted from serial to parallel ad the a verse Fast Fourier Trasform FFT is applied. Fially a cyclic prefi (C) is added to the sigal before trasmittig over the SU-3 multipath chael ith three resolvable paths ith tap spacigs of 5s ad a total delay of s. The model ca be cosidered as a 3-tap trasversal filter ith a fiite impulse respose. The Doppler Effect specified i our chael specificatio is also take ito accout he calculatig the taps coefficiet values. The trasmitted sigal is also corrupted by additive hite Gaussia oise (WG). The total poer of all paths of the chael is ormalized to uity, so that there is o gai provided by the dispersive chael. The chael filter coefficiets are calculated ad the trasmitted sequece from the ateae is multiplied by these coefficiets.

5 There are a umber of SU chael models specified i [] depedig upo the differet terrai coditios. ll of them have three resolvable paths ith either Ricia or Rayleigh amplitude distributios. The amplitude distributio of the SU-3 chael model is Ricia for the first path ad Rayleigh for the other to paths. Hece the received sigal r ca be epressed as r z () here z is the comple additive hite Gaussia oise (WG) ad,, are the comple value coefficiets. The receiver aims to recover the origial iformatio bits from the received samples corrupted by the chael. For the equalised system, the received samples, r, are passed directly to the SRK equaliser to attempt to reduce S. fter equalisatio/demodulatio of the received samples, the log-likelihood (LLR) values, dˆ, of each bit are calculated ad passed to the block de-iterleaver. The decoder applies the message passig algorithm o the de-iterleaved bits, ĉ, i a iterative maer to etract estimates of the origial iformatio bits, bˆ k. For the OFDM system the received samples r are passed directly to the OFDM demodulator. fter OFDM demodulatio the received symbols are passed to the soft demapper here the LLR values of the received symbol are calculated before beig passed to the de-iterleaver ad LDC decoder. 3. Theoretical alysis of BFW Chael [6], iao et al preseted the theoretical aalysis of a BFW system usig coheret ad ocoheret detectio methods for both SK ad M modulatio scheme. The theoretical approach is ecessary to aalyze the effect of S o the performace of the BFW systems, ith a attempt to gai a deep isight ito the physical limitatios of the BFW chaels ith covetioal detectio 5

6 techiques. There are si differet chaels for differet terrai types defied i SU-3 chael specificatio ad e have chose SU-3 chael ith three taps havig tap delay of 5 s betee the adjacet taps. The aalysis assumes that the legth of oe SK/M symbol is equal to the spacig betee the adjacet taps hich gives the data rate of Mbps ith SK modulatio ad 8Mbps ith 6-M. The chael coefficiet is assumed to be costat durig the trasmissio of oe block of data; hoever, they vary from block to block. The trasmitted SK/M symbol at time istat is deoted as variace. j, ad z is the comple additive hite Gaussia oise ith zero mea ad The results i [6] shos there is a closed match betee the simulated ad theoretical results. For both SK ad 6-M modulatio the o-coheret detectio is performed o the origial received sigal ithout correctig the phase shift. The simulatio results of oly o-coheret detectio are preseted ad the authors have also metioed the aalysis of coheret detectio for SK/ 6-M i [6] ad []. The et sub-sectio briefly describes the theoretical approach for o-coheret detectio of a M-SK sigal over the BFW chael. 3. erformace of o-coheret Detectio o the BFW Chael The origial sigal r received from the chael is directly passed to the SK demodulator ithout correctig the phase shift. The received sigal from the chael is give by Eq. () as r o z 3 () Combied S ad oise here z ~ (, ) is combied S ad Gaussia oise ad it is assumed that has a comple Gaussia distributio for a large umber of chael istatiatios. [ ] E[ ] E respectively ad is oise poer., here ad are the poers of the secod ad third taps 6

7 7 3.. o-coheret Detectio for SK Modulatio The probability of a symbol error occurrig, e.g. ˆ give is trasmitted ca be computed as [6] ( ) ( ) ˆ R (3) here R is the decisio regio for symbol i SK costellatio, substitutig j ad j i equatio (3), ( ) R j j > < ( ) ( ) () Similarly, other coditioal probabilities ca be obtaied as ( ) ˆ ( ) ( ) (5) ( ) 3 ˆ ( ) ( ) (6) Let ad here the probability desity fuctio DF of ad is give as ( ) ( ) p, ep σ πσ (7) ( ) ( ) m p, ep σ πσ (8) The sigal costellatio of SK is sho i Figure (a) ad the error evet ( ) ˆ results i bit errors hile ( ) ˆ ad ( ) ˆ result i bit error. The bit error probability is therefore give as ( ) ( ) ( ) [ ] 3 ˆ ˆ ˆ k b (9)

