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2 Combied BER Aalysis for Time-Frequecy Sychroizatio Schemes for MB-OFDM UWB Debarati Se, Member, IEEE, Saswat Chakrabarti, Member, IEEE, ad R. V. Raja Kumar, SM, IEEE Abstract I this paper we preset the closed form expressio of bit-error-rate (BER) of a covolutio coded MB-OFDM Ultra- Widebad (UWB) system. The aalysis cosiders the log-ormal fadig statistics of UWB chaels ad captures the estimatio error variaces of timig ad carrier frequecy offset (CFO) estimatio by our already published sychroizers ATS [] ad MBAFS [] respectively. The derivatio ivokes momet geeratig fuctio (MGF) for log-ormal fadig statistics ad the Gauss-Hermite quadrature itegratio to deliver average BER expressio for rate R c coded QPSK modulated MB-OFDM system with ATS ad MBAFS sychroizers ad least square (LS) chael estimator. The aalytical result is validated with simulatio i the high delay spread UWB chael model CM3. This aalysis helps i thorough uderstadig o the performace of a OFDM based commuicatio system i a Ultra-Widebad eviromet. Idex Terms BER; UWB; sychroizatio I. ITRODUCTIO athematical aalysis of bit-error-rate (BER) performace M boud for coded MB-OFDM Ultra-Widebad (UWB) system with sychroizatio schemes uder realistic UWB chael has immese importace. I this paper, we obtai approximate ad simple aalytical expressio of BER cosiderig a biary covolutio code with soft Viterbi algorithm usig our earlier proposed sychroizatio schemes ATS [] ad MBAFS [] i the receiver. The derivatio associates momet geeratig fuctio (MGF) as the key aalytical tool. We put emphasis o useful assessmet of BER performace ad compare with our simulatio results to validate the aalysis. The aalysis is carried out i three steps: derivig the average BER for ucoded system cosiderig log-ormal fadig statistics, with ideal sychroizatio ad chael estimatio; the, covolutio codig is icorporated to obtai BER expressio for ideal timig, frequecy, ad chael estimatio; fially, the BER is derived for a covolutio coded MB-OFDM system which ivolves timig correctio by ATS method, frequecy sychroizatio by MBAFS method, ad Debarati Se is presetly with the Departmet of Sigals ad Systems, Chalmers Uiversity of Techology, Gotheburg, Swede ( debarati@chalmers.se; phoe: ). Durig this work she was associated with the G. S. S. School of Telecommuicatios, Idia Istitute of Techology Kharagpur, Idia. Saswat Chakrabarti is with the G. S. S. School of Telecommuicatios, Idia Istitute of Techology Kharagpur, Idia. R. V. Raja Kumar is presetly with the R. G. Uiversity of Kowledge Techologies, Hyderabad, Idia. Durig this work he was associated with the Departmet of E&ECE, Idia Istitute of Techology Kharagpur, Idia. Due to page limitatios, readers are requested to follow referece [] ad [] for our earlier proposed timig sychroizer ATS ad frequecy sychroizer MBAFS. chael estimatio by least square (LS) error method. The aalytical results are verified through simulatio. For simplicity of aalysis, the shadowig effect of the chael is excluded ad the statistically idepedet log-ormal distributed multipath gai coefficiets are cosidered. Rest is orgaized as follows: Sectio II presets the BER aalysis for ucoded system, covolutio coded system with ideal sychroizatio, ad with sychroizatio errors. Sectio III briefs about simulatio results ad discussios. Sectio IV cocludes our study with the summary. II. BER AALYSIS A. Ucoded System We aalyze the system i WPA eviromet where chael is expected to be slowly varyig. The fadig statistics govers the probability desity fuctio (PDF) of the istataeous SR per bit ( γ ) which is a time ivariat radom variable. The average BER is give by = Q( a γ ) P ( ) γ γ () where, a depeds o modulatio/detectio combiatio ad is costat. The classical defiitio of Gaussia