8 8 here k is the umber of bits per symbol for SK (k ). Therefore, Eq. (9) becomes ( ) b, ( ) b, () The average bit error probability b is derived by takig the epectatio of ( ) b, epressed i Eq. (9) ith respect to ad. Sice is a zero mea ormal radom variable the ( ) [ ] 5. () E λ. Therefore epectatio of Eq. () ith respect to is equivalet to Eq. () belo ad the proof is give i ppedi ( ) b 3 () f z is a zero-mea, uit-variace, ormal radom variable, the accordig to [8] ( ) [ ] E λ µ λ µ z () ssig ( ) σ / m Z such that Z is a zero-mea, uit-variace, ormal radom variable. By takig the epectatio of Eq. (), e derive the average bit error probability as b m Z m Z E σ σ 3 3 λ µ λ µ b (3) here σ λ, m µ, σ λ, m µ 3.. o-coheret Detectio for 6-M The costellatio of 6-M shoig differet types of decisio regios is sho i Figure (b). Let us cosider is a type a symbol the the received sigal r ca be ritte as

9 9 ( )( ) ( ) ( ) j j j j r () this case, the probability of error evet alog the ais is differet from error evet alog the ais. They are deoted by,, respectively. The probability of a -error evet is deoted as.these probabilities are computed as ( ) { } ( ) { } r r < < ( ) < r ( ) (5) ( ) { } ( ) { } r r < < ( ) < r ( ) (6) ( ) { } ( ) { } < < r r ( ) ( ) (7) The coditioal bit error probabilities a, b, c of decisio regio type a, b ad c respectively is give as [ ] a a a a

10 here a a b c [ ] 8 (8) [ ] b b b [ ] (9) [ ] c c c [ ] (), represets the umber of regios that differ by bit from trasmitted type a ad alog the / ais, ad a is the umber of diagoal regios that differ by bits from a type a symbol. lso, b b,, b,, c, c c are defied similarly for type b ad type c symbols. The bit error probability is thus ber ( ) 6 a a b b c c () here,, is the umber of a, b, c type regios i the costellatio. For the 6-M case e ca a b c see that, 8,. a b c ber ( a 8b c ) () 6 By substitutig values from Eq. (8) () ito the above equatio, e get Deote ( 5 7 ) (3) 6 ber ( ) ( ) 7 6 ( ) ( ) ad (8). The bit error probability ca be epressed as () ith probability desity fuctio DF give by Eq. (7) ad Eq.

11 b 5 7 8, (5) ( ) The epectatio of ( y) b, gives the average bit error probability b ad applyig the same procedure as ith SK modulatio i the previous sub-sectio e have b ( ) 5 3 (6) 5 µ b E (7) [ ( ) ] b 3 λ here λ σ, µ m The simulatio results i Figure sho the theoretical ad simulated results of o-coheret detectio usig SK ad 6-M. s you ca see the simulated BER curve is virtually flat shoig the harsh ature of the dispersive chael ad the theoretical BER curve agrees closely ith it.. Costructio of LDC Codes usig BBD LDC codes are broadly classified by their method of costructio: Radom ad Structured. Radom codes are costructed usig a radom computer search based o certai desig rules or graph structures, such as girth ad degree distributio hereas structured codes are based o algebraic or combiatoric methods. The method preseted here is based o the costructio of structured LDC codes ko as the balace icomplete block (BBD) desig. These LDC codes have a girth of at least 6 ad are cyclic, hich cosequetly simplifies the ecodig procedure compared ith o-cyclic or quasi-cyclic LDC codes. s sho i [] [9] the codes costructed usig the structured method perform as closely or eve better tha radomly costructed codes e.g. MacKay codes. For eample, a LDC (75, 535) code The legth of the cycle is the umber of edges it cotais, ad the girth of a graph is the size of the smallest cycle.