Q-fuctio Qx ( ) = ( ) exp( y ) dy suffers from two x disadvatages: a) for computatio this requires trucatio of upper ifiite limit whe usig umerical itegral evaluatio, b) the presece of argumet of the fuctio as lower limit of the itegral poses aalytical difficulties whe this argumet depeds o aother radom parameter that requires statistical average of their probability distributios. We use the alterative defiitio of Q-fuctio defied by Simo et al. [3] Qx ( ) = exp( x Siθ) for x () The above defiitio of Q-fuctio has fiite itegratio limits which are idepedet of the argumet of fuctio x ad the itegrad has a Gaussia form with respect to x. The derivatio of Eq. () from the classical defiitio of Q-fuctio is give i [3]. Substitutig Eq. () i Eq. () the average BER leads to = exp ( a γ Si θ ) Pγ ( γ) dγ (3) The ier itegral withi the bracket of Eq. (3) ca be expressed i terms of momet geeratig fuctio (MGF) defied as [3]

3 sγ Mγ() s e Pγ ( γ) dγ (4) So, average BER (Eq. (3)) i terms of MGF is expressed as = Mγ ( a Si θ) (5) The UWB chael shows log-ormal fadig with PDF as {( ) } ( ) ( ) = l() exp log (6) P γ γ γ γ μ Here, μ (db) ad (db) are respective mea ad stadard deviatio of log γ. MGF correspodig to log-ormal distributio is obtaied by substitutig Eq. (6) i Eq. (4) ad usig the Gauss-Hermite quadrature itegratio [3, Eq. (5.4.46)] for the ier itegral i Eq. (4) as follows: p ( x + μ) / Mγ ( s) = Hx exp s (7) = x, =,,, p are the zeros of the p -th order where { } Hermite polyomial ( ) p He x ad { H x } ; =,,, p are correspodig weight factors tabulated i Table 5. of [4] for values of p from to. ECMA-368 [5] for MB-OFDM recommeds both QPSK ad Dual Carrier Modulatio (DCM). We cosider QPSK modulatio for our aalysis. For ucoded system, average BER with QPSK modulatio from Eq. (5) is as [6, pp. 4] 3 4 = Mγ ( a Si θ ) ; a = si / M (8) Substitutig the MGF for log-ormal fadig from Eq. (7) ito Eq. (8), average BER for ucoded system with QPSK modulatio ad log-ormal faded chael uder ideal time, frequecy ad chael estimatio coverges to p 3 4 ( x + μ) / = exp ( 3/ Hx E )( si b θ) = where, γ = E (9) b Equatio (9) allows evaluatio of bit error probability i simple way as it ivolves a sigle itegral o θ ad values of { } x ad { } H x ca be substituted from Table 5. of [4]. Our BER aalysis for coded system with ideal sychroizatio ad proposed sychroizers is based o Eq. (9). We do aalysis for covolutio coded system below. B. Covolutio Coded System with Ideal Sychroizatio We ow proceed our BER aalysis further for covolutio coded QPSK modulated MB-OFDM system cosiderig perfect timig ad frequecy sychroizatio. Suppose that trasmitted bits are ecoded by a covolutio ecoder CC (k, ) ad iterleaved with a depth exceedig coherece time of chael. The sequece is bit-wise iterleaved to esure that bits i ay code word fade idepedetly. A soft Viterbi decoder i cojuctio with coheret demodulator retrieves the iformatio at the receiver. Usig uio boud argumets [7], the bit error probability of a covolutio code with code rate R cc (= k/) ca be upper bouded by Pb < c P () d = d Here, free P d is pair-wise error probability with Hammig distace d, c d is sum of bit errors (iformatio error weight) for evets of distace d, ad d free is free distace of the code. For simplificatio, we assume that Hammig distaces d are mutually idepedet to each other. This leads to calculate the pair-wise error probability P usig the error probability expressio for L-diversity receptio with maximum ratio combiig for idetically distributed fadig for all chaels with QPSK modulatio scheme [6, Sectio 9.] may be give as 3 /4 L P e = ( Mγ ( si φ) ) dφ () Followig Eq. () the pair-wise error probability P d _ ideal for covolutio coded (at rate R cc ) QPSK modulated bits trasmitted over log-ormal fadig chael ad decoded by soft Viterbi