12 used i this paper outperforms the equivalet Mackay code at a BER of - []. other e LDC- BBD code of legth 35 ad rate.97 has bee geerated ad the BER has bee simulated over the BFW chael. other mai advatage of usig BBD codes over radomly costructed LDC codes is their loer ecodig compleity. Whereas, quasi-cyclic LDC codes like those geerated by the BBD desig have a ecodig advatage over radom LDC codes as they ca be ecoded usig simple shift registers ith compleity liearly proportioal to their code legth. For eample i [], Li et al shos that i order to implemet the geerator matri of dimesio 5 5 usig quasi-cyclic structure requires flip-flops, 5 to-iput D gates ad 5 to-iput OR gates, hereas the costructio of the same size radom geerator matri requires about 6 to-iput D gates ad 663 to-iput OR gates... Combiatorial Desig Combiatorial mathematics [] basically deals ith the theory of eumeratio, permutatio, combiatio ad the arragemet of objects i order to satisfy certai coditios. formally, oe may defie a combiatorial desig to be a ay of selectig subsets from a fiite set i such a ay that some specified coditios are satisfied. The detail of the costructio of LDC codes are ell documeted i [7], [8] ad [9]. The detailed study of combiatorial mathematics shos that the parity check matri H ca be costructed usig BBD havig covalecy λ. The covalecy coditio λ esures the code is free of cycles of legth ad results i a girth of at least 6. The eample i ppedi B describes the relatioship betee LDC codes ad BBD desig. Over the years may BBD desigs have bee costructed ad they are represeted by mathematical equatios so that variable legth desigs ith differet parameters ca be costructed ith ease. this paper, e have selected the Bose BBD desig method [], [] to geerate high rate LDC codes.

13 . Bose BBD Desig Eample Bose costructed may classes of BBD desig. The folloig eample shos oe of the types of Bose-BBD desig. Let t be a positive iteger that satisfies t p, here p is a prime umber. There eists a Galois Field GF (t) ith t elemets. Suppose GF (t) has a primitive root,, such that t c, here c is a odd iteger less tha t. The there eists a BBD desig ith the folloig parameters: o of ros, v t, Objects o of colums, b t (t) tv, Blocks Colum eight, γ, Ro eight, ρ γt Covalecy, λ The desig block is geerated usig the buildig blocks { i i B t i 8,,, t } i for i < t. From B i e ca form t blocks by addig each elemet i the Galois Field GF (t). The icidet matri of this BBD is a (t ) t (t) matri. t ca be ritte i cyclic form cosistig of t, (t ) (t ) circulat sub matrices as follos: [,,., t ] (8) The icidet matri of Bose-BBD cosists of a ro of t circulats. For z t the parity check matri is H [z] [,,.. z ] (9) ppedi C shos the icidet matri geeratio usig the above method for t. The advatage of BBD scheme is that e ca geerate very high rate LDC codes, hich ca be used to icrease the badidth efficiecy of the overall system. umber of e codes have bee geerated ad the simulatio sectio icludes results usig the ely desiged (35, 3) LDC code of rate.98. 3

14 5. Equalisatio ad LDC Decodig Time domai equalisers are usually of the trasversal filter type (i.e. a tapped delay lie filter) ith coefficiet values adjusted to miimise some error criterio. automatic equalisatio process requires a iitial traiig period durig hich the equaliser reduces the error. this paper e have used the square root Kalma (SRK) algorithm to adaptively determie the equaliser coefficiets, hich is a ehaced versio of the recursive least square (RLS) algorithm. 5.. Recursive least-square (RLS) algorithm The Least Mea Square (LMS) algorithm is the simplest ad most uiversally applicable adaptive algorithm [3]. This algorithm is importat because of its simplicity, ease of computatio, ad because it does ot require off-lie gradiet estimatios or repetitio of data. Hoever, the price paid for its simplicity is slo covergece, especially he the chael characteristics result i a autocorrelatio λ matri (Γ ) hose eigevalues have a large spread, i.e. ma >>, here λ λ ma ad λ mi are maimum ad miimum eigevalues respectively. derivig faster covergig algorithms, a least squares (LS) approach is more appropriate. importat feature of the LS algorithm [] is the utilisatio of iformatio cotaied i the iput data, etedig back to the istat of time he the algorithm as iitiated. This results i faster covergece but at the cost of a icrease i computatioal compleity. The trasmitted sigal corrupted by Gaussia oise ad S, r, arrives at the receiver. The receiver ill the apply equalisatio ad decodig schemes to etract the trasmitted bits, b k,. The adaptive equaliser chose for this applicatio has 5 feed-forard ad 3 feedback filter taps as sho i Figure 3. mi The vector cotaiig the comple coefficiets of` the filters is defied as c [c, T c c c c b b b ] 3 T ad the iput sigal vector is defied as r [r r r r r d d d here T represets 3 3 ]

15 matri traspositio, d represets the output of the decisio device ad represets the discrete time istat. The the output y of the equalizer is give as The error sigal e ca be formed as follos: y c r (3) Durig the traiig period, e d y (3) d is the traiig sequece ad hile receivig data, it is the output of the decisio device (decisio directed mode). The aim of the adaptive equalizer is to adjust the filter coefficiets to move aay from the iitial coditio mea squared error (CMSE) toards the miimum mea squared error (MMSE). The RLS algorithm requires that the iitial values of the iverse correlatio matri Λ esure the osigularity of the correlatio matri ψ []. The coefficiets are cotrolled by the Kalma gai vector resultig i rapid covergece compared to that of the LMS algorithm. But this RLS algorithm has the disadvatage of umerical istability. order to tackle this problem, the RLS algorithm is modified to give the Square Root Kalma (SRK) or Square Root RLS algorithm. 5.. Square Root Kalma (SRK) lgorithm The problem ith the RLS algorithm is that the update formula does ot guaratee Λ is a positive defiite matri [5]. The SRK algorithm overcomes the problem by performig a U-D factorizatio of the matri such that Λ U D U, (3) * T 5