decoder with ideal sychroizatio parameter (timig ad frequecy) estimatio is obtaied by Eq. (9) as Pd _ ideal = 3 /4 p ( x + μ) E () b Hx exp R ( si cc a θ) = As the zeros of Hermite polyomial H x are idepedet of θ, the above expressio ca be rewritte as p P _ ideal = Hx = 3 /4 ( x + μ) exp Rcc ( Eb )( a si θ) (3) I derivig Eq. (3) we have assumed that chael is slow fadig ad chael state iformatio (CSI) is kow. The oise power is the cotributio of AWG with mea zero ad variace awg. Substitutig Eq. (3) i Eq. () the upper boud of BER P b _ ideal with ideal parameter estimatio of chael, timig ad frequecy is obtaied as

4 Pb _ ideal < c P (4) _ ideal = dfree Equatio (4) is plotted i Fig. to verify with simulatio results. As the pair-wise error probability P d _ ideal for fadig chael decreases slower tha the AWG case ( P ( )) d = Q dr cc E b, the iformatio error weight c will have larger ifluece o BER. So, we eed to choose a CC which performs well o both AWG ad fadig chael, ad has maximum free distace with low iformatio error weight o each error path. We have chose the matchig error weights of optical distace spectrum (ODS) CC give i [8]. Cosiderig UWB chael is less severe tha Rayleigh fadig chael [9], ad due to uavailability of error weights for logormal fadig chael, we have used error weights c of Rayleigh fadig from [8] to plot Eq. (4). ext, we derive the BER expressio icorporatig timig ad frequecy estimatio error variaces. C. Covolutio Coded System with Sychroizatio Errors The estimatio errors of timig ad frequecy will cotribute to iter-carrier-iterferece (ICI) which cotrols the BER performace. For the ease of aalysis we exclude the effect of imperfect chael equalizatio. Let, the residual frequecy offset (RFO) after CFO estimatio by MBAFS be δε ad timig jitter for -th sample after timig sychroizatio by ATS be δτ ( ). We evaluate the ICI powers for both the residual errors separately. Let, each sub-carrier has sigal power of s. Assume that sigal power i a specific sub-carrier, ICI power from other sub-carriers due to residual time-frequecy errors, ad additive oise are mutually ucorrelated. The impact of ICI due to timefrequecy offsets o a sub-carrier from all other sub-carriers is modeled as zero mea Gaussia radom variable. Case : Effect of RFO uder the Assumptio of rfect Timig Estimatio First, we compute the ICI power due to RFO δε due to CFO estimatio error of MBAFS algorithm cosiderig perfect timig estimatio. For the system model i [], sampled sigal for the k-th sub-carrier after CFO sychroizatio by MBAFS ad FFT processig at the receiver is Yk ( ) = SkHkI ( ) ( ) δε () + SlHlI ( ) ( ) δε ( l k) + Wk ( ) l k, l= ICIδε (5) Here, k =..., S(k) is trasmitted symbol o k-th subcarrier, H(k) is k-th sample of chael frequecy respose, I δε () is atteuatio of k-th sub-carrier due to RFO, ad W(k) is additive white Gaussia oise with variace awg. Secod term i Eq. (5) is ICI due to RFO δε give as [] ( + ) ( l k+ δε ) ( ) ( δε ) si l k δε j Iδε ( l k) = exp l k + si The correspodig ICI power ca be show as ( l k+ δε) si δε ( k) = s H( l) l, l k = si l k ( + δε ) (6) (7) So, the average SR of the k-th sub-carrier due to residual CFO uder perfect timig sychroizatio is obtaied as { s } { awg} γδε ( k) = H( k) Iδε () δε + (8) where I δε () is obtaied as Iδε () = si ( δε ) si ( δε ) by substitutig l = k i the expressio (5). Case : Effect of Timig Jitter uder the Assumptio of rfect Frequecy Offset Estimatio Assumig frequecy estimatio is perfect, the ICI due to timig algorithm ATS is evaluated. Let, after timig sychroizatio the timig jitter for -th sample is δτ ( ). We assume δτ ( ) as a wide sese statioary Gaussia radom process with zero-mea ad variace τ. So, -th received sample with timig jitter may be show as ( + δτ( )) j k y ( ) = SkHke ( ) ( ) + w ( ) (9) k = Here, =,,... It is kow that the timig jitter at adjacet samples i the receiver becomes more correlated whe samplig rate icreases. To geeralize our aalysis, we icorporate correlatio coefficiet ρ p betwee δτ ( ) ad δτ ( p). For Gaussia correlatio, ρ p ca be expressed as ( ) [] ρ p p e β =. Later o, we have computed the ICI power cosiderig white timig jitter with appropriate coditioig of ρ p. After FFT of Eq. (9) the received sample o the m-th sub-carrier ca be expressed as j m Y( m) = ( ) y e + W( m); ( m=,,... ) = j jkδτ( ) ( k m) = S( k) H ( k) e e + W( m) k= = Expressio () ca be divided ito three parts: ( ) ˆ ( ) () Ym ( ) = Iδτ Sm ( ) + I m + Wm ( ) () ICIδτ Where the first term is the desired part phase rotated by

5 coefficiet I δτ ( ) give as Iδτ ( ) ( j mδτ( ) ) = e () = I δτ ( ) is a small error vector i the complex plae idicatig the effect of residual timig error o the desired sub-carrier. I δτ ( ) is a radom variable ad is assumed that the mea value of its magitude is.. The desired part i Eq. () is also atteuated i magitude due to timig jitter [] by { si( mδτ) } { si ( )( mδτ )}. Here, the effect of this amplitude reductio is eglected assumig that its value will be very small after sychroizatio by ATS. The secod term i equatio () creates the ICI may be give as j j kδτ( ) ( k m ) I( m) S( k) H( k) e e (3) = k=, = k m It is oticeable from equatios () ad (3) that i the absece of timig jitter i.e., for δτ ( ) =, I() becomes uity ad the ICI term I(m) vaishes as expected. ow, the average ICI power ca be computed from Eq. (3) as δτ ( ) = E{ I( m) } = E S() k H() k e e k=, = k m j k m jk δτ () 4 j k τ ( p)( k m) s Hk ( ) e e k m p ( ρ ) p = (4) ow, cosiderig the timig jitter to be white, the correlatio coefficiet is give as, whe m = ρm = (5), whe m I the presece of white timig jitter, where ρ p =, usig Taylor series expasio for the expoetial term i Eq. (4) while omittig the secod ad higher order terms ad usig followig approximatio for small values of θ l : j l e θ jθ, the average ICI power ca be obtaied as l ( 4 k τ ) s δτ = H( k) e (6) k m For large umber of sub-carriers, the ICI power due to timig jitter ca be approximated to [3] ( ) δτ s H k (7) τ k m Case 3 : Effect of Timig Jitter, RFO Error ad AWG I practice, after timig ad frequecy sychroizatio the ICI powers give by equatios (7) ad (7) will cotrol the BER performace of the system. Hece, i presece of residual frequecy error δε ad timig jitter δτ ( ), the best possible k-th sub-carrier amplitude at the output of FFT processor after frequecy ad timig offset correctios may be approximated followig Eq. (5) ad () as Y k S k H k I ICI ICI W k ( ) = ( ) ( ) δε, δτ () + δε + δτ + ( ) (8) where, I δε, δτ () sigifies the magitude atteuatio ad phase rotatio of the desired sigal due to RFO ad timig jitter. Assumig costat timig jitter δτ ad RFO δε, I δε, δτ () ca be evaluated followig Eq. (6) ad usig Eq. () with magitude atteuatio effect as [] ( ) ( + k δτ + δε δτ ) ( ) ( δε kδτ δε δτ ) siδε+ kδτ+ δεδτ j Iδε, δτ () = exp + + si δε (9) It is oticeable from Eq. (9) that whe oly residual CFO is preset, i.e. δτ =, the phase shift [i.e. the expoet i Eq. (9)] is idepedet of sub-carrier idex k ad is idetical i every sub-carrier. But, whe oly timig jitter is preset, i.e. δε =, the phase shift is proportioal to the sub-carrier idex k as well as the timig jitter itself. Uder the assumptio that both ICI power ad oise power are idepedet to each other, the average SR of k-th subcarrier i presece of both residual time-frequecy errors may be obtaied usig equatios (8) with appropriate substitutios from (7), (7), ad (9) as γδε, δτ () k = s H() k Iδε, δτ () δε + δτ + (3) awg { } { } where δε ad δτ are obtaied from Eq. (7) ad (7) respectively, I δε, δτ () ca be computed from Eq. (9). To E the Eq. (9) δε δτ calculate the average SR per bit, ( ) b, is scaled by the factor /logm, where M = 4 for QPSK modulatio. Assumig perfect CSI ( ce =) ad substitutig SR per bit with proper scalig of Eq. (3), pair-wise error probability P d _ syc for covolutio codig with sychroizatio is evaluated from Eq. (3) as d p 3/4 ( x + μ) E d_ syc x exp b a P = H R cc = δε, δτ si θ (3)