16 here U is a upper triagular matri ad D is a diagoal matri. The time updatig is carried out o the matrices U ad D. The procedure of fidig the recursive update of Λ is defied i [5]. This paper also describes the computatioal procedures to carry out the square root formulatio LDC Decodig For our simulatio the equaliser is traied iitially by usig traiig symbols ad the goes ito decisio directed mode. The equalized output is the subsequetly demodulated/de-iterleaved ad the log-likelihood ratios are passed to the LDC decoder to etract the iformatio bits. Let i, ad represet the to bits of a SK symbol. The LLR for the first bit of the symbol i, ca be ritte as l (, ) ( ) Similarly, for secod bit of the symbol l, (, ) ( ), l l j,3 j, j, j,3 ( ) s j c ( s j c) ( ) s j c ( s j c ) (33) (3) here c is the output from the de-iterleaver ad s j is the j th SK symbol. This procedure ca be eteded to higher modulatio schemes such as M. These likelihood ratios are the echaged betee the check ad variable odes of the LDC code. The coectios betee these odes are formed accordig to the positio of s i the parity check matri H. The umber of iteral iteratios as set to 7 i this case. The echage of iformatio betee variable ad check odes is eplaied as follos: The variable ode of the decoder facig toards the chael receives the reliability iformatio i the form of Log-likelihood ratios (LLR). The variable ode edges of the taer graph coected to the check odes are iitialized to these reliability values. So, these messages are passed to the 6

17 correspodig check odes for the etrisic log-likelihood calculatios. The calculatios at oe particular check ode are based o the product of the iformatio from the other variable odes ecludig the oe that particular check ode is directly coected to. These etrisic log likelihood values calculated at the check odes are o beig passed back to the variable odes. So these e messages i the form of LLRs at variable odes are summed together ith the origial LLR (from the chael) ad the the parity check is applied to check that it satisfies the sydrome equatio. f the parity check fails the the e LLRs (or messages) at the variable odes are geerated for the et iteratio ad passed o to the correspodig check odes for etrisic iformatio. t a particular variable ode, the calculatios of e the updated LLR to be passed to the correspodig check ode takes ito accout the messages (LLRs) from all the other check odes ot from the oe it is directly coected to. This echage of iformatio betee variable ad check ode updates the origial LLRs ad at every iteratio a parity check is applied to cofirm the validity of the LLR update. This hole iterative process cotiues util the parity check satisfies the sydrome equatio c T. H or the maimum iteratio is reached. Hece, i this algorithm at the check ode the etrisic iformatio is based o the product of LLRs passed from variable odes ad at the variable ode the e iformatio is based o the summatio of the updated etrisic iformatio from the check ode ad the origial LLR. 6. alysis of Simulatio Results The simulatio results preseted here use LDC codes costructed usig the BBD desig method. To differet codes ere compared; the (75, 535) LDC-BBD code (rate.933) ad a ely desiged (35, 3) LDC-BBD ith rate of.98. The LDC codes are free of cycle four ad the LDC decoder utilizes the message passig algorithm to decode the iformatio bits. The maimum umber of decoder iteratios i all simulatios is set to, sice eperimets shoed that egligible 7

18 additioal codig gai as achieved he iteratios greater tha ere performed. The simulatio results sho i Figure 5 ad 6 shos the performace of (75, 535) LDC-BBD ad (35, 3) LDC-BBD over the WG ad Rayleigh fadig chael respectively. The figure shos the stregth of these high rate codes compared to ucoded SK BER curve over WG ad Rayleigh fadig chael. These codes are tested usig SK ad 6-M modulatio schemes (gray coded) over a chael simulated by the SU-3 BFW model. s metioed earlier, the SU-3 model cosists of three taps ith the first oe Ricia distributed ad the others havig Rayleigh distributios. This model icludes tap delays of µs,.5 µs ad. µs, ith relative poers db, -5 db ad - db, ad ith K-factor, ad, respectively. The Doppler spread is. Hz ad the delay spread is.6 µs. Figure 7 compares the performace of the (75, 535) LDC-BBD code usig equalisatio ad OFDM trasmissio scheme havig 56 sub-carrier ith SK modulatio. The most immediate observatio is that ithout equalisatio, the S domiates the performace ad the LDC codig aloe provides egligible gai over ucoded SK, i cotrast to its very promisig performace over the WG chael. small improvemet i performace over the ucoded system ca be achieved usig either equalisatio or OFDM, ith SRK equalisatio slightly outperformig OFDM, but both schemes suffer from a error floor. By icludig the LDC code the error floor is broke ad sigificat codig gais are achieved over the ucoded equalised ad OFDM systems. The results sho that the LDC code ith SRK equalisatio outperforms the scheme ith the LDC code ad OFDM o the BFW chael, particularly at loer sigal-to-oise ratios, ith a codig gai of aroud db at a BER -3. Figure 8 shos the simulatio results of the (35, 3) LDC-BBD code usig equalisatio ad OFDM ith SK over the BFW chael. Oce agai, this loger code still does ot perform ell 8

19 over the BFW chael o its o ad he used ith SRK equalisatio has approimately db codig gai at a BER -3 over the OFDM system. Figure 9 ad Figure sho the performace compariso of the (75, 535) ad (35, 3) LDC codes respectively usig equalisatio ad OFDM ith 6-M over the BFW chael. The receiver performace usig LDC codig o its o is agai plotted for referece. With OFDM trasmissio e fid the BER performace is agai vastly improved. Hoever, if e compare the equalised system ith the OFDM system for the same code legth, the SRK equalisatio scheme performs slightly better the OFDM scheme ith a improvemet of approimately db at BER Coclusio this paper, high rate variable legth LDC codes have bee costructed usig the BBD method ad their performace over the BFW chael has bee evaluated ith SRK equalisatio ad OFDM modulatio separately to overcome the S itroduced by the chael. The BFW chael is very harsh ad ithout either of these techiques the LDC codes perform poorly. The theoretical aalysis of BFW chael ith a o-coheret detectio techique matches closely ith the results obtaied by simulatios usig SK/6-M. The simulatio results sho that sigificat codig gais ca be achieved usig LDC codes ith SRK equalisatio or OFDM over ucoded systems. Hoever, LDC codes ith SRK equalisatio do outperform LDC codes ith OFDM over the BFW chael, particularly at loer sigal-to-oise ratios. The performace is orse he usig OFDM sice it reduces hat little frequecy diversity is available i the BFW chael resultig i the LDC code beig uable to effectively recover this lost diversity. Cosequetly, for a sigle carrier ith SRK equalisatio e ca still take advatage of more of the available chael frequecy diversity. lso, the use of high rate LDC codes ad high costellatio M ill miimise badidth use ad improve 9

20 the overall data rate of the system. Future ork ill be to ivestigate joit equalisatio ad LDC decodig to improve the performace over the Multi-put-Multi-Output (MMO) BFW chael. ckoledgmet The authors ish to thak the ESRC for their fiacial support of this research. ppedi ccordig to equatio () i sectio 3.., e have b ), ( here ad are defied as { } { } Re m o ad { } { } Re m o. fact, the bit error probability is ot oly a fuctio of ad y, but also a fuctio of ad, i.e., ),,, ( b (35) For a specific data block, the chael coefficiets,, are fied ad the bit error rate of this specific data block ca be computed by (35) provided that the chael coefficiets are ko. Hoever, the aim of this aalysis is to derive the bit error probability averaged over may data blocks, here the chael coefficiets vary from block to block. The average b, deoted as b is derived by averagig ),,, ( b over the distributio of,,, y, i.e, ( ) ) ( ) ( ) ( ) (,,, d d d d p p p p b b (36)

21 here, p ( ), p( ), p( ), p( ) are the DF fuctios of,,, respectively. the derivatio of the average bit error rate,,, should be vieed as radom variables sice they differ from oe block to aother., The amplitude of, i.e., is characterized by a Ricea distributio [3] due to lie-of-sight propagatio (the real ad imagiary parts of, are o-zero mea Gaussia radom variables), ad are still Gaussia distributed ith DFs defied by (7) ad (8). lso, ad are characterized by a Rayleigh distributio [3] due to o-lie-of-sight propagatio (both real ad imagiary parts of ad are zero mea Gaussia radom variables). Therefore, each of the radom variables ad has a cetral chi-square distributio ith degrees of freedom ad DF [] p( ) ep, γ γ p( ) ep, γ γ here, γ [ ] [ ] E E, ad γ [ ] [ ] E E ito (36), e obtai, (37). Substitutig (7), (8) ad (37) ( m ) (,,, ) ep ep b b πσ γ γ σ σ ep ep d d d d γ γ The eact average bit error probability for the SK modulated BFW system should be calculated ith Eq. (38). Hoever, the derivatio of its closed form epressio does ot seem to be tractable, ad umerical evaluatio of this four-fold itegral ould be too tedious. hat follos, e obtai a closed form formula for average bit error probability by usig some approimatio methods. Oe (38)