6 Hece, the simplified BER with timig ad frequecy sychroizatio is obtaied followig Eq. (4) as P b_ syc< c P d _ syc (3) = d We plot the above aalytical expressios (4) ad (3) i Sectio III ad verify with simulatio results. As discussed earlier, we cosider the error weights from Ref. [8]. III. SIMULATIO RESULTS FOR COMBIED ATS AD MBAFS A. Simulatio Eviromet The simulatio cosiders MB-OFDM system with rate-half upuctured covolutio codig ad QPSK modulatio. The geerator polyomial of rate-half covolutio code is 33,7 with costrait legth k = 7. OFDM symbols are trasmitted over 3 frequecy bads of bad group usig TFI patter. IEEE 8.5.3a CM3 is used as propagatio medium. I the receiver timig istat of OFDM symbols i each bad is estimated ad corrected separately usig ATS. Timig sychroizatio is followed by frequecy sychroizatio usig MBAFS algorithm. Simulatio is carried out for, oisy realizatios uder UWB fadig chael. We perform chael estimatio i each bad separately durig chael estimatio sequeces of the frame format by LS method. Relevat parameters are chose from Table. B. Bit-Error-Rate (BER) rformace The theoretical ad simulated BER performace without ad with sychroizatio errors are show i Fig.. The theoretical results are plotted usig aalytical expressios (4) ad (3) with proper substitutio from equatios (8) ad (3) after scalig by the factor /logm. The error weights c cosidered for the same are give i Table [8]. Further, Eq. (4) is plotted cosiderig d = ad evaluatig the summatio up to Hammig distace d =. The zeros { x } ad weight factors { H x } of Hermite polyomial are obtaied from Table 5. of [4] for values of p from to. The aalytical expressio (3) is plotted cosiderig perfect CSI i.e. ce = for two differet coditios: i) with oly ICI due to residual frequecy estimatio error ad AWG i.e. δτ =, free free δε, substitutig the Eq. (8) ito Eq. (3) with proper scalig, ad ii) with ICI due to both the timig jitter ad residual carrier frequecy estimatio error alog with AWG i.e. δτ, δε, substitutig the Eq. (3) ito Eq. (3) with proper scalig. All the aalytical results are verified through simulatio where timig ad frequecy are estimated by ATS ad MBAFS algorithm ad chael is estimated through LS estimate. TABLE : PARAMETERS COSIDERED FOR MBAFS PERFORMACE Parameter Cosideratio Time Frequecy Code Preamble Mode Data Rate stadard Code Rate / Time ad Frequecy domai spreadig o Data carriers i oe OFDM symbol Pilot carriers i oe OFDM symbol umber of modulated bits/ofdm symbol umber of OFDM symbols cosidered for frequecy estimatio UWB Chael Model ormalized Frequecy Offsets TABLE : ERROR WEIGHTS B E R Hammig distace cd 3Mbps, 3, 6 CM-CM4.33 (bad ).38 (bad ).43 (bad 3) cd VALUES FOR HAMMIG DISTACES Hammig distace cd (a) aaly. ideal time, fr., ch. estimate (b) aaly. fr. est. var., ideal timig & ch. est. (c) timig+fr. est. var., ideal ch. est. (d) sim. ATS+MBAFS, ch. est. by LS 5 5 Eb/o i db Fig. : BER vs. E b / for coded MB-OFDM i CM3 for: a) aalysis with perfect estimatio; b) aalysis with frequecy estimatio error; c) aalysis with frequecy ad timig estimatio error; d) simulatio with timig, frequecy, ad chael estimatio error. a b d c