22 simplificatio measure is to remove the depedece of, i (35) by takig the epectatio ith respect to radom variables, E [,,, )] (, ) b ( b (39) The approimatio i (39) is due to the fact that istead of carryig out itegratio to evaluate S epectatio, e simply replace the oise plus iterferece term oise ith its E S mea value [ ] oise. This is so-called stadard Gaussia approimatio (G) i the literature. eample is sho i [3] (see equatios (5), (6)). The eact aalysis is carried out by equatio () i [3]. By compariso, oe ca see that the radom variable ψ (i.e., the K M variace) i [3, Eq ()] is replaced by its mea value E[ ] E[ Z ] ( K ) k k 3 ψ i [3, Eq (5)] ad [3, Eq (6)] so that the itegratio epressed by [3, Eq ()] ca be avoided. t has bee sho i [3], [] that this approimatio method gives close results to that derived by the eact aalysis usig itegratios at high bit error rate. Therefore, it is ell-suited to the performace aalysis of the o-coheret detectio scheme for hich the performace is usually poor (bit error rate is high). t as also cocluded i [3], [] that this approimatio method geerally gives optimistic results. This cocurs ith our results sho i Fig., here e ca observe that theoretical curves lie belo the simulatio curves for both SK ad 6-M systems. more accurate ad still simple approimatio method to evaluate epectatio ithout carryig out itegratio as proposed by Holtzma i [5]. t as sho that the accuracy of aalysis ca be improved if ot oly the first order statistic (mea value), but also the secod order statistic (variace) of the radom variable is take ito accout. our case, it meas e ot oly use E[ ] ad E[ ],

23 but also var[ ] ad var[ ] i the calculatio of (, y) b. Holtzma approimatio has bee adopted i several papers, e.g., i [6] for aalysis of CDM systems ith parallel iterferece cacellatio scheme ad i [7] for aalysis of the BFW system ith coheret detectio. Hoever, this approach is ot cosidered here because the target system to be aalyzed is the BFW system ith o-coheret detectio, for hich the bit error rate is high, ad i such a sceario, stadard G gives sufficiet accuracy as idicated i [3], []. here ( ) ( ) p( ) p( ) b b d d, () b, is give by (39). et, it is sho ho the above itegral ca be avoided by usig the fact that both ad are Gaussia radom variables ad by utilizig some properties of the -fuctio give its argumets are Gaussia radom variables. Sice is a zero-mea Gaussia radom variable, [ ( ) ] ( ). 5 E λ [8, p. ]. Therefore, epectatio of (39) ith respect to is equivalet to 3 ( ) () b hich is equivalet to Eq. () ppedi B BBD Desig Eample Let O {,, 3,, 5, 6, 7 } be a set of seve objects. The folloig blocks: {,, }, {, 3, 5 }, { 3,, 6 }, {, 5, 7 }, { 5, 6, }, { 6, 7, }, { 7,, 3 } form BBD for the set O. The parameters of the BBD desig are as follos: o of objects; v 7 o of blocks; b 7 3

24 The block cosists of three objects, each object appears i three blocks ad every object appears together i pairs i eactly oe block, i.e. covalecy λ. The icidet matri of this BBD is give as follos: The ros are the right cyclic shifts of the ro above ad colums are the doard cyclic shifts of the last colum. is ko as a 7 7 square circulat matri or circulat. Therefore, BBD ith λ has all the structural properties of the LDC parity check matri. ppedi C Eample of Bose BBD desig t BBD Bose parameters ill be as follos GF ( ) GF (3), prime umber p 3 o of ros, v 3, Objects o of colums, b t v 3, blocks Colum eight, γ, Ro eight, ρ, rimitive Root of GF(3), Bi (, i, it, i8t ) for i < t B (,,, 8 ), for i B (,, 3, 9), for i

25 5 B is oe of the blocks of GF (3), the other blocks ca be obtaied by addig all the elemets of GF (3) to B. Blocks of cidet Matri: B (,, 3, 9 ) (,, 3, 9) B (,, 3, 9 ) (,,, ) B (,, 3, 9 ) (, 3, 5, ) B 3 (,, 3, 9 ) 3 (3,, 6, ) B (,, 3, 9 ) (, 5, 7, ) B 5 (,, 3, 9 ) 5 (5, 6, 8, ) B 6 (,, 3, 9 ) 6 (6, 7, 9, ) B 7 (,, 3, 9 ) 7 (7, 8,, 3) B 8 (,, 3, 9 ) 8 (8, 9,, ) B 9 (,, 3, 9 ) 9 (9,,, 5) B (,, 3, 9 ) (,,, 6) B (,, 3, 9 ) (,,, 7) B (,, 3, 9 ) (,,, 8) The icidet matri ca be obtaied by settig i the positio specified by each block as sho belo b v B B B B B B B B B B B B B

26 REFERECES [] W.Webb. Broadbad fied ireless access as a key compoet of the future itegrated commuicatio eviromet, EEE Commuicatio Magazie, vol. 39, pp. 5-, Sept.. [] V. Erceg, K.V.S. Hari, M.S. Smith, D.S. Baum et al, Chael Models for Fied Wireless pplicatios, Cotributio EEE 8.6.3c-/9r, Feb.. [3] EEE Stadard, art 6: ir iterface for fied broadbad ireless access systems medmet : Media access cotrol modificatios ad additioal physical layer specificatios for - GHz, EEE 8.6a, Jauary 3. [] G. L. Stuber, J. R. Barry, S. W. McLaughli,. G. Li, M.. gram, ad T. G. ratt, Broadbad MMO-OFDM ireless commuicatios, roceedigs of the EEE, vol. 9, pp. 7 9, Feb.. [5] T.J.Willik, MMO OFDM for broadbad fied ireless access, EE roceedigs o Commuicatios, vol., issue, pp. 75-8, February 5. [6] C. Berrou,. Glavieu, ad. Thitimajshima, ear Shao limit errorcorrectig codig ad decodig: Turbo-codes, i Coferece Record, EEE teratioal Coferece o Commuicatios, CC-93, May 993, vol., pp [7] S. Beedetto ad G. Motorsi, Uveilig turbo codes: some results o parallel cocateated codig schemes, EEE Trasactios o formatio Theory, vol., o., pp. 9-8, March 996. [8] S. Beedetto, D. Divsalar, G. Motorsi, ad F. ollara, Serial cocateatio of iterleaved codes: performace aalysis, desig, ad iterative decodig, EEE Trasactios o formatio Theory, vol., o. 3, pp , May 998. [9] R.G. Gallager, Lo Desity arity-check Codes. MT ress, Cambridge, M, 963. []. F. Blake ad R. Mulli, The Mathematical Theory of Codig. eork: cademic,

27 [] R. C. Bose, O the costructio of balaced icomplete block desigs,. Eugeics 9, pp , 939. [] B.mmar, B. Hoary, Costructio of Lo-Desity arity-check Codes Based o Balace complete Desigs, i EEE Trasactio o formatio Theory, vol.5, pp , Jue. [3] Widro, B. ad Hoff, daptive sitchig circuits, stitute of Radio Egieers, i WESCO Covetio Record, pages 96 ugust 96. [] Hayki.S, daptive filter theory, 3 rd editio, retice Hall teratioal c., 996. [5] Hsu.FM, Square Root Kalma filterig for high-speed data received over fadig dispersive HF chaels, EEE Trasactio o formatio Theory, vol. T-8, o.5, September 98. [6].iao, R.. Carrasco ad.j. Wassell, erformace alysis of Covetioal Detectio i BFW Systems, roc. Secod F teratioal Coferece o Wireless ad Optical Commuicatios etorks, WOC'5, pp. 7-5, March 5, Dubai, UE. [7] B. Vasic, ad O. Milekovic, Combiatorial Costructios of Lo-Desity arity-check Codes for terative Decodig, EEE Tras. o form. Theory, vol. 5, o. 6, Jue. [8] B. Vasic, E. Kurtas ad. Kuzetsov, LDC Codes Based o Mutually Orthogoal Lati Rectagles ad their pplicatio i erpedicular Magetic Recordig, EEE Tras. Magetics, Vol. 38, o. 5, art:, pp.36-38, Sep.. [9] B. Hoary.. Moiia ad B. mmar, Costructio of Well-Structured uasi-cyclic Lo Desity arity Check Codes, EE roceedigs o Commuicatios, vol. 5, o. 6, pp. 8-85, Dec. 5. [] Z.W. Li, L.Che, S. Li ad W. Fog, Efficiet ecodig of quasi-cyclic lo- desity paritycheck codes, EEE Tras. o Comms., vol. 5, o., Jauary 6. [].iao, R.. Carrasco ad.j. Wassell, Theoretical erformace alysis of Sigle ad Multiple tea FW Systems, ccepted for ublicatio i EE roceedigs of Commuicatios. 7