7 I Fig. plot (a), plot (b), ad plot(c) are the aalytical results obtaied from aalytical expressio (4); Eq. (3) with ideal timig ad chael estimatio (i.e. δε ); ad Eq. (3) with timig ad frequecy estimatio by ATS ad MBAFS algorithms cosiderig ideal chael estimatio (i.e. δτ, δε ) respectively. Simulatio result with parameter estimatio is plotted as plot (d). C. Discussios The BER performace with perfect CSI ad ideal sychroizatio (plot a) is slightly superior tha aalytical result with oly frequecy offset estimatio ad correctio (plot b). This is due to the mea-squared-error (MSE) of frequecy estimatio algorithm. Plot (c) is eve iferior to plot (b) as it depicts the aalytical BER for residual timig ad frequecy estimatio errors assumig perfect chael estimate. It is iterestig to ote that the BER obtaied through simulatio ad aalysis, plot (d) ad plot (c) are cosistet ad close to each other. The chael estimatio error icorporated i simulatio but ot i aalysis sigifies the egligible differece betwee two curves i the figure. [4] M. Abramowitz, I. A. Stegu, Hadbook of Mathematical Fuctios with Formulas, Graphs, ad Mathematical Tables, th Pritig, Dover Publicatios, Y, USA, 97. [5] Stadard ECMA-368, High Rate Ultra Widebad PHY ad MAC Stadard, 3 rd Editio - Dec. 8, Available at: < [6] Itroductio to physical layer specificatios of MB-OFDM UWB proposal Available at: < Semiar/Dowload%files/MB_OFDM_UWB.pdf>, Sept., 4. [7] J. G. Proakis, Digital Commuicatios, 4th Editio. McGraw Hill Iteratioal Editio, ew York, USA,. [8] P. Freger, P. Orte, T. Ottosso, Covolutioal codes with Optimum distace Spectrum, IEEE Commuicatios Letters, vol. 3, o., pp , ov [9] H. Q. Lai, W. P. Siriwogpairat, K. J. R. Liu, rformace aalysis of multibad OFDM UWB systems with imperfect sychroizatio ad itersymbol iterferece, IEEE Joural o Selected Areas i Commuicatios, vol., o. 3, pp , Oct. 7. [] P. H. Moose, A techique for orthogoal frequecy divisio multiplexig frequecy offset correctio, IEEE Trasactios o Commuicatios, vol. 4, o., pp , Oct [] A. V. Balakrisha, O the problem of time jitter i samplig, IEEE Trasactios o Iformatio Theory, vol. 8, o. 3, pp. 6 36, April 96. [] P.-Y. Tsai, H.-Y. Kag, T.-D. Chiueh, Joit weighted least-squares estimatio of carrier-frequecy offset ad timig offset for OFDM systems over multipath fadig chaels, IEEE Trasactios o Vehicular Techology, vol. 54, o., pp. 3, Ja. 5. [3] U. Oukwo, Y. Li, A. Swami, Effect of timig jitter o OFDM-based UWB systems, IEEE Joural o Selected Areas i Commuicatios, vol. 4, o. 4, pp , April 6. IV. COCLUSIOS We provide the BER aalysis of a covolutio coded MB- OFDM system with realizable timig ad frequecy sychroizatio algorithms. The log-ormal fadig statistics of UWB chaels is captured ad the estimatio error variaces of timig ad CFO estimatio by our earlier proposed sychroizers ATS ad MBAFS are cosidered. Derivatio ivokes the momet geeratig fuctio (MGF) as aalytical tool ad uses Gauss-Hermite quadrature itegratio to obtai closed form average BER expressio for rate R cc coded QPSK modulated MB-OFDM system with our earlier proposed sychroizatio algorithms ad a least square chael estimator. The validity of aalysis is cofirmed through simulatio results. V. ACKOWLEDGEMET We would like to ackowledge fudig from VIOVA withi the IKT grat 7-93 for publicatio of this paper. REFERECES [] D. Se, S. Chakrabarti, R. V. Raja Kumar, A Multi-Bad Timig Estimatio ad Compesatio Scheme for Ultra-Widebad Commuicatios, IEEE Globecom-8, LA, USA, pp. -5, ov. 3- Dec. 4, 8. [] D. Se, S. Chakrabarti, R. V. Raja Kumar, A Improved Frequecy Offset Estimatio Algorithm by Multi-Bad Averagig Method for MB- OFDM based UWB Commuicatio for WPA Applicatios, IEEE ATS-8, Bombay, Idia, pp. -3, Dec. 5-7, 8. [3] M. K. Simo, M-S. Alouii, Digital Commuicatio over Fadig Chael, Joh Wiley, J, USA, 5.

Probability of error for LDPC OC with one co-channel Interferer over i.i.d Rayleigh Fading

IOSR Joural of Electroics ad Commuicatio Egieerig (IOSR-JECE) e-issn: 2278-2834,p- ISSN: 2278-8735.Volume 9, Issue 4, Ver. III (Jul - Aug. 24), PP 59-63 Probability of error for LDPC OC with oe co-chael