28 [] J. roakis. Digital Commuicatios, 3rd editio, McGra-Hill, 995. [3] R. Morro ad J. Lehert, Bit-to-bit error depedece i slotted DS/SSM packet systems ith radom sigature sequeces, EEE Trasactios o Commuicatios, vol. 37, o., pp. 5-6, Oct [] R. Buehrer ad B. Woerer, alysis of adaptive multistage iterferece cacellatio for CDM usig a improved Gaussia approimatio, EEE Trasactios o Commuicatios, vol., o., pp [5] J. Holtzma, simple, accurate method to calculate spread-spectrum multiple-access error probabilities, EEE Trasactios o Commuicatios, vol., o. 3, pp. 6-6, March 99. [6]. iao, E. Strom, theoretical evaluatio of parallel iterferece cacellatio i M-ary orthogoal modulated asychroous DS-CDM system over multipath Rayleigh fadig chaels, EEE Trasactios o Vehicular Techology, vol. 5, o., pp. {, July, 5. [7]. iao, R. Carrasco,. Wassell, erformace aalysis of covetioal detectio i BFW systems, roc. Secod F teratioal Coferece o Wireless ad Optical Commuicatios etorks, pp. 7{5, March 5. [8] S. Verdu. Multiuser Detectio, st editio, Cambridge Uiversity ress,

29 FGURE CTOS Fig.: The complete diagram of the sigle-iput sigle-output LDC coded BFW system, (a) ith equalisatio (b) ith OFDM. Fig. : Costellatio diagram of a coded (a) SK (b) 6-M Fig. 3: Structure of daptive Equaliser usig 5 feed-forard ad 3 feedback taps Fig. : o-coheret detectio of SK ad 6-M over the BFW chael. Fig. 5: erformace of the (75, 535) LDC code over the WG ad Rayleigh Fadig Chael Fig. 6: erformace of the (35, 3) LDC code over the WG ad Rayleigh Fadig Chael Fig. 7: Compariso of Simulatio results of the (75, 535) LDC code usig SK over the BFW chael. Fig. 8: Compariso of Simulatio results of the (35, 3) LDC code usig SK over the BFW chael. Fig. 9: Compariso of Simulatio results of (75, 535) LDC code usig M over the BFW chael. Fig. : Compariso of Simulatio results of the (35, 3) LDC code usig M over the BFW chael. 9

30 Figure (a) (b) 3

31 Figure (a) (b) 3

32 Figure 3 3

33 Figure 33

34 Figure 5 erformace of LDC (75,535) code usig SK Modulatio Simulated SK BER over WG Chael Ucoded SK over Rayleigh Fadig Chael LDC(75,535)over WG Chael, teratios LDC(75,535) usig Sumroduct lgorithm, teratios Bit error rate SR [db] 3

35 Figure 6 erformace of LDC (35,3) code usig SK Modulatio Simulated SK BER over WG Chael Ucoded SK over Rayleigh Fadig Chael LDC(35,3)over WG Chael, teratios LDC(35,3) usig Sumroduct lgorithm, teratios Bit error rate SR [db] 35

36 Figure 7 Compariso of LDC (75,535) OFDM ad SRK over BFW usig, SK Bit error rate 3 Ucoded SK over BFW Chael LDC over BFW Chael usig SK modulatio Ucoded SRK Equalised over BFW Chael Ucoded OFDM over BFW Chael LDCBBD (75,535), BFW,OFDM LDCBBD (75,535), BFW,SRK Equalizatio 6 8 SR [db] 36

37 Figure 8 Compariso of LDC (35,3) OFDM ad SRK over BFW usig, SK Bit error rate 3 Ucoded SK over BFW Chael LDC over BFW Chael usig SK modulatio Ucoded SRK Equalised BER over BFW Chael Ucoded OFDM over BFW Chael LDCBBD (35,3), BFW,OFDM LDCBBD (35,3), BFW,SRK Equalizatio 6 8 SR [db] 37

38 Figure 9 Compariso of LDC (75,535) OFDM ad SRK over BFW usig, 6M Bit error rate 3 Ucoded 6M over BFW Chael LDC over BFW Chael usig 6M Ucoded SRK Equalised BER over BFW Chael Ucoded OFDM over BFW Chael LDCBBD (75,535), BFW,OFDM LDCBBD (75,535), BFW,SRK Equalizatio 6 8 SR [db] 38

39 Figure Compariso of LDC (35,3) OFDM ad SRK over BFW usig, 6M Bit error rate 3 Ucoded 6M over BFW Chael LDC over BFW Chael usig 6M Ucoded SRK Equalised BER over BFW Chael Ucoded OFDM over BFW Chael LDCBBD (35,3), BFW,OFDM LDCBBD (35,3), BFW,SRK Equalizatio 6 8 SR [db] 39